chapter 6 polarization and crystal opticsfaculty.washington.edu/lylin/ee485w04/ch6.pdf ·...

EE 485, Winter 2004, Lih Y. Lin

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Chapter 6 Polarization and Crystal Optics

- Polarization ↔ Time course of the direction of ),( tr�EEEE - Polarization affects:

� Amount of light reflected at material interfaces. � Absorption in some materials. � Scattering. � Refractive index (thus velocity) of anisotropic materials. � Optically active materials to rotate polarization.

6.1 Polarization of Light Consider ( ){ }c

ztjtz −ω= (expRe),( AEEEE (6.1-1) With complex envelope yxA ˆˆ yx AA += (6.1-2) Trace the endpoint of ),( tzEEEE at each position z as a function of time. The polarization ellipse )exp( ,)exp( yyyxxx jaAjaA ϕ=ϕ= yx ˆˆ),( yxtz EE +=EEEE (6.1-3)

( )[ ]xxx czta ϕ+−ω= cosE (6.1-4a)

( )[ ]yyy czta ϕ+−ω= cosE (6.1-4b)

ϕ=ϕ−+ 22

2

2

2

sincos2yx

yx

y

y

x

x

aaaaEEEE (6.1-5)

→ An ellipse. The shape of the ellipse depends on x

ya

a and ϕ . The size

of the ellipse determines intensity η+

=2

22yx aa

I (η: impedance of the medium).


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Rotation direction (viewed from the direction towards which the wave is approaching): xy ϕ>ϕ : Clockwise rotation xy ϕ<ϕ : Counter-clockwise rotation • Linearly-polarized light yxa E→= 0 only xya E→= 0 only Or π=ϕ or 0

→

π=ϕ−=

=ϕ=

for

0for

EE

EE

x

yy

xx

yy

aa

aa

�45±→= yx aa polarization

• Circularly-polarized light


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0aaa yx ==

2π=ϕ → Right circularly-polarized

2π−=ϕ → Left circularly-polarized

B. Matrix Representation The Jones Vector Complex envelopes for ),( tr�EEEE : )exp( xxx jaA ϕ= , )exp( yyy jaA ϕ=

≡

y

x

AA

J

• Orthogonal polarizations ( ) 0, *

21*2121 =+= yyxx AAAAJJ (6.1-7)

• Expansion of arbitrary polarization

If J1 and J2 are normalized and orthogonal to each other, then an arbitrary polarization 2211 JJJ α+α= ( )11 , JJ=α , ( )22 , JJ=α


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Example:

−+

=

jj1

211

21

01

Matrix representation of polarization devices

Input:

y

x

AA

1

1 , Output:

y

x

AA

2

2

≡

=

y

x

y

x

y

x

AA

AA

TTTT

AA

1

1

1

1

2221

1211

2

2 T (6.1-9)

T: Jones matrix 12 TJJ =⇒ (6.1-10) • Linear polarizers

=

0001

T (for x polarizer) (6.1-11)

0 , 212 ==⇒ yxx AAA

• Wave retarders

Γ−=

)exp(001

jT (6.1-12)

Γ−=

y

x

y

x

AjA

AA

1

1

2

2

)exp(

→ y component is delayed by a phase Γ. x: fast axis, y: slow axis. Examples: (1) 2/π=Γ (quarter-wave retarder)

−→

j1

11


5

→

111

j

(2) π=Γ (half-wave retarder)

−→

11

11

−→

jj11

• Polarization rotators

θθθ−θ

=cossinsincos

T (6.1-13)

θ+θθ+θ

→

θθ

)sin()cos(

sincos

1

1

1

1

• Cascaded polarization devices

12... TTTT ⋅⋅⋅= M

Coordinate transformation JRJ )(' θ= (6.1-14)

θθ−θθ

=θcossinsincos

)(R (6.1-15)

)()(' θ−θ= TRRT (6.1-16)


