Multilayer Interference Filters with Narrow Stop Bands

Leo Young

Multilayer interference filters having various transmission characteristics are described and compared.The emphasis is on filters with narrow stop bands such as might be used to eliminate the hazard from a laserbeam. Four types of filters are considered: (1) quarter-wave stacks of two dielectric materials havingmatching layers one-eighth wavelength thick; (2) quarter-wave stacks of two dielectric materials havingall layers of the same optical thickness (including the end layers); (3) quarter-wave stacks wherein alllayers are of the same optical thickness, but the refractive indices of the layers may all be different toachieve equal reflection ripples in the passband; and (4) multilayer stacks of two dielectric materials where-in each layer may be of a different optical thickness to achieve nearly equal reflection ripples in the pass-band. The new formulas presented give the bandwidths between nulls of all the various filters as well asthe bandwidths between equal-ripple points of the equal-ripple filters. Explicit formulas are stated forthe ripple envelopes of filter types (1) and (2), and for the ripple heights of equal-ripple filters of types(3) and (4). A first-order design procedure based on the theory of linear arrays is given and evaluated byworking numerical examples; general design criteria are presented to establish the validity of the first-order theory.

I. IntroductionMultilayer interference filters have many applica-

tions and have been extensively described in books.1- 4

Various design methods as well as particular designshave been reported in recent papers.5-14 One of themost useful concepts in the design of multilayer filtersis the Herpin equivalent layer.15-20 Simplified designprocedures are possible when the multilayer stack con-sists of an iteration of layers ABABA---, where Arepresents a dielectric layer of one refractive index, andB represents a dielectric layer of a different refractiveindex but of the same optical thickness.2 1-28

The main purpose of this report is to describe andcompare the designs of various types of multilayer filtersthat have narrow stop bands, such as might be requiredto eliminate the optical hazard of a powerful mono-chromatic source in or near the visible region of thespectrum (without stopping most of the visible radia-tion at the same time); or as might be required to passa wide spectrum without large reflection ripples closeto the stop band. The necessity for a narrow stop bandrequires small changes in refractive index at the inter-faces of a quarter-wave stack, if layers of equal thick-ness are to be used. The further requirement of low

The author is with the Stanford Research Institute, MenloPark, California 94025.

Received 18 April 1966.This work was sponsored by the U.S. Air Force, Air Forces

Systems Command, Air Force Avionics Laboratory, SystemsEngineering Group, Wright-Patterson AFB, Ohio.

transmittance requires a large number of layers, sincethe layers are so nearly alike in refractive index. Theproblem is much like the design of stepped-impedancefilters in transmission line or waveguide. Exact designprocedures exist for this type of microwave filter,29-3 2

including numerical tables. Unfortunately, thesetables either do not apply to a large enough numberof sections (a section corresponds to a layer), or to anarrow enough stop-band bandwidth. Existing syn-thesis procedures require high precision in the numericalworking, and it has been found (both at SRI and else-where3 0) that even a large, high speed, electronic, dig-ital computer is not sufficiently accurate to solve forcases of more than about twenty or thirty sections of astepped-impedance filter.

The behavior of a multilayer filter with small changesin refractive index at the dielectric interfaces, or thebehavior of a stepped-impedance transmission-linefilter with small impedance steps, is analogous to thebehavior of a linear array of small antennas. Thetheory of such linear arrays has been described ex-tensively in the literature.33 -4 5 Numerical tables havebeen published for up to at least ninety elements3 5 (cor-responding to ninety interfaces or eighty-nine layers),and design formulas suitable for programming on adigital computer are available for the design of arraysof many hundreds of elements. One of the purposesof this report is to set up exact correspondences betweenantenna parameters and filter parameters, which areused to carry out the numerical design of particularfilters from existing tables. The accuracy of themethod is demonstrated by subsequent analysis of themultilayer filters.

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 297

Although the exact synthesis of multilayer or stepped-impedance filters is numerically difficult, the optimumperformance that is possible can be predicted fromexplicit formulas. These formulas are presented, andare used to evaluate the analyzed performance of thefilters designed by approximate methods.

Since it is frequently impractical to obtain an arbi-trarily specified set of refractive indices from physicallyavailable materials, an alternative procedure is investi-gated. In this procedure the interfaces in a quarter-wave stack are replaced by thin dielectric layers. Thefilter then consists of a stack of thin and thick layers.The thin layers may all have the same refractive index,and the thick layers may all have the same refractiveindex. Thus, only two refractive indices are required tomake up such a filter.

Numerical examples are worked out at the end of thereport, and it is shown how the numerical performancecan be predicted accurately. The different types offilter are also compared with each other.

In realizing the filter designs practically, it should bepointed out that absorption losses'2 have been neglected,and that some of the designs (as in Sec. VII) may re-quire accurate control of layer thicknesses. Theselimitations may ultimately determine filter perform-ance.

II. Definition of SymbolsThe following symbols will be used and are defined

below in alphabetical order:

a = -(n + 1/n)b = M(n/n2

2) + (n22/nl)

c = (1 + R)/(1 - R) = 2(V + 1/1V)d = spacing between elements of a linear arrayS = excess loss = Lp- 1 = RIT.

= maximum excess loss in stop bandSr = value of excess loss at peak of ripple in passband of

equal-ripple filter= excess loss ratio

Siax/Sr = maximum excess loss ratioI = currents in elements of Dolph-Chebyshev array

LA = attenuation function = -10 logio(Lp) dB. Whenstating the attenuation va]ue, the minus sign is cus-tomarily left out.

(LA)max = maximum numerical value of attenuation (in deci-bels), which occurs at the center of the stop band

(LA)r = attenuation at peak of ripple in passband of equal-ripple filter

Lp = transducer loss ratio = (incident power)/(trans-mitted power) = 1 /T

m = number of layers in a quarter-wave stack, when alllayers are one-quarter wavelength thick at the cen-ter of the stop band

m + 1 = number of layers in quarter-wave stack with half-thickness matching layers, one at each end, includ-ing (m - 1) interior layers one-quarter wavelengththick and two end layers one-eighth wavelengththick, at the center of the stop band

X = refractive indexno = refractive index of end massifs (here assumed to be

the same)nin2 = refractive indices of the layers of a stack consisting

of only two dielectric materials, with n taken as therefractive index of the first layer

R = reflectance = r2R = reflectance at peak of ripple in pass band of equal-

ripple filterRenv = envelope of reflectance peaks outside stop band of a

quarter-wave stack (with or without half-thicknessmatching layers at the ends)

s = 20 logi (SLR), measured in dBSLR = sidelobe ratio (an amplitude ratio)SWR = standing wave ratio, denoted by V

T = transmittance = 1/LpTm(x) = Chebyshev function defined by:

cos(mn cos 1x) when x < 1, and bycosh(m cosh-1 x) when x > 1

u = (ni + n2) cosOv = i -n2

V = standing wave ratio (SWR), equivalent to voltagestanding wave ratio (VSWR) in transmission-linework. It is defined as the ratio of the maximumfield amplitude to the minimum field amplitude inthe standing wave set up by the incident and re-flected waves

V12 = n1/n2 or n2/n, whichever is greater than unityVmx = maximum SWR in stop band. (Vmnx is equal to

the product of all the interface SWR's if the filter isa quarter-wave stack. This quantity has been re-ferred to as R in papers on transmission-linefilters,2 9 -3 2 but Vmax is used here since R usuallydenotes reflectance in optics.)

Vr = peak SWR in passband of equal-ripple filterVen = envelope of SWR peaks outside stop band of a

quarter-wave stack (with or without half-thicknessmatching layers at the ends)

w= fractional bandwidthwo = fractional bandwidth between equal-ripple points

of an optimum equal-ripple filterwoa = fractional bandwidth between equal-ripple points of

a multilayer filter in the limit of vanishingly smallrefractive index changes at the interfaces, whichcan be determined from the corresponding lineararray

w = fractional bandwidth between first nulls of a finitenumber (m/2) of Herpin sandwiches (w1 - wewhen m - co)

w = Herpin fractional bandwidth of a symmetricalthree-layer sandwich (measured between fre-quencies where the equivalent refractive index goesfrom real to imaginary)

w/l = fractional bandwidth between nulls of a multilayerfilter in the limit of vanishingly small refractive in-dex changes at the interfaces, which can be deter-mined from the corresponding linear array

x = sino/ sinoo, [Eq. (32) and Table A-I], or,x = parameter used in designing quarter-wave stack

from Dolph-Chebyshev array [Eq. (48)]y = parameter used in designing thin-and-thick layer

filter from Dolph-Chebyshev array [Eq. (49)]r = magnitude* of reflection coefficient - R'/2

r12 = magnitude of reflection coefficient due to a single in-terface between two semiinfinite media and 2,= (n - n2)/(nl + n2)!

0 = optical thickness of an interior layer (usually indegrees)

*r is used to denote the magnitude of the reflection coefficientvector, since we have no occasion to consider its phase. Thequantities 1? and T are defined in terms of power.

298 APPLIED OPTICS / Vol. 6, No. 2 / February 1967

side. Long wave pass performance (that is, less re-flection on the long wavelength side) results when therefractive index (no) of the two terminating media isequal to the geometric mean of the two refractive in-dices (n, and n2) of the filter layers17 ,

(1)no = nls/In,'/

Shortwave performance results when1 7

no = n/2n2 l

Fig. 1. A dielectric multilayer stack with end layers having halfof the optical thickness of interior layers. The interior layers

have twice the optical thickness of the end layers.

REFLECTANCE I HIGH-REFLECTANCE ZONE

w FIRST FSTZ RIPPLE RI PPPLE

PEAK / PEAK

Lii

90, O- degFIRST NULL 1 FIRST NULL(e e)

9OWH°

90wI.

Fig. 2. Typical envelope of a large number of reflectance curvesof various stacks with different numbers of layers, showing one

typical curve.

00 = value of 0 (less than 90°) at equal-ripple band edge(Fig. 7)

al = value of 0 (less than 90°) at first null (Fig. 2)X = wavelength' = wavelength of radiation from antenna array on

which filter design is based= radiation angle from normal to linear array.

