Gerrard, A._ Burch, J. M. - Introduction to Matrix Methods in Optics-Dover Publications (1975)

328

Appendix F

Connection between Jones
and Mueller Calculi

F.l INTRODUCTION
Since the Mueller and the Jones matrices both des-

cribe the behaviour of optical devices, and for fully
polarized light calculations by the two methods always
lead to the same final results, there must be a close
relationship between the two systems. Nathaniel G.
Parke showed, by using the Wiener coherency matrices,
that formulae can be developed by which the Mueller

matrix elements of the system can be found in terms of
the Jones matrix elements. It is the purpose of this
section to show that this can be done within the form-
alism of the Jones and Mueller systems, and also to
provide formulae by which the process can be reversed,
and the real and imaginary parts of the Jones matrix
elements calculated from a knowledge of the Mueller
matrix (provided that the latter represents a physically
possible polarizing device) •

F.2 DEVELOPMENT OF THE CONNECTION BETWEEN THE STOKES
AND THE MAXWELL COLUMNS

We shall now show that all the Stokes parameters can
be expressed in terms of the Maxwell column, its trans-
posed complex conjugate and some constant two-by-two
matrices, to be introduced in this section.

For a general beam of light, the Maxwell column is

E= [H ] Its transpose is the row matrix

Kexp(illl

[H Kex~ (ii'l)] The complex conjugate of this trans-
pose is LH Kexp(-illl]. (This may be called the

Hermitian conjugate of E. As in section IV.7 it will

be denoted by E.)

329

Then (by matrix multiplication) ,

E fH(a) x E = [H Kexp(-iM] x ]

LKexp(iM

(because exp(iM xexp(-iM = 1). This is I, the first

of the Stokes parameters of the beam. For convenience

later, we shall insert, between the E and the E, the

A,.unit two-by-two matrix [: :] here called which

obviously cannot affect the result. We can thus write

E x Al x E = I ( F.l)

where

(b) Ex [1 0] x E [:expJEx [: _:}

° -1

[H [HKexp (-it,)] x J

-Kexp (it,)

We write this in the form:

Ex A2 x E (F.2)

where

A2

:H:expJ

[H KeXP (i M ]
Kexp (-it,)] x [ H

330

HKexp(itJ.) + HKexp(-itJ.)

HK(cos~ + isin~ + cos~ - isin~)

2HKcos~ = U

We write this in the form ( F.3)

E x As x E = U

where

As

-i](d) E x i0 0 x E -i] lEx 0 [H
[ [i
0 x Kexp (i~>J

[H Kexp(-i~)J x f-iKexp (iM]

LiH

= - iHKexp(i~) + iHKexp{-i~)

= HK{- i(cos~ + isin~) + i{cos~ - isin~)}
HK{- icos~ - i2sin~ + icos~ - i2sin~}

2HKsin~ (because i 2 = - 1) = V

We write this in the form

E x A.. x E = V (F.4)

where

(The above equations (F.1) to (F.4) enable us to rep-
resent each of the four Stokes parameters in terms of

a 'matrix sandwich'. The three matrices A2' As and A..

are closely related to Pauli spin matrices.)

F.3 DEVELOPMENT OF THE SANDWICH MATRICES IN TERMS OF
THE MUELLER AND OF THE JONES MATRIX ELEMENTS

F.3.l Using Mueller elements
The first equation defining the Mueller elements is

Il = MllIo + Ml2QO + MlSUO + Ml.. VO

331

Inserting the expressions developed in the last section
for the Stokes parameter in terms of the Maxwell matrix
elements, this becomes

(Mll xEo x Al x Eo) + (MIz X Eo x Az x Eo)
+ (M13 x Eo x A 3 x Eo) + (Mll+ x Eo x AI+ x Eo)

Substituting the values of the A-matrices and simplifying,
this becomes

(F.5)

The other three defining equations for the Mueller
elements will obviously develop into equations identical
with this except for the subscript on the As on the
left side, and the first subscripts on the Ms on the
right side. Thus:

( F.6)

EI x A 3 x EI ~M31+ M32 (F.7)

Eo x
M33+ iM31+

(F.8)

F.3.2 Using Jones etements
The defining equation for the Jones matrix is El = JEo.

Taking Hermitians on both sides, E 1 = (Hermitian of JEo) .
Now the transpose of a product of two factors is the
product of the transposes of the two factors taken in

reverse order, that is (transpose of AB) = (transpose

of B) x (transpose of A). Since the Hermitian is

merely the complex conjugate of the transpose, a sim-
ilar theorem is clearly true for Hermitians. Thus

(Hermitian of JED) = (Hermitian of EO) x (Hermitian of

J), that is E1 = Eo x J.

