Application of clustering global optimization to thin film design problems Fabien Lemarchand* Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, Marseille 13013, France * [email protected]
Abstract: Refinement techniques usually calculate an optimized local solution, which is strongly dependent on the initial formula used for the thin film design. In the present study, a clustering global optimization method is used which can iteratively change this initial formula, thereby progressing further than in the case of local optimization techniques. A wide panel of local solutions is found using this procedure, resulting in a large range of optical thicknesses. The efficiency of this technique is illustrated by two thin film design problems, in particular an infrared antireflection coating, and a solar-selective absorber coating. ©2014 Optical Society of America OCIS codes: (310.0310) Thin films; (310.4165) Multilayer design; (310.5696) Refinement and synthesis methods; (310.6805) Theory and design.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48(12), 955–957 (1958). C. J. Laan and H. J. Frankena, “Fast computation method for derivatives of multilayer stack reflectance,” Appl. Opt. 17(4), 538–541 (1978). J. A. Dobrowolski and R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29(19), 2876–2893 (1990). J. A. Dobrowolski, “Automatic refinement of optical multilayer assemblies,” J. Opt. Soc. Am. 51, 1475 (1961). Sh. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Ed Frontieres Gif sur-Yvette, 1992). A. V. Tikhonravov, M. K. Trubetskov, and G. W. Debell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35(28), 5493–5508 (1996). A. V. Tikhonravov and M. K. Trubetskov, “Modern design tools and a new paradigm in optical coating design,” Appl. Opt. 51(30), 7319–7332 (2012). W. H. Southwell, “Flip-flop coating synthesis revisited,” Appl. Opt. 53(4), A179–A185 (2014). S. Larouche and L. Martinu, “Step method: a new synthesis method for the design of optical filters with intermediate refractive indices,” Appl. Opt. 47(24), 4321–4330 (2008). P. G. Verly, “Fourier transform technique with refinement in the frequency domain for the synthesis of optical thin films,” Appl. Opt. 35(25), 5148–5154 (1996). K. Hendrix and J. Oliver, “Optical interference coatings design contest 2010: solar absorber and Fabry-Perot etalon,” Appl. Opt. 50(9), C286–C300 (2011). S. F. Masri, G. A. Bekey, and F. B. Safford, “A global optimization algorithm using adaptive random search,” Appl. Math. Comput. 7(4), 353–375 (1980). R. Fletcher, Practical Methods of Optimization, Second Edition (John Wiley & Sons, 1987). J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D. Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson, and R. A. Kemp, “Antireflection coatings for germanium IR optics: a comparison of numerical design methods,” Appl. Opt. 27(14), 2832–2840 (1988). C. G. E. Boender, A. H. G. Rinnooy Kan, L. Strougie, and G. T. Timmer, “A stochastic method for global optimization,” Math. Program. 22(1), 125–140 (1982). T. Csendes, “Nonlinear parameter estimation by global optimization - efficiency and reliability,” Acta Cybernetica 8, 361–370 (1989). T. Csendes, L. Pál, J. Oscar, H. Sendín, and J. R. Banga, “The global optimization method revisited,” Optim. Lett. 2(4), 445–454 (2088). M. Mongeau, H. Karsenty, V. Rouzé, and J.-B. Hiriart-Urruty, “Comparison of public-domain software for black-box global optimization,” Optim. Methods Softw. 13(3), 203–226 (2000). http://link.springer.com/content/pdf/10.1007%2Fs11590-007-0072-3.pdf J. R. Banga, C. G. Moles, and A. A. Alonso, “Global optimization of bioprocesses using stochastic and hybrid methods,” in C.A. Floudas and P.M. Pardalos, eds., Frontiers in Global Optimization (Springer, (2003), pp. 45– 70. C. G. Moles, J. R. Banga, and K. Keller, “Solving non convex climate control problems: pitfalls and algorithm performances,” Appl. Soft Comput. 5(1), 35–44 (2004).
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5166
22. C. G. Moles, G. Gutierrez, A. A. Alonso, and J. R. Banga, “Integrated process design and control via global optimization -A wastewater treatment plant case study,” Chem. Eng. Res. Des. 81(5), 507–517 (2003). 23. http://www.inf.u-szeged.hu/~csendes/index_en.html 24. W. C. Davidon, “Variable metric method for minimization,” SIAM J. Optim. 1(1), 1–17 (1991). 25. T. E. Shoup and F. Mistree, Optimization Methods with Applications to Personal Computers (Springer Verlag, 1985), pp. 110–118.
1. Introduction Since P. Baumeister introduced computer optimization for thin film problems in 1958 [1], many optimization techniques have been proposed in thin film design work. These techniques can be sorted into two subcategories, namely ‘refinement’ and ‘synthesis’ methods. Refinement requires an initial design as a starting solution, and the challenge consists in improving this solution by minimizing a cost function, which is computed as the ‘distance’ between the desired optical properties and those achieved with the optimized design. Usually, and depending on the method used, the optical thicknesses of the initial and final designs are relatively similar, and care needs to be taken in selecting a valid initial solution. Thin film problems are highly nonlinear, and it is important that the optimization technique be carefully chosen. Some techniques involve calculating the derivative of the cost function (these are referred to as gradient methods). It has been shown that these are more effective if the first and second order derivatives can be determined analytically [2]. In an effort to illustrate the potential of various optimization techniques, ten different refinement approaches were compared by applying them to three thin film optimization problems [3]. The fundamental principles applied to the automatic synthesis of thin film systems were first described by J.A. Dobrowolski [4]. With this type of approach, no initial design is required and a multilayer solution is generated. Currently, one of the most commonly used techniques is the so-called needle method, first proposed by [5], who demonstrated this method’s remarkable ability to rapidly solve thin film problems [6,7]. By iteratively inserting a very thin layer at the optimal location inside a stack, and then numerically refining this modified design, it is possible to increase the number of layers and to decrease the cost function at each step of the iteration. Alternative synthesis methods include flip-flop coating synthesis [8], step method [9] and Fourier transform techniques [10]. However, when the absolute lowest value of cost function is needed, or when the design requirements are more complex than simply decreasing a cost function, human judgment and intervention are unavoidable. This challenging issue has been debated at three-year intervals since 1976, when thin film designers are encouraged by the OIC committee to present the developments and potential applications of thin film synthesis techniques, by solving the traditional ‘Design Problem’ presented at the OIC meeting [11]. The variety of methodologies used by the designers suggests that the automatic design of interference coatings remains an open problem, despite the reduced computation times, which can be achieved with modern computers. In the present study, a Clustering Global Optimization (CGO) method for thin film optimization is described. This approach should not be viewed as an automatic design technique, since it makes use of a predetermined number of coating layers, of which only the thicknesses are allowed to vary. In addition, it cannot be thought of as a simple design refinement technique, since it combines both local and global optimization methodologies. In a certain sense, it could be described as a specific type of Adaptive Random Search method [12]. Starting from an initial design, which may be quite different to the final solution, the algorithm refines the thin film parameters by applying the quasi Newton method [13] to various different designs, chosen to lie in the vicinity of the initial design. This process is repeated iteratively on the different solutions, until the cost function no longer improves. Section 2 describes the proposed approach for the design of optimized thin films. In section 3, this method is applied to two different design problems: firstly, an Infrared Antireflection Coating design, already studied in detail in a former publication [14], and secondly, part of the 2010 OIC Design Problem, for which a high-temperature, solar-selective coating was required [11].
