fourier-transform method for the design of wideband ... · fourier-transform method for the design...

Fourier-transform method for the design ofwideband antireflection coatings

P. G. Verly, J. A. Dobrowolski, and R. R. Willey

An iterative correction process, recently incorporated into the National Research Council of CanadaFourier-transform thin-film synthesis program, is applied to the design of wideband antireflectioncoatings. This type of problem is different from those solved in the past by this method. It cannot behandled in a practical way without a correction process. We consider in detail the effects-critical for thisapplication-of constraints on the refractive indices and overall thicknesses of the solutions. Ourgraded-index and multilayer designs have a remarkable resemblance in performance and refractive-indexstructure to results obtained by more conventional techniques. The Fourier-transform method is ofinterest because of its speed and versatility.

I. Introduction

The present Fourier-transform technique has manyfeatures of interest for the design of optical thinfilms.- 10 It frequently yields coatings of the graded-index or multilayer type with remarkable ease, speed,and versatility. However, it is inherently approxi-mate. Furthermore, it has been applied almost exclu-sively to situations in which the indices of the sub-strate and incident medium were identical. The moregeneral case of different external indices was ad-dressed only superficially.3 4 We recently described anumerical correction process that compensates forseveral limitations of the basic method.6 We alsopresented a preliminary description of its applicationto a problem in which the external media weredifferent. 7

This paper is a more complete account of the latterwork. We consider the synthesis of broadband antire-flection (AR) coatings on germanium mostly for thewavelength region 7.7 < X < 12.3 pum or, in wavenumbers, 0.081 < o < 0.13 pLm-'. This particularproblem was chosen because it was recently the objectof a detailed comparison of several refinement tech-niques.11 12 We compare our graded-index and multi-layer designs with the results described in Refs.11-15. We discuss for the first time, to the best of ourknowledge, the effect of several important practical

P. G. Verly and J. A. Dobrowolski are with the Institute forMicrostructural Sciences, National Research Council of Canada,Ottawa, Ontario KlA OR6, Canada. R. R. Willey is with OptoMechanik, Incorporated, P.O. Box 361907, Melbourne, Florida32935.

Received 21 February 1991.

considerations on the Fourier-transform method: theconstraints imposed by the refractive-index rangeavailable for the film materials, and the adjustment ofthe overall optical thicknesses. We describe the incor-poration in the numerical correction process of asubroutine that yields multilayers directly.

In Section II we briefly review the National Re-search Council of Canada (NRCC) implementation ofthe Fourier-transform process, we describe its recentmodifications, and how it can be applied to thepresent class of problems. Sections III and IV aredevoted to numerical examples, with and withoutconstraints on the refractive indices available for thefilm materials. Some limitations of the method arealso discussed.

II. Theory

A. Background

Only a brief review is given below. Additional detailscan be found in the literature.- 10 Essentially, thesynthesis of a graded refractive-index profile n(x) isperformed by computing a Fourier transform:

n n(x)= J r Q(T, exp(-j2'rrcx)da,n o / r _ (1)

where Q is a complex function of the desired transmit-tance T = TD and wave number C. The thicknesscoordinate x is twice the optical thickness, and no is ascaling constant that is used to center the refractiveindices in the specified index range. Dispersion andlosses are neglected.

Several approximate forms of Q (T, ) have been

3836 APPLIED OPTICS / Vol. 31, No. 19 / 1 July 1992

reported, but in principle they are valid only for smallreflectances. This severely limits the accuracy of themethod. Additional errors are often introduced inpractice, for example, when n(x) is truncated in orderto limit the overall thickness, or approximated withdiscrete layers (discretization). Those errors can becompensated by an iterative numerical correctionprocess that was recently described.6 Successive cor-rections,

AQ(u) = w IQ(TD, ) -Q(Tc, )], (2)

are added to the function Q and transferred to n(x) byusing Eq. (1) to remove the residual mismatches inperformance. In Eq. (2) w is an adjustable constant,and T is the calculated transmittance. A properdefinition of the complex phase of Q in Eq. (2) isimportant if proper corrections are to be obtained.

According to Eqs. (1) and (2), the synthesizedrefractive-index profile can also be expressed in theform

n(x) = nA(x)nB(X). (3)

where nA(x) is a starting design and nB(X) is acorrection obtained with our iterative process. Thestarting design can be chosen arbitrarily or calculatedfrom Eq. (1).

B. Design with Different External Media

So far this method has been applied almost exclu-sively to cases in which the refractive indices ns andnM of the substrate and medium were identical. Thisis not a fundamental limitation but only a simplifica-tion. The theory is easily generalized by replacing theamplitude transmission coefficient t by (ns IfnM)"2 t.

Note that the transmittance

ns

involved in Eqs. (1) and (2) is not affected by thischange.

