A p p E

About

N D X I 1

Graphsand Matrices in Mathcad

CHANGING NUMBERS IN A FILE AND PLOTTING A GRAPH

x:=1,2 ... 10 a:=3 b:=4 f(x):=a·x+b 40

f(x) 20

00

2D GRAPH

5

(x)

). := 0.5 A := 1 T = 1 81 = 1 t1 = 0.1

Specification of the number of x and t 1 values

N := 15 i := 0 . .. N j := 0 . .. N

Specification of the range

Xi := -.4 + .025 · i t1j := -.4+, 025 · j

10

425

42 6 APPENDIX A. ABOUT GRAPHSAND MATRICES IN MATHCAD

In the specification of the function only x and t 1 are used

uc(x, tl)

:= [ 2 . A . cos [ 2 . Jr ( 28_\) J . [ cos [ 2 . Jr . (I _ ~) _ 2 . Jr ( 2

8_\) J J r In the plotting function one needs the i and j notation

M;,j := uc(x;, t I i ).

Call on"Surface plof' and type at the place holder just M and push "F9."

M

Go with the mouse on the graph and change the angle of the "point of view." Click twice on the graph and get "3D-Plot Format" for "graph options." Switch to contour plot.

MATRICES

Go to "Insert" and "Matrix" and select 2 by 2

(: :) Type M =• Indicate the matrix and insert

M := 1 to get M := (: :)

APPENDIX A. ABOUT GRAPHSAND MATRICES IN MATHCAD 427

The manipulation of matrices can easily be seen from files containing matrices. Here we give an example of a matrix composed of functions and how to access the matrix elements after a multiplication has been done.

Fill in functions of x directly and call M no M(x) x := 0, .1 ... 5

M(x) := ( co~(x) -sin(x)) + sm(x) cos(x)

One can access the matrix elements separately. Note that in Mathcad one starts with 0. For the 0, 1 and 1, 1 elements one has

M(x)o 1 = M(x)1 1 = 0 1

-0.1 0.995 -0.199 0.98 -0.296 0.955 -0.389 0.921 -0.479 0.878

Consider the matrix product M 1 (x) = M (x )3. After multiplication one can again access the matrix elements

M1(x) := M(x)3 one gets for the 0, 1 element

M1(x)o 1 = 0

-0.296 -0.565 -0.783 -0.932 -0.997

A p p E N D X IJ D

Formulas

CONSTANTS

FORMULAS

100JLm::::} 3000 GHz

100JLm::::} 100 cm- 1

10JLm::::} 1000 cm-1

1 meV = 103 eV = 1.6 X 10-16joule::::} 8.07 cm-1

100 nm = 1000 A 1oooo A = 1JL

1 A = 10- 8 cm = 10-10 m

H=i i 2 =-1

z = a + ib = r(cosqy + i sinqy) = reitP

eix+eix

cosx = -2-eix- eix

sinx = ---2i

x2 x3 ex = 1 +x +- +- + · · ·

2! 3!

x3 sinx = x-- + · · ·

3!

x2 cosx = 1-- + ...