332

Thus (inserting AI' the identity matrix, for consis-
tency with later equations) ,

EI xAI xEI (Eo xJ) xAlx(J xEo)

= Eo x (J xAI xJ) xEo (F.9)

(using the associative property of matrix multiplica-

tion). Similarly, we can obtain

EI xA 2 xE I Eo x (J xA2 xJ) «s, (F.W)

Eo x (J x A 3 x J) x Eo (F.ll)

Eo x (J x Ail x J) x Eo ( F.12)

F.4 COMPARISON OF THE PAIRS OF SANDWICH MATRICES IN

THE TWO SYSTEMS
The left sides of equations (F.5) and (F.9) are

identical, so the right sides must be equal. On each
of the right sides appears Eo x (another matrix) x Eo.

Since the two right sides are equal, the matrices

sandwiched between Eo and Eo must be equal in the two

equations. The same argument applies to the sand-

wiched matrices in equations (F.6) and (F.10), those
in equations (F.?) and (F.ll), and those in equations

(F.8) and (F.12).

The sandwiched matrix in equation (F.9) is

(J x Al x J). J is the matrix obtained by transposing
J, then replacing each of the matrix elements in the

transposed J b its complex conjurate. Since

J= Jll J12 its transpose is Jll J21] , so that

J21 J22 J21 J22
GIl
J= G21
, where the Gs are defined as the com-

Gl2 G22

plex conjugates of the corresponding Js.

Thus, the sandwiched matrix in equation (F. 9)

becomes

fGII

LGl2

fGllJll +G2IJZl

~12Jl1 + G22J:n

333

The elements in the final matrix can now be equated,
each with its corresponding element in the sandwiched
matrix in equation (F.5). Thus:

M11 + M12 G11 J 11 +G21 J21 ( F.l3)

M1 a - iM11t G11 J12 + G21 J22 ( F.14)

M13 +iM11t G12 J 11 + G22J21 ( F.15)

M11 - M12 G12J12 +G22J22 (F.16)

Treating the other pairs of sandwiched matrices in the
same way, we get, from equations ( F.6) and (F.10) ,

M21 + M22 G11 J 11 - G21J21 (F.17)

M23 - iM21t G11 J12 - G21 Ji2 ( F.18)

M23 +iM21t G12 J11 - G22 J21 (F.19)
(F.20)
M21 - M22 G 12 J 12 - G22 J22
(F.21)
from equations (F.7) and (F.ll) , (F.22)

MS1 + MS2 G11 J21 + G21 J11

M33 - iMslt G11 J22 + G21 J12

M33+iM31t G12J21 + G22J11 ( F.23)
Mal - M32 G12 J22 + G22J 12 (F.24)
from equations (F.8) and (F.12) ,
Mit1 + Mlt2 i(G21 J11 G11 J21) ( F.25)
(F.26)
Mit 3 - iM1t1t i (G21J 12 - G11 J22)

Mit 3 + iM1t1t i (G22 J 11 - G1 2J 2 1) (F.27)
i(G22J12 - G12J22) ( F.28)
Mltl - Mlt 2

334

F.5 DERIVATION OF THE MUELLER MATRIX ELEMENTS IN TERMS
OF THE JONES MATRIX ELEMENTS

The derivation of the Mueller from the Jones matrix
elements is now simple: we add together or subtract
suitable pairs of equations from numbers (F.13) to
(F.28) and halve the results. Thus:

(13+16) MIl CGIlJIl +G21J21 +G12J12 +G22J22) /2 (F.29)

(13-16) M12 CG11J11 + G21J21 - G12J12 - G22J22) /2 (F.30)

(14+15) M13 CGIlJ12 +G21J22 +G12JIl +G22J2I> /2 ( F.31)

(15-14) M14 iCGIlJ12 +G21J22 -G12JIl-G22J21) /2 ( F.32)

(17+20) M21 CG11J11 +G12J12 -G21J21-G22J22) /2 ( F.33)

(17-20) M22 (GllJll +G22J22 -G21J21-G12J12) /2 ( F.34)

(18+19) M23 CG12JIl +GllJ12 - G22J21 - G2 1J22) /2 ( F.35)
(19-18) M24 i(GIlJ12 +G22J21 -G21J22 -G12Jll) /2 (F.36)
(21+24) M31 (GIlJ21 +G22JIl +G12J22 +G22J12) /2 (F.37)

(21-24) M32 (G llJ 2 1 +G21Jll -G 1 2J 22 -G22J12) /2 (F.38)
(22+23) M33 CGllJ22 +G21J12 +G12J21 +G22Jll) /2 ( F.39)
(23-22) M34 i(G11J22 +G21J12 -G12J21 -G22J11) /2 (F.40)

( 25+28) M4 1 iCG21JIl +G22J12 -GllJ2 1 -G12J22) /2 ( F.41)
(25-28) M4 2
( 26+27) M43 i CG21J 11 + G12J 22 - GllJ21 - G22J 12) /2 (F.42)

i(G21J12 +G22Jll-GIlJ22 -G12J21) /2 (F.43)

(27-26) M44 (G22Jll +G11J22 -G12J21-G21J12) /2 (F.44)

These expressions are identical with those given by
Parke.

