Introdução à Óptica (4300327)

Prof. Adriano Mesquita Alencar Dep. Física Geral

Instituto de Física da USP

Feixe Gaussiano

D01

1

2

3

Feixe de GaussApesar da natureza da luz deixar a possibilidade de transporte idealizado, sem espalhamento… Na realidade a luz pode ser transportada apenas confinada na forma de feixes, que são quase localizados e quase não divergentes.Os dois extremos do confinamento: (angular) -> a onda plana: As normais de frente de onda (raios) de uma onda plana coincidem com a direção de deslocamento da onda, então não se espalha angularmente, mas a energia se estende espacialmente ao longo de todo o espaço. (espacial) -> a onda esférica: A onda esférica, em contraste, se origina a partir de um único ponto espacial, mas tem as normais de frente de onda (raios) que divergem em todas as direcções angulares. (Lembrando) Ondas cujas normais frente de onda fazer pequenos ângulos com o eixo z são chamados de ondas paraxiais. Eles devem satisfazer a equação de Helmholtz paraxial, que derivamos. O feixe de Gauss é uma solução importante dessa equação que apresenta as características de um feixe óptico, como atestam os seguintes recursos:

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1. A potência do feixe está principalmente concentrada dentro de um pequeno cilindro que envolve o eixo do feixe. 2. A distribuição da intensidade de qualquer plano transversal é uma função de Gauss simétrica circular centrada sobre o eixo do feixe. 3. A largura desta função é mínima na cintura do feixe e gradualmente torna-se maior à medida que a distância entre os aumentos de cintura em ambos os sentidos. 4. As frentes de ondas são aproximadamente planar perto da cintura do feixe, gradualmente se curva com o aumento da distância para a cintura, e em última instância, torna-se aproximadamente esférica longe da cintura. 5. A divergência angular das normais de frente de onda assumem o valor mínimo permitido pela equação de onda para uma determinada largura de feixe. Sob condições ideais , a luz a partir de muitos tipos de laser toma a forma de um feixe de Gauss.

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Raios Paraxiais

(r) = A(r)e�ikz (Eq. do livro 2.2-20)O envelope A(r) é aproximadamente constante dentro da vizinhança de um comprimento de onda. Para a amplitude complexa da função de onda satisfazer a Eq. de Helmholtz, A(r) deve satisfazer:

6

Raios Paraxiais

r2TA� i2k

@A

@z= 0 r2

T =@

2

@x

2+

@

2

@y

2Onde

(Ver B4 Otica Ondulatoria II)

A solução mais simples dessa equação é a onda paraboloide

A paraboloide é uma aproximação paraxial da onda esférica quando x e y são muito menores do que z.

A(r) =A1

z

exp

✓�ik

⇢

2

2z

◆, ⇢

2= x

2+ y

2

7

Uma vez que a Eq. da paroboloide é uma solução da Eq. de Helmholtz,

uma versão transladada por uma constante em z também será:

onde ξ é uma constante. Essa Eq. é uma paraboloide centrada em z = ξ ao invés de z = 0. Isso é verdade mesmo que ξ seja um numero complexo, ou puramente complexo.

r2TA� i2k

@A

@z= 0

A(r) =A1

q(z)

exp

✓�ik

⇢

2

2q(z)

◆, ⇢

2= x

2+ y

2, q(z) = z � ⇠

A(r) =A1

z

exp

✓�ik

⇢

2

2z

◆, ⇢

2= x

2+ y

2

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Raios Paraxiais

A quantidade q é conhecida como parâmetro q e z0 é o Rayleigh-range.

Para separar a parte Real da Imaginaria escrevemos:

1

q(z)=

1

z + iz0=

1

R(z)� i

�

⇡W 2(z)

substituindo

A(r) =A1

q(z)

exp

✓�ik

⇢

2

2q(z)

◆, ⇢

2= x

2+ y

2, q(z) = z + iz0

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Raios Paraxiais

(r) = A0W0

W (z)exp

� ⇢2

W 2(z)

�exp

�ikz � ik

⇢2

2R(z)+ i⇣(z)

�

W (z) = W0

s

1 +

✓z

z0

◆2

R(z) = z

1 +

⇣z0z

⌘2�

⇣ = tan�1 z

z0

W0 =

r�z0⇡

limz!1

W (z) = zW0

limz!1

R(z) = z

limz!1

⇣ =⇡

2

A0 =A1

iz0

Parâmetros independentes A0 e z0

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Intensidade do feixe

(r) = A0W0

W (z)exp

� ⇢2

W 2(z)

�exp

�ikz � ik

⇢2

2R(z)+ i⇣(z)

