Minimax design of optically transparent and reflective coatings

1730 J. Opt. Soc. Am. A/Vol. 21, No. 9 /September 2004 G. Erdmann and F. Santosa

Minimax design of optically transparent andreflective coatings

Grant Erdmann and Fadil Santosa

School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Received October 17, 2003; revised manuscript received April 15, 2004; accepted April 15, 2004

We consider the design problem of creating coatings that are either highly reflective or highly transparent.The goal is to create an optical element, consisting of planar dielectric layers, that reflects (or transmits) en-ergy over a given range of wavelengths and angles of incidence. The approach that we take is to formulate theproblem as a minimax optimization problem. We demonstrate that the approach can be effective in producingcoatings of a few layers with desirable properties. © 2004 Optical Society of America

OCIS codes: 220.4830, 230.4040, 230.4170.

1. INTRODUCTIONThe problem of creating an optical element with desiredproperties by a series of planar coatings is one that hasbeen studied previously. The design of nonreflectivecoatings for acoustic waves was studied by Hager andRostamian,1 Konstanty and Santosa,2 and Hager et al.3

In all three works, a single angle of incidence (in contrastto a multidirectional approach) is chosen. Hager andRostamian1 designed a nonreflective coating by using aminimax criterion and concluded that the minimizer is abang-bang coating, i.e., one that changes from the lowestallowed value to the highest. A specialized algorithm toexploit the minimax structure was used. Konstanty andSantosa2 considered the problem in the time domain andposed it as an optimization problem. Hager et al.3 stud-ied the problem in the frequency domain. Using an ana-lytical approach, they provided a constructive method fordesigning either a nonreflective or a highly reflective slab.

It is known that an infinite-periodic lamination maypossess the photonic bandgap property. That is, planewaves of certain angles of incidence and wavelengths can-not be propagated in such a medium. This property isthe basis of the design presented by Fink et al.4 Thoseauthors showed that a periodic stack can be designed sothat the bandgap lies in the range of wavelengths andangles of incidence of interest. They created a coatingwith a few periods and found that the reflectivity agreedfairly well with predictions for the infinite-periodic case.

Felbacq et al.5 studied the convergence properties of ahalf-space consisting of N periodic units. The normal-incidence reflection and transmission properties of such aslab were studied in the limit as the number of periodicunits approaches infinity. The work is important in thesense that it can be used to estimate the number of layersneeded to approximate well the properties of an infinite-periodic lamination.

We study the problem of designing finite laminationsthat have high reflection coefficients over a range ofangles and wavelengths. The approach can also be usedto produce laminations that have high transmission coef-ficients. The framework for the design procedure is mini-

1084-7529/2004/091730-10$15.00 ©

max optimization, which can be thought of as extremizinga functional under the worst-case scenario.

This paper begins with a review of how reflection andtransmission coefficients are computed in a layered me-dium. We show how the design problem can be posed asan optimization problem. We give an outline of how theresulting minimax optimization problem is solved. Thecomputational approach is based on an interior-pointmethod with the use of a primal-dual formulation.

In the final section, we describe the results of our com-putation. We show that omnidirectional mirrors withgood reflection properties can be achieved with relativelylow-contrast, absorptive material. An appendix is given,showing a derivation of formulas for the derivatives of thereflection and transmission coefficients with respect tothe design parameters.

2. REFLECTION AND TRANSMISSIONCOEFFICIENTSWe consider a set of N planar layers resting on a sub-strate, which is assumed to be a half-space. Air occupiesthe region above the layers. A plane wave at some angleof incidence strikes the layer, as shown in Fig. 1. Wewish to calculate the reflection and transmission proper-ties of the lamination. We will follow the developmentpresented by Chew.6 We give a brief description here forcompleteness.

We will denote the index of refraction in the jth layer asnj and the z-coordinate location of the jth interface as2dj . The layer thicknesses are hj for j 5 1,..., N. Theangle of incidence of the plane wave, u, is measured fromthe normal. We consider only cases with harmonic timedependence. All materials will be nonmagnetic. Thewave number in the jth layer is kj 5 vnj /c, where c isthe speed of light in free space. We define b 5 k0 sin uand a j 5 (kj

2 2 b2)1/2. The permittivity of the jth layeris e j 5 nj

2/(c2m0), where m0 is the permeability of freespace. We can model dissipation in the jth layer by set-ting nj to have a positive imaginary component.

2004 Optical Society of America

G. Erdmann and F. Santosa Vol. 21, No. 9 /September 2004 /J. Opt. Soc. Am. A 1731

We now state the relationship between the electromag-netic field in different layers. The z dependence of the ycomponent of the electric field in the jth layer can beshown to be of the form

ejy 5 Aj@exp~2ia jz ! 1 R̃j exp~2ia jdj 1 ia jz !#. (1)

Here R̃j is a generalized reflection coefficient, which indi-cates the amount of reflection at an interface, taking intoaccount reflections at all interfaces below it.

