Geometric Spin Hall Effect of Light

Der Naturwissenschaftlichen Fakultat derFriedrich-Alexander-Universitat Erlangen-Nurnberg

zur

Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Jan Korgeraus Filderstadt

Als Dissertation genehmigtvon der Naturwissenschaftlichen Fakultatder Friedrich-Alexander-Universitat Erlangen-Nurnberg

Tag der mundlichen Prufung: 11. April 2014

Vorsitzende(r) des Promotionsorgans: Prof. Dr. Johannes BarthGutachter: Prof. Dr. Gerd LeuchsGutachter: Prof. Dr. Nicolas Joly

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Geometrischer Spin-Hall-Effekt des Lichts

Zusammenfassung

Diese Arbeit beschreibt den Weg von der theoretischen Idee bis zum experimentellenNachweis eines neuartigen, optischen Phanomens. Erst vor kurzem, im Jahr 2009,wurde ein Strahlverschiebungseffekt vorausgesagt, der zunachst schwierig einzuord-nen war. Dieser geometrische Spin-Hall-Effekt des Lichts, im Englischen als spin Halleffect of light oder kurz SHEL bezeichnet, fuhrt zu einer Schwerpunktsverschiebungdes Lichtstrahls, die uberraschenderweise nur vom Photonenspin und der Geometriedes Detektionssystems abhangt. Es handelt sich um einen fundamentalen Effekt, dersich bereits bei einer theoretischen Betrachtung der Struktur des zirkular oder el-liptisch polarisierten Lichtstrahls selbst zeigt. Es ist daher umso erstaunlicher, dassdieses Phanomen so lange im Verborgenen geblieben ist.

Eine experimentell Beobachtung des geometrischen SHELs verlangt nach einerphysikalischen Operation, die die Symmetrie des einfallenden Strahls bricht. ImRahmen dieser Arbeit wird hierzu ein Polarisator verwendet. Wahrend der Effekteines Polarisators fur senkrechten Einfall bekannt ist, verdient ein zum Strahlengangverkippter Polarisator besondere Aufmerksamkeit. Um diesen Fall zu verstehen wur-den verschiedene Polarisatoren charakterisiert und ein generisches Modell erarbeitet.Dieses Modell ist geeignet eine Vielzahl an Strahlverschiebungen vorauszusagen.

Die Mehrzahl der in dieser Arbeit dargestellten Experimente verwendet einenGlas-Polarisator in einem mit Immersionsol gefullten Tank. Der Brechungsindex desOls entspricht dem des Glassubstrats, so dass storende Oberflacheneffekte vermiedenwerden und Polarisationseffekte in Reinform untersucht werden konnen. Ohne dieImmersionstechnik, dass heißt mit einem Polarisator in Luft, ware es nicht moglichdie dargestellten Ergebnisse zu wiederholen.

Die Arbeit zeigt, dass der geometrische SHEL genau dann an einem Polarisa-tor auftritt, wenn dieser zu einer vom Wellenvektor abhangigen Phase fuhrt. Diesegeometrische Phase resultiert aus dem Zusammenspiel des Drehimpulses des ein-fallenden Strahls mit dem Symmetriebruch, der durch den verkippten Polarisatoreingefuhrt wird.

Den Hohepunkt dieser Arbeit stellt die direkte Messung solcher polarisations-abhangigen Verschiebungen dar. In der experimentell untersuchten Konfigurationubersteigt die Schwerpunktsverschiebung eine Wellenlange. Diese Arbeit schließtmit einer Diskussion des transversalen Drehimpulses, dem grundlegenden Konzept,das die Arbeit motiviert hat.

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Summary

This thesis describes the journey from the theoretical idea to an experimental proofof a novel optical effect. Only recently, in 2009, an elusive beam shift phenomenonwas predicted. This geometric spin Hall effect of light (SHEL) amounts to a dis-placement of a light beam, which surprisingly depends solely on the photon spin andthe geometry of the detection system. In fact, this very fundamental beam shift hasalways been hidden in plain sight, in the structure of circularly and elliptically po-larized light beams, and can be revealed theoretically by employing a non-standard,i.e. tilted reference frame.

Experimentally, the observation of the geometric SHEL relies on a physical oper-ation breaking the symmetry of a light beam. In this work, we choose to employa polarizer for this purpose. While, the action of a polarizer is obvious for normalincidence, the structure of the light beam transmitted across a tilted polarizing in-terface is worth a closer look. To understand this physically, different polarizers arecharacterized, which leads to a generic polarizer model. This connects polarizinginterfaces to a plethora of beam shifts.

In particular, most experiments in this work use a glass polarizer submerged ina liquid with its refractive index matched to the polarizer’s substrate. This setupavoids detrimental effects occurring at the polarizer surface and allows to studypolarization effects in a particularly pure manner. In fact, without this technique,i.e. using a polarizing interface in air, one cannot reproduce our results.

It is shown that geometric SHEL occurs at a polarizer if the action of this devicesleads to particular wave vector dependent phase term, which can be identified withBerry’s geometric phase. The occurrence of this phase results from the interplaybetween the angular momentum of the incident beam and the symmetry breakinduced by the tilted polarizer.

The pinnacle of this work is the direct measurement of such polarization-dependent beam shifts with nanometre resolution. For the configuration studiedexperimentally, these displacements exceed one wavelength. This thesis concludeswith a discussion of transverse angular momentum, the fundamental idea whichmotivated this work.

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Contents

Summary 5

Introduction 9

1. Properties of light beams 111.1. Laser beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2. The light field in the angular spectrum representation . . . . . . . . 131.3. Understanding beam shifts . . . . . . . . . . . . . . . . . . . . . . . 141.4. Linear and angular momentum of light beams . . . . . . . . . . . . . 15

2. Well-known and novel beam shifts 192.1. Goos-Hanchen shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2. Imbert-Fedorov shift and spin Hall effect of light . . . . . . . . . . . 232.3. Angular momentum conservation and conventional beam shifts . . . 242.4. Geometric spin Hall effect of light and transverse angular momentum 25

3. The geometric spin Hall effect of light at a polarizing interface 273.1. A useful generic polarizer model . . . . . . . . . . . . . . . . . . . . 273.2. Constructing a microscopic polarizer model compatible with empirical

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3. Beam shifts occurring at a tilted polarizer . . . . . . . . . . . . . . . 333.4. Interpretation in terms of the geometric phase . . . . . . . . . . . . . 343.5. Interpretation in terms of transverse angular momentum . . . . . . . 38

Conclusion and outlook 43

A. Publications 51A.1. Geometric Spin Hall Effect of Light at polarizing interfaces . . . . . 51A.2. Observation of the geometric spin Hall effect of light . . . . . . . . . 59A.3. The polarization properties of a tilted polarizer . . . . . . . . . . . . 69A.4. Distributing entanglement with separable states . . . . . . . . . . . . 81

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Introduction

Optics is the scientific discipline studying light and its interaction with matter. Inprinciple, any optical phenomenon can be derived from the quantum theory of thelight field. Practically, many phenomena including the ones discussed in this workcan be understood with a classical theory. Nevertheless, the terms photon andspin, which originate from the quantum mechanical treatment are used in the field[Haus93, Loudon00].

Most of our daily experiences with light and the technical use of optics can beunderstood using a surprisingly simple theory: In geometrical optics, light is thoughtof as a collection of thin rays propagating according to a set of geometric rules[Saleh07, §1].

Physically, light is electromagnetic radiation and its behaviour is governed by elec-tric and magnetic fields. These vector fields are found solving a set of fundamen-tal differential equations attributed to Maxwell. Light beams are special solutionsthereof. Like the hypothetical rays, they travel along a given direction and exhibita finite extent transverse to the direction of propagation. However, internal struc-ture and propagation of even the most basic beams differ from ray optics. Physicallight beams result from the interference of multiple elementary waves and propagateaccordingly [Mandel95].

The topic of this work are beam shift phenomena. Such beam shifts are un-expected deviations from geometrical optics [Bliokh13]. A prime example for thediscrepancy between wave and ray optics is the famous 1943 experiment by Goos andHanchen studying total internal reflection [Goos47]. Inside a glass prism, a physicallight beam is not reflected geometrically at the surface as expected for a ray, butpenetrates into the free space surrounding the prism [Renard64]. Here an evanescentwave carries the energy of the incident beam along the surface for a fraction of thewavelength before it is reflected.

A less-known cousin of this Goos-Hanchen effect, attributed to Imbert and Fe-dorov [Fedorov55, Schilling65, Imbert72], has puzzled researchers for almost 60 years.While the underlying physical problem is similar, here, the vectorial nature of elec-tromagnetic waves manifests itself in a strong dependence on the state of polarization[Liu08]. Since polarized beams can be thought of as an ensemble of photons witha particular spin state, a polarization-dependent beam shift amounts to a couplingbetween the spin and spatial degrees of freedom, referred to as spin-orbit coupling[Bliokh06]. The Imbert-Fedorov shift has recently been rediscovered as the spinHall effect of light [Onoda04, Onoda06, Bliokh06, Hosten08, Menard09, Hermosa11,Yin13] and connected to Berry’s geometric phase [Berry84, Xiao10].

The topic of this thesis is a third kind of beam shift, the geometric spin Hall

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effect of light (geometric SHEL) [Aiello09a, Aiello10, Korger11, Korger13a]. Thisphenomenon is intimately connected to spin and orbital angular momentum of lightbeams. Both properties require a physical descriptions of the light field taking intoaccount the vectorial nature of the problem.

Geometric SHEL shares a number of common characteristics with spin Hall effectsin optics and other branches of physics [Xiao10], most importantly, the connection toBerry’s phase and the dependence on the spin of the incident field. It is important tonote that this geometric shift is practically independent from the physical propertiesof the interface where it occurs, and thus an ideal candidate to study universalfeatures of spin-orbit coupling.

This work is structured as follows: Chapter 1 reviews the physical descriptionof light beams and develops a set of theoretical tools used throughout this thesis.The theory is constructed in an abstract manner, applicable to both conventionaland novel beam shifts. Then, chapter 2 discusses examples of such beam shiftsincluding the geometric spin Hall effect and their relation to angular momentumconservation laws. Finally, chapter 3 deals with the geometric SHEL occurring at atilted polarizer. To this end, a physical description of such polarizing interfaces isestablished. Two different approaches were pursued and eventually both resulted ina suitable physical model. One attempt uses empirical data and a generic projectionformula; the other one builds upon a microscopic description. These models are usedto explain the observed beam shift as well as geometric aspects and fundamentallimits thereof.

The major results of this thesis have been submitted to peer-reviewed journals,reprinted in an appendix to this thesis:

• J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, andG. Leuchs. Geometric Spin Hall Effect of Light at polarizing interfaces.arXiv:1102.1626. Applied Physics B, 102(3), 427–432, 2011

• J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Mar-quardt, and G. Leuchs. Observation of the geometric spin Hall effect of light.arXiv:1303.6974. Phys. Rev. Lett. 112, 113902, 2014

• J. Korger, T. Kolb, P. Banzer, A. Aiello, C. Wittmann, C. Marquardt, andG. Leuchs. The polarization properties of a tilted polarizer. arXiv:1308.4309.Opt. Express 21(22), 27032–27042, 2013

A more detailed discussion of the experiment has been the subject of two relatedBachelor theses:

• Tobias Kolb, Charakterisierung eines zum Strahlengang gekippten Polar-isators, 2010.

• Vanessa Chille, Experimente zum geometrischen Spin-Hall-Effekt des Lichts,2011.

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1. Properties of light beams

The topic of this work is a novel beam shift phenomenon, the geometric spin Halleffect of light at polarizing interfaces. Generally, beam shifts depend on both, theproperties of the incident light field and the operation occurring at the interface.Thus, a proper physical description of both is required to predict and understandthese effects.

This chapter introduces the background relevant to understand beam shifts. Thetheoretical tools presented here will prove useful to calculate established and novelphenomena in the following chapters. First, we establish a physically sound de-scription of Maxwell-Gaussian light beams (section 1.1). These are vectorial beamswith a Gaussian envelope, compatible with Maxwell’s equations. Then, we proceedto calculate in an abstract manner how a physical operation effects the light beamand can yield to a displacement known as a beam shift (section 1.3). Finally, thischapter concludes with a discussion of linear and angular momentum (section 1.4).

Spin Hall effects of light (SHEL) are closely related to the interplay betweenintrinsic and extrinsic angular momenta. For both the geometric SHEL and theconventional spin Hall effect of light, recent experiments [Hosten08, Korger13a] werestimulated by theoretical breakthroughs concerning the angular momentum of thelight field [Onoda04, Bliokh06, Aiello08].

1.1. Laser beams

Since the invention of the laser, collimated light beams are the most ubiquitous toolused in optics. The spatial structure of such laser beams are commonly describedas solutions ψ(r) of the paraxial wave equation. A complete set of such solutions isgiven, for instance, by Laguerre-Gaussian or Hermite-Gaussian laser modes.

In this work, we focus on the fundamental solution of both sets, which yields aGaussian light beam [Saleh07, §3.1]:

ψ(r) =A0

z + izRexp

[−ik(x2 + y2)

2(z + izR)

]=

A0

z + izRexp

−(x2 + y2)

w20

(1− i zzR

)

=A0

z + izRexp

−(x2 + y2)

(1 + i zzR

)

w20

(1 + z2

z2R

)

(1.1)

Here, k = 2π/λ is the magnitude of the wave vector k, w0 is the beam waist,and zR = π w2

0/λ is the Rayleigh length. All of these parameters depend on the

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Figure 1.1.: Reference frames used throughout this work. (a) Sketch of aGaussian light beam described in its natural reference frame x,y, z,where the z is the direction of propagation. (b) Tilted reference framex′,y′, z′ used to describe oblique interfaces relevant to this work. Inparticular, to study the geometric SHEL, we will employ a polarizerwith its absorbing axis oriented in direction of x′.

wavelength λ, the natural length scale in optics. It is convenient to introduce theangular divergence θ0 = λ/(π w0). This dimensionless parameter θ0 is small forwell-collimated beams. Thus, the Taylor expansion

f(θ0) = f (0)︸︷︷︸0th order

+ f (1) θ0︸ ︷︷ ︸1st order

+ f (2) θ20︸ ︷︷ ︸2nd order

+O(θ30) ' f (0) + f (1) θ0, (1.2)

of any observable f(θ0) depending on the light field can usually be truncated afterthe first-order term. In this work, we assert that the beam waist w0 ≈ 100λ exceedsthe wavelength by several orders of magnitude as it is the case in our experiments.Thus, terms proportional to θ20 ≈ 10−5 become negligible.

The electric field of a homogeneously polarized, monochromatic beam

Ein(r) ∝ exp (ikz)

(u+

i θ02z (u · ∇⊥)

)ψ(r) +O(θ20) (1.3)

depends on the spatial profile ψ(r) and a unit vector u describing the state ofpolarization [Haus93]. This prototype light beam Ein(r) will be used throughoutthis work to describe the incident light field. We will refrain from explicitly writ-ing the time-dependence E(r, t) = E(r) exp(i ω t) and it is understood that thephysical field is the real part Re(E) of the complex quantities given here. Unlessstated otherwise, we employ a reference frame x, y, z aligned with the light beamand identify the z-axis with the direction of propagation (Figure 1.1). The fieldcomponents Ex = E · x and Ey = E · y are referred to as horizontal and verticalpolarization respectively.

The magnetic field B(r, t) = iω curlE(r) exp(i ω t) of a light beam can be found

from Faraday’s law of induction [Jackson98, §5.15]. Both, the magnetic and theelectric fields are time-harmonic, and we will later make use of the fact that the time-average of the product of two such fields A1,2(r, t) = A1,2(r) exp(i ω t) is [Jackson98,

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§6.9]

1

T

T∫

t=0

Re(A1(r, t)) Re(A2(r, t)) d t =1

2Re(A1(r)A∗2(r)). (1.4)

1.2. The light field in the angular spectrum representation

The representation of the electric field in the position basis (1.3) is complete andusually intuitive. For the purpose of this work, it is convenient to write the lightbeam in the momentum basis, that is as a function E(k) of the wave vector

k = k(κx, κy,

√1− κ2x − κ2y

)T

︸ ︷︷ ︸κ

. (1.5)

The connection between both representations is given by the Fourier transform

E(κ) =1

2π

∫∫

x,y

E(x, y, z = 0) exp(−i k (κx x+ κy y)) dx d y. (1.6)

In other words, equation (1.6) expresses the light field as a superposition ofplane waves exp(ik · r). Thus, if we understand the effect a physical operation

Ein

(κ) → Eout

(κ) has on a generic plane wave, we can infer the action for anylight field. Furthermore, for well-collimated beams, the first-order approximationκ ' (κx, κy, 1)T is usually sufficient and renders this approach efficient. Generally,the resulting light field

Eoutj (κ) = (α

(0,0)j + α

(1,0)j κx + α

(0,1)j κy)︸ ︷︷ ︸

αj(κ)

ψ(κ) +O(θ20) (1.7)

is a function of six complex amplitudes α(x,y)j and the envelope ψ(κ). The index

j ∈ 1, 2 denotes two orthogonal states of polarization.

This generic light beam (1.7) can equivalently be expressed in position space:

Eoutj (rout) =

(α(0,0)j + i

(α(1,0)j

xout

w0+ α

(0,1)j

yout

w0

)θ0

)ψout(rout) +O(θ20) (1.8)

Throughout this work, equations (1.7) and (1.8) will be used to calculate beamshifts for a number of different physical operations. The reference frame rout can beadopted to suit the problem. If the direction of propagation does not change, as forthe geometric SHEL, we choose rout = rin = r.

As a side note, the plane wave representation (1.6) also explains the physicalstructure of polarized light beams (1.3). Since the electric field of a plane wave is

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strictly transverse to the direction of propagation κ, the orientation

e(κ) =u− (κ · u)κ√

1− (κ · u)= u− (κ · u)κ+O(θ20) =

uxuy−u · κ

+O(θ20) (1.9)

of its electric field must generally differ from the global polarization vector u. Theelectric field of a homogeneously polarized light beam in the angular spectrum rep-resentation

Ein

(κ) = e(κ) ψ(κ) (1.10)

is simply the product of the polarization vector e(κ) introduced above and theFourier amplitude ψ(κ). The inverse Fourier transform of equation (1.10) yields ourprototype light beam in real space (1.3). A rigorous discussion of light beams in theangular spectrum representation is given in [Mandel95, §5.6].

1.3. Understanding beam shifts

Beam shifts are unexpected deviations from geometrical optics. In theory, suchdisplacements are found as first-order corrections to the position of the centre ofmass of a light beam’s energy density. Experimentally, beam shifts are usuallyfound as polarization-dependent displacements, although a prominent example, theGoos-Hanchen effect, also occurs for unpolarized light [Goos47].

To allow for a simple uniform treatment of multiple beam shift phenomena, wecalculate the energy density |Eout(r)|2 = |Eout

1 (r)|2 + |Eout2 (r)|2 once for the generic

first-order light beam (1.8). In terms of the Fourier amplitudes αj introduced above,

the centre of mass⟨r⊥⟩

=∫∫r⊥|Eout(r⊥)|2 dxd y∫∫|Eout(r⊥)|2 dxd y of our generic light beam at the beam

waist (z = 0) is:

⟨r⊥⟩

=(W1

⟨x⟩|E1|2 +W2

⟨x⟩|E2|2

)x

+(W1

⟨y⟩|E1|2 +W2

⟨y⟩|E2|2

)y (1.11a)

Here, we make use of the fact that the energy density of orthogonally polarizedelectric fields can be calculated independently. Both components |E1|2 and |E2|2contribute to the barycentre

⟨r⊥⟩

of the total electric energy density and it is con-venient to introduce relative weights

Wj =

+∞∫∫−∞|Ej(r)|2 dx d y

+∞∫∫−∞|E(r)|2 dx d y

=

∣∣∣α(0,0)j

∣∣∣2

∣∣∣α(0,0)1

∣∣∣2

+∣∣∣α(0,0)

2

∣∣∣2 (1.11b)

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for each polarization component and the so-called relative shifts:

⟨x⟩|Ej |2 =

λ

2π

Im(α(0,0)j

)Re(α(1,0)j

)− Re

(α(0,0)j

)Im(α(1,0)j

)

∣∣∣α(0,0)j

∣∣∣2

=λ

2π

Im(α(0,0)j α

(1,0)j

∗)

∣∣∣α(0,0)j

∣∣∣2 (1.11c)

and⟨y⟩|Ej |2 =

λ

2π

Im(α(0,0)j α

(0,1)j

∗)

∣∣∣α(0,0)j

∣∣∣2 . (1.11d)

The barycentre of our prototype light beam (1.3) coincides, by definition, withthe z axis. Any operation that effects the electric field amplitudes such that thecentre of mass

⟨r⊥⟩6= 0 is displaced is referred to as a beam shift phenomenon. In

particular, we study spatial beam shifts occurring independently of the distance zfrom the beam waist.

Any such spatial displacement in real space is connected to a linear phase factorin k-space. This well-known property of the Fourier transform manifests itself in

equations (1.11). Ignoring an irrelevant global phase (Im(α(0,0)i

)≡ 0), we can verify

that equation (1.11c) reduces to

⟨x⟩|Ej |2 = − λ

2π

∂φj∂κx

∣∣∣∣κx=0κy=0

, (1.12)

where φj is the argument of the electric field amplitude αj(κ) = A(κ) exp(i φj(κ))in Fourier space.

This result is completely general and applies to both well-known and novel beamshifts. In fact, a variant of equation (1.12), appeared in early theoretical work[Artmann48] connected to the Goos-Hanchen shift.

1.4. Linear and angular momentum of light beams

The electric field E(r) (as in equation (1.3)) completely describes a light beam.Furthermore, we have shown in section 1.2 that any physical interaction relevant tothis work can be calculated conveniently using the Fourier transform E(k) thereof.

Nevertheless, connecting the light field to other physical observables, such as linearand angular momentum, proves insightful. Making use of equation (1.4), we findthe time-averaged linear momentum density [Jackson98, §6.7]

p(r) =1

2c2Re(E ×H∗) =

ε02

Re(E ×B∗), (1.13)

which directly leads to the angular momentum density j(r) = r × p(r).

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These densities p(r) and j(r) are vector fields, that is vector-valued functions ofthe position r, just like the electric and magnetic fields. For light beams, the total

linear momentum P =

∫∫

A

p(r) d2 r and (1.14a)

angular momentum J =

∫∫

A

j(r) d2 r (1.14b)

per unit length is found by integrating these densities over a plane A perpendicularto the direction of propagation.