It is known that Eq. (1) yields good approximategraded-index designs when (i) the specified thicknessis large enough, (ii) the reflectance R = 1 - T is nottoo high, and (iii) most of the wave-number regions inwhich R is significant [i.e., Q(T, cr) •f 0] are includedin the target. The reflectance of a typical coating withnM e ns is often nonzero over a spectral region that ismuch broader than the region of primary interest. Inparticular, the reflectance is equal to that of the baresubstrate for a 0. This must be specified in additionto the performance in the region of interest to obtainasymmetrical solutions n(x) with the proper values ofnM and ns.4 The structure of n(x) also depends on theassumed performance in the remaining parts of thespectrum (a default value T = 1 is used in ourprogram). Clearly, additional specifications outsidethe region of interest limit the number of solutionsavailable for the basic problem. The approach out-lined above is thus not attractive in practice.

Another approach proposed by Sossi does not havethe same disadvantage. A primary design was firstcomputed in the usual way without paying attentionto the external media. The constant no was chosen tocenter the indices in the range available for filmmaterials. Next, nM and ns were forced to assume thedesired values. It turned out that this did not signifi-cantly change the performance. Finally, the designwas optimized by adjusting the thicknesses (scalingx)and by performing further corrections.

In this work we use a similar approach in combina-tion with the modified Fourier-transform correctionprocess developed at the NRCC. The problem consid-ered-the design of AR coatings-is particularlydifficult. Note that even the first step in Sossi'sapproach is not directly applicable, since the AR of aninterface between two identical media is trivial. Thewhole object of the design is to eliminate the reflec-tance induced when nM and ns are forced to take onthe required values.

We use arbitrarily predefined starting designs nA(x)with the correct substrate and medium refractiveindices (see, for example, curve 1 in Fig. 1 and thedotted curves in Fig. 3). Successive index correctionsnB(x) are obtained by computing the spectral correc-tions AQ in the AR band [Eq. (2) with TD = 1]. Sincethis region does not normally include low wavenumbers, nB(x) has identical refractive indices-equal to no-in the regions of the substrate andmedium. The proper indices are obtained for thecorrected film n(x) by scaling nB(X) so that no = 1 [Eq.(3)].

We consider several practical points not addressedby Sossi. The. effect of truncating the synthesizedcoatings is particularly important. 6 On truncation,the variation of n(x) is ignored for values of x thatexceed the allowed thickness. We also examine theeffect of forcing n(x) to be in the range of refractiveindices available for film materials. Whenever n(x)moves above the high refractive-index limit nH orbelow the lower limit nL, it is forced to take on thevalue nH or nL. Note that this is a crucial issue for ARcoatings since their performance is limited by theavailability of low-index materials. Parts of n(x) andrefractive-index corrections nB(X) are thus systemati-cally lost. Clearly, this affects the efficiency of theindividual corrections. Nevertheless, good results canbe achieved by performing a sufficiently large numberof iterations. This is acceptable because the computa-tions are fast.

The type of problem considered in this work is quitedifferent from those solved in the past by the Fourier-transform method. So far the specified reflectancehas normally been zero across the whole spectrum,except in a narrow region of interest. Essentially, theopposite is true when AR coatings are designed.

C. Shah Option

Multilayers can be designed in two different wayswith the Fourier-transform method.3 It is possible toapproximate the graded-index coatings with discrete

1 July 1992 / Vol. 31, No. 19 / APPLIED OPTICS 3837

layers. Multilayers can also be obtained directly withour Shah option. (Shah is the name of the Russianletter A, which is used to represent a replicatingfunction in the Fourier theory). The latter approachis often particularly efficient and it has been adaptedin this work for use in the correction process.

Unlike graded-index coatings, multilayers withequal thickness layers have multiple harmonics. Tosimulate such a situation, the complex function Q inEq. (1) is replaced by a periodic repetition of itselfdenoted by Qp, and n(x) is expressed in the form3

rn(x)] m +In[ - = 2 F4~p) < x < (5)noJ PM--. \P P P

where FQ(x) is the Fourier transform of the originalfunction Q and p is the wave-number period. Equa-tion (5) describes a multilayer in which all the layershave the same optical thicknesses 1/2p. A similarequation that involves the Fourier transform of AQ[from Eq. (2)] yields the index corrections nB(X).

To obtain the correct performance in the spectralregion of interest period p must be such that there isno overlap between the harmonics of Qp. Note thatthe initial function Q involves negative as well aspositive wave numbers [see Eq. (1)], with 5

Q(-U) = Q*(U), (6)

where the superscript denotes a complex conjugate.Thus Q already includes two harmonics: a (+1)harmonic that is a function of the desired transmit-tance, and a (-1) harmonic that is obtained bysymmetry using Eq. (6). In our program, p is adjust-able and defaulted to p = 4

.id, where Crinid is themid-wave number in the range of interest. Thus Qpby default consists of evenly spaced harmonics cen-

1.0000

0.8000

0.6000

01.0000

0.9999

0.9998

TRANSMITTANCE

2

1 2

).00 I.05I I I.00 0.05 0.10 0.15 0.20

0.08 0.09 0.10 0.11 0.12WAVENUMBER [ 1- 1]

tered at ±Umid, ±39,mid, etc. The optical thickness of

the layers [1/2p, see Eq. (5)] is then an eighth wave atX = /0rmid. Note that the harmonics of Qp arenormally not all identical. When they are the same,there is an effective period ofp/2 and the layers havea quarter-wave optical thickness. This is an interest-ing special case that applies to AR coatings, as can beseen below.

