= 1+ t D = ρ. = ε χ o

Transcription

1 Optical Diffractin While all electrmagnetic phenmena are escribe by the Maxwell equatins, material interactins, surces, bunaries an receivers iffer acrss the frequency spectrum. This nte briefly reviews the electrmagnetic thery f iffractin an iscusses hw it is applie in ptical analysis. Maxwell Equatins an Wave Equatins The Maxwell equatins are E = B H = J + D D = ρ B = 0 E is the electric fiel, D is the electric isplacement, B is the magnetic inuctin, H is the magnetic fiel, J is the current ensity anρ is the charge ensity. The fiels are further relate by the material equatins D = ε E + P B = µ H + M where P is the plarizatin f material an M is the magnetizatin. In mst ptical materials, M=0 an P is unctin f E. The simplest an mst cmmn case is Then where an P = ε χ e D = ε E ε = ε ε r E ε r = 1+ χ e As a result f the high frequencies an small wavelengths f ptical fiels, charge interactins with the ptical fiel invlve quantum mechanical effects which cannt accurately be analyze by cntinuus mels. In ptical iffractin prblems, ρ is always neglecte. In principle, J is als neglecte, althugh we iscuss belw hw the current

2 ensity can be use frmally t a a lss cmpnent t the pemitivitty, ε. We als nte that E an D nee nt be clinear, meaning that ε may in general be tensr value. Using the material relatins, we substitute in the Maxwell equatins t fin the wave equatins E = µ ε H = µ ε The equatins are reuce t a simpler frm by the vectr ientity Frm the Maxwell equatins we knw that E H A = A A 0, ε E = E ε + ε E = where we have assume fr the mment that ε is scalar value. Thus, an E = E lg ε b g E + E lg ε µ ε E = 0 A meium in which ε = 0 is hmgeneus. Meia in which this is nt the case inclue ptical fiber, grae inex lenses, an vlume hlgrams. In istrpic meia, the wave equatins are Furier Analysis f the Wave Equatins E µ ε E = 0 H µ ε H = 0 In a typical iffractin prblem, ne is given the fiel n a particular bunary r surce an ne wishes t calculate the fiel n a ifferent bunary r in a ifferent regin. This is a linear prblem in the sense that the fiel pruce n the utput bunary by a linear superpsitin f fiels n the input bunary is the linear superpsitin f the utput fiels pruce by the iniviual input fiels. Frmally, ne might express this situatin as fllws:

3 k p frm the input bunary t the utput l f1arf + farfq = l f1arfq + l farfq. Fr the iffractive transfrmatin T L bunary, the transfrmatin f the fiel f r = f1 r + f r satisfies T T T Since the iffractive transfrmatin is linear (an in general als shift invariant), Furier analysis is extremely useful in analyzing iffractin prblems. One slves a iffractin prblem using the Maxwell r wave equatins an bunary cnitins. Nte that iffractin is nt a linear prblem frm all perspectives. Fr example, if ne cnsiers the input t be the bunary cnitin an the utput t be the fiel scattere by the bunary, the utput fr a linear superpsitin f inputs is nt necessarily a linear superpsitin f the iniviual utputs. This situatin arises, as an example, when ne uses arrays f surface features t mulate a scattere fiel. The aitin f new surface features es nt necessarily mulate the scattere fiel in a linear fashin. The ease with which nnlinearity can be intruce in the spatial main cntrasts with the ifficulty f intrucing nnlinearity in the tempral main. The prcess f generating the ptical fiel is highly nnlinear but nce the fiel is generate an prpagating, effects which cuple ifferent ptical frequencies are relatively rare. Fr this reasn, the use f Furier analysis t characterize the tempral an spectral respnse f ptical systems is nearly universal, while the use f Furier analysis in the spatial main is smewhat mre limite. The linearity f the tempral respnse is a mixe blessing, hwever, when ne cnsiers the ifficulty f accurately characterizing tempral aspects f the fiel. Spatial istributins an bunary cnitins can be measure t extrarinary accuracy, but the exact tempral behavir f ptical fiels is funamentally unknwable. Analysis f iffractin prblems generally fcuses n quasi-mnchrmatic fiels, which ne treats as single frequency Furier cmpnents but which ultimately are stchastic. In the Furier main, the wave equatin fr the electric fiel takes the frm c h k k E k, ω = µ ε k, ω ω E where * represents a multi-imensinal cnvlutin peratr. The pssibility f a cnvlutin ver tempral frequency is nt realize in practice an pints ut law in the materials respnse equatins. In general, the ielectric respnse f a linear material is characterize by a linear relatinship in the frequency main an a cnvlutin in the time main, e.g. s that D ω = ε ω E ω t z ε D t = t E t t t (1)

