14 18.matrix methods

7/27/2019 14 18.Matrix Methods

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Chapters 14 & 18: Matr ix methodsChapters 14 & 18: Matr ix methods


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http://www.youtube.com/watch?v=kXxzkSl0XHk

Welcome to the Matrix
http://www.youtube.com/watch?v=kXxzkSl0XHkhttp://www.youtube.com/watch?v=kXxzkSl0XHk


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Lets start with polarization

l igh t is a 3-D vecto r field

linearpolarization circularpolarization

y

x

z
http://en.wikipedia.org/wiki/File:Circular.Polarization.Circularly.Polarized.Light_And.Linearly.Polarized.Light.Comparison.svghttp://en.wikipedia.org/wiki/File:Circular.Polarization.Circularly.Polarized.Light_And.Linearly.Polarized.Light.Comparison.svg


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y

x x

y

Plane waves with kalong zdirectionosci l lat ing(vibrat ing)electr ic field

Any polarization state can be described as linear combination of these two:

complex amplitude

con tains al l polar izat ion info

yeExeE yx

tkzi

y

tkzi

x

00

E

tkziiy

i

x eyeExeEyx 00E


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Vector representation

tkzie 0

~~EE

EE ~

tkziiy

i

x eyeExeEyx 00E

for complex field components

y

x

E

E

0

0

0 ~

~~

E

y

x

i

y

i

x

eE

eE

0

0

0

~E

The state of polarization of light is completely determined bythe relative amplitudes (E0x, E0y) and

the relative phases (D =yx) of these components. The complex amplitude is written as a two-element matrix, or

Jones vector


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Jones calculus (1941)

Indiana Jones Ohio Jones


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The electric field oscillations are only

along the x-axis

The Jones vector is then written,

where we have set the phase x= 0, forconvenience

0

1

00~

~~ 0

0

0

0 AAeE

E

EE

xi

x

y

x

x

y

Normalized form:

0

1

Jones vector forhorizontally polarized light


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The electric field oscillations are only

along the y-axis

The Jones vector is then written,

where we have set the phase y= 0, forconvenience

1

000~

~~

00

0

0 AAeEE

EE

yi

yy

x

x

y

Normalized form:

1

0

Jones vector forvertically polarized light


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sin

cos

sin

cos~

0

0

0 A

A

A

eE

eEy

x

i

y

i

xE

Jones vector forlinearly polarized light at

an arbitrary angle

The electric field oscillations make anangle with respect to the x-axis

If = 0, horizontally polarizedIf = 90, vertically polarized

Relative phase must equal 0:

x= y= 0 Perpendicular component amplitudes:

E0x= A cos E0y = A sin

Jones vector:

x

y


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Circular polarization

Suppose E0x= E0y= A,

and Exleads Eyby 90o = /2 At the instant Exreaches its

maximum displacement (+A),

Eyis zero

A fourth of a period later, Exis

zero and Ey= +A

The vector traces a circular

path, rotating counter-

clockwise


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iA

Ae

A

eE

eEE ii

y

i

x

y

x 1~2

0

0

0

i

1

2

1

Replace /2 with -/2 to get

Circular polarization

The Jones vector for this case where Exleads Eyis

The normalized form is

-This vector represents circularly polarized light, whereErotates counterclockwise, viewed head-on

-This mode is called left-circularly polarized (LCP) light

Corresponding vector for RCP light:

i

1

2

1

LCP

RCP


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iB

ADirection of rotation?

counterc lockwise

c lockwise

Elliptical polarization

IfE0xE0y, e.g. ifE0x= A and E0y= B,the Jone vectors can be written as

iB

A


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General caseresultant vibration due to two perpendicular components

if , a line

else, an ellipse

xy D

nD

Lissajous f igures


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dc

baM

various optical elements modify polarization

2 x 2 matrices describe their effect on light

matrix elements a, b, c, and ddetermine the modificationto the polarization state of the light

matrices


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Optical elements: 1. Linear polarizer

2. Phase retarder

3. Rotator


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Select ively removes al l or most of the E-vib rat ions

except in a given direct ion

TA

x

y

z Linear polarizer

Optical elements: 1. Linear polarizer


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1

0

1

0

dc

ba

10

00M

Consider a linear polarizer with transmission axis along the vertical (y).

Let a 2x2 matrix represent the polarizer operating on vertically polarized light.The transmitted light must also be vertically polarized.

