Chapter 3a

Optically controlled devices

Polarization of lightOptical ModulationOptical Switching

Polarization

• A propagating EM wave has its electric and magnetic field at right angles to the direction of propagation

• If we place a z-axis along the direction of propagation,– Electric field can be any direction in the plane perpendicular to the z-axis

• Polarization of an EM wave describes the behavior of the electric field vector as it propagates through a medium– Linear polarized: the oscillation of the electric field at all time are

contained within a well defined line.– Plane polarized: the field vibrations and the direction of propagation (z)

define a plane of polarization (plane of vibration).

Polarizer

• Unpolarized beam: A beam of light has waves with the E-field in each in a random direction but perpendicular to z.

• A light beam can be linearly polarized by passing the beam through a polarizer– A Polaroid sheet: a device that only passes E-field

oscillations lying on well defined plane at right angles to the direction of propagation.

E-field Components

• Suppose we place x and y axes to describe the E-field in terms of its components Ex and Ey.– Both Ex and Ey can individually be described by a

wave equation which must have the same angular frequency , wavenumber k and phase difference between Ex & Ey . Ex = Exo cos (t – kz) (1)

Ey = Eyo cos (t – kz + ) (2)

x

y

z

Ey

Ex

yEy^

xEx^

(a) (b) (c)

E

Plane of polarization

x̂

y^

E

(a) A linearly polarized wave has its electric field oscillations defined along a lineperpendicular to the direction of propagation, z. The field vector E and z define a plane ofpolarization. (b) The E-field oscillations are contained in the plane of polarization. (c) Alinearly polarized light at any instant can be represented by the superposition of two fields Exand Ey with the right magnitude and phase.

E

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Fig 1: Polarization of light

Linearly Polarized

• The linearly polarized wave in Fig 1(a) has the E oscillations at –45 to x-axis as shown in Fig 1(b) – We can generate this field by choosing Exo=Eyo and = 180 () into eqns (1) & (2)

– Using = , the field in the wave is

direction.- thealong propagates and

axis- the to54at vector that thestate (4) and (3) Eqns

(4) ˆˆ where

(3) cosor

cosˆcosˆˆˆ

z

x

EE

kzt

kztEkztEEE

yoxo

yoxoyx

yxE

EE

yxyxE

o

o

E

y

x

Exo = 0Eyo = 1 = 0

y

x

Exo = 1Eyo = 1 = 0

y

x

Exo = 1Eyo = 1 = /2

E

y

x

Exo = 1Eyo = 1 = /2

(a) (b) (c) (d)

Examples of linearly, (a) and (b), and circularly polarized light (c) and (d); (c) isright circularly and (d) is left circularly polarized light (as seen when the wavedirectly approaches a viewer)


Fig 2: Linearly and circularly Polarized

Circularly Polarized

• There are many choices for the behavior of the E-field besides the simple linear polarization in Fig 1

• If the magnitude of the field vector E remains constant but its tip at a given location on z traces out a circle by rotating in clockwise sense with time– The wave is right circularly polarized– It has Exo = Eyo = A (an amplitude) and = /2– Ex = A cos (t – kz) and Ey = – A sin (t – kz)– It represents a circle as Ex

2 + Ey2 = A2

• If the rotation of the tip of E is counterclockwise,– The wave is left circularly polarized

z

Ey

Ex

EE = kz

z

z

A right circularly polarized light. The field vector E is always at rightangles to z , rotates clockwise around z with time, and traces out a fullcircle over one wavelength of distance propagated.


Fig 3: Right circularly polarized

x

y

Elliptically Polarized

• An elliptically polarized light has a tip of the E-vector trace out an ellipse as the wave propagates through a given location in space– Light can be left or right elliptically polarized depending on

clockwise or counterclockwise rotation of the E-vector– An elliptically polarized light can result for any not zero or

equal to any multiple of and when Exo and Eyo are not equal in magnitude

– Elliptic light can also be obtained when Exo= Eyo and the phase differences is /4 or 3/4 etc

E

y

x

Exo = 1Eyo = 2 = 0

Exo = 1Eyo = 2 = /4

Exo = 1Eyo = 2 = /2

y

x

(a) (b)E

y

x

(c)

(a) Linearly polarized light with Eyo = 2Exo and = 0. (b) When = /4 (45 ), the light isright elliptically polarized with a tilted major axis. (c) When = /2 (90 ), the light isright elliptically polarized. If Exo and Eyo were equal, this would be right circularlypolarized light.


Fig 4: Elliptically Polarized

Example: Elliptical and circular polarization

• Show that if Ex = A cos (t – kz) and Ey = B cos (t – kz + ) that the amplitudes A and B are different and the phase difference is /2 (90), the wave is elliptically polarized.

