Minimax refining of wideband antireflection coatings for wide angular incidence

Minimax refining of widebandantireflection coatings for wide angular incidence

Amedeo Premoli and Maria Luisa Rastello

The design of antireflection coatings at any light incidence is a challenging task in optics. To this aim, aminimax method is presented: it minimizes the maximum deviation of the spectral reflectance from thedesired specifications over the wavelength for a given set of incidence angles. Refining is limited tolossless coatings with assigned refractive indices and undetermined thicknesses; the algorithm consists ofiterating appropriate linear optimization steps. In the examples some minimax-refined coatings arecompared with coatings reported in the literature.

Key words: Thin films, coatings, filters.

Introduction

Intricate structures of many thin dielectric layershave been used even more extensively in optics asbandpass filters and antireflection coatings in theregion from infrared to ultraviolet. In particular,antireflection coatings are often used at nonnormallight incidence or in collimated light beams in whichthe incidence angle 4) varies in a specified wide range.Unfortunately, it is very difficult to design an antire-flection coating that provides low reflectance valuesover a wide wavelength band or a large range ofincidence angles.1 For instance, for )a >> 100 severedeterioration of a good low-reflectance characteristicof normal incidence occurs. On the other hand, bymatching the design to 4) = 4)o, the reflectance for 4) =<)o comes down, but the reflectance is poor at ) << 4o.

This type of problem emerged during a meeting ofthe program committee of the EurOpto Conferenceon optical systems and has been the object of a coatingdesign contest.2 The task was the design of anantireflecting coating in the wavelength range 0.4-0.9 wm at the incidence angles of both 00 and 300.

Refining methods are well known in the literature;see, for instance, Refs. 1 and 3-6. In this paper a

A. Premoli is with the Dipartimento di Elettrotecnica, Elettronicaed Informatica, UniversitA di Trieste, Via A. Valerio, 10, I 34127Trieste, Italy. M. L. Rastello is with the Istituto ElettrotecnicoNazionale G. Ferraris, Strada della Cacce, 91, 1 10135 Torino,Italy.

Received 29 March 1993, revised manuscript received 30 August1993.

0003-6935/94/102018-07$06.00/0.© 1994 Optical Society of America.

minimax refining algorithm is proposed that controlsthe deviation of the spectral reflectance from theassigned specifications at any wavelength and anyincidence angle within some assigned ranges. Thisapproach is a generalization of the minimax refine-ment proposed in Ref. 7 for normal incidence.

The algorithm consists of iterating linear optimiza-tion steps solved by piecewise linear programming.8At present the algorithm takes into account onlyrefinements of multilayers with assigned refractiveindices and undetermined thicknesses of losslesslayers. The refining of some specified antireflectioncoatings is examined in detail and compared with theresults available in the literature. 2

Flat-Reflectance Refinement

Most multilayer stacks are used at normal lightincidence ( = 0) or small deviations from it. Inthis case a unique reflectance R(X, 00, d) is defined forany polarization. Here M vector d groups the physi-cal thicknesses of the M-layer stack. When theincidence angle 4 > 0, reflectances Rp(X, 4), d) andR,(X, 4), d), corresponding top- and s-polarized compo-nents of the incident light, exhibit behaviors versus Xthat are increasingly different for increasing valuesof 4).

For unpolarized incident light, a suitable combina-tion of these reflectances should be taken into ac-count for evaluating the global optical characteristicsof the multilayer:

R(X, 4), d) = yR3 (X, , d) + (1 - y)Rp(X, , d),

withO < y 1. (1)

2018 APPLIED OPTICS / Vol. 33, No. 10 / 1 April 1994

In general, y is chosen equal to 1/2; then, R(X, ¢, d)will denote the average spectral reflectance of theM-layer system, where d contains all the designparameters, while the refractive indices are keptfixed.

In general, R(X, 4, d) fits the spectral reflectanceA(x, +) required by the designer when a suitablemerit function, strictly linked to deviation R(X, 4,, d) -R(x, 4), is minimized over the wavelength range[Xa, Xb] and the incidence angle range [kka, 'kb] ofinterest. In general, 4a is chosen equal to zero toconsider also the usual normal incidence. A com-plete survey of the merit functions that are morefrequently used in the design of multilayer systems isdiscussed in Refs. 9-11. The most popular func-tional is

2(d) = (fb - XA(kb - 4k)

x Jfl [R(x, +, d) - R(X, )12d d. (2)4a ('Xa

In computer practice this integral is approximated bya summation over several discrete values of wave-lengths and incidence angles.

