Minimax refining of optical multilayer systems

Minimax refining of optical multilayer systems

Amedeo Premoli and Maria Luisa Rastello

A new minimax method for refining optical multilayer systems is presented. It minimizes the maximumdeviation of the spectral transmittance from the desired specifications. These are assigned in such ageneral way that any shape can be approximated. The algorithm consists of iterating optimization stepsthat are obtained by developing the transmittance in the Taylor expansion versus the known parametersand solved by piecewise-linear programming. The investigation is limited to the design of losslessmultilayers with assigned refractive indices and undetermined thicknesses. Some minimax-refineddesigns are compared favorably with those reported in the literature.

Key words: Thin films, coatings, filters.

I. IntroductionIn recent times thin dielectric films have been used inoptics even more extensively. Broad and narrowbandpass filters are designed in the IR to UV regionsby considering structures that extend from the sim-plest single layers to the intricate arrangements ofmany layers.

From a mathematical point of view the design ofoptical multilayer systems is rather troublesome,since transcendent functions are involved in themerit function to be minimized. An exact analyticalsolution is available only for simple structures, even iflayer thicknesses are the only design parameters,while refractive indices are fixed. Thus algorithms areneeded to find thicknesses by minimizing a suitablemerit function. In the literature they are usuallycategorized into refinement and synthesis methods.

With a refinement method the design parametersof an assigned system are readjusted to improve itsperformances"13 and are moved from the startingdesign to a close local minimum of the merit function.With a synthesis methods one or more startingdesigns are selected on the basis of approximatedanalytical considerations and/or the designer's expe-rience, and generally a refinement algorithm is usedas a final stage in the overall procedure. If the starting

A. Premoli is with the Dipartimento di Elettrotecnica, Elettron-ica ed Informatica, Universith di Trieste, via A. Valerio, 10, I 34127Trieste, Italy. M. L. Rastello is with the Istituto ElettrotecnicoNazionale G. Ferraris, Strada delle Cacce, 91, I 10135 Torino, Italy.

Received 2 November 1990.0003-6935/92/101597-09$05.00/0.© 1992 Optical Society of America.

design is good enough, the optimal design is expectedto exhibit the global minimum of the merit function.

This paper deals with a new refinement algorithmthat is based on a minimax criterion that allows thedesigner to control the deviation of the spectraltransmittance from the assigned specifications at anywavelength. This feature is the difference betweenthe minimax criterion and the more popular least-squares criterion. The spectral specifications can beassigned in a general form, and any useful shape canbe considered. At any wavelength the transmittanceor the reflectance can be forced to any value betweeno and 1 independently from the specifications that arerequired at different wavelengths.

Moreover this criterion allows one to use piecewise-linear programming (PLP)" to solve the successiveoptimization steps. PLP is a generalization of thewell-known linear programming and can easily dealwith many unknowns and constraints. The optimiza-tion steps are of two types: the coarse step, which ischeap but efficient only when the current solution isfar from the minimum; and the fine step, which ismore expensive but efficient also when the solution isclose to the minimum. Since, to the authors' knowl-edge, general-purpose optimization routines of thistype are not available, an entirely original computercode has been implemented.

At present the algorithm takes into account onlydesigns with assigned refractive indices and undeter-mined thicknesses of lossless layers. The design of anantireflection coating is examined in detail and com-pared with the results available in the literature.'2

Preliminary results about the extension of thisrefinement algorithm to the synthesis have beenpresented in Ref. 13.

1 April 1992 / Vol. 31, No. 10 / APPLIED OPTICS 1597

II. Spectral Specifications and Merit Function

This section is concerned with the spectral specifica-tions and merit function that are used in the presentminimax refinement. Let the spectral transmittanceof an M-layer system be denoted as T(X, d), where theM-vector d groups the thicknesses that constitute theonly design parameters, while the refractive indicesare kept fixed.

The desired wavelength specifications are fitted byminimizing a suitable merit function, which consistsof an appropriate functional of T(X, d) defined overthe wavelength range of interest, Xa-Xb. A completesurvey of the more frequently used merit functions inthe design of optical multilayer systems can be foundin Ref. 14. Most working routines 2 ,""' use the follow-ing integral functional:

22(d) = f [T(X, d) -T(X)]2 dX, (1)

where T(X) is the transmittance to be approximated.In computer practice this integral is approximated bya summation over a discrete set of wavelengths.

Unfortunately, this functional does not allow oneto control the behavior of the approximating functionat the single wavelengths, in particular to the ex-trema Xa and Xb, where the absolute deviation IT(X,d) - T(X) may be excessive. To overcome this theabove squared difference can be multiplied by asuitable weighting function versus . However, thechoice of this function is not straightforward.

On the contrary the minimax criterion assures thatthe deviation between T(X, d) and T(X) is stronglyconstrained at each by minimizing its maximumabsolute value:

Y'(d) = max [jT(X, d) - T() I. (2)

Also in this case a weighting function can be intro-duced to allow a greater/lower deviation in the less/more critical portions of the Xa,-Ab range.