4 The upper limit f integratin at time t is necessary t maintain causality. The time integrate respnse is cnsistent with the physical situatin, in which the plarizatin vectr escribes the micrscpic respnse f the charge in the material t the riving fiel. One es nt expect that this respnse will be instantaneus in the time main. This means, as iscusse abve, tempral aspects f the iffractin prblem are linear an there is n nee t cnsier a cnvlutin ver tempral frequencies. The cnvlutin f the perimittivity an the fiel in the spatial frequency main is physically realistic, hwever. The respnse f the fiel t spatial variatins in ε is lcal in the spatial main an glbal in the spatial frequency main, in cntrast t tempral variatins in ε, where the respnse is lcal in the spectral main an glbal in the time main. The cnvlutin εak, ωf cω Eh thus represents the ptential nnlinearity f the iffractin prblem with respect t variatins in spatial istributins. Limiting urselves t hmgeneus materials, the Furier transfrm f the wave equatin is k k E = µ ω ε ω E. If ε is a scalar, this equatin has slutins nly if k = µ εω. Allwing fr the pssibility that ε is a tensr, slutins crrespn t thse values f k fr which the eterminant f the matrix peratr k k + µ ω ε ω 0 reuces the range f k frm three vanish. The equatin k k + µ ω ε ω = imensins t tw imensins. The surface efine by this equatin is calle the wave nrmal surface. In istrpic materials (materials in which the perimitivitty is a scalar), the wave nrmal surface is a sphere in k-space f raius ω µ ε. In anistrpic materials (crystals), the wave nrmal surface splits int tw sheets, s that there is are tw slutins fr k in almst every irectin. Prpagatin in ansistrpic materials is cnsiere in etail in Optical waves in crystals, by Yeh an Yariv. Each slutin fr k crrespns t an eigenvectr E. The irectin f E is the plarizatin. Fr the istrpic case, tw pssible plarizatins exist fr each k. In the general case, each eigenvectr crrespns t a ifferent value f k. Once the slutin fr Eak,ωf has been etermine, ne may inverse Furier transfrm t fin the general slutin j t k r Ear, tf Eak, fe aω = ω f kω The challenge f this representatin is that, espite the fact that k lies in 3D space, the relatinship between k an ω iscusse abve means that the cmbinatin f k an ω nly spaces 3-space an the Furier representatin nee actually invlve integratin ver three imensins.

5 Furier Analysis f the Diffractin Prblem Returning t the iffractin prblem, ur gal is t use the Furier representatin f the fiel t fin the fiel n the utput bunary given the fiel n the input bunary. f(x,y) g(x,y ) Suppse as an example that we cnsier the iffractin prblem sketche abve. We are given the fiel fax, yf n the input plane an we wish t calculate the fiel gax, y f n the utput plane. The planes cntain the x an y axes an are isplace alng the z axis. Given the linearity f the iffractin prblem with respect t time, as iscusse abve, there is n lss f generality in limiting ur analysis t quasi-mnchrmatic fiels. Once the transfrmatin is knwn as unctin f ω inverse transfrming fr the tempral respnse is straightfrwar. At a single frequency, the Furier representatin f the fiel is = jω t jk r E r,t e E k e k we inclue nly tw imensins in ur integratin fr k in view f the D range fr k iscusse abve. The relatinship between k an ω efine by the wave nrmal surface is calle the ispersin functin. Using the ispersin functin, we can always etermine ne f the fur variables ckx, ky, kz,ωh if the ther three variables are given. Suppse, as an example, that k z is the cnstraine variable (i.e. suppse that we are free t select k x an k y arbitrarily). Drpping the time epenence, the phasr fiel as unctin f space can be expresse Earf j k x j k y x y = Eakfe e exp j kzckx, ky,ω hz kxky () is the z = 0 plane. Then, accring t Eq. (), the fiel in Eq. () prvies an pening fr quick slutin f the iffracitn prblem. Suppse that the input plane cntaining f x, y this plane is