2! 2 n-1 qn- 1

Sn = a + aq + aq + · · · · · · + aq = a -­q-1

iflql < 1, N--+ oo a

Soo = --1-q

429

430 APPENDIX B. FORMULAS

a1b1ct

a2b2c2

a3b3c3

TRIGONOMETRIC FORMULAS

0 30°

sin 0 I 2

cos !v'J 2

tan 0 !v'J 3

cot 00 v'3

sin2 a + cos2 a = 1

1 tana = -­

cosa

45° 60°

!v'2 2 !0 !v'2 I 2 2

1 v'3 !v'J

sma -- =tana cosa

90°

1

0

00

0

1 --2 - = 1 +tan2 a cos a

180° 270° 360°

0 -1 0

-1 0

0 00 0

00 0 00

cosa -.- =cota sma

tana sin a = ----;:===;o=

.J1 + tan2 a 1

cos a = ----;:===;<==== .J1 +tan2 a

sin(90° ±a) = +cosa sin(180° ±a) = =j=sina

cos(90° ± a) = =f sina cos(180° ± a) =- cosa

tan(90° ±a) = =fCOta tan(180° ±a) = ±tana

cot(90° ±a) = =j=tana cot(180° ±a) = ±cota

sin(-a) =- sina

cos(-a) = +cosa

tan(-a) = -tana

cot(-a) = -cota

sin(a ± ß) = sina cosß ± cosa sinß

cos(a ± ß) = cosa cosß =f sina sinß

tana ±tanß tan( +a ± ß) = ------'--

1=j=tana·tanß ( ± ß) cot a · ß =F 1

cota =---­cotß ±cota

APPENDIX B. FORMULAS 431

sin 2a = 2 sin a cos a

cos2a = cos2 a- sin2 a = 1- 2sin2 a = 2cos2 a -1

. 2 tana Sill2a = 2 1+tana

2 2tana 2 tana= - ,

1 - tan2 a cot a - tan a

1 + cosa = 2cos2 ~'

1- cos2a tana =

sin2

1- tan2 a cos 2a = ------=--

1 + tan2 a

cot2 a- 1 1 cot2a = 2 cota = 2(cota- tana)

. 2 a 1 - cos a = 2 Sill 2

2tan ~ 2

1 + cos2a 1 + cos2a 1- cos2a

sin2a 1- tan2 ~ 2

. . . a+ß a-ß silla+sillß = 2sill-2- ·cos-2-

sin a + sin ß a + ß ----- = tan --cosa + cosß 2

. . ß 2 a+ß . a-ß Silla -Sill = cos-- ·Sill--

2 2

a+ß a-ß cos a + cos ß = 2 cos - 2- · cos - 2-

sin a - sin ß a - ß ___ __;,_ = tan --cosa + cosß 2

. a+ß . a-ß COS a - COS ß = -2 Sill-- · Sill --

2 2

tana+tanß ----- = tan a · tan ß cota + cotß

sin(a ± ß) tana+tanß = ---­

cosacosß

±sin(a ± ß) cot a ± cot ß = ----­

sina sinß

1 + tana --- = (tan45° +a) 1- tana

cota + 1 --- = cot(45°- a) cota- 1

cos a + sin a = .Ji sin( 45° + a) = .Ji cos( 45° - a)

cos a - sin a = .Ji cos(45° + a = .Ji sin(45° - a)

2 cota + tana = -­

sin2a cota- tana = 2cot2a

432 APPENDIX B. FORMULAS

DIFFERENTIATION

(u · v)1 = uv1 + u1v I I

( ~)I= U V- V U

v v2

(sinx)1 = cosx

(cosx)1 =- sinx

(eX)I = eX

1 (1nx)1 = -

X

1 (tanx)1 = --2 -

cos X

. 1 ( arcsm x )1 = .J'1=X2

1-x2

-1 (cotx)1 = -.-2-

sm x

INTEGRATION

( arccos x )1 =

I udv = uv - I v du

1

I I xn+l dx = x x"dx = --(n # -1)

n+l

I sinxdx = - cosx

I cotxdx = lnsinx

I d~ = tanx COS X

f ___!!!____ = lnJ1 +x 1- x 2 1- x

I dx . .Jl=X2 = arcsm x

f ex dx = ex J ax dx = ax lna

l dx ~ =lnx

I cosxdx = sinx

I dx -- = -cotx sin2 x

f dx -- = + arctan x 1 +x2

f dx -- = ln(x ±a) x±a

References

M. Born and E. Wolf, Principles of Optics, 2nd ed. Pergarnon Press, New York, 1964.

M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical Phenomena, Springer­Verlag, Heidelberg, 1962.

P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lecture on Physics, 6th ed. Addison-Wesley Publishing Company, Reading, Massachusetts, 1977.

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, lnc., New York, 1988.

J. D. Jackson, Classical Electrodynamics, 2nd ed. John Wiley & Sons, New York, 1975.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. McGraw-Hill lnc., New York, 1976.

C. Kittel, Introduction to Solid State Physics, 3rd ed. John Wiley & Sons, Inc., New York, 1967.

H. Kogelnik and T. Li, Appl. Opt. Vol5, p. 1550 (1966). G. P. Parrent and B. J. Thompson, Physical Optics Notebook, Society ofPhoto­

Optical Engineers, Redondo Beach, California, 1969. F. K. Kneubühl and M. W. Sigrist, Laser, B. G. Teubner, Stuttgart, 1988. E. D. Palik, Handbook of Optical Constants of Solids li, Academic Press, Inc.,

New York, 1991. R. W. Pohl, Einführung in die Optik, Springer-Verlag, Heidelberg, 1948.

