December 1967

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 57, NUMBER 12 DECEMBER 1967

Fourier Synthesis of Multilayer Filters*ERWIN DELANO

Physics Department, St. John Fisher College, Rochester, New York 14618

(Received 21 April 1967)

A brief description is given of a method for approximate synthesis of multilayer filters consisting ofeither homogeneous layers or inhomogeneous films. The method is based on the fact that the amplitudereflectance is the approximate Fourier transform of the function 4U'(p)/U(p), where U(p) is the ef-fective refractive index as a function of the optical thickness p. Two forms of the sampling theorem areapplied to obtain explicit expressions for the film parameters in terms of the specified reflectivity at certainsampling values of the frequency. Numerical examples are included.

THIS paper is a brief summary of recent work byTthe author on the preliminary design of non-

absorbing optical interference filters composed of eitherhomogeneous layers or inhomogeneous films. A detaileddescription of this work, including the derivations whichare omitted here, is given elsewhere.1

The principal motivation for the work was the desireto obtain a simple, although approximate, formulationof the synthesis problem. This would enable themathematically oriented novice to gain useful insightinto the relationship between the spectral characteristicsof an arbitrary filter and its constructional parameters.The various graphical methods,2 and the work ofBrandt,3 Pegis,',' Knittl,6 and others have providedimportant steps in this direction. The method to bedescribed seeks to exploit the important Fourier-

* This work was partially supported by the National ScienceFoundation.

E. Delano, Ph. D. thesis, University of Rochester, June 1966.2 O. S. Heavens, Optical Properties of Thin Solid Films (Dover

Publications, Inc., New York, 1965), Sec. 4.10, p. 80.3 C. H. Greenewalt, WV. Brandt, and D. D. Friel, J. Opt. Soc.

Am. 50, 1005 (1960).4R. J. Pegis, J. Opt. Soc. Am. 51, 1255 (1961).

R. J. Pegis (Private communication, July 1962).Z Z. Knittl, Appl. Opt. 6, 331, (1967).

transform relation obtained by Pegis.5 This relationbrings out the similarity between an optical interferencefilter and a linear filter in communication theory,7 whichsuggests the use of the sampling theorem.8 Applicationof the sampling theorem does, in fact, permit an explicitsolution of the synthesis problem for either homogen-eous layers or inhomogeneous films. The solution is welladapted for preliminary design if used in conjunctionwith a suitable smoothing process.

NOTATION

The notation used is the same as that of Weinstein,9

except that the layers are numbered in the oppositeorder. In addition, several new symbols are introducedto simplify the algebra.

Consider a system of I plane-parallel homogeneous,dielectric films having refractive indices nj and geo-metrical thicknesses hj, where j= 1, 2, * * *, 1. The layersare assumed to be perpendicular to the z axis andnumbered from left to right (see Fig. 1). The bounding

7 H. Wolter, in Progress in Optics, Vol. 1, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, p. 157.

D. A. Linden, Proc. IRE 47, 1219 (1959).9W. Weinstein, Vacuum 4, 3 (1954).

1529

3 ERWIN DELANO

at oblique incidence, so that gl=g2= .. =1g= say,and define x= eig. Putting gl+1= 0 and calculating E -and Eo+ from recursion relations (1), we can show thatp and r will be of the form

INCIDENT I I I I ISUBSTRI

INDEX ° I j- j Q INDEXA. FIG. 1-. ttio sc -h Ie MA

FIG. 1. Notation scheme.

p=Pfl5 ax-2k;k=0

T=f1E akkX,k=0

where

media on the left and right of the multilayer are alsoassumed to be homogeneous dielectrics, having refrac-tive indices no and n,+,, respectively. Assume that planeelectromagnetic waves are incident from the left atangle sco with the z axis and are subsequently refractedat each interface in accordance with Snell's law. Thecorresponding angle within the jth layer is Scj, and theangle with the substrate is ssl+,. The incident light isquasimonochromatic of vacuum wavelength X.