In this paper the minimax criterion is formulatedin a general way by extending the method that isproposed in Ref. 17 for glass filters to the morecomplex case of multilayer structures. Two functions,b U(X, p) and bL(X, p), are introduced by defining theupper and lower bounds of a parametric band (wherep is the parameter) within which the transmittanceT(X, d) of the multilayer stack should be constrained:

bU(X, p) = pbiu(X) + ( -p)boU(x), (3)

bL(X, ) = pbL(X) + ( -p)boL(A), (4)

where the functions bU(X), boU(X), bL(X), and bL(X)satisfy necessarily the following inequalities:

blu(X) > bou(X) boL(X) > bL(X) for X5 • X : X1, (5)

as illustrated in Fig. 1. The width of the band [i.e., thedifference bU(X, p) - bL(X, p)] decreases for any Xaccording to parameterp, since bU(X,p) and bL(, p)

wavelengthFig. 1. Generic transmittance T(X, d) (solid curve) and parametricbands b,L(X) (dotted curve), b () (dotted curve), b(X) (dottedcurve), and b,u(k) (dotted curve) versus X. The dashed curvesdescribe the narrowest band that is associated with T(X, d).

meet the following conditions:

bU(X,pi) > bU(p 2) for p, > p2 andX.<" • A b;

bL(X, p1) < bL(X,P2) forp, > p2 and X, • X < Xb.

(6)

(7)

For example, the p = 1 value characterizes the(wide) band that is delimited by bu(X) and bL(X),while p = 0 characterizes the (narrow) band that isdelimited by boU(X) and b L(X). Note that, if bou(X) =bL(X), the narrow band reduces to a line that iscoincident with T(X), as in the usual minimax approx-imation [see Eq. (2)]. Moreover, when b1u(X) - boU(X) =boL(X) - bL(X) for any X, the proposed formulationcoincides with the usual formulation in which b1u(X) -boU(X) plays the role of a weighting function [see Eq.(2)]. However, the proposed minimax criterion ismore general because b U(X) can be different fromb0L(X) and bU(X) - b(X) can be different fromboL(X) - bL(X).

The above formulation of spectral specifications isappropriate when the designer wishes to force T(X, d)to an assigned value that is greater than 0 and lessthan 1. In other words this case requires bilateralspecifications, since transmittance is necessarily con-trolled on both the lower and upper sides.

However, in many practical cases T(X, d) has to beforced to values as high or low as possible, e.g., in thedesign of antireflection coatings. These cases requireonly lower or upper unilateral specifications; thenonly the lower bound bL(X, p) or the upper boundbU(X, p) is active, respectively. By convention theunilateral specifications are taken into account (forthe inactive bound) by specifying bu(X) = bU(X) 1or bL(X) = bL(X) < 0, since T(X, d) in any case cannotovercome these bounds. Note that a unique design,for example, a passband filter, may involve simulta-neously bilateral, lower unilateral, and upper unilat-eral specifications in different portions of Xa-Xb.

As a practical rule bu(k) and b L(X) are chosen to befar enough apart that the expected optimal transmit-tance is confined between them, while boU(X) andb L(X) are chosen to be so close to each other that thetransmittance cannot be confined between them. Inthis case the minimum value of p will be in the 0-1

1598 APPLIED OPTICS / Vol. 31, No. 10 / 1 April 1992

bU(X, p) ------------

T(X,d)

L .............

b (x),p)

range. However, this choice is not strictly necessary.In practice the user assigns the above four functionsin a piecewise-linear form that is defined over theK-breakpoint mesh X1 Xa < XI< ... <AK --Ab

The previous remarks on spectral specificationsallow one to define the merit function. In fact thelowest value ofp can be associated with a value of d bysatisfying the following constraints:

pb L(X) + (1 -p )boL(X) < T(X, d) • pbU(x) + (1 -p )b0U(x)

for X. S < b (8)

as illustrated in Fig. 1. Then, inequalities (8) yield animplicit definition of the merit function p versus d.The fact that the merit functionp is not written in theform of an explicit function of d may appear mislead-ing compared with other formulations.

An equivalent explicit expression p(d) of the meritfunction can be obtained by introducing the lowernormalized deviation hL(X, d) and the upper normal-ized deviation h U(X d) defined as

hL( T(X, d) - b L(X) (9)b1 L(X) - bo L(X)

h u(x, d) = T(X, d) - bou(X)10b2u(x) - bou(x) (0

If vectors d- and d+ group the lower and upperbounds, respectively, the constraints

d- < d < d, (14)

have to be added to the optimization problem formu-lated in expressions (12) and (13). Thus specific lowerand upper bounds can be fixed for any thickness;however, in the most practical cases the lower (upper)bounds are chosen to be equal for all thicknesses.

Besides physical motivations the introduction ofthis box of constraints is justified also by algorithmicconsiderations. It is clear from the third group ofexamples in Section VI that d- and d+ play the role ofparameters affecting the descent path in the optimiza-tion process and possibly the terminal point of thepath.

