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ABCD Matrix Concepts Ray Description Position Angle Basic Operations Translation Refraction Two-Dimensions Extensible to Three Ray Vector Matrix Operation System Matrix

Lecture 9a 2013_

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Page 1: Lecture 9a 2013_

ABCD Matrix Concepts

Ray Description Position Angle

Basic Operations Translation Refraction

Two-Dimensions Extensible to Three

Ray Vector

Matrix Operation

System Matrix

Page 2: Lecture 9a 2013_

Cascading Matrices

'1

'

12

2

2 1

xx

2

22'

2

'2

xx

1

11'

1

'1

xx

Generic Matrix:

Determinant (You can show thatthis is true for cascaded matrices)

V1 R1 T12 R2 V’2

Light Travels Left to Right, butBuild Matrix from Right to Left

1

1

11122'

2

'2

xx

112212

Page 3: Lecture 9a 2013_

Matrices for Optical Components

Free space

Refraction at a planar boundary

Refraction at a spherical boundary

Transmission through thin lens

Reflection from a planar mirror

Reflection from a spherical mirror

10

1n

d

1

0112

R

nn

1

101

f

10

01

1

201

R

10

01

Page 4: Lecture 9a 2013_

Properties of a system from its matrix

If D = 0, all rays entering the input plane from the same point emerge at the output plane making the same angle with the axis. The input plane must be the focal plane of the system

If B = 0, all rays leaving the point O at the input will pass through the same point I at the output. This is the condition for object-image relationship to occur. In addition, A or 1/D will give the transverse magnification produced by the system

Page 5: Lecture 9a 2013_

Properties of a system from its matrix

If C = 0, all rays which enter the system parallel to one another will emerge parallel to one another in a new direction. This is a telescopic system.

If A = 0, all rays entering the system at the same angle will pass through the same point in the output plane. The system brings a bundle of parallel rays to a focus at the output plane.

Page 6: Lecture 9a 2013_

Matrix representing polarization

A monochromatic plane wave of frequency f traveling in the z direction is completely characterized by the complex envelopes Exo=ax exp (jx) and Eyo=ay exp (jy) of the x and y component of the electric field. It is convenient to write these complex quantities in the form of a column matrix,

jy

xj

y

jx

yo

xo

ea

a

ea

ea

E

EJ

y

x

Jones vector

Where = y - x

Page 7: Lecture 9a 2013_

Matrix representing polarization – degenerate state

Linearly polarized wave in x direction

Linearly polarized wave, plane of polarization making angle θ with x axis

Left circularly polarized

Right circular polarized

01

sincos

j

12

1

j1

2

1

Page 8: Lecture 9a 2013_

Optical Element Jones Matrix

linear horizontal polarizer

linear vertical polarizer

linear polarizer at

linear polarizer at -45°

quarter-wave plate, fast axis vertical

quarter-wave plate, fast axis horizontal

circular polarizer, right-handed

circular polarizer, left-handed

Matrix representing polarization – polarizing element

Page 9: Lecture 9a 2013_

Matrix Representation of Polarization Devices Consider the transmission of a plane wave of arbitrary

polarization through an optical system that maintains the plane wave nature of the wave, but alters its polarization

The system is assumed to be linear, so that the principle of superposition of optical fields is obeyed.

The complex envelopes of two electric –field components of the input (incident) wave, E1x and E1y and those of the output (transmitted or reflected ) wave , E2x and E2y, are in general related by the weighted superpositions

E2x = T11E1x+T12E1y

E2y= T21E1x+T22E1y,

Page 10: Lecture 9a 2013_

Matrix Representation of Polarization Devices

Where T11, T12, T21 and T22 are constants describing the device. The above equation are general relations that all linear optical polarization devices must satisfy.

The linear relations in above equations may conveniently be written in matrix notation by defining a 2 x 2 matrix T with element T11, T12, T21, and T22 so that

2221

1211

2

2

TT

TT

E

E

y

x

y

x

E

E

1

1 The matrix T, called the Jones matrix, describes the optical system, whereas the vectors J1 and J2 describe the input and output waves.J2 = TJ1

Page 11: Lecture 9a 2013_

Optical Material

Anisotropic Material: A dielectric medium is said to be anisotropic if its macroscopic optical properties depends on direction.

The macroscopic properties of matter are of course governed by the microscope properties: The shape and orientation of the individual molecules and the organization of the molecules in space.

Page 12: Lecture 9a 2013_

Optical material – continued..

The following is description of the positional and orientational types of order inherent in several kinds of optical materials .– If the molecules are located in space at totally random

position and are themselves is isotropic or are oriented along totally random direction , the medium is isotropic. Gases, liquids, and amorphous solid are isotopic.

– If the molecule and the orientation are not totally random, the medium is anisotropic, even if the position are totally random. This is the case for liquid crystals, which have orientation order but lack complete positional order.

Page 13: Lecture 9a 2013_

Optical material – continued..

– If the molecules are organized in space according to the regular periodic pattern and are oriented in same direction, as in crystal, the medium is in general anisotropic.

– Polycrystalline materials have a structure in the form of disjoined crystalline grains that are randomly oriented relative to the each other. The grains are themselves generally anisotropic, but their average macroscopic behavior is isotropic.

Page 14: Lecture 9a 2013_

Optical material – continued…

Polaroid Material– Polaroid is the trade name for the most commonly

used dichroic material. – It selectively absorbs light from one plane, typically

transmitting less than 1% through a sheet of polaroid. It may transmit more than 80% of light in the perpendicular plane.

