## Fast numerical methods for the design of layered photonic structures with rough interfaces |

Optics Express, Vol. 19, Issue 6, pp. 5489-5499 (2011)

http://dx.doi.org/10.1364/OE.19.005489

Acrobat PDF (1714 KB)

### Abstract

A multilayer approach (MA) and modified boundary conditions (MBC) are proposed as fast and efficient numerical methods for the design of 1D photonic structures with rough interfaces. These methods are applicable for the structures, composed of materials with an arbitrary permittivity tensor. MA and MBC are numerically validated on different types of interface roughness and permittivities of the constituent materials. The proposed methods can be combined with the 4x4 scattering matrix method as a field solver and an evolutionary strategy as an optimizer. The resulted optimization procedure is fast, accurate, numerically stable and can be used to design structures for various applications.

© 2011 Optical Society of America

## 1. Introduction

4. D. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. **62**, 502–510 (1972). [CrossRef]

6. I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A: Pure Appl. Opt. **1**, 646 (1999). [CrossRef]

7. F. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, and T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. **22**, 831–833 (1997). [CrossRef] [PubMed]

10. S. Martin, J. Rivory, and M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. **34**, 2247–2254 (1995). [CrossRef] [PubMed]

8. T. Yonte, J. Monzón, A. Felipe, and L. Sánchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A: Pure Appl. Opt. **6**, 127–131 (2004). [CrossRef]

9. M. del Rio and G. Pareschi, “Global optimization and reflectivity data fitting for x-ray multilayer mirrors by means of genetic algorithms,” in “Proceedings of SPIE ,”, vol. **4145** (2001), vol. 4145, p. 88. [CrossRef]

10. S. Martin, J. Rivory, and M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. **34**, 2247–2254 (1995). [CrossRef] [PubMed]

12. D. Gerace and L. Andreani, “Low-loss guided modes in photonic crystal waveguides,” Opt. Express **13**, 4939–4951 (2005). [CrossRef] [PubMed]

13. P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. A **71**, 1115–1123 (1981). [CrossRef]

14. I. Ohlídal, “Approximate formulas for the reflectance, transmittance, and scattering losses of nonabsorbing multilayer systems with randomly rough boundaries,” J. Opt. Soc. Am. A **10**, 158–171 (1993). [CrossRef]

15. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. **13**, 1024–1035 (1996). [CrossRef]

17. S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B **66**, 45102 (2002). [CrossRef]

## 2. Optimal design of a dual band omnidirectional mirror

18. A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. **56** (2008). [CrossRef]

19. C. Hafner, C. Xudong, J. Smajic, and R. Vahldieck, “Efficient procedures for the optimization of defects in photonic crystal structures,” J. Opt. Soc. Am. A **24**, 1177–1188 (2007). [CrossRef]

*N*layers, one has (at least)

*N*real-valued parameters (thickness of each layer) that need to be optimized. The search space should be defined for each parameter within a reasonable interval [

*d*

_{min},

*d*

_{max}]. For such optimizations, various algorithms, such as genetic algorithms (GA), micro-genetic algorithms (MGA), or evolutionary strategies (ES) may be applied. The optimization task becomes difficult not only when

*N*is large, but also when the fitness function has a complicated behavior, e.g. when it has many local optima. According to the experience [20], ES is very powerful in real parameter optimization problems and outperforms genetic algorithm (GA), particle swarm optimization (PSO) etc. in most cases. Therefore, we applied it to the multilayer problem without extensive testing. We used an (

*m*+

*n*) ES with adaptive mutation for the optimization in the following example. Here

*m*is the initial number of parents,

*n*is the number of children created in each generation. In the following example we used

*m*= 5 and

*n*= 40.

*R*

_{ss},

*R*

_{sp},

*R*

_{ps}, and

*R*

_{pp}. The first and second indices correspond to the polarization of the incident and reflected light, respectively. The s- and p-polarized light corresponds to an electric or magnetic field parallel to the layers, respectively. The multilayered structure that is to be optimized consists of

*N*=14 layers, embedded between air interfaces. It is composed of two alternating materials with permittivity tensors

*ɛ*̂

_{1}and

*ɛ*̂

_{2}:

*δ*<<

*λ*).