There is however one difficulty with this approachwhen the overall thickness is limited. The summationin Eq. (5) should include all the Fourier componentsFQ(x) for x < 0. In practice, the synthesized multilay-ers are truncated (i.e., values of x that are outside theallowed thickness range are ignored). As a result, theforegoing summation is found to within an additiveconstant, and n(x) to within a multiplicative constant.In our program we assume that ln[n(x)] should beapproximately centered about ln(no) and we add asuitable constant to the right-hand side of Eq. (5).

III. Design without Refractive-Index Constraints

A. Films of Half-Wave Optical Thickness

It is known that step-down AR coatings of thegraded-index or multilayer type can reduce the reflec-tion of a given substrate to an arbitrarily low levelover a broad wavelength region, provided that materi-als with sufficiently low refractive indices are avail-able and the system is thick enough.6' 1 3' 14' 16 -18 Anempirical criterion for the estimation of the opticalthickness is that it should be larger than half of thelargest wavelength H to be suppressed in the reflec-tion spectrum. Several AR coatings designed for thespectral region 0.081 < < 0.13 m-(7.7 < X < 12.3 pm) with overall optical thicknessesInt approximately equal to AH /2 are discussed in thissection. nt is also expressed in terms of quarter

REFRACTIVE INDEX

-5.0 -3.0 -1.0 1.0 3.0OPTICAL THICKNESS [m]

4.0

3.0

2.0

1.0

4.0

3.0

2.0

1.0

5.0

Fig. 1. Step-down AR coatings obtained in the absence of refractive-index constraints: curve 1, exponential sine graded-index profile;curve 2, multilayer designed with Young's method13; curve 3, result obtained with the Fourier-transform method; curve 4, low spatialfrequency content of the latter.


I Il I

(a_

I I I I

2

-5.0 -3.0 -1.0 1.0 3.0 5.C

- I 1 N 2

(b) /n '4A_~~~~1 ;_

waves QW's) at the reference wavelength X0 =l/amid = 9.48 gim. In the diagrams, the origin of theoptical thickness axis [x/2 in Eq. (1)] coincides withthe center of the coatings.

Curve 1 in Fig. 1 is a classical step-down AR coatingof the graded-index type. The refractive index de-creases smoothly from the substrate to the mediumindex with an exponential sine variation {i.e., ln[n(x)]is a sine; the optimum index shape is not important inthis discussion). The optical thickness is exactly equalto AH /2 (Int = 2.6 QW's). The reflectance R = 1 - Tdecreases gradually with increasing a: there is nohigh-wave-number limit UH to the AR band, which istherefore much wider than necessary. The reflectanceis reduced to 6% at the low-wave-number limit UL =0.081 Am-1 .

When the refractive index decreases in steps (curve2), a reflectance band is created in the spectral regionin which the (equal optical thickness) layers arehalf-waves, whereas the AR band is centered on thewave number at which the layers are quarter waves.The present three-layer system (nt = 3 QW's) wasdesigned for the desired AR band with an analyticalprocedure described by Young.13 Its performance,shown on two different scales, is quite good. SimilarAR coatings composed of 1, 2, 4, etc. layers can also besynthesized (nt = 1, 2, 4, . . . QW's). These are be-lieved to be the optimum designs when there are noconstraints on the refractive indices.13

An analogous result was obtained with the Fourier-transform technique (curve 3). The starting designwas curve 1, but the overall optical thickness had tobe subsequently increased from 2.6 to 2.85 QW's, as isdiscussed later. The final performance was essentiallythe same as that of the step-down multilayer and wasnot plotted for the sake of clarity. Note that the indexstructure and performance changed significantly dur-ing the course of the design process. In particular, thefinal index variation is no longer purely gradual.Important refractive-index steps were created at thesubstrate and air interfaces, and a periodic ripple wassuperimposed onto a slow transition from a high to alow index (curve 4). This ripple mimics the layers ofthe multilayer represented by curve 2.

We mentioned before that the position and thewidth of the AR band are related to the thickness ofthe layers. From the point of view of the Fouriertheory, it is seen from Eq. (1) that the Fourierspectrum of the index profile n(x) is closely related tothe transmittance curve. In the present case, n(x) hastwo distinct sets of spatial frequencies that show upin the two reflectance bands on either side of the ARband. The low spatial frequencies (wave numbers)correspond to the slow transition from a high to a lowindex. The high wave numbers correspond to theindex ripples.

The graded-index profile was subdivided into 36layers and refined. The performance after refinementwas still essentially the same as that illustrated inFig. 1. This suggests that the NRCC Fourier-transform process worked well for this problem.

B. Other Thicknesses

We observed that the index structure and perfor-mance of the coatings synthesized with the presentmethod were considerably dependent on the overalloptical thickness. Figure 2 shows the variation withoptical thickness of the values of the merit functionthat characterize the performance. (The transmit-tance was defined at 49 equally spaced wave numberswith a uniform tolerance of 1%. Thus, the values ofthe merit function also represent the average residualreflectance expressed in percent.) Two different start-ing designs were used: an exponential sine refractive-index variation as above and a constant index equal tothe geometric mean (nSnM)112 of the substrate andmedium refractive indices.