6 j k x j k y x y fax, yf = Eak fe e kxky. (3) is essentially the Furier We have rppe the plarizatin vectr fr simplicity, the effect f shifting t scalar fiels is iscusse briefly belw. Accring t Eq. (3), E k transfrm f fax, yf. Nw suppse that the utput plane is the z = plane. Accring t Eq. (), j k y c h j kx x y g x, y = E k e e exp jkz kx, ky, ω kxky. Writing these relatinships in terms f the Furier transfrm f fax, yf, Fau, vf, an k efining u = x k an v = y yiels π π an = = j π ux jπ v y f x, y F u, v e e uv z j ux j v y g x, y F u, v e π e π exp jk u, v, ω uv an gax y Cmparing f x, y, f, we see that the transfrmatin is linear an shiftinvariant with a transfer functin z H u, v = exp jk u, v, ω b g The impulse respnse f the iffractive transfrmatin g x, y = T f x, y is the inverse Furier transfrm f Hau, vf. In an istrpic space the ispersin relatinship is k = k 1 u λ v λ z where k = ω µε = π / λ. The wave nrmal surface in free space is a sphere, as sketche belw.

7 kz k kx ky The wave nrmal surface in free space. The transfer functin fr the free space ispersin relatinship is Hau, vf = expl π j 1 u λ v λ λ NM In many ptical systems, the transfer functin an the impulse can be simplifie by the paraxial apprximatin. This apprximatin states that spatial frequencies in the fiels f interest are such that the angular sprea f the iffracte signal abut the axis f prpagatin is small. The paraxial apprximatin crrespns t limiting k x an k y t the purple regin n the wave nrmal surface. Uner the paraxial apprximatin we assume that ver the spatial banwith range f interest kx, kx << kz. This apprximatin is vali if the separatin,, between the input an utput planes is much greater than aperture f interest in each plane. Uner the paraxial apprximatin, j π c h λ H u, v e exp jπ λ u + v The impulse respnse fr iffractin frm z = 0 t z = is the inverse Furier transfrm f the apprximate transfer functin, π j jπ ux jπ vy h x, y = H u, v e e uv c λ jπ ux jπ vy = e exp jπ λ u + v e e uv Our gal fr the remainer f this nte is t evaluate this integral. The limits f integratin are frm the maximum t the minimum spatial frequency alng each axis ver the small purple circle n the wave nrmal sphere. Despite ur assumptin that uλ, vλ << 1, we will assume that at almst all pints ver the aperture f interest ux, vy >> 1. This assumptin h O QP (4)

8 allws us t exten the limits f integratin in Eq. (4) t plus an minus infinity. x Cmpleting the squares in the expnents with the substitutins u = u + λ an v = v + y λ yiels π x + y i j jπ λ λ jπ λu + v i h x, y = e e e u v π x + y i j jπ λ λ jπ λu + v i = e e e u v 1 = jλ e π j λ e x + y jπ λ The use f this impulse respnse t analyze iffractin an imaging prblems is the fcus f Wilsn an GF. We will buil n this kernel thrughut the term. Finally, we briefly return t the questin f plarizatin, which we rppe miway thrugh ur iscussin. Other than ntatinal simplicity, there was n real reasn fr us t rp the vectr aspect f the fiel an we cul easily revise ur Furier analysis t inclue vectr cmpnents. The ifficulty with vectrs relates t assumptins abut bunary cnitins an spatial nnlinearity which we have avie in this nte. In cmputing the transfrmatin frm ne plane t the next, we have implicitly assume that the fiel in the input plane is a knwable bject. Unfrtunately, this is nt generally the case in ptical systems. T create knwn fiels, ne might try t mulate plane wave inputs using transmissin r reflectin masks. The mulatin f the fiel t create spatial istributins is a very cmplex subject. As iscusse abve, spatial mulatin is nt a linear prcess. The simplest apprximatin t mulatin, knwn as the Kirkhff bunary cnitin, invlves the assumptin that the fiel istributin after a mulatr is equal t the pruct f a transmissin factr fr the mulatr an the input fiel. This apprximatin ften wrks well, but is nt an accurate reflectin f the micrscpic interactin between fiels an surfaces. Mving beyn the Kirhff apprximatin generally invlves numerical meths. The inclusin f vectr plarizatin effects in these mels is cmplex but ften necessary fr accurate results. i