The optimization problem proposed here is nonlin-ear because the transmittance T(X, d) involves nonlin-early the unknown vector d. Since the problem can belinearized in the neighborhood of a specific value of d,it has to be solved by successive iterations. If dl''denotes the reference value of d at the beginning ofthe ith step, a generic value is given by d = di- + Ad.The Taylor expansion of transmittance T(X, d), whichis truncated to the first-order terms, is

Then, the merit functionp(d) coincides with

p(d) = max max[hL(X, d), hU(x, d)]).X. <~X X b

MT(X, d-l + Ad) T(X, di-) + Tm(x, d-')Adm,

r-I(11)

The here-proposed refinement algorithm is based oninequality (8) (see Section III), while the explicit formof Eq. (11) is used in Section IV for sampling appropri-ately the wavelength range.

III. Optimization Problem

By considering the implicit form of the merit functioncontained in inequality (8), the optimization problemconsists in searching for the vector d so that therelated transmittance T(,, d) is confined in thenarrowest possible parametric band. In other wordsone minimizes p by properly choosing d andp itself:

mininize p, (12)

(15)

where Tm(X, d') (m = 1, 2,...,M) are the first-order partial derivatives of transmittance T(X, d)with respect to di, which are calculated at d = di1.Then the optimization problem can be rewritten as

minimizep,(Adpt

(16)

subject to

boL(X) - T(X, di-') < p[boL(X) - bL(X)]

M

+ E Tm(X, d-')Ad for X < X < b, (17)

b,,U(X) - T(X, d-) > p[boU(X) - blu(X)]

M

+ E T ..(X,&di1)Ad . for Xa < X < X,--1lwhere unknowns d and p are linked by the followingconstraints:

pbL(X) + (1 -p)boL(X) < T(X, d) < pblu(X) + (1 -p)boU(x)

for X < X 5 Xb. (13)

In most cases the designer wishes to choose thethicknesses of the multilayer within an assigned box.Specifically an upper bound can help to avoid solu-tions with an overall thickness that is too high, whilea lower bound prevents thicknesses from becomingtoo thin, which are troublesome in practice. In anycase a lower bound that is equal to zero is required toavoid unphysical solutions.

(18)

This truncation should be used only when thefirst-order terms of the Taylor expansion of T(X, d)are dominant with respect to the neglected higher-order terms. This condition is referred to as first-order dominance and is related to the value of thecomponents of the gradient, which in general are highwhen far from the local minima and low within theneighborhood of the minima themselves.

However, the truncation of the Taylor expansionintroduces an approximation error that increasesaccording to the absolute values of the elements ofAd. To control it lower and upper bounds (- Ad.,. and


Ad.., respectively) are imposed on each element ofAd:

-Admit • Adm Adm_,, m = 1, 2,...,M. (19)

These constraints, jointly with inequality (14), consti-tute the current [i.e., they are related to the (i - 1)thstep] box:

max[-Adm_, dn - d-'] Ad • min[Adm_,, d+ - dml]

m = 1, 2, .. .,M. (20)

Note that box constraints can be taken into accountby linear programming (as well as PLP) in a straight-forward way without further calculation.

By choosing a value of Ad., that is high enough,the first-order dominance allows one to implement anefficient (fast and robust) linearized optimizationstep, which causes a remarkable decrease in the meritfunction. Hereafter this specific optimization step isreferred to as a coarse step.

When the first-order dominance is not verified (e.g.,the higher-order dominance appears), the validity ofthe first-order truncated Taylor expansion is confinedwithin too small a neighborhood of d-'. In otherwords undesired oscillations appear in the succes-sive values of the merit function p(d-'), p(d),p(dl), . . . , which are supplied by the linearized steps.The drawback should be overcome by an appropriatevalue of Ad,, which is generally small, but unfortu-nately its selection is critical and could waste theconvergence rate. Then some tricks are exploited toovercome the above undesirable conditions.

First, the hard constraints [expression (19)] on theelements of Ad are omitted, while a unique softrestriction on its squared modulus is introduced tolimit the higher-order terms in the Taylor expansion.More precisely, a penalty function, which is coinci-dent with the squared modulus itself, is added to theoriginal merit function in expression (16):

M4 )2.M

miingizep + (Ad) 2. (21)

The m factor is chosen so that the squared modulus issignificant but not prevalent with respect to p. Sincequadratic terms are not appropriately handled inlinear programming, each of them is approximated by

f(Adm) = (Ad )2, m = 1, 2,... M, (22)

where f (Ad.) denotes a suitable piecewise-linearfunction. This approximation allows one to use a PLProutine, as is explained in Section V.

Second, in spite of the stabilizing action of thepenalty function, the band that confines T(X,di' + Ad) may turn out to be larger (instead ofnarrower) than that confining T(X, di`), that is,p(d- + Ad) may turn out to be higher than p(d&').This possible undesirable result is due to the badchoice of the factor in expression (21). Conse-

quently, when these effects appear, the theoreticalincrement Ad must be reduced by the proper factor a.The optimal a is chosen by investigating the mini-mum value of p(d!-' + aAd), where a is confined inthe 0 < a < 1 interval.