The transverse electric (TE) and transverse magnetic(TM) fields decouple in this geometry and can be treatedseparately. We can use Eq. (1) to describe the TE or TMcomponent of the electric field. The only difference ishow the generalized reflection coefficient is calculated inthe two cases. The computation of these coefficients canbe done by recursion,6 going from the lowest layer andmoving to the top, layer by layer.

The basic ingredient for the calculation is the reflectioncoefficient involving a single interface:

RjTE 5

a j 2 a j11

a j 1 a j11, Rj

TM 5e j11a j 2 e ja j11

e j11a j 1 e ja j11.

The recursive computation goes as follows. We startwith

R̃N 5 RN (2a)

and use

R̃j 5Rj 1 R̃j11 exp~2ia j11hj!

1 1 RjR̃j11 exp~2ia j11hj!for 0 < j , N

(2b)

to calculate the generalized reflection coefficients at eachinterface R̃j . Of particular interest for us is the overallreflection coefficient R̃0 , which is the reflection coefficientof the entire stack. In the computation, we would natu-rally use the coefficients Rj

TE for TE modes and RjTM for

TM modes.In this paper, we will use (as in the graphics of Fink

et al.4) the convention that u , 0 is a TM case and u. 0 is a TE case. This choice is made to make formulasmore compact. It also makes some intuitive and graphi-cal sense, since for u 5 0 the reflection coefficients for theTM and TE modes coincide.

Fig. 1. Problem geometry.

We assume that the properties of the air (z . 2d1) andthe substrate (z , 2dN11) are given. We denote thevariable properties of the layers by vectors:

n 5 @n1 , n2 ,..., nN#

to represent the indices of refraction of each layer and

h 5 @h1 , h2 ,..., hN#

to represent the layer thicknesses. We denote the reflec-tion coefficient of the entire stack, with all its parametricdependences, by

R~n, h, u, v! 5 uR̃0u2. (3)

Similarly, the transmission coefficient for the stack is de-noted by T(n, h, u, v) 5 1 2 R(n, h, u, v). Note that,by this definition, energy absorbed by the lamination isconsidered transmitted energy. We have also indicatedthe coefficients’ dependence on angle of incidence u andfrequency v.

We will compute first and second derivatives ofR(n, h, u, v) and T(n, h, u, v) by using the chain rule andthe recursive definition. One example of this can befound in Appendix A.

3. MINIMAX FORMULATION OF THEDESIGN PROBLEMThe design problem consists of finding properties of thestack of layers that either minimize or maximize the re-flection of the coating. There are many ways in whichone can pose this problem as an optimization problem, de-pending on the goals.

We will use a minimax formulation optimizing over theindices of refraction or the thicknesses of the layers, withthe angles and the frequencies of interest given in ad-vance. That is, we will choose the properties of the lay-ers such that the maximum reflection (or transmission) ofthe coating is minimized.

Consider the class of problems where the thicknesses ofthe layers are fixed; that is, h 5 h0 . Then, in our nota-tion, the minimax problem for minimum reflection is

minnPN

maxvPV,uPU

R~n, h 5 h0 , u, v!, (4a)

whereas for maximum reflection it is

minnPN

maxvPV,uPU

T~n, h 5 h0 , u, v!. (4b)

The set N is the set of feasible indices of refraction, and Vand U are the (discrete or discretized) frequencies and in-cident angles of interest.

We will also consider designing for thicknesses h wherethe index of refraction of the layers n is set to n0 . In thiswork, we consider n0 5 @a, b, a, b,...#; that is, a stack oflayers with alternating index of refraction, which in manyinstances has been shown to be effective.3–5 The desiredunknowns are the thicknesses of the layers. Thus theoptimization problems that we consider are

minhPH

maxvPV,uPU

R~n 5 n0 , h, u, v! (5a)

for minimal reflection and


minhPH

maxvPV,uPU

T~n 5 n0 , h, u, v! (5b)

for maximal reflection. Here H is the set of permissiblethickness sequences.

We assume that N and H are tensor products of one-dimensional (possibly infinite) intervals. For example,N 5 @l, u#N, N 5 RN, N 5 $n0%, and H 5 @0, `)N are allpossible.

One could consider minimizing a single smooth func-tion composed of the reflected energy over all angles andfrequencies of interest. One such optimization problemis the following:

minnPN

(vPV,uPU

R~n, h 5 h0 , u, v!.

This is a smooth optimization problem. However, be-cause the functional is the l1 norm (or l2 norm of R̃0),what tends to happen is that the optimal solution to thisproblem is a case where one or two reflection coefficientsare significantly larger than all others. Thus it producesresults with good performance in an averaged sense, in-stead of considering each angle and frequency combina-tion individually.

On the other hand, the minimax optimization (4a) isthe same as minimizing the l` norm of the reflection co-efficient. This formulation can be interpreted as requir-ing the highest reflection coefficient to be as small as pos-sible. Minimax optimization is a much harder problem,requiring some treatment of the nonsmoothness of thecost functional.