Ill. Multilayer Stacks Having Interior Layersof Equal Optical Thicknesses and Half-ThicknessMatching Layers at the Ends

The type of filter sketched in Fig. 1 is widely used,because it is convenient to evaporate successive layersof the same optical thickness. All the interior layersare of the same optical thickness, while the first andlast layers have half the optical thickness of the interiorlayers. The effect of the half-thickness layers at theends is that they reduce the reflection from the filteron one side of its stop band, which is centered on a wave-length at which the interior layers are one-quarter wave-length thick. On the other side of the stop band thereflection is greater than it would have been withoutthe half-thickness matching layers, since they improvethe match only on one side of the stop band, either onthe long wavelength side or on the short wavelength

(2)

As we increase the number of layers keeping n, andn2 constant, the stop band begins to fill out and to falloff more steeply at the sides, but at the same time thenumber of reflection ripples and their amplitudes in-crease in the passbands. The performance for such afamily of filters can be depicted as shown in Fig. 2.There is a high reflectance zone, or stop band, flankedon both sides by many ripples. If we join the peaksof all the ripples of this family of filters, we obtain asmooth curve, as shown by the broken lines in Fig. 2.High reflectance is maintained over a fractional band-width WH centered on a frequency at which each layeris 900 thick. The fractional bandwidth of the stopband is given by'7

4WH =- [sin-' (n - n2)/(n + n2)|rad]

= I5 [sin-'1(n, - n2)/(nl + n2)|deg], (3)

and is henceforth referred to as the "Herpin equivalentbandwidth". Equation (3) is the lowest curve inFig. 3, enabling one to read off directly the fractionalbandwidth as a function of the refractive index ratio.[The other curves in Fig. 3 are explained in connectionwith Eq. (8).]

Equation (3) and other properties of matchedquarter-wave stacks can be worked out using theconcept of the Herpin equivalent layer.'5 -1 7 The filterin Fig. 1 can be thought of as a cascade of symmetrical,

0.7

0.6

6

3

0.2

0.2

:.

In

10

0

1.5 2.0 2.5V,2 ",'n2 ° 2

I I I I0.2 0.4 0.6 0,8 1.0

3.0

X

Fig. 3. Fractional bandwidth between nulls (wl) as a function ofrefractive index ratio for in = 9, m = 19, m = 29, and m =

infinity (w = WH).

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 299

no

2 t 82 - /, ~ -

H

eno

//

L

(6)= Cos-' [(nl - n2)2 + 4nn2 sin2(r/m)]'/2}

[ n + n2 /

= COS-l [rl2 2 + (1 - r22) sin 2(7r/M) .

DENOTED BY

( L H L )2-- 2~-

(7)

Whereas Eq. (3) can be used to predict the approximatebandwidth () of a filter with specified n and n2

for any number of layers, Eq. (6) or (7) gives preciselythe fractional bandwidth (w1 ) between the first nullson either side for any filter of given n,, n2 and number oflayers, by means of the relation

(b)

Fig. 4. Symmetrical

H L H

w = 2 - [01(deg)/45].

DENOTED BY

( H H )2 2y

three-layer sandwiches forming the basicperiod of many filters.

(8)

The fractional bandwidth between nulls (wi) decreasesas the number of layers increases, as can be seen fromFig. 3, in which w, is plotted against n/n,2 (or n2/ni,whichever is greater than unity) for in = 9, 19, and 29layers.* When in becomes large, w, becomes identicalto the Herpin fractional bandwidth wH.

The position of the ripple peaks is approximatelyhalf-way between the nulls (but slightly towards thestop band from each midway point). When the two endmedia have a refractive index no equal to the geometricmean of ni and n2, as given by Eq. (1), the broken linein Fig. 2, which represents the envelope of the ripplepeaks, is given by the following formulas. In terms ofSWR, the envelope is given by

triple layers, as shown in Fig. 4. At any particularwvavelength, each of these sandwiches is equivalent to asingle layer having a certain equivalent index anda certain equivalent optical thickness. The Herpinequivalent index is a positive real number for layerthicknesses outside the shaded region in Fig. 2, and is apure imaginary number for layer thicknesses inside theshaded region. The Herpin equivalent thicknessincreases from O-180 as the layer thicknesses increasefrom zero to the edge of the stop band. It remainsconstant inside the stop band, and it continues to in-crease beyond 1800 on the other side of the stop band.If the complete filter of Fig. 1 consists of in + 1 layers,which can be decomposed into (m/2) identical triple-layer sandwiches, as shown in Fig. 4, the entire filter isequivalent to a layer having a Herpin index the sameas that of one triple-layer sandwich and a Herpin opticalthickness equal to (m/2) times the thickness of one triple-layer sandwich. If the two end media have the samerefractive index no, the filter is reflectionless when itsequivalent thickness is an integral multiple of 1800.Substituting into the formulas for the Herpin equiva-lent thickness' 5 -'7 yields the following equations forthe optical thickness 0, at which the reflectance nullsoccur:

Oi = Cos- [(n - n2 )2 + 4nn2sin 2(7ri/if)] / (4)n= + n2

= cos[Prl22 + (1 - r122 ) sin 2 (ri/n)], (5)

where F12 is the single-interface reflection coefficientbetween two media of refractive indices n and n.In particular, the first null is given by

where

Venv = [(n 1/n2)(u + v)/(u - v)] 1 > 1,|

u = (n, + n2) cos9

v = ni -n

(9)

Here, the subscript "env" shows that we are referringto the envelope. The quantity in square bracketsmay be greater than or less than unity, and may or maynot have to be inverted (by selecting the sign of theexponent), since the SWR has to be greater than unity.In terms of reflectance, the envelope of the ripple peaksis given by

(10)R = F (n12

- n2')(1 - cost) 12L(n' + n22)2 coSO -(n, - n2)2 j

In general, for any value of no, the corresponding ex-pressions are

V.nv = [(n 1/no)2

(u + v)/(u - v)] -' > 1 (11)

and

(12)Rn= [(n2- n 2)u + (ns + no

2)v]2

L(n2 + no2)U + (n12

- no2)vJ

* The number of layers in Fig. 1 is (m + 1), which must be odd;therefore, m must be even in this case. The curves in Fig. 3 shiftcontinuously with the parameter m, which may be treated mathe-matically as a continuous variable. The same general ideas carryover approximately for the stack of Fig. 5, treated in Sec. IV,where m becomes the number of layers in Fig. 5 and is then an oddnumber. The values in = 9, 19, and 29 were chosen for numericalevaluation and plotting to make it easier to compare with thenumerical examples A through D in Sec. VI.E.

300 APPLIED OPTICS / Vol. 6, No. 2 / February 1967

(a)

Equations (9) through (12) are derived as follows.Consider the maximum SWR that can be obtained whena section of transmission line having an admittanceequal to the Herpin index is inserted between two trans-mission lines having an admittance equal to no. Thismaximum SWR is equal to the ratio of the two admit-tances squared, or the inverse thereof, whichever isgreater than unity. Expressions for the Herpin indexare given in the references (for instance, see Ref. 17), andEqs. (9) through (12) can be reduced from them.

When the number of layers is large, minimum trans-mission through the filter occurs when each layer isnearly 90° long. Then the SWR looking into the filteris a maximum and equal to Vmax, which is given by

Vmac (n,/n 2 )=(m+) > 1.

no no

Fig. . Quarter-wave stack consisting of many layers of identicaloptical thickness. Each layer has the same optical thickness.

(13)

The plus-or-minus sign in the exponent depends uponwhether n or n2 is the greater, since Vmax must begreater than unity. The reflection coefficient r,

the reflectance R, and the transmittance T are relatedto V by

r = ( - 1)/(V + 1) (14)

R= r2)

- >- ~~~~~~~~~(15)T = R

Equations (4) through (15) give the key performancefigures of multilayer filters of the form shown in Fig. 1.They give the bandwidth between the first nulls, thepositions of all the other nulls, the envelope of theripple peaks in the passband, and the peak attenuationin the stop band. Moreover, the given equations areeasy to work out numerically. Equations (4) through(12) are exact for stacks as shown in Fig. 1. Now theyare adapted to yield approximate formulas for stacksof layers, all of which have the same optical thickness.

IV. Multilayer Stacks in which All LayersHave the Same Optical Thickness

Now we consider multilayer filters of the form shownin Fig. 5, which differ from those shown in Fig. 1 onlyin that the two half-thickness matching layers at theends are replaced by full-thickness layers. Let thenumber of layers (all having the same optical thickness)be m, so that the sum of the optical thicknesses of thelayers in Fig. 5, like the sum of the optical thicknessesof the layers in Fig. 1, is mo. Since the matchinglayers in Fig. 1 reduce the reflection only on one sideof the stop band, while they increase the reflection onthe other side of it, filters of the form shown in Fig. 5are to be preferred whenever there is no reason to favorone side of the stop band over the other and a sym-metrical response is desired. These filters, having allm layers of the same thickness, have a symmetricalresponse about the center of the stop band, where eachlayer is one-quarter wave thick. They are of the formABABA ... mentioned in Sec. I. In this report we areprimarily concerned with filters that are to have a nar-row stop band in a wide spectral region (such as thevisible spectrum), where all wavelengths outside thestop band are generally of equal importance. Filters

of the form ABABA... are amenable to analysis, andcalculations involve Chebyshev or similar poly-nomials.21 -28 Simpler formulas, which are close enoughapproximations for a large number of layers, maybe deduced from the Herpin theory using the aboveformulas.

The positions of the close-in nulls are given to a goodapproximation, also for filters of the form ABABA- -,by Eqs. (4) and (5). The position of the first null isgiven closely by Eq. (6) or (7); then the fractionalbandwidth between the first nulls is given by Eq. (8).

The envelope of the ripple peaks on either side of thestop band (Fig. 2) are now symmetrical, unlike the per-formance of the filter shown in Fig. 1. The lack ofsymmetry resulted from the factor n/n 2 in Eq. (9)combined with the fact that u changes sign as one passesthrough the stop band. The expression for V canbe made symmetrical and equal to the geometric meanof the two unsymmetrical curves previously obtained bydropping the factor n,/n 2 . When expressed in termsof the reflection coefficient, this yields the remarkablysimple formula

ren - (n - n2)/(n, + n2) cosO

= r12/cOs6.