The sophisticated adjustments that are required inhigher-order dominance are implemented in the fineversion of the linearized step.

IV. Sampling of the Wavelength RangeAt any step constraints (17) and (18) must be verifiedat any wavelength in the continuous range ak-b,which generates in principle an infinite number ofinequalities, i.e., a pair for any X. In practice, byaccepting some amount of error, these conditions canbe verified in an appropriate discrete set A of wave-lengths X,' (j = 1, 2, . . . , J). Note that the number Jof these wavelengths can depend on the currentlinearized step, that is, on label i.

In creating set A the local maxima of the lower andupper normalized deviations in Eqs. (9) and (10) playa fundamental role. More exactly the sampling erroris necessarily null, when Ai is chosen to be coincidentwith the set FI, which includes the abscissas (touchwavelengths) at which hL(X, d') and hU(X, d) reachtheir maximum value over the Ahab range. Forsecurity all the abscissas of the local maxima of hL(X, d) and hU(X, d) (which are not only the touchwavelengths) are included in i.

Unfortunately at the ith linearized step the exactlocation of the touch wavelengths that are related toT(A, d'-1 + Ad) is not available, since Ad is unknown.Thus the touch wavelengths in IF' are assumed to becoincident with those related to the known transmit-tance T(X, di 1) and are grouped in rill.

By use of the above assumptions constraints (17)and (18) are replaced by the following two sets oflinear inequalities with Xj) A' = -

boL(Xi)j- T(xj', di) p[bL(\i)ji - L(Xji)]

M

+ 2 T.(XjL, d-')Ad,, (j =1, 2...J), (23)

bou(xji) - T(Xji, di) p[bou(\ji) - bu(ji)]

M

+ T.(Xj, d-')Ad (j = 1 2. .. J). (24)

The choice of A' = i-l is efficient in the first-orderdominance and is exploited in the coarse version ofthe linearized step. In the higher-order dominanceparasitic oscillations of the X terms versus i can arisein successive steps. Then the fine version of thelinearized step involves a more sophisticated choice ofA'.

In particular A contains the local maxima of the(i - 2)th step besides those of the (i - 1)th step:

A'= ri- u ri2 (25)

When some wavelengths in AN turn out to be too close,only one of them is considered. In this way the


oscillations of the A' terms are suppressed, while thenumber of wavelengths in A remains moderateenough.

The moderate number of discrete wavelengths in A'is peculiar to the present minimax criterion. In theleast-squares criterion this is not possible because thenumerical integration of the squared deviation re-quires a dense set of sampled wavelengths.

V. Algorithm ImplementationThis section deals with the implementation of theoverall procedure that was described in the previoussections. By setting xm = Adm(m = 1, 2, . . , M) andXM+1 = p and introducing matrix Y and vectors zL, zu,

bL, and bu with elements

Zim = Tm(xjid -1 ), m = 1, 2,..., M, j=1, 2, , (26)

L i - b1i LL(\ji),

zu = boU(Xj) -blu(Xj), j= 1,2,...,J, (28)

bjL = bL(Xi) - T(XjL, di'-), j = 1,2,...,J, (29)

bjU = boU(Xji) - T(j', d1), j=1,2,...,J. (30)

the following PLP problem [see expressions (16) and(22)] can be formulated:

minimize xM+l for a coarse step,

M (31)minimize XM+i + 2 f(x.) for a fine step,

subject to [see expressions (23) and (24)]

[=.z -ZL][-x < b] (32)

and to [see expressions (14 and 20)]

max[-Adm_, d,, - d'-'] < xm < min[Adm,,,, dm+ - dm'-1 ]

for a coarse step,

(dW - dm'-) < xr < (dn - dm 1) for a fine step;

m = 1, 2,. . .,M. (33)

The piecewise-linear terms f(xm) (m = 1, 2, .. ., M)that appear in the merit function (31) require the useof a PLP algorithm. Briefly speaking PLP generalizesthe well-known linear programming in the sense thatthe merit function is constituted by the addition ofpiecewise-linear functions of the single unknownsinstead of a linear combination of the unknowns.This generalization is not expensive from a CPU-timepoint of view, as observed in Ref. 11.

These considerations together with those in theprevious sections suggest the following implementa-tion of the overall algorithm:

(1) Assign the spectral specifications b1U(X), boU (),b1L(X), and boL(X), a starting value d& of the thicknessvector d, and tolerances El and E2; set i = 0.

(2) Calculatep(d&).(3) Seti = i + 1.(4) Calculate T(X, d'),hL(X, dE1),andhu(X, dihl).(5) State the current (first-order/higher-order)

dominance. Initially first-order dominance is as-sumed.

(6) Determine the ri-l set of the local maxima ofhL(X, di ) and hU(X, di ).

(7) Set A = Vil (for first-order dominance) orAi = i-l U i-2 (for higher-order dominance).

(8) Generate the PLP problem that implementsthe linearized optimization step according to expres-sions (31)-(33).