*unpolarized*light in two bands

*B*

_{1}: = 800 – 1300 meV and

*B*

_{2}: = 2000 – 2500 meV. Therefore, we define the fitness function as the averaged reflection: where the reflection coefficients are integrated over the angles of incidence

*θ*∈ [0,

*B*

_{1}and

*B*

_{2}. The thickness of the i-th layer (i = 1,...,

*N*) is

*d*. We now need to maximize the fitness function Eq. (1) of N variables.

_{i}*B*

_{1}and

*B*

_{2}bands. Figure 1(b) shows the geometric profile of the obtained optimal structure.

## 3. Multilayer approach for rough interfaces

### Model

#### Geometry

*z*< −

*δ*,

*ɛ*=

*ɛ*

_{1},

*μ*= 1, and

*z*> +

*δ*,

*ɛ*=

*ɛ*

_{2},

*μ*= 1. The region |

*z*| <

*δ*is occupied with the rough interface, see Fig. 2. Both sizes of the roughness 2

*δ*and

*l*are supposed to be small compared to the wavelength. The volume fractions of two components are

*f*

_{1}=

*V*

_{1}/

*V*and

*f*

_{2}=

*V*

_{2}/

*V*, where

*V*is the total volume of the rough region.

*δ*<

*z*<

*δ*into M layers and assign an effective permittivity

21. O. Wiener, “Die Théorie des Mischkörpers für das Feld des stationären Strömung. Abh. Math,” Physichen Klasse Königl. Säcsh. Gesel. Wissen **32**, 509–604 (1912). [PubMed]

## 4. Modified boundary conditions for rough interfaces

### Model

#### Geometry

*z*< −

*δ*,

*ɛ*=

*ɛ*

_{1},

*μ*= 1, and

*z*> +

*δ*,

*ɛ*=

*ɛ*

_{2},

*μ*= 1. The region |

*z*| <

*δ*is occupied with the rough interface, see Fig. 2. We suppose that the dielectric permittivity is a function of

*z*in this region. Thus, The wave is incident from

*z*= −∞ and propagates in the

*XZ*plane, thus (

*∂/∂y*= 0). The angle of light incidence is

*θ*with respect to the Z axis.

*l*× 2

*δ*contour at the rough interface; then the Maxwell’s equations for the circulation around this contour are: Hereafter, we use the CGS units in the derivations. Note, however, that the results, Eq. (7), Eq. (8), Eq. (10), and Eq. (12), are valid in the both, CGS and SI, unit forms.

**B**

*is continuous at the interface and can be considered as constant there. Thus we obtain: The right part of the first equation (5) can be written as: Assuming*

_{n}*δ*≪

*λ*and

*l*≪

*λ*, we obtain

*z*= 0. The first and second equations originate from the first equation (5) being written for the different components of the magnetic field. The third and forth equations are derived from the second equation of (5).

*ɛ*

_{11}=

*ɛ*

_{22}≠

*ɛ*

_{33}. Then the boundary conditions Eq. (7) and (8) hold with

*δ*/

*λ*needs an additional expansion of the fields in the integrands of Eq. (5). This can be done also by integration of the wave equations over the interface region |

*z*| <

*δ*, see Ref. [22

22. M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. **7**, 1623–1630 (2010). [CrossRef]

*δ*-region at the boundary (see Ref. [22

22. M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. **7**, 1623–1630 (2010). [CrossRef]

*E*

_{1,2}and their derivatives

*E*′

_{1,2}=

*∂E*

_{1,2}/

*∂z*at each side of the boundary (we omit the subscript

*y*for reasons of simplicity): Here

*α*and

*β*are the parameters, which are independent of the wavelength and angle of incidence.

*δ*region is If

*ɛ*

_{33}is independent of

*x*, then Eq. (11) can be written as: Its integration leads to the following BC for

*H*field: Expansion similar to Eq. (6), but to the second order on

_{y}*δ*/

*λ*, leads to

*α*=

*t*

_{1},

*β*= (

*ɛ*

_{2}−

*ɛ*

_{1})

*δ*

^{2}/2 and

*γ*=

*t*

_{2}. Here the parameters

*α*and

*β*are the same as in Eq. (10).