It is interesting that, for Int < 3.75 QW's, i.e., upto a point at which the residual reflectances reachedthe limit of accuracy of our program (R < 0.001%),we obtained designs that compared favorably in per-formance to Young's systems. As in Ref. 14, we foundthat the dependence of our results on the startingdesigns was smaller for Int < 3 QW's, correspondingto optical thicknesses of the order of XH /2 or smaller.For larger thicknesses, the values of the merit func-tion obtained with the exponential sine starting de-signs decreased continuously, whereas those obtainedwith the homogeneous layer starting designs fluctu-ated. They reached successive minima for Int = 3.75,6.75, 9.75 ... QW's. Note the interesting periodicityof these values.

Several designs of various thicknesses are illus-trated in Fig. 3. For easier comparison, Fig. 3(b) is areproduction of Fig. 1(b) on a different scale. Thedotted curves in the other diagrams are the startingdesigns. The final values of the merit function areindicated in Fig. 2.

101

100

w-J

zU-

I-w

101

102

1 0-3

10 o 1 2 3 4 5 6OPTICAL THICKNESS [Ws]

7

Fig. 2. Variation of the merit function values with thickness inthe absence of refractive-index constraints: A, Young's step-downAR multilayers13; + and x, graded-index coatings that evolvedfrom constant index and exponential sine starting designs.


TRANSMITTANCE

I I I0.09 0.10 0.11 0.12

0.08 0.09 0.10 0.11 0.12

WAVENUMBER in-]

REFRACTIVE INDEXI I I I

(a)

I I I I-10.0 -6.0 -2.0 2.0 6.0 10.

-10.0 -6.0 -2.0 2.0OPTICAL THICKNESS [gm]

Fig. 3. Multicycle refractive-index profiles obtained in the absence of refractive-index constraints.

The refractive-index structure of the thinner film[Fig. 3(a), Ynt = 1.5 QW's] has less resemblance to agradual transition from a high to a low index than inFig. 3(b). Southwell obtained, by using a differentapproach, a similar design that also exhibited a thinlayer at the air interface (Ref. 14, Fig. 5). Note that,rated on the basis of a merit function value/thicknessratio, the performance of the present coating is betterthan that of Young's systems (Fig. 2).

The results obtained for nt = 6.75 QW's-corresponding to the second minimum of the solidcurve in Fig. 2-depended on the starting design. Inagreement with Ref. 14, the exponential sine startingdesign was already a good AR coating, and it was onlyslightly modified by the synthesis process. Smallindex steps were created at the interfaces. The finalperformance, indicated in Fig. 2, was quite good. Theother starting design yielded a different type ofsolution [Fig. 3(c)] with cyclic refractive-index varia-tions similar to the multilayers obtained by Willey ina totally different way.17,18 We also designed filmswith different numbers of cycles for the thicknessesthat correspond to other minima of the merit func-tion curve. The multicycle coatings had the samegeneric appearance as those shown in Figs. 3(b) and3(c), namely, a slow transition from a high to a lowindex with superimposed ripples. Their performanceswere also essentially similar: the increased complex-ity of the films did not result in significant improve-

ments (Fig. 2). This is perhaps only of academicinterest in the present case (the residual reflectance isof the order of 0.002%), but we see below thatsolutions of the multicycle type have advantageswhen the refractive indices are forced to lie withinpractically realizable limits. The film illustrated inFig. 3(c) was discretized and refined. As before, nosignificant improvement in performance was ob-served.

Finally, we found that our Shah procedure wassurprisingly effective in synthesizing step-down ARmultilayers that were nearly identical to Young'sdesigns such as curve 2 in Fig. 1. The correctionprocess converged in a matter of seconds. (All thetimes cited in this paper are CPU times. The calcula-tions were performed on a Hewlett-Packard 1000Model A700 computer.) Multilayers of the multicycletype could also be found [e.g., the thick curve in Fig.8(b)]. As in the graded-index case, the actual type ofsolution depends on the starting design and thick-ness. It is remarkable that Shah systems, which wereinitially composed of eighth-wave layers (see Subsec-tion II.C), were reduced to quarter-wave stacks bycombining layers of identical indices. Once again, nosignificant improvement was obtained by refinement.

C. Adjustment of the Overall Optical Thicknesses

One difficulty encountered when a coating is designedwith the Fourier-transform method is that, unlike


1.0000

0.9960

0.9920

0.1.0000

0.9999

08

0 9998 l_

I I

f ' I>

I I II

(b)

I I I_

0.08n

I .uuuu

4.0

3.0

2.0

1.0

0

4.0

3.0

2.0

1.0

0

4.0

3.0

2.0

1.0

0.9999

0.9998

-10.0 -6.0 -2.0 2.0 6.0 10.of1-\_ Y d~ I I

l I I I

(c) \

I I I I~~~~~

6.0 10.0

_ . .

