Optimal single-bandnormal-incidence antireflection coatings

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, Brian T. Sullivan, and P. G. Verly

Mathematical and computational evidence that strongly suggests that optimal solutions exist tosingle-band, normal-incidence antireflection coating problems is presented. It is shown that efficientsynthesis and refinement techniques can quickly and accurately find such solutions. Several visibleand infrared antireflection coating examples are presented to support this claim. Graphs that showthe expected optimal performance for different representative substrates, refractive-index ratios,wavelength ranges, and overall optical thickness combinations are given. Typical designs exhibit apronounced semiperiodic clustering of layers, which has also been observed in the past. Explanationsof this phenomenon are proposed. r 1996 Optical Society of America

1. Introduction

Antireflection 1AR2 coatings are among the mostinvestigated class of optical coatings. They havebeen the primary subject of at least two books and ofnumerous scientific and technical papers.1,2 Aswell,several books on optical coatings deal with thisimportant topic, and original papers on variousaspects of AR coatings continue to be published at asteady rate. Recently there has been a revival ofinterest in the design of AR coatings fueled, in part,by the increasing demands on their performance andby the emergence of new materials and advances inthe deposition of complex multilayer systems. It istherefore important to know if the lowest averagereflectance is being achieved for a given AR coatingproblem.Most of the early AR coatings consisted of quarter-

wave and half-wave layers, i.e., the optical thick-nesses of the layers were an integral multiple of aquarter of the central or design wavelength.3 Thedesign philosophy was to vary the refractive index ofeach layer while keeping the optical thickness fixed.The number of layers in these early AR coatingsrarely exceeded three or four. The advantage of

J. A. Dobrowolski, B. T. Sullivan, and P. G. Verly are with theInstitute for Microstructural Sciences, National Research Councilof Canada, Ottawa, Ontario, Canada K1A 0R6. The other au-thors are with the Research Computer Center, Moscow StateUniversity, Moscow 119899, Russia.Received 25 May 1995; revised manuscript received 8 Septem-

ber 1995.0003-6935@96@040644-15$06.00@0r 1996 Optical Society of America

644 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

this approach is that it is relatively easy to monitorthe deposition of quarter-wave layers. The disad-vantage, however, is that it is frequently difficult tofind materials that have the desired refractive indi-ces. If the refractive indices of the best availablematerials are significantly different from the desiredrefractive indices, then the performance will suffer.One attempt to overcome this limitation involvedcodepositing two materials to produce layers withintermediate refractive indices that are close tothose desired. However, even when such mixturesdid have the desired refractive indices, they mayhave been less than ideal for use on front surfaceoptics because of their chemical or mechanical prop-erties.Within the past decade or two, the design philoso-

phy shifted away from the above approach. It iscustomary now to vary both the refractive index andthe thickness of each layer. Although this can yieldsolutions based on many refractive indices, it is wellknown that for normal-incidence AR coating prob-lems inhomogeneous layers or homogeneous layersystems consisting of many different materials canbe transformed into two-material solutions. Thiscan be accomplished by using either the Herpinequivalent index concept4,5 or by using numericaltechniques developed by Southwell.6 Furthermore,the thin-film maximum principle has shown that asolution to a normal angle of incidence thin-filmproblem based on more than two materials does nothave a better performance than that obtained withtwo materials only, provided that the material indi-ces correspond to the extreme values allowed andthat the total optical thickness is preserved.7 Hence

it is possible to design complex optical coatings thatmeet the desired specifications by the use of only twomaterials. The advantage of this approach is thatoptical coatings can be designed based on the bestavailable materials selected according to their opti-cal, chemical, and mechanical properties. The dis-advantage is that the deposition is more challengingbecause each layer thickness is usually different andsome of the thicknesses are small. However, newadvances in the deposition of complex, stable opticalcoatings have made this approach feasible.8Ultimately the goal of all thin-film design tech-

niques is to obtain an optimal solution. There aredifferent interpretations of what this means. Inthis paper, an optimalAR coating is defined as one inwhich the rms reflectance is as low as possible in agiven wavelength region for a given set of materialsand for an optical thickness close to a specified value.The number and the thicknesses of the individuallayers in the AR coating do not enter into thisdefinition. It is further implied that such optimalsolutions are unique, in the sense that there are noother significantly different solutions that have thesame performance for the same set of specifications.It is well known that excellent AR coatings based

on an inhomogeneous layer can be obtained over anextremely wide spectral region if the refractive indexof the coating is reduced slowly and smoothly fromthe substrate to the ambient medium.9–12 When thewavelength region of interest is limited, Youngshowed that the best possible performance is ob-tained with step-down quarter-wave coatings withsuitable refractive indices.13 However, both of theseapproaches rely on the use of material mixtures orporous materials, which are typically difficult tocontrol. This paper is concerned with the design ofoptimal single-bandAR coatings based on refractive-index constraints that correspond to practical mate-rials.Up until recently, when Willey published two

