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matrices for optics
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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
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
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
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
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.
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
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
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
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,
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
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.
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.
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.
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.
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
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
Law of Malus
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.
•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
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
Polarization by reflection
Glare from water surface Glare blocked by vertical polarizer
Polarization by reflection
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
Polarization by reflection
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
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||||
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
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.
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.
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.
Birefringence
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
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
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