Unfortunately, this functional does not allow oneto control the behavior of the approximating functionat single wavelengths and incidence angles. In par-ticular, close to the extrema XA, Xb, Pan and , theresulting absolute deviation I R(X, , d) -R(X, +) maybe excessive (tail effects), even if it is limited at theinterior values of X and 4. To overcome this draw-back, the above squared difference can be multipliedby a suitable weighting function depending on X andP. However, the choice of this weighting function isnot at all a straightforward task.

On the contrary, the minimax criterion assures thedeviation R(X, +, d) - R(X, 4) to be strongly con-strained at each X and $ by minimizing the maximumabsolute value:

( = max [IR(X, 4, d)-R(X, cff. (3)(X,0)E[Xa,Xf1(ab1

Also, in this case a weighting function can be intro-duced to permit a greater or lower deviation in theless or more critical portions of the rectangle [a, Xbl X[''a, 46i] in the plane (A, ). Here x denotes theCartesian product of two sets.

In this paper we formulate the minimax criterionby generalizing to oblique incidence the method pro-posed in Ref. 7 for the minimax refining of multilayerstructures at normal incidence. Two functionsbU(X, +, p) and bL(X, +, p) are introduced, defining theupper and lower bounds of a parametric band (wherep is the parameter) within which the reflectanceR(X, +, d) should be constrained, respectively:

b u(X, -+, p) R (X, , d) bL(,\, d+, p), (4)

where

bU(X, 4,p) =pblu(X, +) + (1-p)boU(X,( ), (5)

bL(X, i+, p) = pb1 L(X, +) + (1 - p)boL(x, 4). (6)

The functions biu(X, boU(X, 4), blL(X, k) andboL(X, 4)necessarily satisfy the following inequalities:

bju(X,,+) > bou(AX,,) bo L(X, +)> b L(X, )

V(A\, E X b x [x 41b (7)

Because in antireflection coatings the flat reflec-tance is forced to values as low as possible in [Xa, XbJ X[kka, db, the functions bU(k, k, p) and bL(X, +, p) can bechosen to be constant versus X and 4. In this casethe bounds of the parametric band are set up by

boU(X, (k) = b L(x, ) = bL(x, ) = 0,

Ah =- pblu(X, 4k).

(8)

(9)

The refinement consists of searching for the value ofvector d such that R(X, Xk, d) is as low as possible.In other words, one minimizes the ripple A byproperly choosing d and AR itself:

minimize AR,(dAA)

(10)

where unknowns d and AR are linked by the con-straint

R(X, cj, d) < AR, V(X, (i) E [X, x It, &] (11)

When the designer wishes to limit the thicknessesof the multilayer within a box fixed by the lowerbound d- and the upper bound d+, the constraints

d- < d < d+ (m = 1, 2, .. ., M) (12)

are added to the optimization problem formulated inexpression (10) and constraint (11).

In most designs further constraints may be re-quired by specifying the maximum total thickness dtotof the multilayer,

M

X dm < dtot,m=1

(13)

or by constraining the total thickness dmattot of all thelayers of a given material, that is,

ME dmtmat,m < dtmat,

m=l(14)

where parameters .mat,m (m = 1, 2, . . , M) are equalto 1 when the mth layer is made of the controlledmaterial and equal to 0 otherwise. Also, these con-

1 April 1994 / Vol. 33, No. 10 / APPLIED OPTICS 2019

straints are easily handled in the optimization proce-dure.

Sampling over Wavelength and Incidence Angle

At any step, constraint (11) must be verified at anypair (, 4)) in the rectangle [Xa X] X [(4)a, )b]. Thisgenerates, in principle, an infinite number of inequali-ties, one for any pair (X, 4)). In practice, by acceptingsome amount of error, these conditions can be veri-fied in some appropriate discrete subsets A of wave-lengths Xj (j = 1, 2, . . , J) and (P of incidence anglesk (k= 1,2,*. .,K). NotethatthenumberJx Kof

pairs (X, 4)) in A x F may become very high (up to athousand) because the domain [X0, XbI x [4)a, 4)bl is twodimensional.

In creating set A x , the local maxima of thedeviation R(X, 4), d) - R(X, 4)) versus X and 4) play afundamental role. More exactly, the sampling erroris necessarily null when A x is chosen to becoincident with the set including the abscissas (touchwavelengths) at which the deviation reaches its maxi-mum value in the rectangle [Xa, Xbl x [)a, 4b]- Un-fortunately, the exact location of the touch wave-lengths is a rather complicated and expensive taskbecause it consists of searching for many maxima of atwo-variable function.