I

0.09 0.10 0.11 0.12

with most other techniques, the overall optical thick-ness is fixed and must be specified at the start. Thefluctuations of the values of the merit function withthickness in Fig. 2 are a consequence of this. Theyclearly illustrate the effects of a wrong thicknesschoice. Since the optimum thickness is initially un-known, it must be adjusted progressively by examin-ing the design and its performance.

Sossi showed that it is sometimes possible toimprove the fit between the target and calculatedtransmittances by scaling the thicknesses of thesynthesized coating,3 a procedure that we call stretch-ing. For example, the synthesis illustrated in curve 2of Fig. 1 was performed as follows. The thickness ofthe starting design (curve 1) was estimated on thebasis of the empirical criterion mentioned in Subsec-tion III.A (nt = 2.6 QW's). The correction processyielded a primary solution with a performance repre-sented by the thick curve in Fig. 4. The high transmit-tance region was too wide and not correctly centeredin the AR range (denoted by asterisks). A propercentering could be achieved by stretching the synthe-sized coating to nt = 2.85 QW's. The correctionprocess was then invoked again and yielded the finaldesign, whose performance is reproduced in Fig. 4.The solutions that correspond to the other values ofthe merit function plotted in Fig. 2 were obtained inthe same way.

There are other means to adjust the thickness. It ispossible to reoptimize a coating after truncating oneor both of its ends. Conversely, one can append a thinslab of one or both external media to the system. Thislatter operation does not change the performance,but it provides additional degrees of freedom forfurther corrections. An increase in the value of themerit function with thickness, such as observed inFig. 2, is thus impossible. However, graded-indexprofiles extended in this way often have refractive-index discontinuities at the original interfaces, asseen, for example, in Fig. 3. These discontinuitiesmay or may not be smoothed out by the correctionprocess. We found that manual smoothing before theapplication of the correction process could be useful.Smoothing changes the performance somewhat, butusually not beyond recovery. It is performed in ourprogram either by computing a running average (by

wLC.)z

Cl)

1.0000

0.9992

0.9984

0.06 0.10 0.14WAVENUMBER [jInf]

Fig. 4. Performance of the graded-index design shown as curve 3in Fig. 1 before and after thickness adjustments. The asterisksindicate the AR range.

calculating the average refractive index in a smallthickness range for each point of the refractive-indexprofile) or by spatial frequency filtering (by keepingonly the lowest spatial frequencies of the refractive-index profile). These thickness adjustments are illus-trated in Section IV.

IV. Results with Refractive-Index Constraints

A. Comparison with the Results of Refs. 11 and 12

In practice, the performance of AR coatings is limitedby the lowest refractive-index material that is avail-able. The effect of this limitation on calculations withthe Fourier-transform method is discussed below. Weimpose a high and a low limit, nH = 4.2 and nL = 2.2,on the refractive indices, and our designs are com-pared with the best solutions of the same problemobtained in Refs. 11 and 12 by refinement.

Figure 5(a) corresponds to Fig. 3(b) of Ref. 11(ant = 7.56 QW's or 17.9 [m). Our starting design(dotted curve) was a homogeneous layer of refractiveindex equal to (nH nL )112. The synthesis created adeeply modulated graded-index structure (thick curve)that has a remarkable resemblance to the Aguilera etal. multilayer (thin curve) although a totally differentdesign method was used. Both coatings exhibit semi-periodicity such as the one illustrated in Fig. 3(c).Note that in the present case the largest refractive-index changes were clipped to fit the refractive-indexlimits, as explained in Section II. Our iterative correc-tion process behaved quite well in spite of thisadditional constraint, since the two transmittancecurves are nearly identical. As expected, the residualreflectance is much larger than in the case of noconstraints on the refractive indices.

Figure 5(b) corresponds to Fig. 3(f) of Ref. 11(ant = 13.4 QW's). The refractive-index profiles nowhave essentially two more cycles. This is particularlyobvious in the graded-index case. The two profiles areagain in good agreement, especially closer to the airinterface. The performances are also comparable,although that of the multilayer is slightly superior.

Figure 5(c) corresponds to Fig. 6(b) of Ref. 12(ant = 14.1 QW's). The two systems are quite similarto those illustrated in Fig. 5(b) near the air interface,especially in the graded-index case. The multilayerhas the best performance reported thus far for thisproblem. It was designed with a sophisticated processby using random starting designs and combiningseveral refinement routines. Several hours of compu-tation were required. The time required with theFourier-transform method was 15 min. Althoughthis is slow when compared with our past experiencewith this process, it is quite fast when compared withother methods. All the calculations were performedon the same computer.

The transmittance curves of the graded-index de-signs in Figs. 5(b) and 5(c) are slightly tilted inopposite directions. We found a slightly better solu-tion for an intermediate thickness [Int = 13.7 QW's,Fig. 5(d)]. This coating was closely approximated with


- A - - - - I

I I I I

TRANSMITTANCE REFRACTIVE INDEX1.00

0.08 0.09 0.10 0.11 0.12 -20.0 -10.01 .0 0 I I I I I I

0.99

0.98

0.11.00

0.99

.98

0.08 0.09 0.10 0.11

WAVENUMBER [nm1I

0.12

4.0

3.0

0.0 10.0 20.0lR~~~ 4.0

3.0

2.0

1.0

0.0 10.0 20.0

1.0

0. 0.0 20.0

-20.0 -10.0 0.0 10.0OPTiCALTHICKNESS [gm]

Fig. 5. Graded-index and multilayer designs obtained under the same thickness and refractive-index constraints. The multilayers arefrom Refs. 11 and 12.

147 layers and refined without any significant im-provement. The refinement stage took 1 h 40 min.

B. Variation of the Merit Function with Thickness

A number of other graded-index coatings were de-signed and the variation with thickness of the corre-sponding values of the merit function is plotted inFig. 6. For comparison purposes, the values of themerit function of the best solutions from Refs. 11 and12 are also indicated, respectively, by the squares andcircle. Three different design strategies were tried.For each point on curve 1, the starting design was ahomogeneous layer of refractive index (nH nLL)1 12 . Toobtain the starting designs used in curves 2 and 3, thesystem that corresponds to the previous point in thecurve was, respectively, extended or truncated on thesubstrate side. Speed was favored here over accuracy.

Most solutions were not fully optimized, except thosethat could be compared with existing results. Thetriangle corresponds to one such solution that wasobtained in addition to those described in Fig. 5. Theperformances were evaluated at equally spaced wavenumbers instead of equally spaced wavelengths as inRefs. 11 and 12. Several interesting points are apparent.

The merit function values approximately decreasein steps. Each step corresponds to the introduction ofa new cycle into the refractive-index profile. From thepoint of view of performance and apparently up to asaturation point, there is therefore an advantage toinclude a greater number of such cycles. This iscontrary to what was observed in the absence ofrefractive-index constraints.

Note also that, as in Fig. 2, the values of the meritfunction increase between the steps in curve 1. There


0.99

0.98

.8 0.09 0.10 0.11 0.12 -20.0 -10.0

0

I I I I

I I I I

0.09 0.10 0.11 0.12 -20.0 -10.0

0.98

I I I I

I I I I

4.0

3.0

2.0

i_ 1.0

20.0

080.11.00 -

0.99 -

2.5

w

0

C.

I-wc

2.0

1.5

1.0

3 5 7 9 11OPTICAL THICKNESS [OWs]

13

Fig. 6. Variation of the merit function values with thickness inthe presence of refractive-index constraints. The curves correspondto graded-index solutions obtained with different starting designs(see text): O and 0, multilayer solutions from Refs. 11 and 12,respectively; A, optimized graded-index design (see text).

is essentially no such increase in the other cases, aspredicted in Subsection III.C.

Finally, there is an interesting correlation betweenthe steps in the curves and the data points thatcorrespond to the reported multilayer solutions. In allcases but one, there is one such solution at apparentlythe optimum optical thickness, i.e., at the beginningof a step (ant = 4.5, 7.5, and 13.3 QW's). There is alsoa second set of solutions that correspond to the centerof the steps (ant = 11.5 and 14 QW's). Each set has acharacteristic refractive-index structure, as seen inFig. 5 and in the original references. The correspond-ing Fourier-transform designs have similar features.Their performance was always slightly inferior tothat of the multilayers, even when we tried to opti-mize the input parameters of our design process.

C. Adjustment of the ThicknessThe empirical criterion for the estimation of theoptical thickness that was mentioned earlier is validonly if there are no severe refractive-index con-straints. It is not useful for the present case. Figure 6shows that the performance can be improved signifi-cantly when the optical thickness is increased beyonda half-wave XH/2 (nt > 2.6 QW's). In practice, therequired optical thickness is therefore initially un-known. We show below that it can be progressivelyadjusted by using the techniques that were describedin Subsection III.C.

To demonstrate this, we evolved the graded-indexdesigns illustrated in Figs. 5(b) and 5(c)-and whichare reproduced on a different scale in Figs. 7(c) and7(d) for easier comparisons-from a layer of constantrefractive index that was much thicker than neces-sary. The starting design, represented by the dottedline in Fig. 7(a), has an overall thickness set arbi-trarily to Int = 18 QW's. Our correction process wasfirst applied for a small number of iterations (10) toget a quick idea of the refractive-index structure(solid curve). Intuitively, the refractive-index ripplesare similar to layers with interfaces situated at theinflection points. By performing a truncation at aninflection point, we suppress a certain number of suchlayers. The arrow in the figure indicates that the filmwas truncated near a maximum of the refractiveindex. We made this choice because we go from ahigh-index substrate to a low-index medium. Theabove procedure was then repeated: ten more itera-tions yielded another film [Fig. 7(b)] that was alsotruncated at the inflection point indicated by thearrow. Note that the refractive-index periodicity be-came more apparent. The position of the truncationwas chosen to include four refractive-index cycles,starting from the main refractive-index maximum onthe right. Next the correction process was invoked

REFRACTIVE INDEX

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Intermediate solutions produced by the thickness adjustments during the design of the systems shown in Figs. 5(b) and 5(c) (see


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Fig. 7.text).