The integral values P and J are conserved, and thus, obvious candidates for aphysical interpretation of the densities p and j [Haus93, Allen92, Allen00]. For ourprototype light beam, the linear momentum density is

p(r) ∝ k |ψ(r)|2 z︸ ︷︷ ︸pz

+σ

2

∂|ψ(r)|2∂r⊥

Φ︸ ︷︷ ︸

ps

+k r z

zR + z2|ψ(r)|2 r⊥

︸ ︷︷ ︸pd

, (1.15)

where r⊥ = (x, y, 0)T /√x2 + y2 and Φ = (y,−x, 0)T /

√x2 + y2. Using the notion,

that a quasi-monochromatic source emits photons of the energy ~ω, we can statethat the integrated linear and angular momenta of our beam are

P =

∫∫

A

p(r) d2 r =

∫∫

A

pz(r) d2 r = k ~ z and (1.16)

J =

∫∫

A

j(r) d2 r =

∫∫

A

r × ps(r) d2 r = σ ~ z per photon. (1.17)

The divergence term pd does not contribute to either integral.For Laguerre-Gaussian modes ψLG

pl (r), there occurs an additional term of Jo =l ~ z per photon, known as orbital angular momentum [Allen92, Allen00]. Both formsof angular momentum discussed so far are intrinsic, i.e. their values are independentfrom the choice of the reference frame.

In the context of this work, it is important to understand that the angular mo-mentum of a light beam acquires an additional extrinsic contribution if the spatialmode ψ(r −R) is displaced with respect to the z-axis:

J(R) =

∫∫

A

r × p(r −R) d2 r =

∫∫

A

(r′ +R)× p(r′) d2 r′

=

∫∫

A

r′ × p(r′) d2 r′

︸ ︷︷ ︸J(R=0)

+R×∫∫

A

p(r′) d2 r′

︸ ︷︷ ︸Jextrinsic(R)

(1.18)

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For a paraxial beam described in its natural reference frame, i.e. if z is the opticalaxis, the first term J(R = 0) in equation (1.18) yields the intrinsic angular momen-tum [Berry98, O’Neil02], parallel to the direction z of beam propagation. Only thesecond term Jextrinsic(R) depends on the distance R from the origin of the referenceframe. This extrinsic contribution to the angular momentum is perpendicular to z.

Both, for the conventional Imbert-Fedorov shift, as well as for our geometric spinHall effect of light, the occurrence of such extrinsic angular momentum balances theconservation laws for energy, linear and angular momentum.

Optical angular momentum is a rapidly evolving field and the literature goes waybeyond the simple case of free collimated light beams described here [Marrucci11,Franke-Arnold08, Banzer13].

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2. Well-known and novel beam shifts

In this chapter, we introduce a number of beam shift phenomena. First, we show howthe boundary conditions at planar interfaces give rise to Goos-Hanchen and Imbert-Fedorov shifts (sections 2.1 and 2.2). The physical properties of such interfaces areconveniently described using Fresnel’s formulæ and any conventional beam shift isa function of the Fresnel coefficients. Then, we connect those effects to conservationlaws, in particular the conservation of angular momentum (section 2.3). Finally,we conclude with a discussion of transverse angular momentum and the originalproposal of the geometric spin Hall effect of light.

This work focusses on the geometric spin Hall effect occurring at a tilted polarizinginterface. In the following chapter such tilted polarizers are studied, which allows topredict and measure the geometric SHEL analogously to conventional beam shifts.However, unlike conventional beam shifts, geometric SHEL is connected to trans-verse angular momentum and exceeds the magnitude of conventional beam shifts.

2.1. Goos-Hanchen shift

Conventional beam shift phenomena occur for a light beam reflected from or trans-mitted across a planar interface between two homogeneous media. At this boundary,where the properties of the media change sharply (n1 → n2), electric and magneticfields are discontinuous [Born99, Appendix VI.1]. Well-known consequences of theseboundary conditions are the law of reflection, the law of refraction also known asSnell’s law, and Fresnel’s formulæ. Furthermore, all conventional beam shifts emergefrom these relations.

Generally, the centre of mass of a light beam reflected from or refracted at adielectric boundary deviates from the geometrical optics approximation. In theirfamous experiment, Goos and Hanchen observed a longitudinal displacement for alight beam reflected multiple times inside a special glass prism. At that time, itwas already well-known from Maxwell’s theory and experimentally confirmed, thatthere exists an evanescent field beyond the interface where total reflection occurs.While evanescent waves decay rapidly in direction of the surface normal, they arenot negligible as energy is transported along the surface [Renard64, Lai00].

This energy flux explains the observation of the longitudinal displacement ob-served by Goos and Hanchen (Figure 2.1(a)). However, explicitly calculating theevanescent field is never required. To the contrary, Fresnel’s formulæ for the re-flected field are sufficient since they already account for all boundary conditions.As a consequence, today, the term Goos-Hanchen shift refers to any longitudinaldisplacement at an interface, even if no evanescent field is involved [Bliokh13].

19

Figure 2.1.: Illustration of Goos-Hanchen and Imbert-Fedorov shifts occur-ring for a light beam totally reflected inside a glass prism. Thereflected light field is described in a reference frame xR,yR, zR alignedwith its direction of propagation. Here, zR is the geometric reflection ofthe original direction of propagation zin. Beyond this geometrical opticsapproximation, the position of the reflected beam does not coincide withthe zR-axis. (a) For total internal reflection, any light beam undergoes alongitudinal displacement known as the Goos-Hanchen effect. (b) If theincident light beam is polarized, additionally, a transverse shift known asthe Imbert-Fedorov effect can occur. Purely s or p polarized beams areexceptions, where the transverse displacement vanishes. Quantitatively,both effects depedend on the state of polarization (Figure 2.2).

In the previous sections, we have introduced the tools required to calculate the ge-ometric SHEL. Here we connect those to conventional beam shifts using the originalGoos-Hanchen experiment as an example.

For any plane wave component of the incident field, both the reflected and re-fracted fields are also plane waves. The wave vectors of the outgoing waves are bothin the plane of incidence, defined by the incident wave and the surface normal. Ifthe incident wave impinges at an angle θin with respect to the surface normal, thelaws of reflection and refraction require [Born99, §1.5.1]

θR = −θin and (2.1)

sin(θT ) =n1n2

sin(θin). (2.2)

The relation between the field amplitudes is given by Fresnel’s equations [Born99,§1.5.2]. Due to their dependence on the geometry of the problem, it is convenientto express the electric field E = E‖+E⊥ in components parallel and perpendicularto the plane of incidence. The transmitted field

(ET‖ET⊥

)=

(τ‖ 0

0 τ⊥

)(Ein‖

Ein⊥

)(2.3)

20

Figure 2.2.: Beam shifts occurring for a Maxwell-Gaussian light beam to-tally reflected inside a glass prism as a function of the incidentstate of polarization (a,

√1− a2 exp(iδ)). The approximate locations

of linear (H, V, ±45) and circular (R, L) states of polarization are in-dicated by text labels. (a) Longitudinal displacement or Goos-Hanchen(GH) shift. (b) Transverse displacement or Imbert-Fedorov (IF) shift.The angle of incidence θ = 60 is sufficiently far from the critical angleand the refractive indices n1 = 1.5 and n2 = 1 are typical for glass inthe visible and infra-red regime. Both plots share the same color coding,which illustrates that the GH effect is significantly larger then the IFshift. Also, for the GH case, the dependence on the state of polarizationis weak, while the IF effect depends strongly on the relative phase δ.

and the reflected one

(ER‖ER⊥

)=

(ρ‖ 0

0 ρ⊥

)(Ein‖

Ein⊥

)(2.4)

are related to the electric field vector Ein of the incident wave via the so-called

21

Fresnel coefficients:

τ‖ =2n1 cos(θin)

n2 cos(θin) + n1 cos(θT )(2.5a)

τ⊥ =2n1 cos(θin)

n1 cos(θin) + n2 cos(θT )(2.5b)

ρ‖ =n2 cos(θin)− n1 cos(θT )

n2 cos(θin) + n1 cos(θT )(2.5c)

ρ⊥ =n1 cos(θin)− n2 cos(θT )

n1 cos(θin) + n2 cos(θT ). (2.5d)

Both the Goos-Hanchen and the Imbert-Fedorov shifts were first discovered fortotal internal reflection (n1 > n2 and θin > θcritical = arcsin(n2/n1)). Equations(2.5c) and (2.5d) hold true for this case if we take into account that cos(θT ) =√

1− sin(θT )2 is imaginary [Born99, §1.5.4].We explicitly calculate the Goos-Hanchen shift for s polarization (u = y). For this

case, consindering wave vectors in the plane of incidence suffices (κy = 0). Thus, thepolarization vector (1.9) describing the incident beam reduces to e(κx, κy = 0) = y.As a consequence, the relevant electric field vector E‖y is strictly transverse andchanges according to the relevant Fresnel equation (2.5d):

Ein

(κx, κy = 0) = ψ(κ) e(κ) = ψ(κ) y → Eout

(κx, κy = 0) = ψ(κ) ρ⊥ y (2.6)

For total internal reflection, the modulus square |ρ⊥|2 = 1 is unity and the effecton the light field (2.6) amounts to a phase factor [Born99, §1.5.4]

ρ⊥ = exp

−2i arctan

√sin2(θ(κx))− n2

2

n21

cos(θ(κx))

, (2.7)

where

cos(θ(κx)) = κin · z′ = cos(θin) + sin(θin)κinx = cos(θin)− sin(θin)κRx . (2.8)

Computing the Fourier transform or recalling equation (1.12), one immediately rec-ognizes the Goos-Hanchen shift

⟨xR⟩

= +λ

π

n1 sin2(θi)√n21 sin2(θi)− n22

. (2.9)

Our results confirms the formula found 1944 by Artmann [Artmann48, §2] in re-sponse to the original work by Goos and Hanchen [Goos47]. In modern experiments,this displacement is routinely reproduced [Gilles02]. Generally, the Goos-Hancheneffect depends on the state of polarization (Figure 2.2(a))

22

Note that the sign of⟨xR⟩

in (2.9) is a consequence of the choice of our referenceframe (Figure 2.1), which is not consistent across the literature. While our calcu-lation assumes a perfect spatially coherent mode, the degree of coherence actuallypresent in early experiments is not clear. Only recently, it was confirmed experi-mentally that the degree of spatial coherence does not influence such spatial beamshifts [Loffler12].

The angular spectrum method used in this work was didactically presented byMcGuirk and Carniglia [McGuirk77]. It is straightforward to generalize this methodto different media, e.g. metals [Merano07], and include the case of partial reflectionand transmission. The literature building upon this result is extensive [Aiello12,Bliokh13].

Technically, we have found it useful to perform all calculations in Fourier space.However, it is possible to apply the boundary conditions directly in real space[Bekshaev12], which may make these phenomena more accessible for a broader au-dience.

2.2. Imbert-Fedorov shift and spin Hall effect of light

While the Goos-Hanchen effect describes a purely longitudinal displacement⟨x⟩,

light beams interacting with an interface can also undergo a transverse shift⟨y⟩

originally predicted by Fedorov [Fedorov55] and experimentally demonstratedby Imbert [Imbert72]. From a modern, pragmatic point of view, both con-ventional beam shift phenomena result analogously from the boundary condi-tions [Player87, Li07, Aiello08]. It was also shown experimentally that bothshifts occur simultaneously for a single reflection [Pillon04]. Nevertheless, whilethe Goos-Hanchen effect was observed and theoretically explained in the 1940s[Goos47, Artmann48, Goos49], the debate about the Imbert-Fedorov shift was onlysettled recently [Bliokh06, Bliokh07, Aiello12, Bliokh13].

Some of the reasons, why historically the transverse shift was more difficult toasses experimentally can be seen in figure 2.2(b). The magnitude of the displace-ment is typically smaller than the longitudinal shift (Figure 2.2(a)) and dependsstrongly on the state of polarization. Consequently, the Imbert-Fedorov effect van-ishes for unpolarized light. The first successful demonstration of the transverse shift[Imbert72] adopts Goos and Hanchen’s idea to use multiple reflections in order toenlarge the effect. However, due to the fact that the state of polarization generallychanges under reflection, the experiment had to be carefully designed to preservethe handedness of the light beam.

Theoretically understanding the Imbert-Fedorov shift was challenging since it de-pends not only on the properties of the interface, but also on the physical structureof the light beam itself. While the Goos-Hanchen shift can be explained qualitativelywith a two-dimensional scalar theory, both the Imbert-Fedorov effect as well as thegeometric SHEL, require a three-dimensional vectorial treatment, which properlyaccounts for polarization.

23

The angular spectrum method explained in the previous section remains valid forthe transverse shift. Applying the standard Fresnel formulæ (2.5) to our prototypelight beam, predicts the Imbert-Fedorov shift correctly [Li07, Aiello08]. Interest-ingly, Fresnel’s equations can also be rewritten in a basis adopted for circular polar-ization, rendering the calculation of this Imbert-Fedorov shift completely analogousto the Goos-Hanchen case [Player87].

Careful investigation of the electric field reveals that a light beam reflected fromor transmitted across an interface is not exactly equivalent to a displaced beam[Bliokh06, eq. (4)]. In fact, a linearly polarized Gaussian beam transmitted acrossan air-glass interface becomes a superposition of two beams with different helicitiessymmetrically displaced with respect to each other. Due to the striking similarity tothe spin Hall effect experienced by charge carriers, the splitting has become knownas the “spin Hall effect of light” (SHEL) [Onoda04, Hosten08]. This spin-dependentdisplacement amounts only to a fraction of the wavelength. Nevertheless, using acrossed polarization analyzer, this effect is observable [Bliokh06, Hosten08]. Hostenand Kwiat describe this polarization enhancement ingeniously as a quantum weakmeasurement. In their experiment, they additionally make use of beam propagationto enhance the signal [Aiello08].

2.3. Angular momentum conservation and conventionalbeam shifts

When interacting with matter, the angular momentum carried by the light fieldis not necessarily conserved. Generally, energy and linear and angular momentumcan be transferred from photons to other systems. However, the physical nature ofthe interaction and symmetry of the problem imposes constraints. Thus, one canestablish particular conservation laws for a light beam propagating across differentkinds of interfaces.

Traditionally, beam shifts are studied for a planar interface between non-absorbing, isotropic media. This problem is rotationally symmetric with respectto the surface normal z′. As a consequence, the normal component jn = j · z′ of thelight field’s angular momentum is conserved [Player87, Bliokh06, Bliokh07]. Thus,the angular momentum of the incident field, for example the spin of a circularlypolarized beam, determines the normal component jn of the outgoing field, i.e. ofthe refracted and reflected light beams. The latter includes an extrinsic contributionconnected to the position of these beams.

This conservation law for jn is a necessary condition for any beam shift occurringat such an interface. In particular, for the special case of total internal reflection,the transverse Imbert-Fedorov shift can be derived solely from angular momentumconservation [Bliokh13].

If the axial symmetry is broken, the conservation of jn does not hold. In his famousexperiment, Richard Beth showed that transfer of angular momentum from photonsto a birefringent wave plate is indeed possible [Beth36, Beijersbergen05]. Here, the

24

light field exercises an observable torque on the anisotropic plate. The polarizerused in our experiment exhibits a comparable symmetry break. Consequently, theconservation laws governing conventional beam shifts do not apply to the polarizinginterfaces. To the contrary, the geometric SHEL is connected to the conservation oftransverse angular momentum.

2.4. Geometric spin Hall effect of light and transverseangular momentum

Geometric spin Hall effect of light as field of research started with a discussion oflinear and angular momentum in a tilted reference frame x′, y′, z′. In this context,the intensity observed by a detector aligned with the x′-y′-plane was identified withthe longitudinal component pz′ = p · z′ of the linear momentum density or Poyntingvector [Aiello09a].

This “intensity” distribution pz′ has the surprising property that its centre of mass(evaluated at z′ = 0)

⟨r′⊥⟩pz′

=

∫∫r′⊥pz′(r

′⊥) d2 r′∫∫

pz′ d2 r′=

1

Pz′

(−Jy′Jx′

)(2.10)

is given directly by the transverse components Jx′ and Jy′ of the light beam’s angularmomentum (equations (4) and (5) in [Aiello09a]). This fundamental connectionoriginates from the definition of angular momentum and becomes evident if weexplicitly write the components

jx′

jy′

jz′

=

y′ pz′−x′ pz′

x′ py′ − y′ px′

(2.11)

thereof at z′ = 0 and substitute r′⊥pz′ in equation (2.10).

Throughout this work, we employ a tilted reference frame with x′ = cos(θ) x −sin(θ) z and y′ = y. For our prototype light beam (1.3), that is a Maxwell-Gaussianbeam propagating freely in direction of z, the centroid of the linear momentumdensity in the tilted reference frame is

⟨r′⊥⟩pz′

=J inx′

P inz′y′ =

λσ

4πtan(θ) y′, (2.12)

where J inx′ is the projection of the intrinsic spin angular momentum.

Since linear and angular momenta are functions of the electric field, the beamshift (2.12) can also be found by investigating the structure of the electric field. Tothis end, we decompose the electric field and the corresponding energy density

|E(r′⊥)|2 = |Ex′(r′⊥)|2 + |Ey′(r′⊥)|2 + |Ez′(r′⊥)|2 (2.13)

25

into the Cartesian components |El′(r′⊥)|2 = |E(r′⊥) · l′|2 defined with respect to our

tilted reference frame l′ ∈ x′, y′, z′.

The spatial profile of each component is approximately Gaussian. However, therespective barycentres do not coincide [Korger11]. In particular, the centre of massof the x′-component undergoes a displacement

⟨r⊥⟩|Ex′ |2

=λσ

2πtan(θ) y′ (2.14)

resembling the one found for the Poynting vector.This motivates the use of a polarizing element to make the beam shift (2.14)

visible. In the following chapter, the connection between real polarizers, polarizermodels and the geometric spin Hall effect of light is discussed.

26

3. The geometric spin Hall effect of lightat a polarizing interface

This thesis focusses on the geometric spin Hall effect of light. In particular, we studythis fundamental phenomenon occurring for a light beam propagating across a tiltedpolarizer. To this end, a physical description of such polarizing interfaces is vital.

In textbooks, e.g. [Hecht01, §8.2] or [Born99, §15.6.3], polarizers are typicallyintroduced as operating on plane waves. Furthermore, it is implicitly assumed thatthe incident wave impinges perpendicularly onto the entrance face of the polarizer.For our purpose, this approach is insufficient. As shown in chapter 1, physical lightbeams are superpositions of multiple plane waves and beam shifts occur if the lightfield acquires a wave-vector-dependent phase.

In the present work, two different implementations of a polarizing interface werestudied. At early stages, we worked with a polymer film sandwiched between twoglass plates. Then, we proceeded with a Corning Polarcor polarizer, a glass sub-strate with embedded silver nano-particles, which was used for the majority of themeasurements discussed in this work. Both polarizers were extensively characterizedincluding a full Mueller matrix tomography [Kolb10, Korger13b].

Theoretically, we have employed different strategies to find suitable models com-patible with our observation. One way is to start with a model that predicts infiniteextinction ratios and, then, generalize this to include phenomenological parameters(section 3.1). Alternatively, one can make use of a microscopic model for the physi-cal absorption process and describe the polarizer as an ensemble of such elementaryabsorbers (section 3.2). Both approaches yield useful and realistic models.

The pinnacle of this chapter is the calculation of beam shifts including but notlimited to the configuration studied experimentally (section 3.3).

We conclude by connecting this beam shift to Berry’s geometric phase (section3.4) and transverse angular momentum (section 3.5). Both concepts were relevantto the development of the geometric SHEL theory.

3.1. A useful generic polarizer model

It is generally expected that a beam transmitted across a linear polarizer is polarizedalong a given axis and that the transmission through a pair of crossed polarizers van-ishes. This property can be described as a projection onto an effective transmittingaxis t:

Eout = ttTEin. (3.1)

27

Figure 3.1.: Visualization of the projection rule used with our generic po-larizer model. Generally, the action of the polarizer is governed by avector P , either identified with the absorbing or the transmitting axis,and its projection p1 onto the plane of the electric field. (a) For theabsorbing case, the axis P is the orientation of the electric field locallyabsorbed at the polarizer. Since propagation in direction of z requiresthe fields to be in the x-y-plane, the projection p1 can be interpretedas an effective absorbing axis. Consequently, for sufficiently high extinc-tion ratios, the transmitted field Eout is directed along p2 ⊥ p1. (b)Contrarily, for the transmitting case, we choose P perpendicular to thelocal direction of absorption and interpret the projection p1 as an effec-tive transmitting axis. The prediction of both models coincide only ifthe direction of propagation z is perpendicular to the polarizer surface.

In this chapter, we routinely make use of the dyadic product

ttT

=

t2x tx ty tx tzty tx t2y ty tztz tx tz ty t2z

(3.2)

representing a projection operator.For our purpose, we have to account for the three-dimensional geometry of the

problem. Interestingly, there exist two geometric polarizer models [Fainman84,Korger13b], which are closely related from an abstract mathematical point of view,but yield to completely different predictions.

Fainman and Shamir suggested to describe a polarizer by a unit vector P inter-preted as its transmitting axis [Fainman84, Aiello09b]. In their model, the effectivetransmitting axis t = p1(κ) ∝ P − (κ · P ) κ is found by projecting the global vec-tor P onto the local plane of the electric field, i.e. the plane perpendicular to thedirection of propagation κ (Figure 3.1b).

However, the discussion of our experimental results [Kolb10, Chille11, Korger13a]prompted the rejection of Fainman’s transmitting model. We found that the action

28

Figure 3.2.: State of polarization transmitted across two different polariz-ers; each compared to the absorbing and transmitting polarizermodels. We show the polarizance Stokes vector [Lu96], this is the stateof polarization of the transmitted light field if the incident beam is unpo-larized. The experimental data (black circles) was acquired as a part ofour Mueller matrix measurements [Korger13b]. Both theoretical curvesare derived from our generic model (3.3). The solid blue line depictsthe case, where the absorbing axis is projected (τ1 = 0, τ2 = 1). Thedashed red line is constructed analogously using the transmitting axis(τ1 = 1, τ2 = 0), which is equivalent to the model suggested by Fain-man and Shamir. (a) Corning Polarcor, a glass plate embedded withmetal nano-particles with their absorbing axis oriented at φ = 85.5.(b) Techspec NIR linear polarizer, a polymer film layered between twoglass substrates with its absorbing axis oriented at φ = 22.0.

of both polarizers studied experimentally can be approximated using a similarlyconstructed absorbing model [Korger13b] (see figures 3.1(a) and 3.2). In this case,the unit vector P describes the orientation of local absorbers responsible for thepolarization effect. Then, the projection p1 thereof indicates an effective absorbingaxis. And consequently, the effective transmitting axis t = p2(κ) = κ × p1(κ) isperpendicular to p1.