335

F.6 DERIVATION OF THE JONES MATRIX ELEMENTS IN TERMS

OF THE MUELLER MATRIX ELEMENTS

The Jones elements are, in general, complex, and it
is necessary to determine both the real and the imagin-

ary parts. It proves convenient in the derivation to

use the polar form of the matrix elements, expressing

a complex number such as X+iY in the form Rexp(iS) ,
where R = (X2 + y 2) ~ and tanS = YIX. The complex con-
jugate X - iY is then Rexp(-iS). Thus, we express Jll

as Rllexp(iS ll), etc. Thus GllJll = (Rllexp(-iS ll »
etc.
(Rllexp(iS ll » = Rd,

Then, adding equations (F.13) and (F.17),

Mll+ M2 1 + M12+ M22 = 2GllJll = 2

2Rll

Therefore, (F.45)

Rll = {(Mll + M21 + M12 + .M22) /2}~

Similarly, subtracting equation (F.17) from equation

(F.13), we get

R21 = {(Mll + M12 - M21 - M22) /2}~ (F.46)

Adding equations (F.16) and (F.20), (F.47)

R 12 = {(Mll - M12 + M21 - M22) /2}~

Subtracting equation (F.20) from equation (F.16),

R22 = {(Mll - M12 - M21 + M22) /2}~ (F.48)

This gives the magnitudes of the Jones matrix
elements. For the angles, it is sufficient to find
the difference between one of them, chosen arbitrarily,
and the other three, because adding a constant to all
four of the Jones angles is, in effect, only the same
as adding the same constant to wt on the right sides of
equations (F.l) and (F.2), that is in the place of ¢
which, as already remarked, produces no effect on any
intensities.

Adding equations (F.31) and (F.35),

M13 + M23 = GllJ12 + G12Jll

= RllR12 [(exp-iSll) (expi S12)]

+ R12Rll [(exp-iS 12) (expiS 11)]

336

= RllRut[expi(8 11-8 12) + exp-i(8 11-812)]

= RIIR12 x 2cos(811-812)

Therefore,

(F.49)

(on substituting for Rll and R12 from equations (F.45)
and (F.47) respectively). Adding equations (F.32) and
(F.34) ,

Ml'+ + M2'+ i(GllJ12 - G12Jll)
iRllR12 [(exp-i811) (expi 812)

- (exp-i812) (expi811)J

iRIIR12 [expi (612- 611) - exp-i (612-611)J

iRIIR12 x 2isin(812-811)

2RIIR12sin(811-812)

Therefore, as before,

( F.50)

A knowledge of both sin(811-812) and COS(811-812) is
enough to determine the angle completely in the range
from zero to 2'JT.

The sines and cosines of the differences between the
other two angles in the Jones matrix elements and 811
can be found in the same way, by suitable combination
of the equations (F.39) to (F.44). The results are,

adding equations (F.37) and (F.38),

(F. 51)

adding equations (F.4l) and (F.42), 337
sin(821- 811) ( F.52)

adding equations (F.39) and (F.44) ,

( F.53)

sUbtracting equation (F.40) from equation (F.43),

sin (822-811) = -- - - -- - - --- -+-M-12-)2"}~ (F.54)

{(MI l + M2 2 )2 - (M2 1

In all of the ten equations (F.45) to (F.54) any

square root term representing an Rij value should be

taken to have a positive sign.

In using the conversion formulae some caution will be

needed, since the Mueller matrix, with its sixteen real

elements, cannot always be expressed in terms of a
Jones matrix, which contains only four real and four
imaginary elements. It can be shown that to every

Jones matrix which can be written down there corres-
ponds a physically possible polarizing device, so that
the corresponding Mueller matrix also exists. If, on

the other hand, sixteen numbers are arbitrarily written
down to form a Mueller matrix, then it may not be
possible to solve equation~ (F.45) to (F. 54) without

obtaining imaginary answers for what should be reaZ

R-values and 8-differences. Even if the terms within
the square root signs are all positive, the equations
may predict impossible pairs of sine and cosine values,

whose squares do not sum to unity. This happens, for
example, if one considers the Mueller matrix for an

'ideal depolarizer', for which Ml l = 1 and all other
elements vanish. Except in terms of time-averaging,

this is a physically impossible device, for which no

Jones matrix exists.

INDEX

Index Terms Links

A 81 102 119 126
129 131 133 140
ABCD rule 90 263
62 64
Aberration 96 106 140
chromatic
oblique 46 176
spherical 152
17
Absorption, quadratic variation
Adjugate of matrix 40 72 79 90
Afocal system 111 112 137 145
156 266
Airy disc
ALGOL programs 91 159 178
Amplitude, addition of 239
192
complex 84
Amplitude of electric
186 208
field 105
Angle, Brewster 277

field 40 95 136
Angular magnification 247 258
Anisotropic medium

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Index Terms Links

Aperture, numerical 92
Aperture properties 263
Aperture stop 264 268 277 280
Approximation, geometric
24 77 116 157
optics 24 29 46
paraxial 34
thin-lens 85 89
Argand diagram 92
Artificial star
Associative property of 12 14 37 211
332
matrices 106 140