(16)

(17)

Here again it was assumed that the refractive index ofthe two end media is given by Eq. (1). The (power)reflectance envelope is given by squaring expressions(16) and (17). Equations (16) and (17) may be ex-pected to pertain especially to the large, close-in ripples.This was confirmed by checking out several numericalcases. For example, for n2/n, = 2.3/1.38, as in a mag-nesium fluoride-zinc oxide stack, the first ripple peakwas given by Eq. (16) to an accuracy of better than1% for the three cases checked, which had nine, nine-teen, and twenty-nine layers, respectively.

V. Optimum Filters with EqualRipples in the Passband

If we define the optimum design for a stack of layersof equal thicknesses as one that results in the narroweststop band for a given reflection ripple in the pass band,then such a design can be achieved by the properchoice of refractive indices in the filter shown in Fig. 6.

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 301

no n, n2 n3 n4nm-l nm nm-l

Fig. 6. Multilayer filter in which the refractive indices of thelayers may all be different.

Then the ripples in the passband are all of equal ampli-tude, and the filter performance can be simply statedin terms of the excess loss ratio by the equation

a(o) = rTm2(sino/sinoo), (18)

where Tm is a Chebyshev function given by

Tm2(x) = cos

2 (m cos'lx), x < 1 (19)

= cosh 2(?n cosh-'x), x > 1, (20)

and where

DO = sin-'{cosh[(1/in) cosh-'(am.a/ar)/1 } -, (21)

g max being the maximum excess loss (obtained at thecenter of the stop band) and 8 r the excess-loss rippleamplitude in the passband (see Ref. 29 or Chap. 6 ofRef. 32). The transducer loss ratio (see, for instance,Chap. 6, Ref. 32) is denoted by Lp, and is given by

L = + 1. (22)

The excess loss ratio 8 and the transducer loss ratio LPare also related to the reflectance R and the SWR, Vby the following equations:

a = R/(1 - R) (23)

= (V - 1) 2/4V

Lp = (1 - R)-'

(24)

(25)

= (V + 1)2 /4V. (26)

The transducer loss ratio Lp can become a large numberin the stop band, and it is frequently more convenientto express it in decibels. Therefore, we define theattenuation function LA by

LA = -10 logio(Lp) dB. (27)

At the center of the stop band (0 = 90°), the excessloss ratio, the reflectance, and the SWR attain theirmaximum values, related by

gmax = Rmax/(l - Rx)

= (Vmax - 1)2/4V.x.

(28)

(29)

Similarly, the ripple amplitudes of these quantities arerelated by

& = R,/(1 - R,)

= (Vr - 1)2/4V,.

(30)

(31)

The equal-ripple response of such a filter is sketched inFig. 7.

To facilitate numerical design, graphs are availablein filter handbooks, e.g., Chap. 4 of Ref. 32, and can beused to work out the design of Chebyshev filters havingup to fifteen resonators or layers. These numericaldata are useful in practical filter calculations but arenot available for m (the number of layers) greater than15; therefore, the tables in Appendix A were com-puted. Table A-1 gives the Chebyshev functionsquared, which is equal to the excess loss ratio by Eq.(18), as a function of the number of layers m, and thefrequency or wavelength parameter x given by

x = sino/sinoo. (32)

Table A-1 gives all cases likely to be encountered bythe designer, up to one hundred and seventy layers.To convert the excess loss ratio from Table A-1 into anattenuation LA in decibels use the value of 8r from Eq.(30) or (31) if necessary, together with Eqs. (22) and(27). The reflectance or SWR can be determined fromEqs. (25) and (26).

The fractional bandwidth of the stop band betweenequal-ripple points can be seen, by reference to Fig. 7,to be

Oo(deg)wo = 2 -

45(33)

Examiple 1: To deterimine the number of layers requiredto meet a specified performance. It is required to designan optimum multilayer filter having a stop-band band-width between equal-ripple points of 20%, a minimumtransmittance of 0.1%, and a reflectance of less than20% anywhere in the passband.

A minimum transmittance of 0.1% means that (LA)max

is equal to 30 dB, or (Lp)max is equal to 1000. The20% maximum reflectance in the passband means that(LA), is equal to 1.0 dB, or (Lp), is equal to 1.259:1.We establish that

10 logio(amax/ar) = 10 logjO(1000 - 1)/(1.259 - 1) = 35.86 dB.

Since the fractional bandwidth is 20%, substitute 0.2for tv0 in Eq. (33), which yields Oo equal to 80°. Sincethe maximum attenuation occurs when 0 is equal to

9-deg180,

EQUAL-RIPPLE POINTSAT 0 = So ANDAT = 180- So

Fig. 7. Possible equal-ripple response curve obtainable withfilter shown in Fig. 6.

302 APPLIED OPTICS / Vol. 6, No. 2 / February 1967

LA

90°, we obtain from Eq. (32):

x = singO0/sinoo = sin9O0 /sin8l 0 = 1.0125.

Interpolating in Table A-1, we look between the linesx = 1.01 and x = 1.02 until we reach the value 35.8639against the interpolated value x = 1.0125. Thisoccurs close to the column headed m = 30. Thus, tomeet the required specifications, at least thirty-onelayers would be required.

Example 2: In the above example, over what fractionalbandwidth does the transmittance remain below 1.0%?A transmittance of 1% corresponds to Lp equal to 100.Hence,

10 logjo(s/8,) = 10 logio(10O - 1)/(1.259 - 1) = 25.82 dB.

Interpolating for m = 31, in Table A-1, we find x= 1.007 = sinS4.1 0 /sin8l 0 ; therefore, the fractionalbandwidth over which the transmittance remainsbelow 1% is 2(90 - 84.1)/90 = 5.9/45 = 0.131.

Although it is possible to determine what performancecan be achieved and the number of layers that wouldbe necessary, it is not such an easy matter to work outnumerically the values of the refractive indices of theindividual layers (that is, to synthesize the filter).The theory has indeed been completely workedout,29 -32 but the synthesis numerical procedures re-quire a high speed electronic digital computer. Ourexperience has shown that the numerical precisionnecessary is of such an order that even these computerscannot solve for filters with more than about twenty orthirty layers, using the existing synthesis procedures.Fortunately, there exist approximate design proceduresthat can solve many cases of practical interest. Sucha method, based on the theory of linear-array antennas,is now described.

VI. Nonoptimum Filters with Equal Ripples inthe Passband, Using the Theory of LinearArrays

A. Correspondence Between MultilayerFilters and Linear Arrays

It is possible to utilize the theory of antennas con-sisting of linear arrays of radiating elements to com-pute the refractive indices of the layers of an inter-ference filter which will have an equal-ripple character-istic in the passband. The correspondence is true onlyfor the first order, when multiple internal reflections inthe filter may be neglected. Consider a linear arrayas shown in Fig. 8, in which all the radiating elementsare phased so as to produce a wavefront traveling in adirection making an angle VI with the direction of thenormal to the array. If we again denote the opticalthickness of a layer in a filter by , the following cor-respondence exists:

(180d sin#)/X' (0 - 90), (34)

where d is the spacing between elements of the array,A is the wavelength of the radiation from the array,and all angles in Eq. (34) are measured in degrees.

I-NORMAL TO ARRAY

I )\\¾

WAVELENGTH= V

Fig. 8. Schematic of linear array and its radiation pattern.

There is no cross-coupling between the elements of anideal array; therefore, the correspondence is trueexactly only in the limit when the changes in refractiveindex at the interface of the filter are infinitely small.The length of the array from end to end is md, wherethe total number of elements is (m + 1), and corre-sponds to the sum of the optical thicknesses of the layersin the filter. The radiation pattern of such an array isalso shown in Fig. 8, and here the main beam corre-sponds to the stop band of the filter, while the sidelobescorrespond to the passband of the filter. The radiationfield intensity in the space around the antenna cor-responds to the (amplitude) reflection coefficient r

of the filter. In the limit, when the reflection coefficientis small even in the stop band, the sidelobe ratio of theantenna pattern corresponds to the excess-loss ratio ofthe filter; however, such a filter having small reflectioncoefficient even in the stop band would not be usefulin practice, and this correspondence must be modifiedfor practical filters, as explained below.

B. Range of Validity of the First-Order TheoryThe beam width of an antenna array is independent

of the intensity of radiation, but we know that thebandwidth of the filter is a function of the magnitudeof the reflection coefficient at each interface, as can beseen from Eq. (3). Consider a filter consisting of mlayers and having refractive indices that are nearlyalike, so that the reflections at the interfaces are ex-tremely small. It can readily be shown that thebandwidth of the stop band between the first nulls oneither side is given by

bandwidth = 4/m, (35)

where it has been supposed that all interfaces have thesame reflection coefficient. Now let us denote the SWRcorresponding to the reflection coefficient of a singleinterphase by V12, which then is given by

V12 = n/n2 or n2/nli, whichever is > 1. (36)

Then the bandwidth, according to Eq. (3), is givenapproximately by

bandwidth = (4/r)(V2 - 1)/(Vl2 + 1), (37)

provided that the interface reflection coefficients are

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 303

reasonably small so we can replace the sine by the anglein radians. We would expect the first-order theory togive acceptable results until the two bandwidths givenin Eqs. (35) and (37) are approximately equal. Equat-ing Eqs. (35) and (37) and solving for V12 yields

V12 = (m + r)/(rn - r). (38)

We can write down the maximum SWR in the stopband, and when the number of layers is large, it re-duces to

V.. = V1 -( + 27r)- e2 (39)

Numerically, this is equal to approximately 535;it corresponds to a transmittance of approximately0.75%, or an attenuation of about 21.3 dB. This is auseful result. It means that we can use the theory oflinear arrays, for which many numerical tables exist,to design interference filters with many layers andwith a stopping power corresponding to a transmittancedown to about 1%, or an attenuation down to about 20dB. It is seen later from the numerical examples thatthe first-order theory does not break down abruptlywhen we attempt to design filters with greater stoppingpower. It is found that the first-order designs continueto give equal-ripple responses in the passband forattenuations of at least 50 dB (transmittance less thanone-thousandth percent), but that the stop-bandbandwidth increases at the base and exceeds that whichwould be possible with an optimum design (Sec. V),thus rounding out the response shape in the regionwhere the passband goes over into the stop band andreducing the rate of cutoff at the band edges.