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again and allowed to converge completely. A slightimprovement was obtained by smoothing and recor-recting the design. The result is shown in Fig. 7(c).Figure 7(d) was obtained by truncating at the positionindicated by the arrow in Fig. 7(c) and by recorrect-ing. This is essentially an attempt to compress thelast half-cycle of the refractive-index profile. It re-sulted in an increase of the refractive-index modula-tion in this region to compensate for the reduction inoptical thickness.

D. Additional Results

The synthesis exercise illustrated in Fig. 5(a) andreproduced in Fig. 8(a) on a different scale wasrepeated with the Shah option. We obtained a simplequarter- and half-wave stack [thick curve in Fig. 8(b)]after layers of identical indices were combined. Nextwe used a Herpin approximation and refinement totransform the design into a two-material system (thincurve). The latter is similar but not identical to itscounterpart in Fig. 8(a). Note that we now have foursystems with comparable performances. Two systemswere obtained directly with the Fourier-transformmethod and two by refinement. The synthesis withthe Shah procedure was particularly fast and straight-forward (90 s). The Herpin approximation did not

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significantly affect the performance, and refinementwas therefore also quite fast.

Figure 8(c) shows the results obtained by the sameapproach when the high wave-number limit of the ARbandwidth was increased to 0.2 [im-'. As before,there is good agreement between the two designs.Compared with Figs. 8(a) and 8(b), the thicknessesand number of refractive-index cycles (2.5) are almostthe same, but there are now more ripples or layers.This is consistent with the fact that the reflectanceband that limits the AR band on the right wasdisplaced toward higher wave numbers (see Subsec-tion III.A). As expected, the average residual reflec-tance is now larger than in Figs. 8(a) and 8(b). It isonce again approximately the same for both types ofcoating. The transmittance curves are somewhattilted, even in the case of the refined multilayer, andthe tilts are in opposite directions. This is rathersurprising because, as mentioned before, the targettransmittances were defined at equal-wave-numberincrements and the tolerances were uniform acrossthe AR band. This tilt tendency was more pronouncedfor wider bandwidths and larger optical thicknesses.Some transmittance curves of Refs. 11 and 12 are alsotilted.

Finally, the present method readily yielded a graded-

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Fig. 8. Examples of 2 1/2 cycle refractive-index profiles: (a) similar to Figs. 5(a) but shown on a different scale; (b) similar to (a) butobtained with the Shah option (thick curve) and transformed into a two-material system by a Herpin effective-index approximation androfinomont; (c) similar to (a) but for a difforont AR bandwidth.


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Fig. 9. Comparison of graded-index designs obtained by the Fourier-transform method (thick line) and by refinement15 (thin line).

index AR coating similar to a design proposed byBertram et al.15 (Fig. 9). The thin curves are areproduction of the original results, except that thethicknesses-which were not reported-were scaledby us for the sake of comparisons. The sharp refrac-tive-index peak at the substrate interface of ourdesign can be suppressed without much effect on theperformance. The low refractive-index limit and ARbandwidth were, respectively, nL = 1.5 and 4 < X <12 jim. The computations with the Fourier-trans-form method were very fast (140 s).

E. Discussion

It is surprising that, for similar thicknesses, we didnot find graded-index coatings with a better perfor-mance than that of two-material multilayers. Intu-itively one would expect to obtain better results whena whole range of refractive indices is available. Anumber of factors might limit the performance of ourdesigns: (i) the accuracy of our implementation of theFourier-transform method, or (ii) our choice of inputparameters such as the starting designs, or (iii)physical reasons.

It is clear that the Fourier-transform method func-tions here under stringent constraints. Portions ofthe refractive-index profiles-both of the graded-index and of the multilayer types-are systematicallyamputated in order to fit them into the allowedthickness and refractive-index range. These cuts areoften quite severe. Normally, with no truncation, themaximum refractive-index changes occur near thecenter of a film synthesized with this method. It isvisible, for example, in Fig. 5, that the refractive-index variations decrease progressively toward thesubstrate. This is because the refractive-index correc-tions nB(x) are effectively centered at the air interface.Half of nB(X) is in the air region and it is ignored.Other portions of nB(X ), which are in the substrateregion or which result in refractive-index valuesoutside the limits nH, nL, are also clipped. In the end,this design process converges, even if there is still amismatch between the target and the calculatedperformance, because the refractive-index correc-tions necessary for further improvement are outsidethe specified thickness and index limits.

There is no comparable decrease in the amplitudeof the refractive-index modulation of the two materialmultilayers. However the discrepancies between suchsystems, for example, in Figs. 5(b) and 5(c), become

more pronounced closer to the substrate interface.We believe that this is related to the features of thegraded-index profiles. The performance of both typesof design cannot be improved indefinitely by increas-ing the thickness. There appears to be a limit dictatedby the low refractive index. The optimum refractive-index profile is therefore less critical deeper inside thecoating.