(9) Solve the PLP problem with respect to vectorAd.

(10) Calculatep(d&l + Ad).(11) If p(d'-l + Ad) is lower than p(d&'), update

the thickness vector d& = di' + Ad, and go to step(14).

(12) Ifp(d-l + Ad) is greater thanp(d&1), searchfor the arxin value in the 0 < a < 1 range byminimizingp(d&' + aAd) versus a.

(13) Update the thickness vector: d& = di' +a0nAd.

(14) If Idi - dl' I < el or p(dt ) - p(di-l) < estop; otherwise return to step (3).

In general the order of dominance is stated in step(5) according to the trend in the convergence. More-over, when the current order of dominance does notproduce a good convergence, it commutes to the otherone.

VI. Example: Antireflection Coatings for Germanium IROpticsA thorough investigation of the performances ofdifferent optimization algorithms3 was published re-cently. In general a comparison of different refine-ment algorithms is by no means a straightforwardtask. For example, the final (optimal) design stronglydepends on the starting design. Moreover, severaldifferent local minima may be located in the regionnear the starting design. Thus the same startingdesign can produce different solutions depending onthe specific descent path followed by the iterativeoptimization process, since some parameters (i.e., P,dma., d-, and d+ in the present algorithm) affect thepath in an unforeseeable way. Many attempts arerequired, the human factor (affected by the user'sexperience) being relevant in the interactive controlof the process. Final solutions that are related inprinciple to the same local minimum can be character-ized by significantly different numerical values of d asa result of the flatness of the merit function near theminimum and the round-off errors that affect conver-gence.

Furthermore, as observed by Bloom,2 little of thework that concerns computer optimization of thin-film designs is in the literature, and in what has beenpublished almost nothing is said about the specificalgorithm that is used in the procedure.


Recently nine optical technologists 2 participated inthe design of an antireflection coating for germaniumoptics in the 7.7-12.3-pim region, with alternate filmsof germanium (refractive index n = 4.2) and zincsulfide (n = 2.2), regardless of the numbers of layersand their thicknesses. In this exercise reflectanceR(X, d) = 1 - T(X, d) was specified to be zero in theentire range that we considered. With reference to theterminology that was introduced in Section II thesespecifications are lower unilateral in the entire range.

On the basis of the results presented in Ref. 12, thefirst group of examples has been dedicated to thedifferent spectral shapes that are obtainable by thehere-proposed minimax criterion with respect to aleast-squares criterion. With this aim several designs,which are optimized in Ref. 12 according to theleast-squares criterion, have been chosen as startingdesigns for the minimax refinement. The parametricband was defined as blU(X) = bU(X) = b`(W) = 1 and bLover the entire range to maximize the minimum valueof T(X, d) in the 7.7-12.3-[um range. Three cases witha different number M of layers, referred to as 3A(M = 7, designed by Aguilera), 2E (M = 13, designedby Goldstein), and 3C (M = 17, designed by Gold-stein) in Ref. 12, have been considered. Figure 2 andTable I show, respectively, the spectral reflectanceand the physical thicknesses of the optimal designsthat are supplied by the least squares criterion andrefined according to the method proposed here.

A direct inspection shows the expected equiripplebehavior of the minimax-refined reflectance; that is,the reflectance reaches its maximum value at severaltouch wavelengths within the 7.7-12.3-pm range, thetotal number of touch wavelengths depending on thenumber of layers. Generally speaking the resultingtouch wavelengths are about half of the layers. Theminimax-refined reflectance is lower than the least-squares-refined reflectance near the extrema buthigher in the internal portion of the interval. Inparticular in the M = 13 case the minimax-refinedreflectance presents the first and last peaks, whichare lowered with respect to those that appear in theleast-squares solution. This group of examples showsthat the minimax refinement is more capable ofcontrolling T(X, d) at any X in the sense that thenormalized deviation cannot overcome the p value ofthe merit function.

The second group of examples tests the capabilitiesof the minimax criterion in relaxing the spectralspecifications in assigned subranges. For example, inmost practical designs higher tolerances can be ac-cepted near the extrema X and Xb. With this aim theblL(X) curve is modified near the extrema, while thethree curves bL(X), bU(X), and bU(X) stay unmodified.Specifically, bL(X) is relaxed in the 7.7-8.0- and12.0-12.3-pm subranges, while it remains unmodi-fied in the 8.0-12.0-pm internal range. In the 7.7-8.0-pim internal subrange bL(X) varies linearly from asuitable value of bL(7.7) [with bL(7.7) < 0.99] to thevalue of 0.99, and in the 12.0-12.3-pm subrange bL(X) varies from 0.99 to bL(12.3) < 0.99. Thus in the

reflectance

0.025

0.02

0.015

0.01

0.005

OL7 8 9 10 11 12 13

wavelength

(a)reflectan

0.025

0.02

0.015

0.01

0.005 7 8 9 10 11 12 13

wavelength

(b)reflectance

0.01

0.008

0.006

0.004

0 .0027 8 9 10 11 12 13

wavelength

(c)Fig. 2. Spectral reflectances of the starting designs (solid curves)and those of the minimax-refined solutions (dashed curves) for (a)M = 7, (b) M = 13, and (c) M = 17.

entire 7.7-12.3-,um range bL(X) exhibits a three-piecelinear shape. The values of b(7.7) and b(12.3)determine the amount of relaxation of the spectralspecifications at the extrema. By imposing first bL(7.7) = bi' (12.3) = 0.98 and later, b(7.7) = bL(12.3) = 0.96, the refinement of the least-squaressolution in Ref. 12, referred to as the 3D case(M = 15, designed by Aguilera), has been executed.