As stressed by Thelen,I when the incidence angle isbelow 200, only slight changes occur in reflectancewith respect to the case 4) = 0, whereas changes aredramatic beyond 200. A typical behavior versus X isshown in Fig. 1 for several values of 4) in the range[00, 300]. Note that the upper envelope coincideswith either reflectance R(X, 00, d) or R(X, 300, d), bothcorresponding to the extremal incident angles. Thenthe problem can be simplified by considering that fora fixed value of X the behavior of R(X, 4), d) versus 4 issmoother (also monotone in Fig. 1) than the behaviorof R(X, 4), d) versus for a fixed value of 4) (not at allmonotone in Fig. 1). Thus a dense sampling of therange [a, b] (i.e., high value of J) is required, whereasthe range [ka, 4)bJ is sampled at two or three points(i.e., low value of K), extrema included.

reflectance0.01 r

0.004 ...

0 .0 0 2 . ... ... .... .. ...... . .. .. .... . ..... .. I........... ... . ...... ...... .. .

0 I I

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

wavelength

Fig. 1. Spectral reflectances for several incidence angles from 00to 30° of a typical visible-infrared antireflection coating. Solidcurves denote the reflectances for 0° and 300.

Example: Antireflecting Coatings in theVisible-Infrared Range

In general, different refinement algorithms are verydifficult to compare, even if the same starting designis used. In fact several different local minima maybe located in the neighborhood of the starting design,and different solutions can be produced depending onthe specific descent path followed by the optimizationalgorithm. User's experience and luck are relevantin the interactive control of the algorithm. At last,final solutions related, in principle, to the same localminimum can be characterized by significantly differ-ent numerical values of d when final convergence isaffected by round-off errorsjointly with the flatness ofthe merit function in the neighborhood of the minimum.

Recently, 28 optical designers2 participated in theexercise of designing an antireflecting coating forlenses to be used with normal and infrared photo-graphic films in the region limited by Xa = 0.4 i.m andXb = 0.9 m, with unpolarized light for incidenceangles 4 = 0' and 4) = 300. All layers were supposedto be homogeneous and free of losses. The coatingmaterials were MgF2, SiO 2 , A1 2 0 3 , Ta2O5 , and TiO2 ;their indices were linearly interpolated for dispersionwithin the values reported in Table 1. The physicalthickness of all the layers together should not exceed2 m. When TiO2 layers are present, their overallphysical thickness d should not exceed 150 nm toexclude too-high absorption.

Reflectance R(X, 4), d) was specified1% in the whole range 1a, Ar], that is,

to be less than

Rmax,-, = max [R(k, 4), d)] 0.01XEfX0 ,X1bj

for =O0 and =30°. (15)

As referred to in Ref. 2, the lens producers probablywould have preferred 450 as an upper bound butaccepted 30°. In the meantime, the following figureof merit for reflectance was proposed:

F= 2(Xb- Xa)

Xb 11 1/2xI < (R,=Oo) 30)2 + - (R=30.) 2 AX

(16)

where AX = 0.001 [Lm.

Table 1. Refractive Indices n of the Coating Materials

n

Material 0.40 (m) 0.55 (m) 0.90 (l.m)

MgF2 1.39 1.38 1.37SiO2 1.47 1.46 1.45A1203 1.65 1.63 1.60Ta2O5 2.26 2.16 2.09TiO2 2.45 2.32 2.20


reflectance0.01 4 -

0.012

0.01

0.008

0.006

0.004

0.4 0.5 0.6 0.7 0.8 0.9

wavelength(a)

0.3 0.4 0.5 0.6 0.7

wavelength(a)

0.4 0.5 0.6 0.7

wavelength

reflectance0.014

0.012

0.01

0.008

0.006

0.004

0.002

0.8 0.9 1

(b)

Fig. 2. Spectral refiectances for 4 = O° (solid curve) and 4 = 300(dashed curve) of (a) the DOCT1 design (M = 9, designer Dick) inRef. 2, and (b) the design refined according to the method proposedhere.