433

Index

Aberration, 405 achromatic doublet, 422 aplanatic lens, 415 astigmatism of single surface, 458 astigmatism of a thin lens, 419 chromatic aberration, 420 coma, 413 rr -6 equation, 410 spherical of single surface, 405 spherical ofthin lens, 408

absorption, 313 achromatic doublet, 420 achromatic doublet with separated lenses,

422 active medium, 291 Airy disc, 147 Airy function, 114 altemating high and low refractive indices,

254 amplitude division, 94 amplitude grating, 150, 156, 168 amplitude reflection coefficients, 315 angle of deviation, 7 angular magnification, 40 angular momentum quantum number, 278,

279 angular quantum number, 279 antireflectioncoating,248,252 aperture function cx(Tj), 377 aperture plane, 374 aperturein random arrangement, 167 aplanaticlens,415,417

apodization,358,359 Arago, 133 array of source points, 119

nonperiodic, 124 periodic, 119

astigmatic difference ASD, 417,418,419 astigmatism of a single spherical surface,

417 astigmatism of a single surface, 419 astigmatism of a thin lens, 418,419 asymmetric Fourier transform, 362 atornic ernission, 269 atornic energy states, 280

exited states, 282 occupation rule, 279

atornic polarizability, 311, 312

Babinet's theorem, 164, 166 background spectrum, 364 bandpass filter, 356 bandwidth,281,284

doppler, 287 homogeneous,284 Lorentzian, 289 mechanical, 287 natural, 287 quantum,287

beamsplitter, 102 Besselfunction, 147,163,164,381,382 Bessel function Jl(q), 149 Bessel function as transfer function, 386 birefringent, 229

436 INDEX

Blackbody radiation, 201, 269, 270 Boltzmann's, Wien's law, 277 depending on frequency, 275 depending on wavelength, 275 Radiance, Area, solid angle, 277

blackening curves, 368, 369 blocking function, 394 Bohr's model, 278 buildup principle, 279

K-shell, 280 L-shell, 280 M -shell, 280

boundary conditions, 250 Brewster angle, 214, 315, 316

C-ray, 15, 36 calcite, 230, 234 cavity

concentric, 76 condition for stability, 75 confocal, 76,293,295,296,299,304 matrix for eigenvalue problem, 72 reetangular shaped mirrors, 299 stability relation, 73 with round mirrors, 304

changing numbers, 425 charge density, 204, 310 chromatic aberrations, 420,421,422

negative,420 circular aperture, 146 circular mirrors, 304, 305 circular polarized light, 235, 238 Clausius-Mossotti equation, 314 coherence, 183

condition, 187 extended source, 189 intensity fringe pattem, 185 interval, 193 length, 202 two source points, 183 visibility, 192 Young's experiment, 184

coherent light, 390 coma

negative coma, 414 positive coma, 414

commercial microscopes, 44 complementary screens, 164, 166 complex cfft, 372

complex dielectric constant, 309 complex Fourier transformation, 337, 338,

341,362 complex notation, 85 complex refractive index, 309, 315, 320,

321 concave mirror, 70 concave spherical mirror, 69 condition for laser action, 289 confocal cavity, 293

beam parameters, 293 beam waist, 295-297 wavelength at center, 295 wavefront at mirror, 296 wavefront of beam and mirror, 295

confocal resonator, 300, 305 conjugate points, 9 constructive interference, 89, 91, 92, 96,

97, 102, 111, 114 convex single refracting surface, 12, 14 convex spherical mirror, 71 convex spherical surfaces, 9, 14, 19 convex-plane lens, 66 convolution

integral342, 379 product of two functions, 342 spread function, 389

convolution with the spread function, 389 coordinates for the derivation of Fresnel's

formulas, 210 Comu's spiral, 180 critical angle

phase shift, 217 reftected and transmitted intensity, 221 total reftection, 225

crossed polarizers, 236, 237, 243, 244 current density, 320, 321 current density vector, 204, 310 curvature of the wavefront, 295 cylindrical coordinates, 305 cylindricallens, 380

Damping term, 312 denominator, 121 dense medium, 214,217, 313 density of the oscillators, 313 dependence on (}, 107 destructive interference, 89, 91, 92,96-98,