C. Bandwidth and Beam WidthThe bandwidth of the stop band of a filter can be

deduced from the beam width of the array to which itcorresponds. For example, in Table 1 of Ref. 34 thebeam width of many arrays are calculated. This tableincludes the beam width between the first nulls oneither side of the beam, and corresponds to the band-width of the stop band between the two nulls on eitherside of it. They are related by

bandwidth, wi = 2 sin(beam width/2), (40)

which can be proven from Eq. (34).

D. Sidelobe Ratio and BandwidthAntenna tables usually measure beam width between

points a stated number of decibels below the peak of thebeam down to the null points, which are an infinitenumber of decibels below the peak of the beam. Infilter theory, the bandwidth between the equal-ripplepoints is perhaps the most useful, and it can be ob-tained from Eq. (21) by replacing the excess-loss ratioby the sidelobe ratio (SLR), giving

wo = 2- sin-If cosh[(1/m) cosh-'(SLR)] 'deg}. (41)

One can also obtain w0 from Table A-1 of Appendix A,as shown by the following example.

Example 3: What is the equal-ripple fractional band-width (wo) of the stop band of a filter consisting of twenty-five layers and having a maximum excess-loss ratio of 47dB? Referring to Table A-1, we find that x = 1.030.As before, this is the value of 1/sin0o. From sinetables we obtain 00 = 76.15°. Hence, from Eq. (33)the equal-ripple bandwidth is 30.7%.

Recall that the sidelobe ratio of a Dolph-Chebyshevarray" is the ratio of the field intensity in the directionof the peak of the main lobe to the field intensity in thedirection of the peaks of one of the sidelobes. To ob-tain the resultant field intensity in any direction from anantenna array, one adds vectorially the contributionsfrom the individual elements of the array to the fieldintensity in the direction of interest. In the first-ordertheory for filters, one sets up a correspondence betweenthe reflection coefficient of one interface with the fieldintensity produced by the corresponding element ofthe array. However, it is clear that the reflectioncoefficients of all the interfaces of a filter having anappreciable stopping power do not merely add numer-ically in the stop band; otherwise, the reflection co-efficient of the filter as a whole might add up to greaterthan unity, which is impossible. There is, however, aquantity which does add numerically in the stop bandof a filter, as can be seen from the following argument.At the peak reflection in the stop band, where the max-imum SWR (Vmx) is obtained, Vmx is given by theproduct of all the interface SWRs. The quantity thatis additive is therefore the logarithm of the SWRs.Moreover, in the passband, where reflection coefficientsmay still be used because the net reflection coefficientis also small, these reflection coefficients may also bereplaced by the logarithm of the SWRs, since loge (1 +6) 6, when a << 1. Therefore, it is more appropriate toset up the following correspondence with the sideloberatio of antenna theory:

sidelobe ratio, SLR > log(Vmax)/log(Vr), (42)

where Vmax is again the maximum SWR in the centerof the stop band, and Vr is the SWR at the peak of theripples in the passband. Using natural logarithms,Eq. (42) can be rewritten as

Vr = 1 + [log,(Vm.,)]/SLR. (43)

In antenna terminology, the sidelobe ratio is usuallyexpressed in decibels, so

s = 20 logio(SLR). (44)

For a sufficiently large number of layers and for a suf-ficiently high sidelobe ratio, the equal-ripple bandwidthis given by

I2 { sin' 1 + 2 (0.69 + 0.115s) 2/m

21 degas

(45)

as follows from Eqs. (33) and (41).is quite accurate whenever

m > s > 1

Equation (45)

(46)

304 APPLIED OPTICS / Vol. 6, No. 2 / February 1967

Table I. Current Distributions on Array Elements for 40-dBSidelobe Ratio (Normalized with Respect to the Center

Element)

Element number Currents for Currents for(i) 20 elements, i 30 elements, It

i 0.1177 0.13832 0.1656 0.13133 0.2636 0.18724 0.3812 0.25295 0.5117 0.32746 0.6458 0.40897 0.7724 0.49528 0.8802 0.58379 0.9586 0.6712

10 1.0000 0.754411 0.830012 0.894913 0.946214 0.981815 1.0000

is satisfied, and when the refractive index changesat the interfaces are sufficiently small.

When the changes in refractive index from layer tolayer are no longer small enough to neglect, it has beenfound that the following formula is quite accuratelytrue for the equal-ripple bandwidth of the filter:

W (w02

+ W12)1/2, (47)

where wo is given by Eq. (41) and wv is given by Eq. (8),using the refractive indices at the center of the multi-layer filter. [To the accuracy of Eq. (47), wH fromEq. (3) may be substituted for wi, or woa from Eq. (45)may be substituted for w0, or both, and similar resultswill be obtained.] It should be pointed out that Eq.(47) applies only to the bandwidth between equal-ripple points, and not necessarily to the bandwidthbetween other levels on the stop-band curve. It isfound that most of the stop-band curve follows closelythat of the optimum filter (See. V), and that the widen-ing occurs mainly at the base of the stop band, as al-ready mentioned.

E. Numerical Examples

Five numerical examples are described and discussed.In two examples there were nineteen layers (twentyinterfaces); in three examples there were twenty-ninelayers (thirty interfaces). They were based on thenumerical tables in Ref. 34, using a 40-dB sidelobe ratio.The antenna element currents are reproduced here inTable J.34 [The required refractive indices vary;for instance, in one example (Example C), the refrac-tive index ratios at the interfaces vary from close tounity up to 1.22. Such a filter might be constructedby varying the proportion of silicon to oxygen fromlayer to layer, each layer being a different mixture ofsilicon oxide.]

Since the array is symmetrical from either end, it isonly necessary to tabulate half the current values.We notice in passing that the currents decrease from

the center to the end, except that sometimes the currentin an end element is greater than the current in theelement next to it.

The refractive index of the ith layer was then setequal to ni, given by*

ni/no = 1 - E (-1 )kjk

k= I

The following five cases were(48) and Table I:

Case A:

Case B:

Case C:

Case D:

Case E:

= 19,

= 19,

m = 29,

m = 29,

m = 29,

(i = 1 to n). (48)

computed using Eq.

x = 0.4

x = 0.8

x = 0.2

x = 0.4

X = 0.8

These five cases were analyzed on a digital computer,and the results are shown in Figs. 9 through 13. Anexisting computer program was used to calculate theattenuation (in decibels) and SWR. Then the com-puter output was plotted by an automatic plottingmachine, and the reflectance and transmittance scaleswere subsequently added to the ordinate axis. Thequantity on the abscissa is proportional to 0, the opticalthickness of any one layer, but is scaled in such a waythat the scale reads unity at the center of the stop band,where each layer is one-quarter wave thick; therefore,

= 900. This scale is referred to as the normalizedthickness, and measures 0 (deg)/90.

A few general remarks are required before proceedingto a more quantitative description and comparison ofthese five figures. We note first of all that Cases A andC (plotted in Figs. 9 and 11) meet our criterion for apeak attenuation less than about 20 dB; Case D (Fig.12) has a peak attenuation close to 20 dB; and CasesB and E (Figs. 10 and 13) have a peak attenuationappreciably greater than 20 dB. We would thus ex-pect Cases A and C to follow the first-order theory well,Case D to be marginal, and Cases B and especially Eto be beyond the region of validity of the first-ordertheory. This is borne out by counting the number ofzeros of reflectance between a normalized thickness ofzero (infinite wavelength) and the stop band, or be-tween the stop band and a normalized thickness of 2.0.(All the curves are symmetrical about the center of thestop band, as they are plotted against optical thickness

* Equation (48) ensures that the interface reflection coefficientsare approximately proportional to the element currents, at leastwhen x is small. This equation was also partly chosen becausethese computations were made by hand and the arithmetic wasquite simple. Upon further thought, it is considered that thefollowing equation might have given somewhat better results:log Vi I. This is also in keeping with the same practiceadopted in Ref. 29 to extend the first-order theory. In addition,this modified equation for Vi enables one to work out quickly thestop-band peak attenuation, which is determined by V, theproduct of all the Vi, since this computation involves adding allthe log Vi.

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 305

.4 - -z

- ~0.02'1 CA t

0.05

0.005

0.0001i ---F-----7---------T----I ---- 7I

o 0.5 1.0 1.5 2.0

o 0.5 1.0 1.5 2.01 t - I I I I I I I- II I I I I I I

1.0

z -1 I0 01

z

-20 - I0 1

CASE A , UARTER -WAVE STACK19 LAYERS X 0.4

I _ I I I I I I I I I I I I I I 10.5 1.0 1.5

e8(DEGREES)90

Fig. 9. Response curvefor the filter Case A.

0.5 I.0 1.5 2.0I I I I I I I I I I I I I I I I I -I

I II _1 I 10.5

I I I I I I I I I I I1.0 1.5

08(DEGREES)90

Fig. 10. Responsecurve for the filter

Case B.

2.0

306 APPLIED OPTICS / Vol. 6, No. 2 / February 1967

IA

nf

E

0 2.0

1.4

-~: 1.2UA

1.0

051

0 2

z-20 -

wI- -

30 -

-40 -

0

1.0

0.1I~

wz

z

4r

4 CASE B ,OUARTER-WAVE STACK

-001 19 LAYERS X�0.8

0.001 _

A (11001

i

i

C

ICASE 8 QUARTER -WAVE STACK

19LAYERS X=0.8

_ - I 1.0 1.5

0.5I I I I I I

1.0 1.51 I 1 1 I .

1.0

0.5-

UzI- 0 /CASE C, QUARTER-WAVE STACK

zA 29 LAYERS X=0.2z

0.2

0.1 I

2.0

Fig. 11. Responsecurve for the filter

Case C.

0 0.5 1.0 1.5 2.00 (DEGREES)

90

I I T I i T T0.5

I I I I I I I I I1.0 1.5 2.0

ID 1.51 I C I 1

I I I I I I I I I I I I I I I 0.5 1.0 I.5

e (DEGREES)90

2.0

Fig. 12. Responsecurve for the filter

Case D.

2.0

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 307

1.4 -Lj

UO 002

a 0.01

0.005

0.0020.001

0.00011.0

0I I I I I5

0 5 2.C

0

- 1

1.4 -LU

I-

LI 0.01

0.005

0.0020.00 I

0.000 100I ,ry.N/ \

0.5I 1 1 0

02 -10 -

z0I-

4

z

! -20 -

1.0

0.I FLIUzI--

4)I-

CASE D , QUARTER-WAVE STACK29 LAYERS X 0.4

0.01 -

0.0 01

0

- 5 -5 -I i l s v --v y -

s I

l l l

I . I . . . . . . . . . . . . . . . . . I_,

.,

I

I

I

-,u

Fig. 13. Responsecurve for the filter

Case E.

0 0.5 1.0 1.5 2.08 (DEGREES )

90

rather than wavelength.) An optimum filter (Sec.V) would have (m + 1)/2 zeros of reflectance in theseintervals. Thus, Case A should have ten zeros, buthas only eight; the two zeros closest to the stop bandhave evidently merged and become a minimum orpoint of inflection on the edge of the stop band (Fig.9). In Case B there are also eight zeros, but there is nosign of the two zeros closest to the stop band, as theyhave evidently been swallowed by the widened stopband, leaving no trace. Turning now to the filters withtwenty-nine layers, Case C should show fifteen zeros,but shows only thirteen zeros; the two zeros nearest thestop band have evidently combined to form a deepminimum. In Case D there are also thirteen zeros,but there is no trace of the two lost zeros. In Case Ethere are only eleven zeros, but there is also one mini-mum showing that there had been two zeros there;the other two zeros to account for a total of fifteen zeroshave completely disappeared. It is possible that, if themodified Eq. (48) in the previous footnote had been usedin place of Eq. (48) to design these filters, better agree-ment with the first-order theory might have been ob-tained.

F. Further Discussion of Numerical ResultsTo evaluate the numerical designs of Cases A through

E (Figs. 9 through 13), and to compare them with theperformance of an optimum design (Sec. V) as well as

with the performance of a regular quarter-wave stack(Sec. IV), Table II was composed. Table II summar-izes the principal numerical results and, where neces-sary, additional explanations are supplied below.

Lines 1 through 6 have already been commentedon, and would seem to show that the 20-dB rule is agood indicator concerning the applicability of the first-order theory based on linear arrays. Lines 7 and 8show that Eq. (43) is remarkably accurate. Line 10gives the Herpin bandwidth of a hypothetical multi-layer that consists of a uniform quarter-wave stackdetermined by Line 9. Lines 11 and 12 come out asexpected, since the equal-ripple bandwidth should beslightly less than the bandwidth between nulls. Line13 shows how narrow the bandwidth could have beenfor the same peak attenuation in the stop band and forthe same ripples in the passband. Line 14 gives thebandwidth computed by Eq. (47), and by comparisonwith Line 15 shows that Eq. (47) is only moderatelyaccurate.

A comparison with regular quarter-wave stacksas described in Sec. IV is made on Lines 16(a) through(e). These quarter-wave stacks were all designedto have the same peak attenuation in the stop band asthe equal-ripple filters (Line 4). To meet this require-ment, the refractive indices n and n2 (Fig. 5) and theirratio are as given on Lines 16(a) and (b). The stop-band bandwidths of these quarter-wave stacks are, inall cases, appreciably less than even the optimum equal-

308 APPLIED OPTICS / Vol. 6, No. 2 / February 1967

In

I 1 0E

0 05 1.0

[4

'3: 1 2V)

I

0

I-20

C

z~- 30

<

-40

-50

1.5 2 0I

Table II. Summary of Numerical Results and Comparisons for the Five Filter Cases A, B, C, D, E (All Designs are Based on a40-dB Sidelobe Ratio, as in Table I)

Line Case A B C D E

I Number of layers (n) 19 1!) 29 29 292 Number of zeros that should be 10 10 15 15 15

between 0 = 0 and 0 = 9003 Number of zeros that are between 8 8 13 13 11

0 = 0 and 0 = 90°4 Maximum attenuation (dB) 13.59 33.10 8.99 23.13 51.685 Minimum transmittance 0.044 4.9 X 10-4 0.126 4.9 X 10-a 6.8 X 10-66 Vmax 89.41 8160 29.67 819.7 589,1947 V, predicted from Eq. (43) 1.045 1.09 1.03 1.07 1.138 V computed by analysis 1.045 1.10 1.03 1.07 1.139 Refractive index ratio (ni/n2) in 1.49 2.25 1.22 1.49 2.25

center of filter10 Bandwidth to nulls (we) of stacks 0.328 0.542 0.190 0.285 0.520

with ni/n2 as on Line 9, fromEq. (8)

11 Bandwidth to nulls (wie) from 0.366 0.366 0.242 0.242 0.242beam width, determined fromEq. (40) and Ref. 34

12 Equal-ripple bandwidth (wo") 0.352 0.352 0.232 0.232 0.232from Eq. (41)

13 Equal-ripple bandwidth (wo) of 0.400 0.494 0.258 0.295 0.407"optimum" filter from Eqs. (21)and (33)

14 Bandwidth predicted by Eq. 0.51 0.73 0.32 0.41 0.66(47), with Lines 10 and 13

15 Equal-ripple bandwidth computed 0.48 0.72 0.28 0.43 0.61by analysis

16 Quarter-wave stack (Fig. 5)consistent with Lines 1 and 4,5 or 6:

(a) ni and n2 of quarter-wave f0.890 0.790. 0.944 0.892 0.795stacks(no= lineachcase) 11.125 1.268 1.060 1.122 1.259

(b) Ratio n/n2 1.267 1.607 1.124 1.260 1.585(c) Bandwidth to first nulls 0.26 0.36 0.16 0.20 0.32(d) First attenuation ripple height (dB) 1.0 2.7 0.6 2.0 5.0(e) First reflectance ripple height 0.2 0.5 0.1 0.4 0.7

ripple filters. This is to be expected from antennatheory, where it is well known that a uniform lineararray has the narrowest beam width.* However, nowthe ripples in the passband are high, again correspond-ing to the well-known result in antenna theory of highsidelobes associated with equal-element arrays. Theattenuation at the peak of the first ripple is given onLine 16(d), and the corresponding reflectance height ofthe first ripple is given on Line 16(e). It is seen thatan appreciable amount of power is lost by reflection dueto the high ripples in the passband [Lines 16(d) or(e)]. The ripples decrease away from the stop band,but it was found by direct analysis on a digital computerthat the ripples fall off in such a way that nearly all ofthem remain higher than the equal ripples of the corre-sponding one of the five filters with the same peak at-tenuation shown in Figs. 9 through 13.

* The linear array is supposed to consist of radiators that arein phase and spaced one-half wavelength apart.

VII. Filters with AlternatingThin and Thick Layers

The underlying idea in designing narrow stop-bandfilters has been the iteration of a large number of inter-faces at regular separations and with small but con-trolled changes in refractive index at each interface.Since it may not be easy to control the refractive indexaccurately, we now consider replacing the interfaces byother small but sharp discontinuities. Figure 14shows a stack of dielectric layers that are alternativelythin and thick. The thick layers all have the samerefractive index (n = 1.38 or 2.3), and the thin layersall have the same refractive index (n2 = 2.3 or 1.38).*The thin layers represent fairly sharp planar discon-tinuities with their centers spaced at regular intervalsof 1800 within another and larger dielectric medium.

* Only the ratio na/n2 or n2/nl = 1.38/2.3 = 0.6 figured in thecalculations.

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 309

Fig. 14. Multilayer filter consisting of alternate thin and thicklayers.

The reflection coefficient due to a thin layer is approx-imately proportional to its thickness; therefore, thefollowing formulas were used together with Table I ofSec. VI to test some numerical examples:

01 = YJ1I

03 = ?YF2

03 = yla3 (49)

a, = y11(i + 1)/2, i = 1, 3, 5, ... , 2in +

and

1.4

3 1.2

1.0

0

M -10

0

I-

1~ -20

-30

I02 = 180° - - (a, + 03)2

... (50)

i= 180° - 2(0j- + j+l) = 2, 4, ,2m

The odd 0's represent the optical thicknesses of thethin layers, and the even O's represent the optical thick-nesses of the thick layers. A filter based on an arrayof twenty elements, as in the first column of currentsin Table I, now results in a filter with thirty-nine layers(compared with nineteen layers in Sec. VI).

Figures 15 and 16 give the analyzed performance oftwo cases, Cases F and G, which were designed fromEqs. (49) and (50) applied to the two sets of array cur-rents in Table I, with y assigned the numerical value40. (Cases with other values of y were also workedout, but Figs. 15 and 16 are representative of themalso.)

First we make some general comments concerningFigs. 15 and 16. One notices immediately that theenvelope of the ripples is nearly a straight line thatpasses through the origin; this effect is attributableto the fact that the periodically spaced discontinuitiesare no longer single interfaces but are thin layers thatreflect increasingly as the frequency increases, sincetheir optical thickness increases in direct proportion tofrequency. In Figs. 15 and 16, points were only com-puted for normalized frequencies lying between 0.5 and

zW

W 0.02

0005-0.002-0.001

0.000I __________0 0.20.5 2.0

0 0.5 1.0 1.5 2.0

1.0

0.I

Cfd

z CASE F, THIN AND THICK LAYERS4

0.01 ~~~~~~~~~~~~39 LAYERS Y=40

0.0i1

I I I I I

0 0.5 1.0O (DEGREES)

90

1.5

Fig. 15. Responsecurve of filter Case F.

2.0

310 APPLIED OPTICS / Vol. 6, No. 2 / February 1967

w

0 0.02

W 0.01cr

'000

_ s0002 9LYR =4

0.0010.0001

0 0.5 1.0 1.5 2.0

0 0.5 1.0 1.5 2.0

1.0

0.1

w'-' 0.01-

2 ~~~~~~~~~~~~~~~CASE G, THIN AND THICK LAYERS

'40.00 V 59 LAYERS Y= 40

0.0001L

0.00001

V.0 I"~~~~~~~~~~~~~~~~~~~~~~~~~~~5 OrU

Fig. 16. Responsecurve of filter Case G.