The choice of starting points used by the designers2is not discussed here, because our aim in this paper isto show the capabilities of a new refining algorithm.In this sense the proposed examples are of limitedinterest for the more general problem of antireflec-tion coating synthesis, because starting designs are

0.3 0.4 0.5 0.6 0.7

wavelength0.8 0.9 1

(b)

Fig. 3. Spectral reflectances for 4) = 0 (solid curve) and 4) = 300(dashed curve) of (a) the HACK3 design (M = 8, designer Hacker) inRef. 2, and (b) the design refined according to the method proposedhere.

chosen to be coincident with some of the final designsreported in Ref. 2. This choice is motivated by theneed to compare the different effects of least-squaresand minimax criteria on the shape of the refinedreflectance.

Table 2. Physical Thicknesses and Characteristics of Starting Designs DOCT1, HACK3, HACK4, and Related Minimax-Refined Designs

DOCTi (Fig. 2) HACK3 (Fig. 3) HACK4 (Fig. 4)

Thickness (pLm) Thickness (*im) Thickness (.Lm)

Layer Starting Refined Layer Starting Refined Layer Starting Refined

Glass - - Glass - - GlassMgF2 0.049880 0.050169 Ta2O5 0.01383 0.014482 Ta2 0 5 0.01691 0.015521Ta2 O5 0.014324 0.013412 MgF2 0.04346 0.042215 MgF2 0.04112 0.042439MgF2 0.058657 0.057829 Ta2 0 5 0.03841 0.040096 Ta2 05 0.04374 0.042240Ta2 O5 0.033605 0.033746 MgF2 0.02049 0.018906 MgF2 0.01836 0.018835MgF2 0.022212 0.021246 TiO 2 0.14290 0.145112 TiO2 0.15020 0.149195Ta2O5 0.091452 0.090095 MgF2 0.02203 0.022721 MgF2 0.02135 0.022508MgF2 0.017138 0.017797 Ta2 O5 0.03075 0.030602 Ta2 O5 0.03231 0.030810Ta2 O5 0.032685 0.030617 MgF2 0.11900 0.116788 MgF2 0.11760 0.116100MgF2 0.110371 0.110087 - - - - - -Air - - Air - - Air - -dtot 0.4303 0.4250 dtot 0.4309 0.4309 dtot 0.4413 0.4376dtT~ot 0.0 0.0 dtot 0.1429 0.1451 dt 2 0.1502 0.1492Rmax,O 1.966% 0.802% RmaxO' 1.241% 0.818% RmaxO' 1.158% 0.798%Rmax,30° 0.964% 1.036% Rmax,30' 0.859% 0.926% Rmax,30 1.092% 1.006%F 0.534% 0.568% F 0.725% 0.704% F 0.680% 0.696%


reflectance0.025 -

0.02

0.015

0.01

0.005

0 0.3

reflectance0.025

0.02 _.

0.015 _ .. ............. ............................... ...................................... ...... .......................... ....

0.8 0.8 1

0.01

0.005

0 0.3

. ........ . ........ .. . . . . . . . .

... .. < /~~ ~~~~~~~~~~~~~~~~~~..

_ .. ......... ..... ............................ .. .................. ...... ......... ........ .......

_ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........ .........

.. .. .. . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. ................ .. ................ .. ...........

. ................... ........ . ........

. ......... .......... . .. .. . . ........ - .. ..... ....-, ............... .... .. .

.............

V.VU;1

At first, two designs, optimized in Ref. 2 withrespect to F but not complying with all the requiredconditions, have been chosen as starting designs forthe minimax refinement. The maximum value ofreflectance over the range 0.-0.9 jim has beenminimized for 00 and + = 30°.

Figure 2(a) reports the spectral reflectance of thedesign referred to as DOCTi (M = 9, designer Dick),2and Fig. 2(b) reports the spectral reflectance refinedaccording to the method proposed here. UnderDOCTI, Table 2 shows the starting physical thick-nesses of the design presented by Dick,2 the thick-nesses refined according to the method proposedhere, and the design characteristics required by Thelenand Langfeld.2 That is, the overall thickness dtot,the overall thickness d' 2 of TiO2 layers, the maxi-mum reflectances Rn,0° and Rma, 30o, and the figureof merit F.

Figure 3(a) reports the spectral reflectance of thedesign referred to as HACK3 (M = 8, designerHacker),2 and Fig. 3(b) reports the spectral reflec-tance refined according to the method proposed here.Under HACK3, Table 2 shows the starting physicalthicknesses of the design presented by Hacker, therefined thicknesses, and the design characteristicsrequired by Thelen and Langfeld.