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5167
2. Clustering global optimization applied to thin film optimization 2.1 Description of the global optimization method Global optimization approaches to the solution of an N-dimensional problem can be classified into two categories: deterministic and stochastic. The former approach involves making an exhaustive search over the entire ‘solution space’. Unfortunately, the computation time increases exponentially with N (the dimension of the problem), making such methods entirely unsuitable for all but the most simple thin film design problems. Stochastic methods start from several random points. If the number of starting points is increased, the probability of finding the global minimum is improved, at the expense of increased computing time. The nonlinear clustering optimization described in the present study is derived from the ‘stochastic method for global optimization’ initially developed by C.G.E Boender et al. [15], which was later improved by T. Csendes [16,17]. The stochastic method has several advantages: - Firstly, it has been shown to be one of the most efficient methods for solving black box optimization problems [18]. - It has demonstrated true efficiency when applied to practical problems in various practical fields, such as: theoretical chemistry problems [19], the real life optimization of industrial applications such as bioprocess analysis [20], climate control [21], and integrated process design for wastewater treatment plants [22]. - It is freely available online, in various programming languages (Fortran, C, Matlab) via the following link [23]. The optimization algorithm is designed to find all of the local minima, and the one with the lowest cost function could potentially be the global minimum. These are found using a local search procedure, starting from suitably chosen starting points. The local search is made with a 2nd order quasi-Newton procedure based on the Davidon Fletcher Powell (DFP) update formula [24]. The quasi-Newton method gives a good approximation to the inverse of the Hessian matrix, and is widely considered to be one of the most sophisticated methods for solving unconstrained problems. The DFP update formula, also referred as the variable metric method, simultaneously generates the directions of the conjugate gradient while constructing the inverse Hessian matrix. One important aspect of this method is the judicious choice of starting points. Although all local minima should be investigated, in order to optimize computing time the same minima should not be repeatedly searched for. The aim of this method is thus to identify ‘regions of attraction’, defined by the set of all points leading to the same local minimum when the local procedure is applied. This step is called a clustering procedure. The construction of clusters, described in detail in [15], is the key characteristic of this method. Clusters are built in such a way that the probability that a local method will not be applied to any given point, which could possibly lead to an undiscovered local minimum, tends to zero when the size of the sample increases. If a starting point is identified as belonging to a cluster, the optimization procedure is stopped. The overall procedure used by this algorithm can be summarized as follows: 1. Consider m uniformly distributed points in an initially N-dimensional space S, and add them to the current cumulative sample C. Then refine this distribution to a smaller selection of only p points in C having the best merit function. 2. Apply the clustering procedure, one by one, to each of the p points. If all of these points belong to an existing cluster, go to step 4. Let p’ be the number of points which are not clustered. 3. Apply a local search to these p’ points. If a new local minimum is located, return to step 1 with m new starting points surrounding the local minimum. #204786 - $15.00 USD (C) 2014 OSA
Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5168
4. Determine the value of the smallest of all the local minima. Since step (3) allows a new starting process to be initiated in the vicinity of the previously computed local minimum, it is possible for the local procedure to find solutions outside the starting interval. Although the algorithm has not been demonstrated to be fully “global”, this specific characteristic led to it being defined as a “global optimization procedure”. Optimal use of this algorithm requires some of its parameters to be carefully adjusted: - The grid density, corresponding to the ratio m / p, is usually adjusted to lie between 102 and 104. - The distance of the N-dimensional space boundaries: this depends on the complexity of the optimization problem. If a small starting interval is set, it is generally difficult to jump to another minimum. For a given grid density (ratio m / p), a broader starting interval leads to a greater distance between evaluated points. Under such conditions, the algorithm may be unable to detect very sharp minima. A good compromise can be found when the user has acquired a certain degree of experience with the routine. This is mandatory when a high number of variables (typically > 15) is used. All of the minima can be found whenever the number of variable parameters is less than 15. More specifically, for the optimization of a thin film design, a high number of layers should be associated with small boundary distances. Depending on the complexity of the design requirements, the difference between the smallest and largest starting optical thickness (OT) of each layer is usually set to lie in the range between λ/10 and λ/2. 2.2 CGO applied to thin film optimization problems When applying this method to thin film optimization, an initial thin film design must be defined, involving a fixed number of layers with given values of refractive index. Each layer can be characterized by a fixed or variable thickness, such that in the latter case it is able to be modified by the optimization process. The cost or merit function, given by the distance between the calculated and desired values of various optical properties – which is generally defined by a set of reflectance and/or transmittance values for a given set of incidence angles and wavelengths – can be written as follows: 1/2
2 OPi Tgt − OPi Cal 1 N data (1) MF ( d1 ,...d n ) = N data i =1 Δi where MF represents the merit function, Ndata is the number of quantities under consideration, Δ i is the tolerance of the ith quantity, OPi Tgt is the ith desired value, OPi Cal is the ith numerically calculated optical property, and dj is the jth thickness of the n-layer stack. The definition given in Eq. (1) for the MF is the mathematical formulation of an unconstrained optimization problem. If, however, the designer wishes to bound some thicknesses by minimum or maximum values, the MF should include such constraints. It can do this, for example, by adding a penalty when these thicknesses are out of range. In refinement problems, the choice of starting solution is crucial. There can be many local minima for a thin film optimization problem, and the efficiency of the local solution is strongly related to the initial design’s ability to converge towards an efficient solution. Conversely, the CGO method is able to considerably modify an initial design by iteratively modifying the region over which local optimization is performed. If no obvious starting design is known, one acceptable compromise is to set the initial optical thicknesses to random values ranging between zero and one half wavelength, with the latter belonging to the desired spectral range.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5169
3. Results and discussion
3.1 Infrared antireflection coating on a germanium substrate In 1988, six different institutions used refinement and thin film synthesis methods to design antireflection coatings for germanium substrates, and their solutions were presented in Aguilera et al. [10]. The design problem was to find the best antireflection coating for the spectral range between 7.7µm and 12.3µm, under normal incidence. The relevant refractive indices are: ns = 4 for the substrate, and nH = 4.2, nL = 2.2 for the two materials under consideration. MF, defined as the RMS value of reflectance expressed as a percentage, is given by: 1 MF ( d1 ,...d n ) = 100 N data
1/2
Ndata i =1
Ri2 (λi )
(2)
