COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.

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1 C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH Introduction Th comb function is a rgularly-spacd array of Dirac dlta functions. W will first discuss th dlta function thn mov on to th comb function. W hav put som intrsting, but for us lss important, proprtis of th dlta function into a st of appndics at th nd of this chaptr. Th Dirac dlta function Th simpl dfinition is that th Dirac dlta function, dnotd by δ ( x), has th proprtis δ ( 0) = δ = 0 for 0 x0 x 0 δ x dx= A mor carful considration of its proprtis coms through considring a family of functions whos proprtis approach thos dsird of th Dirac dlta function in a limit. On such family is that of x x normalizd Gaussian functions σ σ. Th normalization is such that th intgral dx σ π σ π for all valus of th varianc σ. W can think of th dlta function as a vry tall and narrow Gaussian, δ ( x) Lt s s how it masurs up to our rquirmnts, δ ( x ) 0 δ x0 σ σ 0 σ π ( 0) lim 0 = x lim σ σ 0 σ π = = On dividd by zro is a vry larg numbr. σ σ π = lim = 0, whr x 0 is a constant. Th xponntial dis much fastr than σ grows. x σ 0 σ π σ 0 δ x dx= lim dx= lim = Thr is a littl mor to this story. σ Notic that w movd th limiting procss outsid th intgral. W ar not taking th limit of th function to gt th dlta function thn intgrating. W ar looking at th valu of th intgral for ach of th valus of th paramtr as th limit is takn. This is known as taking a limit in th distribution sns. W ar looking at a family of limits to th intgration. For furthr rading, look into (wll-)tmprd distributions. =

2 C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 On of th important proprtis of th Dirac dlta function is th sifting proprty, δ ( ) = f x x x dx f x Lt s look at th sifting proprty in our limiting sns, ( xx ) 0 σ f x x x dx f x dx δ ( ) = lim σ 0 σ π Th Gaussian is vry narrow and cntrd on x. Th only position whr th dlta function has a valu is at x0 = x. Thrfor, f ( x 0 ) can b rplacd in th intgrand by f ( x) which w can immdiatly factor out of th intgral. ( 0) ( 0) ( 0) x x x x x x lim σ lim σ lim σ σ 0 σ π σ 0 σ π σ 0 σ π f x dx = f x dx = f x dx = f x Th nd rsult coms bcaus th valu of th final intgral is unity. Th sifting intgral has a vry simpl but powrful intrprtation that coms from idntifying th dlta functions with points. What th intgral shows is that a function can b xprssd as a wightd sum of shiftd points. Fourir transform of th Dirac dlta function For convninc, w rpat th dfinitions hr, and also introduc som shorthand notation π Th forward transform: F.T. f ( x) = F f ( x) = f ( x) dx= F( ) π Th invrs transform: I. FT.. F( ) = F F( ) = F( ) d = f( x) π F δ x = δ x dx= δ x 0 dx= = If w tak th complx conjugat w s that π 0 { F * δ } δ δ F δ δ x = x dx= x dx= x = x 0 dx= = * + π + π + π 0 Summarizing: F δ ( x) = and F δ ( x) = W can crat mor rlations by transforming th rsults: δ ( x) { f x } = f ( x) But, for any function F F, so that { } = () F F F.

3 C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc3 of 8 F () = δ ( x) and, from similar argumnts, = δ ( x) Th Comb Function F. Thr ar svral applications that can b simply xplaind using th dlta comb dfind as comb( x) = δ ( xm) m= It will hlp to dfin comb math hr. This abstraction is usful for daling with priodic and point-form, i.. punctil, objcts in continuous Fourir transforms. Th rlations will allow us to writ down, in compact form, dscriptions that w will latr xpand upon. It is also straight forward to xpand th rlations from compact form into thir full form. Th -D comb function (brush?) is Fourir Rlations of Comb Functions ( x y) = ( x) ( y) = δ ( xm) δ ( yn) comb, comb comb A comb with unit-spacd tth transforms into itslf, m= = FT.. comb x comb n= On important Fourir rlation is that a shift transforms into multiplication by a linar phas factor ± πix0 ( ± ) ( ) FT.. f x x F 0 this shift thorm givs anothr usful form for th transform of th comb. ± im = = FT.. comb x π comb m= Anothr important Fourir rlation is scaling: magnification transforms into minification Shifting and scaling th comb function givs us, x x comb Using th prvious rlations w hav that x FT.. f = bf b b ( ) 0 δ 0 b = n= b x x nb xx0 FT.. comb = bcomb b b π ix0 ( )