We believe that the norm to be optimized should be cho-sen to match the application whenever possible. Sincewe consider the l` norm to be the correct norm to usewhen evaluating the quality of the solution, we will use amethod that finds minimax optima.

4. MINIMAX OPTIMIZATIONThe goal of this section is to describe a method for solvingthe minimax design problems formulated in problems (4)and (5).

A. ReformulationIn this subsection, we will restate the design problems ina simpler form. First, we index the angles and the fre-quencies together, so that all combinations are consideredwith one index variable. That is, v P V and u P U ifand only if there is a j, 1 < j < m, such that v j 5 v andu j 5 u. Since we are using the maximum norm, the or-dering of these variables is unimportant. Next, we definethe functions $ fj% j51

m to be the functions on which the op-timization is performed. That is,

fj~n, h! 5 R~n, h, u j , v j!

if the maximum reflection is to be minimized and

fj~n, h! 5 T~n, h, u j , v j!

if the maximum transmission is to be minimized.When we solve for indices of refraction with thick-

nesses fixed [problem (4)], the only design variable is n.Similarly, if we solve for thicknesses of a stack of alternat-

ing index of refraction, the design variable is h. In eithercase, we denote the unknown by x P RN. The feasibleregion for x, defined by upper and/or lower bounds oneach component, is denoted by X.

Therefore our design problem can be written as theclassic discrete minimax problem

minxPx

max1<j<m

fj~x!. (6)

It is well-known7 that this problem is equivalent to theoptimization problem

min~z,x!PR3x

z subject to z > fj~x!

for all j P $1, 2,..., m%. (7)

The reason is that the smallest upper bound of all thefunctions [found by using expression (7)] necessarily oc-curs at the same point x as the smallest maximum func-tion [found by using expression (6)].

We now wish to combine the notation for the con-straints of the form z > fj(x) with the constraints thatdefine X. First, let cj(z, x) 5 z 2 fj(x), so that z> fj(x) for all j P $1, 2,..., m% is equivalent to cj(z, x)> 0 for all j P $1, 2,..., m%.

Next, recall that constraint set X is defined by one-dimensional intervals. In this work, no component willhave a feasibility region of a single point, although inpractice they can be treated in our approach. Thereforethe constraint x P X can be written as a list of upper andlower bounds. If xi > li Þ 2`, then we will add a newconstraint of the form

cj~z, x! 5 sgn~xi 2 li!uxi 2 liun.

Likewise, if xi < ui Þ `, then we will add a new con-straint of the form

cj~z, x! 5 sgn~ui 2 xi!uui 2 xiun.

In both expressions, n is a positive constant used for scal-ing, which does not change the feasible region. Rewrit-ing the constraints and including the variable n was madenecessary by the tendency of the method to jump imme-diately away from a good initial guess when n 5 1. Wechoose n ! 1, n } 1/N, which allows us to find perturba-tions of the good initial guesses. If p such constraints de-fine the feasible region X, then we index them from m1 1 to m 1 p and define cj(z, x) to be the correspondingconstraint. Therefore x P X is equivalent to cj(z, x)> 0 for all j P $m 1 1, m 1 2,..., m 1 p%.

Then we can rewrite problem (7) as

min~z,x!PRN11

z subject to cj~z, x! > 0

for all j P $1, 2,...,m 1 p%. (8)

One major difficulty that arises in the discrete minimaxproblem is that the maximum function in expression (6) isnonsmooth and expression (8) involves nonlinear con-straints. There are two main approaches to suchproblems—working-set8–11 and primal(-dual)12–15 meth-ods. The method that we will use is a primal-dualinterior-point trust-region method, which is adapted fromthat used by Conn et al.12 We refer the reader to that pa-


per for a general presentation of the method and ananalysis of the approach. Details of our implementationhave been described by Erdmann.16

B. Penalty and BarriersThe approach that we take to solving problem (8) is towrite a merit function

f~z, x, m! 5 z 2 m (j51

m1p

log@cj~z, x!#, (9)

with m . 0. The constraints in problem (8) are incorpo-rated as penalty terms, which provide a barrier to keepthe optimization process from violating any one of the in-equality constraints. For each m, the solution to Eq. (9)can be viewed as an approximate solution to problem (8).These barriers decrease in severity as m goes to zero, andthe accuracy of the approximation improves.

We will solve the problem

min~z,x!PRN

f~z, x, m!, (10)

choosing a sequence of positive penalty parameters $mk%such that limk→` mk 5 0.

C. Inner and Outer IterationsWe provide a very-high-level view of the algorithm forsolving problem (8) using the merit function (9).

Start with a feasible initial guess (z0 , x0) and initialpenalty parameter m0 ; set k 5 0.

(i) Inner iterations: Minimize f(z, x, mk) to somelevel of accuracy, starting from (zk , xk). Denote this so-lution by (zk* , xk* ).

(ii) Set xk11 5 xk* , update mk11 , mk , and find a fea-sible zk11 from xk11 and mk11 .

(iii) Increment k by 1 and return to (i).