where n = 21 is the number of layers, Ri is the calculated reflectance (in the range between 0 and 1) at wavelength λi, Ndata = 47, λi are linearly distributed over the range between 7.7µm and 12.3µm, and dj is the thickness of the jth layer of the n-layer stack. The initial design (called ‘DesA’) was proposed by Baumeister using the Chebyshev method. It comprises 21 layers (see Fig. 1(a)), and has an Optical Thickness (OT) close to 27 µm. The corresponding MF is approximately 10.58. Figure 1(a) plots the computed reflectance as a function of wavelength, and the refractive index profile inside the stack. As stated in section 2.2, many local solutions can be found for the optimisation of a thin film design, and in the case of the present 21-parameter problem, the solutions are innumerable. This is illustrated in Fig. 2, where MF is plotted as a function of two arbitrary layer thicknesses. When the 20th and the 21st layer thicknesses are allowed to vary over the range: 0-4µm, 10 local minima are found, among which four lead to an improvement in the MF with respect to the starting initial design. The strength of CGO is its ability to jump from one local minimum to another, and to then investigate solutions further than a simple refinement program. The results presented by Aguilera et al. are summarized in the following: Six refined designs are presented, each of which is reached using a different method. For five of these, the MF lies in the range between 1.35 and 1.51, and the optical thickness lies in the range between 24.7 and 31.6 µm. The authors noticed a remarkable resemblance between these five solutions and the small difference between the OT of the starting design and that of the solutions. The sixth solution, found using the Hooke and Jeeves pattern search optimization procedure which allows significant changes to be made to the construction parameters [25], led to an effective improvement with MF = 0.94 and OT = 31.6 µm and corresponds to design called ‘DesB’(see Fig. 1(b)). By changing the initial design to a stack with 69 layers and a 3.4 µm OT, the pattern change optimization procedure converged to a 17-layer solution with an OT of 31.3 µm and a MF equal to 0.66.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5170
Fig. 1. Reflectance vs wavelength (left), and refractive index profile vs optical thickness (right), for six thin-film designs. (a) DesA, MF = 10.58, initial design given by Baumeister. (b) DesB, MF = 0.94, resulting from the refinement of DesA using Hooke and Jeeves pattern search optimization procedure. (c) DesC, MF = 1.12, first local minimum using DesA as starting design. (d) DesD, MF = 0.55, best solution with DesA as starting design. (e) DesE, MF = 0.61, and (f) DesF, MF = 0.70, compromise between a good MF and a relatively small OT.Nota: DesA and DesB formula are given by [14], DesC, DesD, DesE, and DesF are calculated using CGO.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5171
Fig. 2. Merit Function as a function of the thicknesses of layers 20 and 21.
When applying CGO to this 21-parameter optimization problem, the first task consists in defining the initial boundaries for the 21 different thicknesses. As stated in section 2, this choice is not very critical, since CGO allows the initial space S to be modified, thereby influencing the routine’s ability to find reasonably sharp minimum peaks and investigate solutions far from the initial design. The average OT of a layer from initial design was approximately 1.3 µm, and the initial boundaries were given by: LB (i ) = max 0; d i − 1.3 4ni UB (i ) = d i + 1.3
(3)
4ni
where LB(i) and UB(i) are the lower and upper boundaries of the ith parameter, and ni is the refractive index of the ith layer. With such values, the cumulative OT of the initial upper and lower boundary limits are close to OTmin = 20 µm, and OTmax = 34 µm, respectively. The number of scanned points belonging to the starting interval was set to m = 10 000, and the number of selected points was set to p = 50. In accordance with the procedure described in section 2, after finding a local solution [ s (i ) ]i =1,n , CGO iteratively builds a new starting interval [LB’(i); UB’(i)], where LB '(i ) = max 0; si − 1.3 and UB '(i ) = si + 1.3 . With this updated sliding interval, 4ni 4ni CGO thus investigates a large panel of designs, with various index profiles and values of OT. The first local minimum, found in just a few seconds by the routine, is referred to as DesC (see Fig. 1 (c)), with an OT close to 31.5µm and MF = 1.12. This solution is similar, in terms of its structure and OT, to those presented by Aguilera et al. At the end of the computation, i.e. when all of the detected local minima belong to an existing cluster, the 50 best solutions are listed, of which that with the best MF, referred to as DesD, is shown in Fig. 1(d). Since this design has an OT = 45 µm and MF = 0.55, CGO clearly makes full use of its ability to investigate parameters which are very different from their initial values. The increase in OT is certainly beneficial in terms of improving the MF. If these solutions were evaluated on the basis of their merit functions only, it is likely that CGO would not find the best global solution to the problem. In the case of problems with more than 15 parameters, CGO certainly #204786 - $15.00 USD (C) 2014 OSA
Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5172
provides very efficient solutions. However, only an exhaustive study making use of a large set of starting designs can lead to a truly global solution. The solutions presented by Aguilera et al. [10], which were produced using DesA as the initial design, have an OT lying in the range [24.3 µm; 31.6 µm] and a minimum value of MF equal to 0.94, corresponding to DesC. Among the fifty solutions found by CGO, two designs are of particular interest, and have an OT lying in the same range as that of the solutions of Aguilera et al. As shown in Fig. 1, these correspond to DesE, with OT = 33.9 µm and MF = 0.61, and DesF, with OT = 27.0 µm and MF = 0.70. These two solutions thus represent a considerable improvement with respect to the designs proposed by Aguilera et al. The remarkable efficiency of CGO is thus illustrated by the three designs DesD, DesE and DesF, of which the solution given by DesD has a very high performance in terms of MF. The other two designs - DesE and DesF - represent an excellent compromise between a good MF and a relatively small OT.
3.2 Design of a solar-selective absorber coating with the smallest possible number of layers In the traditional design contest proposed and presented at the OIC 2010 Conference, two design problems were posed. The first of these concerned a high temperature solar-selective coating, for which 42 solutions were submitted, by 14 designers from different institutions. The problem statement and analysis of the results are described by K. Hendrix and J. Oliver [7]. In the present study, it is of interest to demonstrate the potential of CGO, when applied to the same design problem. The main requirement of the solar-selective coating was to absorb as much incident solar energy as possible, and to re-emit as little IR as possible from a normalized 450 °C blackbody spectrum. The blackbody emissivity was defined, in the range from 400 to 35720 cm−1, by the BB450C function and the solar absorbance was defined according to the standard table ASTM173 table, over the range from 2480 to 35720 cm−1. All of the data were defined at regular 40 cm−1 intervals. Finally, the Merit Function was given by:
MF = MFsolar absorbance − MFbb emittance
(4)
where MFsolar absorbance =
1 ASTM 173(σ i )
2480≤σ i ≤ 35720
MFemittance =
400 ≤σ i ≤ 35720
σ i2
1 BB 450C (σ i )
σ i2
A(σ ) ASTM 173(σ i ) 2 i , σi i ≤ 35720
σ
2480 ≤
A(σ ) BB 450C (σ i ) 2 i , σi i ≤ 35720
σ
400 ≤
and A(σ i ) = 1 − R (σ i ) − T (σ i ) is the absorbance of the design coating at the wavenumber given by: σ i = 1 / λi . ASTM173 is the standard reference solar spectral irradiance (provided in the form of a table at 0.5 nm intervals), and BB450C refers to the blackbody radiation at 450 °C. One aspect of the problem involved finding the simplest design simultaneously providing MFsolar absorbance > 94% and MFbb emittance < 10% . In particular, the winning design was that having the smallest number of layers, and the tiebreaker was that having the smallest physical thickness. A total of 14 different coating materials, with their corresponding refractive index dispersions (n and k as a function of wavenumber), were made available to the designers: Al2O3, SiO2, SiO, TiO2, ZrO2, HfO2, AlN, TiN, ZrN, HfN, Si, Ti, Zr, and Hf. The substrate was considered to have a constant index equal to 1.45.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5173
The winning design was determined through the use of CGO and consisted in a 6-layer stack with a total thickness equal to 265 nm. This design, together with its absorbance as a function of wavenumber is shown in Fig. 3.
