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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 19 — Sep. 14, 2009
  • pp: 16920–16926
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Jump method for optical thin film design

Lei Li, Qiong-Hua Wang, Da-Hai Li, and Hua-Rong Peng  »View Author Affiliations


Optics Express, Vol. 17, Issue 19, pp. 16920-16926 (2009)
http://dx.doi.org/10.1364/OE.17.016920


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Abstract

This paper proposes a method called Jump method for optimization of optical thin films. The method is the combination of local search strategy using modified Coordinate-Wise Algorithm and global search strategy using modified Evolutionary Algorithm. Jump method can evolve the designs of optical thin films for good performances. The design of a dielectric beam splitter and an edge filter as examples is carried out and the results indicate that Jump method is a very robust algorithm for optical thin film designs.

© 2009 OSA

1. Introduction

The optical thin film is very important to modern optics. It not only makes great progress in traditional optical precision measurement technology, but also is widely used in high-tech fields such as laser technology, simulation technology, guidance technology, aerospace technology and so on. The problem of optical thin film design can be formulated as an optimization problem based on the use of merit functions [1

1. J. A. Dobrowolski, F. C. Ho, A. Belkind, and V. A. Koss, “Merit function for more effective thin film calculations,” Appl. Opt. 28(14), 2824–2831 (1989). [CrossRef] [PubMed]

]. A merit function is a complicated Multi-modal function, when the number of the layers for thin films is more than 8, the peak values of the merit function surge. There are several basic approaches to the design of thin films. They can be divided roughly into three categories: analytical, graphical, and numerical methods [2

2. A. V. Tikhonravov and M. K. Trubetskov, “Modern status and prospects of the development of methods of designing multilayer optical coatings,” J. Opt. Technol. 74, 845–850 (2007). [CrossRef]

].

In this paper, we propose a new method for optical thin film designs.

2. Theory of Jump method

2.1 Theory of thin film design

In this paper, we denote the reflectance as R(η, d, λΚ), where λΚ is the discrete wavelength value. The desired spectral reflectance profiles are fitted by minimizing a suitable merit function which composes of an appropriate function of R(η, d, λΚ) defined within the wavelength range. The merit function can be defined in the following equation:
F=K=1W{[R(η,d,λK)R(λK)]2}1/2,
(1)
where W represents the number of wavelength used in calculation, R(λΚ) is the target reflectance.

The most general method of calculating R(η,d,λΚ) is based on a matrix formulation [9

9. Q. H. Wang, D. H. Li, B. J. Peng, Y. H. Tao, and W. X. Zhao, “Multi-layer dielectric color filters for optically written display using up-conversion of near infrared light,” J. Disp. Technol. 4(2), 250–253 (2008). [CrossRef]

]. Therefore, the coating problem can be formulated as a constraint optimization problem.

2.2 Principle of Jump method

Figure 2
Fig. 2 Flow chart of Jump method.
shows the basic steps of Jump method.

In the following discussion, we use di to represent the ith thickness of M layers, and use u and v to denote the step sizes of Modified Coordinate-Wise Procedure and Modified Evolutionary Procedure, respectively.

2.2.1 Modified Coordinate-Wise Procedure

In order to find the local minimum effectively, we decrease the step-size vector u when each layer is self-optimized for a round:
u=Cu
(6)
where u’ is the value of step-size after change, C is a constant whose value is between 0 and 1 so as to converge. In this paper, C is set to 0.97.

2.2.2 Modified Evolutionary Procedure

For easy description of the operators, we use a = (da, va) to represent a parent, and b =(db, vb) as another parent. The offspring, c = (dc, vc), is generated by a genetic operation.

Recombination: the two parents of the recombination operator are randomly selected. The child inherits genes from the two parents with a probability Pc . The operator is as follows:
dic=dia(withtheprobabilityPc),
(7)
dic=dib(withtheprobability1Pc),
(8)
In this paper, Pc is set to 0.4.

Mutation: after recombination a mutation operator is applied. We adapt Schwefel’s proposal [10

10. H.-P. Schwefel, “Numerical Optimization of Computer Models,” (Wiley, New York, 1981).

] to use self-adaptive Gaussian mutation:
vic=viaexp[τN(0,1)+τNi(0,1)],
(9)
dic=dia+vicNi(0,1),
(10)
where N(0,1) is the standard normal distribution, Ni(0,1) is a new value with distribution N(0,1) that must be regenerated for each index i. We set τ and τ’ as follows [6

6. T. Bäck and H. P. Schwefel, “An overview of evolution algorithms for parameter optimization,” Evol. Comput. 1(1), l–23 (1993). [CrossRef]

]:

τ=(2n)1,
(11)
τ=(2n)1.
(12)

Selection: of all the solutions including the parents and children, we evaluate the merit function of each solution. The one with minimum value survives.

For each loop, we compare the solution selected from Modified Evolutionary Procedure with the solution from the Modified Coordinate-Wise Procedure. If the solution selected from Modified Evolutionary Procedure performs better than the solution from the Modified Coordinate-Wise Procedure, the solution will return to the beginning to start a second loop, or else the solution will continue doing Modified Evolutionary Procedure until it turns out to be satisfactory one.

3. Design examples and results discussion

Jump method can be used to design a wide variety of optical thin films from the ultraviolet to the far infrared wavelength. As design examples, we use it to design a broadband multilayer dielectric film beam splitter and an edge filter in the visible range.

The first design example is to design a beam splitter. The dielectric film is composed of alternating TiO2 (ηh =2.40) and SiO2 (ηl = 1.46) layers deposited on a substrate. The transmittance of the dielectric beam splitter can be varied from 0 to 100% by adjusting the individual dielectric layer thickness.