8( DEGREES )90

1.5. Therefore, it is not evident that another stopband exists at a normalized frequency of 2.0, that is,at the second harmonic frequency of the desired stopband. In the case of quarter-wave stacks, including thefilters shown in Figs. 9 through 13, the next stop bandoccurs much farther out, namely, at the third harmonicfrequency of the desired stop band. The thin andthick layer filters have stop bands at all frequencies thatare integral multiples of the first stop-band frequency.Moreover, as long as the thin layers remain opticallyfairly thin, these successive stop bands increase inattenuation, and when plotted in the manner of thelower half of Figs. 15 and 16, the peaks of the successivestop bands also lie approximately on a straight linethat passes through the origin.

Again, the ripple heights can be calculated accuratelyusing Eq. (43). The ripple height thus calculatedshould be plotted as a single point above 0 = 900 in theupper half of Figs. 15 or 16, and a straight line shouldbe drawn from the origin through that point. It wasfound that this line coincides within slide rule accuracywith the envelope of the ripples.

The bandwidth may be determined in a way similarto that which was used in connection with quarter-wave stacks in Sec. VI. The Herpin bandwidth ofEq. (3) is obtained by first substituting the SWR of athin layer for the refractive index ratio of the corre-sponding interface of a quarter-wave stack. The band-width thus obtained has to be halved, since the spacing

between centers of thin layers is twice the spacing be-tween interfaces (1800 instead of 900); therefore,the new filter is twice as frequency sensitive. (An-other manifestation of this effect is that the separationbetween stop bands is only one-half the separation forquarter-wave stacks. That is why the second stopband now occurs at the second harmonic frequencyinstead of at the third harmonic frequency.) Againthe optimum filter bandwidth is determined from Eq.(41) (with mT being the number of thick layers), butis divided by 2, because of the increased frequencysensitivity alluded to. The actual filter bandwidth ofthe approximate design can be obtained from Eq. (47)with a fair degree of accuracy.

Fig. 17. Dielectric layer of refractive index n2 between twosemiinfinite media of refractive index n1.

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 311

1.4

1.0

0

-10

m

| -20z0

W_ -30

-40

-50I.J C<.VToU.D

The following formulas determine the reflectance of athin layer of refractive index n2 separating two mediaof refractive index ni, as shown in Fig. 17. Let

a = (n i +-) (51)

2 (n22 ni

then

a cos20 + b sin2o -

a COS20 + b inO - I

where R is the reflectance of the layer of refractive indexn2 in the medium of refractive index i1 shown in Fig.17. The SWR () presented by the layer can be de-termined from

V = [1 + (R)'1'l/[1 - (R)/2. (53)

Similarly, if R or V is given, 0 can be determined from

0 = sin-'[(c - a)/(c - )], (54)

where

c = 2 (V + 1/V) = (1 + R)/(1 - R). (55)

In Cases F and G shown in Figs. 15 and 16, the thick-nesses of the two centermost thin layers were madeequal to 40°. From Eqs. (52) and (53) we can deter-mine that V = 1.96. Similarly, if V or R had beengiven first, 0 could have been determined from Eqs.(54) and (55).

VII. ConclusionsFour types of multilayer interference filters have been

considered. First, there is the multilayer stack withinterior layers of equal optical thicknesses and withhalf-thickness matching layers at the ends. Thistype of filter was investigated using the concept of theHerpin equivalent layer. New formulas for the stop-band width and for the envelope of the ripple peaks inthe passband were presented.

Second, multilayer quarter-wave stacks, in which alllayers have the same optical thickness, were treated.Although this type of filter can be analyzed exactlyby means of Chebyshev functions, simpler approximateformulas were presented here, which should prove usefulfor quick numerical calculations by slide rule. Numer-ical checks seem to indicate that these formulas areremarkably accurate.

Third, equal-ripple filters were investigated in detail.The optimum performance of these filters can be pre-dicted by exact formulas. Although exact synthesisprocedures exist, they are not suitable for numericalcalculations, even on a large electronic digital computer.For this reason, an approximate design procedure waspresented, which is based on the theory of linear arrays.Numerical examples were worked out and explained.

Fourth, filters with alternating thin and thick layerswere described. They have the advantage that onlytwo refractive indices are required. Numerical ex-

amples were worked out, in which the two refractiveindices are assumed to be in the ratio 1.38/2.3 = 0.6,corresponding to the conveniently available materialsmagnesium fluoride and zinc sulfide. It was shownthat the layer thicknesses can be designed to yield anequal-ripple response.

Design formulas were given, many of which have notbeen stated before. It is hoped that this presentationwill enable the designer not only to choose the bestfilter for his application by showing him what is possible,but in many cases will also help him to design it.

Appendix A: Tables for Calculating Charac-teristics of Chebyshev (Equal-Ripple) Filters

Table A-1 gives the square of the Chebyshev functionfor filters with up to m = 170 layers. This functiongives the excess loss ratio (/,), and its use is illus-trated in Examples 1, 2, and 3 in the text.

References1. 0. S. Heavens, Optical Properties of Thin Solid Films

(Butterworth, London, 1955).2. A. Vasicek, Optics of Thin Films (North-Holland Publish-

ing Company, Amsterdam, 1960).3. M. Born and E. Wolf, Principles of Optics (Pergamon Press,

New York, 1964) Sec. 1.6.4. P. W. Baumeister, Handbook of Optical Design, Chap. 20,

MIL-Handbook No. 141 (5 October 1962), obtainable fromControl Center 550, Frankford Arsenal, Philadelphia, Pa.

5. A. F. Turner and P. W. Baumeister, Appl. Opt. 5, 69(1966).

6. J. S. Seeley and S. D. Smith, Appl. Opt. 5, 81 (1966).7. L. Young and E. G. Cristal, Appl. Opt. 5, 77 (1966).8. J. A. Dobrowolski, Appl. Opt. 4, 937 (1965).9. J. S. Seeley, J. Opt. Soc. Am. 54,342 (1964).

10. G. Tricoles, J. Phys. 25, 262 (1964).11. A. Thelen, J. Opt. Soc. Am. 53, 1266 (1963).12. L. Young, J. Opt. Soc. Am. 52, 753 (1962).13. L. Young, J. Opt. Soc. Am. 51, 967 (1961).14. R. J. Pegis, J. Opt. Soc. Am. 51, 1255 (1961).15. A. Herpin, Compt. Rend. 225, 182 (1947).16. L. I. Epstein, J. Opt. Soc. Am. 42, 806 (1952).17. L. Young and E. G. Cristal, IEEE Trans. MTT-14, 75

(1966).18. P. H. Berning, J. Opt. Soc. Am. 52, 431 (1962).19. X. Iwata, S. Katsube, and T. Fukuda, Sci. Light Tokyo

7, No. 2, 33 (1958).20. M. Iwata, Sci. Light Tokyo 2, 116 (1953).21. F. Abels, Compt. Rend. 226, 1872 (1948).22. A. lerpin, Compt. Rend. 17 (1947).23. F. Abelhs, Ann. Phys. 5, 596, 706 (1950).24. K. D. Mielenz, J. Res. Natl. Bur. Std. 63A, 297 (1959).25. K. D. Mielenz, J. Opt. Soc. Am. 50, 1014 (1960).26. C. D. and A. Herpin, Rev. Opt. 32,321 (1953).27. P. G. Kard, Opt. Spectry. 11, 49 (1960).28. P. G. Kard, Opt. Spectry. 14, 121 (1963).29. L. Young, IEEE Trans. MTT-10, 339 (1962).30. R. Levy, IEEE Trans. MTT-13,514 (1965).31. L. Young, IEEE Trans. MTT-13, 488 (1965).32. G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave

Filters, Impedance Matching Networks, and Coupling Struc-tures (McGraw-Hill Book Co., Inc., New York, 1964).

312 APPLIED OPTICS / Vol. 6, No. 2 / February 1967

Table A-I. Excess Loss Ratio in Decibels, 10 log (sg) = 10 loglo [Tm2 (X)]a

2

0 .00000.03470.06930 .10380. 13810. 17240.20660.24070.27470.30860.34230.67490.99831.31311.61971. 91872.21042.49532.77363. 04585.48327.53159.3077

10. 881412.297913. 588614.775615. 875816. 902024. 609029.827233.803937.025239.735442.076144. 136545.9771

3

0.00000. 07790. 15540. 23230.30880.38480.46040.53550.61010.68430.75811.47312.15042.79423.40823.99554.55865.09995.62126.1242

10.401813.782616.619519.084921.277223.258525.070626.743628.299539.912747.747853.714858.547462.613166.124269.215071.9758

4

0 .00000.13820.27490.41020.54410.67650.80770.93751.06601.19331.31932.51783.61904.63995.59366.48987.33668.14008.90549.6369

15.662120.281024.111027.421430.356833. 005535.426037.659239.735455.223565.670573.626680. 070185.491090.172594.293597.9746

5

0 .00000.21530.42710.63550.84051. 04251.24131.43731.63031.82072.00843.75685.31456.72598.02119.221A

10.343811.399212.397313.345621.025526.835431.634935.777539.448942.760845.786948.578851.174270.534583.593393.5384

101.5927108.3689114.2207119.3720123.9734

6

0 .000 00.30900.61070.90571.19421.47661.75322. 02422.29002.55082.80675.14237.16428.9611

10.588012.081313.466614.762515.982817.138426.419033.402139.164444.136548.542552.517056.148459.498662.613185.8456

101.5161113.4503123.1154131.2468138.2690144.4505149.9722

7

0 .0 0000.41880.82451.21821. 60061.97252.33462.68753.03173.36773.69616.63459.1150

11.286013.232315.007016.645918.173919. 609420.966331. 821239.971746.695052.496057.636362.273266.50970.418474.0521

101.1566119.4389133.3621144.6381154.1247162.3173169.5290175.9710

8

0 .00000.54431. 06701.57002. 05502.52352.97693.41623.84254.25674.65968.2026

11.130413.663415.918717.966619.852321.607223.253824.808937.226046.541854.225860.855466.730272.029576.871381.338385.4910