Solution

polarized. circularly

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find, we,1cossin Using,

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22

22

BEEtAEEtz

BA

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EE

B

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A

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kztkzt

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yx

yx

yx

yx

y

x

Malus’s Law

• A linearly polarized light from Polarizer 1 is now incident on Polarizer 2 (also called Analyzer)– The transmission axis of the Analyzer is at an angle q to the

E-field of the incident beam – Only component E cosq of the field will be allowed to pass

through the Analyzer– The irradiance (intensity) of light passing through the

analyzer (E cosq)2 providing all the E-field will pass when q = 0.

• The irradiance I at any other angle q is then given by Malus’s Law: I (q)= I (0) cos2q

Polarizer 1

TA 1

Polarizer 2 = Analyzer

TA 2

Light detectorE

Ecos

Unpolarized light

Linearlypolarized light

Randomly polarized light is incident on a Polarizer 1 with a transmission axis TA1. Lightemerging from Polarizer 1 is linearly polarized with E along TA 1, and becomes incidenton Polarizer 2 (called "analyzer") with a transmission axis TA 2 at an angle to TA 1. Adetector measures the intensity of the incident light. TA 1 and TA 2 are normal to the lightdirection.


Fig 5: Malus’s Law

The intensity of light transmitted through two polarizers depends on the relative orientation of their

transmission axes.

Light propagation in an anisotropic medium: birefringence

• An important characteristic of crystals is that many of their properties depend on the crystal direction– That is crystals are generally anisotropic

• The dielectric constant r depends on electronic polarization which involves the displacement of electrons with respect to positive atomic nuclei– Electronic polarization depends on the crystal direction

• Refractive index n of a crystal depends on the direction of electric field in the propagating light beam– The velocity of light in a crystal depends on the direction of propagation

and on the state of its polarization

Optically Isotropic and Anisotropic

• Most non-crystalline materials e.g. glasses and liquids and all cubic crystals are optically isotropic– The refractive index is the same in all directions

• For all classes of crystals excepts cubic structure, the refractive index depends on the propagating direction and the state of polarization– Except along certain special directions, any unpolarized light

ray entering such a crystal breaks into two different rays with different polarizations and phase velocities

Optically anisotropic crystals

• Most anisotropic crystals: the highest degree of anisotropy– We can describe light propagation in term of three principal refractive

indices (n1, n2, n3) along three mutually orthogonal directions in the crystal (x, y, z) called principal axes.

• Biaxial Crystals – Have three distinct principal indices and also have two optic axes

• Uniaxial crystals– Have two of principal indices the same (n1= n2) and only have one optic

axis– Positive Uniaxial Crystals such as quartz have n3> n1

– Negative Uniaxial Crystals such as calcite have n3< n1

Biaxial Crystal

Uniaxial Crystal

A line viewed through a cubic sodium chloride (halite) crystal(optically isotropic) and a calcite crystal (optically anisotropic).

• When we view an image through a calcite crystal (an optically anisotropic crystal), we see two images each constituted by light of different polarization passing through the crystal.

• Whereas there is only one image through an optically isotropic crystal

• Optically anisotropic crystals are called birefringent because an incident light beam may be doubly refracted.

Uniaxial crystals and Fresnel’s Optical Indicatrix

• Any EM wave entering an uniaxial crystal splits into two orthogonal linearly polarized waves – which travel with different phase velocities and experience

different refractive indices• These two orthogonally polarized waves are– Ordinary Wave (o-wave)

• Has the same phase velocity in all directions • behaves like an ordinary wave in which the field is perpendicular to the

phase propagation direction.– Extraordinary Wave (e-wave)

• Has a phase velocity that depends on its direction of propagation and its state of polarization

• The E-field is not necessarily perpendicular to the phase propagation direction

Optic axis

• These two waves (o-wave & e-wave) propagate with the same velocity only along a special directions called optic axis– The o-wave is always perpendicularly polarized to

the optic axis and obeys the usual Snell’s Law

Optical properties of crystal

• It can be represented in terms of three refractive indices along three orthogonal axes, the principal axes of the crystal shown as x, y & z in Fig 6(a).– These are special axes along which the polarization vector and

the E-field are parallel

• The refractive indices along these x, y & z axes are the principal indices n1, n2 & n3 respectively for E-field oscillations along these directions– For a wave with a polarization parallel to the x-axis, the

refractive index is n1

Optical indicatrix

• The refractive index associated with a particular EM wave in a crystal can be determined by using Fresnel’s refractive index ellipsoid called optical indicatrix.– A refractive index surface placed in the center of the

principal axes (see Fig 6(a))– If n1= n2 = n3 , we would have a spherical surface and

all E-field polarization directions would experience the same refractive index, no .

n2

Opticaxis

n1

x

y

zn3 O

B

A

P

B

A

z

k

O

(b) An EM wave propagating along OP at anangle to optic axis.