Astigmatism 96
Astronomical cameras 184 322
Attenuation due to polaroid 192 295 296
Averages, statistical 98 129
Axial modes 107
Axis, effective 187
187 251 324
fast 190
optic
Axis ratio of ellipse 169 202
192 202 207 294
B 133 283

Back-to-front optical
system

Beam-energy measurements
Beam-expander

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Index Terms Links

Beam, Gaussian 116 118
geometry 139 150
neck 130
radius 127 129 130 158
transformation 139 141
192
Beams, combination of 178
Boys point 208 234 292
Bracket product 195
Brewster 105
Brewster angle 25 338
Brouwer
184
C 92 96 134 267

Calcite crystal 277
Camera 111 173 175

Carbon dioxide laser 57
Cardinal planes: Table of 25 54 60

positions 215 218 75 171
Cardinal points of system 67 70
Cartesian representation 73 103

of Jones matrix 337
Catadioptric systems 1
Cat’s-eye reflector
Caution about Mueller to

Jones conversion
Cayley

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Index Terms Links

Characteristic equation of 19 21
a matrix 266
202
Chief ray 194 212
Circular polarizer 181 189 200 217
Circular retarder 299 300
Circularly polarized light 183 186 209
293 328 340
Coherence 87 92 180
Coherency matrix, Wiener 195
Coherent illumination 181 186 192 207
Coker 209 222 297
Column, Maxwell 192 194 197 202
293 297
Stokes 202

Stokes, experimental detmination 19
Column eigenvectors 2
Column matrix 2 181
Column vector 192
Combination of beams
Compatibility of matrices 8
296
for multiplication 30
Compound disturbance
Compound layers 84
Complex 323
146 332
amplitude
amplitude transmittance
conjugate

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Index Terms Links

Complex (Cont.) 293 328
conjugate of transpose 118 124 158
curvature parameter 114 119 122
eigenvector ratio 181
exponential convention 140
q-value 69
107
Concave mirror
Concentric system 123 125 127 141
Confocal parameter of 144 147 174
328
Gaussian beam
40 79 89
Conjugate, Hermitian
Conjugate planes 328
Connection between Jones 140
69
and Mueller matrices 118
Contours, equi-energy 69 108
Converging mirror 86
Converging power 26 179
Convex mirror 85
Convolution 240 258
Coordinate system 184
Cornu spiral 184 187 240 249
Crystal, anisotropic 251 259 308 327
73 103
calcite
uniaxial

Cube-corner reflector

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Index Terms Links

Curvature, centre of 32
reduced 80
wavefront 102 117 126 141
148 151
Curvature
of equi-phase surfaces 123

Curvature 124 158 63 150
parameter, complex 25 32
of surface 121

Cylindrical waves 259
115
D
193
D, transverse nature of 114 128
De Moivre’s theorem 184 192
Decomposition of partially
248
polarized beam
Degenerate eigenvalues 39
Degree of polarization 95 135 163 165
Derivation of matrix form 166
1 14 15 28
of wave equation 313 318
Derivation of properties

of a system from its
matrix
Determinant of ray pair

Determinants

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Index Terms Links

Diagonal, principal 10
Diagonalization of matrix 18 101 114
Diagonalizing transformation 18 22
Diaphragms 263 283
Dielectric breakdown 112 139 152 178
Dielectric properties 240 248
Differential operator 243
Diffraction 77 84 119
86 90 144
Fraunhofer 85
Fresnel 84 91 133
Diffraction integral 110 112 146 178
Diffraction limit 34 49
Dioptre 27
Direction cosine, optical 255 258 259
Directional derivatives 93 110 179 209
Directionality 54
Distances, reduced 284
Distortion 243
Distributivity of operators 118 127 141 150
Divergence 166
69
Diverging mirror 16
Division of matrices 240 253
Double refraction
107 290
E

Effective axis

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Index Terms Links

Efficiency of polarizer 184 215
Eigenfunctions 173
Eigenvalue
19 21 101 108
Eigenvalues, degenerate 113
Eigenvector 114 116 128
Eigenvector ratios 19 22 110 113
102 108 112 126
complex 159 161 340
Eigenvectors of 2 × 2 uni- 115 122

modular matrix 19
Eikonal 24 82 85 120
133
three-dimensional 90
Electric field 186 208 210 252
Electric field vector 179 182 183 292
Electromagnetic theory of light 240 247 251
Ellipse 190 191 228
Ellipsoidal Huygens’ wavelet 251
Elliptical polarization 139 181 187 226
234 300
Entrance port 267
Entrance pupil 94 134 264 267
270 274 275
Entrance window 94 134 135 266
267 272 282 285
Envelope calculation 260
Equi-energy contours 140

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Index Terms Links

Equi-phase surfaces 117 123 127
Equivalent focal length 63 161
Etendue 91 95 107 134

Exit port 183
Exit pupil 267
Exit window 134 265 270 278
Experimental determination of 134 267 272