�

I(r) = | (r)|2 é uma função de z e de ρ

I(⇢, z) = I0

W0

W (z)

�2exp

� 2⇢

2

W

2(z)

�, ⇢ =

px

2+ y

2

I(⇢, z)|⇢=0 = I0

W0

W (z)

�2=

I0

1 +⇣

zz0

⌘2

I(0, 0) = I0

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I(⇢, z)|⇢=0 = I0

W0

W (z)

�2=

I0

1 +⇣

zz0

⌘2

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Potência

P =

Z 1

0I(⇢, z)2⇡⇢d⇢

P =1

2I0(⇡W

20 ) Independente de z

Feixes ópticos são usualmente caracterizados pela potência P então é util escrever I0 em temos de P e então:

I(⇢, z) =2P

⇡W 2(z)

exp

� 2⇢2

W 2(z)

�

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Potência

1

P

Z ⇢0

0I(⇢, z)2⇡⇢d⇢ = 1� exp

� 2⇢20W 2

(z)

�

A fração da potência dentro de um circulo de raio ⇢0

Para ⇢0 = W (z) 89% da potênciaPara ⇢0 = 1.5W (z) 99% da potência

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PotênciaPara ⇢0 = W (z) 89% da potênciaPara ⇢0 = 1.5W (z) 99% da potência

Em qualquer plano transverso, a intensidade tem o seu pico no eixo do feixe, diminuindo por um fator de 1/e2 a uma distância radial ⇢0 = W (z)

Chamamos então W(z) de largura do feixe

W (z) = W0

s

1 +

✓z

z0

◆2

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Largura do Feixe e Divergência

W (z) = W0

s

1 +

✓z

z0

◆2

Para z ≫ z0 W (z) ⇡ W0z

z0= ✓0z

✓0 =W0

z0=

�

⇡W0(Paraxial)

Divergência

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Lasers became the first choice of energy source for a steadily increasing number of applications in science, medicine and industry because they deliver light energy in an exceedingly useful form and set of features. A comprehensive analysis of lasers and laser systems goes far beyond the measurement of just output power and pulse energy. The most commonly measured laser beam parameters besides power or energy are beam diameter (or radius), spatial intensity distribution (or profile), divergence and the beam quality factor (or beam parameter product). In many applications, these parameters define success or failure and, therefore, their control and optimization seems to be crucial.

Na pagina de um fabricante de Laser:

2z0 =2⇡W 2

0

�

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Fase

(r) = A0W0

W (z)exp

� ⇢2

W 2(z)

�exp

�ikz � ik

⇢2

2R(z)+ i⇣(z)

�(ver Pag. 8 e 9)

Onde a fase é: ' = kz + k⇢2

2R(z)� ⇣(z)

No eixo do feixe: ' = kz � ⇣(z)

Fase de uma onda plana Retardo da fase em relação a uma

onda plana, ou onda esférica, Efeito Gouy

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FaseOnde a fase é: ' = kz + k

⇢2

2R(z)� ⇣(z)

No eixo do feixe: ' = kz � ⇣(z)

Fase de uma onda plana

Retardo da fase em relação a uma onda plana, ou onda esférica, Efeito Gouy

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April 15, 2001 / Vol. 26, No. 8 / OPTICS LETTERS 485

Physical origin of the Gouy phase shift

Simin Feng* and Herbert G. Winful

Department of Electrical Engineering and Computer Science, University of Michigan,1301 Beal Avenue, Ann Arbor, Michigan 48109-2122

Received October 10, 2000

We show explicitly that the well-known Gouy phase shift of any focused beam originates from transversespatial confinement, which, through the uncertainty principle, introduces a spread in the transverse momentaand hence a shift in the expectation value of the axial propagation constant. A general expression is givenfor the Gouy phase shift in terms of expectation values of the squares of the transverse momenta. Our resultalso explains the phase shift in front of the Kirchhoff diffraction integral. © 2001 Optical Society of America

OCIS codes: 050.5080, 050.1940, 050.1960.