If Ej+ and Ej- are the tangential components of theelectric field for the progressive and recessive waves,respectively, then Weinstein's recursion Eqs. (12a)become

E~j-f-= tfl-1-1(e-ioiEJ:-+rj_,ei iE J+) (1)

Rams,+= tj-1-1 (rj-1e-i9iE-+ eigiEj+),

where

gj= 2 7rX-1nj1 zj cosvSj;

nj cos Sj (TE wave)

nj secSvj (TM wave);

rj-l= (uj-n-"j) (1ij-1+2j)-';

j-= I-+j- r = 2uj.1 (- 1+ u 3j)-1.

Moreover, E1+±-= 0 and El+,+= 1. If a and q' denotethe amplitude reflectance and transmittance and ifR and T denote the corresponding flux reflectance andtransmittance, then,

I-= ;Eo+

R=16112; T= H'1r2 , (2)Ito

where the T used here is Weinstein's Tb.

FOURIER TRANSFORM RELATION

For the purpose of Fourier synthesis, it is mathe-matically convenient to introduce two new quantitiesp and T to replace a and T. These are defined as

p= (uo/u,+,)4eiGEo; 7= (uo/ul±,le iGEo+, (3)

where G=2;j.1 gj. From Eqs. (2) and (3), it followsthat

R/T= !p1 2 ; 1/T= 1T12 . (4)

Assume that all the layers are of equal optical thickness

f = II (1 -rj2)j=o

and where ak and a,, are functions of the rj.Recursion relations for the ak and a, are given by

Delano' (p. 27). If P denotes the effective total opticalthickness of the multilayer at oblique incidence, and ifA= 2/X, then

so thatwritten

P = E nj.,j cos.j = X(2r)-1G = GIrj=1

x= exp(i7rAiP/1). Therefore Eq. (5) may be

P (A) = a yZe72,rik,,11;k=0

(14) = E 0k 2rik/0k=O

(6a)

(6b)

where yk= ak/f, ok7 = ak/f and 0= 1/P. Thus p and T arefinite complex Fourier series in pu with period 0 andhaving only I harmonics. The expression for p is especi-ally useful because to a first approximation yk = rk whererk is just the Fresnel reflectance for the kth surface. Infact, the familiar vector diagram2 may be interpretedas an approximate graphical construction for p insteadof (R. In the case of many layers or large rk, the ordinaryvector diagram sometimes yields 1 611 > 1 which cannotbe interpreted physically, since (R is subject to theconstraint 61 (R 1. Since p is not subject to any suchconstraint, it is preferable to interpret the resultant asa p vector instead of an (R vector in order to increase theusefulness of the vector diagram.

The case of an inhomogeneous film of total opticalthickness P may be easily treated by letting I -> oo.This implies that 0= 1/P -- oo, so that p(,4) becomes anaperiodic function of Au for an inhomogeneous film.Applying a heuristic procedure similar to that used inthe elementary derivation of the Fourier integralformula,10 we can show that for an inhomogeneous film

p(A)=y F(p)e7 211iAPdp;

F(p) = a p(1.)e`XiyPd1.,

(7a)

(7b)

10 E. H. Linfoot, Fourier Methods in Optical Image Evaluation(Focal Press, London, 1964), Appendix, p. 75.

(5)

1530 Vol. 57

FOURIER SYNTHESIS OF MULTILAYER FILTERS 1531

where

p= n(z) c9s n(z)dz

is the effective optical path.Equation (7a) and (7b) are valid for homogeneous

layers as well as inhomogeneous films if F (p) is regardedas a distribution"' instead of as an ordinary function.Relations similar to Eqs. (7a) and (7b) but involvingT(,I) may be derived, but these are not useful in approx-imate synthesis. Comparing Eq. (6a) with Eq. (7a) wesee that yk for a homogeneous layer corresponds toF(p)dp for an inhomogeneous film. To a first approxi-mation,

yk= rk= (u, 1 - U)(Uk-1+Uk)' = -2 (U'/U)dp,where U(p) is the functional form of u(z) when thelatter is expressed as a function of p. The prime denotesdifferentiation with respect to p. Therefore, to the firstorder of approximation

F(p)= -PU(p)/U(p) (8)

from which

U(p)= U(O) exp[-2f F(p)dp]- (9)

Essentially this same result was originally obtained byPegis5 by finding an approximate solution for (R fromthe Riccati differential equation for the inhomogeneousfilm.