4 C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc4 of 8 aking a continuous function into a punctil on is don by multiplication: comb = δ ( ) = δ ( ) = δ ( ) f x x f x x m f m x m f x m m= m= m= Priodic continuation is usually dscribd for functions of boundd support. It is important that w undrstand th mor gnral cas covring functions of larg xtnt or vn unboundd support. Th rason for this considration is th fact that a function cannot b of boundd support both in th objct and in th Fourir domain. W will concntrat on objcts of boundd support. Ths functions ncssarily hav Fourir transforms of unboundd support. Priodically continuing a function is don by convolving with a comb. comb = δ ( ) = δ ( ) = ( ) f x x f x x m f x x m f x m m= m= m= W s that nowhr in this dfinition is boundd support rquird. W should b awar of th fact that contributions from ach rptition of th function will contribut to th valu at ach point. This fact is, howvr, for us not important. Th Fourir transforms of th rlations ar m and [ ] FT.. f( x) comb( x) = F( )comb( ) = FT.. f x comb x F comb Rlation of th Fourir Sris to th Fourir Transform W bgin with th Fourir transform of an objct of boundd support. W thn crat a priodic function rplicating that objct. u x =comb x * u x Fourir transforming th priodic function givs a function that is punctil U( ) = U 0 ( ) comb W now brak th transform into priods and sum th priods, m m π i x π i ( ) = = m= m= 0 π U u x dx u x dx Using th handy rlation givs comb = m = π im

5 C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc5 of 8 π πimx comb U( ) = comb u( x) dx= u( x) dx Th factor lading th last xprssion coms from th scaling rlations whn applying a dlta function. Th ffct can b viwd as a consqunc of th tooth-thicknss in th comb. W s that th wights applid to th dlta functions in th comb ar th Fourir transform valuatd at spcific positions, πimx Th Fourir sris cofficints on th othr hand ar U m = u x dx m πimx C = u x dx Comparing ths cofficints to th wights of th dlta functions in th Fourir transform w hav Cm = U m Rlation of th Discrt Fourir Transform (DFT) to th Fourir Transform W again start with an objct of boundd support. This tim w crat a function that is both priodic and punctil with a priod tims th point spacing and tak th Fourir transform Th rplication rturns th Fourir sris rlation Th punctility forcs priodicity of th transform u( x) =comb( x) comb x u0 ( x) π ( ) U = u x dx m x π i U( m ) = u( x) dx

6 C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc6 of 8 ε j x jm πi i comb m π ε m= Cj = U( j ) = ( x) u( x) dx= u Notic th slight chang of th intgration rang. Th Discrt Fourir Transform cofficints on th othr hand ar U j = m= u m jm π i Comparing ths cofficints to th wights of th dlta functions in th Fourir transform w hav Encoding Phas in Fring Shifts U j = U j U j = ( ) ( ) W hav sn that th wav-front phas is ncodd in fring shifts for intrfromtric holograms. W will not xamin phas ncoding by using comb functions. W will us gnralizd harmonic analysis. Th dfinition of th comb function is ± comb( x) = π m= W mak th substitution φ ( η, ) x + π Th rsult is that an ffctiv phas function is associatd with diffraction ordrs φη (, ) φ (, η im ) π + imφη (, comb ) + = = π m m imx π π im Th rsult is a suprposition which w can intrprt as consisting of phas distributions associatd with various diffraction ordrs. APPENDIX A THE GAUSSIAN CONVERGENCE FACTOR AND THE DELTA FUNCTION In som cass whn intgrals don t convrg, w can still gt a maningful rsult by multiplying th intgrand by a convrgnc factor that contains som adjustabl paramtr. Th intgral is thn calculatd and th valu of th intgral takn to b that for som limit of th paramtr. Whn doing Fourir transforms, a handy convrgnc factor is th Gaussian. Lt s tak a look at th intgral

7 C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc7 of 8 π This is th Fourir transform of on. If w do th intgral in th classic fashion, w find that dx π π dx= π i At this point, w ar facd with valuating an oscillating xponntial with infinit argumnt. Instad of looking dirctly at this intgral, lt s look at th intgral π πa x dx Th Fourir cookbook tlls us that this intgral is πa x a dx= a π π Now w look at th limit as a 0. W s that on th lft sid, th intgrating factor gos to on and th intgral approachs our targt. Th right sid gos to our dfinition of th Dirac dlta function. lim a x π π π π dx= dx = lim a = δ ( ) a 0 a 0 a APPENDIX B FOURIER TRANSFORS AND THE GAUSSIAN CONVERGENCE FACTOR Th rason that th Gaussian convrgnc factor is so important is that w oftn dal with functions that ar not wll-bhavd nough to hav Fourir according to th strict dfinition. On of th common problms is that th function undr considration dos not di-off rapidly nough at infinity. W can gt around this problm by introducing a convrgnc factor and a limit. In th gnral sns, suppos w f x, but th function dos not di off rapidly want to tak th Fourir transform of th function nough. Instad of taking th transform dirctly by using th formula x π lim σ σ 0 σ π π f x dx, w calculat f x dx. As σ sts small, th convrgnc factor approachs unity. Howvr, for any finit valu of σ and if f ( x ) is not incras for larg valus of x, th Gaussian-rat di-off is sufficint to allow th xistnc of a Fourir transform. Th convrgnc factor is xactly th tool w usd to calculat th Fourir transform of unity. Unity dos not di off at all at infinity. APPENDIX C THE DELTA FUNCTION OF A FUNCTION

8 C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc8 of 8 A rlation that w will nd latr is that for a dlta function whos argumnt is a function. Th formula is ( xi ) '( x ) δ δ f ( x) = whr f ( xi ) = 0 f i A particularly simpl xampl is whn f ( x) = ax with a bing a positiv constant. Th formula applid =. a to that cas tlls us that δ ( ax) δ ( x) i

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation. Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform