Interference Filters for the Far Infrared

Reinhard Ulrich

Perforated metal plates (grids) of various kinds are used for the construction of transmission filters forthe far ir. Examples are given of filters with low pass, high pass, bandpass, and bandstop characteristicswith steep slopes. These filters are the optical equivalents of microwave waveguide filters. They canbe designed by the same theoretical procedures as those to meet a wide variety of different specifications,at least in principle. Actually, losses and constructional tolerances limit the performance. Measure-ments at oblique incidence indicate that the filters will find useful application in light pipes too. Further-more, they may prove advantageous in short millimeter wave systems that employ optical techniques.

1. IntroductionA metal plate, perforated in a two-dimensional

periodic pattern of square symmetry, can be consideredin many respects as the optical equivalent of an iris in awaveguide.' Such structure will be called a grid, andthe simplest examples for grids are metal mesh and itscomplementary structure. 2 One important difference,however, between a grid and a waveguide iris is theabsence of a cutoff frequency for the grid. Thus, gridsare useful over much broader frequency ranges thanirises. Especially this property makes it appear at-tractive to use grids for the construction of filters thatare the optical equivalent of iris coupled waveguidefilters. These waveguide filters, widely employed inmicrowave techniques, can be designed to meet a greatvariety of different specifications.3 It is to be expectedthat the optical filters, constructed from grids, willhave a comparable performance.

While the main objective of this paper is to demon-strate that high performance filters of various typesactually can be constructed of grids, it must be em-phasized here that the use of many grids in one filteris essential to achieve such performance. One possiblemeasure of the performance is the slope of the filtercharacteristic at the transition from a passband to thestopband, e.g., at a transmission T = 0.1. As gen-erally in the combination of filters, this slope [defined asd(logT)/d(logv) and expressed, e.g., in dB/octave] isroughly proportional to the number of sections in thefilter (Ref. 3, p. 156). This is one reason why manygrids are of advantage. Another reason is that, byadding more grids to the simplest possible two-gridfilter,4 the number of design parameters increases that

The author was with the Ohio State University, Columbus,Ohio, when this work was done; he is now with Bell TelephoneLaboratories, Inc., Holmdel, New Jersey 07733.

Received 8 April 1968.

can be adjusted in order to shape the filter characteristicin a desired way, e.g., to produce an approximately flatpassband or a very wide stopband.

11. Construction and Measurement of the FiltersAll filters that are discussed here have been con-

structed for the spectral region of about 30-200 cm-',since this region was covered best by the availablespectrometer.5 6 In the construction of filters for thesehigh frequencies, some basic difficulties are encounteredthat play only a minor role at low microwave frequen-cies. Both the resistive losses (skin effect) in the metalparts of the grids and the dielectric losses in the filmssupporting some types of grids increase with frequencyand begin to affect seriously the performance of thefilters. The other difficulty is that of the construc-tional tolerances. Their influence increases propor-tional to the frequency. In this respect, the most criti-cal parts in the construction of the filters (Fig. 1) arethe spacers that determine the parallelism and flatnessof the grids. In order to avoid any burrs, these spacerswere photoetched from brass shim stock and werecarefully inspected for dust particles before the finalassembly of a filter. Thus, it was possible to keep twoadjacent grids flat and parallel to within 1 At. Thiscould be checked by observing Haidinger's fringes inmonochromatic light reflected from both grids. Theabsolute values of the spacings are believed to be wellwithin 4 2 ut of the values quoted below. The con-struction of the filters permitted them to be reassembledrepeatedly, e.g., after changing the spacers. For apermanent filter, the construction used in Ref. 7 seemspreferable. When this is applied to grids on a filmsubstrate in a filter that is to operate in vacuum, how-ever, provision must be made to vent the spaces betweenthe grids.

Another difficulty in the construction of a filteraccording to a given specification is the limitednumber of different grids that are commerciably

October 1968 / Vol. 7, No. 10 / APPLIED OPTICS 1987

, _=_ _ _ _ _ -pcr

0Siid} A = ==>|||A1\11

,1 cm

Fig. 1. Construction of multigrid filters (schematic). Each gridis glued to a ring and stretched like a drumskin by means of threescrews and nuts (only one shown). Surface A is optically flat.

available, and the insufficient knowledge of their opticalproperties, especially of their reflection phase +(P). Apossible way to solve this latter problem would be toconstruct high order, two-grid filters from a pair ofeach of the grids in question, with accurately knownspacings. From the measured frequencies of the trans-mission peaks of these filters (e.g., like Fig. 5 of Ref. 7),the required reflection phase q(v) could be directlydetermined at a number of discrete frequencies. Ifsmall adjustments of the normalized impedance of ametal mesh are necessary in order to meet the require-ments of a design procedure, one may consider thepossibility of reducing the thickness and strip width ofthe mesh by etching, or to increase them8 by electro-plating.

Except for the metal meshes, all grids to be discussedconsist of a 1-,u thick copper layer supported by a 2.5-athick PTER* film. They were prepared by a photo-lithographic process, mentioned briefly already in Ref.2, that will be described in detail elsewhere9 (comparealso Ref. 10).

All measurements were made with the asymmetricMichelson interferometer described by Bell' and Russeland Bell.6 In this instrument, the sample under in-vestigation is placed in one arm of the interferometerand not, as usual in other instruments, at a place com-mon to both beams. Consequently, the Fourier trans-form of the interferogram is the complex amplitudetransmission spectrum r(v) of the sample. 5 This modeof operation has two important advantages for themeasurements presented here.

(1) In addition to the absolute value of the trans-mission, the phase , = argr(v) is given by the in-strument, too. For thick and/or lossy grids, thisphase differs from the value arccosirj valid for thin,lossless grids.",2 The knowledge of the true phases

P6T(v) was of great help in the design of the filters.Actually, as mentioned, the reflection phases are ofprimary importance. These could also be measured bythe instrument at normal incidence. It was, however,not possible to obtain sufficiently accurate values of thereflection phases of grids in this way, since such mea-surements require that the grid be replaced by a refer-ence mirror whose position coincides with the surfaceof the grid within 1 or better. Dust particles andplasticity prevented this for the very thin and flexible

* Polyethylene terephthalate resin. This film (trade nameHOSTAPHAN RE 2.5) was kindly supplied by KALLE A.G.,6202 Wiesbaden-Biebrich, Germany.

grids. Therefore, the reflection phase k had to beapproximated via the transmission phase that could bemeasured accurately.

(2) In normal operation, the spectral signal/noiseratio was of the order of 100. This limits the smallestdetectable transmission to about 10-2. Since in theasymmetric interferometer this figure is an amplitudetransmission 17|, the power transmission T = I r|2 couldbe determined down to values of the order of 10-4.

Therefore, asymmetric Fourier spectroscopy isuniquely suited for the measurement of the high stop-band attenuation of some of the filters. The achieve-ment of this low minimum measureable power trans-mission depends, of course, on many factors likespectral intensity of the source, efficiency of the beam-splitter, chosen spectral resolution, and scanning speed.Since these conditions were not constant, not all givenspectra have the quoted high accuracy in all theirspectral regions. Therefore, some (especially therapidly oscillating, e.g., in Fig. 3) spectral structures inthe range T = 10-' to 10-4 of the given spectra are notreproducible, but rather represent noise. All powertransmissions T > 10-3 are believed to be real. Onespecial feature of the interferometric method must bementioned, however, in this connection. All measuredtransmissions refer only to radiation that passescoherently through the samples, i.e., without distortionof the wavefronts. Any energy that is scattered intoother directions is not detected. It could, however,be detected in a grating spectrometer, for example. Ifthe filters are to be applied there, care has to be taken ofthis scattered radiation which may appear in the diffrac-tion region of the filters.

All filters were measured at normal incidence unlessotherwise stated. Because of the square symmetry ofall grids, their properties are independent of polariza-tion at normal incidence, and unpolarized radiationcould be used in all normal incidence measurements.The radiation had an f/10 aperture at the position ofthe sample. The equivalent slit width indicated in the

1.0

n

I. lo-

U 100 UU

FREOUENCY CM-'

Fig. 2. Measured transmission T of low pass filters composed offour identical grids (capacitive squares g = 51 p, a/g = 0.18) withthree equal spacers. The thickness of the spacers is the identify-ing parameter. The transmission of a single grid is shown for

reference.

1988 APPLIED OPTICS / Vol. 7, No. 10 / October 1968

ingle grid /

25

18

T

measured spectra is the half-value width of the apodizedapparatus function for the amplitude measurement.

Ill. Low Pass Filters

A. Low Pass Filters Made from Identical GridsThe construction and behavior of some low pass

filters made from identical grids of capacitive squareshad already been described in Ref. 11. At this time,more experimental data are available: Fig. 2 shows themeasured transmission characteristics of three low passfilters. Each filter consists of four identical grids(capacitive squares, grid constant g = 51 A, half rela-tive gap width a/g = 0.18). The grids are equallyspaced in each filter, but the value of the spacing sis different for the three filters.

The low pass characteristic of such four-grid filtercan be understood by considering the filter as a com-bination of two two-grid subfilters that are separatedby the central spacer and that act as the reflectors of themain interference filter. Each two-grid subfilter hasa low pass characteristic itself, the transition frompassband to stopband is, however, a rather smooth one(compare Fig. 3 of Ref. 11). Such a main interferencefilter, combined properly from two identical subfiltersof poor filter characteristic but of low losses, hasgenerally a filter characteristic that is better than theproduct of the two subfilter characteristics. Thereason for this is that the subfilters have high reflectivi-ties in their stopband, resulting in a high finesse andlow minimum transmission of the main filter. Thus,any transmission maximum of the main filter occuringin the stopband of the subfilters will be very sharp andis moreover likely to be spoiled in its peak transmissionby losses and imperfect adjustment. Such maximatherefore appear only as spurious peaks in the stop-band of the main filter. In the passband of the sub-filters, on the other hand, the main filter has low finesse.Consequently, any interference maxima there arebroad and appear only as smooth ripples of the pass-band transmission. If the spacing is chosen properly,an interference maximum is obtained right at the edgeof the subfilter passband. The wing of this maximumat the side of the subfilter stopband represents thetransition to the stopband of the main filter. It hasa much steeper slope than have the subfilters, sincethe slope of the wing of the Airy function adds to thesum of the subfilter slopes.

This somewhat coarse explanation is illustrated bythe dependence of the filter characteristic on the thick-ness of the spacers in Fig. 2. The filter with s = 30-Aspacers shows a narrow spurious transmission peak ofthe type mentioned at 126 cm-' in the stopband.When the thickness of the spacers is reduced to s =25 , this peak is shifted to 141 cm-'. Its height isreduced, since, at the higher frequency, the transmis-sion of the single grid (also given in Fig. 2, dotted line)is lower so that the finesse and with it all peak reducinginfluences are increased. For 18-/u spacing, the spur-ious peak has disappeared below the threshold of de-tectability. With decreasing the spacing from 30 u to

25 and to 18 , the cutoff frequency of the filter,measured at T = 0.1, increases from 71 cm-' to 80cm- 1 and to 86 cm-', and the cutoff slope at the T =0.1 point increases from 58 dB/octave to 64 dB/octaveand to 71 dB/octave, respectively.

The spacing of s = 25 tz corresponds to the spacingXR/

2 recommended in Ref. 11, if for the resonant wave-length XR the measured wavelength (60 A) of the trans-mission minimum of the single grid is used. The re-maining difference of 5 , is allowed for the thickness ofthe grids, since in Ref. 11 infinitely thin grids were as-sumed [compare also Eq. (3) below]. This s = 25 characteristic agrees roughly with the curve calculatedin Ref. 11 for a four-grid filter from a model equivalentcircuit if Z = 0.8 is used there for the characteristicimpedance of the grid. This value of Zo is consistentwith the Z0 calculated from the transmission of a singlegrid in the 80-90-cm- 1 region. The slope of 64 dB/oc-tave measured for this four-grid filter is, however, muchsmaller than the approximately 104 dB/octave expectedfrom the model. This discrepancy is certainly due tolosses that are higher than was assumed for the modeland to imperfect adjustment of the parallelism. Theseeffects also greatly reduce the peaks of the ripple in thepassband.

One major flaw of the filters in Fig. 2 is their hightransmission peak at about 194 cm-'. Several otherpeaks exist also between 250 cm-' and 400 cm-', theirpeak transmissions ranging from 0.02 to 0.08. In thisregion v > /g = 196 cm-', the grids act as two-di-mensional diffraction gratings, in addition to theirsimple semitransparent action at v < 1/g. Thus, thefunction of the filters is a qualitatively different oneabove and below v = 1/g. At low frequencies, all energythat is not transmitted by the filters is either reflectedor absorbed, whereas at high frequencies v > 1/g, partof the energy may also be diffracted into various direc-tions of space. In this frequency region, the filtersact partially as scatter filters. Thus, the mentionedpeaks between 250 cm-' and 400 cm-' are probablycaused by diffraction and subsequent rediffraction ofradiation into its original direction.

The peaks at 194 cm-', however, are of a slightlydifferent nature, since they occur partially inside thenondiffraction region. They cannot be spurious peaksin the stopband of the type discussed above, since theirposition is practically independent of the spacing. Thefact that they occur slightly below the wavenumber1/g suggests the explanation that they are caused byinteraction of the evanescent waves of the grids. Thisis a limiting case of rediffraction: at v = 1/g, the four(1,0) diffraction orders become propagating. At lowerfrequencies, they are evanescent and their amplitudeis attenuated to 1/e at a distance (g/ 2 7r)(1 - g2zA)-lfrom the grid. For v > 190 cm-', this distance islarger than the spacings of the grids, and the evanescentwaves can transfer substantial amounts of energy fromone grid to the next. At the last grid, the incidentevanescent waves can be diffracted to form propagat-ing waves, emerging normally from the surface. Thisexplanation is supported by the fact that the s = 30-y

October 1968 / Vol. 7, No. 10 / APPLIED OPTICS 1989

filter, where the interaction should be weakest, has thenarrowest peak, and the s = 18-,u filter has the broadest.It is not understood, however, why the heights of thethree peaks are equal. A further check on the in-fluence of the evanescent waves was made by rotatingthe grids of the s = 5 -,u filter in their planes. In allfilters of Fig. 2, the symmetry axes of the four gridsformed the angles 00, 350, 6, 280 with some referencedirection. When this orientation was changed to 00,450, 150, 600, the peak at 194 cm-' was reduced to T =0.02, and in the case that all axes were parallel, it wasreduced to T = 0.01. These orientation effects alsoindicate that higher order modes are involved.

The s = 2 5 -u filter has also been tested at oblique in-cidence with unpolarized radiation, in order to exploreits suitability for use in a light pipe. At an angle ofincidence of 0 = 27°, which corresponds to the marginalrays in an f/1 optic, the cutoff frequency was increasedby about 2 cm-'. The whole cutoff characteristic inthe T < 0.1 range was shifted by the same amount, andthree spurious peaks of T < 0.004 showed up in thestopband. The corner of the passband at T > 0.1 wasrounded off. Since at small angles of incidence allthese effects must depend on 02, the low pass filters seemwell applicable in a light pipe at moderate apertureratios.

B. Low Pass Filters Made from Different Grids

The problems of interaction of the evanescent wavesand of rediffraction can be greatly reduced by combininga filter from grids of different grid constants g. In suchfilters, the effects mentioned are unlikely to occur simul-taneouslyat all grids. Thus, the heights of the spuriouspeaks can be expected to be lower. An additional ad-vantage of such a filter arises from the fact that thedifferent grids have their transmission minima at dif-ferent frequencies. Instead of one very deep minimumat the single frequency , as in the curves of Fig. 2,there will now be several minima. Both effects assistin keeping the stopband attenuation more uniformlyhigh.

The problem of determining the best combinationof grids and the required spacings is unsolved. None ofthe existing design procedures3 is rigorously applicableto low pass filters of capacitive grids, especially not inthe diffraction region. Probably not a best, but never-theless useful, low pass filter can be designed by knownprocedures if attention is focused at the edge of thepassband. In order to demonstrate this, it is assumedthat a maximally flat or equal ripple bandpass filter hasbeen designed by a standard procedure3 for a broadbandwidth A and for a center frequency Pi, which isAv/2 below the cutoff frequency v desired for the lowpass filter. This filter can be realized with capacitivegrids. *

* Figure 8.06-1 in Ref. 3 gives the necessary design pro-cedure for inductive shunts, but this procedure is equally wellapplicable to capacitive shunts.

Since the design procedure specifies only the nor-malized susceptances of the grids at the midband fre-quency Pi, one may choose the grid constants conve-niently distributed within a reasonable range (but allg, < l/VP) and adjust the susceptances by means of theparameters a/q of the grids. This filter will not havethe bandpass characteristic it was designed for, sincethe susceptances of the grids change within the speci-fied bandwidth AP, in contrast to the assumptions of thedesign procedure. If Av was chosen to be sufficientlybroad, the lower stopband of the bandpass characteristicwill be vestigial, since for v - 0 all capacitive grids be-come completely transparent. Therefore, this filterwill in effect be a low pass filter.

Because of this effect, and because of the mentionedinfluences of losses and constructional tolerances athigher frequencies, such design procedure for equalripple or maximally flat response is, in practice, only oflimited value for the design of a low pass filter fromdifferent grids. A few significant features of such de-sign procedure are, however, independent of the specialtype of response, and they may be stated in the formof the following rules.

(a) The thickness of the spacer between any twosuccessive grids must be3

S = (4

7rvi)- (j,@(Vi) + kij+1,j(V) + 2rm). (1)

Here, Of ,(v) is the reflection phase of the ith grid (i =1,2,3,..., n) measured at and referred to that surfaceof the grid that faces the kth spacer (k = i, i - 1). Forsymmetric or thin grids, o. = do -1. The frequencyvl is the midband wavenumber and the integer m isusually adjusted so that 0 < sj < (2 vp)-'. Equation(1) means that all spacings must be, in effect, a halfwavelength wide at the midband frequency.

(b) The less transparent grids are to be used in thecenter sections of the filter, and the more transparentgrids at both ends. In a filter of capacitive grids, there-fore, the finest grids must be at the ends, and in a filterof inductive grids the coarsest. This assumes approxi-mately equal a/g ratios.

(c) The filter should be constructed symmetricallywith respect to the center spacer, if the total number nof grids in the filter is even, and with respect to thecenter grid for an odd number of grids.

As a substitute for a sophisticated design procedurewith a well specified response, the following rule (d)may be added. The only design idea leading to thisrule is the suppression of the second order response ofa two-grid, low pass filter as discussed in Ref. 11. Itapplies to the central two-grid part of a larger filterconsisting of an even number n of grids.

(d) An advantageous value for the thickness of thecentral spacer in a symmetric filter with even n is

Sn/2 = (27rPVR) 1" 0n/2,n2(PR)' (2)

Here, vi, is the measured wavenumber of the trans-mission minimum of a single one of the central grids.For thin grids, () = r, and Eq. (2) simplifies to

Sn/2 = (2

vR)-'. (2a,

1990 APPLIED OPTICS / Vol. 7, No. 10 / October 1968

Z0

7

-1

I

z?.0

10

0 00o 200 400 600 800

FREQUENCY CM-'

Fig. 3. Measured transmission T of four-grid ow pass filterscombined from grids (capacitive squares) of different grid con-stants. Most of the fine structure of these curves in the T =10-1O"0- range is not reproducible and is given here only inorder to demonstrate the noise level of the measurements. Data:

..=filter of three grids g g 2 3 = 102 y, a/g = 0.08 plusone grid g4 = 51 pu, ag= 0. 18, spacers s, = 8 50 u, 53 = 40 1A.

The two other filters are symmetrical and have two central gridsof g = 51 pu, a/g = 0.18 and two outer grids of g = 25 A, ag =0.10. Their spacers are .s = 83 28 /1, 82 2CI u (-), and s =

82 83 = 2 u(-

The application of this rule (d) to a given pair ofcentral grids roughly specifies the whole filter.Through Eq. (1), the midband frequency Pi of the filteris fixed indirectly, and thus the approximate cutofffrequency is determined too. Small deviations of thecentral spacer from the value of Eq. (2) are tolerable.They result in shifted midband and cutoff frequencies.But the deviation should be so small that the secondorder peak of the central filter remains in the vicinityof R, where the height of this peak is small due to thehigh finesse in that region. If the chosen spacingSnI2 is smaller than given by Eq. (2), the cutoff will besteeper. This may be desirable, and it also helps toreduce spurious peaks that may result in the stopbandbelow P'R from the addition of the other grids. Whilethe cutoff frequency v, is determined roughly by thecenter section of the filter, its exact value and the slopeof the cutoff depend on the parameters of the other,finer grids. In order to design a low pass filter by therules (a)-(d) for a given cutoff frequency, therefore,some trial and error procedure is necessary.

As examples for filters consisting of diff erent grids,in Fig. 3, the measured transmission characteristics ofsome four-grid low pass filters are given. These filtersare composed in approximate accordance with theabove rules. As expected, the spurious peaks at thefrequencies = 1g, and especially in the diffractionregion are considerably reduced in comparison with thefilters of Fig. 2. The two filters with the high cutofffrequencies in Fig. 3 use the same central grids (2 =

9 = 51 Aut, R2 = 166 cm-') as the filters of Fig. 2.Their outer grids have g = 3 = 25 Au and i= 300cm-,'.

For an analysis of the measured curves in terms ofthe above rules, a main problem is the determinationof the reflection phases 4)(v). They were determinedapproximately from the more accurately measurable

transmission phases it' 7(P) by

0(v) "'' 4(v) ± ir/2 - 2rvKt (.3)

The idea on which this approximation is based is thatfor any (thin or thick) symmetric, lossless grid, Vt', 47r/ 2 is the exact reflection phase referred to the centerplane of the grid. The plus sign applies for inductivegrids, the minus sign for capacitive ones. The lastterm in Eq. (3) approximately represents the shift ofthe reference plane from the center of the grid to itssurface, with t = actual thickness of the grid (3-4 jg) andK = refractive index of substrate (1.71 -1.75, seeRef. 5). The approximation made in using Eq. (3) is,of course, the neglect of the losses and of the actualasymmetryc of the grids. With the phase of Eq. (3)and R2 = 166 cm-', Eq. (2) yields 24 1A as the thicknessof the central spacer. For the reasons mentionedabove, the smaller value S = 20 /2 was chosen. FromEq. (1), then, the midband frequency i = 59 cm-'' isfound by using Eq. (3) and the measured transmissionphase. At this frequency, 0 = 3.736 rad for the innergrids. For the outer grids, Eq. (3) yields = 4.236rad from the measured 4. With these values, Eq. (1)gives the required thickness of the outer spacers s =

s3= 23 A. The values actually used in the filters ofFig. 3 are s = 3 = 20 A (broken line) and .s = 3

28 A (continuous line).Both filters show a spurious transmission peak of

T = 0.007 in the stopband below V12. The thickerspacers result in a slightly lower cutoff frequency (64cm-'" at T = 0.1) and a steeper cutoff slope (59 dB/oc-tave) but a broader spurious band, compared with thefilter with the thinner spacers (68 c-"' and 44 dB/-octave). These differences illustrate the degree of ac-curacy required in the construction of the filters. Thetransmission peak to be expected at P, is hardly de-veloped and seems to occur at a lower frequency thanthe theoretical 59 cm-'. This shift is probably due tothe approximation Eq. (3) made for the phases. Thetransmission in the passband is T > 0.55 for 57 cm--'.

In order to test the possibility of scaling a low passfilter to other wavelengths, a 2: 1 enlarged version ofthe filter just described was built. It had a poor trans-mission characteristic, however, that was not well inscale with the previously discussed curves. One reasonfor this is that the respective ag ratios of the gridswere not the same in both filters. Another reason isthat the PTER film was not scaled up, but was thesame (thinnest available) for all grids. That this filmhas a marked influence on the optical properties of thegrids can be recognized from a comparison of thenormalized resonant frequencies c, = g (= 0.94,0.85, and 0.75) of the three sizes of grids used ( =102 Au, 51 u, and 25 u). The low co, of the finest grid isbelieved to be caused mainly by the relatively largestfraction of space in the vicinity of this grid, which isoccupied by the dielectric film. Probably, this in-fluence of the substrate is also a reason why the mea-sured power transmission curves of these fine gridscannot be represented satisfactorily over the whole

October 1968 / Vol. 7, No. 10 / APPLIED OPTICS 1991

:... 1 1 I

10" ____ _______-_______ ________________________________

10' � T

0" ___

I�-J

,�I

z0

0

"I

To

1°

U0_ - ! _ _ -_-

diffroction

10.2 -

IT~ ____

I10-4 L

0 100 200 300

FREQUENCY CM-I

Fig. 4. Measured transmission of a high pass filter. Four metalmeshes (g = 51 u, a/g = 0.11, nickel), and three equal spacers of

20 u.

(o < R range by an equivalent resonant circuit, whichwas shown in Ref. 2 to be possible for much coarsergrids.

In order to improve the cutoff slope of the enlargedversion, a combination of three identical coarse grids(g = 102 ) with only one finer grid ( = 51 u) waschosen (Fig. 3, dotted line). This violates the sym-metry [rule (b) ] of the filter, but it effectively reducesthe spurious peaks in the stopband to T < 0.004. The50-u spacers between the coarser grids are in agreementwith Eq. (2), whereas the 40-,u spacing of the finer gridwas determined experimentally. Its theoretical valuefrom Eqs. (1) and (3) would be 65 Ai.

IV. High Pass FiltersIn Fig. 4, the measured transmission characteristic

is given for a filter composed of four grids of inductivesquares,' i.e., of ordinary metal meshes. This filterhas a steep cuton at 108 cm-' (T = 0.1). The slopethere is approximately 136 dB/octave. Because ofthese features, the filter may well serve as a high passfilter. Actually, however, its passband extends onlyup to 200 cm-'. Therefore, this pseudo high passfilter' is rather a bandpass filter with a bandwidth ofabout 1 octave. The explanation for this large band-width is that: the peak at 135 cm-' in Fig. 4marks the midband frequency v. At this frequency,the three cavities of the filter are in resonance, as canbe concluded from Eqs. (1) and (3), from the thicknessof the meshes of about 5 , and from the measuredtransmission phase V'T = 0.796 rad of a single mesh atvl. The finesse of the total filter, considered as acombination of two two-grid subfilters, is moderate atvj, since there each grid has high transmission r|2 =

0.67. The peak at v is therefore broad. The otherpeaks in Fig. 4 at 114 cm-' and 155 cm-' are passbandripple, typical for a filter of equal grids. The 114 cm-'peak has a reduced peak transmission, since for v <vi, the reflectivity of the grids, the finesse of the filter,and, thus, the influences of loss and inaccuracies are allincreased. Inversely, the 155-cm-' peak is higher thanthe midband peak. The peak at v = 170 cm-' iscaused by the resonance' of the single grids. At thisfrequency R, each grid has |r12 0.98 and all inter-

ference action of the bandpass filter practically vanishes.Above 1/a = 196 cm-', the transmission falls offgradually, owing to the combined effects of the stop-band interference action in the zeroth order and themultiple diffraction possible at > 1/g. Thus, thelarge bandwidth is due to the special choice of v some-what below the resonance VR of the single grid. Thefact that the cutoff slope of this filter is so much steeperthan the cutoff slope of the comparable low pass filtersof Fig. 2 may be the result of the lower losses of theunsupported metal meshes compared with the supportedcapacitive squares and of the higher perfection anduniformity of the commercial metal mesh comparedwith the laboratory made grids. This would indicatethat the low pass filters may be substantially improvedby chosing a thinner substrate than the present 2.5-uPTER film and/or to use a material of lower loss, e.g.,polyethylene, for it, and by improving the perfectionof the fabrication technique.

V. Bandpass FiltersThe use of metal mesh for the construction of band-

pass filters was the first application of grids in far irfilters.4 Recently, Raweliff and Randall7 have demon-strated that with this simplest type of filter, combinedfrom two metal meshes, one is able to achieve a band-width in the far ir that is narrower than the resolutionof even the best existing ir spectrometers. For anumber of applications, however, broader bandpassfilters are needed with bandwidths ranging from a fewpercent up to one octave. Such filters cannot be con-structed as two-grid filters from metal mesh, if, inaddition to the broad bandwidth, a high attenuation inthe stopbands is required. The reason is that thetransmission characteristic of a filter consisting of twometal meshes is the Airy function with a frequencydependent finesse. The bandwidth and the stopbandattenuation are not independent, and a broad band-width implies a low attenuation.

The following examples show that such broad filterswith high stopband attenuation can be constructed byusing more than two metal meshes, or by using severalgrids of a new, resonant type. '

A. Broad Bandpass Filters Made from MetalMesh

Figure 5 shows the measured transmission of three-grid and four-grid filters combined from metal meshes.For comparison, the (calculated) transmission of atwo-grid filter with the same half-value bandwidth(5 cm-') as the three-grid filter is also given. It isapparent how the attenuation in the stopbands (es-pecially in the upper one) increases rapidly with thenumber of grids. The two-grid filter would probablynot be useful for many applications, since its transmis-sion in the upper stopband is too high (T > 0.05).For the three-grid filter, the minimum transmissionin the upper stopband is about T = 0.0004, and for thefour-grid filter, it is below 10-4. The left half of Fig.5 illustrates well what had been mentioned in Sec. I

1992 APPLIED OPTICS / Vol. 7, No. 10 / October 1968

"I

Z 1.0

0:

2

ii0'

0 1

Z o-

10-4 L60 70 80 90 100 1

Fig. 5. Measured transmission of band flmetal mesh. By using more than two grids,and high stopband attentuation can be achi-= filter of three grids, gi = 93 = 51 A, 2 =

= filter of four grids, gi = 94 = 51 p, 92 =

3= 51 u. All meshes are nickel. The 51-p,0.11, the 25-a meshes a/g = 0.16. ... =sion of a two-grid filter of two thin inductivesame bandwidth (5 cm-') as the three-grid fi(compare Ref. 2) are g = 51 , wo = 1, Zo

55 p.