– Polaroid materials accomplish polarization by dichroism. At angles other than 90°, the transmitted intensity is given by the Law of Malus.

Page 15: Lecture 9a 2013_

Optical material – continued..

Dichroism– Causes visible light to split into distinct

beams of different wavelengths, or– One in which light rays having different

polarizations are absorbed by different amount

There is a distinct difference between dichroism and dispersion

Page 16: Lecture 9a 2013_

Law of Malus

When unpolarized light passes through a polarizer, the light intensity—proportional to the square of its electric field strength—is reduced, since only the E-field component along the transmission axis of the polarizer is passed.

When linearly polarized light is directed through a polarizer and the direction of the E-field is at an angle to the transmission axis of the polarizer, the light intensity is likewise reduced. The reduction in intensity is expressed by the law of Malus, I=I0cos2θ

Page 17: Lecture 9a 2013_

Law of Malus

Page 18: Lecture 9a 2013_

Polarization by reflection

Unpolarized light can also undergo polarization by reflection off nonmetallic surfaces. The extent to which polarization occurs is dependent upon the angle at which the light approaches the surface and upon the material which the surface is made of.

A person viewing objects by means of light reflected off nonmetallic surfaces will often perceive a glare if the extent of polarization is large.

Metallic surfaces reflect light with a variety of vibrational directions; such reflected light is unpolarized.

Page 19: Lecture 9a 2013_

•For normal incidence case, the reflection coefficient and transmission coefficient is independent of polarization, because the electric and magnetic fields are both always tangential to the boundary

•This is not the case for wave with an oblique angle, because the polarization is not always tangential to the surface or boundary

Polarization by reflection

Page 20: Lecture 9a 2013_

Polarization by reflection

Brewster’s angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent nonmetalic surfaces.

B = arctan (n2/n1) The angle of reflection and angle of refraction adds

up to be 90o

Light that reflects from a surface at this angle is entirely polarized perpendicular to the incident plane - GLARE

If the angle is not exactly Brewster’s angle the reflected ray will only be partially polarized

Page 21: Lecture 9a 2013_

Polarization by reflection

Page 22: Lecture 9a 2013_

Glare from water surface Glare blocked by vertical polarizer

Polarization by reflection

Page 23: Lecture 9a 2013_

There are two types of polarization relative to the plane of incidence

1. Parallel polarization (TM polarization)2. Perpendicular polarization (TE polarization)

Plane of polarization is defined by the plane containing the normal of the boundary and the direction of propagation of the incident wave.

Relative to E

Polarization by reflection

Page 24: Lecture 9a 2013_

Polarization by reflection

Page 25: Lecture 9a 2013_

Perpendicular polarization

The reflection and transmission coefficients are given by

coscos

coscos

12

12

i

tiio

ro

E

E

ti

iio

to

E

E

coscos

cos2

12

2

1

Page 26: Lecture 9a 2013_

it

itio

ro

E

E

coscos

coscos

12

12

||

||||

it

iio

to

E

E

coscos

cos2

12

2

||

||||

Parallel polarization

The Fresnel reflection and transmission coefficients are given by

1||||

Page 27: Lecture 9a 2013_

Polarization by refraction

•Polarization can also occur by the refraction of light. Refraction occurs when a beam of light passes from one material into another material. •At the surface of the two materials, the path of the beam changes its direction. The refracted beam acquires some degree of polarization

Page 28: Lecture 9a 2013_

Birefringence

Birefringence, or double refraction, is the division of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on the polarization of the light.

This is explained by assigning two different refractive indices to the material for different polarizations. The birefringence is quantified by:

Δn = ne - no where no is the refractive index for the ordinary ray and ne is the

refractive index for the extraordinary ray.

Page 29: Lecture 9a 2013_

Birefringence

When a beam of ordinary unpolarized light is incident on a calcite or quartz crystal, there will be, in addition to the reflected beam, two refracted beams in place of the usual single one observed, e.g., in glass.

This phenomenon is called double refraction or birefringence.

Upon measuring the angles of refraction for different angles of incidence, one finds that Snell's law of refraction holds for one ray but not for the other. The ray for which the law holds is called the ordinary ray and the other is called the extraordinary ray.

Page 30: Lecture 9a 2013_

Birefringence

Unpolarized light entering a birefringent crystal is split into two linearly polarized beams which are refracted by different amounts.

There are two refractive indices

Optic axis: this is the direction within the crystal along which there is no double refraction.

Page 31: Lecture 9a 2013_

Birefringence

Page 32: Lecture 9a 2013_

Application of polarization

Linear polarizar– The linear polarizer selectively removes all

or most of the E fields in a given direction, while allowing fields in the perpendicular direction to be transmitted.

– In most cases, the selectivity is not 100% efficient, so the transmitted light is partially polarized

Page 33: Lecture 9a 2013_

Application of polarization

Phase retarder– The phase retarder does not remove either

of the component orthogonal E fields but introduces a phase difference between them.

– If light corresponding to each vibration travels with different speeds through such a retardation plate, there will be cumulative phase difference between the two waves

Page 34: Lecture 9a 2013_

Application of polarization

Rotator– The rotator has the effect of rotating the

direction of linearly polarized light incident on it by some particular angle.

– The effect of the rotator element is to transmit linearly polarized light whose direction of vibration has rotated counterclockwise or vice versa, by an angle