Jones matrix 215
Maxwell column 211
Mueller matrix 202 205
ray-transfer matrix 41 52
Stokes column 202
Exponential, complex 181
Extinction 186 187 221 228
Extraordinary wave 240 251 254 258
259
Eye hazard 139
Eye-lens 278
Eyepiece 62 64 284

F 118 125 127
187
Far field 24
Fast axis 267 273 276 282
Fermat’s principle 257
Field, illuminated

magnetic

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Index Terms Links

Field lens 138 283
Field stop 266 273 276 277
Filon 195
Filters, spatial 97 183
Focal length
Focal planes 55 57 63 161
39 41 55 59
ray-transfer matrix
between 59 79 145
25 54 57
Focal point 34 156 161
Focal power 157
Focus geometric 110 176
Focused image 239
FORTRAN programs 86 120 144 145
Fourier transform 86 90 144
Fraunhofer diffraction, 179 180
Frequency, optical
Fresnel 85

diffraction 240
extension of Huygens 84 132
85
construction 97 111 125 126
-Huygens equation 128 133
integrals
number 116 122 150 158
159 174
Fundamental Gaussian
mode

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Index Terms Links

G

G-parameters 142
Gabor 86
Gain coefficient 175
Gain in discharge-tube laser 124
Gas laser, 98 151
Gas laser windows 105
Gauss 25 116
Gaussian beam 116 118
118 130
beam-neck 125 127 129 141
confocal parameter 144 147 174
125 127 143 166
far-field semi-angle, 125 127
location of neck 125 127 129 141
neck radius 147
129
phase shift
propagation in lens-like 157
173
media 125 146 148 173
radius of curvature in 175
spot radius
119
Gaussian beams, verifica- 25
tion of ABCD rule for 116 121
116 158 174
Gaussi.an brackets
Gaussian function
Gaussian mode, fundamental

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Index Terms Links

Gaussian optics 116
Geometric
113
curvature radius 157
focus 24 110 157
optics 116 144 157
optics approximation 107 178
Giant pulses 184
Glan-Foucault polarizer 60
Glass hemisphere 98 105
Glass laser 90
Gradient of eikonal 106
Gyroscope, laser
258
H 258
25 339
H-part of 188
extraordinary wave 212
ordinary wave 196 305
281 284 285
Halbach
Half-wave plate 1 209
97 100 113 128
Jones matrix for 138 178
Mueller matrix for 128 164
Head position 173
Heisenberg
Helium-neon laser

Hemi-confocal resonator
Hemispherical resonator

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Index Terms Links

Hermitian conjugate 222 328 331
Hermitian matrix 293
High-power lasers 139 151
Horizontal polarization 181 298 303 306
310 316 325
Holography 86 96
Huygens 77
Huygens’
24 240
construction 64
eyepiece 251 261
wavelet 117
Hyperbola 115
Hyperbolic functions
332
I 273
87
Identity matrix 90 92 179
Illuminated field 110 176
Illumination, coherent 25
54 64
incoherent 152
Image, focused 153 154
Imaging, paraxial 96
Immersed systems 37 287
Infinitesimal lens
Infinitesimal matrix
Information processing
Input ray

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Index Terms Links

Instantaneous state of 180
polarization 192
188 207 242
Intensities, addition of 295
Intensity
208
mutual 202 207
Intensity calculations in
85
Jones calculus 295
Intensity measurements
Integrals, Fresnel’s 69 176
Interference term 94 95
Internal reflections 16
Invariant, Lagrangian 264 268 273 274
Inversion of a matrix 277
Iris 87 130 188 292
199 201
Irradiance 199 201
Isochromatics
Isoclinics 207 339
207 210
J 208
225 328 335
Jones 212
Jones calculus 210 212 222
215
intensity calculation in
relation with Mueller calculus
Jones matrices: Table of
Jones matrix
experimental determination

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Index Terms Links

Jones matrix of 324 327
linear retarder 322
polarizer 326
rotator

K

Kerr 195
Ket 181
Kogelnik 26 116 129 339

L

Lagrangian invariant 94 95
Laser 97 180 183 209
111 173 175
carbon dioxide 98 151
gas 98 100 178
glass 97 100 113 128
helium-neon 138 178
111 139 151 178
high-power 138
low-power 139
medium-power 151 178
pulsed 99 105
ring 178
ruby
Laser 98 124 133
amplifier 139
applications

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Index Terms Links

Laser (Cont.) 106
gyroscope 122
output, prediction 99
resonator 139
safety 104
systems, bent and folded
49 95
Lateral magnification 106
Lateral misalignment 195
Laws of photoelasticity
Left-handed circularly pol- 235 300
138 283
arized light 152
Lens, field 142
62
infinitesimal 119 141 162
spectacle 151 160
telephoto
thin 54
Lens-like media 263
Lens-system, negative
stops of 93
Light beam, directionality 92 180
90 179
of 183
Light, coherent 185

incoherent 179
partially coherent 183
plane-polarized
polarized, production and

analysis
quasi-thermal

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Index Terms Links

Light, coherent (Cont.) 179 180 297
unpolarized
40 136
Linear 196 212
magnification 194 236
polarizer 294
retarder 182
superposition 269 271 278