Figure 3 and Table II show, respectively, thespectral reflectances and the thicknesses of the start-ing design supplied by the least-squares criterion inRef. 12 and those that are minimax refined withdifferent spectral bounds. As expected, in the internalsubrange of 8.0-12.0 pm, the relaxed-extrema specifi-cations permit the reflectance to be lowered in acontrolled way with respect to the case in which norelaxation in the extrema is considered as in the first


,e

I _,�-

..,

.U.UIZ

Table . Physical Thicknesses (in Micrometers) of Starting Designs and Minimax-Refined Solutions Reported in Fig. 2

Case 3A, M = 7 Case 2E, M = 13 Case 3C, M = 17

Layer n Starting Refined Starting Refined Starting Refined

Substrate 4.00 - - - - -

1 2.20 0.34773 0.36641 0.35595 0.34521 0.23368 0.25089

2 4.20 0.18274 0.17603 0.26952 0.27476 0.23429 0.22884

3 2.20 1.23068 1.24937 0.55336 0.54262 1.30259 1.27245

4 4.20 0.20298 0.21931 1.20929 1.21779 0.07831 0.09618

5 2.20 0.34318 0.32539 1.87832 1.87274 0.68545 0.65532

6 4.20 0.64048 0.62179 0.20769 0.19766 0.60493 0.59735

7 2.20 1.09205 1.10436 0.28827 0.29407 0.31877 0.34891

8 4.20 - - 0.39907 0.40553 0.24248 0.22464

9 2.20 - - 2.46973 2.47313 1.22859 1.25765

10 4.20 - - 0.21021 0.21967 0.33338 0.33733

11 2.20 - - 2.59586 2.59468 0.15673 0.12767

12 4.20 - - 0.27867 0.28184 0.64671 0.67372

13 2.20 - - 1.28759 1.29332 1.25682 1.28696

14 4.20 - - - - 0.11507 0.11419

15 2.20 - - - - 0.50886 0.47957

16 4.20 - - - - 0.60648 0.63306

17 2.20 - - - - 1.08609 1.07404

Air 1.00 - - - -

group of examples. Note that a similar relaxation ofthe reflectance at the extrema is obtained by theleast-squares criterion. In this case, however, theamount of relaxation cannot be controlled in astraightforward way.

In particular, the results shown here allow one tosay that the relaxations in the specifications at theextrema can supply minimax-refined reflectance witha much flatter behavior with respect to the least-squares-refined solutions. These flat solutions can beappreciated in several practical problems (see, forexample, Refs. 18 and 19).

The results that are supplied by the above twogroups of examples are of limited interest for thegeneral problem of antireflection coating design, be-cause the starting designs are chosen to be coincidentwith the final designs that are reported in Ref. 12.This choice is motivated by the need to compare thedifferent effects on the shape of the refined reflec-tance caused by least-squares and minimax criteria.

reflectance0.025 , I

0.02

0.015

0.01

0.005

08 9 10 11 12 13

wavelength

Fig. 3. Spectral reflectances of the starting design (solid curve) andthose of the minimax-refined solutions for M = 15 for b,'(7.7) =b,L(1 2.3) = 0.99 (dashed curve), bL(7.7) = b,'(1 2.3) = 0.98 (dottedcurve), and b,L(7.7) = b,' (12.3) = 0.96 (long-dashed curve).

The next and last group of examples compares theglobal capabilities of the present minimax refinementwith those of the least-squares refinements that arediscussed in Ref. 12. Thus the same standard startingdesign, with M = 21 layers, that is used in Ref. 12 hasbeen considered in the minimax method (Table III).The parametric band is used as in the first group ofexamples.