103

dielectric circular waveguide, 264 dielectric constant, 312 dielectrics, 310 differentiation, 432 differentiation "space", 206 differentiation "time" a;at, 206 differentiation operation, 206 diffraction

amplitude grating, 153 circular aperture, 146 circular opening, 131 circular stop, 133 echelette grating, 157 on an edge, 174 factor, 151 far field, 136 Fraunhofer, 136 generated wavelets, 129 grating, 150 incident light under an angle, 156 Kirchhoff-Fresnel Integral, 129 losses, 298 on slit, 138-140, 142, 170, 172 3-D graph of reetangular aperture, 145 3-D graph of round aperture, 149

discrete Fourier transform, 358 discrete 1ength coordinates, 348 dispersion of light, 7 Doppler broading, 385

line shape, 286 line width, 292

double slit, 155 double-sided step function, 336, 338, 339 drift velocity, 320 Drude model, 320

Rehelette grating, 157, 159 eigenvalue problem, 72 Einstein coefficient, 285

coefficient of stimulated absorption, 288

probability coefficient, 272 electrical field vector, 204, 310 electrical polarizability, 310 electromagnetic theory, 203 ellipsometry, 329 elliptically polarized light, 235, 238, 239,

328 emission of light from 86Kr, 200

INDEX 437

energy density per frequency, 270 energy levels, 280 energy state, 280

transitions between states, 282 evanescent wave 226, 227

attenuation factor, 227 penetration depth, 227, 325 and index of refraction, 230

excited state, 282 extended source, 189, 194 extinction index, 312 extraordinary indices of refraction, 229 eye,2

Fabry-Perot, 76, 113-115, 245 cavity, 290 etalon, 113 plates, 245 spectrometer and resolution, 116 transmission depending on D, 115

far field approximation, 134 Fast Fourier transformation, 333, 334 Fermat's princip1e, 2, 5

law of refraction, 5 optimum path, 3 optimum time, 6 velocity for travel, 4

Fiber optics waveguide, 262 Bessel function solution, 263 determination of k, 264 periodic exponential solution, 262

Fizeau fringes, 104 focallength, 34 folding of the Fourier transform spectrum,

351 folding of the spectrum, 352 formula for summation, 120 formula for the summation process, 110 formulas, 429 Fourier integrals, 331 Fourier series, 370 Fourier transform integral, 364 Fourier transform spectrometer, 201 Fourier transform spectroscopy, 331, 346,

347 apodization, 360 folded spectrums, 356 high resolution, 357 large optical path difference, 355

438 INDEX

Fourier transform spectroscopy (cont.) Michelson interferometer, 347

Fourier transformation, 142, 331, 342, 368,384

asymmetric, 362 discrete length and frequency

coordinates, 350 fast, 335 functions 1/(1 + x2 ) and ne-2"", 333 gauss function, 332 general, 343 numerical, 333 real, 334, 341, 358 sample interval, 350, 357 two transformations, 374

Fourier transformation using analytical functions, 332

Fraunhofer diffraction, 136, 137 Fraunhofer observation, 134 frequency Coordinates, 348 frequency domain, 393 frequency spectrum, 359 Fresnel, 93, 136 Fresnel diffraction 134, 170

on an edge, 173 integrals, 172 on a slit, 171

Fresnel number, 298 Fresnel's double mirror experiment, 91 Fresnel's formulas, 78, 209, 314

as function of angle, 213, 215 parallel case 209, 315 perpendicular case, 212 transrnission coefficient, 216

Fresnel's mirror, 93 fringe pattem, 87 fundamental mode, 295, 304 fused quartz, 319

Gain of the beam, 290 Galilean telescope, 46, 48 Gaussfunction,332,342,343 Gaussian beam, 293 Gaussian line shape, 286 generated wavelet, 129 Geometrical construction, 18 geometrical construction, 15, 17, 18, 29,

31,69, 71 geometrical optics, 1

graphical constructions, 25, 30, 36 graphical method, 69 graphs and matrices in Mathcad, 425 grating, 150

number of lines N, 153 openings d, 153 periodicity constant, 157 resolution, 152 side maxima, 152 side rninima, 150, 154

Green's function, 130 guided waves, 255

Half-wave plate, 231, 236, 243 phase shift cp, 231

harmonic waves phase factor, 78 Superposition, 80, 204 two depending on space and time

coordinates, 80 Heidinger interference fringes, 101, 104,

105, 106 high frequency region, 322 high resolution spectroscopy, 355 Holography, 395

different waves, 397 hologram, 396 product of real image, 397 real image, 396 recovery, 396, 397 size of hologram, 398 transmission curve, 396 under an angle, 397 virtualimage,396,397

homogeneous equation, 321 Huygens' principle, 127, 129 Huygens' wavelets, 127, 367