Our target design is the reflectance R(λΚ) specified to 0.5 at 10nm increment between 400nm and 700nm. The incident medium is air (η0 = 1) and the substrate is glass (ηg = 1.5).

The initial number of layers is randomly chosen from 10 to 30. The initial thickness of each layer is uniformly selected from the region from 0 to 200nm.

Table 1

Table 1. Parameters of Jump method used for the simulation of the dielectric beam splitter.

table-icon
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shows the parameters of Jump method used for the simulation of the beam splitter. Table 2

Table 2. Some parameters of the dielectric beam splitter using Jump method.

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gives some solutions of Jump method for the beam splitter and Fig. 3
Fig. 3 The wavelength-dependent transmittance of the dielectric beam splitter for different number of layers using Jump method.
shows the wavelength-dependent transmittance of the beam splitter for different number of layers using Jump method. We can see the results are good.

The second design example is to produce an edge filter in visible range. It is also composed of alternating TiO2 (ηh=2.40) and SiO2 (ηl=1.46). Our target design is to transmit blue and green, but reflect red. So the reflectance R(λΚ) is specified to 0 between 400nm and 550nm, and specified to 1 between 550nm and 700nm at 10nm increment. The incident medium is air (η0 = 1) and the substrate is glass (ηg = 1.5).

Table 3

Table 3. Some parameters of the edge filter using Jump method.

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gives the parameters of the edge filter using Jump method. The transmittance curves of the edge filter for Jump method is shown in Fig. 4
Fig. 4 The transmittance of the edge filter using Jump method.
. It is obvious that the performance of the edge filter using Jump method is very good.

From the examples above, we can make a conclusion that Jump method is very robust in designing thin film. At present, no one technique appears ideal for all design problems, and the proposed method is a useful addition to those available.

4. Conclusion

Acknowledgments

The work was supported by Program for New Century Excellent Talents in University in China under Grant No. NCET-07-0582.

References and links

1.

J. A. Dobrowolski, F. C. Ho, A. Belkind, and V. A. Koss, “Merit function for more effective thin film calculations,” Appl. Opt. 28(14), 2824–2831 (1989). [CrossRef] [PubMed]

2.

A. V. Tikhonravov and M. K. Trubetskov, “Modern status and prospects of the development of methods of designing multilayer optical coatings,” J. Opt. Technol. 74, 845–850 (2007). [CrossRef]

3.

J. A. Dobrowolski and R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29(19), 2876–2893 (1990). [CrossRef] [PubMed]

4.

J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photonics News 8, 24–33 (1997). [CrossRef]

5.

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46(5), 704–710 (2007). [CrossRef] [PubMed]

6.

T. Bäck and H. P. Schwefel, “An overview of evolution algorithms for parameter optimization,” Evol. Comput. 1(1), l–23 (1993). [CrossRef]

7.

J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. 2, 978–985 (2000).

8.

V. Yakovlev and G. Tempea, “Optimization of chirped mirrors,” Appl. Opt. 41(30), 6514–6520 (2002). [CrossRef] [PubMed]

9.

Q. H. Wang, D. H. Li, B. J. Peng, Y. H. Tao, and W. X. Zhao, “Multi-layer dielectric color filters for optically written display using up-conversion of near infrared light,” J. Disp. Technol. 4(2), 250–253 (2008). [CrossRef]

10.

H.-P. Schwefel, “Numerical Optimization of Computer Models,” (Wiley, New York, 1981).

OCIS Codes
(310.0310) Thin films : Thin films
(310.4165) Thin films : Multilayer design

ToC Category:
Thin Films

History
Original Manuscript: July 17, 2009
Revised Manuscript: July 29, 2009
Manuscript Accepted: August 28, 2009
Published: September 8, 2009

Citation
Lei Li, Qiong-Hua Wang, Da-Hai Li, and Hua-Rong Peng, "Jump method for optical thin film design," Opt. Express 17, 16920-16926 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16920


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References

  1. J. A. Dobrowolski, F. C. Ho, A. Belkind, and V. A. Koss, “Merit function for more effective thin film calculations,” Appl. Opt. 28(14), 2824–2831 (1989). [CrossRef] [PubMed]
  2. A. V. Tikhonravov and M. K. Trubetskov, “Modern status and prospects of the development of methods of designing multilayer optical coatings,” J. Opt. Technol. 74, 845–850 (2007). [CrossRef]
  3. J. A. Dobrowolski and R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29(19), 2876–2893 (1990). [CrossRef] [PubMed]
  4. J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photonics News 8, 24–33 (1997). [CrossRef]
  5. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46(5), 704–710 (2007). [CrossRef] [PubMed]
  6. T. Bäck and H. P. Schwefel, “An overview of evolution algorithms for parameter optimization,” Evol. Comput. 1(1), l–23 (1993). [CrossRef]
  7. J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. 2, 978–985 (2000).
  8. V. Yakovlev and G. Tempea, “Optimization of chirped mirrors,” Appl. Opt. 41(30), 6514–6520 (2002). [CrossRef] [PubMed]
  9. Q. H. Wang, D. H. Li, B. J. Peng, Y. H. Tao, and W. X. Zhao, “Multi-layer dielectric color filters for optically written display using up-conversion of near infrared light,” J. Disp. Technol. 4(2), 250–253 (2008). [CrossRef]
  10. H.-P. Schwefel, “Numerical Optimization of Computer Models,” (Wiley, New York, 1981).

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