The above procedure, which does not deal with the ac-tual minimization of f(z, x, mk), is called the outer itera-tion. The details of step (i), which is called the inner it-eration, will be described next.

D. Trust-Region MethodWe will use a trust-region method to perform the minimi-zation in step (i) of the algorithm in Subsection 4.C. Thecentral idea behind this method is to approximate thepenalty function (9) in the neighborhood of a point (z, x)by a quadratic model:

m~z 1 t, x 1 s! 5 f~z, x, m! 1 ^ g, ~t, s!&

112 ^~t, s!, H~t, s!&. (11)

We look for the minimum inside a bounded region calledthe ‘‘trust region.’’ That is, we solve

mini~t,x!i<D

m~z 1 t, x 1 s!. (12)

The size of the trust region is controlled by D, and thistrust-region radius is increased or decreased dependingon the ratio of the actual decrease in the functionalf(z, x, m) after updating to the decrease predicted by thequadratic model.

The gradient of the model, g, is merely the gradient off, but we will alter the Hessian of the model, H, to ac-count for optimality conditions of problem (10), which wegive in Subsection 4.E. The optimality conditions will al-low us to motivate our choice for the model Hessian,which we will state in Eq. (18) below.

E. Optimality ConditionsA primal-dual approach is adopted for solving problem(10). The method starts by looking at the necessary con-dition for a minimizer of problem (10), which is that thegradient of the merit function (9) is zero at a minimum.Recall that cj(z, x) 5 z 2 fj(x) for j 5 1,..., m. There-fore we have

]f

]z5 1 2 m(

j51

m 1

cj~z, x!5 0,

]f

]xk5 2m (

j51

m1p 1

cj~z, x!

]cj~z, x!

]xk5 0.

Writing the Jacobian of the vector with elements cj(z, x)as

J̃~x! 5 F ]c1 /]x1 ]c1 /]x2 ¯ ]c1 /]xN

] ] � ]

]cm1p /]x1 ]cm1p /]x2 ¯ ]cm1p /]xN

G T

and letting

yT 5 F m

c1,

m

c2,¯,

m

cm1pG ,

we can rewrite the above conditions as

1 2 (j51

m

yj 5 0, (13)

J̃y 5 0. (14)

We view the variables in y as dual variables, and theirdefinition as necessary conditions. Therefore we have

yj . 0 for all j 5 1, 2,..., m 1 p, (15)

cjyj 5 m for all j 5 1, 2,..., m 1 p. (16)

Moreover, inherited directly from the log-barrier func-tions, we have the conditions

cj . 0 for all j 5 1, 2,..., m 1 p. (17)

Expressions (13)–(17) are the first-order necessary condi-tions. Note that these conditions bear some resemblanceto the well-known Karush–Kuhn–Tucker conditions17 ap-plied to the constrained minimization (8), the only differ-ence being that the minimax Karush–Kuhn–Tucker con-ditions allow equality in conditions (15) and (17) andreplace Eq. (16) with cjyj 5 0.

F. Model DefinitionAlthough the goal of the inner iterations is to solve prob-lem (10), it has been shown to be beneficial12 to choose amodel Hessian that is not the Hessian of the merit func-tion f. In this subsection, we define this model Hessianand give a motivation for it.


Define the matrix Y (or C) as the diagonal matrix withthe components of y [or c(x)] as the diagonal entries.Also, define J to be the Jacobian of c with respect to bothz and x, i.e.,

J~z, x! 5 F ]c1 /]z ]c1 /]x1 ¯ ]c1 /]xN

] ] � ]

]cm1p /]z ]cm1p /]x1 ¯ ]cm1p /]xN

G T

,

and define ¹(z,x)(z,x)ci to be the Hessian of ci(z, x) with re-spect to both z and x.

Then we can define our model Hessian to be

H 5 JC21YJT 2 (i51

m1p

yi¹~z,x!~z,x!ci . (18)

Note that H is the Hessian of the merit function f if andonly if yi 5 m/ci .

The inclusion of the dual variables y allows us the free-dom to include optimality condition information from Eqs.(13), (14), and (16). In short, these conditions are used todetermine a likely value for the optimal yi* 5 mk /ci(xk* )and to perturb yi from m/ci toward the guessed yi* . Notethat yi linearly scales the first and second derivatives of ciin the definition of H. Therefore, making a good guessfor yi* allows one to weigh the different constraints betterin early iterations, allowing xk, j to converge more quicklyto xk* .

Another common motivation for the Hessian definition(18) relates to using Newton’s method on Eqs. (13), (14),and (16) in all three variables z, x, and y.12,16

G. Inner-Iteration AlgorithmThe approach to minimizing Eq. (9) for a fixed mk consistsof the following inner iterations. Here we assume thatwe are at the kth outer iteration and enter the inner it-erations with initial primal values zk and xk . An initialtrust-region radius is set to Dk,0 .

Set j 5 0, and start with zk,0 5 zk , xk,0 5 xk , andDk,0 . From these find a good choice for the dual vari-ables yk,0 .