Lissajous figure 49
L-matrices 98 107 111
Longitudinal magnification 138
Losses in resonator 110 113
Low-power laser
L-plane 247 252 253 257
40 95 136
M 64

Magnetic field 49
Magnification, angular 40 49 95 113
132 136
chromatic difference of 83 89
Magnification, 94 163 274
134
longitudinal 12
transverse 12

Malus theorem
Marginal ray
Matching of optical systems
Matrices, addition of

associative property

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Index Terms Links

Matrices, addition of (Cont.) 286 287
augmented 15
2 × 2, determinants of 16
division of 212
Jones: Table of 269 271 278
L- 196
Mueller: Table of
ray-transfer, separate for 140
xz- and yz-planes 12
subtraction of 17

Matrix, adjugate of 19
characteristic equation 293 328 340
of
coherency, Wiener 137
derivation of optical 10 249
system from 293
diagonal 332
Hermitian 153 154
identity 210 212 222
infinitesimal
Jones 215
experimental determina- 195 196
tion
Mueller 202
experimental determina- 9
tion
null 330
Pauli spin

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Index Terms Links

Matrix, adjugate of (Cont.) 251
permittivity 271 281 289
Q- 26
ray-transfer
reciprocal 13 17 22 136
169
reflection
refraction 67
Matrix, 31 268
resonator
97 100 109 167
rotation 171
rotator 156 310 325 326
sandwich 196
singular 330
spur of
square 16
symmetric 21 126
thin-lens
trace of 2 249
translation, 249
transpose 34
unimodular 21
29 30 268
unit 13
Z- 18 19 27 171
287
10 33
293 294 296 297
299

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Index Terms Links

Matrix 18 19 101 114
diagonalization
elements, suffix notation 5
for
form of Maxwell’s equa- 247
tions 240
form of vector operations 248
form of wave equation 16
inversion
multiplication 4 6 38 287
associative property 12 37 332
non-commutative property 7 35
product 4
determinant of 15
reciprocal of 18 38
181 186 192 207
Maxwell column 209 222 297
211
experimental determination
Maxwell column represent- 328
240
ation of Stokes para-
meters 247
Maxwell’s equations
Maxwell’s equations in 192 202 207 294
matrix form 248
Measurements of beam 91 267
energy
Medium, anisotropic
Microscope objective

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Index Terms Links

Mirror, converging 69 111
diverging 69 108
output 99 108 124 132
167 173 174
rotating 107
Mirror curvature 150 174
Misalignment effects 106 175 286
Mode discrimination 124 126
Mode-matching, application
133
of ABCD rule 122
Mode(s), fundamental 98
97 179
axial 179
transverse 92
Modes of radiation field 72
Modulation transfer function
Monocentric system 225 328 334
Mueller and Jones calculus 194
196
relation between 195 196
Mueller calculus
Mueller matrices: List of 202 205
Mueller matrix
experimental determina- 196 315 320
302 305
tion 11
Mueller matrix of linear
4 8 38 287
retarder
polarizer
Multiple products
Multiplication, matrix,

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Index Terms Links

Mutual intensity 295
Mutually coherent beams, 192
295
combination of
Mutually incoherent oscil- 118
118
lations
125 127 129 141
N 147

Near field 147
Neck of Gaussian beam
Neck radius of Gaussian 160 57
54
beam 67
59 143
Neck-size transformation 184 302
formula 54
25 56
Negative
lens-like distribution 7 12 35
lens-systems 243
refractive index 133 139 209
92 147 268
Newton’s equation
Nicol prism
Nodal planes
Nodal points
Non-commutativity of

matrix multiplication
operators
Non-linear effects
Numerical aperture

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Index Terms Links

O 51
40 41 110
Object-image distance minimum 91 267
Object-image relationship 265
Objective, microscope 184 185
photographic 281 284
Oblique reflection
Observer 121
One-dimensional ray-trans- 25 338
243
fer matrix
O’Neill, E.L. 184 193
Operator, differential 187 195 251
Opposite states of polar-
107 290 291
ization 144 151
Optic axis 27
Optical 151
129 132 179 180
axis, effective 92 183 263 283
communication 97
direction-cosine 290
fibre 202
frequency 137
pinhole 135
resonator

misalignment of
system, ‘back-to-front’

derivation from matrix
systems, matching of

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Index Terms Links

Optics, geometric 24 140 157
Gaussian 116
261
Ordinary Huygens wavelet 187
Ordinary (O) vibration 251
Ordinary wave
Orientation of 190
185
Ellipse 180 185 193
pass-plane
Orthogonal states 295
Oscillations, mutually in-
122 124
coherent 99 108 124 132
Output 167 173 174
37 135 287
of laser, prediction 290
mirror 35 38 278

