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Hecht; 10/12/2010; 8-1
Chapter 8. Polarization 8.1 The Nature of Polarized Light A. Linear Polarization Sum of two waves ( ) ( ) ( )ω ω ε= − + − +ˆ ˆ, cos cosox oyE z t xE kz t yE kz t For 0, 2 ...ε π= ± , ( ) ( )ˆ ˆ cosox oyE xE yE kz tω= + −
For ...ε π= ± , ( ) ( )ˆ ˆ cosox oyE xE yE kz tω= − −
B. Circular Polarization Right-circularly polarized
For o ox oyE E E= = and /2 2 mε π π= − +
( ) ( )ˆ ˆcos sinoE E x kz t y kz tω ω= − + −⎡ ⎤⎣ ⎦ E is constant but direction of E changes in space and in time.
At oz z= and t=0 → ( )ˆ cosx o oE xE kz= , ( )ˆ siny o oE yE kz=
At oz z= and /ot kz ω= → ˆx oE xE= , 0yE = E is rotating clockwise in time Left-circularly polarized
For o ox oyE E E= = and /2 2 mε π π= +
( ) ( )ˆ ˆcos sinoE E x kz t y kz tω ω= − − −⎡ ⎤⎣ ⎦
C. Elliptic Combin xE
yE
→
It is an
tan
Various
States o mI=== b -state
Right-c
cal Polarizane two wave
(cosoxE kz=
(cosoyE k=2
x
ox
EE E
⎛⎛ ⎞+ ⎜⎜ ⎟ ⎜⎝ ⎠ ⎝
ellipse mak
2
2n2 ox
ox
E EE
α =
s polarizatio
of polarizatioI===iI===b -st
e is given by
circularly po
ation es
)kz tω−
)kz tω ε− + 2
2y
oy
E EE E
⎞ ⎛−⎟ ⎜⎟ ⎝⎠
king an ang
2
cosoy
oy
EE
ε−
on configura
ion tates
y a sum of
olarization
coyx
ox oy
EEE E
⎛ ⎞⎞⎜ ⎟⎟ ⎜ ⎟⎠ ⎝ ⎠
gle α with x
ations
o -state an
2os sinε ε=
x-axis
nd i -state
e.
Hechht; 10/12/22010; 8-2
8.2 Pola Natural (Random Principl Dic
A linear The elec TE The tra (I θ
arizers l light mly polarize
les of polarichroism(Sele
r polarizer h
ctric field cocosoE θ=
↑ ↑ T The input
ansmitted i) ( )0 cosIθ =
ed)
izer ective absor
has transmi
omponent t
The angle bt electric fie
intensity 2s θ
→ ↑ Polariz
rption), Ref
ission axis a
that is paral
between oE eld.
Pol
zer
flection, Bir
and extinct
llel to the tr
and the tra
: Malus’s L
larized light
refringence(D
tion axis.
ransmission
ansmission
Law
t
Double refr
n axis can o
axis.
Hech
raction)
only be tran
ht; 10/12/2
nsmitted.
2010; 8-3
8.3 Dich Dichroic A. The Wi y-polari → Cur → Rer → For Bac x-polari B. Dichro Absorpt → Ani Tourma
Electric → Tra Some ab (Tourma C. Polaro J-sheet H-sheet
hroism c material a
ire-Grid Po
ized light (Vrrent in y-dradiation inrward radiackward rad
ized light pr
oic Crystalstion and conisotropic cry
aline crystalc field perpeansmission
bsorption aalin is usua
id : Herapathi ↑ Dichro
t: Long mole
absorbs one
olarizer
Vertical diredirection (Son forward anation canceliation form
ropagates th
s nductivity dystals
ls endicular toalong optic
along the opally green co
ite needles a
ic crystal
ecules, not
e of the two
ction) ome energy nd backwars the incides a reflected
hrough the
depend on d
o the optic acs axis.
ptic axis depolored)
aligned in o
dichroic, fo
orthogonal
loss by jou
rd directionsent wave. d wave.
grid with n
direction
axis is stron
pends on wa
one directio
orm the wire
l m -states
ule heat). s.
no change
ngly absorb
avelength.
n
e grid.
bed.