FOURIER SYNTHESIS OF IN-HOMOGENEOUS FILMS

It would appear as if Eqs. (7b) and (9) directlyprovide a first-order solution to the synthesis problem,i.e., given p(gi) we could calculate F(p) from Eq. (7b)and then obtain U(p) from Eq. (9). This is not the case,because (a) F(p) must be real and (b) F(p) is subject tothe constraint that F(p)_=0 for p<0 and p>P. There-fore p(,u) cannot be chosen arbitrarily. Condition (a)implies that p(-,i)=p(,) where the bar denotes acomplex conjugate. Condition (b) implies that thesampling theorem is applicable, so that the functionp(,u) is completely determined by its value for certainvalues of g, i.e., the "sampling points."

For the case of an inhomogeneous film, the samplingtheorem is slightly modified from the usual form8' 12

because the frequency spectrum F(p) of p(,A) is notcentered at p= 0. For inhomogeneous films (see Ref. 1,p. 87)

p(,A)=e-7'i1 E (-1)-p(mP1) sinc(,uP-m), (10)

1l A. Papoulis, The Fourier Integral and its Applications(McGraw-Hill Book Co., New York, 1962), Appendix 1, p. 269.

12 S. Goldman, Information Theory (Prentice-Hall, Inc., NewYork, 1954), Secs. 2.1, 2.2; see also Appendix XII, p. 368.

where sincy= (7rA)-l sinmrg. In order for Eq. (10) to bevalid, the values p(mP') must satisfy the conditionsp(-mP') = (mP-') and 2m-o0 1 p(mP') I < oo . Sub-stituting Eq. (10) into Eq. (7b) we obtain the corre-sponding expression for F(p),

F(p) = P-' rect(pP-'- ') E p(mPl)e 27rimpIP

where{ IP <12

rectp = >2

(11)

It is convenient to express p(li) in the form p(,4)-A (,)eiOP(). Then A (/i) and ,6(,u) must satisfy A (-i1)=A(,u))0 and V(-,u)=- '(,u)+2k7r, k=0, At1, **on account of p(- A))= P(A). Substituting into Eq. (11)and simplifying, we obtain

F(p) = P-' rect (pP-'-4 ) { A (0) cosq, (0)

00

+2 57 A (mP-') cos[2rmpP'+V/(mnP-1)]). (12)m=-

Equation (12) and (9) solve the problem of synthesis tofirst order for an inhomogeneous film, except for theelimination of Gibbs's oscillations which appear atdiscontinuities in the function p(,u). Such oscillationscan be greatly reduced by using the smoothing tech-nique to be described in another section.

Although only the case of equally spaced samplingpoints is discussed in this paper, it is possible that otherkinds of sampling8"13 may prove more useful in design.The resulting formulas, however, will probably be morecomplex.

FOURIER SYNTHESIS OFHOMOGENEOUS LAYERS

For the case of homogeneous layers of equal opticalthickness, the sampling theorem for the periodic func-tion p(,u) takes the form (see Ref. 1, p. 137)

Nip(,U)= e-ri,-P E erriM1Z+lp (m0/l+ 1)

mu N

sin ( (1+1)w7r[El'-was (1+ 1)-1]}X- n

(1+1) sin(7r[,uS-1-m(1+1)-l2}'(13)

where N, is taken to be N if 1= 2N (even number oflayers) and taken to be (N+ 1) if I= 2N+ 1 (odd numberof layers). The function F(p) obtained by substitutingEq. (6a) into Eq. (7b) is