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)(')( θθ−= RTRT (6.1-17) Normal modes (of a polarization system) States of polarization that are not changed when the wave is transmitted through the optical system. JTJ µ= (6.1-19) µ: eigenvalue J: eigenvector If T is Hermitian, i.e., T12 = T21*, the normal modes are orthogonal to each other, and can be used as an expansion basis. The response to the system can be evaluated more easily if the input wave is decomposed into the two normal modes: 2211 JJJ α+α= 2221112211 )( JJJJTTJ µα+µα=α+α= Examples (Exercise 6.1-4): (a) The normal modes of the linear polarizer are linearly polarized waves. (b) The normal modes of the wave retarder are linearly polarized waves. (c) The normal modes of the polarization rotator are right and left circularly

polarized waves. 6.2 Reflection and Refraction

=

=

=

y

x

y

x

y

x

EE

EE

EE

3

33

2

22

1

11 , , JJJ

ttJJ ,12 = : 2×2 Jones matrix for transmission r rJJ ,13 = : 2×2 Jones matrix for reflection

=

=

y

x

y

x

rr

tt

00

,0

0rt


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yyyxxx EtEEtE 1212 , == (6.2-2)

yyyxxx ErEErE 1313 , == (6.2-3) For transverse electric (TE) polarization, yHxE �HE == ,ˆ

2211

2211

coscoscoscos

θ+θθ−θ=

nnnnrx (6.2-4)

xx rt +=1 (6.2-5) For transverse magnetic (TM) polarization, xHyE ˆ , HE == �

2112

2112

coscoscoscos

θ+θθ−θ=

nnnnry (6.2-6)

)1(2

1yy r

nnt += (6.2-7)

rx, tx, ry, ty can be complex numbers → )exp(|| ,)exp(|| yyyxxx jrrjrr ϕ=ϕ= TE polarization • External reflection ( 21 nn < )

π=ϕx

12

12||nnnnrx +

−= at θ1 = 0º (θ2 = 0º)


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• Internal reflection ( 21 nn > ) For θ1 < θc, 0=ϕx

At θ1 = 0º (θ2 = 0º), 21

21

nnnnrx +

−=

At θ1 = θc, 1=xr For θ1 > θc, total internal reflection, 1|| =xr ,

1

21

2

cossinsin

2tan

θθ−θ

=ϕ cx (6.2-9)

TM polarization • External reflection ( 21 nn < )

ry is real

At θ1 = 0º (θ2 = 0º), 12

12

nnnnry +

−= is positive

At θ1 = θB1

21tannn−= , 0=yr (θB: Brewster angle)

For θ1 > θB, ry becomes negative, π=ϕ y At θ1 = 90º, ry = -1, π=ϕ y


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• Internal reflection ( 21 nn > )

At θ1 = 0º (θ2 = 0º), ry is negative, 12

12||nnnnry +

−= , π=ϕ y

For θ1 < θB, |ry| decreases with θ, π=ϕ y At θ1 = θB, 0=yr For θ1 > θB, ry becomes positive and increases with θ, 0=ϕ y At θ1 = θc, 1=yr For θ1 > θc, total internal reflection, 1|| =yr ,

c

cx

θθθ−θ

=ϕ2

1

21

2

sincossinsin

2tan (6.2-11)


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Power reflectance and transmittance || 2rR = (6.2-12) RT −=1 (6.2-13) At normal incidence, for both TE and TM,

2

21

21

+−=

nnnnR

Examples: (a) Glass (n = 1.5) and Air (n = 1) interface R = 0.04 for normal incidence (b) GaAs (n = 3.6) and Air (n = 1) interface R = 0.32 for normal incidence


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6.3 Optics of Anisotropic Media A. Refractive Indices Permittivity tensor ∑ε=

jjiji ED , i, j = 1, 2, 3 (6.3-1)

{ } ε≡εij : Electric permittivity tensor (3 × 3) εED = εεεε is symmetrical, jiij ε=ε → only 6 independent numbers. Principal axes and principal refractive indices Choose a coordinate system such that

εε

ε=

3

2

1

000000

ε

333222111 , , EDEDED ε=ε=ε= (6.3-2) → This coordinate system defines the “principal axes” of the crystal. Principal refractive indices:

0

33

0

22

0

11 , ,

εε=

εε=

εε= nnn (6.3-3)

Isotropic, uniaxial, and biaxial crystals Isotropic: 321 nnn == Uniaxial: onnn == 21 (ordinary index, ordinary axes) enn =3 (extraordinary index, extraordinary axis) Positive uniaxial: oe nn > Negative uniaxial: oe nn < Biaxial: 321 nnn ≠≠ The index ellipsoid

123

23

22

22

21

21 =++

nx

nx

nx

(6.3-7) Direction of D determines the refractive index from the index ellipsoid.