(i) Compute the quadratic model mk, j , and solve

mintk, j ,sk, j

mk, j~zk, j 1 tk, j , xk, j 1 sk, j!

subject to i~tk, j , sk, j!ik, j < Dk, j .

(ii) If desired accuracy is achieved, set zk* 5 zk, j andxk* 5 xk, j and stop.

(iii) Update primal variables. If the step reduced f[i.e.,

f~zk, j 1 tk, j , xk, j 1 sk, j! , f~zk, j , xk, j!

is satisfied], then zk, j11 5 zk, j 1 tk, j and xk, j11 5 xk, j1 sk, j . Otherwise, zk, j11 5 zk, j and xk, j11 5 xk, j .

(iv) Update trust-region radius Dk, j11 .(v) Update dual variable yk, j11 .(vi) Increment j and return to (i).

Notes:1. In step (iii), it is possible that we may also reject

the update on account of constraint violation. tk, j can beadjusted to fix some instances of this. In other cases, wewould decrease the trust-region radius and return to (i).

2. The trust-region radius is made larger (smaller) ifthe actual reduction in the function f is sufficiently large(small) in comparison with the reduction predicted by thequadratic model mj,k .

3. The norm that appears in step (i) is an energy normbased on the positive-definite matrix Mk, j : ivik, j

2

5 vTMk, jv. Mk, j is based on the model Hessian H.16

4. There is a lot of freedom in how the dual variablesare updated. The necessary condition is that (yk, j) i2 mk /ci(zk, j , xk, j) → 0. We form a quadratic program-ming problem in y to minimize sensibly the violation ofEqs. (13), (14), and (16).

5. A stopping rule based on the level of first-order con-dition satisfaction is implemented.

Details of the algorithm have been given by Erdmann.16

H. Algorithm and ConvergenceWith the above description of the inner iterations, theminimax algorithm is complete. The outer iterations arestopped when mk is small enough that violation of thefirst-order conditions for the minimax problem is also suf-ficiently small.

Following the arguments given by Conn et al.,12 we canshow that the algorithm described in this section con-verges to a solution of problem (8) under mild assump-tions. The main steps of the proof12 carry over to thepresent case, and particular proofs related to the mini-max problem or the algorithmic implementation are givenin detail by Erdmann.16 For the applications in this pa-per, the mild assumptions are met by the following facts:

• cj(z, x) is infinitely differentiable in the interior ofthe feasible region for every j P $1, 2,..., m 1 p%.

• fj(x) is bounded for every j P $1, 2,..., m%.• An a posteriori check verifies that x remains in a

bounded region. A restriction that x be finite wouldguarantee this condition a priori, but it proved unneces-sary.

Owing to the nonconvex and complicated nature of theexpressions for the reflection coefficients, we use optimal-ity in a necessarily weaker sense than we would prefer.There are an unknown number of local optima for prob-lem (8). The algorithm will converge to local, second-order optima for problem (8). Global optimality is notguaranteed, and the optimum found by the algorithm de-pends heavily on the initial guess.

5. DESIGN PROBLEMSThis section contains results from numerical computa-tions using the minimax algorithm presented for design-ing planar coatings. We will design both transparentand reflective slabs.


A. Minimizing Reflection over Refraction IndicesFor transparent slabs, the setup is such that the lamina-tion is sandwiched between air above and substrate be-low. The index of refraction of air is unity, while the sub-strate has n 5 1.5. The coating is meant to perform wellfor angles between normal incidence and angle of inci-dence of 50° and for both TE and TM modes. The anglesare sampled at 2.94°. The range of wavelengths is from9.8 to 14.6 mm. A total of 13 samples is taken in thiswavelength range. The layer thickness is fixed at 0.8mm, while the index of refraction is allowed to vary be-tween 1.2 and 5. The design problem is to find a lamina-tion that produces the least amount of reflection over therange of wavelengths and angles of incidence.

As an initial design, we assign a linear profile. Threecases were considered: 12, 18, and 36 layers. The resultof our computation is summarized in Table 1. The firstcolumn in the table is the number of layers in the slab.The initial maximum reflection, computed for the linearprofile, is given for comparison purposes in column 2.The minimized maximum reflection is given in column 3.Once the optimal profile is obtained, we recalculate thereflection coefficient sampled ten times more finely inboth wavelengths and angles to ensure that the discreti-zation used in optimization did not miss any feature ofthe problem. The maximum resampled reflection valuesare displayed in column 4.

We investigated the robustness of the optimal designby introducing errors into the indices of refraction of thelayers and computing the maximum reflection of the per-turbed layers. The errors are multinormally distributedwith the componentwise standard deviation of 1% and 5%of the optimal variable value. The number of trials takenis the cube of the number of layers. The worst reflectionis recorded in each run and averaged over the number oftrials. The results of this study are displayed in columns5 and 6 of Table 1 and lead us to conclude that the optimaldesign is quite stable when the number of layers is smallbut that better transparency is achieved when the num-ber of layers is large. We can control trade-off betweenrobustness of design and smallness of the objective func-tion through the number of layers.