In both cases the maximum reflectance accom-plishes the required specifications at both 0° and 300.The minimax-refined reflectance is lower than the

least-squares-refined reflectance near the extrema Xaand Xb but higher in the internal portion of theinterval. These two cases show the capabilities ofthe minimax refinement in controlling R(X, d) at anyX, in the sense that deviation cannot overcome thevalue AR of the merit function V(X, 4) [, Xb] X[ka, tb]

The solution referred to as HACK4 (M = 8, de-signer Hacker2), also characterized by a too-highoverall thickness of TiO2, has then been refined, and anew solution has been obtained, complying with thespecifications on both maximum reflectance and maxi-mum overall thickness of TiO2 layers. Figure 4(a),reports the spectral reflectance of the examined de-sign, and Fig. 4(b) reports the spectral reflectancerefined according to the method proposed here.Under HACK4, Table 2 shows the starting physicalthicknesses of the design HACK4, the refined thick-nesses, and the design characteristics.

Some comments on F are needed; as shown in theproposed examples, its value may be lower or higherthan the value related to the starting design. This isdue to the fact that the proposed algorithm does nottake into account the figure of merit F.

The last example compares the global capabilitiesof the present minimax refinement with those of theminimax refinement presented in Ref. 7. To thisaim, the solution proposed by us, with M = 13 layers,

reflectance0.008 reflectance

3 0.4 0.5 0.6 0.7 0.8 0.9 1

wavelength(a)

:tance

0.3 0.4 0.5 0.6 0.7

wavelength(b)

O.8 0.9 1

Fig. 4. Spectral reflectances for 4 = O° (solid curve) and 4 = 30°(dashed curve) of (a) the HACK4 design (M = 8, designer Hacker) inRef. 2, and (b) the design refined according to the method proposedhere.

0.007

0.006

0.005

0.004

0.003

0.002

° °°0o 0.4 0.5 0.6 0.7

wavelength(a)

O.S 0.9 1

reflectance0.008 r-

0.006

0.004

0.002

0 L.0.3 OA 0.5 0.6 0.7 0.8 0.9 1

wavelength(b)

Fig. 5. Spectral reflectances for 4) = O° (solid curve) and 4 = 30°(dashed curve) of (a) the IEN2 design (M = 13, designers Rastelloand Premoli) in Ref. 2, and (b) the design refined according to themethod proposed here.


0.014

0.012

0.01

0.008

0.006

0.004

0.002

00.

reflec0.014

0.012

0.01

O.OOB

0.006

0.004

0.002

....................... ... . . ........... . ... ... ....

- . ---............. ........ . ..... ............................ ...

l

............ .. . ... . ......... ..... . .......... .. .. .....................

. . ............. ...................... ...... .. .. .............

. ........... ....................... .... . .. . ...... .............. .......... .. .. .. .....

. ............. ... .........

...........

............................... ..... ..... .....................

........... ..................

............. . . .............

.... . ......

............. .. ... ........

,3

............

I.

referred to as IEN2 in Ref. 2, has been considered.The theoretical bases of the computer code used forthese designs are described in Refs. 7 and 12. Designspecifications for incidence angle 4) = 30° were takeninto account by means of some tricks, because thatversion of the computer code cannot consider simulta-neously specifications at several different incidenceangles. Also, specifications on the total thicknesshave been taken into account by some heuristicconsiderations.

Figures 5 report reflectances versus X. Note thatthe value of the minimax merit function AR may bedetermined from Figs. 5, because the minimax valuecoincides with the maximum value of reflectanceversus X. The new minimax-refined design exhibitsa value of the least-squares merit figure F lower thanthe solution presented in Ref. 2. This result wasexpected, because IEN2 was the consequence of heu-ristic considerations.

Figure 6(a) shows the behavior of the solutionnamed IEN2 versus X for three values of 4). Notethat the spectral reflectance at 4) = 40 > 1%. Then,IEN2 has been considered as the starting design for anew minimax refinement controlled within the range0-400. Figure 6(b) shows the spectral reflectancesversus 4 of the obtained solution. Note that thereflectances at 0° and 300 are poor but still satisfy thespecifications.