Hechht; 10/12/22010; 8-4
8.4 Bire Birefrin Electron → z-ax A. Calcite 3CaCO Calcite
[Fig 8.16
A single → Two The Principa
Principa The ordina The sec
efringencngent crysta
n binding foxis become
e has 3-fold
in a cleavag6] [Fig 8.18
e beam into o beams aftey are linea
al plane al section
ary ray condary wav
ce al has two d
orces are ids optic axis
symmetry a
ge form form8]
Calcite ter the crys
arly polarize
: A plane c: A princip
velets have
different ind
entical in xs
about the o
ms rhomboh
tal: Ordinaed and ortho
containing tpal plane tha
the speed o
dices of refra
x- and y-axi
ptic axis
hedron
ary ray, Extogonal to ea
the optic axat is norma
of v⊥
action.
s
raordinary ach other
xis al to two sur
ray.
rfaces of th
o-ray ⊥ k-e-ray ⊥ k-
Hech
e cleavage f
-vector and-vector and
ht; 10/12/2
form
d optic axis.d o-ray.
2010; 8-5
The extrao The elec
Ray dire From It’s p It is n B. Birefrin Cubic c Optic Singl Hexagon Symm Unia Orthorh Biax Uniaxia
Birefrin
ordinary rayctric field =
ection : m the origin parallel to thnot normal
ngent Crystcrystal cally isotrople index of r
nal, Tetragometric in xy
axial crysta
hombic, Molxial crystals
al crystal hangence is de
y perpendicu ↑ =v⊥
of a wavelehe energy fl to the wave
tals
pic. refraction.
onal, Trigony-plane. als with opti
loclinic, Tris with two o
as two indicefined as nΔ
ular + paral ↑=
et to the tanlow directioefront.
nal
ic axis alon
iclinic optic axes
ces of refrac
e on n n≡ −
Neg
llel to optic v . (Two di
ngent point n (Poynting
g z-axis.
ction: on c=
gative and p
axis ifferent spee
of the waveg vector).
/ and ec v n⊥
positive unia
eds)
elet with the
/e c v=
axial crysta
Hech
e wavefront
als
ht; 10/12/2
t.
2010; 8-6
C. BirefrinNicol prism Split ca ↑ on = The inci o-ray is
Glan-Fouc Same p o-ray is Field o Max in Cemen Field o Max in
Wollaston o- and The dev
ngent Polarm alcite rhomb
1.6584, en
ident beam totally inte
cault polarizpropagations totally intf view = 10
ntensity = 10
nted calcite f view = 30
ntensity = 1W
prism e-rays sepaviation ang
rizers
bohedron is
1.4864e =
m refracts internally refle
zer (or Glann direction oernally refle degree,
200 /W cm
prisms is ca degree,
2/W cm
arate at the
gle ranges fr
cemented b
to o- and e-cted at the
n-Air) of o- and e-ected at the
alled Glan-
e interface. rom 15 to 4o
by Canada ↑ n=1.55
-rays in the interface.
rays in the e interface.
-Thompson
45o .
balsam
e first prism
first prism.
n polarizer
m.
.
Hech
ht; 10/12/22010; 8-7
8.6 Pola When θ Using S tan
At pθ , n Plate pola If iθ = θ
Polarizing Multipl
Degree of
V =
max, I I
→ pI
→ V =
arization 90o
i tθ θ+ =
Snell’s law n /p t in nθ =
no reflection
arizer pθ , a single
g Cube le layers at
Polarizatiop
p n
II I
=+
minI are mea
max minI I= −
max min
max min
I II I
−=
+
by Refle, iθ is call
sini pn n=θ
n of parallel
plate is a p
the diagona
on
asured afte
n , nI
n
n
ction ed Brewste
sint t tn n⇒θ
l polarizatio
perfect pola
al surface (
, p nI I
r the analyz
min2I=
er angle (or
sin(90op−θ
on beam.
arizer.
Similar as p
: Intensitie
zer
polarization
)p
pile-of-plate
es of polariz
n angle, pθ
es)
zed and unp
Hech
)
polarized lig
ht; 10/12/2
ght.