Fig.
3.
vs wavenumber (cm−1) given by the winning design. > 94% , MFbb emittance < 10% , with a minimal number of layers (6) and
Absorbance
MFsolar absorbance
minimal total thickness (265 nm).
Two other designers provided different 6-layer designs, also having MFsolar absorbance > 94% and MFbb emittance < 10% . The thinnest reported design was a 20-layer design with a total thickness equal to 226.5 nm. The particularity of this design problem is the difficulty in choosing the best materials, among the 14 proposed coating materials. Over the very wide spectral range required by the solar-selective absorber coating, the correspondingly broad refractive-index dispersion of each material makes it difficult for the designer to intuitively select the most efficient materials. It should be noted that the low blackbody emissivity requirement can be achieved with high reflectance coatings, for which ZrN is a priori the best candidate, with a reflectance of approximately 95% in the low emissivity region. When CGO is applied to such a design problem, it is perfectly capable of finding the optimal solution. Indeed, for a 6-parameter problem with given material indices and initial thickness limits arbitrarily set to: LB = 0 , and UB = 50nm for each layer, if a solution simultaneously providing MFsolar absorbance > 94% and MFbb emittance < 10% exists, it will certainly be found by CGO. After having found this solution (if it exists), the designer must then minimize the total thickness in order to satisfy all of the design problem requirements. This procedure was iteratively applied to all of the proposed materials, in order to ensure that the 6-layer solution with the smallest physical thickness would be found. All of the 6layer solutions having a total thickness ≤ 320 nm are listed in Table 1.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5174
Table 1. Ten 6-layer Designs, Each Having a Total Thickness ≤ 320 nm, MFsolar absorbance > 94% , and MFbb emittance < 10% Layer number
1
2
3
4
5
6
Design 1
ZrN
Hf
Si
ZrO2
Hf
SiO2
265.0 nm
69.3
44.2
10.9
51.9
4.6
84.2
Design 2 268.0 nm
ZrN 72.6
Hf 40.5
Si 10.6
HfO2 56.3
Hf 5.1
SiO2 82.9
Design 3 281.0 nm
ZrN 88.6
Hf 24.2
ZrO2 51.9
Hf 11.8
SiO 44.3
SiO2 60.2
Design 4 283.0 nm
ZrN 83.0
Hf 43.3
Si 13.3
SiO 58.0
Hf 4.3
SiO2 81.1
Design 5 288.0 nm
ZrN 94.7
Hf 25.1.
ZrO2 47.9
Hf 12.2
ZrO2 38.4
SiO2 69.7
Design 6 291.0 nm
ZrN 98.5
Hf 22.3.
HfO2 50.8
Hf 12.5.
ZrO2 37.1
SiO2 69.9
Design 7 300.0 nm
ZrN 111.7
Si 27.0
Hf 28.8
Si 20.7
SiO 49.7
SiO2 62.1
Design 8 315.0 nm
ZrN
Hf
SiO
Hf
SiO
SiO2
118.9
24.4
55.7
11.4
41.0
63.7
Design 9 315.0 nm
ZrN 128.4
ZrO2 50.8
Hf 19.6
ZrO2 30.0
Si 4.2
SiO2 82.0
Design 10 320.0 nm
ZrN
Hf
SiO
Hf
ZrO2
SiO2
123.6
25.5.
51.8
11.5
35.4
72.2
Four of these ten solutions were submitted, by several designers, to the OIC contest. Taking the specific properties of the various materials into account, ZrN was always used for the first layer (immediately above the substrate), whereas SiO2 was always selected as the last layer, in order to maximize the global solar absorbance. Hf was used in each design, in most cases for the 2nd layer, whereas Si, HfO2, SiO, ZrO2 appeared in layers 3, 4 or 5. In practical terms, all of these ten designs are quite similar, with only 7 out the 14 proposed materials being used to produce the desired antireflective effect in the high solar absorbance region. By allowing the number of layers in the design to increase, CGO also found a 7-layer solution with a total thickness equal to 230 nm, and an 8-layer solution with a total thickness equal to 216 nm (see Table 2). This latter solution has a total thickness smaller than any of those presented for the OIC contest. Although, in all of the solutions ZrN was found to be the best material for the first layer (from those listed in Table 1), this was not a design requirement. The numerical optimizations did not converge to a solution when any another material was selected for the first layer in the 6-layer design.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5175
Table 2. Seven- and eight-layer designs with a total thickness ≤ 230 nm, MFsolar absorbance > 94% , and MFbb emittance < 10% Layer number
1
2
3
4
5
6
7
Design 11
ZrN
Hf
Si
Hf
ZrO2
Si
SiO2
230nm
49.1
38.7
22.0
8.2
27.8
5.4
78.8
8
Design 12
ZrN
Hf
Si
Hf
ZrO2
Si
SiO2
ZrO2
216nm
53.2
38.0
22.6
9.3
25.0
6.5
49.4
12.0
4. Conclusion
Clustering Global Optimization is more than a refinement technique, since it involves a stochastic procedure which, when provided with an initial design, generates a sliding interval in which it is able to search for new solutions with potentially very different optical thicknesses. The large number of iterations involved in this procedure, and its poor performance in terms of computation time, are largely outweighed by the very good performance of its solutions. The combination of a clustering method with a quasi-Newton descent algorithm delivers design solutions which are close to the global minimum. Once the number of layers and the refractive index of the materials have been set, CGO provides a wide panel of local solutions, characterized by a large variety of optical thicknesses. In the present study, this approach demonstrates its ability to improve on the solutions found for two previously studied design problems. In particular, CGO is used to exhaustively extract all possible 6-layer solutions for a high temperature solar-selective coating problem. When the challenge of developing an optical design is simply a preliminary step in an industrial process, for which the manufacture of a real thin-layer coating is the final goal, CGO can also be helpful. In such situations, all of the solutions given by CGO, which from the end user point of view have an acceptable error function value, are potentially suitable designs. The designer can then use computational simulations to test the each solution’s sensitivity to the expected manufacturing tolerances (errors in layer thickness, or refractive index variations), in order to select a design resulting in a manufactured thin film having the best potential performance and robustness.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5176
Abstract: Refinement techniques usually calculate an optimized local solution, which is strongly dependent on the initial formula used for the thin film design. In the present study, a clustering global optimization method is used which can iteratively change this initial formula, thereby progressing further than in the case of local optimization techniques. A wide panel of local solutions is found using this procedure, resulting in a large range of optical thicknesses. The efficiency of this technique is illustrated by two thin film design problems, in particular an infrared antireflection coating, and a solar-selective absorber coating. ©2014 Optical Society of America OCIS codes: (310.0310) Thin films; (310.4165) Multilayer design; (310.5696) Refinement and synthesis methods; (310.6805) Theory and design.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48(12), 955–957 (1958). C. J. Laan and H. J. Frankena, “Fast computation method for derivatives of multilayer stack reflectance,” Appl. Opt. 17(4), 538–541 (1978). J. A. Dobrowolski and R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29(19), 2876–2893 (1990). J. A. Dobrowolski, “Automatic refinement of optical multilayer assemblies,” J. Opt. Soc. Am. 51, 1475 (1961). Sh. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Ed Frontieres Gif sur-Yvette, 1992). A. V. Tikhonravov, M. K. Trubetskov, and G. W. Debell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35(28), 5493–5508 (1996). A. V. Tikhonravov and M. K. Trubetskov, “Modern design tools and a new paradigm in optical coating design,” Appl. Opt. 51(30), 7319–7332 (2012). W. H. Southwell, “Flip-flop coating synthesis revisited,” Appl. Opt. 53(4), A179–A185 (2014). S. Larouche and L. Martinu, “Step method: a new synthesis method for the design of optical filters with intermediate refractive indices,” Appl. Opt. 47(24), 4321–4330 (2008). P. G. Verly, “Fourier transform technique with refinement in the frequency domain for the synthesis of optical thin films,” Appl. Opt. 35(25), 5148–5154 (1996). K. Hendrix and J. Oliver, “Optical interference coatings design contest 2010: solar absorber and Fabry-Perot etalon,” Appl. Opt. 50(9), C286–C300 (2011). S. F. Masri, G. A. Bekey, and F. B. Safford, “A global optimization algorithm using adaptive random search,” Appl. Math. Comput. 7(4), 353–375 (1980). R. Fletcher, Practical Methods of Optimization, Second Edition (John Wiley & Sons, 1987). J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D. Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson, and R. A. Kemp, “Antireflection coatings for germanium IR optics: a comparison of numerical design methods,” Appl. Opt. 27(14), 2832–2840 (1988). C. G. E. Boender, A. H. G. Rinnooy Kan, L. Strougie, and G. T. Timmer, “A stochastic method for global optimization,” Math. Program. 22(1), 125–140 (1982). T. Csendes, “Nonlinear parameter estimation by global optimization - efficiency and reliability,” Acta Cybernetica 8, 361–370 (1989). T. Csendes, L. Pál, J. Oscar, H. Sendín, and J. R. Banga, “The global optimization method revisited,” Optim. Lett. 2(4), 445–454 (2088). M. Mongeau, H. Karsenty, V. Rouzé, and J.-B. Hiriart-Urruty, “Comparison of public-domain software for black-box global optimization,” Optim. Methods Softw. 13(3), 203–226 (2000). http://link.springer.com/content/pdf/10.1007%2Fs11590-007-0072-3.pdf J. R. Banga, C. G. Moles, and A. A. Alonso, “Global optimization of bioprocesses using stochastic and hybrid methods,” in C.A. Floudas and P.M. Pardalos, eds., Frontiers in Global Optimization (Springer, (2003), pp. 45– 70. C. G. Moles, J. R. Banga, and K. Keller, “Solving non convex climate control problems: pitfalls and algorithm performances,” Appl. Soft Comput. 5(1), 35–44 (2004).
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22. C. G. Moles, G. Gutierrez, A. A. Alonso, and J. R. Banga, “Integrated process design and control via global optimization -A wastewater treatment plant case study,” Chem. Eng. Res. Des. 81(5), 507–517 (2003). 23. http://www.inf.u-szeged.hu/~csendes/index_en.html 24. W. C. Davidon, “Variable metric method for minimization,” SIAM J. Optim. 1(1), 1–17 (1991). 25. T. E. Shoup and F. Mistree, Optimization Methods with Applications to Personal Computers (Springer Verlag, 1985), pp. 110–118.
1. Introduction Since P. Baumeister introduced computer optimization for thin film problems in 1958 [1], many optimization techniques have been proposed in thin film design work. These techniques can be sorted into two subcategories, namely ‘refinement’ and ‘synthesis’ methods. Refinement requires an initial design as a starting solution, and the challenge consists in improving this solution by minimizing a cost function, which is computed as the ‘distance’ between the desired optical properties and those achieved with the optimized design. Usually, and depending on the method used, the optical thicknesses of the initial and final designs are relatively similar, and care needs to be taken in selecting a valid initial solution. Thin film problems are highly nonlinear, and it is important that the optimization technique be carefully chosen. Some techniques involve calculating the derivative of the cost function (these are referred to as gradient methods). It has been shown that these are more effective if the first and second order derivatives can be determined analytically [2]. In an effort to illustrate the potential of various optimization techniques, ten different refinement approaches were compared by applying them to three thin film optimization problems [3]. The fundamental principles applied to the automatic synthesis of thin film systems were first described by J.A. Dobrowolski [4]. With this type of approach, no initial design is required and a multilayer solution is generated. Currently, one of the most commonly used techniques is the so-called needle method, first proposed by [5], who demonstrated this method’s remarkable ability to rapidly solve thin film problems [6,7]. By iteratively inserting a very thin layer at the optimal location inside a stack, and then numerically refining this modified design, it is possible to increase the number of layers and to decrease the cost function at each step of the iteration. Alternative synthesis methods include flip-flop coating synthesis [8], step method [9] and Fourier transform techniques [10]. However, when the absolute lowest value of cost function is needed, or when the design requirements are more complex than simply decreasing a cost function, human judgment and intervention are unavoidable. This challenging issue has been debated at three-year intervals since 1976, when thin film designers are encouraged by the OIC committee to present the developments and potential applications of thin film synthesis techniques, by solving the traditional ‘Design Problem’ presented at the OIC meeting [11]. The variety of methodologies used by the designers suggests that the automatic design of interference coatings remains an open problem, despite the reduced computation times, which can be achieved with modern computers. In the present study, a Clustering Global Optimization (CGO) method for thin film optimization is described. This approach should not be viewed as an automatic design technique, since it makes use of a predetermined number of coating layers, of which only the thicknesses are allowed to vary. In addition, it cannot be thought of as a simple design refinement technique, since it combines both local and global optimization methodologies. In a certain sense, it could be described as a specific type of Adaptive Random Search method [12]. Starting from an initial design, which may be quite different to the final solution, the algorithm refines the thin film parameters by applying the quasi Newton method [13] to various different designs, chosen to lie in the vicinity of the initial design. This process is repeated iteratively on the different solutions, until the cost function no longer improves. Section 2 describes the proposed approach for the design of optimized thin films. In section 3, this method is applied to two different design problems: firstly, an Infrared Antireflection Coating design, already studied in detail in a former publication [14], and secondly, part of the 2010 OIC Design Problem, for which a high-temperature, solar-selective coating was required [11].