ray 24 29 46
Overall misalignment terms
Overall ray-transfer 156
328 334
P 183

Paraxial approximations 267
Paraxial rays, propagation

in lens-like media
Parke, N.G.
Partially coherent light
Partially obscured exit

pupil

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Index Terms Links

Partially polarized light 184 193 207
Pass-plane 185 233 234 302
323 332
Pauli spin matrices 330
Pencils, ray 163
Pendlebury 163
Penumbral region 267 273
Permittivity 240 248 251
Phase-angles in Jones
144 217
matrix elements 181 210 292 295
Phase differences 186 208 210
Phase of electric field 187 194 195
Phase plate 187
212 324 327
convention about sign in
general Jones matrix for 196 315 320
general Mueller matrix 188
188
for 116 121 129 140
half-wave 277 282
quarter-wave 195 235
Phase shift 294
ø-n diagram 265
Photoelastic effect 90 209
Photoelectric measurements 92 183 263 283
Photographic objective 30
185 298
momentum vector
Pinhole, optical
Plane-parallel plates
Plane-polarized light

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Index Terms Links

Plane, reference 26
vibration 185

Plane waves through uni- 251 254
axial crystal 40
54
Planes, conjugate 25 54
principal 193

Points, cardinal 215
Poincaré sphere 292
Polar representation of 184 192
180
Jones matrix 185 193
Polarization, complete 184 207

degree of 189
instantaneous state of
orthogonal states 189
partial 184 185
specification by Stokes 184
181 189 200 217
parameters 299 300
Polarization 181 187 226 228
234 235
ellipse 179
by oblique reflection 189
by scattering
Polarized light, circular

elliptical

production and analysis
rough experimental analysis

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Index Terms Links

Polarizer, circular 202
Glan-Foucault 184 215
linear 184 196 212 302
322
Polarizer efficiency 184 215
Polaroid 184 185 215
Port, entrance 267
267
exit 17
Postdivision
Postmultiplication 8 12 17
Power, converging 118
34 156 161
focal
refracting 33 34
resolving 91
Poynting vector 24 259
Predivision 17
Premultiplication
Principal 8 12 17
diagonal
planes 10
54 55
ray-transfer matrix
between 58 79
25 57
points 94 266 282
rays 251
section 184
Prism, Nicol 208 292
Product, bracket 319 331
transpose of

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Index Terms Links

Product, bracket (Cont.) 241
scalar 242
vector 11

Products, multiple 239
Programs for polarization 151

calculations 94 134 264 267
Pulsed lasers 270 274 278
Pupil

Q

Q-matrix 271 281 289
Q-switch 105 108
Quadratic variation of
152
absorption 151 152
refractive index 188 199 204 213
Quarter-wave plate 228 231 237 299
212
Jones matrix for 196 308 315
Mueller matrix of 119 140 143 145
q-value 158 161

q-values, separate for xz- 140
and yz-planes

R

Radiant sterance 96

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Index Terms Links

Radius of curvature 32 118 126 165
173
geometrical 113
reduced 80 178
Ramsden disc 281
Ramsden eyepiece 64
Ratios, eigenvector 102 108 115
Ray, chief 266
marginal 94 163 274
paraxial 24
principal
skew 94 266
Ray pair, determinant of 26
95 135 163 165
Ray-paths, reversibility of 166
Ray pencil 168
Ray-transfer matrices, 77 94 163

summary of 78
Ray-transfer matrices sep-
140
arate for xz- and yz- 26
planes
Ray-transfer matrix 52
experimental determina-
tion 151
Ray-transfer matrix for 68 78
lens-like media 33 78
reflection
refraction

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Index Terms Links

Ray-transfer matrix for (Cont.) 35
a system 35 78
thin lens 30 78
translation 37 275 287
13 17 22 136
Ray-vector 169
Reciprocal matrix 18 38
14
Reciprocal of matrix product 30 31 54
Rectangular matrix 80 178
Reduced distances 30 31
Reduced radius of curvature 26
Reduced thickness
Reference plane (RP) 68
Reflecting 67
69 176
surface 184
systems, rule for
Reflection, internal
oblique, polarization by
Reflection

matrix 67 78

in sphere 70
Reflector, ‘cat’s-eye’ 103

cube-corner 73 103
resonant 132
Refracting power
Refraction 33 34
double 28
240

Refraction matrix 31 78

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Index Terms Links

Refractive index 25
discontinuity in 151
negative 67
quadratic variation 151 152

Refractive index distribu- 160
tion 90 138 143 144
151
Relay system
5
Repeated suffix notation 91 95
Resolution criterion 91 92
Resolving power 132
Resonant reflector 133
Resonator, concentric 107
128
effective axis 173
hemi-confoca1 97 99
hemispherical 133
optical 113 122 124 126
plane-parallel 172 174
stable 291
108 110 113 126
two-mirror 160 173
unstable
98 107 111 133
Resonator 100 109 167 171
Losses
matrix 73
Retroreflector 168
Reversibility of ray-paths

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Index Terms Links

Richards 25 339
Right-handed circularly-
300 314 78
polarized light 189 232
Right-handed state 99 105
Ring lasers 33 68