It is not absolutely obvious that graded-index filmsshould have an advantage in optical performance overconventional multilayers when there are strong refrac-tive-index constraints. One would expect that somegraded-index designs would require a larger refractive-index range than multilayers. This is the case forrugate filters, compared with quarter-wave stackswith identical rejections and half widths (Fig. 7 inRef. 5). Furthermore, the graded-index coatings dis-cussed in this section are in fact hybrid systems sincethey contain several abrupt refractive-index steps,especially near the air interface.

Note that the multilayers used in the above compar-isons were the best from Refs. 11 and 12. They were,at times, obtained with quite sophisticated designprocedures. Our designs have a better performancethan a number of other reported solutions. Thegraded-index coatings were also compared with multi-layers obtained with the various options available inour computer program: the Shah routine and Herpinequivalent-index approximation, with or without sub-sequent refinement. Several of the graded-index coat-ings were subdivided into many layers and refined,without a significant improvement in performance.This seems to indicate that the Fourier-transformprocess worked well.

V. Conclusions

We have shown that the NRCC modified Fourier-transform techniques could be applied to the designof wideband AR coatings even when strong con-straints are imposed on the overall optical thicknessand on the refractive-index range available for thefilm materials. Such constraints taxed the speed ofthe computations, which nevertheless remained re-spectable. We obtained graded-index and multilayerdesigns that had a remarkable resemblance in refrac-tive-index structure and performance to some of thebest systems found by more conventional methods.However, in most cases the optical performance of thegraded-index coatings was not better than that of the


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multilayers. Only those examples that pertain tohigh-index substrates have been shown, but themethod also performed well for other substrates.

This work was first presented at the Annual Meet-ing of the Optical Society of America, Boston, Massa-chusetts, 4-9 November 1990.

References and Notes1. E. Delano, "Fourier synthesis of multilayer filters," J. Opt.

Soc. Am. 57, 1529-1533 (1967).2. L. Sossi, "A method for the synthesis of multilayer interfer-

ence coatings," Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23,229-237 (1974). An English translation of this paper isavailable from Translation Services of the Canada Institute forTechnical & Scientific Information, National Research Councilof Canada, Ottawa, Ontario KIA OR6, Canada.

3. L. Sossi, "On the synthesis of interference coatings," EestiNSV Tead. Akad. Toim. Fuus. Mat. 26, 28-36 (1977). AnEnglish translation is available; see Ref. 2.

4. J. A. Dobrowolski and D. Lowe, "Optical thin film synthesisprogram based on the use of Fourier transforms," Appl. Opt.17, 3039-3050 (1978).

5. P. G. Verly, J. A. Dobrowolski, W. W. Wild, and R. L. Burton,"Synthesis of high rejection filters with the Fourier transformmethod," Appl. Opt. 28, 2864-2875 (1989).

6. P. G. Verly and J. A. Dobrowolski, "Iterative correctionprocess for optical thin film synthesis with the Fourier trans-form method," Appl. Opt. 29, 3672-3684 (1990).

7. R. R. Willey, P. G. Verly, and J. A. Dobrowolski, "Synthesis ofwide band AR coatings with the Fourier transform method,"in Optical Thin Films and Applications, R. Herrman, ed.,Proc. Soc. Photo-Opt. Instrum. Eng. 1270, 36-44 (1990).

8. B. G. Bovard, "Fourier transform technique applied to quarterwave optical coatings," Appl Opt. 27, 3062-3063 (1988).

9. B. G. Bovard, "Derivation of a matrix describing a rugatedielectric film," Appl. Opt. 27, 1998-2005 (1988).

10. B. G. Bovard, "Rugate filter design: the modified Fouriertransform technique," Appl. Opt. 29, 24-30 (1990).

11. J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D.Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson,and R. A. Kemp, "Antireflection coatings for germanium IRoptics: a comparison of numerical design methods," Appl. Opt.27, 2832-2840 (1988).

12. J. A. Dobrowolski and R. A. Kemp, "Refinement of opticalmultilayer systems with different optimization procedures,"Appl. Opt. 29, 2876-2893 (1990).

13. L. Young, "Synthesis of multiple antireflection films over aprescribed frequency band," J. Opt. Soc. Am. 51, 967-974(1961).

14. W. H. Southwell, "Gradient index antireflection coatings,"Opt. Lett. 8, 584-586 (1983).

15. R. W. Bertram, M. F. Ouellette, and P. Y. Tse, "Inhomoge-neous optical coatings: an experimental study of a newapproach," Appl. Opt. 28, 2935-2939 (1989).

16. J. A. Dobrowolski and F. C. Ho, "High performance step-downAR coatings for high refractive-index IR materials," Appl. Opt.21,288-292 (1982).

17. R. R. Willey, "Rugate broadband antireflection coating design,"in Current Developments in Optical Engineering and Commer-cial Optics, R. E. Fischer, H. M. Pollicove, and W. J. Smith,eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1168, 224-228(1989).

18. R. R. Willey, "Another viewpoint on antireflection coatingdesign," in Optical Systems for Space and Defence, A. H.Lettington, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1191,181-188 (1989).


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