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5167
2. Clustering global optimization applied to thin film optimization 2.1 Description of the global optimization method Global optimization approaches to the solution of an N-dimensional problem can be classified into two categories: deterministic and stochastic. The former approach involves making an exhaustive search over the entire ‘solution space’. Unfortunately, the computation time increases exponentially with N (the dimension of the problem), making such methods entirely unsuitable for all but the most simple thin film design problems. Stochastic methods start from several random points. If the number of starting points is increased, the probability of finding the global minimum is improved, at the expense of increased computing time. The nonlinear clustering optimization described in the present study is derived from the ‘stochastic method for global optimization’ initially developed by C.G.E Boender et al. [15], which was later improved by T. Csendes [16,17]. The stochastic method has several advantages: - Firstly, it has been shown to be one of the most efficient methods for solving black box optimization problems [18]. - It has demonstrated true efficiency when applied to practical problems in various practical fields, such as: theoretical chemistry problems [19], the real life optimization of industrial applications such as bioprocess analysis [20], climate control [21], and integrated process design for wastewater treatment plants [22]. - It is freely available online, in various programming languages (Fortran, C, Matlab) via the following link [23]. The optimization algorithm is designed to find all of the local minima, and the one with the lowest cost function could potentially be the global minimum. These are found using a local search procedure, starting from suitably chosen starting points. The local search is made with a 2nd order quasi-Newton procedure based on the Davidon Fletcher Powell (DFP) update formula [24]. The quasi-Newton method gives a good approximation to the inverse of the Hessian matrix, and is widely considered to be one of the most sophisticated methods for solving unconstrained problems. The DFP update formula, also referred as the variable metric method, simultaneously generates the directions of the conjugate gradient while constructing the inverse Hessian matrix. One important aspect of this method is the judicious choice of starting points. Although all local minima should be investigated, in order to optimize computing time the same minima should not be repeatedly searched for. The aim of this method is thus to identify ‘regions of attraction’, defined by the set of all points leading to the same local minimum when the local procedure is applied. This step is called a clustering procedure. The construction of clusters, described in detail in [15], is the key characteristic of this method. Clusters are built in such a way that the probability that a local method will not be applied to any given point, which could possibly lead to an undiscovered local minimum, tends to zero when the size of the sample increases. If a starting point is identified as belonging to a cluster, the optimization procedure is stopped. The overall procedure used by this algorithm can be summarized as follows: 1. Consider m uniformly distributed points in an initially N-dimensional space S, and add them to the current cumulative sample C. Then refine this distribution to a smaller selection of only p points in C having the best merit function. 2. Apply the clustering procedure, one by one, to each of the p points. If all of these points belong to an existing cluster, go to step 4. Let p’ be the number of points which are not clustered. 3. Apply a local search to these p’ points. If a new local minimum is located, return to step 1 with m new starting points surrounding the local minimum. #204786 - $15.00 USD (C) 2014 OSA
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4. Determine the value of the smallest of all the local minima. Since step (3) allows a new starting process to be initiated in the vicinity of the previously computed local minimum, it is possible for the local procedure to find solutions outside the starting interval. Although the algorithm has not been demonstrated to be fully “global”, this specific characteristic led to it being defined as a “global optimization procedure”. Optimal use of this algorithm requires some of its parameters to be carefully adjusted: - The grid density, corresponding to the ratio m / p, is usually adjusted to lie between 102 and 104. - The distance of the N-dimensional space boundaries: this depends on the complexity of the optimization problem. If a small starting interval is set, it is generally difficult to jump to another minimum. For a given grid density (ratio m / p), a broader starting interval leads to a greater distance between evaluated points. Under such conditions, the algorithm may be unable to detect very sharp minima. A good compromise can be found when the user has acquired a certain degree of experience with the routine. This is mandatory when a high number of variables (typically > 15) is used. All of the minima can be found whenever the number of variable parameters is less than 15. More specifically, for the optimization of a thin film design, a high number of layers should be associated with small boundary distances. Depending on the complexity of the design requirements, the difference between the smallest and largest starting optical thickness (OT) of each layer is usually set to lie in the range between λ/10 and λ/2. 2.2 CGO applied to thin film optimization problems When applying this method to thin film optimization, an initial thin film design must be defined, involving a fixed number of layers with given values of refractive index. Each layer can be characterized by a fixed or variable thickness, such that in the latter case it is able to be modified by the optimization process. The cost or merit function, given by the distance between the calculated and desired values of various optical properties – which is generally defined by a set of reflectance and/or transmittance values for a given set of incidence angles and wavelengths – can be written as follows: 1/2
2 OPi Tgt − OPi Cal 1 N data (1) MF ( d1 ,...d n ) = N data i =1 Δi where MF represents the merit function, Ndata is the number of quantities under consideration, Δ i is the tolerance of the ith quantity, OPi Tgt is the ith desired value, OPi Cal is the ith numerically calculated optical property, and dj is the jth thickness of the n-layer stack. The definition given in Eq. (1) for the MF is the mathematical formulation of an unconstrained optimization problem. If, however, the designer wishes to bound some thicknesses by minimum or maximum values, the MF should include such constraints. It can do this, for example, by adding a penalty when these thicknesses are out of range. In refinement problems, the choice of starting solution is crucial. There can be many local minima for a thin film optimization problem, and the efficiency of the local solution is strongly related to the initial design’s ability to converge towards an efficient solution. Conversely, the CGO method is able to considerably modify an initial design by iteratively modifying the region over which local optimization is performed. If no obvious starting design is known, one acceptable compromise is to set the initial optical thicknesses to random values ranging between zero and one half wavelength, with the latter belonging to the desired spectral range.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5169
3. Results and discussion
3.1 Infrared antireflection coating on a germanium substrate In 1988, six different institutions used refinement and thin film synthesis methods to design antireflection coatings for germanium substrates, and their solutions were presented in Aguilera et al. [10]. The design problem was to find the best antireflection coating for the spectral range between 7.7µm and 12.3µm, under normal incidence. The relevant refractive indices are: ns = 4 for the substrate, and nH = 4.2, nL = 2.2 for the two materials under consideration. MF, defined as the RMS value of reflectance expressed as a percentage, is given by: 1 MF ( d1 ,...d n ) = 100 N data
1/2
Ndata i =1
Ri2 (λi )
(2)
