## Design of binary diffractive microlenses with subwavelength structures using the genetic algorithm

Optics Express, Vol. 18, Issue 8, pp. 8383-8391 (2010)

http://dx.doi.org/10.1364/OE.18.008383

Acrobat PDF (10700 KB)

### Abstract

We present a method to design binary diffractive microlenses with subwavelength structures, based on the finite-difference time-domain method and the genetic algorithm, also accounting for limitations on feature size and aspect ratio imposed by fabrication. The focusing efficiency of the microlens designed by this method is close to that of the convex lens and much higher than that of the binary Fresnel lens designed by a previous method. Although the optimized structure appears to be a binary Fresnel lens qualitatively, it is hard to quantitatively derive directly from the convex Fresnel lens. The design of a microlens with reduced chromatic aberration is also presented.

© 2010 OSA

## 1. Introduction

3. H. Kikuta, H. Toyota, and W. Yu, “Optical Elements with Subwavelength Structured Surfaces,” Opt. Rev. **10**(2), 63–73 (2003). [CrossRef]

4. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**(12), 971–973 (2003). [CrossRef] [PubMed]

7. D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-Based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A **15**(6), 1599–1607 (1998). [CrossRef]

8. D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A **11**(6), 1827–1834 (1994). [CrossRef]

10. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**(5), 1068–1076 (1995). [CrossRef]

11. S. Kirkpatrick, C. D. Gelatt Jr, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science **220**(4598), 671–680 (1983). [CrossRef] [PubMed]

13. E. G. Johnson and M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A **12**(5), 1152–1160 (1995). [CrossRef]

15. H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS **127**(9), 1298–1303 (2007) (Japanese). [CrossRef]

## 2. Computational approach

### 2.1 Bodies of revolution (BOR) FDTD method

17. J. P. Berenger, “A perfectly matched layer for the absorption of electro-magnetic waves,” J. Comput. Phys. **114**(2), 185–200 (1994). [CrossRef]

*z*-axis, and the propagation direction of the incident plane wave is parallel to the axis (Fig. 1 ).

### 2.2 Genetic Algorithm (GA)

*Coding #1*(Fig. 2 ): The first 200 bits express the binary relief pattern in the

*r*direction. One bit corresponds to a width Δ

*d*of 20 nm and the lens radius is 4 μm. Then, the grating height is coded by 4 bits from 200 to 500 nm with a step of 20 nm. An advantage of this coding is that it is simple and general, requires no prior knowledge, and, thus, can also be applied to different kinds of DOEs.

*Coding #2*(Fig. 3 ): After some trials, we notice that phenotypes of better performance are composed of inner and outer parts separated by a space in general, as we will see later in Sec. 3. Thus, let us reflect this in the coding by assigning additional 7 and 5 bits to the width of the inner part and the space, respectively. Then those of the first 200 bits that correspond to the space are disregarded when mapped to the phenotype. While this coding is inspired a posteriori by the feature observed during the optimization using Coding #1, it should be noted that any arbitrary phenotype expressed by Coding #1 can also be expressed by Coding #2, since the space width can be zero. Thus, the solution space is not reduced by the introduction of Coding #2 and as general as for #1.

*F*of each genotype using the focusing performance as,where

*S*denotes the

_{z}*z*-component of the Poynting vector (intensity),

*f*and

_{0}*f*the targeted and actual focal distance, respectively, and

*f*and

_{d}*f*the length and radius, respectively, of the evaluation domain (shown in Fig. 1). The stronger the focusing around the targeted focal point, the higher the fitness value takes.

_{r}### 2.3 GA-FDTD method

- (1) Randomly prepare sixteen GTYPEs as the initial generation.
- (2) Evaluate the fitness of each GTYPE belonging to the generation, by FDTD calculation for the corresponding PTYPE.
- (3) Select eight pairs of GTYPEs according to their fitness by the roulette-wheel selection
- (4) Apply the uniform crossover (crossover rate: 0.95) and mutation (mutation rate: 0.01) to create new GTYPEs, which form the next generation
- (5) Then start again from (2)

*M*, crossover rate

*P*, and mutation rate

_{c}*P*) by comparing the fitness values after 1000 generations with Coding #1. We first varied

_{m}*P*between 0.005 and 0.2, while fixing

_{m}*M*= 16, and

*P*= 0.3 and 0.9. For both values of

_{c}*P*the fitness value was highest at

_{c}*P*= 0.01. Next, by fixing

_{m}*M*= 16 and

*P*= 0.005 and 0.01, we varied

_{m}*P*between 0.1 and 1, and found that the fitness value was generally high for

_{c}*P*= 0.9 - 1. Finally, we varied

_{c}*M*between 8 and 40. The obtained fitness value was low at

*M*= 8 but approximately equally high between

*M*= 16 and 40. Based on these observations, we have chosen, as GA parameter values,

*M*= 16,

*P*= 0.95, and

_{c}*P*= 0.01 to obtain the results presented in what follows. We have also confirmed that all the tested parameter combinations yield similar structures in the end.