2010; 8-8
8.7 Reta Two m
→ Chan A. Wave P After th
( ,E z
The pha
ϕΔ =
Fast axi The Half-W ϕ πΔ = It rotate It conve The Quart
/ϕ πΔ = It conve Retarders Single-o Relat Thin Multiple Relat Less Compou Fast Less B. Compe A comp Babinet Optic Soleil C
arders -states expenge of the p
Plates and he plate
) ˆ, co= oxt xE
ase differen
(2πλ
= −eo
d n
is has smal
Wave Plate
es the polarerts o stat
ter-Wave Pl
/2 erts m stat
in generalorder retardtive phase o and difficu
e-order retative phase os expensive
und zero-or axis of one sensitive to
ensators anensator can
t Compensac axes of we
Compensato
erience diffepolarization
Rhombs
s ω⎛
− +⎜⎝kz t
ce
)on
ller refractiv
rization direte to i sta
late
te to o /i
l der : of 2Δ <ϕ π . ult to make
arder : of 2 mϕ πΔ +, but narrow
rder retardee plate is oveo temperatu
nd Variablen control th
ator edge calcite
r
ferent phase state.
2 ˆπλ
⎞+⎟
⎠e
o
dn y
ve index tha
ection by 2θate and vice
i state and
but large fi
m w FOV, sen
er : erlapped wiure but still
e Retardershe amount o
es are shown
e changes in
ˆ cos⎛⎜⎝
oyyE kz
an that of s
θ . e versa.
d vice versa
eld-of-view
nsitive to inc
ith slow axil narrow fiel
s of the retard
n by dots
n the retard
2πωλ
− +o
t dn
low axis
.
cident angle
is of the othld-of-view.
dance.
der.
⎞⎟⎠
on
e, λ and T.
her.
Hech
ht; 10/12/22010; 8-9
8.8 Circ Linear Left-cir 8.9 PolaA. Bandw Monoch Polychr Quasim B. Interfe
Phase d Blue ma 8.10 Op Opticall Looking Dext Levor A. A Usefu A wave → Elec → Rota B. Optica Synthes But one Natu Prote Antib Five am They co
cular Pola polarizer rcular polar
arization width and Chromatic wa
omatic wav
monochroma
erence Colo
difference, Δakes constr
ptical Actly active: A
g in the diretrorotatory (rotatory (l-r
ful Model into helical
ctric and maated polariz
ally Active sized organi
e is dominanural sugar iseins consistbiotics cont
mino acids inontain an eq
arizers + 90o retrizer allows
of PolychCoherence Tave
ve
atic wave
ors
↑ Polarizer .
ϕΔ , after thructive inter
tivity linear plan
ection of the(d-rotatory) rotatory)
l molecules agnetic dipozation
Biological ic molecule
nt in naturas always d-rts of amino tain amino a
n a meteoritqual numbe
tarder. only the lef
hromaticTime of a P
: Single
: A ran Const
: A nar
↑ Wave plat
he wave platrference wh
e wave app
e source : Clock
: Coun
ole moment
Substances have an e
al organic mrotatory. acids(compacids that a
te in Austraer of l- and
ft-circular l
c Light Polychromae fequency.
nge of frequetant polariz
rrow bandw
↑ te. Analyz
te can depehile yellow d
ears to rota
kwise nter-clockwi
ts
s equal numb
molecules.
pounds of Care d-rotato
alia. d-rotatories
light.
atic Wave
encies. zation state
width .
zer.
nds of λ. destructive i
ate.
ise
ber of l- and
C, H, O, N), ory.
s
only during
interference
d d-isomers.
generally l-
Hecht
g a short tim
e at the out
.
-rotatory
t; 10/12/20
me.
tput.
010; 8-10
Hecht; 10/12/2010; 8-11 8.11 Induced Optical Effects-Optical Modulators A. Photoelasticity Isotropic substances can be made optically anisotropic by mechanical stress. B. The Faraday Effect Linear polarization rotates in a magnetic field applied along the beam direction. Bdβ = s : Rotation angle in minutes Verdet constant, magnetic flux density, length. Other magneto-optic effects Voigt and Cotton-Mouton effects occur for a magnetic field perpendicular to the beam direction. ↑ ↑ Vapors Liquids C. The Kerr and Pockels Effects Some isotropic substances become birefringent in an applied electric field. The birefringence in this case. 2
on KEλΔ = K is Kerr constant E is electric field in statvolts( ≈300V) Pockels Effect It is called linear electro-optic effect ~n EΔ . 20 out of 32 crystal symmetry groups with noncentrosymmetry
8.12 Liq Long cig Positive Alignme No posit ↑ lik Parallel Nema Ra Two Rubb
The Liqu The b
The r
(1) In
(2) (3) LC
Twiste
Two The i Mole Incid Appl → R Voltage LC ce
quid Crysgar-shaped e uniaxial bient directiontional order ke liquid
l nematic catic crystalandom in p
opposite gla
bed ITO sur
uid Crystal birefringenc
retardance
ncident E p“ n
C cell btwn
ed nematic glass platesinput direct
ecules are grdent E paral
ied electric Reduced bir
controlled ell between
stals molecules.irefringent wn of LC molr, but large-
cell s:
position but
ass plates a↑ ~ 10 mμ
rface →
Variable Rce : Δ
: Δ
parallel to dinot parallel 45o± cross
cell s are relativtor is horizoradually rotllel to the in
field tilts mrefringence.