interesting papers,14,15 no systematic study had beenmade to determine the optimum performance thatan AR coating can achieve when the refractiveindices are constrained. Willey derived two empiri-cal equations that give estimates of the lowestnormal-incidence reflectances that can be achievedfor different substrates. These two equations werederived fromnumerical simulations involving a num-ber of different examples and are valid for a certainrange of refractive indices, widths of the spectralregion, and overall coating thicknesses.Several new advanced synthesis design tech-

niques have been developed recently.16,17 As is seenbelow, there is good reason to believe that these andother techniques that are discussed allow one toobtain optimal AR coatings. The aim of this paperis, therefore, first to justify the claim that suchoptimal AR coatings actually exist and, second, todetermine their performance for a given set of speci-fications. Because it is impossible to cover all typesofAR coatings in one paper, this discussion is limited

to normal-incidence single-band coatings based ontwo nonabsorbing materials. This is, of course, themost technologically important application of ARcoatings.Section 2 describes the design techniques used in

this study and justifies the claim that optimal ARcoatings were obtained. In Sections 3 and 4, de-tailed results are presented for glass and Ge sub-strates, which are important representative sub-strates for the visible and the infrared spectralregions, respectively. Graphs giving the optimumrms reflectance versus optical thickness are shownfor various design parameters. In Section 5, theclusters of layers evident in the optimal AR coatingsare discussed, and a physical explanation for theirexistence is presented. Finally, in Section 6, someconclusions are given along with suggestions forfurther investigations.

2. Optimal Antireflection Solutions

In a single-band normal-incidence AR problem, theusual specifications are

1a2 the wavelength region of interest 1lL to lU2;1b2 the substrate and the ambient refractive indi-

ces, nS and na, respectively;1c2 the highest and the lowest refractive indices,

nH and nL, respectively;1d2 the total maximum optical thickness.

Throughout the rest of the paper, the ambientmedium is assumed to be air, i.e., na 5 1. lL and lUare the lower and the upper wavelength limits,respectively, of the spectral region of interest forwhich a low reflectance is required, and let lj, j 51, . . . , m, be the wavelengths within this regionwhere the reflectance is specified to be zero. If R1lj2is the reflectance of anAR coating at a wavelength ljand DRj is the associated design tolerance, then amerit function f can be defined,

f 5 51m oj51

m

3R1lj2

DRj42

61@2

, 112

to evaluate the performance of the AR coating. Ifthe tolerances are all set equal to 0.01, then themerit function f is equal to the rms residual reflec-tance 1in percent2 of theAR coating RAV.Merit functions similar to Eq. 112 can be defined for

different synthesis or refinement techniques. Forinstance, many techniques use the square of theabove merit function f 2, whereas the quadratic pro-gramming method17 uses the following merit func-tion:

F 5 elL

lU R1l2

1 2 R1l2dl. 122

However, the differences between these variousmeritfunctions are, in practice, not that significant whenused to evaluate the performance of anAR coating.

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Within certain restrictions, the existence of anoptimal or a global solution to the present class ofARcoating problems has been justifiedmathematically.17Briefly, there is a Fourier-transform relationshipbetween the logarithm of a refractive-index profile,ln3n1x24, and a complex spectral function of the ampli-tude reflectance and transmittance coefficientsknown as the Q-function.18 For small reflectances,the magnitude of the Q-function is simply the reflec-tance. The analytical forms of the Q-function usedin practice become less accurate as the reflectanceincreases. For AR problems, this limitation meansthat the Fourier-transform relationship holds whenthe reflectance adjacent to the AR spectral region isnot too high.19 Under these conditions, the ARoptimization problem belongs to a mathematicalclass of convex problems as the merit function speci-fied in Eq. 122 can be approximated by a quadraticfunction. This is important because convex func-tions possess a single global minimum. The solu-tion of this quadratic problem is an inhomogeneouslayer that requires only minor modifications to betranslated into a two-material system. This multi-layer, in turn, is only slightly modified by refine-ment.17 Even when the Fourier-transform theorydoes not strictly hold, it is likely that there is still anoptimal solution to this class of AR problems.However, there is as yet no absolute proof of this.Another strong argument that optimal AR coat-

ings do exist is provided by numerical simulations.Many powerful thin-film design methods generatethe same excellent AR solutions in a multidimen-sional parameter space, regardless of the startingdesigns. Although numerous methods have beenused in the past for designing AR coatings,20–24 twodesign techniques, in particular, were used exten-sively in this paper: 1a2 the needle optimizationtechnique24 and, 1b2 the quadratic programmingmethod.17 Care must be taken when investigatingother design methods as not all of them will yield anoptimal solution from any starting point.The needle optimization technique25 involvesmak-

ing needlelike variations in the refractive indexthroughout a layer system to determine where thebest position for inserting a new layer is. After thislayer has been inserted, the resulting system is fullyrefined. The needle insertion and refinement cycleare repeated until no further layers can be inserted.Several thin-film design programs incorporate thistechnique. Results presented in this paper wereobtained with two different versions of the needlemethod as implemented in the programs OPTILAYER25and TFDESIGN.26The needle method does not require an elaborate

starting design. As is seen below, for this techniqueonly the total optical thickness of the starting designis important. The number of layers in the finaldesign depends on the optical thickness of the start-ing design. In general, the final designs obtainedby the needle optimization method are different ifthe starting design’s total optical thickness is signifi-