Table 3. Physical Thicknesses and Characteristics of the StartingDesign IEN2 and the Minimax-Refined Design in Figs. 5 and 6

Physical Thickness (,.m)

Refined for Refined forLayer Starting ) = 00, 300 4 = )0, 40

Glass - - _MgF 2 0.03061 0.032243 0.031448TiO 2 0.00622 0.006588 0.005408MgF 2 0.08679 0.087130 0.087628TiO 2 0.00728 0.008913 0.008118MgF2 0.08887 0.087237 0.088032Ta 2O5 0.02126 0.022368 0.021647MgF 2 0.04160 0.043233 0.042438Ta 2O5 0.05603 0.055766 0.055218MgF 2 0.00624 0.007873 0.007052TiO 2 0.08342 0.082474 0.082608MgF2 0.02209 0.021509 0.022074TiO2 0.02498 0.025390 0.024698MgF 2 0.11397 0.113052 0.114782Air - - _

dtot 0.5894 0.5938 0.5912dt~ot 0.1219 0.1234 0.1208Rma,0° 0.425% 0.564% 0.987%Rmax,30' 0.726% 0.564% 0.727%Rmax,40' 1.205% 1.007% 0.889%F 0.507% 0.474% -

reflectance0.014

0 ,OA 0.5 0.6 0.7

wavelength(a)

reflectance0.014 -

0.3 OA 0.5 0.6 0.7wavelength

(b)

Fig. 6. Spectral reflectances for 4) = 00 (soliccurve), and 40° (dotted curve) of (a) the IEIdesigners Rastello and Premoli) in Ref. 2 and (according to the method proposed here, forcinO0 and 4)b = 400.

0.8 0.9 1

Table 3 shows the starting physical thicknesses ofthe design IEN2, the thicknesses refined according tothe method proposed here for 4 = 0, 300, thoserefined for 4) = 00, 400, and the related characteristics.

The favorable results shown in these examplescannot be considered definitive and repeatable in anypractical design. In fact, because a method maysupply better results with a given starting point andworse results with another starting point, generalconclusions should be drawn only after deep andaccurate comparisons, which are outside the scope ofthis paper.

Conclusions

A minimax refinement method is proposed for wide-band antireflection coatings with oblique incidence.By a proper choice of the layer thicknesses, themethod minimizes the maximum reflectance overwavelength in a range of incidence angles. Theminimax criterion assures that the reflectance iscontrolled at each wavelength in the range of interest,in particular near the extrema. A specialized codebased on piecewise linear programming has beenimplemented on purpose.

The refining of antireflection coatings for lenses tobe used with normal and infrared photographic films

0.8 0.9 1 in the 0.4-0.9-jm region is presented for 4) = 0° and30°. A refinement is also proposed within the range4 = 040'. The results compare favorably with

I curve), 300 (dashed those supplied by the more popular least-squaresN2 design (M = 13, refinements.b) the design refinedg R(X, 4, d) for 4) = The authors acknowledge the use of computer

facilities freely offered by the Centro Studi per la


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0.01

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0.012

0.01

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..

, ,

.S

Televisione, Consiglio Nazionale delle Ricerche,Torino, Italy.

References1. A. Thelen, Design of Optical Interference Coatings (McGraw-

Hill, New York, 1989).2. A. Thelen and R. Langfeld, "Coating design context: antire-

flecting coating for lenses to be used with normal and infraredphotographic films," presented at the International Sympo-sium on Optical Systems Design, Berlin, 14-18 September1992.

3. H. Zycha, "Refining algorithm for the design of multilayerfilters," Appl. Opt. 12, 979-983 (1973).

4. P. Baumeister, R. Moore, and K. Walsh, "Application of linearprogramming to antireflection coating design," J. Opt. Soc.Am. 67, 1039-1045 (1977).

5. A. L. Bloom, "Refining and optimization in multilayers," Appl.Opt. 20, 66-73 (1981).

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

7. A. Premoli and M. L. Rastello, "Minimax refining of opticalmultilayer systems," Appl. Opt. 31, 1597-1605 (1992).

8. A. Premoli, "Piecewise-linear programming: the compact(CPLP) algorithm," Math. Programm. 36, 210-227 (1986).

9. 0. S. Heavens and H. M. Liddell, "Least squares method forthe automatic design of multilayers," Opt. Acta 15, 129-138(1968).

10. J. F. Tang and Q. Zheng, "Automatic design of opticalthin-film systems-merit function and numerical optimiza-tion method," J. Opt. Soc. Am. 72, 1522-1528 (1982).

11. J. A. Dobrowolski, F. C. Ho, A. Belkind, and V. A. Koss, "Meritfunctions for more effective thin-film calculations," Appl. Opt.28,2824-2831 (1989).

12. A. Premoli and M. L. Rastello, "Continuation method forsynthesizing antireflection coatings," Appl. Opt. 31, 6741-6746(1992).


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Minimax refining of wideband antireflection coatings for wide angular incidence