116.4676137.3617153.2739166.1607177.002618o.3656194.6075201.9698

9 10

0. 00000.68521.33671.95812.55273.12313.67164. 20024.71075.20445.68279.8236

13.186416.071218.628020.943223. 071825.050626.906028.657442.631553.112061.756569.215075.824081.785787.232892.258296.9300

131.7786155.2845173.18571A7.6R34199.8805210.4138219.6861227.9686

0.00000.841n1.63192.37943.08903.76504.41125. 03065.62596.19936.752A

11. 481n15.267q18.496521.349423.928626.297528.498530.561F32.508948.037259. 682369.287377.574984.917991.542097.5943

103.1780108.3689147.0897173.2073193.0975209.2061222.7584234.4621244.7646253.9674

All 11 12 13 14 15 16 17 18 19 20

0 .00001. 19482.29213. 30944.25975.15315. 99746.79917.56328.29418.9952

14.862719.471823.372926.808829.910032.756135.399237.876040.213058.848772.822984.348994.2935

103.1056111.0545118.3173125.0178131.2468177.7117209.0528232.9211252.2514268.5142282.5586294.9216305.9650

0.00001. 39182.65383.81234.88615.88936.83267.72458.57169.3795

10.152616.573421.584625.817229.542032.903035.986838. 850541.533844. 065764.254479.393291.8797

102.6530112.1994120.8108128.6788135.9376142.6858193.0228226.9756252.8329273.7741291.3921306.6069320.0001331.9638

0.00001.60143. 03444.33695.53516.64797.68948.67049.5994

10.483011.326918.291923.701128.263432.276335. 896539.217842.301945.191747.918469.660285.963599.4105

111.0125121.2933130. 5670139. 0403146.8575154.1247208.3338244.8984272.7447295.2968314.2700330.6552345.0786357.9626

0. 00001.82303. 43224.88076. 20357.42548.56409.6329

10.642411.600612. 514020. 015625.819830. 710635.011138. 890242.449045.753548.849751.771275.066092.5338

106.9413119.3720130.3871140.3233149.4018157.7774165.5637223.6448262. 8212292.6565316.8194337.1479354.7034370.1572383.9614

0.00002. 05603.84585.44146.88868.21859.4531

10.608711.697512.729013.710921.742727.940033.158437.746241.884145.680249.205052.507755.624080.471899.1041

114.4721127.7315139.4810150.0796159.7633168.6972177.0026238.9558280.7440312.5683338.3421360.0258378.7517395.2357409.9603

0. 00002.29984. 27386.01707.58809.0248

10.354111.595112.762013.865814.915223.472230. 060935.606640.481444.878148.911552.656656.165759.476785.8775

105.6744122. 0029136. 0910148.5748159.8358170.1247179.6171188.4416254.2669298.6668332.4801359.8648382.9037402.8000420.3142435.9591

0.00002.55394.71476.60578.29949.8420

11.264712.589913.833915. 009116.125225. 203332.182438.055043.216747.872152.142856.108259.823763.329591.2833

112.2447129.5337144.4505157.6687169.5921180.4862190.5370199. 8805269.5779316.5896352.3920381.3874405.7816426.8483445.3927461.9579

0.00002.81765.16737.20589.0212

10.668212.183113.591414.911516.157317.339426.935434.304240.503545.952150.866155.374059.559863.481767.182396.6891

118.8150137.0645152.8100166.7625179.3483190.8477201.4568211.3195284.8889334.5123372.3038402.9101428.6595450.8965470.4712487.9567

0.00003.09045.63067.81589.7517

11.501913.107914.598315.993717.309318.556828.668336.426242.952048.687553.860158.605363.011467.139871.0351

102.0948125.3853144.5853161.1695175.8564189.1046201.2092212.3767222.7584300.199Q352.4351392.2156424.4328451.5374474.9448495.5497513.9555

a See Eqs. (18) and (32).

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 313

1

1.0001.0011.0021.0031 . 0041 . 0051. 0061.0071.0081.0091.0101.0201.0301.0401.0501.0601.0701.0801.0901.1001.2001.3001 .* 001.5001.6001.7001.8001.9002.0003.0004.0005.0006.0007.0008.0009.000

10. 0 00

0 . 00000.00870 01740.02600. 03470. 04330. 05200.06060 .06920.07780.0 8640.17200.25670.34070.42380.50610.58770.66850.74850 82791.58362.27892.92263.52184.08244.60905.10555. 57516.02069.5424

12. 041213.979415. 563016.902016 0b1819. 084920.0000

1.0001.0011.0021.0031.0041 0051. 0061.0071.0081. 0091.0101 . 0201.0301.0401.0501 .0601.0701 .0801.0901.1001.2001.3001.4001.5001.6001.7001.8001.9002.0003.0004.0005.0006.0007.0008.00G9.00010 . 0 0 0

0. 00001.01101.95092. 83093.65944.44345. 18855. 89926.57927. 23197.8597

13.163417. 365020.931824.077326.918229.526131.948434.218436.360553. 442966.252676.818185. 934094. 0117

101.2983107.9558114. 0979119.8079162.40 07191 .1301213.0093230.7287245.6363258.5104269.8431279.9662

continued

Table A-I. Continued

35 40 45 50 55 60 65 70

0 .00009.7553

16.004520.902625.051428.711432.021435.065337.898040.558143.073563.351278.872191.9244

103.3955113.7408123.2313132.0434140.3001148.0908210.2103256.7911295.2112328.3596357.7333384.2298408.4390430.7740451.5374606.4205710.8909790.4517854.8861909.0955955.9102997.1201

1033.9315

0 .000011.611818.725024.256328 930633. 050736. 775640.200443.387546. 380249.210172. 022789.4836

104.1675117.0725128.7110139.3878149.3014158.5902167.3547237.2391289.6426332.8651370.1572403.2025433.0111460.2465485.3733508.7322682.9756800.5048890.0108962.4995

1023.48501076.15151122.51271163.9255

0.0 oo13.498721.457627.615132.812137.391241. 530445.336048.877352.202555.346980.6942

100. 0952116.4106130.7495143.6812155.5442166.5594176.8803186.6186264.2680322.4940370.5191411.9547448.6718481.7924512.0539539.9726565.9269759.5308890.1187989.5698

1070.11281137.87451196.39291247.90521293.9196

0 .000015.405324.196630.976236.694641.732146.285450.471654.367058.024861.483689.3657

110.7068128.6537144.4265158.6514171.7007183.8174195.1704205.8826291.2969355.3455408.1731453.7522494.1410530.5737563.8614594.5719623.1217836.0859979.7327

1089.12891177.72611252.26401316.63421373.29781423.9136

0 .0 00017.324626.938934.338540.577446.073151.040455.607259.856863.847167.620398.0371

121.3184140.8969158.1035173.6215187.8572201.0754213.4605225.1465318.3257388.1970445.8271495.5497539.6103579.3550615.6689649.1713680.3165912.6410

1069.34661188.68791285.33951366.65351436.87561498.69041553.9076

0 .000019.252129.683137.701244.460550.414255.795560.742865.346669.669473.7571

106.7086131.9300153.1400171.7805188.5917204.0137218.3334231.7505244.4104345.3546421.0484483.4810537.3473585.0795628.1363667.4763703.7706737.5112989.1962

1158.96051288.24701392.95281481.04301557.11691624.08301683.9016

0.oon21.184832.428341.064248.343654.755360.550665.878570.836475.491879.893R

115.3801142.5415165.3831185.4575203.561Q220. 170?235.5915250.0406263.6741372.3834453.8999521.1350579.144A630.5487676.9176719.2839758.3699794.7060

1065.75131248.57451387.806n1500.56621595.43251677.35831749.47561813.8956

.80 90 100 110 120 130 140 150 160 170

0.0000 0.000032.8218 36.705248.9047 54.397261.2432 67.969671.6425 79.408880.8021 89.484489.0810 98.591296.6924 106.9637

103.7751 114.7546110.4256 122.0702116.7143 128.9878167.4090 184.7519206.2110 227.4342238.8418 263.3281267.5196 294.8736293.3830 323.3233317.1091 349.4220339.1395 373.6555359.7812 396.3613379.2579 417.7857534.5566 588.6143651.0087 716.7116747.0588 822.3668829.9300 913.5250903.3642 994.3027969.6054 1067.1680

1030.1285 1133.74341085.9658 1195.16451137.8745 1252.26401525.0821 1678.19241786.2581 1965.48591985.1602 2184.27832146.2462 2361.47292281.7696 2510.54862398.8064 2639.28912501.8311 2752.61632593.8597 2853.8478

0 .000040.589159.889774.696087.175198.1667

108.1014117.2350125. 7342133. 7149141.2613202.0949248.6574287.8143322.2276353.2637381.7350408 1715432.9415456.3136642.6720782.4145897.6747997.1201

1085. 24111164.73061237.35831304.36311366.65351831 .30272144. 71382383.39642576. 69962739.32762879. 77183003.4014

0.000044.473165.382281.422394.9414

106.8490117.6115127.5063136.7138145.3595153.5348219.4378269.8805312.3006349.5816383.2040414.0480442.6875469.5217494.8414696.7298848.1175972.9826

1080.71511176.17961262.29321340.97321413.56171481.04301984.41302323.94172582.51452791.92622968.10663120.25453254.1866

3113.8358 3373.8238

314 APPLIED OPTICS / Vol. 6, No. 2 / February 1967

25 30

1.0001.0011.0021.0031.0041.0051.0061.0071.0081.0091.0101.0201.0301.0401.0501.0601.0701.0801.0901.1001.2001.3001.4001.5001.6001.7001.8001.9002.0003.0004.0005.0006.0007.0008.0009.0100

10.000

0 .0 0004. 57198. 0710

10.974313.494015 743517.791119.681121.443923. 101624.670537. 337247.037455. 195062.364468.830374.761880.269485.429890.2990

129.1237158.2367182. 2493202.9670221.3256237.8859253.0167266.9760279.9532376.7551442.0491491.7746532.0461565.9269595.1861620.9423643.9495

0 .00006.2061

10.650314. 241517.318920.047822.522124.800526.922528. 916130.801746. 008357.648967.438176. 041583. 800590.918397. 5274