646 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

cantly different. As a rule, if the number of layersin the final designs is the same, then the designs areidentical, even if different starting designs are em-ployed.The quadratic programming method, recently de-

scribed by Tikhonravov and Dobrowolski, is a quasi-optimal synthesis method developed specifically forAR coatings.17 The term quasi-optimal was appliedto this method because it is based on the quadraticfunction approximation mentioned above. Thismethod does not require any starting design, and itgenerates an inhomogeneous layer system boundedby two indices. This solution is transformed into atwo-material system and is then numerically re-fined.The spatial frequency filtering Fourier-transform

technique27 is also successful in generating optimalAR coatings. This method relies on the suppressionof the thin-film system’s spatial frequencies in theAR spectral region. It is similar to the quadraticprogramming method in that both methods mini-mize the Q-function, yield similar results, and havesimilar limitations. As with the quadratic program-ming method, the final solution is an inhomogeneouslayer that can be quickly transformed into a two-material solution and that is easily refined.To illustrate that different synthesis or refinement

techniques generate the same optimum solution fora given total optical thickness, the following ARproblem was specified: nS 5 1.52, nH 5 2.35, nL 51.38 with zero reflectance specified from lL 5 0.40µm to lU 5 0.80 µm 1lU@lL 5 2.02 at Dl 5 0.010 µmintervals. In this and all other numerical calcula-tions in this paper, it has been assumed that all thematerials are nonabsorbing and nondispersive. InFig. 1, the spectral reflectance of each layer systemdiscussed and its associated refractive-index profileare shown in the first and the second columns,respectively. In Fig. 1A, the solid curves correspondto an inhomogeneous layer system that was gener-ated by the quadratic programming method, assum-ing a desired optical thickness of 1.700 µm. As canbe seen, this solution is good except near the bound-aries of the AR region, with RAV 5 0.2% beingachieved. This inhomogeneous layer was thentransformed into a two-material system consisting of20 layers with RAV 5 8.4% 1dotted curves, Fig. 1A2.After numerical refinement, the resulting system isshown in Fig. 1E. As can be seen, there are onlyminor differences between the inhomogeneous andthe refined refractive-index profiles. The rms reflec-tance over the spectral region of interest is now onlyRAV 5 0.06% for a total optical thickness of 1.643 µm.The performance of the needle optimization tech-

nique is illustrated with two different starting de-signs shown in Figs. 1B and 1C, along with theirinitial spectral reflectance. The first starting de-sign 1Fig. 1B2 corresponds to a single layer with arefractive index of nH and an optical thickness of1.250 µm, whereas the second starting design 1Fig.1C2 corresponds to an 18-layer quarter-wave stack

Fig. 1. Performance and refractive-index profiles of four different starting designs: A, multilayer approximation 1dotted curves2 of aninhomogeneous layer solution 1solid curves2 obtained by a quadratic programming method; B, C, a single-layer and an 18-layerquarter-wave stack; D, best 8-layer solution to the problem. After refinement and, in the case of D, the addition of layers at the glassinterface, all calculations converged to the system shown in row E.

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system, centered at 0.50 µm, with a total opticalthickness of 2.250 µm. After the needle optimiza-tion procedure was applied to these starting designs,the resulting systems were identical to the layersystem obtained by the quadratic programming ap-proach 1Fig. 1E2. As a further test of the needleoptimization method, a starting design similar tothat shown in Fig. 1B, but with an optical thicknessof 0.50 µm, was used to generate the layer systemshown in Fig. 1D. With a variation of the gradualevolution method,28 a layer with a low refractiveindex nL and an optical thickness of 0.55 µm wasadded at the substrate interface of the layer systemcorresponding to Fig. 1D. The needle optimizationmethod was then applied to this system to obtain anintermediate 1optimal2 solution. After this gradualevolution–needle optimization procedure was re-peated a second time, the 20-layer system depictedin Fig. 1E was once again obtained. The samesolution is also obtained if the new layers are addedat the air interface in the gradual evolution proce-dure.Other optimization techniques such as ordinary

refinement alone and the Fourier-transform methodfollowed by refinement were also used with a varietyof starting designs. When the initial starting thick-nesses were varied, a series of solutions were gener-ated with different final optical thicknesses. Forsystems with an initial optical thickness of ,1.6 µm,the final solutions agreed fully with the layer systemshown in Fig. 1E. It is worth pointing out that allthe optimal solutions generated had optical thick-nesses that agreed with one another to 1 part in 105.The above mathematical arguments and computa-

tional experiments provide convincing evidence thatoptimal solutions have been found. This conjectureis further supported in the sections below by addi-tional numerical examples involving different sub-strates, coating materials, and spectral regions ofinterest.