The refinement has been executed in two succes-sive phases: first, as is appropriate and common toany layer, lower and upper bounds (see vectors d- andd+) have been used to control the descent paths. Notethat the (in general, local) minimum that is reached

Table 1. Physical Thicknesses (in Micrometers) of the Starting Designand Minimax-Refined Solutions Reported in Fig. 3

Case 3D, M = 15

Refined

Layer n Starting 0.99-0.99 0.98-0.98 0.96-0.96

Substrate 4.00 - - - -

1 2.20 0.09091 0.09985 0.17026 0.122692 4.20 0.34702 0.32890 0.28440 0.306943 2.20 0.45682 0.47839 0.59380 0.480894 4.20 0.14702 0.13632 0.08318 0.126215 2.20 1.65795 1.64571 1.77477 1.660176 4.20 0.04940 0.05806 0.07770 0.044877 2.20 2.71477 2.69320 2.61833 2.690338 4.20 0.40000 0.42157 0.43856 0.394219 2.20 0.04318 0.02327 0.02191 0.07197

10 .4.20 0.69524 0.69568 0.68000 0.6558811 2.20 1.22955 1.25112 1.22705 1.2380812 4.20 0.12440 0.12966 0.12774 0.1165813 2.20 0.48750 0.46855 0.48380 0.5026714 4.20 0.60893 0.60618 0.60694 0.6111515 2.20 1.08977 1.08730 1.08867 1.08495Air 1.00 - - - -


I ~~~~~~~~~~~~~I.I

Ii

Tabla l11. Physical Thicknesses (in Micrometers) of the Starting Designand Minimax-Refined Solutions with Related Merit Functions p as

Reported in Fig. 4

Case 1B, M = 21

Refined

Layer n Starting M= 17 M= 19 M =21

Substrate 4.00 -

1 2.20 0.325 0.17613 0.16209 0.254812 4.20 0.183 0.17984 0.35899 0.308773 2.20 0.332 0.09397 0.45049 0.471794 4.20 0.186 0.10921 0.35678 0.251195 2.20 0.332 0.11709 0.27288 0.209076 4.20 0.183 0.00390 0.57617 0.329197 2.20 0.325 0.77683 0.01638 0.239578 4.20 0.183 0.11362 0.23691 0.265169 2.20 0.500 0.78684 0.46127 0.33496

10 4.20 0.035 0.58795 0.17445 0.1291111 2.20 0.500 0.17785 1.03605 0.8456012 4.20 0.073 0.37695 0.47231 0.2537113 2.20 0.748 1.32691 0.11767 0.4642914 4.20 0.073 0.15474 0.56271 0.3453515 2.20 0.500 0.41518 1.17951 0.4371216 4.20 0.035 0.62475 0.14321 0.0756917 2.20 1.966 1.09321 0.48314 0.9602118 4.20 0.139 - 0.62660 0.1690919 2.20 2.733 - 1.09109 0.4182220 4.20 0.330 - - 0.5689521 2.20 1.272 - - 1.11728Air 1.00 - - - -

p 1.051 0.862 1.108

often depends on the chosen values of the abovebounds.

Then the final designs that are supplied by the firstphase are refined further (in the second phase) byrelaxing the bound vectors, namely, d = 0 anddm+ = 3 um for m = 1,2, ... , M. Consequentlydifferent final designs have been obtained. They canexhibit some null layers, because the lower bound isrelaxed. The best solutions with 17, 19, and 21nonnull layers are reported in Table III jointly withthe value of the merit function. Figure 4 reports thespectral reflectance versus X. Note that the value ofthe merit functionp may be read from Fig. 4 since it isstrictly related to the maximum value of the reflec-tance versus X.

These minimax-refined designs exhibit better ap-proximation performances than the best final solu-tion presented in Ref. 12, which is referred to as case2F (M = 21, designed by Kemp), whose reflectance isalso shown by the solid curve in Fig. 4..

The refined designs, as reported in Table III, withM = 17 and M = 19 layers both exhibit one very thinlayer (< 0.05). Consequently they were refined againby suppressing it. Thus, two slightly different designswith M = 15 and M = 17 layers, respectively, wereobtained and are reported in Table IV.

The favorable results supplied by the present mini-max refinement cannot be considered definitive andrepeatable with any other practical design. In fact, asmentioned at the beginning of this section, a method

reflectance0.025

0.02

0.015

0.01

0.005

017

reflectance0.025 -

0.02

0.015

0.01

0.005

0

8 9 10 11 12 13

wavelength

(a)

7 8 9 10 11 12 13

wavelength

(b)

reflectance

0.02

0.015

0.01

0.005

n7 8 9 10

wavelength11 12 13

(c)Fig. 4. Spectral reflectances (dashed curves) of the minimaxsolutions refined from the standard starting design in Ref. 12 with (a)M = 17, (b) M = 19, and (c) M = 21 layers. The solid curverepresents the solution referred to as 2F (M = 21, designed byKemp) in Ref. 12.

with a given starting point may provide better resultsand worse results when a different starting point isused. Thus general conclusions cannot be drawn;they may be reached only after an extensive andtime-consuming comparison, which is outside thescope of this paper.

VII. Conclusions

A new minimax-refinement method has been pro-posed. By properly selecting the layer thicknesses, wecan minimize the maximum absolute deviation of themultilayer transmittance from the desired specifica-tions. The minimax criterion assures us that thedeviation is controlled at each wavelength in therange of interest, particularly near the extrema.


_ . L

* * L | I

nrx

v

Table IV. Physical Thicknesses (in Micrometers) and the Merit Function Referencesof Minimax-Refined Solutions Obtained from the Optimal Solutions of 1. H. Zycha, "Refining algorithm for the design of multilayer

Table by Suppressing the Layers that are Too Thin filters," Appl. Opt. 12, 979-983 (1973).