(14)I

F(p) = E yka(p-k6-'),k=O

13 R. Bracewell, The Fourier Transform and its Applications(McGraw-Hill Book Co., New York, 1965), Ch. 10.

December 1967 1531

ER\VIN DELANO

where 3(p) is the Dirac 8 function." The yM can beexpressed in terms of the sampling values of A (,M) and4'(,u) by the equations (see Ref. 1, p. 137)

IV

Yk= (I+))' (A (0) cos#(0)+2 E A (mO/l+ 1)

XcosE27rkm (1+ 1)-'+#(mO/l+ 1)]} (ISa)

for l-= 2N and

N= (1+ 1)-1((0) cos& (0) + 2 E A (nz0/l+ 1)

X cos[27rkmt ( 1+1)-'+V(melO/l+ 1)]

+ ( 1)kA (0/2) cosVt(0/2)} (1Sb)

for 1=2N+1. Because of symmetry, 4'(0) and xt (0/ 2)must be (generally different) multiples of 7r. Equations(15) solve the problem of synthesis to first order forhomogeneous layers. Substituting Eq. (14) into Eq. (9)and formally integrating, we obtain

1tk+1 =e 102yk, k = 0 1, ) 1. (16)

SMOOTHING PROCESS

The sampling theorems (10) and (13) are essentiallyinterpolation formulas. They are mainly useful to thedesigner if p(p) varies "smoothly" between samplingvalues. Otherwise, the smooth curve connecting thesampling values will have little resemblance to theactual curve. For this reason a smoothing technique isneeded to reduce undesirable behavior such as theGibbs phenomenon (see Ref. 14, p. 217) which appearsat "discontinuities" in p(,Q). Although the smoothingprocess must necessarily change the sampling values,it should keep the changes small. In addition, it mustkeep A (0) rigorously fixed, since incident medium andsubstrate are usually specified.

A smoothing technique which is often (but notalways) useful is described in Ref. 14 (p. 225). Considerthe function

P (,) =P rectpzP*pQ(,), (17)

where the star (*) denotes the convolution integral ofthe two functions. The function $(g) is smoother thanp(,p) because it averages out the frequencies correspond-ing to p= 1/P and higher, which are the ones responsiblefor Gibbs's oscillations. Evaluating the convolutionintegral in Eq. (17), we obtain

(A) = F(p) sinc(pP')e- 27i-"dp,

R

(18)

so that the smoothing process replaces the functionF(p) by F(p) sinc(p/P). The smoothing process shifts

11 C. Lanczos, Applied A nalysis (Prentice-Hall, Inc., EnglewoodCliffs, N. J., 1964) Ch. IV.

A (0) fromn its specified value, but this shift can becompensated for. In the case of an inhomogeneous film,the final function is (see Ref. 1, p. 100)

PF(P )= {F(p)+1.696foP F(p)[1-sinc(pP-h)]dp}Xsinc(pP1) rect(pP-'-1). (19)

In the case of homogeneous layers, the final values PFk

are (see Ref. 1, p. 96)

1-1YFkM= (Yk+[ Z sinc(ml11)j-1

m=0

I

X { E ym[l-sinc(rn-h)]}) sinc(kh-'),

k==0, 1, *--,1 (20)

Note that since sinci = 0, this smoothing processreduces the number of layers by one.

NUMERICAL EXAMPLES

As a first example, consider a 9-layer filter which isto have a reflectance peak of 90% at Xo= 5000 A andlow reflectance elsewhere except for periodic repetition.For computational simplicity assume that light isincident normally on the film. Also assume thatno= n1 0= 1.52. Since N= 4, there are six samplingpoints. Putting the reflectance peak at po=0/ 2 , thesampling values of A(,A) are chosen as follows: A(O)= | p(0) = 0 since the incident medium and substratehave the same refractive index; A (0/2)= I p(0/2)1= (R/T)i= (0.90/0.10)'= 3. For the remaining ampli-tudes, we choose A (mO/10) = 0, rn= 1, 2, 3, 4 in order tominimize the reflectance elsewhere. The only samplingvalue of #(ti) which must be specified is #(0/2) whichcan be set equal to either zero or 7r. If 4'(0/2)=0,application of Eq. (15b) yields

yS= (-1)iE3/10, k=g, 1, appoxa9.