where n = 21 is the number of layers, Ri is the calculated reflectance (in the range between 0 and 1) at wavelength λi, Ndata = 47, λi are linearly distributed over the range between 7.7µm and 12.3µm, and dj is the thickness of the jth layer of the n-layer stack. The initial design (called ‘DesA’) was proposed by Baumeister using the Chebyshev method. It comprises 21 layers (see Fig. 1(a)), and has an Optical Thickness (OT) close to 27 µm. The corresponding MF is approximately 10.58. Figure 1(a) plots the computed reflectance as a function of wavelength, and the refractive index profile inside the stack. As stated in section 2.2, many local solutions can be found for the optimisation of a thin film design, and in the case of the present 21-parameter problem, the solutions are innumerable. This is illustrated in Fig. 2, where MF is plotted as a function of two arbitrary layer thicknesses. When the 20th and the 21st layer thicknesses are allowed to vary over the range: 0-4µm, 10 local minima are found, among which four lead to an improvement in the MF with respect to the starting initial design. The strength of CGO is its ability to jump from one local minimum to another, and to then investigate solutions further than a simple refinement program. The results presented by Aguilera et al. are summarized in the following: Six refined designs are presented, each of which is reached using a different method. For five of these, the MF lies in the range between 1.35 and 1.51, and the optical thickness lies in the range between 24.7 and 31.6 µm. The authors noticed a remarkable resemblance between these five solutions and the small difference between the OT of the starting design and that of the solutions. The sixth solution, found using the Hooke and Jeeves pattern search optimization procedure which allows significant changes to be made to the construction parameters [25], led to an effective improvement with MF = 0.94 and OT = 31.6 µm and corresponds to design called ‘DesB’(see Fig. 1(b)). By changing the initial design to a stack with 69 layers and a 3.4 µm OT, the pattern change optimization procedure converged to a 17-layer solution with an OT of 31.3 µm and a MF equal to 0.66.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5170
Fig. 1. Reflectance vs wavelength (left), and refractive index profile vs optical thickness (right), for six thin-film designs. (a) DesA, MF = 10.58, initial design given by Baumeister. (b) DesB, MF = 0.94, resulting from the refinement of DesA using Hooke and Jeeves pattern search optimization procedure. (c) DesC, MF = 1.12, first local minimum using DesA as starting design. (d) DesD, MF = 0.55, best solution with DesA as starting design. (e) DesE, MF = 0.61, and (f) DesF, MF = 0.70, compromise between a good MF and a relatively small OT.Nota: DesA and DesB formula are given by [14], DesC, DesD, DesE, and DesF are calculated using CGO.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5171
Fig. 2. Merit Function as a function of the thicknesses of layers 20 and 21.
When applying CGO to this 21-parameter optimization problem, the first task consists in defining the initial boundaries for the 21 different thicknesses. As stated in section 2, this choice is not very critical, since CGO allows the initial space S to be modified, thereby influencing the routine’s ability to find reasonably sharp minimum peaks and investigate solutions far from the initial design. The average OT of a layer from initial design was approximately 1.3 µm, and the initial boundaries were given by: LB (i ) = max 0; d i − 1.3 4ni UB (i ) = d i + 1.3
(3)
4ni
where LB(i) and UB(i) are the lower and upper boundaries of the ith parameter, and ni is the refractive index of the ith layer. With such values, the cumulative OT of the initial upper and lower boundary limits are close to OTmin = 20 µm, and OTmax = 34 µm, respectively. The number of scanned points belonging to the starting interval was set to m = 10 000, and the number of selected points was set to p = 50. In accordance with the procedure described in section 2, after finding a local solution [ s (i ) ]i =1,n , CGO iteratively builds a new starting interval [LB’(i); UB’(i)], where LB '(i ) = max 0; si − 1.3 and UB '(i ) = si + 1.3 . With this updated sliding interval, 4ni 4ni CGO thus investigates a large panel of designs, with various index profiles and values of OT. The first local minimum, found in just a few seconds by the routine, is referred to as DesC (see Fig. 1 (c)), with an OT close to 31.5µm and MF = 1.12. This solution is similar, in terms of its structure and OT, to those presented by Aguilera et al. At the end of the computation, i.e. when all of the detected local minima belong to an existing cluster, the 50 best solutions are listed, of which that with the best MF, referred to as DesD, is shown in Fig. 1(d). Since this design has an OT = 45 µm and MF = 0.55, CGO clearly makes full use of its ability to investigate parameters which are very different from their initial values. The increase in OT is certainly beneficial in terms of improving the MF. If these solutions were evaluated on the basis of their merit functions only, it is likely that CGO would not find the best global solution to the problem. In the case of problems with more than 15 parameters, CGO certainly #204786 - $15.00 USD (C) 2014 OSA
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provides very efficient solutions. However, only an exhaustive study making use of a large set of starting designs can lead to a truly global solution. The solutions presented by Aguilera et al. [10], which were produced using DesA as the initial design, have an OT lying in the range [24.3 µm; 31.6 µm] and a minimum value of MF equal to 0.94, corresponding to DesC. Among the fifty solutions found by CGO, two designs are of particular interest, and have an OT lying in the same range as that of the solutions of Aguilera et al. As shown in Fig. 1, these correspond to DesE, with OT = 33.9 µm and MF = 0.61, and DesF, with OT = 27.0 µm and MF = 0.70. These two solutions thus represent a considerable improvement with respect to the designs proposed by Aguilera et al. The remarkable efficiency of CGO is thus illustrated by the three designs DesD, DesE and DesF, of which the solution given by DesD has a very high performance in terms of MF. The other two designs - DesE and DesF - represent an excellent compromise between a good MF and a relatively small OT.
3.2 Design of a solar-selective absorber coating with the smallest possible number of layers In the traditional design contest proposed and presented at the OIC 2010 Conference, two design problems were posed. The first of these concerned a high temperature solar-selective coating, for which 42 solutions were submitted, by 14 designers from different institutions. The problem statement and analysis of the results are described by K. Hendrix and J. Oliver [7]. In the present study, it is of interest to demonstrate the potential of CGO, when applied to the same design problem. The main requirement of the solar-selective coating was to absorb as much incident solar energy as possible, and to re-emit as little IR as possible from a normalized 450 °C blackbody spectrum. The blackbody emissivity was defined, in the range from 400 to 35720 cm−1, by the BB450C function and the solar absorbance was defined according to the standard table ASTM173 table, over the range from 2480 to 35720 cm−1. All of the data were defined at regular 40 cm−1 intervals. Finally, the Merit Function was given by:
MF = MFsolar absorbance − MFbb emittance
(4)
where MFsolar absorbance =
1 ASTM 173(σ i )
2480≤σ i ≤ 35720
MFemittance =
400 ≤σ i ≤ 35720
σ i2
1 BB 450C (σ i )
σ i2
A(σ ) ASTM 173(σ i ) 2 i , σi i ≤ 35720
σ
2480 ≤
A(σ ) BB 450C (σ i ) 2 i , σi i ≤ 35720
σ
400 ≤
and A(σ i ) = 1 − R (σ i ) − T (σ i ) is the absorbance of the design coating at the wavenumber given by: σ i = 1 / λi . ASTM173 is the standard reference solar spectral irradiance (provided in the form of a table at 0.5 nm intervals), and BB450C refers to the blackbody radiation at 450 °C. One aspect of the problem involved finding the simplest design simultaneously providing MFsolar absorbance > 94% and MFbb emittance < 10% . In particular, the winning design was that having the smallest number of layers, and the tiebreaker was that having the smallest physical thickness. A total of 14 different coating materials, with their corresponding refractive index dispersions (n and k as a function of wavenumber), were made available to the designers: Al2O3, SiO2, SiO, TiO2, ZrO2, HfO2, AlN, TiN, ZrN, HfN, Si, Ti, Zr, and Hf. The substrate was considered to have a constant index equal to 1.45.