Refined 2. A. L. Bloom, "Refining and optimization in multilayers," Appl.Opt. 20, 66-73 (1981).

Layer n M= 15 M= 17 3. J. A. Dobrowolski and R. A. Kemp," Refinement of optical

multilayer systems with different optimization procedures,"Substrate 4.00 - - Appl. Opt. 19, 2876-2893 (1990).

1 2.20 0.17813 0.17331 4. S. D. Smith, "Design of multilayer filters by considering two2 4.20 0.17325 0.35185 effective interfaces," J. Opt. Soc. Am. 48, 43-50 (1958).3 2.20 0.08018 0.48176 5. P. Baumeister, "Design of multilayer filters by successive

5 2.20 0.89065 0.28966 approximations," J. Opt. Soc. Am. 48, 955-958 (1958).6 4.20 0.11159 0.80248 6. J. A. Dobrowolski, "Completely automatic synthesis of optical7 2.20 0.78481 0.44737 thin film systems," Appl. Opt. 4, 937-946 (1965).8 4.20 0.60938 0.18432 7. J. A. Dobrowolski, "Subtractive method of optical thin-film9 2.20 0.15930 1.03913 interference filter design," Appl. Opt. 12, 1885-1893 (1973).

10 420 037930 045996 8. P. Baumeister, R. Moore, and K. Walsh, "Application of linear

11 2.20 1.31473 0.10086 programming to antireflection coating design," J. Opt. Soc.12 4.20 0.13 0.59457 Am. 67, 1039-1045 (1977).12 4.20 0.41628 0.59457 9. J. A. Dobrowolski and D. Lowe, "Optical thin film synthesis

14 4.20 0.62221 0.1353 program based on the use of Fourier transform," Appl. Opt.15 2.20 1.02781 0.47979 17, 3039-3050 (1978).

15 2.20 1.09278 *677 10. W. J. Wild and H. Buhay, "Thin-film multilayer design16 4.20 - 0.61877 optimization using a Monte Carlo approach," Opt. Lett. 11,17 2.20 -1.09150 745imz747o (1986).e al apoah"Op.Ltt 1Air 1.00 - -745-747(1986).Air 1.00 0-82 11. A. Premoli, "Piecewise-linear programming: the compactp 1.094 0.88 (CPLP) algorithm," Math. Programm. 36, 210-227 (1986).

12. J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D.Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson,

Since the transmittance is nonlinear versus the and R. A. Kemp," Antireflection coatings for germanium IR

thicknesses, it is developed by the Taylor expansion. optics: a comparison of numerical design methods," Appl.Conseqently he optmizaton conists i a suces- 0Ot. 27,2832-2840 (1988).Consequently the optimization consists in a succes 13. A. Premoli and M. L. Rastello, "Stochastic synthesis of

sion of linearized descent steps, which are imple- multilayers," in Optical Thin Films and Applications, R.mented in two (coarse and fine) vesonProc. S6&. Photo-Opt. Instrum. Eng. 1270,

overall procedure more efficient. A specialized code 18-27(1990).

that is based on PLP has been implemented on 14. J. A. Dobrowolski, F. C. Ho, A. Belkind and V. A. Koss, "Merit

purpose. functions for more effective thin film calculations," Appl. Opt.

The design of antireflection coatings for germa- 28,2824-2831 (1989).

nium optics in the 7.7-12.3-ptm region of the IR with 15. 0. S. Heavens and H. M. Liddell, "Least squares method for

films of germanium (n = 4.2) and zinc sulfide (n = 2.2) the automatic design of multilayers," Opt. Acta 15, 129-138

only is presented. The results compare favorably with (1968).those supplied by the more popular least-squares 16. J. F. Tang and Q. Zheng, "Automatic design of opticalthoesuppiednby the menpopuar easts.uares thin-film systems-merit function and numerical optimizationrefinements method," J. Opt. Soc. Am. 72, 1522-1528 (1982).

The authors are grateful for the freely offered 17. V. Falletti, A. Premoli, and M. L. Rastello, "Optical glass

computer facilities of the Centro Studi per la Televi- filters: an algorithm for optimum design," Appl. Opt. 21,sion, CnsilioNazionale delle Ricerche, Torino, 1.4345-4350 (1982).

sione Consiglio nebtedlt I. Montre and i.oP 18. E. Pelletier, M. Klapisch, and P. Giacomo, "SyntheseItaly. They are indebted to I. Montrosset and G. P. d'empilements de couches minces," Nouv. Rev. Opt. Appl. 2,Bava, Dipartimento di Elettronica, Politecnico di 247-256 (1971).

Torino, for stimulating discussions. They also appre- 19. Z. Les and J. Kuros, "Method for the synthesis of semitranspar-

ciate the criticisms and suggestions for further work ent wide band dielectric mirrors," Thin Solid Films 46,

of two unknown reviewers. 117-126 (1977).


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Minimax refining of optical multilayer systems