Image of one bar, 380 one round object, 381 two round objects, 382, 389 two bars, 381

Image formation using wave theory, 367 amplitude function, 377 aperture function, 377 circular lens, 381-383 convolution, 379 image formation, 378, 390

image forming process, 374 impulse response, 379 one bar, two bars as object, 381 one round, two round as object, 381,

382 pair of Fourier transformations, 377 resolution, 391 spread function, 378 Summation process, 375, 376 transfer function, 393

imaginary part, 312 imaging with coherent light, 392 irnpulse response, 379 incident intensity, 111 incident light under an angle 1/1, 156 incoherent light, 378 induced absorption, 271 induced dipoles, 310 induced emission, 271 inhomogeneousequation,321 intensity, 86, 124

complex notation, 85 normalization, 86 time average, 85

intensity fringe patterns, 185 intensity pattern, 379 interference, 77 (minima)

air gaps, 98, 99 factor, 151 fringes, 111 maxima, minima, 81 pattern generated by two sources, 81 pattern of N sources, 122

interferngram function S(y), 348 interferograms, 345 interferometry, 87

amplitude dividing, 94 Fresnel double mirror, 91 Lloyd,90 Michelson, 101 model, 87 Newton's Rings, 99 plane parallel plate, 95 wavefront dividing, 87 wedge shaped, 97 Young,88

internally reflected components, 242 inverse transformation, 334, 336, 338, 339 inverted irnage, 2

INDEX 439

isotropic medium, 79 isotropic nonconducting medium, 208

Jones matrices, 242, 243 Jones vectors, 242

Kepler telescope, 45, 46, 48 Kirchhoff-Fresnel integral, 129, 130, 137,

368 Kramer-Kroning model, 319

Labels for energy Ievels, 279 Iaser beam expander, 48 Iasers, 269

active medium, 291 amplification factor, 289, 290 gain of the beam, 290 spontaneous transition, 288 stimulated emission, 287, 288, 289 stimulated transition, 289 two-level systems, 288

lateral magnification, 17 lateral spherical aberration, 408, 410 law of reflection, 2 law ofrefraction, 1, 2, 5, 220, 314 left polarized light, 235 length of wavetrains, 200 length units, 12 Lens

negative f, 31 plane-convex, 66

lenses, 1 less dense medium, 227, 311 life-time, 282 lifetime T, 282, 284 linear polarized light, 238 Lloyd's mirror, 90, 93,94 longitudinal modes, 291 longitudinal spherical aberration, 405,

407,409 Lorentz correction, 314 Lorentzian line shape, 282, 284, 286 Lorentzian line width, 291 lossless dielectrics, 328 low frequency region, 321

Magnetic field vector, 204, 310 magnetic quantum number and

degeneracy, 278

440 INDEX

rnagnification, 17, 18,25,36,37,43,44, 46,68

rnagnifier, 37, 47 angular rnagnification, 40 rnagnifying power, 40 virtual irnage at infinity, 39, 41 virtual irnage nearpoint, 39

rnagnifying power, 40, 43 rnany electron atorns, 278 Mathcad

plotting a graph, 425 rnatrices, 426 rnatrix elernents, 53 rnatrix rnethod, 49

application to two-and three-lens systerns, 47

glass sphere, 58 hernisphericallens, 58 principal planes, 51 refraction rnatrix, 49 thick lens, 55 translation rnatrix, 49 two lenses in air, 59 two thick lenses, 62

rnaxirna, but only N- 2, 123 Maxwell's equations, 203, 309, 320 meridional (vertical) plane, 418 Michelson interferorneter, 104

dependence on 0, 107 nonnormal incidence, 104 nonnormal incident light, 104 normal incident light, 101 Superposition of two cosine waves, 344