103.7199109. 5629156. 1526191. 0882219.9032244.7646266.7948286.6672304.8241321.5753337.1479453.3102531.6630591.3337639.6594680.3165715.4275746.3349773.9435

0 .00007.9454

13.306517.560121.178124.375327.269129.931132.409134.736336.937154.679868.260579.681389.718598.7706

107. 0748114.7854122.0100128.8269183.1814223.9396257.5572286.5621312.2641335.4485356.6316376.1746394.3427529.8654621.2769690.8927747.2728794.7060835.6688871.7275903.9375

\m

1.0001.0011.0021 0031. 0041.0051.0061 .0071.0081.0091. 0101.0201.0301.0401.0501.0601.0701.0801.0901 . 1001.2001.3001.4001.5001 .6001. 7001.8001.9002. 0 003. 0004.0005.0006. 0007.0008. 0009. 000

10. 000

0 .000025.059237.919947.790556.109963.437670.060776.149881.815987. 136492.1673

132.7230163.7647189.8693212.8115233.5022252.4831270.1075286.6208302.2022426.4412519.6028596.4429662.7398721.4872774.4802822.8987867.5685909.0955

1218.86161427.80231586.92411715.79281824.21161917.84102000.26072073.8837

0 .000028. 939343.412254.516863.876272.119979.570986.421192. 795598.7810

104. 4408150.0 660184.9879214.3556240.1656263.4426284.7961304.6235323.2010340.7300480.4989585.3058671.7509746.3349812.4257872.0428926.5136976.7672

1023.48501371.97191607.03021786.04221931.01952052.99062158.32372251.04592333.8717

0.000048.357270.874788.1487

102.7077115.5312127.1217137.7776147. 6933157. 0041165 8083236.7808291 . 1037336.7868376.9356413.1444446.3609477.2035506.1019533.3693750.7875913.8204

1048.29061164.31021267. 11811359.85581444.58811522.76041595.43252137. 52322503.16952781.63263007.15293196. 88573360.73723504.97183633.8119

0.000052.241376.367394.8751

110.4740124.2135136.6319148.0489158.6729168.6487178.0817254.1237312.3269361.2730404.2897443.0847478.6739511.7195542. 6821571.8971804.8452979.5233

1123.59851247.90521358.05661457.41851548.20301631.95901709.82202290.633526P2.39742980.75073222.37963425.66473601.21993755.75693893.7999

0.000056.125481.8598

101.6015118.2403132.8958146.1420158. 3202169.6525180.2933190.3552271.4667333.5500385.7593431.6437473.0251510.9869546.2355579.2622610.4250858.9029

1045.22621198.90651331.50031448.99501554.98111651.81801741.15771824.21162443.74382861.62533179.86883437.60633654.44373841.70264006.54214153.7879

o. 00on60.009587.3523

108.3279126.0066141.5781155.6522168.5915180.632n191.9380202.6287288.8096354.7732410.245F458.9977502.9654543.2998580.7515615.8424648.952P912.9606

1110.92921274.21441415.0Q541539.93351652.54371755.43201850.35631938.60112596.85413040.85313378.98683652.83303883.222740A2.18534257.32734413.7760

-

33. C. L. Dolph, Proc. Inst. Radio Engrs. 34, 335 (1946). Seealso H. J. Riblet and C. L. Dolph Proc. Inst. Radio Engrs.35, 489 (1957).

34. L. B. Brown and G. A. Sharp, "Tchebyscheff Antenna Dis-tribution, Beamwidth, and Gain Tables", NAVORD Report4629, NOLC Report 383, Naval Ordnance Laboratory, Co-rona, Calif. (February 1958).

35. M. L. Reuss, Jr., "Some Design Considerations ConcerningLinear Arrays Having Dolph-Tchebycheff Amplitude Dis-tributions", NLR Report 5240, Project NE 120-000-27,U. S. Naval Research Laboratory, Washington, D. C.(12 February 1959) (AD No. 212 621).

36. G. J. Van der Maas, J. Appl. Phys. 25, 121 (1954).

37. G. J. Van der Maas, J. Appl. Phys. 21, 1250 (1953).38. C. J. Drane, Jr., Proc. IEEE, 110, 1755 (1963).39. R. S. Elliott, IEEE Trans. AP-li, 707 (1963).40. R. S. Elliott, IEEE Trans. AP-li, 378 (1963).41. R. J. Stegen, Proc. Inst. Radio Engrs. 41, 1671 (1953).42. R. J. Stegen, Inst. Radio Engrs. Trans. AP-8, 629 (1960).43. D. Barbiere, Proc. Inst. Radio Engrs. 40, 78 (1952).44. L. L. Bailin and M. J. Ehrlich, Proc. Inst. Radio Engrs. 41,

235 (1953).45. H. E. Salzer, "The Use of Poisson's Formula in Pattern

Synthesis", DOFL Report TR-28, Project No. 4104-10-6448, Diamond Ordnance Fuze Laboratories, WashingtonD.C. (19 January 1954).

Meeting Reports continued from page 296

on radiation disordering of crystalline quartz, G. J. DienesBrookhaven on radiation enhanced diffusion in metals, R. W.Douglas University of Sheffield on spectroscopy of glasses, J. H.Van Vleck Harvard on crystal field parameters of rare earthions, W. A. Weyl Penn State on optical phonon interactions,and C. J. Delbecq Argonne on electron and hole reactions inKCl:Tl. A number of contributed papers on a wide varietyof topics were presented by Egyptian scientists who work in thelaboratories of American University, the University of Cairo, theAtomic Energy Establishment, and the National ResearchCouncil.

In addition to the scientific program, our Egyptian hostsgraciously gave the conferees an opportunity to discover thegrandeur of Egyptian history. Tours were arranged for visits inthe vicinity of Cairo (the pyramids at Giza and Sakkara, themarvelous museum, and the sites of Old Cairo), and also in thesouth (Luxor and Aswan); the latter tour took place aftercompletion of the formal program. The kind hospitality andthoughtful attention made the conference both professionallyprofitable and personally enjoyable.

During the conference it was announced that two devicesinvented by Thomas A. Edison had been donated to the Uni-versity. The items, an incandescent lamp employing a filamentof carbonized bamboo, and the Edison Electromotograph, weredisplayed in the Science Building. Edison developed the Bam-boo Filament Lamp in 1880, and the type remained in use duringthe subsequent decade. The particular lamp donated to theUniversity was constructed in 1884; it had been stored formost of the intervening years at Edison's winter home in FortMyers, Florida. The Electromotograph, consisting of an elec-trical contact touching a rotating chalk cylinder, demonstratesEdison's discovery that changes in current produce changes inthe friction between sliding electrical contacts. Edison firstdescribed the phenomenon in 1874. The device displayed inCairo apparently was used in Edison's experiments with thephonograph about 1897 to 1900.

The proceedings of the conference will be published in bookform, and will include the discussion. Inquiries should beaddressed to Adli Bishay, Chairman, Department of PhysicalSciences, American University in Cairo, 113 Sharia Hasr El Aini,Cairo, UAR.

51st Annual Meeting of the Optical Societyof America, San Francisco, 18-21 October 1966Reported by Jurgen R. Meyer-Arendt, Pacific UniversityThe OSA 51st Annual Meeting was part of the San FranciscoScience Symposium, sponsored jointly by the OSA, the AmericanChemical Society, and the Northern Calif. Society for Spectro-scopy. In terms of attendance and number of papers presented,the meeting has broken all records-about 2200 registrants-but

final figures were not available at the time it ended. Most of the31 invited papers and addresses and all of the 201 contributed pa-pers were delivered in simultaneous sessions, without distractingfrom the general satisfaction. Contributing to this feeling was thesmooth handling of the details and the favorable location of theexhibition area, just outside the meeting rooms.

The meeting began in an applied way, with a symposium oit fe-mote sensing. Sensing the earth's surface may involve portable,self-contained spectrophotometers, the use of Ektachrome film>.and of a special aerial camouflage detection film making possible-color translation from the infrared into the visible, and a combi-nation of images taken in the visible and the infrared (R. N-Colwell and G. A. Thorley Berkeley). Sensing planetaryatmospheres is greatly facilitated by Fourier spectroscopy, basedin part on the use of a modified Michelson interferometer madevirtually insensitive to rotation, vibration, and other causes ofmechanical misalignment (R. Beer JPL).

The first full day of the meeting opened with a broadly based re-view of spatial filtering, Fourier transforms, phase contrast, andcharacter recognition (A. W. Lohmann and D. P. Paris IBM),followed by a particularly well-illustrated survey of known andpotential applications of holography (J. W. Goodman NASA).Among such applications are microscopy, imaging through aber-rating media, information storage, cryptography, doppler mappingof moving objects, displays in commercial advertising, the mea-surement of particle sizes in aerosols, vibration analysis, mappingof planetary surfaces, and bubble chamber photography. Ofparticular interest were the results of inserting distorting media,such as shower glass, when making the hologram. Then decod-ing is possible only by using an identical distorting plate in thereconstruction process.

Out of the many contributed papers only a few can be listed.There were talks on synthetic antennas-one of the important de-velopments in radar technology-(E. N. Leith, L. J. Cutrona, andL. J. Porcello University of Michigan), on eigenvalue problems incoherence theory and its mechanical analog (E. Wolf and D. Dia-letis University of Rochester), and a demonstration of the nowfamiliar full-color hologram reconstructions (L. H. Lin and C. V.LoBianco BTL). Fresnel holograms of a hair with a knot tied intoit can be obtained conveniently (G. 0. Reynolds Tech Ops), theuse of transfer functions has led to still another practical measur-ing device (N. S. Kapany Optics Technology).

The same evening most of the Technical Groups met. Duringthe meeting of the Atmospheric Optics Technical Group, for in-stance, reports were given on the Woods Hole Summer Study onRestoration of Atmospherically Degraded Images. There aretwo main approaches to restoring an image that had been de-graded by transmission through atmospheric turbulence: (1)passive image processing (J. Harris), which is facilitated greatly ifa known point source is located near the object, giving a pointspread function that can be utilized in processing and restoringthe image of the more complex object; and (2) active techniques

February 1967 / Vol. 6, No. 2 / APPLIED OPTICS 315