Both models introduced above, referred to as transmitting and absorbing polariz-ers respectively, describe polarizing elements as projectors. As a consequence, bothmodels predict that the extinction ratio of a polarizer is only limited by the beam’sdivergence.

In this work, we make use of a generic polarizer model, which includes as spe-cial cases, both transmitting and absorbing polarizers. Additionally, our univer-sal approach allows to include phenomenological parameters accounting for im-perfect extinction ratios and other deviations from the simplified models (as in[Korger13a, Korger13b]).

29

In any case, the polarizing device is described by a three-dimensional unit vectorP = (− cos θ sinφ, cosφ, sin θ sinφ)T projected onto the electric field plane spannedby two unit vectors p1(κ) and p2(κ). While, the physical interpretation of thesevectors differs, the transmitted electric field can always be written as

Eout

(κ) =(τ1(θk) p1p

T1 + τ2(θk) p2p

T2

)E

in(κ). (3.3)

Here, the electric fields E is given in the angular spectrum representation (1.6). Thecoefficients τ are arbitrary real-valued functions of the tilting angle

θk(κ) = arccos(κ · z′) ' arccos(cos θ + κx sin θ) ≈ θ. (3.4)

Technically, the angle θk introduced above is the angle between the wave vector andthe polarizer normal z′. For collimated light beams, the parameters κx and κy aresmall and higher-order terms are negligible as discussed in section 1.2.

This universal approach is valid for a wide range of real and idealized polarizers.For example, if we set τ1 = 1 and τ2 = 0, equation (3.3) reduces to the transmittingmodel. Analogously, setting τ1 = 0 and τ2 = 1 yields the absorbing model used toqualitatively describe our observation.

For our shift measurements, we have employed a polarizer with its absorbingaxis P oriented horizontally. In this configuration, the modulus squared of the twophenomenological parameters can be directly measured. For horizontally polarizedincident light, the transmittance is τ21 (θ) and for vertical polarization, the transmit-tance is τ22 (θ).

In the context of our shift measurements [Korger13a], we have established the setof parameters

τ2(θ) = 1− 1.2 exp(−12 cos θ) and (3.5)

τ1(θ) = 0.51 exp(−1.3 cos θ), (3.6)

to adequately describe the polarizer used. A complete Mueller matrix tomographyof this device was reported in [Korger13b]. This includes a physical discussion ofthe result noted above.

3.2. Constructing a microscopic polarizer model compatiblewith empirical data

Practically, the phenomenological model described in the previous section sufficesto predict both intensity and centre of mass for a beam transmitted across ourpolarizer. Nevertheless it is insightful to connect the observed behaviour to a physicaldescription of the interaction. The Polarcor polarizer uses polarizing layers on bothfaces of the glass substrate. These 25 to 50µm thick layers are made of elongated andoriented silver nano-particles embedded in glass. For our purpose, we have chosen aproduct without the usual anti-reflection coating since large angles of incidence aredesired for our experiment and require the use of an index-matching liquid.

30

Figure 3.3.: Illustration of a light beam propagating across a tilted polar-izer with multiple layers of absorbing nano-particles orientedhorizontally. Assuming the refractive index of the polarizer substratematches its environment as in our experiment, the tilted entrance facehas no effect on the field Ein of the incident beam. Locally, at the po-sition where the light interacts with a nano-particle, the electric fieldcomponent parallel to the particle’s absorbing axis vanishes. Conse-quently, the local field Elocal after the interaction is no longer trans-verse with respect to the direction z of wave propagation. However,after propagating for a distance of multiple wavelengths, the transver-sality of the electric field E1 is restored. Then, when the light beamencounters a another such particle, the process is repeated. In each stepEin → E1 → E2 → · · · → EN = Eout the the orientation of the electricfield vector changes slightly. For sufficiently large N , the transmittedfield is polarized almost perpendicularly to the absorbing axis.

In this section, we attempt to construct a simple microscopic model compatiblewith our observation. The embedded nano-wires can be thought of as microscopicabsorbers interacting locally with the electric field. The orientation A of theseabsorbers define the component A ·E of the electric field which couples to the nano-particles and is consequently scattered or absorbed. As illustrated in figure 3.3, thisresults in a local electric field vector Elocal, which is no longer transverse with respectto the wave vector k. The transmitted wave is observed to propagate along theoptical axis κ in the far field. Thus, we conjecture that the longitudinal componentElocal · κ is lost after propagating a distance exceeding the order of the wavelength.

The two steps described above can be expressed as a projection rule

E1(κ) =

(1− κκT

)(1− AAT

)E

in(κ) =

(1−A⊥AT

⊥)E

in(κ), (3.7)

where A⊥ = A − (κ · A) κ. Generally, this differs from our generic model (3.3)since A⊥ 6= p1. An exception to this rule is normal incidence. In this case, the

31

Figure 3.4.: State of polarization transmitted across the nano-particle po-larizer compared to our microscopic model. Black circles showthe experimentally observed polarizance vector as in figure 3.2(a). Alltheoretical curves are calculated from our microscopic model (3.8) fordifferent values of N . Assuming only one elementary absorption process,the predicted extinction ratio is much smaller than observed. One candirectly see from the plot that S2

1 + S22 + S2

3 < 1 for θ > 20. For largernumbers of N , the agreement with the experimental data is better, andfor N → ∞ the microscopic model approaches the absorbing limit ofour generic model (blue line in figure 3.2).

microscopic absorbers are oriented perpendicularly to the wave vector κ and equa-tion (3.7) coincides with both the absorbing and transmitting limits discussed inthe previous section. However, when tilted, the behaviour differs and the extinctionratio decreases significantly – faster than observed.

The size and density of the nano-particles embedded in our polarizer is not knownprecisely. A description of the manufacturing process [Borrelli93], x-ray scatteringdata [Polizzi98], and TEM images [Polizzi97] suggest that the nano-particle’s extentperpendicular to the elongated axis does not exceed 50 nm. On one hand, this allowsfor up to 1000 of such elementary absorbers, one after another interacting with alight beam propagating across a single polarizing layer. On the other hand, thepropagation distance between these absorbers can easily exceed multiple wavelengthsand grows if the polarizer is tilted. The latter justifies treating each absorptionprocess individually.

Thus, it is reasonable to assume that in both polarizing layers combined, a totalof N elementary absorption processes (3.7) occur (as illustrated in figure 3.3):

Eout

(κ) =[(

1− κκ)(

1− AA)]N

Ein

(κ) (3.8)

This effectively increases the extinction ratio. Choosing N = 35 yields good agree-ment with empirical data as shown in figure 3.4.

While the model (3.8) described in this section is constructed differently fromour generic polarizer (3.3), both allow for a phenomenological description of ourobservation. For sufficiently large extinction ratios N →∞, this microscopic model

32

agrees with our absorbing polarizer model introduced in section 3.1. The similaritybetween both approaches indicates that both are equally suitable for our purpose.

3.3. Beam shifts occurring at a tilted polarizer

A polarizer described by the projection rule (3.3) with the axis P =(− cos θ, 0, sin θ)T oriented horizontally gives rise to a vertical displacement calledthe geometric spin Hall effect of light [Korger11]. We can understand this by study-ing how a beam of light interacts with such a polarizer.

For any wave vector k ' k(κx, κy, 1), our universal polarizer model defines a

basis of two unit vectors, p1 = (−1,−κy tan θ, κx)T and p2 = (κy tan θ,−1, κy)T.

Generally, the effect of the polarizer is governed by projections onto these vectors.Applied to the electric field of the light beam (1.10), we obtain up to the first

order of the k-vector components:

Eout

(κ) '

τ1

a+ beiδκy tan θ

aκy tan θ−aκx

+ τ2

−beiδκy tan θbeiδ − aκy tan θ−beiδκy

ψ(κ)

=

aτ1beiδτ2

0

︸ ︷︷ ︸α(0,0)

+

beiδ(τ1 − τ2) tan θa(τ1 − τ2) tan θ−beiδτ2

︸ ︷︷ ︸α(0,1)

κy +

00

−aτ1κx

ψ(κ), (3.9)

where u = (a, beiδ) is the incident state of polarization. Here, the terms dependingon κy indicate a coupling between polarization and spatial degrees of freedom andhint at the occurrence of transverse beam shifts.

Applying the generic formulas (1.11) to this expression yields the centroid ofthe beam. The vector field Eout has both horizontally and vertically polarizedcomponents. It is insightful to calculate the two corresponding beam shifts,

⟨r⊥⟩|Ex|2 =

λ

2π

b sin δ

a

(1− τ2(θ)

τ1(θ)

)tan θ y and (3.10a)

⟨r⊥⟩|Ey |2 =

λ

2π

a sin δ

b

(1− τ1(θ)

τ2(θ)

)tan θ y, (3.10b)

individually. Experimentally, both shifts described by equations (3.10) can be ob-served employing an additional polarization analyser or polarizing beam splitter.If no analyser is used, both components of the energy density, |Ex|2 and |Ey|2,contribute to the observed displacement. In this case, the total shift

⟨r⊥⟩

=τ21 (θ)a2

τ21 (θ)a2 + τ22 (θ)b2︸ ︷︷ ︸Wx

⟨r⊥⟩|Ex|2 +

τ22 (θ)b2

τ21 (θ)a2 + τ22 (θ)b2︸ ︷︷ ︸Wy

⟨r⊥⟩|Ey |2 . (3.11)

33

is a weighted average of both relative shifts (3.10a) and (3.10b). The weightingfactors Wx and Wy are given by the fraction of the energy in each polarization mode(equation (1.11b)).

From this calculation, we learn that real polarizers with finite extinction ratiosgive rise to a plethora of beam shift phenomena. The predicted displacements de-pend on both the incident state of polarization and the polarization-dependence ofthe detection system (Figure 3.5). Nevertheless, these shifts have a common phys-ical origin, that is the projection rule (3.3) and the intrinsic structure of polarizedlight beams. This manifests itself in the tan(θ)-dependence characteristic for thegeometric spin Hall effect of light (Figure 3.6).

Comparing the above-mentioned formulas or figures to the Imbert-Fedorov effect(as shown in figure 2.2) reveals the striking difference to conventional beam shiftphenomena. For example, the geometric spin Hall effect of light is directly propor-tional to the helicity of the incident beam. Beyond the diameter of the incident beamitself, there appears to be no fundamental limit to the magnitude of the geometricSHEL (Figure 3.7).

3.4. Interpretation in terms of the geometric phase

As already noted at the beginning of this chapter, our generic polarizer model in-cludes two important special cases dubbed transmitting and absorbing polarizers.Both allow for a geometric interpretation of the transmitted state of polarization(as shown in figure 3.1). In this section, we connect those geometric aspects to theoccurrence of beam shifts. This helps to see in which configuration the geometricSHEL is observable and underlines the importance of choosing a suitable polarizermodel.

The geometric SHEL is a consequence of the geometric phase exp(i κy tan θ), alight beam acquires when passing through a tilted polarizing element [Bliokh12a,Korger13a]. To understand the origin of this phase term, consider a plane wavewith its wave vector k ' k (κy, 0, 1)T in the y-z-plane. For circular polarization, the

corresponding electric field is E(k) ∼ (1, i,−i κy)T, perpendicular to k.

Originally, the geometric spin Hall effect of light was connected to the transmittingpolarizer model suggested by Fainman and Shamir [Fainman84, Korger11]. Here,the sought phase term occurs if the transmitting axis P is oriented horizontally.In this case, the projection of the light field onto the effective transmitting axisp1 = (−1,−κy tan θ, 0)T yields (as in (3.9) with τ1 = 1 and τ2 = 0):

Eout

(κ) ' 1√2

1 + iκy tan θκy tan θ−κx

ψ(κ) ' 1√

2

exp(iκy tan θ)κy tan θ−κx

ψ(κ). (3.12)

As expected, the horizontally polarized component is dominant for well-collimatedbeams with small κ and has acquired a k-vector dependent phase (illustrated in

34

Figure 3.5.: Geometric spin Hall effect as a function of the incident andobserved state of polarization as predicted by our phenomeno-logical polarizer model. (a) Scheme of the setup. We assume theabsorbing axis is horizontal, and choose a moderate tilting angle ofθ = 60. (b) Centroid

⟨y⟩

of the total energy density. (c) Centroid⟨y⟩|Ex|2 of the horizontally polarized component thereof. (d) Centroid⟨

y⟩|Ey |2 of the vertically polarized component. Experimentally,

⟨y⟩

and⟨y⟩|Ey |2 were confirmed for circular states of polarization [Korger13a].

All plots share the same color coding.

35

Figure 3.6.: Geometric spin Hall effect of a light as a function of the tilt-ing angle as observed with an optional analyzer in front of thedetector. The relative shift when switching the polarization of the in-cident beam (λ = 795 nm) between the two states u = (a,±i

√1− a2)

is calculated from our phenomenological model and compared to ex-perimental data where available. (a) Displacement

⟨y⟩|Ex|2 of the hori-

zontally polarized field component. This beam shift contributes to thetotal displacement

⟨y⟩

as observed in our experiment without employ-ing an analyzer. In principle, this “giant” geometric SHEL can also beobserved directly. (b) Displacement

⟨y⟩|Ey |2 of the vertically polarized

field component as observed in our experiment with the optional ananalyzer oriented vertically.

figure 3.8). According to the Fourier transform shift theorem, the correspondingelectric field Eout(r) in the position space is displaced [Goodman05, Korger11]:

⟨r⊥⟩

=λ

2πtan θ y. (3.13)

For any other orientation φ of the polarizing axis, the predicted displacement issmaller and vanishes if the transmitting axis is vertical (Figure 3.9(a)).

However, as we learned from early experiments employing a polymer film[Kolb10, Chille11] and confirmed for our nano-particle polarizer later, this modeldoes not adequately describe polarization effects based on local absorption. Tounderstand, why this is significant for beam shifts, let us describe the situation ex-plained above with the absorbing model. Clearly, the absorbing axis P = y of apolarizer transmitting horizontally polarized light is oriented vertically. Indepen-dent of the rotation angle θ around the y-axis, for this configuration, the relevantprojection onto p2 = −x never yields a κy-dependent phase factor in first order(Figure 3.10(a)):

Eout

(κ) ' 1√2

10−κx

ψ(κ) (3.14)

36

Figure 3.7.: Fundamental limits of the geometric spin Hall effect of light.The energy density of a circuarly polarized light beam transmittedacross a tilted absorbing polarizer (θ = 89) is calculated for differ-ent values of the beam diameter w0. The dashed red line depicts the1/e2-width of the incident light beam. Here, terms up to θ20 relevant forthe distortion of the beam have been included. (a) For a reasonably col-limated beam, the beam waist, here w0 = 30λ, is large compared to thewavelength λ. Thus, the expected displacement

⟨y⟩≈ 9λ is relatively

small and the transmitted beam remains approximately Gaussian. (b)At w0 = 20λ, the beam becomes visibly distorted. (c) For w0 = 10λ, oneclearly sees that geometric SHEL causes a redistribution of the energydensity within the boundaries of the incident beam. Here, the first-orderapproximation breaks down and the distortions becomes dominant.

Thus, a collimated light beam will not undergo a displacement.

Nevertheless, as demonstrated by our experiment [Korger13a], absorbing polariz-ers are equally suitable to study the geometric spin Hall effect of light. Here, thegeometric phase term occurs if the absorbing axis is oriented horizontally (Figure3.10(b)):

Eout

(κ) ' 1√2

−κy tan θ

1 + iκy tan θ−κy

ψ(κ) ' 1√

2

−κy tan θ

exp(iκy tan θ)−κy

ψ(κ). (3.15)

For this configuration, an absorbing polarizer with a sufficiently high extinctionratio, result in a displacement

⟨r⊥⟩

=λ

2πtan θ y. (3.16)

resembling the one found originally with Fainman model. Figure 3.9(b) illustrateshow the predicted shift changes for an arbitrary orientation of the absorbing axis.

37

Figure 3.8.: Light field in the angular spectrum representation before andafter interacting with a transmitting polarizer oriented hor-izontally. (a) The phase of the electric field of a circularly polar-ized Gaussian light beam as a function of the transverse wave vector

k = k (κx, κy,√

1− κ2x − κ2y). (b) Visualization of the electric field vec-

tor before and after interaction with the polarizer (τ1 = 1, τ2 = 0,P = x′). Note that the phase of the transmitted field depends on theorientation of the k-vector. The phase offset between k0 and k′ is the ge-ometric phase term relevant for the geometric SHEL. (c) Electric fieldof the transmitted horizontally polarized light field. The color-codedphase depends linearly on κy. The Fourier transform thereof yields anelectric field in position space, which is displaced with respect to theoriginal one.

3.5. Interpretation in terms of transverse angularmomentum

Historically, the proposal of the geometric spin Hall effect of light was connectedto the occurrence of transverse angular momentum Jx′ in a tilted reference frame(section 2.4). Here, we demonstrate the relevance of this quantity for our experiment.To this end, we choose a reference frame x′, y′, z′ aligned with the polarizer suchthat x′ = P = (− cos θ, 0, sin θ)T is the absorbing axis and z′ is the surface normal.

The angular momentum J in of the incident light field, a circularly polarized Gaus-sian beam, is parallel to the direction z of beam propagation. Formally, in the tiltedreference frame, this yields to a transverse angular momentum

J inx′ = J in · x′ = P in

z

λσ

4πsin θ = P in

z′λσ

4πtan θ, (3.17)

consistent with (2.10). In the last step above, we have employed the relation P inz′ =

P inz cos θ, which results from the rotation of the reference frame.

After interaction with the polarizer, the resulting light field is linearly polarizedand does not carry intrinsic angular momentum: Jout · z = 0. However, as shown byour experiment [Korger13a] and the corresponding theory (section 3.3), the trans-mitted beam is displaced with respect to the optical axis of the incident beam. Forthe idealized case, assuming an infinite extinction ratio (τ1 = 0, τ2=1), the centre

38

Figure 3.9.: Geometric spin Hall effect of light as a function of the ori-entation of the polarizer used for two special cases of ourgeneric model. (a) Beam shifts occurring at a hypothetical trans-mitting polarizer (τ1 = 1, τ2 = 0) for an arbitrary orientation P =(− cos θ sinφ, cosφ, sin θ sinφ)T of its transmitting axis. Here, the dis-placement is most pronounced if the polarizer is oriented to trans-mit horizontal polarization. (b) Beam shifts occurring at an absorb-ing polarizer (τ1 = 0, τ2 = 1) for an arbitrary orientation P =(− cos θ sinφ, cosφ, sin θ sinφ)T of its absorbing axis. The displacementis perpendicular to its effective absorbing axis p1 and the magnitudereaches its maximum if the horizontal field component is absorbed. Ex-perimentally, we studied a configuration where the transmitted state ofpolarization is vertical and confirmed the prediction of the absorbingmodel (b).

of mass is:

⟨rout⊥

⟩=λσ

2πtan(θ) y. (3.18)

As discussed in section 1.4, this – like any beam shift – results in the occurrence ofextrinsic angular momentum

Jout = P outz

⟨rout⊥

⟩x = P out

z

λσ

2π

(sin(θ)x′ − sin(θ)

cos2(θ)z′)

. (3.19)

39

Figure 3.10.: Light field in the angular spectrum representation before andafter interacting with an absorbing polarizer. (a) The phaseof the electric field of a circularly polarized Gaussian light beam as

a function of the transverse wave vector k = k (κx, κy,√

1− κ2x − κ2y).(b) Visualization of the electric field vector before and after interactionwith the polarizer (τ1 = 0, τ2 = 1, P = y) aligned such that horizon-tally polarized light is transmitted. Note that, here, the transmittedfield is in-phase and no k-vector depend effect occurs. (c) Electric fieldof the transmitted horizontally polarized light field. The color-codedphase is completely flat. As a result, the field in position space is notdisplaced. (d) Visualization of the electric field vector before and af-ter interaction with the polarizer (τ1 = 0, τ2 = 1, P = x′) alignedsuch that vertically polarized light is transmitted. As in figure 3.8,the phase of the transmitted field depends on the orientation of thek-vector. (e) Electric field of the transmitted vertically polarized lightfield in k-space. The color-coded phase depends linearly on κy and,thus, the transmitted field in real space is displaced in the y-direction.

Taking into account that P inz = 2P out

z , one sees that the transverse angular momen-tum

Joutx′ = Jout · x′ = P in

z

λσ

4πsin(θ) = J in

x′ (3.20)

in the reference frame aligned with the polarizer is conserved. This shows thatthe beam shift, we have observed is indeed a manifestation of transverse angularmomentum.

40

Figure 3.11.: Angular momentum of a light beam evaluated in a tilted refer-ence frame before and after interaction with a tilted polarizer.The absorbing axis x′ of the polarizer is oriented horizontally at an an-gle of θ = 60 with respect to the direction z of beam propagation.The incident beam is circularly polarized and its angular momentumJ in is directed along the z-axis. After interaction with the polarizer,the extrinsic angular momentum Jout is perpendicular to propagationdirection z. However, the transverse angular momentum J in

x′ = Joutx′

relevant for the geometric SHEL is conserved.

41

42

Conclusion and outlook

This thesis is centred around the experimental demonstration of the geometric spinHall effect of light. In particular, this novel beam shift phenomenon has been studiedfor a beam of light passing across a tilted polarizing interface.

To this end, we have designed and set up a physical implementation of such aninterface. A glass polarizer made of oriented, elongated metal nano-particles wasfound suitable for the purpose. This polarizer was submerged in a tank filled withan index-matching liquid and characterized extensively.

Since the sought beam shift depends subtly on the type and orientation of the po-larizer used, it was of utmost importance to establish a reasonable model compatiblewith the observation. In this work, three relevant models were presented: A simplegeometric model provided a useful qualitative approximation for both the nano-particle polarizer and other polymer-film based products. Our phenomenologicalmodel accurately predicted the observed beam shift. And, finally, the microscopicapproach connected the observation to a physical description of the interaction.

A polarizer was found suitable for our purpose if its operation on a circularlypolarized beam yields a geometric phase term in Fourier space. This wave vectordependent phase was shown to result in a displacement in real space. In principle,a wide range of possible polarizers share this property including those describedby the aforementioned models. Nevertheless, those models differ and our genericcalculation of beam shifts stresses that it is vital to choose a suitable model in orderto predict beam shifts occurring at a real-world polarizer correctly.

After having established a suitable polarizer model, an experiment was set upto measure the polarization-dependent displacement of the transmitted beam. Theobservation confirmed the geometric spin Hall effect of light as predicted by ourtheory. In particular, the characteristic dependence on the incident polarization andthe tangent of the tilting angle was demonstrated.