A sample of an optimal design is given in Fig. 2. Theupper figure is the initial design, and the lower figure isthe optimized design. The index of refraction is given ingrayscale as well as in height.

B. Minimizing Transmission for High-ContrastMaterialsIn the design of reflective slabs, the lamination is sand-wiched by air above and below. The coating is meant toperform well for all angles and for both TE and TMmodes. The angles are sampled at 6.07°, stopping at

Table 1. Transparent Film Results

No. ofLayers

Initialmax R

Final max R(Sampled)

Finalmax R

1% Error(Sampled)

5% Error(Sampled)

12 0.0388 0.00245 0.0246 0.00624 0.044418 0.0379 0.00199 0.00205 0.00768 0.076136 0.0346 0.00179 0.00184 0.0109 0.141

685°. We wish to compare our results with the results ofFink et al.,4 so the initial design, angles, and frequenciesconsidered will be similar. The range of wavelengths isfrom 9.8 to 14.6 mm. A total of 13 samples is taken inthis wavelength range.

The initial design repeats a 0.8-mm layer with index ofrefraction 4.6 followed by one 1.6-mm layer (or two 0.8-mmlayers if the indices of refraction vary) with index of re-fraction 1.6. We note that this design, from Fink et al.,4

is a good starting point, which is a great advantage infinding a better design. If n is the design variable, theindices of refraction are allowed to vary between 1.2 and5. If h is the design variable, we simply require that thelayer thicknesses remain nonnegative. The design prob-lem is to find a lamination that produces the most reflec-tion over the range of wavelengths and angles of inci-dence.

Three main cases were considered: 12, 18, and 36 lay-ers (or the corresponding cases if n is the design variable).The result of our computation is summarized in Table 2.The first column in the table is the number of layers inthe slab. The second column tells whether n or h is thedesign variable. The third column says whether dissipa-tion was included or not. If it is included, dissipation is10 dB/m for the high-index layers and 105 dB/m for thelow-index layers. The air half-spaces above and belowthe slab are considered nondissipative. The initial maxi-mum transmission is given in column 4 for comparisonpurposes. The minimized maximum transmission isgiven in column 5. Once the optimal profile is obtained,we recalculate the transmission coefficient sampled tentimes more finely in both wavelengths and angles to en-sure that the discretization used in optimization did notmiss any feature of the problem. The maximum resam-pled reflection values are displayed in column 6.

We ran the optimization for dissipative 18- and 36-layer examples with the same frequency and angle dis-cretization. We found that the sampling was not suffi-ciently fine. There was a discrepancy between the

Fig. 2. Optimal design of an 18-layer transparent film. Theinitial design (top) has a maximum reflection of 0.0379 for the TEor TM mode over the wavelength range 9.8–14.6 mm and theangle range 0°–50°. The optimized design (bottom) has a maxi-mum reflection of 0.00205.


Table 2. High-Contrast Mirror Results

No. ofLayers

DesignVar. (x) Dissipated

Initialmax T

Final max T(Sampled)

Finalmax T

1% Error(Sampled)

5% Error(Sampled)

18 n N 0.0243 0.00497 0.00530 0.00572 0.014027 n N 0.00526 0.000434 0.000443 0.000505 0.0073854 n N 0.0000847 0.00000468 0.00000572 0.00000580 0.0234

12 h N 0.0243 0.00350 0.00354 0.00431 0.066918 h N 0.00526 0.000329 0.000351 0.000354 0.045936 h N 0.0000847 0.0000161 0.0000178 0.0000179 0.0280

12 h Y 0.175 0.115 0.117 0.138 0.308

Fig. 3. Optimal design of a 27-layer nondissipative reflectivefilm. Here the layer thickness is fixed, and the indices of thelayers are optimized. The initial design (top) has a maximumtransmission of 0.00526 for the TE or TM mode over the wave-length range 9.8–14.6 mm and the angle range 0°–85°. The op-timized design (bottom) has a maximum transmission of0.000443.

Fig. 4. Optimal design of an 18-layer nondissipative reflectivefilm. Here the indices of refraction are fixed, and the layerthicknesses are optimized. The initial design (top) has a maxi-mum transmission of 0.00526 for the TE or TM mode over thewavelength range 9.8–14.6 mm and the angle range 0°–85°. Theoptimized design (bottom) has a maximum transmission of0.000351. Note that the structure is no longer periodic and that,in the interior, the ratio of layer thicknesses has changed.Moreover, the first high-index layer is now somewhat thinner.

predicted response of the optimized design at the coarseand fine samplings. This discrepancy could be elimi-nated if we performed the optimization with finer sam-pling, which would come at an increased computationalcost.

In general, optimizing dissipative layers is more diffi-cult because energy that is absorbed can never be re-flected. Our best result for a 12-layer case produced animprovement of approximately 30%.