3. Visible Antireflection Coatings AnS 5 1.52B

In this section and Section 4, the needle optimizationtechnique was used to study the dependence of theoptimum reflectance on the primary specifications ofthe AR coating. In this section, visible AR coatingswere generated for a representative substrate refrac-tive index of nS 5 1.52. First the effect of differentrefractive-index pairs 1nL and nH2 for the coatingmaterials was investigated for a wavelength ratio oflU@lL 5 2.0. For AR problems, the best results areachieved with the pair of coating materials featuringthe lowest and the highest refractive indices. How-ever, various practical demands often prescribe thechoice of low- and high-index materials that may notnecessarily be the best choice theoretically. Forexample, oxide coatings, in which the lowest refrac-tive index available is 1.45, are favored whenever ahard coating is essential. For this reason, it isworthwhile to examine optimal AR coatings forvarious combinations of low- and high-index materi-

648 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

als. Shown in Figs. 2A, 2B, 2C and 2D are 20-layersolutions generated for the 1nL, nH2 refractive indexpairs 11.45, 1.92, 11.45, 2.102, 11.45, 2.352 and11.38, 2.352, respectively. As expected, the perfor-mance of the solutions improves or the overall opti-cal thicknesses of the systems decrease as the speci-fied refractive index ratio nH@nL increases. Thus,for the highest refractive-index ratio, RAV 5 0.06%1Fig. 2D2. It has been pointed out previously thatthe refractive index of the layer next to the ambientmedium is especially important for wideband ARcoatings.14,29 This can be demonstrated by replac-ing all the nL 5 1.38 low-index layers in Fig. 2D by nL5 1.45 except for the outermost layer and thenrefining the layer thicknesses. The resulting sys-tem 1Fig. 2E2 has a performance that is equivalent tothe original layer system, and the total opticalthickness is only slightly greater.The next example illustrates the optimal solutions

that can be obtained for a given nH@nL index ratio1nH 5 2.35, nL 5 1.382 and different optical thick-nesses. In Fig. 3 are shown four solutions obtainedfor increasing optical thicknesses and, hence, thenumber of layers. The needle optimization methodwas able to generate all these solutions automati-cally by using the gradual evolution approach out-lined in Section 2. The starting design consisted ofa high-index layer with an optical thickness of 0.50µm. Applying the needle optimization techniqueuntil no more layers could be added resulted in thefirst solution shown in Fig. 3A. A low-index half-wave layer at l 5 0.55 µm was then automaticallyadded to the substrate interface, and the process wasrepeated until the remaining solutions shown inFigs. 3B to 3D were generated. As might be ex-pected, the rms reflectance is reduced as the numberof layers and the overall optical thickness are in-creased. However, the reduction in reflectance isasymptotic, and there is less and less improvementwith an increase in thickness. In practice onewouldexpect to reach a point at which the losses that aredue to residual absorption and light scattering wouldexceed the increase in transmission. Correspond-ing results for the nH 5 2.35 and nL 5 1.45 refractive-index pair are shown in Fig. 4. The cluster of layersevident in both Figs. 3 and 4 is discussed in moredetail in Section 5.Similar optimal solutions were obtained for the

refractive-index pair 12.35, 1.452 for different spec-tral regions of interest starting from lL 5 0.40 µm.To present all the data as in Figs. 3 and 4 would beunwieldy. Instead the data presented in Fig. 5show the rms reflectance plotted on a logarithmicscale as a function of the total optical thickness Snddivided by the upper wavelength limit lU for ratioslU@lL 5 1.25, 1.5, 2.0, 2.5, 3.0, and 3.5. The justifi-cation of this normalization of the optical thicknessis given in Section 5. The calculated systems corre-spond to filled circles in the figure. The number oflayers in each solution is also indicated. The curveswere generated by interpolation between the calcu-

Fig. 2. Optimal 20-layer AR coatings on glass for the 0.4–0.8-µm spectral region based on refractive-index pairs: A, 1.90 and 1.45; B,1.45 and 2.10; C, 1.45 and 2.35; D, 1.38 and 2.35. E, as in C, except that the refractive index of the outermost layer is 1.38.