switch crossed po
with optic alecules is ca-scale orien
nearly para
are coated w
ParalleMolecu
Retarder ( )en n V= −
2
o
d n= Δπϕλ
irector “ s polarizers
vely rotated ontal and thtated betwenput directo
molecules pe.
olarizers
axis in the ealled directontational ord
↑ like crysta
allel to each
with indium ↑ Con Tra
el microgrooules parallel
on
→ phase m→ Retarde
→ Irradian
by 90o. he output deen plates. or is rotated
erpendicula
elongated dor. der.
al
h other.
m tin oxide ↑ nducting mansparent fr
oves. l to the glas
modulator er. nce modula
director is ve
d by 90o .
ar to the ele
irection.
metallic film.
rom 450nm
ss and to ea
ator.
ertical.
ectrode.
Hecht
. m to 1800nm
ach other
t; 10/12/20
m.
010; 8-12
8.13 A MA. The St B. The Jo They ar The elec
E⎡
= ⎢⎣
m -stat
→ E =
→ E E=
o -stat
E =o
i -stat
E =i
The sum
Elliptica
15⎡⎢⎣
Two ligh if their J
E •o
hE •
Note E •o
E •o
Mathematokes Param
ones Vectorre used only
ctric field ve( )( )
ox
y o
EE tE t E
⎡⎡ ⎤= ⎢⎥⎢⎣ ⎦ ⎣
Comple
e at 45o
( )( )
x
y
E t EE t E
⎡⎡ ⎤= ⎢⎢ ⎥
⎣ ⎦ ⎣
11
oioE e ⎡ ⎤
⇒⎢ ⎥⎣ ⎦
ϕ
e light
( /2)
o
o
i
o i
eEe −
⎡= ⎢
⎣
ϕ
ϕ π
te light 11
2 i⎡ ⎤
= ⎢ ⎥⎣ ⎦
m E E+ =o i
al polarizati2i⎤⎥− ⎦
ht polarizatJones vecto
( )1 12
E ⎡= ⎣Gi
( ) ( )v 1 0E ⎡= ⎣G
E E• = •G
o i
E E• = •Gi i
atical Desmeters
rs y for polariz
ector x
y
iox o
ioy
e Ee
⎤ ⎡⇒⎥ ⎢
⎥ ⎣⎦
ϕ
ϕ
↑ ex form H
o
o
io
io
E eE e
⎤⎥⎦
ϕ
ϕ
1112⎡ ⎤⎢ ⎥⎣ ⎦
)
112 i
⎤ ⎡ ⎤⇒⎥ ⎢ ⎥−⎣ ⎦⎦
1202⎡ ⎤
= ⎢ ⎥⎣ ⎦
ion for exam
ions are ortors are orth
( ) ( ) ( )*1 i i+ −
) ( ) ( )* *0 1 ⎤+ =⎦
1E =Gi
0E =Go
scription
zed lights.
xiox
oy
oeEo
⎡⎤+ ⎢⎥⎢⎦ ⎣
ϕ
↑ Horizontal m
Ve
: Norm
⎤
⎦
mple
thogonal ogonal.
)* 0⎤ =⎦
0=
of Polari
yi
o
e⎤⎥⎥⎦
ϕ
↑ m -state ertical m -s
malized to ha
ization
tate
ave unity irrradiance
Hechtt; 10/12/20010; 8-13
Hecht; 10/12/2010; 8-14 C. The Jones and Mueller Matrices The input and output Jones vectors are related by a transmission matrix A t iE E= A
m -state at 45o is incident into a quarter-wave plate with vertical fast axis
1 0 1 10 1
tx
ty
EE i i⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
= ⇒⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ : o -state
A wave passing through a series of optical elements 1, 2, 3…n 3 2 1..t n iE E= A A A A
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