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The winning design was determined through the use of CGO and consisted in a 6-layer stack with a total thickness equal to 265 nm. This design, together with its absorbance as a function of wavenumber is shown in Fig. 3.
Fig.
3.
vs wavenumber (cm−1) given by the winning design. > 94% , MFbb emittance < 10% , with a minimal number of layers (6) and
Absorbance
MFsolar absorbance
minimal total thickness (265 nm).
Two other designers provided different 6-layer designs, also having MFsolar absorbance > 94% and MFbb emittance < 10% . The thinnest reported design was a 20-layer design with a total thickness equal to 226.5 nm. The particularity of this design problem is the difficulty in choosing the best materials, among the 14 proposed coating materials. Over the very wide spectral range required by the solar-selective absorber coating, the correspondingly broad refractive-index dispersion of each material makes it difficult for the designer to intuitively select the most efficient materials. It should be noted that the low blackbody emissivity requirement can be achieved with high reflectance coatings, for which ZrN is a priori the best candidate, with a reflectance of approximately 95% in the low emissivity region. When CGO is applied to such a design problem, it is perfectly capable of finding the optimal solution. Indeed, for a 6-parameter problem with given material indices and initial thickness limits arbitrarily set to: LB = 0 , and UB = 50nm for each layer, if a solution simultaneously providing MFsolar absorbance > 94% and MFbb emittance < 10% exists, it will certainly be found by CGO. After having found this solution (if it exists), the designer must then minimize the total thickness in order to satisfy all of the design problem requirements. This procedure was iteratively applied to all of the proposed materials, in order to ensure that the 6-layer solution with the smallest physical thickness would be found. All of the 6layer solutions having a total thickness ≤ 320 nm are listed in Table 1.
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Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5174
Table 1. Ten 6-layer Designs, Each Having a Total Thickness ≤ 320 nm, MFsolar absorbance > 94% , and MFbb emittance < 10% Layer number
1
2
3
4
5
6
Design 1
ZrN
Hf
Si
ZrO2
Hf
SiO2
265.0 nm
69.3
44.2
10.9
51.9
4.6
84.2
Design 2 268.0 nm
ZrN 72.6
Hf 40.5
Si 10.6
HfO2 56.3
Hf 5.1
SiO2 82.9
Design 3 281.0 nm
ZrN 88.6
Hf 24.2
ZrO2 51.9
Hf 11.8
SiO 44.3
SiO2 60.2
Design 4 283.0 nm
ZrN 83.0
Hf 43.3
Si 13.3
SiO 58.0
Hf 4.3
SiO2 81.1
Design 5 288.0 nm
ZrN 94.7
Hf 25.1.
ZrO2 47.9
Hf 12.2
ZrO2 38.4
SiO2 69.7
Design 6 291.0 nm
ZrN 98.5
Hf 22.3.
HfO2 50.8
Hf 12.5.
ZrO2 37.1
SiO2 69.9
Design 7 300.0 nm
ZrN 111.7
Si 27.0
Hf 28.8
Si 20.7
SiO 49.7
SiO2 62.1
Design 8 315.0 nm
ZrN
Hf
SiO
Hf
SiO
SiO2
118.9
24.4
55.7
11.4
41.0
63.7
Design 9 315.0 nm
ZrN 128.4
ZrO2 50.8
Hf 19.6
ZrO2 30.0
Si 4.2
SiO2 82.0
Design 10 320.0 nm
ZrN
Hf
SiO
Hf
ZrO2
SiO2
123.6
25.5.
51.8
11.5
35.4
72.2
Four of these ten solutions were submitted, by several designers, to the OIC contest. Taking the specific properties of the various materials into account, ZrN was always used for the first layer (immediately above the substrate), whereas SiO2 was always selected as the last layer, in order to maximize the global solar absorbance. Hf was used in each design, in most cases for the 2nd layer, whereas Si, HfO2, SiO, ZrO2 appeared in layers 3, 4 or 5. In practical terms, all of these ten designs are quite similar, with only 7 out the 14 proposed materials being used to produce the desired antireflective effect in the high solar absorbance region. By allowing the number of layers in the design to increase, CGO also found a 7-layer solution with a total thickness equal to 230 nm, and an 8-layer solution with a total thickness equal to 216 nm (see Table 2). This latter solution has a total thickness smaller than any of those presented for the OIC contest. Although, in all of the solutions ZrN was found to be the best material for the first layer (from those listed in Table 1), this was not a design requirement. The numerical optimizations did not converge to a solution when any another material was selected for the first layer in the 6-layer design.
#204786 - $15.00 USD (C) 2014 OSA
Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5175
Table 2. Seven- and eight-layer designs with a total thickness ≤ 230 nm, MFsolar absorbance > 94% , and MFbb emittance < 10% Layer number
1
2
3
4
5
6
7
Design 11
ZrN
Hf
Si
Hf
ZrO2
Si
SiO2
230nm
49.1
38.7
22.0
8.2
27.8
5.4
78.8
8
Design 12
ZrN
Hf
Si
Hf
ZrO2
Si
SiO2
ZrO2
216nm
53.2
38.0
22.6
9.3
25.0
6.5
49.4
12.0
4. Conclusion
Clustering Global Optimization is more than a refinement technique, since it involves a stochastic procedure which, when provided with an initial design, generates a sliding interval in which it is able to search for new solutions with potentially very different optical thicknesses. The large number of iterations involved in this procedure, and its poor performance in terms of computation time, are largely outweighed by the very good performance of its solutions. The combination of a clustering method with a quasi-Newton descent algorithm delivers design solutions which are close to the global minimum. Once the number of layers and the refractive index of the materials have been set, CGO provides a wide panel of local solutions, characterized by a large variety of optical thicknesses. In the present study, this approach demonstrates its ability to improve on the solutions found for two previously studied design problems. In particular, CGO is used to exhaustively extract all possible 6-layer solutions for a high temperature solar-selective coating problem. When the challenge of developing an optical design is simply a preliminary step in an industrial process, for which the manufacture of a real thin-layer coating is the final goal, CGO can also be helpful. In such situations, all of the solutions given by CGO, which from the end user point of view have an acceptable error function value, are potentially suitable designs. The designer can then use computational simulations to test the each solution’s sensitivity to the expected manufacturing tolerances (errors in layer thickness, or refractive index variations), in order to select a design resulting in a manufactured thin film having the best potential performance and robustness.
#204786 - $15.00 USD (C) 2014 OSA
Received 15 Jan 2014; revised 17 Feb 2014; accepted 17 Feb 2014; published 26 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005166 | OPTICS EXPRESS 5176