, ....... ... bandwidth of 5.4 cm-'. The measured bandwidth is6 cm-', using an 1.2 cm-' resolution. This filter has

/_ 9 i\-a peak transmission of 0.46 and the passband shows aslight ripple.

All spacers used in these filters were of equal thick-ness 51 . This is appropriate for the three-grid filterand for the outer spacers of the four-grid filter. Thecentral spacer of the four-grid filter, however, should beslightly thicker according to Eq. (1) and to the generaldependence of on the parameters of the grids. Since

1 ' 200 the additional thickness was estimated to be only aboutFREQUENCY CM-' 3 A, i.e., in the order of the constructional tolerances, it

was neglected. This probably accounts for the slightlyaters bndwfrom higher midband frequency of the four-grid filter in

, a broad bandwidth comparison with the three-grid filter. In this designved simultaneously. method, no attention is paid to the response of the- 252, Si = 2 = 511 filter at frequencies far away from the passband.93 25 A, S = 2 =

meshes have a/g = Typically,3 there is a second order response to be ex-calculated transmis- pected at roughly twice the frequency of the funda-grids which has the mental passband. This second order is very pro-

Iter. Its parameters nounced in all filters of Fig. 5. It is obvious, however,- 1, R = 0.02, how the width of this order is reduced along with the

general transmission in the upper stopband when thenumber of grids increases. A filter that is free of thisstrong second order response can be constructed fromresonant grids, as is discussed in Sec. V. C.

about the steepness of the filter slopes depending on thenumber of sections in the filter.

A straightforward design to meet given specifica-tions is possible for this type of filter at moderate band-widths by the general design procedure for electricalladder filters. A convenient form of it is given in Ref.3, Fig. 8.06-1. From this procedure, the rules (a)-(c)of Sec. III.B were taken. They apply here too, butthe practical difficulties of realization are the same asthere. The measured filter curves of Fig. 5 are in fairagreement with the quoted design procedure. Inorder to demonstrate this, those filter specifications arederived here that would lead via the design procedureto the actual filters of Fig. 5. These specifications canthen be compared with the measured characteristics.

For the calculation of the necessary impedance in-verter3 parameters Kj +l (but not for the calculationof the spacings), the grids are considered as thin andlossless. At the measured midband frequency of P z79 cm-', the Kj j+l of the 51-u grids and 2 5-Au grids be-come 0.214 and 0.070, respectively, as determined fromthe measured power transmissions of 0.165 and 0.0197.From these parameters, the filter specifications arefound by interpolation from the relevant tables of Ref.3. For the three-grid filter: Tschebycheff responsewith 0.8-dB ripple and 3-dB bandwidth of 4.9 cm-',centered at v, = 79 cm-'. The measured 3-dB band-width of this filter is 5 cm-', which is in better agreementwith the specification than should be expected in viewof the wide equivalent slit width of 2.5 cm-' used inthis measurement. The measured peak transmission,however, is only 0.45, and the expected ripple is not ob-served, perhaps because of the moderate resolution.

For the four-grid filter, the specifications are these:Tschebycheff response with 1.3-dB ripple and a 3-dB

B. Narrow Band Filters Made from Metal MeshAll filters considered so far operated in first order at

the design frequency v. The band filters discussedabove can, of course, also be operated in a higher orderby choosing a higher value for the integer m in Eq. (1).If all spacers are designed for the qth order, the band-width of the filter response at the design frequency vi isreduced by a factor q, and q - 1 new transmission peaksappear below v. This possibility is of interest for theconstruction of narrow band filters, either from twogrids if the Airy function is an acceptable transmission

Z 1.0O0(n

z 1

B Q'2

00-

70 75 80 85 120 125 130 135 140

FREQUENCY CM-'

Fig. 6. Measured characteristic of a three-grid band filter oper-ating in higher order. Same grids as in the three-grid filter of Fig.5, but spacers 5 = 2 = 919 pa (glass rings). The interferenceorders are given in parentheses. The indicated points are theresult of the Fourier transform of the interferogram, and thesmooth curve has been drawn through them to average theirregularities due to noise. Especially in the left half the shape of

the transmission peaks is not resolved.

October 1968 / Vol. 7, No. 10 / APPLIED OPTICS 1993

l0-4

zIO

102

104

0 60 80 100 150 200

FREQUENCY CM-l

Fig. 7. Measured transmission of a bandpass filter composed offour resonant grids. Inductive crosses, g = 102 , alg = 0.06,b/g = 0.14 (for a, b, g see insert in Fig. 8), St = 3 =

5 0 a, S2 =100 pA. - = normal incidence, --- = oblique incidence at 270and M polarization. ... = transmission of a single resonant

grid (inductive crosses).

characteristic, or from more than two grids if a morebox-shaped characteristic or a higher attenuation in thestopbands is desired. If such filters were designed tooperate in first order at Pi, the required flatness of thegrids would become prohibitively small.7 Therefore,it is preferable to achieve the narrow bandwidth bydesigning the filter for a high order q, which relaxes theflatness tolerances by a factor q, and to use an auxiliarybroad bandpass filter to suppress the undesired otherorders of the narrow bandfilter. This is exactly what isbeing done in Fabry-Perot spectroscopy in the visible.Several two-grid filters operating in higher orders haverecently been demonstrated by Raweliffe and Randall.7

An example for a three-grid filter is given here in Fig. 6.In the low frequency section of Fig. 6, the transmissionpeaks are so narrow that their shape is not resolved.They all show the same width of 0.6 cm-', which isequal to the instrumental resolution used in thismeasurement. Therefore, their peak transmissions areprobably considerably higher than is apparent from Fig.6. It is not unreasonable to expect the true peaktransmissions to be approximately equal to that of thethree-grid filter in Fig. 5, i.e., about T = 0.4.