_{m}## 3. Results

### 3.1 Optimally-designed microlens and its focusing performance

*z*= 9.3 μm) for 660 nm wavelength, respectively. The refractive index of the material (Ta

_{2}O

_{5}) is assumed to be 2.1466 at this wavelength. The grid size in BOR-FDTD calculations is 20 nm, unless otherwise stated. Figure 4 displays the obtained structure and the intensity distribution of the beam incident from the bottom of the structure as a plane wave. We define the focal distance from the bottom (

*z*= 1 μm) of the gratings. The actual focal distance of the obtained structure is 8.3 μm as targeted. The red curve of Fig. 6(a) shows the intensity distribution in the focal face. The spot radius given by the half width at half maximum (HWHM) is 0.40 μm, and the spot shape is almost circular [Fig. 6(b)]. The focusing efficiency of the obtained structure, defined as the ratio of the beam power at

*r*< 1 μm to the incident power, is 59.3%. Figure 6(a) also compares the results obtained with two different values of grid size (the red curve for 5 nm and the black dashed curve for 20 nm) for FDTD calculations. The difference between the two curves turns out to be less than 0.1%. This result indicates that the FDTD calculation with a grid size of 20 nm is sufficiently accurate.

### 3.2 Comparison with the convex lens, the convex Fresnel lens, and the binary Fresnel lens

*z*= 9.3 μm) for the four lenses. The focusing efficiency of each structure is 59.3%, 79.1%, 70.4%, and 28.0%, respectively. It is remarkable that the performance of the binary microlens designed in this study is more than twice as high as that of the binarization of the Fresnel lens reported previously, and is close to that of the lenses with a curved surface. Moreover, the relation between the widths of neighboring gratings cannot be expressed in a simple way.

### 3.3 Effect of fabrication error (trapezoidal gratings)

### 3.4 Comparison of the coding methods

### 3.5 Design of a binary lens with reduced chromatic aberration at three wavelengths

## 4. Conclusion

## Acknowledgment

## References and links

1. | V. A. Soifer, |

2. | D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, |

3. | H. Kikuta, H. Toyota, and W. Yu, “Optical Elements with Subwavelength Structured Surfaces,” Opt. Rev. |

4. | G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. |

5. | Y. Okuno, M. Fujimoto, and T. Matsuda, “Numerical evaluation of diffractive optical elements with binary subwavelength structures,” IEIC Tech. Rep. |

6. | J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Binary subwavelength diffractive-lens design,” Opt. Lett. |

7. | D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-Based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A |

8. | D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A |

9. | A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans |

10. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

11. | S. Kirkpatrick, C. D. Gelatt Jr, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science |

12. | M. Mitchell, |

13. | E. G. Johnson and M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A |

14. | C. F. Huang and H. M. Li, ““Design optimization of chip antennas using the GA-FDTD approach”, Int. J. RF Microw,” Computer-Aided Engineering |

15. | H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS |

16. | A. Taflove, and S. C. Hagness, |

17. | J. P. Berenger, “A perfectly matched layer for the absorption of electro-magnetic waves,” J. Comput. Phys. |

18. | K. Ono and K. Eriguchi, “Modeling of plasma-surface interactions and profile evolution during dry etching,” J. Plasma Fusion Res. |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(050.1965) Diffraction and gratings : Diffractive lenses

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: February 9, 2010

Revised Manuscript: April 4, 2010

Manuscript Accepted: April 5, 2010

Published: April 6, 2010

**Citation**

Tatsuya Shirakawa, Kenichi L. Ishikawa, Shuichi Suzuki, Yasufumi Yamada, and Hiroyuki Takahashi, "Design of binary diffractive microlenses with subwavelength structures using the genetic algorithm," Opt. Express **18**, 8383-8391 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8383

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### References

- V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley Series in Lasers and Applications) (John Wiley & Sons, 2002).
- D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).
- H. Kikuta, H. Toyota, and W. Yu, “Optical Elements with Subwavelength Structured Surfaces,” Opt. Rev. 10(2), 63–73 (2003). [CrossRef]
- G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28(12), 971–973 (2003). [CrossRef] [PubMed]
- Y. Okuno, M. Fujimoto, and T. Matsuda, “Numerical evaluation of diffractive optical elements with binary subwavelength structures,” IEIC Tech. Rep. 100, 157–162 (2001) (Japanese).
- J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Binary subwavelength diffractive-lens design,” Opt. Lett. 23(17), 1343–1345 (1998). [CrossRef]
- D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-Based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15(6), 1599–1607 (1998). [CrossRef]
- D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11(6), 1827–1834 (1994). [CrossRef]
- A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans 22, 191–202 (1980).
- M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]
- S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983). [CrossRef] [PubMed]
- M. Mitchell, An Introduction to Genetic Algorithms (MIT Press, 1998).
- E. G. Johnson and M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12(5), 1152–1160 (1995). [CrossRef]
- C. F. Huang and H. M. Li, ““Design optimization of chip antennas using the GA-FDTD approach”, Int. J. RF Microw,” Computer-Aided Engineering 15, 116–127 (2005).
- H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS 127(9), 1298–1303 (2007) (Japanese). [CrossRef]
- A. Taflove, and S. C. Hagness, Computional Electrodynamics: The Finite-Difference Time-Domain Method, Chap. 12 (Artech House, 2005).
- J. P. Berenger, “A perfectly matched layer for the absorption of electro-magnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
- K. Ono and K. Eriguchi, “Modeling of plasma-surface interactions and profile evolution during dry etching,” J. Plasma Fusion Res. 85, 165–176 (2009) (Japanese).

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