## 5. Numerical results of multilayer approach and modified boundary conditions

### 5.1. Sinusoidal roughness profile

*ɛ*= 5. The structure is embedded in air. The dielectric permittivity in the region −

*h*/2 <

*z*<

*h*/2 is: The period and height of the grating are much smaller than the considered wavelength. As a consequence, the dependence of the reflectivity on the azimuthal angle

*φ*is almost negligible, despite the absence of the symmetry with respect to

*φ*.

*L*+

*h*/2 = 105 nm. It can be seen that the reflection from a flat interface deviates significantly from the “reference solution”. This is especially pronounced for higher energies. The blue curve corresponds to the multilayer approach, with

*M*= 6 layers in the sinusoidal region −

*h*/2 <

*z*<

*h*/2. The green curve corresponds to the MBC solution (Eq. (10) and Eq. (12)), with optimal parameters

*α*[cm],

*β*[cm

^{2}] and

*γ*[cm].

*θ*. MBC is also in good agreement. The parameters

*α*,

*β*and

*γ*of the MBC are independent of the frequency and incidence angle. These parameters can be found either by fitting to the reference solution, as done in this paper, or by fitting to experimental reflection/transmission spectra of multilayers.

*ɛ*

_{11}=

*ɛ*

_{22}= 7,

*ɛ*

_{33}= 9. It should be noted that the application of MBC is very much simplified in the case of isotropic, or uniaxial crystals (

*ɛ*

_{11}=

*ɛ*

_{22}≠

*ɛ*

_{33}), whereas MA is equally easy to apply for arbitrary permittivity tensors.

### 5.2. Random hilly roughness

*h*

_{max}= 10 nm. The roughness profile is periodic, with periods

*d*=

_{x}*d*= 50 nm. The characteristic size of the roughness is much smaller than the wavelength, therefore the dependence of the reflectivity on the azimuthal angle

_{y}*φ*is practically negligible.

*ɛ*= 5 are shown in Fig. 5(a) and (b). The results for an uniaxial crystal with

*ɛ*

_{11}=

*ɛ*

_{22}= 7,

*ɛ*

_{33}= 9 are shown in Fig. 5(c) and (d). Evidently, a good agreement of MA and MBC with the reference solution is observed.

### 5.3. Multilayers with rough interfaces

*ɛ*

_{1}= 2 and

*ɛ*

_{2}= 7, whereby the high index material is on top. Interfaces 1–3 are rough, with random hilly roughness profiles, as depicted in Fig. 3(b). The roughness of each individual interface was generated randomly. The maximum height of the hills does not exceed

*h*

_{max}= 10 nm. The structure is periodic with periods

*d*=

_{x}*d*= 50 nm.

_{y}*M*= 6 layers in the interface regions. The green curve on the plot corresponds to MBC at each rough interface. It can be seen that MA and MBC in general demonstrate a better agreement with “reference solution”. This is especially pronounced in two regions, which are marked with arrows in Fig. 6.

## 6. Conclusions

## References and links

1. | J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, |

2. | A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. |

3. | V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. |

4. | D. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. |

5. | P. Yeh, |

6. | I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A: Pure Appl. Opt. |

7. | F. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, and T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. |

8. | T. Yonte, J. Monzón, A. Felipe, and L. Sánchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A: Pure Appl. Opt. |

9. | M. del Rio and G. Pareschi, “Global optimization and reflectivity data fitting for x-ray multilayer mirrors by means of genetic algorithms,” in “Proceedings of SPIE ,”, vol. |

10. | S. Martin, J. Rivory, and M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. |

11. | D. Bose, E. McCorkle, C. Thompson, D. Bogdanoff, D. Prabhu, G. Allen, and J. Grinstead, “Analysis and model validation of shock layer radiation in air,” VKI LS Course on hypersonic entry and cruise vehicles, Palo Alto, California, USA (2008). |

12. | D. Gerace and L. Andreani, “Low-loss guided modes in photonic crystal waveguides,” Opt. Express |