Michelson's stellar interferorneter, 195, 196, 197

two patterns, 195 rnicroscope, 42, 47,48

rnagnification 43 rnagnifying power, 43 near point configuration 41-43 slides, 98 virtual irnage at infinity, 43

rninirnurn deviation, 8 rnirror equation, 67 rnirrors, 1 rnirrors for laser cavities, 248 rnode

in a dielectric waveguide, 262 formation, 258

nurober of nodes, 246 propagation,245,257 reetangular box, 247 restrictive conditions, 257 (TM) rnodes or p-polarization, 263 (TM) rnodes or s-polarization, 263

rnodes in a dielectric waveguide, 262 rnodes of the reetangular box, 247 rnonochrornatic light, 198 M Wilson observatory, 196 multiple layer filters, 254 multiple lens systern, 49 mutual orthogonal triad, 207

Natural ernission line width, 285 negative chrornatic aberration, 420 negative corna, 414 negative crystal, 230, 234 negative lens, 33 Newton, 7 Newton's rings, 99, 100 Newton's work, 77 node lines, 246 noncornrnutation of rnatrices, 65 nonconductive medium, 208 normal and anornalous dispersion, 313 nurober of nodes, 246

Object arnplitude function h(y), 377 object focus, 12, 16, 24, 32 object point, 10 object positions, 37 occupation rule, 279 one electron atorn, 278 one oscillator, 316 one round object, 381 optical axis, 230 optical axis is the fast axis, 230 optical constants, 309, 310, 320, 330, 364 optical constants of rnetals, 322

high frequency region, 324 low frequency region, 323 skin depth, 326

optical constants n and K, 362 optical instrurnents, l, 35 optical rnaterials, 229 optical path difference

Fresnel's double rnirror, 92 Lloyd's rnirror, 91

Newton's rings, 101 Michelson interferometer, 102 plane parallel plate, 96 wedge, 97 Young's experiment, 89

optically denser medium, 78 order of interference, 81 ordinary index n0 , 230 oscillator expressions, 316 oscillator model, 311, 312

damping term, 312

Pair of Fourier transform integrals, 348 pair of Fourier transformation, 377 paraxial approximation, 15 paraxial theory, 1, 10 paraxial wave equation, 293 Pauli principle, 279 periodic arrangement, 73 periodic array, 124, 169 periodic set of slits, 150 perrnittivity, 204, 310, 311 perpendicular case, 212 PF-ray, 15, 36 phase difference /1, 109 phase factor, 78 phase jump, 78 phase velocity, 208 phase velocity in vacuum, 79 planar waveguide, 255, 259 Planck, 269 Planck's radiation law, 271, 276, 288 plane mirror, 66

virtual image, 66 plane parallel plate, 95, 108, 111, 112

normal incidence, 112 phase difference, 8, 109 summation of the reftected amplitude,

109 transmission, 111 transmitted amplitudes, 109 transmitted intensity, 112, 114

plane plate waveguide, 266 characteristic deterrninant, 267 traveling waves, 255

plane wave, 204 depending on space and time

coordinates, 80 plasma frequency, 312, 322

Poisson spot, 133 polarization, 310 polarized light, 228

INDEX 441

linear circular elliptical, 238 population inversion, 287, 288 positive chromatic aberration, 420 positive coma, 414 positive crystal, 230, 232, 234 positive lens, 25, 28, 30 Poynting vector S, 220 Poynting vector in vacuum, 207 principal angle, 315 principal axis transformation, 241 principal planes, 56, 60, 62, 63, 66 principal quantum number, 278, 279 prism, 7

angle of deviation, 7

Quality factor, 284 quantum emission, 269 quantum mechanical model, 285 quarter-wave plate, 237, 244 quartz,230,232,234 quasimonochromatic light, 198

Radius of curvature, 68, 70, 296, 407 radius of curvature p, 407 Raleigh-Jean law, 271 random arrangement, 167 random arrangement of source points

array, 123 grating, 167

random array, 168, 170 random phase angles, 124 randomly distributed, 124 rate equations, 288 ratio rs/rp, 316, 329 Rayleigh criterion, 162, 387 Rayleigh distance, 163, 387 Rayleigh-Jeans law, 270, 272, 276 Real Fourier transformation, 341 real image, 29, 396 real object, 16, 29, 32 real object function, 372 real object point, 11 real objects, 15, 20 recording of the interferogram, 395 recovery of image, 396 reetangular aperture, 143, 145