For a linear polarizer with TA vertical.

Operating on horizontally polarized light,

Jones matrix for a linear polarizer

0

0

0

1

dc

ba

1)1()0(

0)1()0(

dc

ba

1

0

d

b

0)0()1(

0)0()1(

dc

ba

0

0

ca


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Jones matrix for a linear polarizer

10

00MFor a linear polarizer with TA vertical

For a linear polarizer with TA horizontal

00

01M

11

11

2

1M

2

2

sincossin

cossincosM

For a linear polarizer with TA at 45

For a linear polarizer with TA at


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Introduces a phase difference () between orthogonalcomponents

The fast axis (FA) and slow axis (SA) are shown

when D /2: quarter-wave plateD : half-wave plate

FA

x

y

z Retardation plate

SA

Optical elements: 2. Phase retarder


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/2

and wave plates

net phase difference:

= /2 quarter-wave plate = half-wave plate


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We wish to find a matrix which will transform the

elements as follows:

It is easy to show by inspection that,

Here xand yrepresent the advance in phase ofthe components

yyy

xxx

i

oy

i

oy

i

ox

i

ox

eEeE

eEeE

into

into

y

x

i

i

e

eM

0

0

Jones matrix for a phase retarder


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ie

e

eM

i

i

i

0

01

0

0 44

4

Jones matrix for a quarter wave plate

Consider a quarter wave plate for which || = /2 Fory-x= /2 (Slow axis vertical) Let x= -/4 and y= /4 The matrix representing a quarter wave plate, with its

slow axis vertical is,

QWP, SA vertical

ieM

i

0

014

QWP, SA horizontal


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10

01

0

0

10

01

0

0

2

2

2

2

2

2

i

i

i

i

i

i

ee

eM

ee

eM

HWP, SA vertical

HWP, SA horizontal

Jones matrix for a half wave plate

For|| =


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x

y

Rotator

SA

Optical elements: 3. Rotator

rotates the direction of linearly polarized light by a


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sin

cos

sin

cos

dc

ba

cossin

sincosM

Jones matrix for a rotator

An E-vector oscillating linearly at is rotated by an angle Thus, the light must be converted to one that oscillates

linearly at (+)


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To model the effects of more than one component on the

polarization state, just multiply the input polarization Jones

vector by all of the Jones matrices:

1 3 2 1 0E E A A A

Remember to use the correct order!

A single Jones matrix (the product of the individual Jones

matrices) can describe the combination of several components.

Multiplying Jones matrices


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Crossed polarizers:

0 0 1 0 0 0

0 1 0 0 0 0

y x

A A

x

y z

1 0y xE E A A0E 1E

x-pol

y-pol

so no light leaks

through.

Uncrossed polarizers:

0 0 1 0 0

0 1 0 0

y x

A A

0E

1E

rotatedx-pol

y-pol

00 0

0

x x

y y x

E E

E E E

y xA A SoIout 2Iin,x

Multiplying Jones matrices


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In dealing with a system of lenses,

we simply chase the ray throughthe succession of lens.

That is all there is to it.

Richard Feynman

Feynman Lectures in Physics

Matrix methods for complex optical systems


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The ray vector

A light ray can be defined by two coordinates:

position (height), y

islope, optical axis

y

These parameters define a ray vector, which will change

with distance and as the ray propagates through optics:

To chase the ray, use ray transfer matrices that characterize

translationrefraction

reflection

etc.

y

Note to those using Hecht: vectors are formulated as

y


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A B

C D

Optical system 2 x 2 ray matrix

System ray-transfer matrices

in

iny

out

outy


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DC

BAM

2 x 2 matrices describe the effect of many elements

can also determine a composite ray-transfer matrix

matrix elements A, B, C, and Ddetermine useful properties

matrices

fn

nBCADMDet 0


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O1 O3O2

Multiplying ray matrices

Notice that the order looks

opposite to what it should be,

but it makes sense when you

think about it.

in

iny

out

outy

in

in

in

in

out

out yOOO

yOOO

y

123123


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Exercises

You are encouraged to

solve all problems in thetextbook (Pedrotti3).

The following may be

covered in the werkcollege

on 20 October 2010:

Chapter 14:

2, 5, 13, 15, 17

Chapter 18:3, 8, 11, 12

14 18.matrix methods

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