The theory presented in this work shows that real-world polarizers with finite ex-tinction ratios give rise to a plethora of beam shift effects. As a consequence of theimperfect nature of the polarizer used, the displacements observed experimentallyso far are slightly smaller than expected from an idealized calculation. Interestingly,such imperfect polarizers can also incur significantly larger beam shifts if used in adifferent configuration (as shown in section 3.3). Using a suitable combination ofincident and detected state of polarization, a “giant” geometric spin Hall effect ex-ceeding 10 wavelengths was predicted (Figure 3.5(c)) and is waiting for experimentalconfirmation.

While this theses focusses on collimated light beams, where transverse angularmomentum manifests itself as a beam shift, related phenomena can be observed

43

with a tightly focussed spot. In fact, there exists a state of the light field withpurely transverse angular momentum [Banzer13]. Here, the interplay between thepolarization structure of a suitable tailored vector beam yields a distorted field inthe focal plane connected to the geometric SHEL [Neugebauer13].

Despite the novelty of the topic, there exist already theoretical works buildingupon the geometric SHEL. For example, Kong et. al. study the effect of orbitalangular momentum on this geometric Hall effect of light [Kong12]. And, Bliokhand Nori, propose a relativistic Hall effect, occurring for a light beam in free space,observed from a moving reference frame [Bliokh12b]. They note that this relativis-tic effect is formally equivalent to the geometric SHEL and provide a convenientgeometric interpretation.

44

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50

A. Publications

A.1. Geometric Spin Hall Effect of Light at polarizinginterfaces

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs.Geometric Spin Hall Effect of Light at polarizing interfaces. arXiv:1102.1626. Ap-plied Physics B, 102(3), 427–432, 2011

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52

Geometric Spin Hall Effect of Light at Polarizing Interfaces

Jan Korger · Andrea Aiello · Christian Gabriel · Peter Banzer · Tobias Kolb ·Christoph Marquardt · Gerd Leuchs

February 6, 2011

Abstract The geometric Spin Hall Effect of Light (geomet-ric SHEL) amounts to a polarization-dependent positionalshift when a light beam is observed from a reference frametilted with respect to its direction of propagation. Motivatedby this intriguing phenomenon, the energy density of thelight beam is decomposed into its Cartesian components inthe tilted reference frame. This illustrates the occurrence ofthe characteristic shift and the significance of the effectiveresponse function of the detector.

We introduce the concept of a tilted polarizing interfaceand provide a scheme for its experimental implementation.A light beam passing through such an interface undergoes ashift resembling the original geometric SHEL in a tilted ref-erence frame. This displacement is generated at the polarizerand its occurrence does not depend on the properties of thedetection system. We give explicit results for this novel typeof geometric SHEL and show that at grazing incidence thiseffect amounts to a displacement of multiple wavelengths, ashift larger than the one introduced by Goos-Hanchen andImbert-Fedorov effects.

PACS 42.25.Ja · 42.79.Ci · 42.25.Gy

Jan Korger · Andrea Aiello · Christian Gabriel · Peter Banzer · TobiasKolb · Christoph Marquardt · Gerd LeuchsMax-Planck-Institute for the Science of Light, Guenther-Scharowsky-Str. 1/Bau 24, 91058 Erlangen, GermanyTel: +49 9131 6877 125Fax: +49 9131 6877 199E-mail: jan.korger@mpl.mpg.de

Jan Korger · Andrea Aiello · Christian Gabriel · Peter Banzer · TobiasKolb · Christoph Marquardt · Gerd LeuchsInstitute of Optics, Information and Photonics, University Erlangen-Nuremberg, Staudtstr. 7/B2, 91058 Erlangen, Germany

1 Introduction

It is well-known that a beam of light transmitted throughor reflected from a dielectric interface undergoes a polar-ization-dependent shift of its spatial intensity distribution.The so-called Goos-Hanchen (GH) [1,2] effect amounts toa longitudinal shift, i.e. a displacement in the plane of in-cidence, while the Imbert-Fedorov (IF) shift [3] can be ob-served transverse to this plane. These positional shifts areconnected with angular counterparts [4,5]. Both, the GH [6–10] and the IF shift [11–14] have been verified experimen-tally in a number of configurations while also the theoreti-cal understanding of those effects has advanced significantly[15–17].

The IF shift is also known as the Spin Hall Effect ofLight (SHEL) [18–20] due to its resemblance to the SpinHall Effect in solid state physics. It amounts to a displace-ment of a circularly polarized beam perpendicular to theplane of incidence, where the direction depends on thebeam’s helicity or photon spin. Consequently, a linearly po-larized beam will split into components of different helicity.

The geometric Spin Hall Effect of Light [21] is a novelphenomenon, which like SHEL amounts to a spin-depen-dent shift or split of the intensity distribution of an obliquelyincident light beam. This effect depends significantly on thegeometric properties of the detection system and, beyond thedetection process, no light-matter interaction is required.

This article is structured as follows: In the following sec-tion the original geometric SHEL is reviewed and results notexplicitly given in [21] are provided. In sections 3 and 4 weintroduce a theoretical model for an arbitrarily oriented pla-nar polarizing interface [22,23] and provide a scheme for itsexperimental realization. Finally, we find that a light beamcrossing a tilted polarizing interface undergoes a shift twiceas large as the one found in the case of a tilted referenceframe. Therefore, the methods presented in this article lead

2 Jan Korger et al.

θz´

x´

z

x

Fig. 1 Geometry of the problem: A Gaussian laser beam (red) propa-gating in direction of z′ is observed in a plane (x, y) tilted with respectto z′. The direction y = y′ is perpendicular to the drawing plane.

to a straightforward measurement of the geometric Spin HallEffect of Light.

2 Geometric Shift in a Tilted Reference Frame

The geometric Spin Hall Effect of Light [21] occurs whena circularly polarized beam of light is observed in a planenot perpendicular to its direction of propagation. This effectamounts to a spatial shift of the intensity distribution withthe intensity being defined as the flux of the Poynting vec-tor through the detector plane. We stress that the geometricSHEL only depends on the geometry of the setup (Fig. 1),the state of polarization and the effective response functionof the detector.

Here we give explicit results for a fundamental Gaussianlight beam traveling in direction of k = sin(θ) x+ cos(θ) zdetected in the plane (x, y). The normal z to the detectorsurface and the propagation vector k ∦ z unambiguously de-fine a plane of incidence. As shown by Aiello et al. [21],the intensity barycenter or centroid of a circularly polarizedbeam is shifted in direction of y perpendicular to the planeof incidence. This displacement is equal to

〈yTD〉S =λ4π

σ tan(θ). (1)

The superscript S indicates that the centroid was evaluatedwith respect to the Poynting vector flux while the subscriptTD refers to a tilted detector. The shift depends on the helic-ity σ = ±1 (for left or right hand circular polarization) andis prominent at grazing incidence θ → 90 where it amountsto a displacement larger than the wavelength λ .

In [21] it was noted that the energy density (ED) distri-bution exhibits no such effect

(〈yTD〉ED = 0

), which under-

lines the dependence on the detector response. The apparentdiscrepancy can be understood by decomposing the electricfield energy density

u(r⊥) = |E(r⊥)|2 = ∑l=x,y,z

|El(r⊥)|2 (2)

into terms depending on one Cartesian component of theelectric field in the detector reference frame (x, y, z) only.u(r⊥) is a distribution in the observation plane and r⊥ =

x x+ y y is a two-dimensional position vector. This decom-position is depicted in Fig. 2.

Analogously to (2), we decompose the barycenter〈yTD〉ED of the energy density as

〈yTD〉ED =

∫∫y|E(r⊥)|2 dxdy∫∫ |E(r⊥)|2 dxdy

= ∑l=x,y,z

wl ∆l , (3)

where we define

wl :=∫∫ |El(r⊥)|2 dxdy∫∫ |E(r⊥)|2 dxdy

(4)

as the relative weight of the field component |El |2 and

∆l =

∫∫y|El(r⊥)|2 dxdy∫∫ |El(r⊥)|2 dxdy

(5)

as the contribution of this component to the total shift. Froma straightforward application of equations (4) and (5) tothe electric field distribution E(r) of a circularly polarizedGaussian light beam one finds

∆x =λ σ tan(θ)

2π+O(θ 2

0 ), (6a)

∆y = 0+O(θ 20 ), (6b)

∆z =−λ σ cot(θ)

2π+O(θ 2

0 ), (6c)

and within the same approximation

wx =12

cos2(θ), (7a)

wy =12

, (7b)

wz =12

sin2(θ), (7c)

where θ0 = 2/(k w0) = λ/(π w0) is the angular divergenceof the beam [24].

Substituting equations (6) and (7) into (3) we verify thatthe barycenter of a light beam’s energy density

〈yTD〉ED = ∑l=x,y,z

wl ∆l = 0 (8)

does not shift under rotation of the reference frame. Thisresult underlines the scalar nature of the energy density.

We remind the reader that the Poynting vector fluxthrough the detector surface

sz(r⊥) = s(r⊥) · z ∝ [E(r⊥)×B∗(r⊥)] · z, (9)

where z is the surface normal, is a distribution different fromu(r⊥) and exhibits a net shift

〈yTD〉S ∝ ∆x, (10)

where ∆x is given in (6a).

Geometric Spin Hall Effect of Light at Polarizing Interfaces 3

1

0

−1

y/w

0

−10 0 10x/w0

1

0

−1

y/w

0

−10 0 10x/w0

1

0

−1

y/w

0

−10 0 10x/w0

1 2 30 1 2 3 40 1 2 3 40101 · |Ey|

2 / |E0|2 101 · |Ez|

2 / |E0|2103 · |Ex|

2 / |E0|2

(a) (b) (c)

Fig. 2 Electric field energy density distribution of a Gaussian light beam (circular polarization) impinging obliquely (θ = ∠(k, z) = 85) ona detector. The components |Ex(r⊥)|2, |Ey(r⊥)|2, and |Ez(r⊥)|2 in the detector reference frame (x, y, z) are shown on color scales. |E0|2 is acommon normalization constant and w0 is the beam waist. (a) |Ex(r⊥)|2 is clearly shifted in the positive y direction. (b) |Ey(r⊥)|2 exhibits no suchshift. (c) While not visible in this pictorial representation, |Ez(r⊥)|2 is shifted in the negative y direction. Note that the relative weight wz of thiscomponent is more than two orders of magnitude larger than wx.

This connects the geometric SHEL with the fundamen-tal question about the local response of a position-sensitivedetector. The definition of the Poynting vector flux as theintensity is motivated by Poynting’s theorem. However, thischoice is debatable as the theorem does not define the lo-cal Poynting vector unambiguously [25] and the definitiongiven in equation (9) depends on the state of polarization.Contrarily, the response function of a real polarization-inde-pendent detector is more likely to be isotropic, i.e. to dependon u = |Ex|2 + |Ey|2 + |Ez|2, and thus yield 〈yTD〉ED = 0.

The shift ∆x can be detected directly if a detectionscheme is used where a plane of observation (x, y) can bechosen arbitrarily and the effective response function de-pends on |Ex|2 but not on |Ez|2. The polarization-dependentabsorption in semiconductor quantum wells [26–28] or sin-gle molecules [29,30] can in principle be used to build asuitable detector.

In the remaining part of this article we develop an alter-native strategy to measure the characteristic shift caused bythe geometric SHEL.

3 The Ideal Polarizer Model

The operation performed by a polarizing optical elementis commonly only determined for normally incident beamswhich can be approximated as planar wave fronts. In thiscase the action of a polarizer is described as a projectionwithin a plane perpendicular to the direction of propagationk = 1

|k| k. This simple model fails to describe the operationof a polarizing element when no such assumptions about thelight field are made, as it is the case in this article where wedeal with obliquely incident beams.

z´

x´

θP

k

Ex´

k

Ex´(a)

(b)

z

x

Index Matching Liquid

Dielectric PlatePolarizing Interface

incidentbeam

Fig. 3 (a) Interaction of a plane wave propagating in direction of z′with a tilted polarizer described by P = x = cos(θ) x′+ sin(θ) z′. Be-fore and after passing the polarizing element the electric field is per-pendicular to k. (b) Thin film polarizer submerged in a tank of indexmatching liquid. This scheme allows to study a tilted polarizing inter-face eliminating effects of physical boundaries.

To overcome this limitation, Fainman and Shamir (FS)have proposed a model describing an arbitrarily orientedideal polarizer [22]. They introduce a three-dimensionalcomplex-valued unit vector P and describe the action of apolarizer as follows: The electric field vector E(k) of eachplane wave of the incident beam’s angular spectrum is pro-jected onto eP, a unit vector perpendicular to k:

eP(k) =k× (P× k)√

1− (k · P)2=

P− k(k · P)√1− (k · P)2

(11)

E(k)→ eP(k)[e∗P(k) ·E(k)

](12)

The model is constructed such that the operation is idempo-tent and does not change k (Fig. 3a).

A remarkable characteristic of the FS polarizer is thatrotation (around y) has no effect on a single plane wave if P

4 Jan Korger et al.

is parallel or perpendicular to y. Hence, an ideal polarizingelement cannot be used to cause an imbalance between theweights wx and wz of the electric field component paralleland perpendicular to the polarizer surface.

However, as we will show in section 5, the rotation givesrise to an effect on bounded beams similar to the geometricSHEL in a tilted reference frame. In the following sectionwe will propose an experimental realization of an arbitrarilyoriented polarizing interface.

4 Experimental realization of a universal polarizinginterface

We propose a scheme using only off-the-shelf optical com-ponents to study the interaction of a light beam with a polar-izing element in any geometry (Fig. 3b). For this purpose wemodel a real polarizer as a composite device consisting ofan infinitely thin polarizing interface sandwiched betweendielectric plates with a refractive index n > 1. Since com-mercial thin film polarizers are typically protected from theenvironment by either a substrate or a coating on each face,our model is close to the realistic scenario.

The interaction of the light field with an air-dielectricboundary is polarization-dependent and changes the direc-tion of propagation k of a plane wave. Those well-knowneffects, described by Snell’s law of refraction and Fresnel’sformulas [31], are caused by the refractive index step. Atgrazing incidence refraction is so severe that inside the frontdielectric plate the angle between the direction of propaga-tion and the surface normal is always significantly smallerthan the corresponding angle θ in air. Therefore it is de-sirable to eliminate the change of the refractive index atthe physical boundary. This can be done, for example, byembedding the glass-polarizer-glass system in an index-matched environment. As a side effect this also eliminatesunwanted Imbert-Fedorov and Goos-Hanchen shifts.

It is not common for vendors to specify the behavior ofpolarization optics under non-normal incidence. Measure-ments in our laboratory using the proposed scheme indi-cate that the FS model is suitable to describe a tilted po-larizing interface. A detailed investigation will be reportedelsewhere.

5 Geometric Shift at a Polarizing Interface

In section 2 we described the geometric SHEL as an effectwhich occurs for a light beam in vacuum when the plane ofobservation is tilted with respect to the direction of propa-gation. The predicted shift depends on specific assumptionsabout the detection process and, therefore, cannot be easilyverified. In this section we shall show that an ideal polar-izer performs an operation on a light beam that amounts to

the characteristic geometric SHEL shift independent of theproperties of the detection system.

As in the case of the tilted detector, we assume the in-cident beam to travel in direction of k =: z′ and to have afundamental Gaussian profile with its barycenter at 〈x′〉= 0and 〈y′〉= 0, where (x′, y′, z′) is the beam’s natural referenceframe and y′ coincides with y (geometry as in Fig. 3). Thepolarizer shall be oriented along

P = x = cos(θ) x′+ sin(θ) z′. (13)

Using dimensionless coordinates x = x′/w0, y = y′/w0,z = z′/L and r′ = (x, y, z) where w0 denotes the beam waistand L = k w2

0/2 the Rayleigh length, the fundamental solu-tion of the paraxial scalar wave equation is:

ψ(r′) =1

1+ izexp(− x2 + y2

1+ iz

)(14)

Let u = 1√2(x′± i y′) be a complex unit vector denoting left

or right hand circular states of polarization. The electric fieldof a fundamental Gaussian beam can be written as

E(r′) ∝ exp(

i2zθ 2

0

)(u+

iθ0

2z(u ·∇⊥)

)ψ(r′), (15)

where θ0 = 2/(k w0) = λ/(π w0) is the angular spread of thebeam and ∇⊥ = (∂x,∂y) is the transverse gradient operator[32].

To apply the FS polarizer model (12), the electric field(15) must be expressed in its angular spectrum representa-tion E(k) and the beam after interacting with the tilted po-larizing element becomes:

E(r′) =∫∫∫

exp(ik ·r′)eP(k)[e∗P(k) ·E(k)

]d3k (16)

From the electric field distribution (16), we calculate theenergy density of a light beam after passing through a tiltedpolarizer. The Cartesian components thereof are depicted inFig. 4. Unlike in section 2, in this case the evaluation is per-formed in the beam reference frame (x′, y′, z′) where z′ = k.Decomposing the energy density barycenter 〈yTP〉ED as in(3) one finds

∆x′

w0=

θ0

2σ tan(θ)+O(θ 2

0 ), (17a)

∆y′

w0= 0+O(θ 2

0 ), and (17b)

∆z′

w0= 0+O(θ 2

0 ), (17c)

where σ =±1 is the helicity of the beam. Since we observea collimated light beam in its natural reference frame afterpassing through a linear polarizer, the weights

wy′ = 0+O(θ 20 ) and (18a)

wz′ = 0+O(θ 20 ) (18b)

Geometric Spin Hall Effect of Light at Polarizing Interfaces 5

1

0

−1

y´/w

0

−1 0 1x´/w0

1

0

−1

y´/w

0

−1 0 1x´/w0

1

0

−1

y´/w

0

−1 0 1x´/w0

(a) (b) (c)

0 5 10 15104 · |Ey´|

2 / |E0|2

0 5 10 15 20106 · |Ez´|

2 / |E0|2

0 2 4 86101 · |Ex´|

2 / |E0|2

Fig. 4 Electric field energy density distribution of a Gaussian light beam (circular polarization) after interacting with a tilted polarizer(P = cos(85) x′+ sin(85) z′

). The components |Ex′ (r⊥)|2, |Ey′ (r⊥)|2, and |Ez′ (r⊥)|2 in the beam’s natural reference frame (x′, y′, z′) are shown

on color scales. |E0|2 is a common normalization constant and w0 is the beam waist. (a) |Ex′ (r⊥)|2 is shifted as in Fig. 2a. (b), (c) |Ey′ (r⊥)|2 and|Ez′ (r⊥)|2 are not shifted and their relative weights are negligible.

vanish and, consequently,

wx′ = 1+O(θ 20 ). (18c)

The centroid of a circularly polarized beam transmittedacross a tilted polarizer is thus

〈yTP〉ED = 〈yTP〉S = ∆x =λ2π

σ tan(θ)+O(θ 20 ) (19)

and can be measured with any detector sensitive to anyweighted sum of |Ex′ |2, |Ey′ |2, and |Ez′ |2 if the weight of thefirst term does not vanish. Standard detectors such as photo-diodes and CCD cameras certainly meet this requirement.

We stress that equation (19) resembles the original result(1). The displacement introduced by the tilted polarizer

〈yTP〉S = 2〈yTD〉S (20)

is twice the one found for the Poynting vector flux througha tilted plane of observation. Furthermore, this article givesa straightforward recipe to measure the shift.

Since equation (5) from [21] is generally valid, bothshifts are connected to a transverse angular momentum,which occurs when the angular momentum calculated in thelocal frame attached to the light beam is projected upon aglobal frame tilted with respect to the former. For the geo-metric SHEL (occurring at a tilted detector), the projectionis implicitly given by the definition of intensity as the fluxSz = S · z of the Poynting vector across the detector surface.Conversely, in the case of the tilted polarizing interface, asdescribed in this section, the projection is caused by the po-larizer. Therefore, we can conclude that both shifts arise be-cause of the projection of the intrinsic longitudinal angularmomentum of a circularly polarized light beam onto a tiltedreference frame. We remind the reader that beyond those ge-ometric projections, no physical interaction occurs.

6 Conclusion

First, the original geometric Spin Hall Effect of Light, asdescribed by Aiello et al., was illustrated using an explicitdecomposition of a light beam’s energy density in a tiltedreference frame. We showed that in order to observe thiseffect, which occurs in vacuum and amounts to a polariza-tion-dependent shift, a suitably tailored detection system isrequired.

Then, a novel type of geometric SHEL occurring at a po-larizing interface was introduced. To this end we discusseda theoretical model for an ideal polarizer and suggested anexperimental implementation thereof. The light field of acollimated laser beam transmitted across such a polarizerwas evaluated. In the case of the polarizing interface beingtilted with respect to the direction of propagation, a beamdisplacement resembling the original geometric SHEL wasfound. This shift does not depend on the detection processand can be measured in a straightforward way by using thescheme proposed in this article.

The effect derived in our work is unavoidable when acircularly polarized light beam passes through a polarizinginterface tilted with respect to the direction of propagation.This underlines the importance of the geometric SHEL aspolarization is a fundamental property of the light field andnumerous optical devices are polarization-dependent.

Acknowledgements

A.A. acknowledges support from the Alexander von Hum-boldt foundation.

The final publication is available at www.

springerlink.com.

6 Jan Korger et al.

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A.2. Observation of the geometric spin Hall effect of light

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Mar-quardt, and G. Leuchs. Observation of the geometric spin Hall effect of light.arXiv:1303.6974. Phys. Rev. Lett. 112, 113902, 2014

59

60

Observation of the geometric spin Hall effect of light

Jan Korger,1, 2 Andrea Aiello,1, 2, ∗ Vanessa Chille,1, 2 Peter Banzer,1, 2 Christoffer

Wittmann,1, 2 Norbert Lindlein,2 Christoph Marquardt,1, 2 and Gerd Leuchs1, 2

1Max Planck Institute for the Science of Light, Erlangen, Germany2Institute for Optics, Information and Photonics, University Erlangen-Nuremberg, Germany

(Dated: October 1, 2013)

The spin Hall effect of light (SHEL) is the photonic analogue of the spin Hall effect occurringfor charge carriers in solid-state systems. This intriguing phenomenon manifests itself when a lightbeam refracts at an air-glass interface (conventional SHEL), or when it is projected onto an obliqueplane, the latter effect being known as geometric SHEL. It amounts to a polarization-dependentdisplacement perpendicular to the plane of incidence. Here, we experimentally demonstrate thegeometric SHEL for a light beam transmitted across an oblique polarizer. We find that the spatialintensity distribution of the transmitted beam depends on the incident state of polarization and itscentroid undergoes a positional displacement exceeding one wavelength. This novel phenomenonis virtually independent from the material properties of the polarizer and, thus, reveals universalfeatures of spin-orbit coupling.