The Gouy phase shift is the well-known np!2 axialphase shift that a converging light wave experiencesas it passes through its focus in propagating from 2`to 1`. Here the dimension n equals 1 for a line fo-cus (cylindrical wave) and equals 2 for a point focus(spherical wave). This phase anomaly was first ob-served by Gouy1 – 3 and was shown to exist for anywaves, including acoustic waves, that pass through afocus. The Gouy phase shift plays an important rolein optics. It explains the phase advance for the sec-ondary Huygens wavelets emanating from a primarywave front. It also determines the resonant frequen-cies of transverse modes in laser cavities.3 In non-linear optics the Gouy shift affects the efficiency of thegeneration of odd-order harmonics with focused beams.It also plays a role in the lateral trapping force at thefocus of optical tweezers and leads to phase velocitiesthat exceed the speed of light in vacuum. Recentlywe pointed out the effect of the Gouy phase shift onthe temporal profile of a single-cycle electromagneticpulse4,5 and made a direct observation of the polar-ity reversal that results from a Gouy phase shift ofp.6 Another direct observation of a p!2 Gouy phaseshift of terahertz beams in a cylindrical focusing ge-ometry was reported recently.7

Although Gouy made his discovery more than 100years ago, efforts are still being made to providea simple and satisfying physical interpretation ofthis phase anomaly. An earlier paper8 provided anintuitive explanation of this phase anomaly based onthe geometrical properties of Gaussian beams. How-ever, that argument cannot explain the p!2 phaseshift for cylindrical focusing. In a recent paper aninterpretation of the Gouy phase shift as a geometricalquantum effect was also proposed.9 Whereas thisinterpretation is satisfying in its simplicity, the con-nection to quantum mechanics appears unnecessarybecause Gouy showed that the phase jump exists forall waves, including sound waves. It has also beensuggested that the Gouy phase shift is a manifestationof a general Berry phase, which is an additional geo-metric (topological) phase acquired by a system after acyclic adiabatic evolution in parameter space.10 Theparameter that is cycled in the case of the Gouyphase is the complex wave-front radius of curvature

q associated with a Gaussian beam.11,12 This sophis-ticated modern interpretation requires knowledge ofsuch concepts as anholonomy and is far from beingintuitive.

In this Letter we provide a simple intuitive explana-tion of the physical origin of the Gouy phase shift. Weshow explicitly that the Gouy phase shift of any focusedbeam originates from the transverse spatial confine-ment, which, through the uncertainty principle, intro-duces a spread in the transverse momenta and hencea shift in the expectation value of the axial propaga-tion constant. A general expression is given for theGouy phase shift in terms of the expectation valuesof the transverse momenta. It yields the correct val-ues for both line and point focusing and also explainsthe phase shift in front of the Kirchhoff diffractionintegral.

Consider a monochromatic wave of frequency vand wave number k ! v!c propagating along the zdirection. For an infinite plane wave, the momentum,which is proportional to k, is z directed and has notransverse components. The spread in transversemomentum is zero and hence, by the uncertaintyprinciple, the spread in transverse position is infinite.A finite beam, however, will have a spread in trans-verse momentum because it is made up of an angularspectrum of plane waves obtainable by means of aFourier transform. The wave number is related tothese transverse components through

k2 ! kx2 1 ky

2 1 kz2, (1)

where kx, ky , and kz are the wave-vector componentsalong the coordinate axes. Inasmuch as k "!v!c# isconstant, the presence of the transverse componentsreduces the magnitude of the axial component from itsvalue of kz ! k for an infinite plane wave propagatingalong z. Because of the f inite spread in wave-vectorcomponents, it is appropriate to deal with averages orexpectation values defined by

$j% &R1`

2` jjf "j#j2djR1`

2` jf "j#j2dj, (2)

where f "j# is the distribution of the variable j.Then from Eq. (1) we can define an effective axial

0146-9592/01/080485-03$15.00/0 © 2001 Optical Society of America

April 15, 2001 / Vol. 26, No. 8 / OPTICS LETTERS 485

Physical origin of the Gouy phase shift

Simin Feng* and Herbert G. Winful

Department of Electrical Engineering and Computer Science, University of Michigan,1301 Beal Avenue, Ann Arbor, Michigan 48109-2122

Received October 10, 2000

We show explicitly that the well-known Gouy phase shift of any focused beam originates from transversespatial confinement, which, through the uncertainty principle, introduces a spread in the transverse momentaand hence a shift in the expectation value of the axial propagation constant. A general expression is givenfor the Gouy phase shift in terms of expectation values of the squares of the transverse momenta. Our resultalso explains the phase shift in front of the Kirchhoff diffraction integral. © 2001 Optical Society of America

OCIS codes: 050.5080, 050.1940, 050.1960.