(a) Fresnel's ellipsoid


Fig 6: Optical properties of crystal

The refractive index associated with a particular EM wave in a crystal can be determined by using Fresnel’s refractive index ellipsoid (Optical Indicatrix)

Birefringence of Calcite

• Consider when unpolarized light enters a calcite crystal at normal incidence and thus also normal to a principal section to this surface, but at an angle to the optic axis– The ray breaks into ordinary (o) and extraordinary (e)

waves with mutually orthogonal polarizations– The wave propagates in the plane of the principal

section as this plane contains the incident light

e-wave

o-wave

Optic axis(in plane of paper)

Optic axis

Principal section

e-ray

o-ray

Principal section

A calcite rhomb

E

E/ /Incident ray

Incident wave

An EM wave that is off the optic axis of a calcite crystal splits into two waves calledordinary and extraordinary waves. These waves have orthogonal polarizations andtravel with different velocities. The o-wave has a polarization that is alwaysperpendicular to the optical axis.


Fig 7: Birefringence of Calcite

Two polaroid analyzers are placed with their transmission axes, alongthe long edges, at right angles to each other. The ordinary ray,undeflected, goes through the left polarizer whereas the extraordinarywave, deflected, goes through the right polarizer. The two wavestherefore have orthogonal polarizations.

Ordinary and Extraordinary waves

• The o-wave has its field oscillations perpendicular to the optic axis, E (oscillating into & out of the paper)– It obeys Snell’s Law which means that it enters the crystal

undeflected• The e-wave has a polarization orthogonal to the o-

wave, E║, and in the principal section– It travels with a different velocity and diverges from the o-

wave.– The e-wave does not obey Snell’s Law because the angle of

refraction is not zero.

Birefringent optical devices: Retarding plate

• Consider a positive uniaxial crystal such as quartz (ne>no) plate that has optic axis (taken along z) parallel to the plate faces.

• Suppose that a linearly polarized wave is incident at normal incidence on a plate face• For E║ parallel to the optic axis,

• this wave will travel through the crystal as an e-wave• With a velocity c/ne slower than the o-wave since ne>no• Thus optic axis is the “slow axis” for waves polarized parallel to it

• For E right angle to the optic axis,• this wave will travel through the crystal as an o-wave• with a velocity c/no (the fastest velocity in the crystal)• Thus the axis perpendicular to the optic axis will be the “fast axis” for

polarization along this axis.

Retarding plate, 2

• When a light ray enters a crystal at normal incidence to the optic axis and plate surface as shown in Fig.8– The o-wave and e-wave travel along the same direction

• A linear polarization at an angle to z can be resolved into E║ and E – Two components would have been phase shifted by when

the light comes out at the opposite face. – The total phase shift through the plate depends on the

initial angle and the length of crystal L.– The emerging beam can have its initial polarization rotated,

or an elliptically or circularly polarized light as shown in Fig.8.

x = Fast axis

z = Slow axis

E//

E

E//

E

E

L

y

no

ne = n3

Optic axis

L

y

no

ne = n3

A retarder plate. The optic axis is parallel to the plate face. The o- and e-waves travelin the same direction but at different speeds.


Fig. 8: Retarding plate

Light ray propagates inside half wave plate retarder

Retarding plate: Phase different

• If L is the thickness of the plate,– o-wave experiences a phase change ko-waveL = (2/) noL ,

where ko-wave the wavevector of o-wave and is the free space wavelength

– Similarly, the e-wave experiences a phase change (2/)neL

through the plate

• Thus, the phase different between two orthogonal components of E║ and E the emerging beam is

= (2/ ) (ne – no) L

Types of plate retarder

1. A half-wave plate retarder– It has a thickness L such that the phase difference is or

180, corresponding to /2 of retardation.– The emerging E║ and E with this phase shift are added, E

would be at an angle of – to the optic axis and still linearly polarized -> see Fig.9

2. A quarter-wave plate retarder– It has a thickness L such that the phase difference is /2 or

90, corresponding to /4 of retardation.– The emerging E║ and E with this phase shift are added, E

will be elliptically polarized if 0 < < 45 and circularly polarized if = 45 -> see Fig.9

Fig 9: Retarding plate

x

a = arbitrary

(b)

Input

z

xE

z

x

(a)

Output

Optic axis

a

a

Half wavelength plate: =f Quarter wavelength plate:

x

a

0<a < 45°

E

z

x

E

E

x

z z

a = 45°

45°

Input and output polarizations of light through (a) a half-wavelengthplate and (b) through a quarter-wavelength plate.

=f /2

Example: Quartz half-wave plate

• What should be the thickness of a half-wave quartz plate for a wavelength 590nm given the ordinary and extraordinary refractive indices are 1.5442 and 1.5533 respectively.

Solution

thickness.

limpracticaan in result tolarge too is that difference a havethey

because platesretarder as used are substances polymericor quartz

mica,Typically practical.not very is which cknessmicron thi 1.7

about find would wecalcite,for n calculatio repeat the to were weIf

4.325442.15533.1

10590

giving

2

that so of difference phase a isn retardatio wavelength-Half

921

21

notoe

oe

oe

nn

mm

nnL

Lnn

Science

Chapter 3a