Already in 1943, Goos and Hanchen observed that theposition of a light beam totally reflected from a glass-airinterface differs from metallic reflection [1]. This is themost well-known example of a longitudinal beam shiftoccurring at the interface between two optical media. Inhonour of their seminal work, any such deviation fromgeometrical optics occurring in the plane of incidence,is referred to as a Goos-Hanchen shift. Conversely, asimilar shift occurring in a direction perpendicular to theplane of incidence is known as Imbert-Fedorov shift [2, 3].These phenomena generally depend both on properties ofthe incident light beam and the physical properties of theinterface [4]. Goos-Hanchen and Imbert-Fedorov shiftshave been observed at dielectric [5], semi-conductor [6]and metal [7] interfaces.

The Imbert-Fedorov shift [2, 3, 8] is an example ofthe spin-Hall effect of light (SHEL), the photonic ana-logue of spin Hall effects occurring for charge carriers insolid-state systems [9–11]. SHEL is a consequence of thespin-orbit interaction (SOI) of light, namely the couplingbetween the spin and the trajectory of the optical field[9, 10, 12–22]. All electromagnetic SOI phenomena invacuum and in locally isotropic media can be interpretedin terms of the geometric Berry phase and angular mo-mentum dynamics [23]. For a freely propagating paraxialbeam of light, SOI effects vanish unless a relevant break-ing of symmetry occurs. A typical example of such asymmetry break is the interaction of the beam with anoblique surface as in ordinary light refraction processes(Figure 1).

Although the resulting phenomena essentially dependon the type of the interaction with the surface [24], thereare some common characteristics that reveal universalityin SOI of light due to the geometry and the dynamicalangular momentum aspects of the problem. Amongstthe various observable effects resulting from the beam-surface interaction, the so-called geometric Hall effect oflight is virtually independent from the properties of the

quadrantdetector

QWP

polarizer

Figure 1. Pictorial representation of the geometric SHEL.After the quarter wave-plate (QWP) the beam is circularlypolarized and passes through a polarizer tilted by an angleθ. The center of the intensity distribution of the transmittedbeam appears shifted with respect to the axis of the incidentbeam. Such a shift can be measured by a quadrant detectorput behind the polarizer. The reference frame x′,y′,z′ isaligned with the polarizer surface.

surface [19, 25–27] and, therefore, represents the idealcandidate for studying above-mentioned universal fea-tures. It amounts to a shift of the centroid of the in-tensity distribution represented by Poynting-vector flowof the beam across the oblique surface of a tilted de-tector. Direct observation of this effect as originally pro-posed depends critically on the detector’s response to thelight field. However, the question whether the responsefunction of a real detector is indeed proportional to thePoynting vector density is subject to a long-standing de-bate [28–30].

In this work, we implement an alternative scheme [25],in which the occurrence of the beam shift is independentof the detector response. To this end, we send a circularly

2

δ

y' z'

x'0k

k

ˆ x' a

y

x

a

δ tanθ

(a)

(b)

ˆ x

zy

x

θ

Figure 2. The geometric SHEL at a polarizing interface. (a)The horizontal dark grey plane represents the polarizing in-terface with the Cartesian reference frame x′,y′ ≡ y,z′attached to it. The axis x′ is taken parallel to the absorbingaxis of the polarizing interface. θ ∈ [0, π/2) denotes the angleof incidence. The central wave vector k0 of the incident beamdefines the direction of the axis z of the frame x,y,z at-tached to the beam. It is instructive to study a wave vectork in the z-y-plane, rotated by an angle δ 1 with respect tok0. (b) The unit vector a ⊥ k, representing the direction ofthe absorbed component of the incident field, lies in the com-mon plane of k and x′ (coloured light brown in the figure)and can be obtained, in the first-order approximation, fromthe rotation by the angle δ tan θ around k, of the unit vectorx.

polarized beam of light across a tilted polarizing inter-face to demonstrate a novel kind of geometric Hall effect,in which the centroid of the resulting linearly polarizedtransmitted beam undergoes a spin-induced transverseshift up to several wavelengths. This is a spatial shift,which is independent of the distance the beam propa-gated after interaction with the polarizer. Our approachis different from the early proposal [19] in that it is ob-servable with standard optical detectors. Furthermore, itis different from the SHEL occurring in a beam passingthrough an air-glass interface [10] since this geometricSHEL is practically independent of Snell’s law and theFresnel formulas for the interface.

As for conventional SHEL, the physical origin of thisgeometric version resides in the SOI of light. In fact, wecan describe this effect in terms of the geometric phasegenerated by the spin-orbit interaction as follows: Thinkof the monochromatic beam as a superposition of manyinterfering plane waves with the same wavelength and dif-ferent directions of propagation. When the beam passesthrough the polarizing interface, the plane wave com-ponents with different orientations of their wave vectorsacquire different geometric phases determined by distinctlocal projections yielding to effective “rotations” of the

polarization vector around the wave vector. The interfer-ence of these modified plane waves produces a redistribu-tion of the intensity spatial profile of the beam resultingin a spin-dependent transverse shift of the intensity cen-troid.

This effect can be better understood with the help ofFigure 2 that illustrates the geometry of the problem.The incident monochromatic beam is made of many planewave components with wave vectors k spreading aroundthe central one k0 = kz, which represents the main di-rection of propagation of the beam, where |k0| = k = |k|.For a well-collimated beam, the angle δ between an ar-bitrary wave vector k and the central one k0 is, by def-inition, small: δ 1. In the first-order approximationwith respect to δ, we consider k = k(z cos δ − y sin δ) ∼=k(z + κy y), where κy ≡ ky/k = − sin δ ∼= −δ. Thiswave vector is not the general one, but, due to the trans-verse nature of the phenomenon, it is sufficient to restrictthe discussion to wave vectors lying in the yz plane. Theeither left- (σ = +1) or right-handed (σ = −1) circu-lar polarization of the incident beam is determined bythe unit vector uσ = (x+ iσy)/

√2 globally defined with

respect to the axis z of the beam frame x, y, z. How-ever, from Maxwell’s equations it follows that the diver-gence of the electric field of a light wave in vacuum iszero. This requires that the polarization vector eσ(k)of each plane wave component of wave vector k mustnecessarily be transverse, namely k · eσ(k) = 0. Thisrequirement is clearly not satisfied by uσ for which onehas k · uσ/k ∼= iσκy/

√2 6= 0. Anyhow, the transverse

nature of the light field can be easily restored by subtract-ing from uσ its longitudinal component: uσ → eσ(k) ∝uσ − k (k · uσ) /k2 ∼= uσ − (iσ κy/

√2)z. Now that we

have properly modeled the polarization of the incidentfield, let us see how it changes when the beam crossesthe polarizing interface.

A linear polarizer is an optical device that absorbs ra-diation polarized parallel to a given direction, say x′,and transmits radiation polarized perpendicular to thatdirection. The electric field of each plane wave com-ponent of the beam sent through the polarizing inter-face, changes according to the projection rule eσ(k) →eσ(k)− a(a · eσ(k)) = t(t · eσ(k)), where a = x′−k(k ·x′)/k2 is the effective absorbing axis with a = a/|a| ∼=x− y κy tan θ, and t = a× k/k is the effective transmit-ting axis. The amplitude of the transmitted plane waveis t·eσ(k) ∝ (1−iσκy tan θ)/

√2 ∼= exp(−iσκy tan θ)/

√2,

where an irrelevant κy-independent overall phase factorhas been omitted. Therefore, as a result of the trans-mission, the amplitude of each plane wave component isreduced by a factor 1/

√2 and multiplied by the geometric

phase term exp(−iσκy tan θ). Here, the “tan θ” behav-ior of the phase, characteristic of the geometric SHEL[19], is in striking contrast to the typical “cot θ” angu-lar dependence of the conventional SHEL as, e. g., theImbert-Fedorov shift [31]. However, the spin-orbit in-

3

teraction term σκy, expressing the coupling between thespin σ and the transverse momentum κy of the beam,is characteristic of both phenomena, thus revealing theircommon physical origin.

According to the Fourier transform shift theorem [32],a linear phase shift in the wave vector domain introducesa translation in the space domain. Using this theorem, wecan show that the geometric phase term exp(−iσκy tan θ)leads to a beam shift along the y-direction. This canbe explicitly demonstrated by writing the incident cir-cularly polarized paraxial beam as Ψin(y) = uσψ

inσ (y),

where only the dependence on the relevant transversecoordinate y has been displayed. With ψin

σ (κy) we de-note the Fourier transform of ψin

σ (y). The two func-tions are connected by the simple relation ψin

σ (y) =∫ψinσ (κy) exp(ikκyy)dκy. After transmission across the

polarizing interface, the Fourier transform ψinσ (κy) of

ψinσ (y) changes to ψin

σ (κy) exp(−iσκy tan θ)/√

2 and theoutput field can be written as

ψoutσ (y) =

1√2

∫ψinσ (κy) exp[ikκy(y − σ tan θ/k)]dκy

=1√2ψinσ (y − σ tan θ/k). (1)

This last expression clearly shows that the scalar ampli-tude of the output field is equal, apart from the 1/

√2

factor, to the amplitude of the input field transversallyshifted by σ tan θ/k. Finally, the position of the centroidof the transmitted beam is given by

〈y〉 =

∫y |ψout

σ (y)|2dy∫|ψoutσ (y)|2dy

=σ

ktan θ, (2)

where k = 2πλ .

Eq. (2) as derived above is valid for an ideal polarizerwith a high extinction ratio. In this case, the descriptionof the polarizer as a projection onto the effective trans-mission axis t is adequate. At normal incidence (θ = 0)such polarizers are readily available. However, workingwith tilted polarizing interfaces, we have to account forthe fact that the efficiency of the polarizer diminisheswith larger angles θ. As a result of such loss of efficiency,the “tan θ”-dependence of the shift is modified as illus-trated in Fig. 5. This is explained in detail in the Supple-mental Material, where we introduce a phenomenologicalmodel for our real-world polarizer.

In the experiment, we use a Corning Polarcor polarizer,made of two layers of elongated and oriented silver nano-particles embedded in a 25 mm × 25 mm × 0.5 mm glasssubstrate. Directed absorption from these particles effec-tively polarizes the transmitted beam. In order to avoidparasitic effects from the glass surfaces (nG = 1.517),we have submerged the polarizer in a tank with indexmatching liquid (Cargille laser liquid 5610, nL = 1.521).Without this liquid, the effective tilting angle inside

Stokesmeasurement

state preparation shift measurementpiezo

HWP

HWP SMFBSPBS

QWP

QWPM

HWP

MQWPPBS

It

Ib

A

I–I+

Figure 3. Experimental Setup: Simultaneous measurement ofthe incident state of polarization and the position of a lightbeam transmitted across a tilted polarizer. State prepara-tion: The relative phase between horizontally and verticallypolarized components is modulated using a polarizing beamsplitter (PBS), a piezo mirror (M), quarter and half waveplates (QWP, HWP). The beam is spatially filtered using asingle-mode fibre (SMF). Stokes measurement: The transmit-ted port of a beam splitter (BS) is used to monitor the state ofpolarization. We use the Stokes parameters S3 to distinguishleft- and right-hand circular polarization. Shift measurement:The beam is propagated across our sample, a tank contain-ing a glass polarizer and an index-matching liquid, and itsposition is observed using a quadrant detector. An optionalPBS (A) can be employed as an analyzer in front of the de-tector. The photo currents I+, I−, It, and Ib are amplifiedand digitally sampled for 1 s at 50 kHz.

the glass polarizer would be limited by Snell’s law toarcsin(1/nG) ≈ 41.

In our setup (Figure 3), a fundamental Gaussian lightbeam (λ = 795 nm) is prepared with the state of polar-ization alternating between left- and right-hand circular.To avoid spatial jitter, the spatial mode is cleaned usinga single mode fibre and no active elements are used afterthe fibre. We collimate the light beam using an asphericlens (New Focus 5724-H-B) aligned such that the beamwaist is at the position of the detector.

In order to simultaneously measure the beam position⟨y⟩

and the incident state of polarization, we employa dielectric mirror (Layertec 103210) as a non-polarizingbeam splitter at an angle of incidence of 3. The reflectedand transmitted states of polarization coincide within ex-perimental accuracy. The beam centroid

⟨y⟩

= f It−IbIt−Iband Stokes parameter S3 = I+−I−

I++I−are calculated from

the digitized photo currents. We can measure the calibra-tion factor f in-situ by translating the quadrant detectorusing a micrometre stage.

Since the signal is periodic with the modulation fre-quency fmod = 29 Hz, we can filter technical noise in apost-processing stage. To this end, the discrete Fouriertransform is computed and only spectral componentswith frequencies equal to fmod and higher harmonics

4

Tilting Angle θ [°]0 20 40 60 80

shift

[μm

]1.0

0.8

0.6

0.4

0.2

0.0

(a) total shift ΔT

Tilting Angle θ [°]0 20 40 60 80

shift

[μm

]

1.0

0.8

0.6

0.4

0.2

0.0

(b) relative shift Δy(vertical polarization)

Figure 4. Polarization-dependent beam shifts occurring at atilted polarizer. Measurement data and theoretical predic-tions are shown. Both series of shift measurements were re-peated five times. We report the mean value and standarddeviation of the mean. The dashed blue line shows the the-ory for a perfect polarizer, while the solid lines were calculatedfrom our phenomenological polarizer model. (a) Overall dis-placement ∆T of the intensity barycentre after transmissionacross the polarizer. (b) Displacement ∆y of the verticallypolarized intensity component solely. The two measurements∆T and ∆y differ due to the imperfect nature of our polarizeras discussed in the SM.

thereof are passed.

We identify S3 = 0.99 ± 0.01 and S3 = −0.99 ± 0.01with the circular states of polarization σ = +1 andσ = −1 respectively. For both states, we calculate themean of all corresponding beam positions

⟨y⟩

(σ) and the

helicity-dependent displacement ∆R =⟨y⟩

(σ = +1) −⟨y⟩

(σ = −1).

An extensive characterization of statistical and system-atic errors revealed that the observed position of the lightbeam depends slightly on the state of polarization even ifno sample is present in the beam path. The magnitude ofthis spurious beam shift is typically much smaller thanthe phenomenon, we intend to study. In the Supple-mental material, we discuss that this amounts to a smalloffset on the raw data points ∆R(θ), independent from

geometric SHELconventional SHEL

Tilting Angle θ [°]

shift

[λ]

0.5

0.4

0.3

0.2

0.1

0.0

0

(c) (b)

(a)

(d)

(e)

20 40 60 80

Figure 5. Theory for the conventional spin Hall effect of lightcompared to the geometric SHEL. A light beam transmit-ted across an interface between two media can undergo atransverse displacement known as the Imbert-Fedorov effector conventional SHEL. Here, we plot this displacement for aleft-hand circularly polarized beam (σ = +1) for two differentcases and compare this with the geometric SHEL studied inthis work. (a) and (b) Geometric SHEL 1

2∆T and 1

2∆y for

the two configurations studied experimentally (Figure 4). (c)Geometric SHEL as predicted for an ideal polarizing interface.(d) Conventional SHEL occurring at an air-glass interface(n1 = 1, n2 = 1.5). (e) Conventional SHEL expected for theentrance face of our submerged polarizer (n1 = nL = 1.521,n2 = nG = 1.517).

the action of the sample. Thus, the shift measurements∆(θ) = ∆R(θ)−∆R(0) reported here are corrected withrespect to the raw data.

We investigate beam shifts in two different config-urations. First, we measure the displacement ∆T =2⟨y⟩|ET |2 of the total transmitted energy density distri-

bution |ET |2 when switching the incident state of polar-ization from σ = +1 to σ = −1 (Figure 4(a)). Then,we employ an additional polarization analyzer in front ofthe detector and observe the shift ∆y = 2

⟨y⟩|Ey|2 of the

energy density |Ey|2 = |ET · y|2 of the vertically polar-ized field component solely (Figure 4(b)). These variantsof the experiment coincide for polarizers with perfect ex-tinction ratios but can differ significantly for real-worldpolarizers with minor deficiencies.

The beam shift observed in the latter case in-creases proportionally to the tangent of the tiltingangle, exceeding one wavelength. This characteristic“tan θ”-behaviour (and “real-world-polarizer” modifica-tion thereof) is unique to the geometric spin Hall effectof light. In both cases, the measurement agrees wellwith theoretical predictions using our phenomenologicalmodel.

To the best of our knowledge, this is the first directmeasurement of this intriguing phenomenon. The geo-metric spin Hall effect of light should not be confused

5

with the conventional SHEL or Imbert-Fedorov shift.The latter occurs at a physical interface and, while suchinterfaces are present in our experimental setup, they canonly give rise to beam shifts significantly smaller than theobserved effect. In Figure 5, we compare our results tothe Imbert-Fedorov shift, which could occur at the polar-izer substrate, for a set of realistic parameters [16, 31].This illustrates that the beam shifts measured in thiswork constitute a novel spin Hall effect of light, virtuallyindependent from surface effects.

In conclusion, we have demonstrated the geometricspin Hall effect of light experimentally by propagatinga circularly polarized laser beam across a suitable polar-izing interface. The centre of mass of the transmittedlight field was found to be displaced with respect to po-sition of the incident beam as predicted by the theory.While a Gaussian light beam itself is invariant with re-spect to rotation around its axis of propagation, the ge-ometry induced by the tilted polarizer, breaks this sym-metry. The resulting displacement can be interpreted asa spin-to-orbit coupling characteristic for spin Hall effectsof light.

ACKNOWLEDGEMENTS

The authors thank Christian Gabriel for fruitful dis-cussions and for his contribution in the initial stage ofthe experiment.

∗ andrea.aiello@mpl.mpg.de[1] F. Goos and H. Hanchen, Ann. Phys. (Leipzig) 436, 333

(1947).[2] F. I. Fedorov, Dokl. Akad. Nauk. SSSR 105, 465 (1955).[3] C. Imbert, Phys. Rev. D 5, 787 (1972).[4] A. Aiello, New J. Phys. 14, 013058 (2012).[5] F. Pillon, H. Gilles, and S. Fahr, Appl. Opt. 43, 1863

(2004).[6] J.-M. Menard, A. E. Mattacchione, M. Betz, and H. M.

van Driel, Opt. Lett. 34, 2312 (2009).[7] M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter,

E. R. Eliel, and J. P. Woerdman, Opt. Express 15, 15928(2007).

[8] H. Schilling, Ann. Phys. (Leipzig) 471, 122 (1965).[9] M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev.

Lett. 93, 083901 (2004).[10] O. Hosten and P. Kwiat, Science 319, 787 (2008).[11] X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, Science

339, 1405 (2013).[12] A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and

B. Zel’dovich, Phys. Rev. A 45, 8204 (1992).[13] C. C. Leary, M. G. Raymer, and S. J. van Enk, Phys.

Rev. A 80, 061804 (2009).[14] V. S. Liberman and B. Y. Zel’dovich, Phys. Rev. A 46,

5199 (1992).

[15] K. Y. Bliokh and Y. P. Bliokh, Phys. Lett. A 333, 181(2004).

[16] K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96,073903 (2006).

[17] A. Aiello and J. P. Woerdman, Opt. Lett. 33, 1437(2008).

[18] K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, NaturePhoton. 2, 748 (2008).

[19] A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs,Phys. Rev. Lett. 103, 100401 (2009).

[20] N. B. Baranova, A. Y. Savchenko, and B. Y. Zel’dovich,JETP Lett. 59, 232 (1994).

[21] K. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman,Phys. Rev. Lett. 101, 030404 (2008).

[22] P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt,N. Lindlein, T. Bauer, and G. Leuchs, J. Europ. Opt.Soc. Rap. Public. 8, 13032 (2013).

[23] K. Y. Bliokh, A. Aiello, and M. A. Alonso, in The an-gular momentum of light, edited by D. L. Andrews andM. Babiker (Cambridge University Press, 2012).

[24] A. Bekshaev, K. Y. Bliokh, and M. Soskin, Journal ofOptics 13, 053001 (2011).

[25] J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb,C. Marquardt, and G. Leuchs, Appl. Phys. B 102, 427(2011).

[26] A. Y. Bekshaev, J. Opt. A: Pure Appl. Opt. 11, 094003(2009).

[27] K. Y. Bliokh and F. Nori, Phys. Rev. Lett. 108, 120403(2012).

[28] M. V. Berry, J. Opt. A: Pure Appl. Opt. 11, 094001(2009).

[29] J. Durnin, C. Reece, and L. Mandel, J. Opt. Soc. Am.71, 115 (1981).

[30] J. Braat, S. van Haver, A. Janssen, and P. Dirksen, J.Europ. Opt. Soc. Rap. Public. 2, 07032 (2007).

[31] B.-Y. Liu and C.-F. Li, Opt. Commun. 281, 3427 (2008).[32] J. W. Goodman, Introduction to Fourier optics, 3rd ed.

(Roberts & Co, Colorado, USA, 2005).[33] Y. Fainman and J. Shamir, Appl. Opt. 23, 3188 (1984).[34] H. A. Haus, American Journal of Physics 61, 818 (1993).[35] A. Aiello, C. Marquardt, and G. Leuchs, Opt. Lett. 34,

3160 (2009).

6

SUPPLEMENTAL INFORMATION

Phenomenological polarizer model andcharacterization of our real-world polarizer

In this section, we describe a geometric polarizermodel, analogously to the work by Fainman and Shamir[33], and determine empirical parameters relevant for theactual polarizer used in our experiment. Here, the inter-action of an arbitrarily oriented polarizer with a planewave is discussed while we deal with real light beams inthe subsequent section on beam shifts.

A polarizer is an optical device that alters the state ofpolarization and intensity of a plane wave without effect-ing its direction of propagation κ = k/k. The polarizerused within this work can be described by a real-valuedunit vector Pa describing its absorbing axis. ProjectingPa onto the transverse plane of the electric field yieldsan effective absorbing axis

a(κ) =Pa − (Pa · κ) κ√

1− (Pa · κ)2

. (S1)

Thus, the electric field transmitted across an idealizedpolarizer is

ET = EI − a(a · EI

)= t

(t · EI

), (S2)

where EI = EI(k) is the amplitude of the plane waveexp(i κ · r), and

t(κ) = a(κ)× κ =Pa × κ|Pa × κ|

(S3)

is the effective transmitting axis.However, real-world polarizers have finite extinction

ratios and an experimental characterization shows thatthe effectiveness of our polarizer decreases when tilted(Figure S1). Thus, we have found it convenient to phe-nomenologically describe the transmitted light field as

ET = τt t(t · EI

)+ τa a

(a · EI

). (S4)

Here, we employ two empirical parameters, τa(θ) andτt(θ), depending on the angle θ between the propagationdirection κ and the unit vector perpendicular to the sur-face of the polarizer. We have found the following setof parameters to be in good agreement with both theobserved transmission and beam shifts:

τt(θ) = 1− 1.2 exp(−12 cos θ) (S5)

τa(θ) = 0.51 exp(−1.3 cos θ) (S6)

A more detailed study of tilted polarizers will be pub-lished elsewhere.