-matrix 212
Rotation of axes, Jones 196
156 310 325 326
matrix for 196
Mueller matrix for 326
Rotation matrix 111 113
Rotator, Mueller matrix for
Jones matrix for 19
Round-trip gain 178
Row eigenvectors
Ruby laser 18
Rule for inversion of 2 × 2 uni- 6
67
modular matrices 13
matrix multiplication 80 102 109 110
reflecting systems 112 117 118 161
transposing a product 178
R-value
139
S 83 163 165

Safety code
Sagitta

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Index Terms Links

Sampson 25
Sandwich matrices 330
Scalar multiplication 12 186 198 209
Scalar product 241
Searchlight
Section, principal 92
Self-focusing effect 251
Semi-angle of divergence 139 151 160 176
Shadow transform 129
Sheet polarizer 86
Sinclair 185
Single-mode output
Singular matrices 26 339
Skew rays 133
Slit-shaped aperture Fraunhofer
16
diffraction at 26
Snell’s law
Space-bandwidth product 86
Spatial filter
Speckle 21
Spectacle lenses
Spectroscopic filter 91 95 134
Sphere
reflection in 97 183 263 283
Spherical
aberration 96 183
mirrors
wave 142

183

72 240

70

46 64 73 176
150
121

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Spheroid 240 262
S-plane 110 113
Spot radius of Gaussian
116 123 146 173
beam
Spot size transformation 131 147 158
21 108 126 172
formula
Spur of matrix 2 249
Square matrix 113 122 124 126
Stable resonator 172 174
92
Star, artificial 192
Statistical averages 192 194 195 197
Stokes column 202 293

experimental determina- 202
tion
294
Stokes column calculations 184 189 192 194
assumptions in 200 292
264 277
Stokes parameters 266 273 276 277
263
Stop, aperture 195
field 197
196
Stops of lens system 12
Strain, direction of
Strain distribution
Strain optical coefficient
Subtraction of matrices

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Index Terms Links

Suffix notation for matrix 5
elements
78
Summary of 126
ray-transfer matrices 33
resonator properties
25 32 63
Surface, reflecting 25
Surface 115 153 177
40 72 90 111
curvatures 112 137 145 156
separations 266
Sylvester’s theorem 141
System, afocal 54
67 70 75 171
beam-transforming 54 64
cardinal points of 169
catadioptric 54
immersed 72 107
lens, back-to-front 67
lens, negative 90
monocentric 266
reflecting, rule for 40 112
relay
telecentric
telescopic

T

Table of

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Index Terms Links

Table of (Cont.) 212
Jones matrices 196
Mueller matrices
positions of cardinal 57
planes 266

Telecentric system 62
Telephoto lens 134 267 277 283
Telescope
62
eyepiece for
Telescope objective, resol- 91
40 112
ution limit of 134
Telescopic system 240
Television camera 160
Tensor 30 31
Thermal lensing 119 141 161
Thickness, reduced 34
Thin lens 34 78
Thin-lens approximation 90
Thin-lens matrix 90
Third-order aberrations 95
Three-dimensional eikonal 267 273 276
30 78
I Throughput’ 21
Total field 86 120 144 145
86
-matrix 18 22
Trace of a matrix
Transform, Fourier

shadow
Transformation, diagonalizing

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Index Terms Links

Translation matrix 29 30 78
Transpose 13 241 242 312
328
complex conjugate of 293 328
Transpose of a product 13
Transverse
62
chromatic aberration 252 253
field 97 129 179
modes 133
Truncating effects 174
Tube diameter

U

Uniaxial crystal 184 187 240 249
251 259
negative 327
Unimodular matrix 18 27 171 287
Unit matrix 10 33
Unitary transformation 196 212
Unpolarized light 180 224 297 302
306 309 316
Unstable resonators 108 110 113 126
160 173

V

Vandermonde 1
Vector, column 2 181

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Index Terms Links

Vector, column (Cont.) 241
defining equation for 179 182 183 292
electric field
latent 19
photon momentum 90
Poynting 24 259
ray 35
row
2
Vector
analysis 240
operations in matrix
form 240
product 242
252 254 259
Velocity, wave 187
Vibration, extraordinary (E) 187
185
ordinary (O) 107 134 267 273
Vibration-plane 282 285
Vignetting
110 114
W 32

Walk-off 247 250
Water
Wave equation for electro- 248

magnetic field
Wave equation in matrix

form, derivation of

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Index Terms Links

Wave, extraordinary 240 251 254 258
259
ordinary 251
Wavefront 77 117 148 163
240 259
Wavefront
curvature 102 117 126 141
148 151
velocity 259
Wavelength, effect of 174
Wavelets, Huygens 261
Waves, plane, through uni-
254
axial crystal 328 340
Wiener coherency matrices 94 134 135 266
Window, entrance 267 273 282 285
134 267 272
exit 105
Windows, gas laser 151
Wood, R.W., Physical Optics
274 276
Y 25

y-n diagram 229
‘ynv’ method

Z

Zero column matrix

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Index Terms Links
Z-matrix
averaged 293 294 296 297
299
296

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