13. | P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. A |

14. | I. Ohlídal, “Approximate formulas for the reflectance, transmittance, and scattering losses of nonabsorbing multilayer systems with randomly rough boundaries,” J. Opt. Soc. Am. A |

15. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. |

16. | D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B |

17. | S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B |

18. | A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. |

19. | C. Hafner, C. Xudong, J. Smajic, and R. Vahldieck, “Efficient procedures for the optimization of defects in photonic crystal structures,” J. Opt. Soc. Am. A |

20. | J. Fröhlich, “Evolutionary optimization for computational electromagnetics,” Ph.D. thesis, ETH Zurich, IFH Laboratory (1997). |

21. | O. Wiener, “Die Théorie des Mischkörpers für das Feld des stationären Strömung. Abh. Math,” Physichen Klasse Königl. Säcsh. Gesel. Wissen |

22. | M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(310.4165) Thin films : Multilayer design

(160.5298) Materials : Photonic crystals

(310.6845) Thin films : Thin film devices and applications

**ToC Category:**

Optical Devices

**History**

Original Manuscript: January 28, 2011

Revised Manuscript: March 1, 2011

Manuscript Accepted: March 1, 2011

Published: March 8, 2011

**Citation**

Nikolay Komarevskiy, Leonid Braginsky, Valery Shklover, Christian Hafner, and John Lawson, "Fast numerical methods for the design of layered photonic structures with rough interfaces," Opt. Express **19**, 5489-5499 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5489

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### References

- J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic crystals: molding the flow of light (Princeton Univ Pr, 2008).
- A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996). [CrossRef] [PubMed]
- V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. 5, 862 (2008).
- D. Berreman, “Optics in stratified and anisotropic media: 4× 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972). [CrossRef]
- P. Yeh, Optical waves in layered media (Wiley New York, 1988).
- I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646 (1999). [CrossRef]
- F. K¨artner, N. Matuschek, T. Schibli, U. Keller, H. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, and T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997). [CrossRef] [PubMed]
- T. Yonte, J. Monz’on, A. Felipe, and L. S’anchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A, Pure Appl. Opt. 6, 127–131 (2004). [CrossRef]
- M. del Rio and G. Pareschi, “Global optimization and reflectivity data fitting for x-ray multilayer mirrors by means of genetic algorithms,” in “Proceedings of SPIE,”, vol. 4145 (2001), vol. 4145, p. 88. [CrossRef]
- S. Martin, J. Rivory, and M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34, 2247–2254 (1995). [CrossRef] [PubMed]
- D. Bose, E. McCorkle, C. Thompson, D. Bogdanoff, D. Prabhu, G. Allen, and J. Grinstead, “Analysis and model validation of shock layer radiation in air,” VKI LS Course on hypersonic entry and cruise vehicles, Palo Alto, California, USA (2008).
- D. Gerace and L. Andreani, “Low-loss guided modes in photonic crystal waveguides,” Opt. Express 13, 4939–4951 (2005). [CrossRef] [PubMed]
- P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981). [CrossRef]
- I. Ohlídal, “Approximate formulas for the reflectance, transmittance, and scattering losses of nonabsorbing multilayer systems with randomly rough boundaries,” J. Opt. Soc. Am. A 10, 158–171 (1993). [CrossRef]
- L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. 13, 1024–1035 (1996). [CrossRef]
- D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999). [CrossRef]
- S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002). [CrossRef]
- A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. 56 (2008). [CrossRef]
- C. Hafner, C. Xudong, J. Smajic, and R. Vahldieck, “Efficient procedures for the optimization of defects in photonic crystal structures,” J. Opt. Soc. Am. A 24, 1177–1188 (2007). [CrossRef]
- J. Fröhlich, “Evolutionary optimization for computational electromagnetics,” Ph.D. thesis, ETH Zurich, IFH Laboratory (1997).
- O. Wiener, “Die Théorie desMischkörpers für das Feld des stationären Strömung. Abh.Math,” Physichen Klasse Königl. Säcsh. GeselWissen 32, 509–604 (1912). [PubMed]
- M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. 7, 1623–1630 (2010). [CrossRef]

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