C. Bandpass Filter Made from Resonant GridsIn transmission line language, the band filters of the

preceding sections can be generally described as asequence of resonant circuits alternating with im-pedance inverters to provide the necessary coupling.3

In this picture, the resonant circuits represent thecavity resonators formed by the spaces between thegrids. According to Eq. (1), all cavities are tuned tothe midband frequency v. The impedance invertersrepresent the grids, which are considered as essentiallynonresonant, i.e., their transmission is assumed to beconstant over the bandwidth of the filter (thereforethe design procedure is restricted to moderate band-widths).

In the band filters to be discussed here, this assign-ment is inverted: the grids act as resonant shunt cir-

cuits and the spaces between them play the role of theimpedance- (or now better, admittance- ) inverters.This is possible through the use of resonant grids.'These are the optical equivalent of a resonant iris in awaveguide. The resonant grids used in the bandfilterof Fig. 7 are similar to metal mesh, except that the openareas are not square but have the shape of a cross.This pattern is indicated in the insert in Fig. 8.

The measured transmission of a single one of thesegrids is shown by the dotted line in Fig. 7. At low fre-quencies va << 1, this transmission can be representedby a constant inductance shunting an equivalent trans-mission line, exactly as for metal mesh. Therefore, thenew structure will be called inductive crosses, in orderto distinguish it from the capacitive crosses that will bediscussed in connection with the bandstop filter inSec. VI.

At higher frequencies, the representation of a metalmesh requires a capacitance connected in parallel tothe (constant) inductance.2 This capacitance may beunderstood from the fact that at higher frequencieselectrical charges will exist on opposite sides of thesquare holes, just like on the plates of a capacitor. Thetransmission peak of the metal mesh at about g =0.85 can then be interpreted simply as the resonance ofthe LC circuit. Thus, the ordinary metal mesh is infact a resonant grid, but in a marginal way, sinceits resonant frequency v, is only slightly below theonset of diffraction at v = 1/g. Distorting the squareholes to cross-shaped ones brings the opposite sides ofthe holes closer together and thus increases theircapacitance (while the square symmetry is maintainedto prevent polarization dependence). As a consequence,the resonant frequency of the inductive crosses is onlyabout gR = 0.5 (see Fig. 7).

This reduction of the resonant frequency to a valuefar below the onset of diffraction makes it possible toemploy this resonance directly for filtering, withoutdisturbance by diffraction. A single resonant gridmay serve already by itself as a bandpass filter intransmission. The single grid of Fig. 7 has a centerfrequency of ve = 51 cm-i and the bandwidth between

U, \ -if DIFFRACTION

<:10-l21

0

10 3 2 b 0

0 50 00 150 200

FREQUENCY CM-

Fig. 8. Measured transmission characteristic of bandstop filters.--- single resonant grid (capacitive crosses), g = 102 , a/g =0.13, b/g = 0.06. - = filter of two of these grids, spacer s-

38 p.

1994 APPLIED OPTICS / Vol. 7, No. 10 / October 1968

the half-value points is AP = 23 cm-i. As in the otherfilters, the slopes of the filter characteristic can be verymuch improved by combining a filter from severalresonant grids. All grids of the filter must have thesame resonant frequency , which becomes the mid-band frequency. It is determined only by the detailsof the geometry of the grid, i.e., by the dimension a,and b, and not by the spacing of the grids as in the pre-viously discussed filters. A filter consisting of severalresonant grids can be designed under idealizing assump-tions for various responses by a straightforward pro-cedure (Ref. 3, Chap. 8.10). Its realization wouldrequire, however, that resonant grids be available withspecified vR and specified width Aiv of their resonance.The adjustment of these parameters is possible to awide extent by proper choice of the dimensions g, a,and b. This can be concluded from a comparison ofthe limiting cases 2b -# (g - 2a) in which the crossdegenerates to a square, and 2b - 0 in which the crossdegenerates to thin crossed dipoles.i3.i4 So far, how-ever, the dependence of R and Av. on (, a, and b isnot known quantitatively.

The four-grid filter of Fig. 7 is composed of fouridentical resonant grids. At normal incidence (fullline), it has a peak transmission of T = 0.54 at thecenter frequency of R = 51 cm-i. Its half-valuebandwidth is 13 cm-' or 0.26 Vs. The outer spacersof this filter have thicknesses very close to XR/4, as isrequired by the above mentioned design procedure, andthis is also the thickness required for the central spacer.Actually, however, the central spacer of the filter ofFig. 7 has a thickness of S2 X AR/2. It was found thatthis increase of 2 only slightly deteriorated the shapeof the passband (it caused the shoulder at 37 cm-i), butit improved considerably the attenuation in the stop-band at v > 240 cm-'. In this region (not shown inFig. 7), the transmission is T < 0.001 up to at least 380cm-i. The structures at about 75 cm-i in Fig. 7 arecaused by the hump in the transmission of the singlegrids. The nature of this hump is not fully under-stood at present.

This filter has also been measured at oblique inci-dence in polarized light, in order to test its suitability foruse in a light pipe. The broken line in Fig. 7 shows thetransmission for an angle of incidence of 270 (corre-sponding to the marginal rays of an f/1 optic) and TMpolarization. The main difference of this curve com-pared with normal incidence is the reduction of thepeak transmission to T = 0.40. For TE polarization,this reduction is even more severe (T = 0.30). Otherdifferences are a slight reduction of the bandwidth andof the center frequency for both polarizations. Thestructures near 75 cm-' were not observed in TEpolarization. These facts indicate that this filter wouldbe useful in a light pipe at the usual aperture ratios.

One main advantage of this band filter of resonantgrids in comparison with the band filters of metal meshis the complete absence of higher order responses neartwo and three times the fundamental passband fre-quency.

