## Jump method for optical thin film design

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

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]

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

## 2. Theory of Jump method

### 2.1 Theory of thin film design

*M*), the thickness (

*d*), refractive indices (

*η*), and extinction coefficients (

*k*) of the medium, substrate and layer. Generally, the task of thin film design is to find the thicknesses (

*d*, …,

_{1}*d*) and indices (

_{M}*η*

_{1}, …,

*η*) of the thin film. However, it has no advantages to use more than two materials that have low

_{M}*η*and high

_{l}*η*refractive indices, respectively. Usually the two materials are given. Therefore, our goal is to search the thicknesses (

_{h}*d*, . . .,

_{1}*d*) in order to obtain the best performance.

_{M}*R(η, d, λ*, where

_{Κ})*λ*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

_{Κ}*R(η, d, λ*defined within the wavelength range. The merit function can be defined in the following equation:where

_{Κ})*W*represents the number of wavelength used in calculation,

*R(λ*is the target reflectance.

_{Κ})*R(η,d,λ*is based on a matrix formulation [9

_{Κ})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]

### 2.2 Principle of Jump method

*n*solutions are randomly generated. Combined with the previous solution from Modified Coordinate-Wise Procedure, they make up an evolution population,

*n*represents population size. Then, the procedure is realized by doing recombination, mutation, and selection. The best one survives. If the solution is better than the previous one from the Modified Coordinate-Wise Procedure, it means the solution escapes from local minimum successfully. Then the output becomes the input of next loop. If not, this procedure will cycle for several times until it turns out to be satisfactory one.

*d*to represent the

_{i}*i*th 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

*u*to change the thickness of each layer as follows: where

*f*represents the value of the merit function before change,

*f’*represents the value of the merit function after change. Similarly,

*T*represents a variable before change,

*T’*represents the variable after change. Initial value of

*T*is 0. And

*d*is the

_{i}’*i*th variable of

*d*after change. We note that if

*T’*is larger than 1, we change

*i*into

*i+*1. Meanwhile, we reset

*T’*as 0.

#### 2.2.2 Modified Evolutionary Procedure

*n*solutions are first generated randomly. The

*n*solutions combined with the previous solution from Modified Coordinate-Wise Procedure make up a population. Then, the procedure is realized by doing recombination, mutation and selection.

*a*= (

*d*) to represent a parent, and

^{a}, v^{a}*b*=(

*d*) as another parent. The offspring,

^{b}, v^{b}*c*= (

*d*), is generated by a genetic operation.

^{c}, v^{c}**:**the two parents of the recombination operator are randomly selected. The child inherits genes from the two parents with a probability

*P*

_{c}_{.}The operator is as follows: In this paper,

*P*is set to 0.4.

_{c}**:**after recombination a mutation operator is applied. We adapt Schwefel’s proposal [10] to use self-adaptive Gaussian mutation: where

*N(0,1)*is the standard normal distribution,

*N*is a new value with distribution

_{i}(0,1)*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]

## 3. Design examples and results discussion

_{2}(

*η*=2.40) and SiO

_{h}_{2}(

*η*= 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.

_{l}*R(λ*specified to 0.5 at 10nm increment between 400nm and 700nm. The incident medium is air (

_{Κ})*η*= 1) and the substrate is glass (

_{0}*η*= 1.5).

_{g}_{2}(

*η*=2.40) and SiO

_{h}_{2}(

*η*=1.46). Our target design is to transmit blue and green, but reflect red. So the reflectance

_{l}*R(λ*is specified to 0 between 400nm and 550nm, and specified to 1 between 550nm and 700nm at 10nm increment. The incident medium is air (

_{Κ})*η*= 1) and the substrate is glass (

_{0}*η*= 1.5).

_{g}## 4. Conclusion

## Acknowledgments

## 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. |

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. |

3. | J. A. Dobrowolski and R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. |

4. | J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photonics News |

5. | A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. |

6. | T. Bäck and H. P. Schwefel, “An overview of evolution algorithms for parameter optimization,” Evol. Comput. |

7. | J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. |

8. | V. Yakovlev and G. Tempea, “Optimization of chirped mirrors,” Appl. Opt. |

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. |

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

- 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. 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]
- 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]
- J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photonics News 8, 24–33 (1997). [CrossRef]
- 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]
- T. Bäck and H. P. Schwefel, “An overview of evolution algorithms for parameter optimization,” Evol. Comput. 1(1), l–23 (1993). [CrossRef]
- J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. 2, 978–985 (2000).
- V. Yakovlev and G. Tempea, “Optimization of chirped mirrors,” Appl. Opt. 41(30), 6514–6520 (2002). [CrossRef] [PubMed]
- 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]
- H.-P. Schwefel, “Numerical Optimization of Computer Models,” (Wiley, New York, 1981).

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