V polarization:

H polarization:

shift

[μm

]

1.0

0.8

0.6

0.4

0.2

0.0

Tilting Angle θ [°]0 20 40 60 80

Figure S1. Transmittance of light beams across the po-larizer as a function of the tilting angle θ and the in-cident state of polarization. The polarizer is aligned suchthat at normal incidence (θ = 0) vertical (V) polarization istransmitted and horizontal (H) polarization is blocked. Weobserve how the transmittance changes when the polarizer isrotated around the vertical axes and compare the experimen-tal data (circles) to our phenomenological model (S4) (solidlines).

Calculation of beam shifts occurring at an obliquepolarizer according to our phenomenological model

In this section, we adapt the theory for the geometricspin Hall effect of light originally calculated for a differenttype of polarizer [25] to our phenomenological model. Tothis end, we express the polarizer’s absorbing axis

Pa = x′ = cos θ x+ sin θ z (S7)

in the global reference frame x, y, z, aligned with thedirection z of beam propagation.

The incident beam EI(r) is circularly polarized andwell-collimated, i.e. it has a low divergence θ0, with aGaussian transverse intensity profile. This is expandedin a plane wave basis with amplitudes EI(κ) such thatthe κ-dependent projection (S4) can be applied.

As a consequence of our polarizer model, the electricfieldET (r) after transmission across such a polarizer, is asuperposition of two orthogonally polarized field compo-nents, Ex(r) and Ey(r). Thus, the electric field energydensity distribution (at the detection plane z = 0)

|ET (x, y)|2 = |ET · x|2︸ ︷︷ ︸=|Ex|2

+ |ET · y|2︸ ︷︷ ︸=|Ey|2

+ |ET · z|2︸ ︷︷ ︸≈0

(S8)

can be decomposed analogously.Geometric SHEL manifests itself as a transverse dis-

placement⟨y⟩

of the transmitted light beam’s barycen-

tre. The total electric energy density |ET |2 and bothof its non-vanishing components undergo such a shift.Since this spatial displacement is independent from the

7

z coordinate, we restrict the discussion to the detectionplane at z = 0. It is convenient to write the total energydensity’s barycentre⟨y⟩|ET |2 = wx

⟨y⟩|Ex|2 + wy

⟨y⟩|Ey|2 (S9)

as a weighted sum of the relative shifts occurring for thehorizontally and vertically polarized components respec-tively. Here,

⟨y⟩u

=

∫∫y u(x, y) dx d y∫∫u(x, y) dxd y

(S10)

denotes the centre of mass along the y direction calcu-lated with respect to a scalar distribution u(x, y). Theintegration spans the whole detection plane.

The weights

wx(θ) =

∫∫E2x(x, y) dxd y∫∫

|E|2(x, y) dx d y

=τ2a (θ)

τ2a (θ) + τ2

t (θ)+O(θ2

0) and (S11)

wy(θ) =τ2t (θ)

τ2a (θ) + τ2

t (θ)+O(θ2

0) (S12)

introduced in (S9) depend on the empirical parameters τtand τa (S5). Finally, we can calculate the relative shifts

⟨y⟩|Ei|2

λ=

∫∫x,y

y |Ei|2 dxd y

∫∫x,y

|Ei|2 dxd y

= σtan θ

2πfi(θ) +O(θ2

0), (S13)

where the factors

fx(θ) = 1− τt(θ)

τa(θ)< 0 and (S14a)

fy(θ) = 1− τa(θ)

τt(θ)> 0 (S14b)

depend critically on the performance of the polarizer.Note that, since the transmission coefficients are real

and positive and 1 ≥ τt > τa (Figure S1), these relativeshifts have opposite signs. Consequently, the displace-ment of the total energy density⟨y⟩|ET |2

λ= σ

tan θ

2π(wx fx + wy fy)︸ ︷︷ ︸

<1

(S15)

is smaller than the relative shift⟨y⟩|Ey′ |2

of the verti-

cally polarized component solely. For our realistic polar-izer model, both shifts are smaller than expected for theideal case. This matches very well with our experimentalobservation.

For a perfect polarizer with τa = 0 and τt = 1, equation

(S15) reduces to eq. (2) since wx fx =τ2a

τ2t

(1− τt

τa

)→ 0

for τa → 0. This is exactly the same expression that wasoriginally found for the polarizer model by Fainman andShamir [25].

Calculation of the geometric spin Hall effect of lightat a polarizing interface

In this part, we derive detailed eq. (2) for a parax-ial fundamental Gaussian beam incident at an angle θupon the surface of an absorbing polarizer whose absorb-ing axis is directed along x′. The Cartesian coordinatesystems attached to the polarizing interface and to thebeam are still defined as in Fig. 2A in the main text.However, now the axis x′ is taken parallel to the absorb-ing axis of the polarizer, in order to reproduce the actualexperimental conditions.

Consider the fundamental solution of the scalar parax-ial wave equation [32] that we indicate with ψ(r):

ψ(r) =

(k

πL

)1/2i

1 + iz/Lexp

(− 1

w20

x2 + y2

1 + iz/L

),

(S16)

where the Rayleigh range L of the beam can be expressedin terms of the beam waist w0 as L = kw2

0/2, with θ0 =2/(kw0) 1 denoting the angular spread of the beam.In the first-order approximation with respect to θ0, theelectric vector field of the incident beam can be writtenas [34]:

Ψin(r) = uσ ψ(r) + zi

kuσ ·∇ψ(r)

=

[uσ − iz

θ0 (x+ iσy)√2w0 (1 + iz/L)

]ψ(r), (S17)

where uσ = (x + iσy)/√

2 and an overall (irrelevant)multiplicative term has been omitted. In Ref. [35] it wasdemonstrated that the field transmitted by an arbitrar-ily oriented polarizer can be written as a perturbativeexpansion of the form

Ψout(r) = G00Ψin(r)

− i

k

[G10

∂Ψin

∂x(r) + G01

∂Ψin

∂y(r)

]

+O(θ20), (S18)

where the 3× 3 matrices Gnm are defined as

Gnm =kn+m

n!m!

∂n+m(nn)

∂knx∂kmy

∣∣∣∣kx=0, ky=0

. (S19)

For our absorbing polarizer, the dyadic form

nn =

n2x nx ny nx nz

ny nx n2y ny nz

nz nx nz ny n2z

(S20)

is defined in terms of the effective-transmission unit vec-tor n = a× κ = nxx+ nyy + nzz, where

a =x′ − κ(κ · x′)√

1− |κ · x′|2, (S21)

8

and κ = k/k. A straightforward calculation furnishes

G00 =

0 0 00 1 00 0 0

, (S22a)

G01 =

0 tan θ 0tan θ 0 −1

0 −1 0

, (S22b)

G10 =

0 0 00 0 00 0 0

. (S22c)

Substitution of equations (S22) and (S17) into equa-tion (S18) yields to the following first-order expressionfor the transmitted field:

Ψout(r) =ψ(r)√

2

[− x θ0

y

w0

σ tan θ

1 + iz/L

+ y iσ

(1 + θ0

y

w0

σ tan θ

1 + iz/L

)

+ z θ0y

w0

σ

1 + iz/L

]+O(θ2

0). (S23)

The electric field energy density, the physical quantityactually measured by a standard optical detector, is pro-portional to |Ψout(r)|2 which can be calculated from Eq.(S21) as:

|Ψout(r)|2 = |ψ(r)|2(

1

2+ θ0

y

w0

σ tan θ

1 + iz/L

)+O(θ2

0).

(S24)

Finally, the sought shift is calculated as the first momentof the electric field energy density distribution, namely

〈y〉 =

∫∫y |Ψout(r)|2dxdy

∫∫|Ψout(r)|2dxdy

=σ

ktan θ, (S25)

where the integration is evaluated over all the xy planeat z = 0. Equation (S25) reproduces the result given byeq. (2) in the main text.

Discussion of spurious beam shifts

In our experiment, we prepare left and right circularlypolarized Gaussian light beams and intend to study howa sample effects the position of the transmitted beam inboth cases. The spatial modes prepared in each case arealmost, yet not exactly identical. Despite being filteredusing two metres of single-mode fibre, there can be a tinyoffset between the centre of mass of the left versus theright circularly polarized beam prepared in this manner.While this effect is too small to be imaged on a CCDcamera, it can be revealed in a precision measurement asemployed in this work.

Theoretically, we can account for such spurious beamshifts by substituting the ideal Gaussian profile EI(κ)with a generic paraxial beam such as a power series orsuperposition of Hermite-Gaussian laser modes. Thischanges the result (S13) to

⟨y⟩|Ei|2

λ= σ

tan θ

2πfi(θ) + ηi(σ) +O(θ2

0), (S26)

where ηi(σ) is an offset independent of the tilting angle θ,which can also be understood intuitively: The deviationbetween two almost identical beam profiles amounts toeither a spatial or an angular beam shift. First, since thesample is spatially homogeneous, a spatial displacementof the whole beam, does not change the interaction atall. And second, while the interaction with the samplegenerally depends on the angle of incidence, a slightlydifferent direction of propagation, here on the order of10−6 rad, is practically negligible.

In our experiment, we measure the helicity-dependentbeam shift

∆R =⟨y⟩

(σ = +1)−⟨y⟩

(σ = −1)

= 2tan θ

2πfi(θ)

︸ ︷︷ ︸∆(Θ)

+ ηi(σ = +1)− ηi(σ = −1)︸ ︷︷ ︸∆R(θ=0)

, (S27)

which contains a spurious offset ∆R(θ = 0) ≈ 60 nm.The data shown in Fig. 4 is corrected for this offset.

A.3. The polarization properties of a tilted polarizer

J. Korger, T. Kolb, P. Banzer, A. Aiello, C. Wittmann, C. Marquardt, andG. Leuchs. The polarization properties of a tilted polarizer. arXiv:1308.4309. Opt.Express 21(22), 27032–27042, 2013

69

70

The polarization properties of a tiltedpolarizer

Jan Korger∗, Tobias Kolb, Peter Banzer, Andrea Aiello, ChristofferWittmann, Christoph Marquardt, and Gerd Leuchs

Max Planck Institute for the Science of Light, Guenther-Scharowsky-Str. 1/Bldg. 24, 91058Erlangen, Germany

Institute for Optics, Information and Photonics, University Erlangen-Nuremberg, Staudtstr.7/B2, 90158 Erlangen, Germany∗jan.korger@mpl.mpg.de

Abstract: Polarizers are key components in optical science and technol-ogy. Thus, understanding the action of a polarizer beyond oversimplifyingapproximations is crucial. In this work, we study the interaction of a po-larizing interface with an obliquely incident wave experimentally. To thisend, a set of Mueller matrices is acquired employing a novel procedure ro-bust against experimental imperfections. We connect our observation to ageometric model, useful to predict the effect of polarizers on complex lightfields.OCIS codes: 120.5410;260.2130;260.5430.

References and links1. Q. Hong, T. Wu, X. Zhu, R. Lu, and S.-T. Wu, “Designs of wide-view and broadband circular polarizers,” Optics

Express 13, 8318–8331 (2005).2. Q. Hong, T. X. Wu, R. Lu, and S.-T. Wu, “Wide-view circular polarizer consisting of a linear polarizer and two

biaxial films,” Optics Express 13, 10777–10783 (2005).3. J.-W. Moon, W.-S. Kang, H. yong Han, S. M. Kim, S. H. Lee, Y. gyu Jang, C. H. Lee, and G.-D. Lee, “Wideband

and wide-view circular polarizer for a transflective vertical alignment liquid crystal display,” Applied Optics 49,3875–3882 (2010).

4. Y. Fainman and J. Shamir, “Polarization of nonplanar wave fronts,” Applied Optics 23, 3188 (1984).5. A. Aiello, C. Marquardt, and G. Leuchs, “Nonparaxial polarizers,” Optics Letters 34, 3160–3162 (2009).6. A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,”

Optics Letters 31, 817 (2006).7. D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable

from a Jones matrix,” Journal of the Optical Society of America A 11, 2305 (1994).8. A. Aiello and J. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scatte-

ring,” Physical Review Letters 94, 090406 (2005).9. R. M. A. Azzam and A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect

polarizing optical elements,” Journal of the Optical Society of America A 6, 1513 (1989).10. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Applied Optics 31, 6676 (1992).11. B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,”

American Journal of Physics 75, 163 (2007).12. A. M. Braczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating

quantum state tomography,” New Journal of Physics 14, 085003 (2012).13. J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect

of Light at polarizing interfaces,” Applied Physics B 102, 427–432 (2011).14. J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation

of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).15. R. C. Jones, “A New Calculus for the Treatment of Optical Systems,” Journal of the Optical Society of America

31, 488 (1941).16. J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005).17. M. Born and E. Wolf, Principles of optics (Pergamon Pr., Oxford, 1999), 7th ed.

1

18. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” Journalof the Optical Society of America A 4, 433 (1987).

19. P. Yeh, “Generalized model for wire grid polarizers,” Proceedings of SPIE 0307, 13–21 (1982).20. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” Journal of the

Optical Society of America A 13, 1106 (1996).21. J. J. Gil and E. Bernabeu, “Depolarization and Polarization Indices of an Optical System,” Optica Acta: Interna-

tional Journal of Optics 33, 185–189 (1986).22. A. Beer, “Bestimmung der Absorption des rothen Lichts in farbigen Flussigkeiten,” Annalen der Physik und

Chemie 162, 78–88 (1852).23. S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn Phase-Separated Borosilicate

Glasses: A TEM Investigation,” Microscopy Microanalysis Microstructures 8, 157–165 (1997).24. S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elon-

gated and oriented phase-separated crystalline particles,” Journal of Non-Crystalline Solids 232-234, 147–154(1998).

25. R. C. Thompson, J. R. Bottiger, and E. S. Fry, “Measurement of polarized light interactions via the Muellermatrix,” Applied Optics 19, 1323 (1980).

26. E. Compain, S. Poirier, and B. Drevillon, “General and Self-Consistent Method for the Calibration of PolarizationModulators, Polarimeters, and Mueller-Matrix Ellipsometers,” Applied Optics 38, 3490 (1999).

1. Introduction

Electromagnetic radiation is described as a vector field and, thus, the orientation of the elec-tric field vector, known as polarization, is of great importance, both in classical and quantumoptics. Polarized states of the light field are often prepared and measured using polarizers andanalyzers, respectively, which can refer to the same device. The physical implementation ofsuch polarizing elements can be very different according to the application the device is de-signed for. For example, the liquid crystal display (LCD) industry has refined their polarizerdesign over the past decades to achieve the outstanding performance that these devices showtoday [1–3].

From a more fundamental point of view, it is desirable to work with a generic polarizermodel, which is computationally convenient and takes into account the geometric nature of theproblem while being suitable to describe a wide range of polarizers. Such geometric modelsare currently used in the theoretical literature [4, 5]. However, to the knowledge of the authors,they lack experimental validation, in particular for unusual corner cases. Even for wide-viewLCDs, the propagation angle inside the polarizing element is not as steep as in our measure-ments. Since any device which qualifies as a polarizer acts similarly on a normally incidentlight beam, these obliquely incident waves can be used to establish a realistic geometric modeland demonstrate its validity.

In this article, the Mueller matrix of a commercial polarizer made of elongated nano-particlesshall be measured. Reconstructing such a matrix from potentially noisy experimental data ischallenging and prone to errors [6]. We solve this problem using a self-calibrating polarimeter,which additionally warrants that the result is physically acceptable [7,8]. Our method combinesa number of ideas discussed in the literature [9–12].

This article is structured as follows: First, we introduce and illustrate polarization and po-larizer models. Then, we propose a Mueller matrix polarimeter, which is robust against exper-imental imperfections and does not rely on precision optics nor calibrated reference samples.Finally, we employ this setup to obtain Mueller matrices for a commercial polarizer and connectthe observation to its microscopic structure. Motivated by recent theoretical and experimentalwork connecting the action of a tilted polarizer to a beam shift phenomenon [13,14], we extendour studies to include the unusual case of almost grazing incidence.

2

2. Polarization of a light beam

In this work, we use both, Jones and Mueller-Stokes calculi, to represent the polarization prop-erties of the light field. There are two key differences between both approaches. First, the for-mer method works with the electric field, while the latter depends only on intensities, whichcan be directly measured. And second, the Mueller-Stokes representation is more general sinceit allows for describing unpolarized states of light and depolarizing optical elements.

For our purpose, a collimated, polarized light beam can be approximated as a planar wavefield. A plane wave is completely determined by the complex envelope JJJ = Exxxx+Eyyyy of itselectric field EEE(rrr, t) = Re [JJJ exp(i(kkk ·rrr−ωt))], where kkk = kzzz is the wave vector. The complexcolumn-vector JJJ has become known as the Jones vector [15, 16].

Alternatively, the state of polarization of any light beam can be described by a set of four realStokes parameters [17]

S0S1S2S3

=

I0 + I90

I0 − I90

I+45 − I−45

IR− IL

=

|Ex|2 + |Ey|2|Ex|2−|Ey|2ExE∗y +E∗x Ey

i(ExE∗y −E∗x Ey)

, (1)

where Iα is the intensity of the light beam transmitted across a linear polarizer oriented atan angle α with respect to the xxx axis and IR,L is the right- or left-handed circularly polarizedcomponent of the intensity.

The four Stokes parameters Sµ are related to the Jones vector JJJ through the dyadic prod-uct of the Jones vector and its conjugate transpose JJJ† multiplied with the Pauli matrix σ (µ)

corresponding to each Stokes parameter [18]:

SSSµ = Tr[(

JJJ⊗ JJJ†)σ (µ)]

. (2)

The trace operation is in general irreversible. Obviously, every Jones vector JJJ = (Ex,Ey)T can

be converted into a set of Stokes parameters, whereas the reverse is not true. In this article, wechoose a basis

σ (0) =1√2

(1 00 1

),σ (1) =

1√2

(1 00 −1

),σ (2) =

1√2

(0 11 0

),σ (3) =

1√2

(0 −ii 0

),

consistent with the definition of the Stokes parameters used in popular textbooks [17].Both approaches allow for a matrix calculus to describe linear operations affecting the state

of polarization [16],

JJJin→ JJJout = T JJJin, (3)

SSSin→ SSSout = M SSSin, (4)

where T and M are called Jones and Mueller matrices, respectively.

3. Geometric Polarizer Models

Generally, a polarizer is understood to project the light field onto a particular state of polariza-tion. For a plane wave impinging perpendicularly onto a linear polarizer, this state is triviallygiven by the orientation of the polarizing axis. In any other case, we need to work with a suit-able model taking into account the physical nature of the interaction. For polarizers, for whichthe polarizing effect takes place at an interface between two media, e.g. reflection at the Brew-ster angle, this problem is solved by applying the well-known boundary conditions or Fresnelformulas.

3

Fig. 1. Geometric interpretation of polarizer models. A plane wave with its electric fieldEEE in in the xxxyyy-plane interacts with a tilted polarizer not parallel to this plane. Our goal is toconnect the orientation θ , φ of the polarizer to the direction of the transmitted field compo-nent EEEout. (a) Fainman and Shamir [4] suggested to find this direction tttFS by projecting avector PPPT interpreted as the polarizer’s transmitting axis onto the xxxyyy-plane. (b) The polar-izer in question is made of elongated particles, all with their long axes oriented in directionof PPPA. Thus, our absorbing model makes of use of the projection aaa of the absorbing axisPPPA. The field component parallel to aaa is scattered and eventually absorbed. Consequently,the transmitted field is polarized in direction of ttt, orthogonal to aaa.

Fainman and Shamir (FS) have constructed a convenient geometrical model applicable topolarizers that do not change the direction zzz of wave propagation [4]. They allow for an arbi-trary orientation of the polarizer and assert that it can be completely described with a three-dimensional unit vector PPPT . FS make use of the transversality of the electric field vector andconclude that the effect of a polarizer reduces to the projection onto an effective transmittingaxis tttFS (illustrated in Fig. 1(a)). In their model, the unit vector tttFS ∝ PPPT − (zzz · PPPT )zzz is found byprojecting the polarizer’s transmitting axis PPPT onto the plane of the electric field perpendicularto the direction of wave propagation zzz.

Fainman and Shamir’s approach is practically useful since establishing an effective transmit-ting axis reduces the complexity of the intrinsically three-dimensional problem to an operationon the two-dimensional Jones vector JJJ. For any orientation of the polarizer, the resulting Jonesmatrix TFS = tttFStttT

FS is a projector as expected for an ideal polarizer. However, their recipe doesnot take into account the physical nature of the interaction.

In this work, we attempt to adapt FS’s approach to our observation. In particular, we studya polarizing element made of anisotropic absorbing and scattering particles. The ensemble ofthese elementary absorbers shall be oriented with their absorbing axes PPPA parallel to each other.Analogously to the transmitting case, we interpret the projection of this unit vector PPPA as aneffective absorbing axis

aaa =PPPA− (PPPA · zzz)zzz√

1− (PPPA · zzz)2. (5)

If this interpretation holds true, the light field after transmission across an absorbing polarizerbecomes

EEE in→ EEEout = EEE in− (EEE in · aaa)aaa. (6)

As above, the corresponding Jones matrix TA = 1− aaaaaaT = ttttttT is a projector, where ttt = zzz× aaacan be interpreted as the effective transmitting axis as illustrated in Fig. 1(b). While Eq. (5) is

4

Fig. 2. State of polarization transmitted across a polarizer rotated around the vertical axis yyyby an angle θ , keeping the angle φ = 94.5 between the absorbing axis and yyy constant. (a)Visualization of the FS model [4]: The projection of the polarizer’s transmitting axis PPPT(red arrow) onto the plane of the electric field (green plane) determines the transmitted fieldcomponent (green arrow). (b) Visualization of the absorbing polarizer model (6): The pro-jection of the polarizer’s absorbing axis PPPA (blue arrow) onto the plane of the electric field(green plane) determines the absorbed field component. (c) Experimental data points (blackcircles) compared to both models. The dashed red line depicts the original FS model, whilethe solid blue line describes the analogously constructed absorbing model. The data showsthe polarizance vector Mi0 [20] acquired as a part of our Mueller matrix measurement. Thisis the state of polarization after transmission across the polarizer if the incident wave isunpolarized. Only the absorbing model explains the drastic change of the transmitted stateof polarization observed when the polarizer is tilted.

structurally equivalent to Fainman and Shamir’s construction, our model coincides with theirapproach only for normal incidence. Generally, our absorbing model TA = 1− aaaaaaT differs fromthe FS case TFS = tttFStttT

FS. We want to note that the absorbing model can be found alternativelyby treating the sub-wavelength structure of the polarizer as a composite material, which behavesas an anisotropic absorbing crystal [19].