Robustness of the optimal design was investigated byintroducing errors into the design variable and computingthe maximum reflection of the perturbed layers. The re-sults for 1% and 5% perturbations are displayed in col-umns 7 and 8 of Table 2. We conclude that the optimaldesign is fairly stable.

Samples of optimal designs are given in Figs. 3, 4, and5, which represent optimizing indices of refraction, opti-mizing thicknesses without dissipation, and optimizingthicknesses with dissipation, respectively.

C. Minimizing Transmission of Low-Contrast Materialswith BandgapWe also considered minimizing transmission when low-contrast materials are used. There are several reasonswhy mirrors made of dielectrics with low contrast perform

Fig. 5. Optimal design of a 12-layer dissipative reflective film.Here the indices of refraction are fixed, and the layer thicknessesare optimized. The initial design (top) has a maximum trans-mission of 0.175 for the TE or TM mode over the wavelengthrange 9.8–14.6 mm and the angle range 0°–85°. The optimizeddesign (bottom), which is no longer periodic, has a maximumtransmission of 0.117. The improvement is less dramatic incomparison with that for the nondissipative cases.


more poorly than high-contrast mirrors. The size of thebandgap of an infinite-periodic stack depends on the con-trast. The lower the contrast, the smaller the bandgap.Bandgaps are produced when total internal reflection oc-curs, and this happens more easily when the contrast islarge. When the range of frequencies and angles is con-tained in the bandgap, improvement is again quite sub-stantial. However, when it is outside the bandgap, im-provement is less spectacular.

The problem setup and angles of interest used for thelow-contrast case are the same as those used for the high-contrast case. The initial design, angles, and frequenciesconsidered will be similar to those given by Temelkuranet al.18 The range of wavelengths is from 11.8 to 12.6mm. A total of 11 samples is taken in this wavelengthrange.

The initial design repeats a 0.8444-mm layer with indexof refraction 2.8 followed by one 2.8146-mm layer (or two1.4073-mm layers if the indices of refraction vary) with in-

dex of refraction 1.55. We note that this design, from Te-melkuran et al.,18 is a good starting point. Our task is toimprove upon this design. If n is the design variable, theindices of refraction are allowed to vary between 1.2 and3. If h is the design variable, we simply require that thelayer thicknesses remain nonnegative. The design prob-lem is to find a lamination that produces the most reflec-tion over the range of wavelengths and angles of inci-dence.

Three main cases were considered: 12, 18, and 36 lay-ers (or the corresponding cases if n is the design variable).The result of our computation is summarized in Table 3,which is organized identically to Table 2. We see across-the-board improvement of approximately one order ofmagnitude over the initial designs, and the sensitivity re-sults lead us to conclude that the optimal design is quitestable.

Samples of optimal designs are given in Figs. 6, 7, and8, which represent optimizing indices of refraction, opti-

Table 3. Low-Contrast Bandgap Mirror Results

No. ofLayers

DesignVar. (x) Dissipated

Initialmax T


Finalmax T

1% Error(Sampled)

5% Error(Sampled)

18 n N 0.0743 0.0218 0.0219 0.0243 0.048027 n N 0.0224 0.00241 0.00244 0.00288 0.015254 n N 0.00241 0.00000724 0.00000724 0.00000984 0.00324

12 h N 0.0743 0.0280 0.0284 0.0302 0.058118 h N 0.0224 0.00302 0.00313 0.00340 0.017636 h N 0.00241 0.0000245 0.0000245 0.0000260 0.0102

12 h Y 0.278 0.153 0.155 0.155 0.16918 h Y 0.270 0.120 0.121 0.124 0.21336 h Y 0.288 0.112 0.122 0.125 0.258

Fig. 6. Optimal design of a 27-layer nondissipative low-contrastreflective film. Here the layer thickness is fixed, and the indicesof the layers are optimized. The initial design (top) has a maxi-mum transmission of 0.0224 for the TE or TM mode over thewavelength range 11.8–12.6 mm and the angle range 0°–85°.The optimized design (bottom) has a maximum transmission of0.00244.

Fig. 7. Optimal design of an 18-layer non-dissipative low-contrast reflective film. Here the indices of refraction are fixed,and the layer thicknesses are optimized. The initial design (top)has a maximum transmission of 0.0224 for the TE or TM modeover the wavelength range 11.8–12.6 mm and the angle range 0°–85°. The optimized design (bottom) has a maximum transmis-sion of 0.00313. Note that the structure is no longer periodicand that, in the interior, the ratio of layer thicknesses haschanged. Moreover, the first high-index layer is now somewhatthinner.


mizing thicknesses without dissipation, and optimizingthicknesses with dissipation, respectively.

D. Minimizing Transmission of Low-Contrast Materialswithout BandgapThe final example that we will give is the design of a re-flective slab that has a small bandgap, but we will be op-timizing its performance over a range of frequencies thatis much larger than the bandgap. Again, we ask that theslab perform well for TE and TM modes at angles from 0to 85°. The range of wavelengths is from 9.8 to 14.6 mm.The angles are sampled at 6.07°, while 13 samples of thewavelengths are taken.