Substituting, into Eq. (16) glives the approximate

FIG. 2. Nine-layer, unsmoothed, high-reflectance filter. Solidcurve: exact evaluation using refractive indices obtained fromfirst-order synthesis. Dashed curve: first-order prediction basedon sampling procedure.

1532 Vool. 57

FOURIER SYNTHESIS OF MULTILAYER FILTERS

R a

0.4 a

00 2 4 6aIN MICRONS-

FIG. 3. Nine-layer, smoothed, high-reflectance filter. Solidcurve: exact evaluation using modified refractive indices toeliminate secondary maxima. Dashed curve: same as solid curvein Fig. 2. Shows effect of smoothing in this case.

refractive indices

1l1= n3= n5`= n7= n9 = 2.7712= N 4=h n6=n= 1.52.

Figure 2 shows a plot of R(,A) vs j for the resultingmultilayer. In order to reduce the secondary maxima,Eq. (20) can be applied to calculate the smoothedvalues yFh. These turn out to be

go=0.2744 Pi -0.3191 /2-0.2373 P= -0.2693

y4 0.1935 9s -0.1837 Q6 0.1135 y7 -0.0805

ps=0.0335 /9=O.

Substituting into Eq. (16) gives the new values of theapproximate refractive indices

n1 =2.00 n2 =1.45 n3=1.84 n4=1.41 n5=1.71n6=1.42 n7 =1.59 n8 =1.47 n9 =1.52.

The smoothing process has greatly reduced the second-ary maxima, but at the cost of broadening the high-reflectance band and reducing the peak reflectance(see Fig. 3).

As a second example, consider an inhomogeneous filmwith just one reflectance peak of 90% at Xot= 5000 A andlow reflectance everywhere else. Assume that the opticalthickness P1 of the inhomogeneous film is to be approxi-mately the same as for the multilayer above. Then sincethe reflectance peak of the film is at po=0/2 for themultilayer, the value of m=k for the nonvanishingamplitude A(k/Pl) in Eq. (12) is determined by the

0.4-

0.2-

0) 2 4 6IN MICPONS "

FIG. 4. Inhomogeneous, high-reflectance filter. Solid curve:exact evaluation using inhomogeneous index obtained from first-order synthesis. Dashed curve: first-order prediction based onsampling procedure.

condition

k/PI=0/22= 1P.

Since I= 9, k = 4 or 5 depending on whether we need aslightly wider or narrower peak than before. Choosek= 5, then since P= /oX 4= 0.001125 mm for the multi-layer, Pi= 1.111P= 0.001250 mm. Put A (m/P) =0 formnz5, A (5/P1 ) = 3, and arbitrarily set i4(5/P1) =0 forconvenience. Substituting the numerical values into(12) and (9), we obtain

F(p) = 4800 rect (800p- ) cos8OOO7rp

N (p) = 1.52 exp(-9600 cos8OO0rPdp)

= 1.52 exp (-0.382 sin80007rp), 0 G p < Pi,

to the first-order approximation. A plot of R(g) vsp inthis case is given in Fig. 4.

ACKNOWLEDGMENTS

The author wishes to express his gratitude andacknowledge his deep indebtedness to P. W. Baumeisterof the University of Rochester and R. J. Pegis ofSt. John Fisher College for generously sharing theirknowledge with the author and for their kind encourage-ment. The author also wishes to thank Robert Riley andDouglas Mosey, students at St. John Fisher College,for two of the computer programs which were used inthe calculations.

1533December 1967