We rely on empirical evidence to decide, whether any of those two geometric models ad-equately describes our real-world polarizer. To this end, we compare the state of polarizationtransmitted across the polarizer to the one predicted by both models [Fig. 2]. This shows thatour polarizer can be approximated as a projector. When tilted, the state of polarization, thedevice projects onto, is given by Eq. (6).

This simple absorbing model, Eq. (6), is the first main result of this article. Using the Muellermatrix measurements reported in the following sections, we can establish a phenomenologicalmodel and connect the observation to a physical picture of the light field’s interaction with thenano-particles.

4. Mueller Matrix measurement

In this section, we present a method to measure the Mueller matrix of an arbitrary optical ele-ment, which is robust against experimental imperfections, such as noise and systematic errors.With this least squares based estimation, we can gain full information about the polarizationproperties of the device-under-test performing only a limited number of intensity measure-ments. The method employs a polarizer, a polarization beam splitter as an analyzer, and two

5

rotating wave plates to select the states of polarization [Fig. 3(a)].In any of these measurements, the observed intensities

Ii j =12(SSSout

j )T M SSSini (7)

depend on the first waveplate, which prepares a state of polarization SSSini , the unknown Mueller

matrix M describing the device-under-test, and the state of polarization SSSoutj , we project onto at

the detection stage. Here, the row vector 12 SSST describes the action of an analyzer transmitting

the state of polarization given by SSS. Applied to any Stokes vector, this yields the transmittedintensity.

In principle, a generic real-valued 4×4 matrix M is unambiguously determined by 16 equa-tions like Eq. (7). However, the measured intensities IE

i j, where the superscript E denotes experi-mental values, can be noisy. Thus, acquiring more than 16 values helps to reduce both statisticaland systematic errors significantly. To this end, instead of solving a linear system of equations,we pick the Mueller matrix MLS from the set of all possible Mueller matrices, such that

ε(MLS) = ∑i, j

∣∣∣∣12(SSSout

j )T MLS SSSini − IE

i j

∣∣∣∣2

(8)

becomes minimal.A Mueller matrix M is physically acceptable [6–8] if and only if its matrix elements

Mab = Tr[H(

σ (a)⊗σ (b)∗)]

(9)

are a function of a Hermitian matrix H with non-negative eigenvalues [6]. Any such matrixH = H† can be expressed using a set of 16 real numbers h1, . . . ,h16. Therefore, these 16parameters span the vector space of physical Mueller matrices and the set hLS

1 , . . . ,hLS16 which

minimizes Eq. (8) yields to the best estimate MLS for the actual Mueller matrix.If the states of polarization SSS are not known precisely, we can find these parameter em-

ploying a procedure similar to the one described above. Interestingly, this requires no a prioriinformation beyond the knowledge that the polarization states SSS are prepared using polarizersand birefringent retarders. To this end, the device-under-test is removed from the beam path.Using the same procedure as for the actual measurement, a set of intensities Ical

i j is acquired,which characterizes the setup.

Theoretically, this calibration run corresponds to substituting MLS in Eq. (8) with the identitymatrix (Mueller matrix of empty space). Additionally, we express the abstract Stokes vectors(SSSout

j )T = (SSSH,V )T Mout

j and SSSini = Min

i SSSH in terms of Mueller matrices Moutj and Min

i , whichphysically describe our measurement device. The Stokes vectors SSSH,V represent horizontally orvertically polarized states, respectively. In our experiment, those are the states transmitted orreflected by a polarizing beam splitter [Fig. 3(a)]. This yields:

ε(MLS) = ∑i, j

∣∣∣∣12(SSSH,V )

T Moutj Min

i SSSH − Icali j

∣∣∣∣2

. (10)

In particular, Mini (α in

i ,ρ in) represents the first wave plate with the retardation ρ in and its fastaxis oriented at an angle α in

i with respect to the xxx axis. Analogously, Moutj (αout

j ,ρout) describesthe second wave plate.

Since we cannot fully rely on the manufacturer to specify retardation and orientation ofthe fast axis with the desired accuracy, those values are treated as unknown. Nevertheless,

6

Fig. 3. (a) Scheme of the Mueller matrix measurement. Using a collimated light beam(wavelength λ = 795nm), polarizing beam splitters (PBS), quarter wave plates (QWP),and two photo detectors IH and IV, the effect of an unknown sample on the polarizationcan be measured. For both QWPs, we use 6 different settings α in/out of their fast axes. Oursample is a commercial glass polarizer submerged in an index-matching liquid, which canbe rotated around the vertical axis such that the incident beam impinges under an angle θ .This setup allows to study the polarizing effect of the metal nano-particles, the polarizer ismade of, without interference from the glass surfaces. (b) Observed depolarization indexPD [21] as a function of the orientation φ , θ of the polarizer relative to the incident beam.PD = 1 describes a non-depolarization sample while PD = 0 indicates a total depolarizer.

employing motorized rotation stages, we can precisely reproduce relative movements ∆αi =αi+1−αi of both wave plates, where, for example, ∆αi = ∆α = 22.5. Thus, our measurementsetup is completely described by four parameters, α in

0 , ρ in, αout0 , and ρout, which are to be

found with this calibration procedure. The set of parameters which minimizes Eq. (10) yieldsthe states of polarization SSSin

i and SSSoutj relevant for our experiment.

As soon as these calibration parameters are known, Eq. (8) only depends on properties ofthe device-under-test. Minimizing Eq. (8) yields to the best experimental estimate for the actualMueller Matrix MLS describing the device.

5. Experiment

In our experiment, we study a polarizing interface, made of anisotropically absorbing nano-particles. To this end, we employ a commercial “Corning Polarcor” polarizer, made of a glasssubstrates with 25 to 50µm thick polarizing layers on each face. These layers contain embed-ded, elongated and oriented silver particles.

We are particularly interested in the interaction beyond the trivial case of normal incidence.However, at larger angles of incidence θ , the existence of surfaces becomes problematic sincea light beam propagating across an interface experiences both, a change of its direction ofpropagation (Snell’s law) and of its polarization (a consequence of Fresnel’s formulas) [17].These well-known effects are unrelated to the action of the actual polarizing layer inside theglass substrate. Thus, we avoid such surface effects by submerging the polarizer in a tank filledwith an index matching liquid (Cargille laser liquid 5610). The refractive index of this liquid(nL = 1.521) matches with the one of the polarizer’s substrate (nG = 1.517).

Each measurement run consists of 6× 6 steps, acquiring two intensity values per step. Ev-ery step uses a different combination of the two wave plates’ angles. For the required cal-ibration run, we remove the polarizer from the beam path, but keep the container with theindex-matching liquid. Neither the liquid nor the glass windows were observed to affect thestate of polarization. From this calibration data, we learn that both of our quarter-wave plates

7

Fig. 4. Jones matrix representation of the operation a light beam experiences when pass-ing across our polarizer. The polarizer’s absorbing axis PPPA is oriented almost horizontally(φ = 89.2) and rotated around the vertical axis yyy by an angle θ . The experimental datapoints (black circles) are calculated from our measured Mueller matrices. Ignoring an ir-relevant global phase, we set Im(J11) = 0. Our phenomenological model, described by TP,is depicted using solid green lines. Dashed blue lines show the geometric absorbing modelgiven by TA.

perform within their specifications (α in0 = 2.57, ρ in = π/2 + 0.008rad, αout

0 = 0.89, andρout = π/2+ 0.019rad). Nevertheless, knowledge of these parameters is crucial to performa highly accurate Mueller matrix reconstruction.

Our goal is three-fold: First, we attempt to establish a phenomenological model taking intoaccount the finite extinction ratio exhibited by real-world polarizers. Then, we demonstrate thatthis model accurately predicts the behaviour for a wide range of parameters. And finally, weconnect the observation to the interaction of the light field with the ensemble of nano-particles.

To this end, we perform three series of measurement runs for different orientations φ of thepolarizer’s absorbing axis, each for a large number of tilting angles 0 ≤ θ ≤ 82. We apply theleast-squares method described above to find the Mueller matrices describing our polarizer.

Results acquired with this method can be reproduced precisely. Comparing independentmeasurements for the same configuration shows that the statistical error of any Mueller ma-trix element Mab is less then 10−3. Furthermore, our data indicates that the results are alsoaccurate. The sample, we have studied is a linear polarizer. For normal incidence (θ = 0),the transmittance across such a polarizer does not depend on the helicity of the incident beamand the transmitted beam is linearly polarized. The corresponding Mueller matrix elements|M03|< 0.01 and |M30|< 0.01 clearly vanish for all relevant measurements. Thus, we estimatesystematic errors to be below 10−2.

All measured Mueller matrices are practically non-depolarizing [Fig. 3(b)]. This means that aJones matrix representation suffices to describe our sample. The data series with the absorbingaxis oriented almost horizontally [Fig. 4] shows clearly that both the transmittance and theextinction ratios decreases with the tilting angle θ . This behaviour cannot be described by aperfect projector as in the geometric models discussed earlier. Thus, we propose to generalizethe projection rule, Eq. (6), to include two transmission coefficients τa and τt for states ofpolarization parallel and perpendicular to the effective absorbing axis:

EEE in→ EEEout = TP EEE in with TP = τa aaaaaaT + τt ttttttT. (11)

8

Fig. 5. Reduced Mueller matrices M′ = 1M00

M describing the tilted polarizer for two differ-ent orientations of its absorbing axis φ . Our polarizer model (solid lines) agrees well withthe experimental data (markers). The model, we have employed, is deterministic. The smalldeviation from the model occurs for large tilting angles θ , where the devices is slightlydepolarizing (compare Fig. 3(b)). Depolarization effects cannot be modelled using Jonescalculus as employed by our model.

Using an ansatz implied by the qualitative behaviour of the data set shown in Fig. 4, we applya curve-fitting algorithm to this data set, which yields:

τt(θ) = exp(−0.025/cos(θ)) and (12a)τa(θ) = 0.89 exp(−6.70 cos(θ))− i0.62 exp(−13.6 cos(θ)). (12b)

Equations (11) and (12) constitute a phenomenological model for our polarizer suitable to pre-dict Jones and Mueller matrices for any choice of the parameters θ and φ . In Fig. 5, we demon-strate that this model accurately agrees with our observation for different configurations.

Equation Eq. (12a) is a variant of Beer’s law [17,22] and describes how the absorption scaleswith the increasing effective thickness of the sample when tilted. The modulus square |τa|2 > 0

9

of Eq. (12b) accounts for the transmittance for crossed polarization, i.e. the fact that even if theelectric field is polarized parallel to the effective absorbing axis, the absorption is not 100%.The phase of the complex parameter τa indicates that this field component is scattered with aphase determined by the orientation of the nano-particles relative to the incoming wave.

For small tilting angles θ < 45, the observation agrees with the prediction of the geometricabsorbing model TA. Close to grazing incidence θ → 90, the latter deviates, which we canunderstand in a physical picture. The particles embedded in our polarizer are cigar-shaped [23,24]. Relevant for the polarization effect is the coupling of the light field to their long axesPPPA. By design, the wavelength is close to the resonance of the particles’ long axes. At normalincidence, the scattering and absorption is strong for states of polarization parallel to the longaxis and negligible in the orthogonal case.

When the polarizer is tilted, only the component of the electric field vector directed alongthe particles’ absorbing axis PPPA takes part in the interaction. Thus, the effect of a single particledecreases proportionally to cos(θ) as the coupling becomes less efficient.

The thickness of the polarizing layer guarantees that a light beam interacts with multipleparticles while propagating across the device. Consequently, the observed extinction ratio issignificantly larger than expected for a single particle. Our phenomenological model subsumesthe sophisticated effect of this ensemble using only two functions τa(θ) and τt(θ), which canbe directly measured.

6. Conclusion

We have presented a Mueller matrix polarimeter making use of inexpensive linear polarizersand arbitrary retarding elements. Our least squares optimization approach is fast, yet accurateand precise. In particular, we have used this setup to study the effect a tilted polarizer has onthe light field.

Incidentally, linear polarizers are also popular as reference samples to characterize suchmeasurement devices. Our data indicates that the combined statistical and systematic error ofany matrix element is less than 0.01, while for polarimeters of comparable speed and feasibil-ity, deviations between 0.03 and 0.10 per matrix element are typical [25]. In fact, our method iscomparable with the accuracy achieved by more sophisticated calibration techniques requiringthe use of multiple reference samples [26].

Finally, we have shown that a real-world polarizer, even when tilted, can be modeled geo-metrically. Using only the projection of the absorbing axis yielded already to an acceptableapproximation for the collective action of the nano-particle ensemble. It was demonstrated thatthe finite extinction ratio of realistic polarizers can be taken into account phenomenologically,including configurations close to grazing incidence. We are confident that future work will con-nect the observation to a detailed microscopic study of such nano-particles and their interactionwith the light field.

Acknowledgments

The authors would like to thank Norbert Lindlein and Vanessa Chille for useful discussions,and the anonymous Referees for insightful comments.

10

A.4. Distributing entanglement with separable states

Beyond the first-author publications summarized in this thesis, I was involved in aproject related to the distribution of entanglement with separable quantum statesof light. This work will be an essential part of Christian Peuntinger’s dissertation.

C. Peuntinger, V. Chille, L. Mista, N. Korolkova, M. Fortsch, J. Korger,C. Marquardt, and G. Leuchs. Distributing entanglement with separable states.arXiv:1304.0504. Phys. Rev. Lett. 111, 230506, 2013

81

82

Distributing entanglement with separable states

Christian Peuntinger,1, 2, ∗ Vanessa Chille,1, 2, ∗ Ladislav Mista, Jr.,3 Natalia Korolkova,4

Michael Fortsch,1, 2 Jan Korger,1, 2 Christoph Marquardt,1, 2 and Gerd Leuchs1, 2

1Max Planck Institute for the Science of Light, Gunther-Scharowsky-Str. 1 / Bldg. 24, Erlangen, Germany2Institute of Optics, Information and Photonics,

University of Erlangen-Nuremberg, Staudtstraße 7/B2, Erlangen, Germany3Department of Optics, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

4School of Physics and Astronomy, University of St. Andrews,North Haugh, St. Andrews, Fife, KY16 9SS, Scotland

(Dated: August 26, 2013)

We experimentally demonstrate a protocol for entanglement distribution by a separable quantumsystem. In our experiment two spatially separated modes of an electromagnetic field get entangledby local operations, classical communication, and transmission of a correlated but separable modebetween them. This highlights the utility of quantum correlations beyond entanglement for estab-lishment of a fundamental quantum information resource and verifies that its distribution by a dualclassical and separable quantum communication is possible.

PACS numbers: 03.65.Ud, 03.67.Hk

Like a silver thread, quantum entanglement [1] runsthrough the foundations and breakthrough applicationsof quantum information theory. It cannot arise from lo-cal operations and classical communication (LOCC) andtherefore represents a more intimate relationship amongphysical systems than we may encounter in the classicalworld. The ‘nonlocal’ character of entanglement man-ifests itself through a number of counterintuitive phe-nomena encompassing Einstein-Podolsky-Rosen paradox[2, 3], steering [4], Bell nonlocality [5] or negativity ofentropy [6, 7]. Furthermore, it extends our abilities toprocess information. Here, entanglement is used as a re-source which needs to be shared between remote parties.However, entanglement is not the only manifestation ofquantum correlations. Notably, also separable quantumstates can be used as a shared resource for quantumcommunication. The experiment presented in this pa-per highlights the quantumness of correlations in sepa-rable mixed states and the role of classical informationin quantum communication by demonstrating entangle-ment distribution using merely a separable ancilla mode.

The role of entanglement in quantum information isnowadays vividly demonstrated in a number of exper-iments. A pair of entangled quantum systems sharedby two observers enables to teleport [8] quantum statesbetween them with a fidelity beyond the boundary setby classical physics. Concatenated teleportations [9] canfurther span entanglement over large distances [10] whichcan be subsequently used for secure communication [11].An a priori shared entanglement also allows to double therate at which information can be sent through a quan-tum channel [12] or one can fuse bipartite entanglementinto larger entangled cluster states being a ‘hardware’ forquantum computing [13].

∗ contributed equally to this work

The common feature of all entangling methods usedso far is that entanglement is either produced by someglobal operation on the systems that are to be entangledor it results from a direct transmission of entanglement(possibly mediated by a third system) between the sys-tems. Even entanglement swapping [9, 14], capable ofestablishing entanglement between the systems that donot have a common past, is not an exception to the rulebecause also here entanglement is directly transmittedbetween the participants.

However, quantum mechanics admits conceptually dif-ferent means of establishing entanglement which are freeof transmission of entanglement. Remarkably, creation ofentanglement between two observers can be disassembledinto local operations and the communication of a sepa-rable quantum system between them [15]. The impossi-bility of entanglement creation by LOCC is not violatedbecause communication of a quantum system is involved.The corresponding protocol exists only in a mixed-statescenario and obviously utilizes less quantum resources incomparison with the previous cases because communi-cation of only a discordant [16–18] separable quantumsystem is required.

In this paper, we experimentally demonstrate the en-tanglement distribution by a separable ancilla [15] withGaussian states of light modes [19]. The protocol aimsat entangling mode A which is in possession of a senderAlice with mode B held by a distant receiver Bob by lo-cal operations and transmission of a separable mediatingmode C from Alice to Bob. This requires the parties toprepare their initial modes A, B and C in a specific cor-related but fully separable Gaussian state. Once the re-source state ρABC is established no further classical com-munication is needed to accomplish the protocol. To em-phasize this, we attribute the state preparation processto a separate party, David. Note, that this resource statepreparation is performed by LOCC only. No global quan-tum operation with respect to David’s separated boxes

2

FIG. 1. Sketch of the Gaussian entanglement distribution pro-tocol. David prepares a momentum squeezed vacuum modeA, a position squeezed vacuum mode C and a vacuum modeB. He then applies random displacements (green boxes) ofthe x quadrature (horizontal arrow) and the p quadrature(vertical arrow) as in (1), which are correlated via classicalcommunication channel (green line). David passes the modesA and C to Alice, and mode B to Bob. Alice superimposesthe modes A and C on a balanced beam splitter BSAC andcommunicates the separable output mode C′ to Bob (red lineconnecting Alice and Bob). Bob superimposes the receivedmode C′ with his mode B on another balanced beam splitterBSBC , which establishes entanglement between the outputmodes A′ and B′ (black lemniscata). Note the position ofthe displacement on mode B. In the original protocol thedisplacement is performed before BSBC , which is depictedby the corresponding box with dashed green line. Equiva-lently, this displacement on mode B can be performed afterBSBC (dashed arrow indicates the respective relocation ofdisplacement) on mode B′, and even a posteriori after themeasurement of mode B′.

is executed at the initial stage and no entanglement ispresent.

Protocol. The protocol [19] depicted in Fig. 1 con-sists of three steps. Initially, a distributor David pre-pares modes A and C in a momentum squeezed and po-sition squeezed vacuum state, respectively, with quadra-

tures xA,C = e±rx(0)A,C , pA,C = e∓rp(0)A,C , whereas mode

B is in a vacuum state with quadratures xB = x(0)B and

pB = p(0)B . Here r is the squeezing parameter and the su-

perscript “(0)” denotes the vacuum quadratures. Davidthen exposes all the modes to suitably tailored local cor-related displacements [20]:

pA → pA − p, xC → xC + x,

xB → xB +√

2x, pB → pB +√

2p. (1)

The uncorrelated classical displacements x and p obey azero mean Gaussian distribution with the same variance(e2r−1)/2. The state has been prepared by LOCC acrossA|B|C splitting and hence is fully separable.

In the second step, David passes modes A, C of theresource state to Alice and mode B to Bob. Alice su-perimposes modes A and C on a balanced beam split-

ter BSAC , which output modes are denoted by A′ andC ′. The beam splitter BSAC cannot create entanglementwith mode B. Hence the state is separable with respectto B|A′C ′ splitting. Moreover, the state also fulfils thepositive partial transpose (PPT) criterion [21, 22] withrespect to mode C ′ and hence is also separable acrossC ′|A′B splitting [23] as required [24].

In the final step, Alice sends mode C ′ to Bob whosuperimposes it with his mode B on another balancedbeam splitter BSBC . The presence of the entanglementbetween modes A′ and B′ is confirmed by the sufficientcondition for entanglement [25, 26]

∆2norm(gxA′ + xB′)∆

2norm(gpA′ − pB′) < 1, (2)

where g is a variable gain factor. Minimizing the lefthand side of Ineq. 2 with respect to g we get fulfilmentof the criterion for any r > 0, which confirms successfulentanglement distribution.Experiment. The experimental realization is divided

in three steps: state preparation, measurement, and dataprocessing. The corresponding setup is depicted in Fig. 2.From now on, we will work with polarization variables de-scribed by Stokes observables (see, e.g., [27, 28]) insteadof quadratures. We choose the state of polarization suchthat mean values of S1 and S2 equal zero while 〈S3〉 0.

This configuration allows to identify the “dark” S1-S2-plane with the quadrature phase space. Sθ, Sθ+π/2 in this

plane correspond to S1, S2 renormalized with respect toS3 ≈ S3 and can be associated with the effective quadra-tures x, p. We use the modified version of the protocolindicated in Fig. 1 by the dashed arrow showing the alter-native position of displacement in mode B: The randomdisplacement applied by David can be performed afterthe beam splitter interaction of B and C ′, even a posteri-ori after the measurement of mode B′. This is technicallymore convenient and emphasizes that the classical infor-mation is sufficient for the entanglement recovery afterthe interaction of both modes A and B with the ancillamode C and C ′, respectively.

David prepares two identically, polarization squeezedmodes [26, 27, 29, 30] and adds noise in the form ofrandom displacements to the squeezed observables. Thetechnical details on the generation of these modes can befound in the Supplemental Material [24]. The modula-tion patterns applied to modes A and C to implement therandom displacements are realized using electro-opticalmodulators (EOMs) and are chosen such that the two-mode state ρA′C′ is separable. By applying a sinusoidalvoltage Vmod, the birefringence of the EOMs changes ata frequency of 18.2 MHz. In this way the state is modu-lated along the direction of its squeezed observable.