The Gouy phase shift is the well-known np!2 axialphase shift that a converging light wave experiencesas it passes through its focus in propagating from 2`to 1`. Here the dimension n equals 1 for a line fo-cus (cylindrical wave) and equals 2 for a point focus(spherical wave). This phase anomaly was first ob-served by Gouy1 – 3 and was shown to exist for anywaves, including acoustic waves, that pass through afocus. The Gouy phase shift plays an important rolein optics. It explains the phase advance for the sec-ondary Huygens wavelets emanating from a primarywave front. It also determines the resonant frequen-cies of transverse modes in laser cavities.3 In non-linear optics the Gouy shift affects the efficiency of thegeneration of odd-order harmonics with focused beams.It also plays a role in the lateral trapping force at thefocus of optical tweezers and leads to phase velocitiesthat exceed the speed of light in vacuum. Recentlywe pointed out the effect of the Gouy phase shift onthe temporal profile of a single-cycle electromagneticpulse4,5 and made a direct observation of the polar-ity reversal that results from a Gouy phase shift ofp.6 Another direct observation of a p!2 Gouy phaseshift of terahertz beams in a cylindrical focusing ge-ometry was reported recently.7

Although Gouy made his discovery more than 100years ago, efforts are still being made to providea simple and satisfying physical interpretation ofthis phase anomaly. An earlier paper8 provided anintuitive explanation of this phase anomaly based onthe geometrical properties of Gaussian beams. How-ever, that argument cannot explain the p!2 phaseshift for cylindrical focusing. In a recent paper aninterpretation of the Gouy phase shift as a geometricalquantum effect was also proposed.9 Whereas thisinterpretation is satisfying in its simplicity, the con-nection to quantum mechanics appears unnecessarybecause Gouy showed that the phase jump exists forall waves, including sound waves. It has also beensuggested that the Gouy phase shift is a manifestationof a general Berry phase, which is an additional geo-metric (topological) phase acquired by a system after acyclic adiabatic evolution in parameter space.10 Theparameter that is cycled in the case of the Gouyphase is the complex wave-front radius of curvature

q associated with a Gaussian beam.11,12 This sophis-ticated modern interpretation requires knowledge ofsuch concepts as anholonomy and is far from beingintuitive.

In this Letter we provide a simple intuitive explana-tion of the physical origin of the Gouy phase shift. Weshow explicitly that the Gouy phase shift of any focusedbeam originates from the transverse spatial confine-ment, which, through the uncertainty principle, intro-duces a spread in the transverse momenta and hencea shift in the expectation value of the axial propaga-tion constant. A general expression is given for theGouy phase shift in terms of the expectation valuesof the transverse momenta. It yields the correct val-ues for both line and point focusing and also explainsthe phase shift in front of the Kirchhoff diffractionintegral.

Consider a monochromatic wave of frequency vand wave number k ! v!c propagating along the zdirection. For an infinite plane wave, the momentum,which is proportional to k, is z directed and has notransverse components. The spread in transversemomentum is zero and hence, by the uncertaintyprinciple, the spread in transverse position is infinite.A finite beam, however, will have a spread in trans-verse momentum because it is made up of an angularspectrum of plane waves obtainable by means of aFourier transform. The wave number is related tothese transverse components through

k2 ! kx2 1 ky

2 1 kz2, (1)

where kx, ky , and kz are the wave-vector componentsalong the coordinate axes. Inasmuch as k "!v!c# isconstant, the presence of the transverse componentsreduces the magnitude of the axial component from itsvalue of kz ! k for an infinite plane wave propagatingalong z. Because of the f inite spread in wave-vectorcomponents, it is appropriate to deal with averages orexpectation values defined by

$j% &R1`

2` jjf "j#j2djR1`

2` jf "j#j2dj, (2)

where f "j# is the distribution of the variable j.Then from Eq. (1) we can define an effective axial

0146-9592/01/080485-03$15.00/0 © 2001 Optical Society of America

We show explicitly that the Gouy phase shift of any focused beam originates from the transverse spatial confinement, which, through the uncertainty principle, introduces a spread in the transverse momenta and hence a shift in the expectation value of the axial propagation constant. A general expression is given for the Gouy phase shift in terms of the expectation values of the transverse momenta. It yields the correct values for both line and point focusing and also explains the phase shift in front of the Kirchhoff diffraction integral.

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Frente de Ondaz +

⇢2

2R⇡ ⇣�

2⇡+ q�

z

c

=x

2

a

2+

y

2

b

2

Paraboloide

Essa é a equação da paraboloide com R sendo o Raio de Curvatura

Raio de curvatura aproximado da frente de onda

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Frente de OndaOnde a fase é: ' = kz + k

⇢2

2R(z)� ⇣(z)

Representa a curvatura da frente de onda para pontos fora do eixo

Uma superfície de fase constante satisfaz: k

z +

⇢2

2R(z)

�� ⇣(z) = 2⇡q

variam lentamente, e são praticamente constantes nos pontos dentro da largura do feixe

z +⇢2

2R⇡ ⇣�

2⇡+ q�

z

c

=x

2

a

2+

y

2

b

2

Paraboloide

22

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