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B. Propagation Along a Principal Axis Consider a plane wave propagating along z-direction.

Normal modes For xDxE ˆ ,ˆ 111 EE ε== ,

1

001 n

ccknk =→=

For yDyE ˆ ,ˆ 212 EE ε== ,

2

002 n

ccknk =→=

Polarization along an arbitrary direction Decompose the electric field: yxE ˆˆ 21 EE += The x- and y-components travel with different speeds. The phase retardation dknn 012 )( −=ϕ → Linearly-polarized wave becomes an elliptically-polarized wave. The crystal acts like a wave retarder.

C. Propagation in an Arbitrary Direction


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Determine the polarizations and refractive indices an and bn of the normal modes of a wave traveling in u direction. (1) Draw a plane passing thru the origin of the index ellipsoid, normal to u . The

intersection of the plane with the ellipsoid is an ellipse, called the index ellipse.

(2) The half-width of the major and minor axes of the index ellipse are the refractive indices an and bn of the two normal modes.

(3) The directions of the major and minor axes of the index ellipse are the directions of the vectors aD and bD for the normal modes. These directions are orthogonal.

(4) The vectors aE and bE may be determined from aD and bD by εED = . DHk ω−=× (6.3-8) HEk 0ωµ=× (6.3-9)

Example: Uniaxial crystal For a wave traveling at an angle θ with the optical axis (z-axis), the index ellipse has half-length on and )(θn .

2

2

2

2

2

sincos)(

1

eo nnnθ+θ=

θ (6.3-15)

The normal modes have refractive indices oa nn = for ordinary wave, and )(θ= nnb for extraordinary wave. onn =θ)( when θ = 0º, and enn =θ)( when θ =

90º. For ordinary wave, D is along the principal axis, D and E are parallel. D is perpendicular to k and z . For extraordinary wave, D and E are not parallel (unless D is along z ). D is in the k-z plane.


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D. Double Refraction In the crystal, the optical wave can be decomposed into the two normal-mode components, propagating in two different directions: For ordinary wave (TE, E ⊥ optical axis), oon θ=θ sinsin 1 For extraordinary wave (TM), een θθ=θ sin)(sin 1

6.4 Optical Activity and Faraday Effect A. Optical Activity

Certain materials act naturally as polarization rotators. The normal modes are circularly-polarized waves. The phase velocity for the right-circularly-polarized wave is +nc0 , and the phase velocity of the left-circularly-polarized wave is

−nc0 .


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The material rotates a linearly-polarized wave by 0

)(2λ−π=φ +− dnn after a

propagation distance of d. 0

)(λ−π≡ρ +− nn is called “rotatory power”.

Medium equation: EεED ×∇ξε−= 0 (6.4-2) For a plane wave )(exp)( rjkErE ⋅−= , kG ξ≡ , EGεED ×ε+= 0j (6.4-3) → Depends on k direction. Therefore reversal of the propagation

reverses the sense of rotation of the polarization plane.

B. Faraday Effect The material acts as a polarization rotator when placed in a static magnetic field. The rotatory power is proportional to the component of B in k-direction. Medium equation: EBεED ×γε+= 0j (6.4-8) → Independent of k, but dependent on B. Therefore reversal of the

propagation does not reverse the sense of rotation of the polarization plane.

Optical isolators Acting as a “one-way valve” for the optical wave.


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6.5 Liquid Crystals The molecules are elongated, with ne for the long axis and no for the short axis. The twist angle of the molecules: zα=θ (6.5-1) Phase retardation coefficient: 0)( knn oe −=β (6.5-2) In general, α>>β . If the incident wave at z = 0 is linearly polarized, the wave maintains its linear polarization, but the plane of polarization rotates in alignment with the molecular twist. → Output wave polarized at dα=θ .


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• Applications: Optical switch and display (LCD).

chapter 6 polarization and crystal opticsfaculty.washington.edu/lylin/ee485w04/ch6.pdf ·...

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