1 February 1996 @ Vol. 35, No. 4 @ APPLIED OPTICS 649

Fig. 3. Optimal AR coatings on glass based on the refractive-index pair 1.38 and 2.35 for different overall thicknesses of the systems.

lated reflectances. These curves are meant to be aguide to the eye, and they should not be interpretedas meaning that there is an optimal solution for anyoptical thickness. There may be intermediate opti-mal solutions that actually lie away from this inter-polated curve, so these curves should not be taken torepresent absolute boundaries. Note that most ofthe optimal solutions lie close to a set of vertical,equispaced lines on the Snd@lU axis. It is shownbelow that the spacing between these lines, which inthis diagram is equal to 0.625, is related to the periodof the clusters.As expected, the number of layers and the overall

thickness required for achieving a certain low rms

650 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

reflectance decreases rapidly as the ratio lU@lLdecreases. It is common knowledge, for example,that a zero reflectance at one wavelength can beobtained with a one- or a two-layer system with atotal optical thickness of less than a half-wave.1It is clear from this graph, and has been observedpreviously,15 that for givenwavelength and refractive-index ratios there is a limit to the minimum averagereflectance that can be achieved even for very largeoptical thicknesses.In Fig. 6, the curve and the points for the case

lU@lL 5 2.0 are replotted versus a total opticalthickness scale and are compared with the lowestaverage reflectances predicted from Willey’s formu-

Fig. 4. Optimal AR coatings on glass based on the refractive-index pair 1.45 and 2.35 for different overall thicknesses of the systems.

las.14,15 Willey’s equations are based on themean orthe average reflectance, whereas the data in thispaper are based on the rms reflectance. However,for low reflections the relative difference between themean and the rms reflectances is less than 5%. Theresults are thus in qualitative agreement.

4. Infrared Antireflection Coatings AnS 5 4.0B

Next, optimal AR coatings for the infrared regionwere investigated by the use of a representativesubstrate with a refractive index of nS 5 4.0. Thelow- and the high-index coating materials were ZnSand Ge with refractive indices of nL 5 2.2 and nH 54.2, respectively. In Fig. 7, the refractive-index

profiles and spectral reflectances are shown for thisproblem for a wavelength ratio of 1.6. The clustersevident in Figs. 3 and 4 are also visible here albeitthey are less periodic. This is discussed in Section5.The rms reflectance of the optimal AR coatings

is plotted on a linear scale in Fig. 8 as a functionof optical thickness for different wavelength ratioslU@lL 5 1.4, 1.6, 1.8, and 2.0. The upper wave-length limits for these curves were lU 5 11.2, 12.3,12.6, and 14.0 µm, respectively. Once again, theabscissa of the x axis is Snd@lU and a set of verticallines is shown with an equal spacing of 0.598. Withthis figure it is easy to estimate the number of layers

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and optical thicknesses required for achieving acertain rms reflectance over a specified wavelengthregion.For the case lU@lL 5 1.6, a careful search was

made to try to determine all the optimal solutions foroptical thicknesses varying from 10 to 60 µm. Thesolutions from Fig. 8 are replotted in Fig. 9A to-gether with additional points produced by either theordinary refinement of suitable starting designs orby the needle method used in conjunction withgradual evolution steps of L@2, L, and 2L, where Lrepresents a quarter-wave optical thickness of low-index material at l 5 10.0 µm. The dashed curve isonce again drawn to provide a guide to the eye.Note that many of the points coincide exactly withprevious results. These data indicate that there isonly a finite number of optimal solutions with mono-tonically better rms reflectance for this AR problem.In other words, given an optimal solution at oneoptical thickness, to generate an optimal solution

Fig. 5. Lowest rms reflectances of a glass surface that can beachieved with AR coatings made of films of refractive indices 1.45and 2.35 for different values of the ratio lU@lL. The integers nextto the calculated points correspond to the number of layers in thesolutions. The spacing between the vertical lines is 0.625.

Fig. 6. Lowest rms reflectances of a glass surface over a lU@lL 5

2.0 spectral range that can be achieved with AR coatings made offilms of refractive indices 1.45 and 2.35. For comparison, theaverage reflectance values predicted byWilley are also shown.14,15

652 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

with a lower reflectance it is necessary to have asignificant jump in the optical thickness.We believe that the discrete set of solutions ob-

served in the above specific example is characteristicof this class of AR problems. Although there is nostrict proof for this conjecture, the clustering oflayers discussed in Section 5 could be an explanationfor the finite optical thickness jump between eachoptimal solution.It has been previously shown that the merit func-

tion versus thickness curve obtained by a Fourier-transform approach is a pronounced steplike curve.16However, the points in Ref. 16 were calculated withthe total optical thickness kept fixed. This is notthe case with the optimization techniques used inthis study. A further refinement of these solutionsin which the total optical thickness is now allowed tovary leads to a finite number of optimal solutions.The problem discussed above was the subject of a

thin-film design competition in 1988.29 Since thattime, at least 10 papers containing at least 57different solutions have been published.16,17,29–36This is therefore an excellent example to check if thesolutions obtained in the present work are indeedoptimal. Some of the solutions from the literatureare shown in Fig. 9B. Several, but by no means all,of these previously published solutions lie close tothe optimal curve from Fig. 9A and none lie below it.It is actually interesting that a number of thesesolutions are relatively far from this curve. Thisindicates the importance of using good startingdesigns or using the best design techniques avail-able.Also shown in this graph is the optimum reflec-

tance predicted by one of Willey’s empirical equa-tions.15 In a general sense, the predicted values arereasonably close to the actual optimal reflectancesfor optical thicknesses greater than 25 µm. How-ever, these equations can also give rise to erroneouspredictions. For example, Willey’s equation pre-dicts that an optical thickness of 14 µm would benecessary to achieve an rms reflectance of 0.80%.According to the actual optimal solutions obtained,the optical thickness required would be closer to 22µm. Although it is very likely that the initialsolutions obtained byWilley are optimal, the validityof the subsequently derived empirical equations islimited, we believe, to the range and the combinationof parameters used in their derivation.

5. Discussion

A series of layer clusters could be observed in therefractive-index profiles of the optimal AR coatingsshown in Figs. 3, 4, and 7. Such clusters, cycles orsemiperiodic refractive-index profiles have been pre-viously observed in wideband AR coatings.14,15,37In this study clusters were observed in all series ofoptimal solutions generated for particular sets ofrefractive indices andwavelength ratios. It is inter-esting and suggestive that most of the time theneedle optimization technique generated such re-

Fig. 7. Optimal AR coatings on a nS 5 4.0 substrate based on the refractive index pair 2.2 and 4.20 for different overall opticalthicknesses of the systems.

sults automatically when operating in a gradualevolution approach. Graded-index profiles resem-bling the layer-cluster solutions were also obtainedwith the Fourier-transform method.16 This sug-gests that there is something fundamental andimportant about the cluster formation and its period-icity.The first cluster solution shown in Fig. 3A consists

of eight layers with an optical thickness L1 < 0.600µm. This same cluster, with minor variations inlayer thickness, is evident at the substrate end of theoptimal solutions in Figs. 3B–3D. Starting fromthe second design 1Fig. 3B2, there is a second cluster

consisting of six layers with an optical thicknessL2 < 0.530 µm. This second cluster is repeated insuccessively generated solutions, i.e., this cluster isrepeated two and three times in Figs. 3C and 3D,respectively. From induction, after one- and two-cluster solutions have been obtained, the approxi-mate number of layers and period thickness forhigher-order cluster solutions can be easily estimated.Hence it is possible to generate AR coatings withlower reflectances simply by adding clusters andthen numerically refining all the layers. Designsidentical to those shown in Fig. 3 were synthesizedwith this technique.

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From a comparison of Figs. 7 and 9 it is seen thatsolutions with a nonintegral number of clustersexist. Such a situation can arise, for example, whenthe refractive index of one of the coating materials isclose to that of the substrate. As an example, thepoints at abscissa values 2.55 and 5.45 in Fig. 8correspond to systems in which a high refractive-index layer 1nH 5 4.22 is next to the substrate1nS 5 4.02. These solutions tend to lie above thedashed curve that links the optimum solutions, andin this respect they appear to be inferior.The Fourier-transform theory provides a simple

intuitive interpretation of the layer clustering inwideband AR coatings. In spite of the fact that,as discussed above, the analytical forms of theQ-function used in practice become less accurate asthe reflectance increases, practical experience hasshown that the Fourier-transform method is useful

Fig. 8. Lowest calculated rms reflectances of a Ge surface thatcan be achieved with AR coatings made of films of refractiveindices 2.2 and 4.2 for different values of the ratio lU@lL. Thenumber of layers in the solution is displayed next to calculatedpoints. The spacing between the vertical lines is 0.598.

654 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

even for quite high reflectances. It is without adoubt able to predict approximately the regionswhere the reflectance is high or low, as long as theprecise value of the reflectance is not needed.Therefore the Fourier analysis can qualitativelyexplain the clustering of layers as follows.The primary objective in the design of AR coatings

is to suppress the reflectance of an arbitrary startingdesign in a specified spectral region by the introduc-tion of a suitable modulation of the refractive index.As seen in the above sections, this is not a trivialproblem when the AR region is wide, as a pro-nounced refractive-index modulation is often neces-sary. This index modulation then gives rise in theFourier domain to significant spectral components inthe frequency 1or wavelength2 region of interest.In other words, the Fourier spectrum of ln3n1x24 musthave a significant amplitude at frequencies in thevicinity of the AR region. Because the Fourier andthe reflectance spectra are closely related and be-cause the reflectance must be small in the AR band,the next available regions where both spectra canhave a significant amplitude is just outside the ARband. Figures 10A–10D show that two reflectancepeaks develop in these precise locations as theaverage reflectance decreases inside the AR bandand the refractive-index modulation is in-creased.16,19,37 Figure 11 demonstrates the connec-tion between the reflectance peaks and the layerclusters. The dotted curves represent the AR coat-ing from Fig. 3C, and its reflectance is plotted on awave-number scale. The solid curves in Figs. 11Aand 11B show that the main reflectance peaksadjacent to the AR region correspond to two rugatefilters with different refractive-index periodicities.The superposition of these rugates, through themultiplication of their index profiles, gives rise to abeating pattern in the resulting refractive-indexprofile. This combined index profile, represented

Fig. 9. Optimal rms reflectances of a nS 5 4.0 substrate over a lU@lL 5 1.6 spectral range that can be achieved withAR coatings made offilms of refractive indices 2.2 and 4.2: A, optimal solutions obtained during this study. The integers represent the number of coincidentpoints. B, Results obtained by others20,21,25–32 for the same problem, including average reflectance values predicted by Willey.14,15

by a solid curve in Fig. 11C, is a smoothed version ofthe optimal multilayer 1dashed curve2. The reflec-tance peak on the low-frequency 1upper wavelength2side of the AR band corresponds to a slow refractive-index modulation and gives rise to the clusters.The other reflectance peak corresponds to a fasterindex modulation and basically determines the loca-tion of the thinner layers within each cluster.Additional reflectance peaks at higher frequencies in

Fig. 10. Reflectance of the one- to four-cluster optimal solutionsof Figs. 3A–D are plotted in A–D for a larger wavelength region.As can be seen, reflectance peaks adjacent to the AR band arevisible, and the amplitudes of these peaks increase as the numberof clusters increases.

the spectrum correspond to the fine or sharp featuresin the index profile, i.e., abrupt interfaces. It isseen that the low-frequency reflectance peak is essen-tially isolated. This is a consequence of the indexconstraints, as other significant peaks at lower fre-quencies would create a slow, large-amplitude refrac-tive-index variation that would violate the indexconstraints. The presence of this isolated peakexplains why the layer clusters are so well defined.As well, the beating between the two main reflec-tance peaks explains why the clusters are not neces-sarily perfectly periodic. In addition, it has beenobserved, in Figs. 3, 4 and 7, that as the number ofclusters increases there is a corresponding increasein the steepness of the wideband AR boundaries.This too can be explained by the fact that as thenumber of clusters increases, the two main reflec-tance peaks sharpen up, giving rise to the steeperARboundaries.The graded refractive-index profile shown in Fig.

11C is unrealistic because it violates the refractive-index limits. The application of such limits modi-fies the performance and, in particular, increases theripples in the AR band. There are actually ripplespresent in the AR band in Fig. 11C, although theyare barely visible on this scale. The ripples haveamplitudes that are 15 times larger than thosepresent in the optimal multilayer and are due toconstraints on the overall coating thickness. Clearlythe role of the much deeper index modulation foundin the optimal multilayer solution is to reduce suchripples.For different parameters, such as the wavelength

and the refractive-index ratios, the position and themagnitudes of the reflectance peaks will be different.Hence the slow and the fast modulations of the ARrefractive-index profile will vary accordingly. Forsome specific values of the parameters, the variousclusters will appear to be nearly identical to eachother, as in Figs. 3 and 4, whereas for other param-eters the positions of the thin layers within eachcluster will vary from cluster to cluster 1Fig. 72. Inthe former case, the optimal solutions for widebandAR problems are of the form, ABN, where the A-typecluster is next to the substrate interface and theB-type cluster is next to the air or ambient medium.In view of the above discussion, it is possible to

come up with good starting designs that can be usedto generate a series of optimal solutions by ordinaryrefinement alone. One possible starting design is asimple quarter-wave stack with a reflectance bandsituated just outside the AR band on the high-frequency side. This is consistent with the factthat, according to the above results, the final solu-tion should have a reflectance peak in this position1Fig. 112. In a number of tests, an optimal solutioncould be obtained quickly by using this startingdesign in conjunction with a Levenberg–Marquardtor a quasi-Newton optimization algorithm. It waspossible to generate a series of solutions by changingthe number of periods in the quarter-wave stack.

1 February 1996 @ Vol. 35, No. 4 @ APPLIED OPTICS 655

Fig. 11. Demonstration that the reflectance peaks adjacent to theAR band determine the layer clustering evident in the refractive-indexprofile of optimal solutions. The dashed curves correspond to the optimal multilayer solution shown in Fig. 3C. A, The solid curvecorresponds to a low-frequency rugate filter; B, the solid curve corresponds to a high-frequency rugate filter; C, the solid curvecorresponds to the superposition of the previous two rugate filters, obtained when their respective refractive-index profiles are multiplied1see text2.