Such two identically prepared modes A and C areinterfered on a balanced beam splitter (BSAC) with afixed relative phase of π/2 by controlling the optical pathlength of one mode with a piezoelectric transducer and alocking loop. This results in equal intensities of both out-put modes. In the final step, Bob mixes the ancilla modeC ′ with a vacuum mode B on another balanced beam

3

splitter, and performs a measurement on the transmit-ted mode B′.

function generator

WSHWP

HWP WS

HWP QWP

polarizationsqueezer A

polarizationsqueezer C

HWP QWP

BSAC

BSBC

State Preparation Measurement Process Data Acquisition

Down-mixer

Amplifier

AD sampling10Msamples/s

vacuum mode B

FIG. 2. Sketch of the experimental setup. Used abbrevi-ations: HWP: Half-wave plate; QWP: Quarter-wave plate;EOM: Electro-optical modulator; BS: Beam splitter; WS:Wollaston prism; (State Preparation) The polarization of twopolarization squeezed states (A and C) is modulated usingEOMs and sinusoidal voltages from a function generator (dot-ted lines). The HWPs before the EOMs are used to adjustthe direction of modulation to the squeezed Stokes variable,whereas the QWPs compensate for the stationary birefrin-gence of the EOMs. The such prepared modes interfere witha relative phase of π/2 on a balanced beam splitter BSAC . Inthe last step of the protocol, the mode C′ interferes with thevacuum mode B on a second balanced beam splitter BSBC .(Measurement Process) A rotatable HWP, followed by a WSand a pair of detectors, from which the difference signal istaken, allows to measure all possible Stokes observables in theS1-S2-plane. To determine the two-mode covariance matrixγA′B′ all necessary combinations of Stokes observables aremeasured. Removing the second beam splitter of the statepreparation allows us to measure the covariance matrix ofthe two-mode state ρA′C′ . (Data Acquisition) To achieve dis-

placements of the modes in the S1-S2-plane we electronicallymix the Stokes signals with a phase matched electrical localoscillator and sample them by an analog-to-digital converter.

The states involved are Gaussian quantum states and,hence, are completely characterized by their first mo-ments and the covariance matrix γ comprising all secondmoments [24]. To study the correlations between modeA′ and C ′ after BSAC , multiple pairs of Stokes observ-ables (SA′,θ, SC′,θ) are measured. The covariance ma-trix γA′C′ is obtained by measuring five pairs of observ-ables: (SA′,0 , SC′,0), (SA′,90 , SC′,0), (SA′,0 , SC′,90),

(SA′,90 , SC′,90) and (SA′,45 , SC′,45), which determineall its 10 independent elements. Here, θ is the angle inthe S1-S2-plane between S0 and Sθ.

For the measurements of the different Stokes observ-ables, we use two Stokes measurement setups, each com-prising a rotatable half-wave plate, a Wollaston prismand two balanced detectors. The difference signal of onepair of detectors gives one Stokes observable Sθ in the S1-S2-plane, depending on the orientation of the half-waveplate. The signals are electrically down-mixed using anelectric local oscillator at 18.2 MHz, which is in phasewith the modulation used in the state preparation step.With this detection scheme, the modulation translatesto a displacement of the states in the S1-S2-plane. Thedifference signal is low pass filtered (1.9 MHz), amplified

and then digitized using an analog-to-digital convertercard (GaGe Compuscope 1610) at a sampling rate of 10Msamples/s. After the measurement process we digitallylow pass filter the data by an average filter with a windowof 10 samples.

Due to the ergodicity of the problem, we are able tocreate a Gaussian mixed state computationally from thedata acquired as described above. By applying 80 dif-ferent modulation depths Vmod to each of the EOMs weacquire a set of 6400 different modes. From these set ofmodes we take various amounts of samples, weighted bya two dimensional Gaussian distribution.

The covariance matrix γA′C′ for the two-mode stateafter BSAC has been measured to be:

γA′C′ =

20.90 1.102 −7.796 −1.6791.102 25.30 1.000 14.63−7.796 1.000 20.68 0.8010−1.679 14.63 0.8010 24.65

. (3)

The estimation of the statistical errors of this covariancematrix γA′C′ can be found in the Supplemental Mate-rial [24]. A necessary and sufficient condition for sepa-rability of a Gaussian state ρXY of two modes X andY with the covariance matrix γXY is given by the PPTcriterion:

γ(TY )XY + iΩ2 ≥ 0, Ω2=

2⊕

i=1

(0 1−1 0

)(4)

where γ(TY )XY is the matrix corresponding to the partial

transpose of the state ρXY with respect to the modeY [24]. Effects that could possibly lead to some non-Gaussianity of the utilized states are discussed in detailalso in [24]. The state described by γA′C′ fulfils the con-dition (4) as the eigenvalues (39.84, 28.47, 13.85, 9.371)

of (γ(TC′ )A′C′ + iΩ2) are positive, hence mode C ′ remains

separable after BSAC .The measured two-mode covariance matrix of the out-

put state γA′B′ is given by:

γA′B′ =

19.95 1.025 −4.758 −1.0631.025 22.92 0.9699 9.153−4.758 0.9699 9.925 0.2881−1.063 9.153 0.2881 11.65

. (5)

The statistical error of this measured covariance matrix isgiven in the Supplemental Material [24]. The separabilityis proven by the PPT criterion (eigenvalues 28.24, 21.79,8.646, 5.756).

The post-processing for the recovery of the entangle-ment is performed on the measured raw data of modeB′. Therefore, the displacement of the individual modescaused by the two modulators is calibrated. By means ofthis calibration, suitable displacements are applied dig-itally. The classical noise inherent in the mode B′ iscompletely removed. A part of the classical noise asso-ciated with SA′,0 is subtracted from SB′,0 , while the

same fraction of the noise in SA′,90 is added to SB′,90 .

4

FIG. 3. Entanglement distributed between modes A′ and B′.The experimental values for the criterion (2) are depicted independence of the gain factor g. Due to the attenuation ofthe mode B by 50 %, a gain factor about 0.5 yields a valuesmaller than 1, i.e. below the limit for entanglement (redline). The inset zooms into the interesting section around theminimum. The depicted estimated errors are so small becauseof the large amount of data taken.

In this way, the noise partially cancels out in the calcula-tion of the separability criterion (2) and allows to revealthe entanglement. We chose the fraction as in (1), whichis compatible with the separability of the transmittedmode C ′ from the subsystem (A′B) in the scenario withmodulation on mode B before the beam splitter BSBC .

Only as Bob receives the classical information aboutthe modulation on the initial modes A and C from David,he is able to recover the entanglement between A′ andB′. Bob verifies that the product entanglement crite-rion (2) is fulfilled as illustrated in Fig. 3. That provesthe emergence of entanglement. The used gain factor gconsiders the slightly different detector response and theintentional loss of 50 % at Bob’s beam splitter. The clear-est confirmation of entanglement 0.6922 ± 0.0002 < 1 isshown for gopt = 0.4235 ± 0.0005 (Fig. 3). This is theonly step of the protocol, where entanglement emerges,thus demonstrating the remarkable possibility to entan-gle the remote parties Alice and Bob by sending solely aseparable auxiliary mode C ′.

Discussion The performance of the protocol can beexplained using the structure of the displacements (1).Entanglement distribution without sending entanglementhighlights vividly the important role played by classicalinformation in quantum information protocols. Classicalinformation lies in our knowledge about all the correlateddisplacement involved. This allows the communicatingparties (or David on their behalf) to adjust the displace-ments locally to recover through clever noise additionquantum resources initially present in the input quan-tum squeezed states. Mode C ′ transmitted from Alice toBob carries on top of the sub-shot noise quadrature ofthe input squeezed state the displacement noise which

is anticorrelated with the displacement noise of Bob’smode. Therefore, when the modes are interfered on Bob’sbeam splitter, this noise partially cancels out in the out-put mode B′ when the light quadratures of both modesadd. Moreover, the residual noise in Bob’s position (mo-mentum) quadrature is correlated (anticorrelated) withthe displacement noise in Alice’s position (momentum)quadrature in mode A′, again initially squeezed. Due tothis the product of variances in criterion (2) drop belowthe value for separable states and thus entanglement be-tween Alice’s and Bob’s modes emerges. The differencebetween the theoretically proposed protocol [19] and theexperimental demonstration reported in this paper liesmerely in the way how classical information is used. Inthe original protocol, the classical information is retainedby David and he is responsible for clever tailoring of cor-related noise. Bob evokes the required noise cancellationby carrying out the final part of global operation via su-perimposing his mode with the ancilla on BSBC . In theexperimentally implemented protocol, David shares partof his information with Bob, giving Bob a possibility toget entanglement a posteriori, by using his part of clas-sical information after the quantum operation is carriedout. Thus entanglement distribution in our case is trulyperformed via a dual classical and quantum channel, viaclassical information exchange in combination with thetransmission of separable quantum states.

There are other interesting aspects to this protocol,which may open new, promising avenues for research.Noise introduced into the initial states by displacementscontains specific classical correlations. On a more fun-damental level, these displacements can be seen as cor-related dissipation (including mode C into ”environ-ment”). It is already known, that dissipation to a com-mon reservoir can even lead to the creation of entangle-ment [31, 32]. Our scheme can be viewed as anothermanifestation of a positive role dissipation may play inquantum protocols.

The presence of correlated noise results in non-zeroGaussian discord at all stages of the protocol, a moregeneral form of quantum correlations, which are be-yond entanglement [33]. The role of discord in entan-glement distribution has been recently discussed theo-retically [16, 17]. The requirements devised there arereflected in the particular separability properties of ourglobal state after the interaction of modes A and C onAlice’s beam splitter. The state ρA′BC′ contains discordand entanglement across A′|BC ′ splitting and is sepa-rable and discordant across C ′|A′B splitting as requiredby the protocol. Our work thus illustrates an interplayof entanglement and other quantum correlations, such ascorrelations described by discord, across different parti-tions of a multipartite quantum system..

L. M. acknowledges project P205/12/0694 of GACR.N. K. is grateful for the support provided by the A. vonHumboldt Foundation. The project was supported bythe BMBF grant “QuORep” and by the FP7 projectQESSENCE. We thank Christoffer Wittmann and

5

Christian Gabriel for fruitful discussions.

Note. Recently, an experiment has been presentedin [34], which is based on a similar protocol. The maindifference consists in the fact that it starts with entangle-ment which is hidden and recovered with thermal states.For this implementation no knowledge about classical

information has to be communicated to Bob, besides theused thermal state. By contrast the setup presented inthis work exhibits entanglement only at the last step ofthe protocol. Thus both works give good insights on dif-ferent aspects of the theoretically proposed protocol [19].Another independent demonstration of a similar protocolbased on discrete variables was recently presented in [35].

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Distributing entanglement with separable statesSupplementary Information

Christian Peuntinger,1, 2, ∗ Vanessa Chille,1, 2, ∗ Ladislav Mista, Jr.,3 Natalia Korolkova,4

Michael Fortsch,1, 2 Jan Korger,1, 2 Christoph Marquardt,1, 2 and Gerd Leuchs1, 2

1Max Planck Institute for the Science of Light, Gunther-Scharowsky-Str. 1 / Bldg. 24, Erlangen, Germany2Institute of Optics, Information and Photonics,

University of Erlangen-Nuremberg, Staudtstraße 7/B2, Erlangen, Germany3Department of Optics, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

4School of Physics and Astronomy, University of St. Andrews,North Haugh, St. Andrews, Fife, KY16 9SS, Scotland

(Dated: August 26, 2013)

I. PREPARATION OF POLARIZATIONSQUEEZED STATES

To prepare two identically, polarization squeezedmodes we use a well known technique like in [1–4]. Eachof these modes is generated by launching two orthog-onally polarized femtosecond pulses (∼200 fs) with bal-anced powers onto the two birefringent axes of a polar-ization maintaining fiber (FS-PM-7811, Thorlabs, 13 m).The pump source is a soliton-laser emitting light at acenter wavelength of 1559 nm and a repetition rate of80 MHz. By exploiting the optical Kerr effect of thefibers, the orthogonally polarized pulses are individuallyquadrature squeezed and subsequently temporally over-lapped with a relative phase of π/2, resulting in a circu-lar polarized light beam. The relative phase is activelycontrolled using an interferometric birefringence compen-sator including a piezoelectric transducer and a lockingloop based on a 0.1 % tap-off signal after the fiber. Interms of Stokes observables (see [1, 5]) this results in

states with zero mean values of S1 and S2, but a bright〈S3〉 0 component. These states exhibit polarization

squeezing at a particular angle in the S1-S2-plane.

II. GAUSSIAN STATES

We implement the entanglement distribution proto-col using optical modes which are systems in infinitely-dimensional Hilbert state space. An N -mode system canbe conveniently characterized by the quadrature oper-ators xj , pk, j, k = 1, 2, . . . , N satisfying the canonicalcommutation rules [xj , pk] = iδjk which can be expressedin the compact form as

[ξj , ξk] = iΩNjk. (1)

∗ contributed equally to this work

Here we have introduced the vector of quadratures ξ =(x1, p1, . . . , xN , pN ) and

ΩN =N⊕

i=1

J, J =

(0 1−1 0

), (2)

is the symplectic matrix.The present protocol relies on Gaussian quantum

states. As any standard Gaussian distribution, a Gaus-sian state ρ is fully characterized by the vector of its firstmoments

d = Tr(ρξ), (3)

and by the covariance matrix γ with elements

γjk = Tr[ρξj − dj11, ξk − dk11], (4)

where A, B = AB+ BA is the anticommutator. A realsymmetric positive-definite 2N × 2N matrix γ describesa covariance matrix of a physical quantum state if andonly if it satisfies the condition [6]:

γ + iΩN ≥ 0. (5)

The separability of Gaussian states can be tested us-ing the positive partial transpose (PPT) criterion. Asingle mode j is separable from the remaining N − 1modes if and only if the Gaussian state ρ has a posi-tive partial transposition ρTj with respect to the modej [7, 8]. On the level of the covariance matrices, thepartial transposition is represented by a matrix Λj =(⊕N−1

i 6=j=1 11(i))⊕ σ(j)

z , where σ(j)z = diag(1,−1) is the di-

agonal Pauli z-matrix of mode j and 11(i) is the 2 × 2identity matrix. The matrix γ(Tj) corresponding to apartially transposed state ρTj reads γ(Tj) = ΛjγΛTj . Interms of the covariance matrix, one can then express thePPT criterion in the following form. A mode j is sep-arable from the remaining N − 1 modes if and only if[7, 8]

γ(Tj) + iΩN ≥ 0. (6)

The PPT criterion (6) is a sufficient condition for sepa-rability only under the assumption of Gaussianity. In our

2

experiment, however, non-Gaussian states can be gener-ated for which this criterion represents only a necessarycondition for separability. Therefore it can fail in detect-ing entanglement.

III. ANALYSIS OF NON-GAUSSIANITY

There are two sources of imperfections in our ex-perimental set up that are potential sources of non-Gaussianity. These are phase fluctuations and the modu-lation of the initial squeezed states before the first beamsplitter. They are discussed in the following sections.

A. Phase fluctuations

The experiment includes an interference of the modesA and C on a beam splitter, which is the first beamsplitter BSAC in the protocol. Imperfect phase lockingat this beam splitter might cause a phase drift result-ing in a non-Gaussian character of the state ρA′C′ afterthe beam splitter. The phase fluctuations can be mod-elled by a random phase shift of mode A before the beamsplitter described by a Gaussian distribution P (φ) withzero mean and variance σ2. Denoting the operator cor-responding to a beam splitter transformation as U andthe phase shift φ on mode A as VA(φ), the state ρA′C′can be linked to the state ρAC before the onset of phasefluctuations as

ρA′C′ =

∫ ∞

−∞P (φ)U VA(φ)ρAC V

†A(φ)U†dφ. (7)

Hence we can express the measured covariance matrixγA′C′ given in Eq. (3) of the main letter, and the vec-tor of the first moments d′ of the state ρA′C′ in termsof the covariance matrix γAC and the vector of thefirst moments d of the input state ρAC . For this itis convenient to define matrices D and D′ of the firstmoments with elements Dij = didj and D′ij = d′id

′j ,

i, j = 1, . . . , 4. Using Eq. (7) and after some algebra,one gets the transformation rule for the matrix of thefirst moments in the form D′ = UΣDΣUT , where U de-scribes the beam splitter on the level of covariance ma-

trices. Σ = diag(e−σ2

2 , e−σ2

2 , 1, 1) is a diagonal matrix.Similarly we get the covariance matrix

γA′C′ = U (ΣγACΣ + π ⊕ 0)UT , (8)

where 0 is the 2× 2 zero matrix and

π =(1− e−σ2

)2

2(A+ α) +

1− e−2σ2

2J(A+ α)JT . (9)

Here the matrix A is the 2 × 2 matrix with elementsAij = (γAC)ij , i, j = 1, 2, α is the 2 × 2 matrix withelements αij = 2Dij , i, j = 1, 2, and J is defined inEq. (2). Similar to Ref. [9] we can now invert the relation(8) and express the input covariance matrix γAC via the

output covariance matrix γA′C′ and the first momentsafter the beam splitter BSAC as

γAC = Σ−1UT γA′C′UΣ−1 + π ⊕ 0, (10)

where

π =(1− eσ2

)2

2(A+ α) +

1− e2σ2

2J(A+ α)JT . (11)

The 2 × 2 matrices A and α possess the elements Aij =(UT γA′C′U)ij and αij = 2(UTD′U)ij , i, j = 1, 2.

Our estimate for the variance of the phase fluctuationsis σ2 = 0.02 and the vector d′ of the measured meanvalues of the state ρA′C′ reads

d′ = (−0.208, 9.876, 13.32, 1.78). (12)

By substituting these experimental values for σ2 and d′ inEq. (10) and using the beam splitter with the measuredtransmissivity T = 0.49 we get a legitimate covariancematrix γAC before the phase fluctuations as can be easilyverified by checking the condition (5).

Provided that the state with the covariance matrix γACis classical it can be expressed as a convex mixture ofproducts of coherent states. Gaussian distributed phasefluctuations and a beam splitter preserve the structureof the state, hence the state after the first beam splittercannot be entangled. The covariance matrix γAC deter-mines a physical Gaussian quantum state. Moreover, thecovariance matrix possesses all eigenvalues greater thanone and therefore the state is not squeezed [6] which is ina full agreement with the fact that modulations of modesA and C completely destroy the squeezing. It then fol-lows that this Gaussian state is classical and it thereforetransforms to a separable state after the first beam split-ter.

The inversion (10) thus allows us to associate a Gaus-sian state before the phase fluctuations with the covari-ance matrix γA′C′ measured after the first beam splitter.The separability properties of the state after the beamsplitter can then be determined from the non-classicalityproperties of this Gaussian state.

B. Gaussianity of the utilized states

We have paid great attention on the modulations onmodes A and C to preserve Gaussian character of thestate ρA′C′ . Our success can be visually inspected atthe examples in Fig. 1, which illustrates that both themodulation and the subsequent Gaussian mixing faith-fully samples the required Gaussian shape. Besides thisraw visual check we have also tested quantitatively Gaus-sianity of the involved states by measuring higher-ordermoments of the Stokes measurements on modes A′ andC ′. Specifically, we have focused on the determinationof the shape measures called skewness S and kurtosis K

3

FIG. 1. Histogram plots for SA′,0 of the Gaussianmixed state (blue) and three exemplary individualmodes. This figure illustrates the preparation of the Gaus-sian mixed state via post processing. Exemplarily, three of the6400 displaced individual modes are visualized by their his-tograms (in green, black and red colour). The normalizationis chosen such that they can be depicted in the same plot asthe histogram of the mixed state (blue), which is normalizedto its maximum value. By merging the data for all individ-ual modes using a weighting with a two dimensional Gaussiandistribution, the mixed state is achieved. Its Gaussianity isvisualized by the Gaussian fit (red curve).

defined for a random variable x as the following third andfourth standardized moments

S =µ3

s3, K =

µ4

s4, (13)

where µk = 〈(x−〈x〉)k〉 is the kth central moment, 〈x〉 isthe mean value and s =

õ2 is the standard deviation.

Skewness characterizes the orientation and the amountof skew of a given distribution and therefore informs usabout its asymmetry in the horizontal direction. Gaus-sian distributions possess skewness of zero. The exem-

plary values of skewness for various measurement settingsare summarized in the Table I.

TABLE I. Skewness S for Stokes measurements on modes A′

and C′ in different measurement directions.Measurement SA′,0 SA′,90

Skewness×103 6.240 ± 0.781 −1.478 ± 0.563

Measurement SC′,0 SC′,90

Skewness×103 10.123 ± 0.727 1.106 ± 0.830

The skewness can vanish also for the other symmetricaldistributions, which may, however, differ from a Gaussiandistribution in the peak profile and the weight of tails.These differences can be captured by the kurtosis whichis equal to 3 for Gaussian distributions. The exemplaryvalues of kurtosis for various measurement settings aresummarized in the Table II.

TABLE II. Kurtosis K for Stokes measurements on modes A′

and C′ in different measurement directions.Measurement SA′,0 SA′,90

Kurtosis 2.971 ± 2.211 × 10−3 2.986 ± 1.852 × 10−3

Measurement SC′,0 SC′,90

Kurtosis 2.972 ± 1.978 × 10−3 2.992 ± 1.568 × 10−3

The tables reveal that the measured probability dis-tributions satisfy within the experimental error the nec-essary Gaussianity conditions S = 0 and K = 3. Moresophisticated normality tests can be performed, which isbeyond the scope of the present manuscript.

IV. STATISTICAL ERRORS OF THEMEASURED COVARIANCE MATRICES

By dividing our dataset in 10 equal in size parts wecan estimate the statistical errors of our measured co-variance matrices γA′C′ and γA′B′ given in Eqs. (3) and(5) of the main letter. We calculate the covariance ma-trix for each part and use the standard deviation as errorestimation. The covariance matrix γA′C′ including thestatistical error turns out to be

γA′C′ =

20.90± 0.0087 1.102± 0.0091 −7.796± 0.0069 −1.679± 0.00761.102± 0.0091 25.30± 0.013 1.000± 0.0071 14.63± 0.0091−7.796± 0.0069 1.000± 0.0071 20.68± 0.0093 0.8010± 0.011−1.679± 0.0076 14.63± 0.0091 0.8010± 0.011 24.65± 0.0073

. (14)

Similarly, the covariance matrix γA′B′ including the sta-tistical error reads as

4

γA′B′ =

19.95± 0.011 1.025± 0.016 −4.758± 0.0050 −1.063± 0.00511.025± 0.016 22.92± 0.012 0.9699± 0.0047 9.153± 0.0058−4.758± 0.0050 0.9699± 0.0047 9.925± 0.0048 0.2881± 0.0047−1.063± 0.0051 9.153± 0.0058 0.2881± 0.0047 11.65± 0.0038

. (15)

We could achieve such small statistical errors by record-ing sufficient large datasets.

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