442 INDEX

rectangular-shaped mirrors, 299 references, 433 retlectance R, 224 reftected intensity, 112, 114 reftecting cavity, 72 reftection, 111

amplitude, 109 intensity, 112, 114, 222, 225 power, 51 reftected and transrnitted, 270

reftection coefficients, 215, 218,314 reftection coefficients with absorption, 327 reftection in a rnirror cavity, 73 reftection measurements, 316 refracting powers, 51 refraction, 49 refraction matrix, 50 refractive index, 2, 4, 310 relaxationtime-r, 321 resolution

coherent light, 391 grating, 160 incoherent light,, 386

resolving power, 117, 162 resonance

condition,300,304 mode, 114 wave numbers, 317

resonance of vibrations, 317 resonator parameters, 73 restricting, 280 right polarized light, 235 rotation, 65 rotation of the coordinate system, 241 round aperture, 14 7 round object, 382 round objects, 383

Sagittal (horizontal) plane, 418 sagittal coma Cs, 413, 415 sampling interval, 351 scalar wave equation, 77, 129 Schawlow and Townes, 291 selection rules, 280 Sellmeier formula, 318 shape factor, 411 sign convention, 11, 68 simulations of interferograms, 350

sine function and apodization, 359 single refracting surface, 405 single surface, 406 single-sided step function, 335, 338 size of hologram, 398 skin depth, 325, 327 slow axis, 230 small angle approximation, 10, 16, 135,

152 solution ofthe eigenvalue problem, 241 spatial coherence, 183

frequencies, 368 wavelength, 370 waves, 368

spherical aberration, 405, 406, 408, 411, 412

spherical rnirrors, 67 concave spherical rnirror, 69 convex spherical rnirror, 71 graphical method, 69 magnification, 68 virtual image, 72

spherical surfaces, 1, 9, 11, 15 conjugate points, 9 geometrical construction, 16 image focus, 12 image forrning equation, 11 image point, 10 magnification, 16, 18 real object, 16 sign convention, 11 virtual image, 16, 18, 19 virtual object, 17, 18, 19

spherical thick lens, 58 sphericalwave, 78,129 spin states, 279 spontaneaus ernission A2~o 271, 285, 289 spread function, 378, 384, 390 standing wave conditions, 245 static conductivity, 321

gold, 324 Iead, 324 nicke!, 324 silver, 324

Stefan-Boltzmann law, 273, 274 step function, 332 step grating, 176 stratified media, 248

antireflection coating, 254 plate ofthickness d = J...f2n 2 , 253 two interfaces at distance d, 249

Superposition of two cosine waves, 82 of two double slit pattems, 186 of wavetrains, 199 principle, 78, 206

susceptibility x, 311

Tangential coma Cr, 414, 415 TE modes, 258, 266 telescope, 44

Galilean, 46 Kepler's, 44

temporal coherence, 198 length of wavetrains, 200 quasimonochromatic light, 198 Superposition, 198 wavetrains, 198

Theory of Color, 7 thick lens, 51, 54

concentration lens, 66 focallength, 54 matrix, 51 two hemispherical 62, 63 two thick lenses, 61, 63 virtual image, 66

thin lens, 24, 33, 53, 54, 408, 412 different media, 34 equation,23,30 image focus, 24 magnification, 25 matrix, 52 model, 1 negative lens, 31 object focus, 24 positive lens, 26 transformation to principal planes, 54 two different media, 33 two thin lenses, 36 two thin lenses in air, 59 virtual image, 29, 32 virtual object, 29, 32

INDEX 443

three lens system, 42 three-levellaser, 292 threshold condition, 290-292 time average, 85 time-dependent, 86 TM modes, 261 total intemal reflection, 255, 259 total reflection, 224, 226, 227 transmission coefficient, 219 transfer function, 384, 393

function for (Bess/arg), 394 function for (sinx/x), 394

translation matrices, 49, 51 transmission intensity, 223, 225 transmittance T, 221 transposed matrix, 65 trigonometric formulas, 430 two lens system, 36 two lenses in air, 59 two round apertures, 383, 387 two-level system, 288

Uniaxial crystals, 229 unit matrix, 65

Velocity for travel, 4 vertex of the spherical surfaces, 56 violet catastrophe, 271 virtual image, 16, 29, 32, 67, 396, 397 virtual object, 30, 32 visibility, 192-194 visibility for two point sources, 192

W ater waves, 81 wave equation, 240, 311 wavefront division, 87 wavelength, 275, J..., 140 wavetrains, 198, 283 wedge shaped air gap, 97 Wien's displacement law, 274

Young's experiment, 88, 89, 94 Young, Thomas, 77