VI. Bandstop FilterThe grids used for the bandstop filter consist of

capacitive crosses, i.e., of insulated, cross-shaped, thinmetal pieces arranged in the pattern shown in the insertof Fig. 8. This structure is complementary to theinductive crosses of Sec. V in the sense discussedin Ref. 2. As is to be expected from Babinet's theorem,the transmission characteristic of this grid (Fig. 8,broken line) has a pronounced minimum (T < 0.01) atthe resonant frequency R = 59 cm-'. This i isslightly different from the R = 51 cm-i of the inductivecrosses, since the dimensions of the capacitive crossesare different and the dielectric influence of the sup-porting film may also be different. At frequenciessufficiently far away from R the grid of capacitivecrosses is fairly transparent, and even diffraction (atv > 98 cm-i) does not remove much energy from thezero order transmitted beam. Again, as in the induc-tive crosses, a hump is found near 72 cm-i, this timeinverted, of course.

For designing a bandstop filter from several of thesegrids, the general procedure of Ref. 3, Fig. 12.04-1(a)can be used under some idealizing assumptions. Butagain the realization of such design would require thatgrids can be made with a resonance of specified widthat the desired stop frequency. All spacings in thebandstop filter must be XR/4 if the grids are thin. Thetwo-grid filter of Fig. 8 is in accordance with this if acorrection of 4 A is allowed for the finite thickness of thegrids. If the grids of this filter were lossless, one shouldexpect two transmission peaks with T = 1.0 near 40cm-l and 70 cm-i, since there the reflection phases ofthe grids fulfill the condition for maximum transmissionof the two-grid interference filter. Actually these peaksare not observed because of the losses, but the measuredtransmission of the filter near these two frequencies isconsiderably higher than the product of the trans-missions of the two grids. The transmission of thetwo-grid filter of Fig. 8 is T > 0.36 if v < 48 cm-i andif v > 66 cm-i. The narrow stopband minimum at 59cm-l may actually be slightly wider than is apparentfrom Fig. 8 since the instrumental resolution in thismeasurement was 2.5 cm-'.

VI1I. DiscussionIt has been demonstrated that several important

filter characteristics can be realized with grids: lowpass, high pass, bandpass, and bandstop response. Asoptical equivalents of waveguide filters, these filterscan be designed fairly well by existing design proce-dures. For application in far infrared spectroscopy,the low pass filters must have extremely wide stop-bands. For such filters it has been shown that the useof grids of different grid constants in one filter cansubstantially improve the attenuation in the diffrac-tion region of the stopband.

The main advantage of these filters in comparisonwith many other ir filters is, of course, that they operateon a purely electromagnetic principle, if losses are ex-cepted. Therefore, they can be scaled to any desired

October 1968 / Vol. 7, No. 10 / APPLIED OPTICS 1995

frequency range. Scaling up to lower frequencies(millimeter waves) should not produce any difficultiesbut only improve the general performance of a filter.In scaling down, however, losses and constructionaltolerances deteriorate the performance of the filters,especially at frequencies of the order of 100 cm- andhigher. This situation may be improved by the de-velopment of grids with low loss substrate. The lossesare also expected to be greatly reduced if the filtersare used at low temperatures. Under such low lossconditions it is no question that filters constructed withultimate mechanical precision will have a performanceat far (and possibly even at near) ir wavelengths thatapproximates the performance of waveguide filters atlonger microwaves. A natural upper frequency limitof the applicability of the filters would be only where themetal parts of the grids no longer behave like goodconductors, i.e., in the visible.

The measurements of the low pass and of the resonantbandpass filters with quasi-parallel radiation at obliqueincidence strongly indicate that these filters may beused directly in a light pipe for incoherent radiation atmoderate aperture ratios. A similar application seemspossible and promising in coherent oversize waveguidesystems.' 5

The author wishes to thank Ely E. Bell for hishospitability and the many stimulating discussions dur-ing the author's stay at the Ohio State University. The

Graduate School of the Ohio State University made thiswork possible through a postdoctoral fellowship whichis gratefully acknowledged.

References1. R. Ulrich, Symposium on Molecular Spectroscopy, Sept.

1967, Columbus, Ohio.2. R. Ulrich, Infrared Phys. 7, 37 (1967) [Errata: Eq. (3a)

must read sin'kr = ,12. In Eq. (13), the first 2 must beomitted from the denominator.]

3. G. L. Matthaei, L. Young, and E. M. T. Jones, MicrowaveFilters, Impedance-Matching Networks, and Coupling Struc-tures (McGraw-Hill Book Company, Inc., New York, 1964).

4. K. F. Renk and L. Genzel, Appl. Opt. 1, 643 (1962).5. E. E. Bell, Infrared Phys. 6, 57 (1966).6. E. E. Russel and E. E. Bell, Infrared Phys. 6, 75 (1966).7. R. D. Rawcliffe and C. M. Randall, Appl. Opt. 6, 1353

(1967).8. K. D. Moeller, Fairleigh Dickenson University, Teaneck,

N. J. (private communication).9. R. Ulrich (to be published).

10. J. P. Auton, Appl. Opt. 6, 1023 (1967).11. R. Ulrich, Infrared Phys. 7, 65 (1967).12. M. Born and E. Wolf, Principles of Optics (Pergamon Press,

Inc., New York, 1964), p. 327.13. A. F. Wickersham, Jr. Appl. Phys. 29, 1537 (1958).14. R. H. Ott, "The Scattering by a Two-Dimensional Periodic

Array of Plates," Tech. Rep. 2148-2, June 1966, AntennaLaboratory of The Ohio State University, Columbus, Ohio.

15. J. Taub and J. J. Cohen, Proc. IEEE 54, 647 (1966).

International Atomic Absorption Spectroscopy

Sheffield, England

14-18 July 1969

The Atomic Absorption Spectroscopy Group of the Society for Ana-lytical Chemistry, in association with the Spectroscopy Group of theInstitute of Physics, will hold an International Conference on AtomicAbsorption Spectroscopy in Sheffield next July, covering all aspectsof atomic absorption and atomic fluorescence spectroscopy and in-cluding invited papers from internationally known workers in thefield. An instrumentation exhibition, technical visits, and a ladies'program will be arranged, as well. Further information can be ob-tained from the AAS Conference Secretary, Society for Analytical

Chemistry, 9 Savile Row, London, W.1., England.

1996 APPLIED OPTICS / Vol. 7, No. 10 / October 1968