The initial design repeats a 0.8444-mm layer with indexof refraction 2.8 followed by two 1.4073-mm layers withindex of refraction 1.55. This is not as good of an initialdesign as those used in Subsections 5.B and 5.C, since itdoes not correspond to a bandgap for this frequencyrange. Because of this, we expect the resulting designsto perform more poorly. If n is the design variable, theindices of refraction are allowed to vary between 1.2 and3. The design problem is to find a lamination that pro-duces the most reflection over the range of wavelengthsand angles of incidence.

Two cases were considered: 18 and 27 layers. The re-sult of our computation is summarized in Table 4, whichis organized like Tables 2 and 3, except that n is always

Fig. 8. Optimal design of an 18-layer dissipative low-contrastreflective film. Here the indices of refraction are fixed, and thelayer thicknesses are optimized. The initial design (top) has amaximum transmission of 0.270 for the TE or TM mode over thewavelength range 11.8–12.6 mm and the angle range 0°–85°.The optimized design (bottom), which is no longer periodic, has amaximum transmission of 0.121. The improvement is less dra-matic in comparison with that for the nondissipative cases.

Table 4. Low-Contrast Nonbandgap MirrorResults

No. ofLayers

Initialmax T


Finalmax T

1% Error(Sampled)

5% Error(Sampled)

18 1.00 0.360 0.399 0.402 0.61127 1.00 0.0987 0.129 0.125 0.468

the design variable and there is no dissipation. Notethat, while we are unable to get nearly perfect reflection,we are still able to make the transmission smaller. Asample of an optimal design is given in Fig. 9.

6. CONCLUSIONSThis paper introduces the use of a new minimax optimi-zation algorithm as a tool for optimizing the reflectionand transmission properties of a finite stack of optical lay-ers. We present the interior-point algorithm used for theoptimization and evaluate the designs obtained with thismethod. We first give an example of creating highlytransparent coatings. We then turn our attention to de-signing reflective coatings. When optimizing within abandgap, our approach finds an optimal design with verygood reflective properties, even with a small number oflayers. When the layers are dissipative, optimization isimportant because the optimal design has qualitativeproperties significantly different from those of the peri-odic design. Finally, we show an example of optimal de-sign in which the range of wavelengths is much largerthan the bandgap. This also leads to a design that is notclose to periodic. We believe that the minimax approachpresented here, which in principle considers the worst-case scenario as cost function, has broader applications inphotonics.

APPENDIX A: RECURSIVE GRADIENTCALCULATIONSHere we show how we compute first and second deriva-tives of R with respect to h 5 @h1 , h2 ,..., hN#T. To sim-plify notation, we will use the abbreviations RjªRj, j11

and R̃jªR̃j, j11 .First, we have, from the definitions of the two-layer re-

flection coefficients,

Fig. 9. Optimal design of a 27-layer nondissipative reflectivefilm. Here the layer thickness is fixed, and the indices of thelayers are optimized. The initial design (top) has a maximumtransmission of 1.00 for the TE or TM mode over the wavelengthrange 9.8–14.6 mm and the angle range 0°–85° (at some angleand wavelength, we have near-perfect transmission). The opti-mized design (bottom) has a maximum transmission of 0.129.


¹hRjTE 5 ¹hRj

TM 5 0, ¹hhRjTE 5 ¹hhRj

TM 5 0.

This implies that

¹hR̃N 5 0, ¹hhR̃N 5 0.

Now, starting with j 5 N 2 1 and descending to j5 0, we can evaluate the following relations for the de-rivatives of R̃j recursively. We label the numerator andthe denominator of the definition of R̃j as fj and gj , re-spectively:

R̃j 5Rj 1 R̃j11 exp~2ia j11hj!

1 1 RjR̃j11 exp~2ia j11hj!5..

fj

gj.

Taking derivatives, we have

¹hfj 5 ~2ia j11R̃j11ej 1 ¹hR̃j11!exp~2ia j11hj!,

¹hhfj 5 $~2ia j11!2R̃j11ejejT 1 2ia j11@¹hR̃j11ej

T

1 ej~¹hR̃j11!T#

1 ¹hhR̃j11%exp~2ia j11hj!,

¹hgj 5 Rj¹hfj .

The derivatives of R̃j can now be expressed succinctly as

¹hR̃j 5~1 2 Rj

2!¹hfj

gj2

,

¹hhR̃j 51 2 Rj

2

gj3

@ gj¹hhfj 2 2Rj¹hfj~¹hfj!T#.

Using these expressions, we can find that

¹hR~n, h, u, v! 5 2 Re~R̃0¹hR̃0!,

¹hhR~n, h, u, v! 5 2 Re@R̃0¹hhR̃0

1 ¹hR̃0~¹hR̃0!T#.

The authors may be reached by e-mail as follows: GrantErdmann, [email protected]. Fadil Santosa,[email protected].

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Minimax design of optically transparent and reflective coatings

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