Usually, after refinement, several layers have col-lapsed to zero thickness and could be eliminated.Another successful method used to generate a

series of optimal solutions relies on the clustersystems ABN. For the AR problem of Fig. 3, thespectral performances of such systems correspond-ing to N 5 1, 2, 3, and 4 are shown in Fig. 12, wherethe thickness of the layers were obtained by averag-ing the thicknesses of the clusters shown in Figs.3A–3D. It can be seen that the maximum reflec-tance throughout the AR band is 1.5% or less beforethe individual layers are refined. If these systemsare used as starting designs, numerical refinementof all the layers quickly leads to same optimalsolutions shown in Fig. 3 as well as to other solutionsnot shown. This method can be used for generatingnew solutions after the one- and the two-clustersolutions have been found.It would be useful to have an estimate of the

cluster period thickness required for the above start-ing designs. It is clear that there is a fundamental

656 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

relationship between the AR problem specifications1the wavelength limits and the refractive-index ra-tios2 and the cluster period and the number ofclusters. Essentially, the reflectance peak beyondthe upper wavelength boundary of the AR band iscentered at a wavelength l0 that is twice the opticalthickness of the cluster period L. If Dg is the fre-quency bandwidth of this reflectance peak, then

L0 5l0

251

2lU11 1 Dg2. 132

For a quarter-wave stack, the frequency bandwidthis given by22

Dg 52

parcsin1nH@nL 2 1

nH@nL 1 1 2 . 142

As a first approximation, Eq. 142 can be used in Eq.132 to obtain an initial estimate of the optical thick-

ness required for a starting design. Note that thisestimate does not depend on the lower wavelengthAR boundary because this wavelength mostly affectsthe modulation of the refractive index within a givencluster. For theAR problem discussed in the begin-ning of this section, the Eqs. 132 and 142 predicted acluster period of ,470 nm. This estimate is within10% of the observed period, ,530 nm, of the B-typecluster.The normalization of the total optical thicknesses

by the upper wavelength limit lU in Figs. 5 and 8 cannow be justified by Eq. 132. Essentially, this normal-ization highlights the number of clusters in thesolution. Each vertical line in these figures corre-sponds to a different number of clusters. This iseasily confirmed by an examination of the refractive-index profiles in Figs. 3 and 7. The expression

Lj 5lU, j

lU,iLi, 152

which follows from Eq. 132, shows that, for a given setof materials, the cluster thickness L for one upperwavelength limit lU,i can be easily scaled for anyother upper wavelength limit lU, j.

6. Conclusions

It has been shown in this paper that, with highprobability, optimal solutions exist to single-band,normal-incidence AR coatings, even when the math-ematical theory underlying this assertion is no longerstrictly valid.17 It has also been shown that severalavailable synthesis and refinement techniques canquickly and accurately find such optimal solutions.Representative designs have been generated fordifferent combinations of coating and substrate ma-terials, wavelength regions in the visible and theinfrared, and optical thicknesses. Graphs that showthe expected optical performance have been calcu-lated for several combinations of the design param-eters.No attempt was made in this paper to fit the data

with a theoretical expression. Willey previouslyderived two empirical equations that provide a first

Fig. 12. Theoretical calculation of the performance of nonrefinedABN cluster systems with different values ofN.

approximation of the performance that can be ex-pected for certain AR specifications.14,15 Given thelarge range of variables, this is a difficult task. Ifmore accurate estimates are required, then it hasbeen shown in this paper that graphs similar tothose shown in Figs. 5 and 8 can be quickly generated.These yield a better insight into what performance isreasonable for a given set of specifications.An intuitive explanation of the layer clusters has

been presented. It is based on the interactions ofthe layer systems that are responsible for the reflec-tance peaks that occur just beyond the upper and thelower wavelength limits of the antireflection region.It has been shown that the layer system correspond-ing to the long-wavelength-region reflectance peakgoverns the cluster period L, whereas the structurewithin the clusters is determined by the systemresponsible for the short-wavelength reflectancepeak.To demonstrate that the optimal solutions ob-

tained by the above methods are indeed practical, an18-layer Nb2O5@SiO2 AR coating was designed for aquartz substrate. The design took into account thedispersion of the optical constants of the substrateand coating materials. For this reason, the systemdoes not exactly correspond to any of the multilayersshown in the earlier diagrams, although it is quitesimilar to that of Fig. 3C. The multilayer wasdeposited by reactive rf magnetron sputtering by theuse of system and process control techniques thathave been previously described.8,38 The calculatedand the experimental performances of this 18-layerAR coating are compared in Fig. 13. This was a firstattempt at fabricating this coating, and the agree-ment between the theoretical and the measuredcurves is good.The authors acknowledge the useful comments

made by the referees. This work was first pre-

Fig. 13. 18-layer Nb2O5@SiO2 AR coating applied to one side of aquartz substrate: A, calculated and measured reflectance; B,refractive-index profile of theAR coating.

1 February 1996 @ Vol. 35, No. 4 @ APPLIED OPTICS 657

sented at the Sixth Topical Meeting on OpticalInterference Coatings, Tucson, Arizona, 5–9 June1995 and has been supported by NATO linkage grantHTECH.LG 930841-12841932.

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