## A rigorous unidirectional method for designing finite aperture diffractive optical elements

Optics Express, Vol. 7, Issue 6, pp. 237-242 (2000)

http://dx.doi.org/10.1364/OE.7.000237

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### Abstract

We have developed a rigorous unidirectional method for designing finite-aperture diffractive optical elements (DOE’s) that employs a micro-genetic algorithm (µGA) for global optimization in conjunction with a 2-D Finite-Difference Time-Domain (FDTD) method for rigorous electromagnetic computation. The theory and implementation of this µGA-FDTD design method for normally incident TE illumination are briefly discussed. Design examples are presented, including a micro-lens, a 1-to-2 beam-fanner and a 1-to-3 beam-fanner.

© Optical Society of America

## 1. Introduction

1. G. Nordin, J. Meier, P. Deguzman, and M. Jones, “Micropolazier array for infrared imaging polarimetry,” J. Opt. Soc. Am. A , **16**, 1168–1174 (1999). [CrossRef]

3. D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodectors,” Opt. Eng. , **38**, 870–878 (1999). [CrossRef]

4. D. A. Pommet, M. G. Moharam, and E. Gram, “Limits of scalar diffarction theory for diffractive phase elements,” J. Opt. Soc. Am. A , **11**, 1827–1834 (1995). [CrossRef]

3. D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodectors,” Opt. Eng. , **38**, 870–878 (1999). [CrossRef]

5. 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**, 1599–1607 (1998). [CrossRef]

6. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A , **14**, 34–43 (1997). [CrossRef]

9. J. N. Mait., “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A , **12**, 2145–2158 (1995). [CrossRef]

## 2. Implementation of the µGA-FDTD design method

10. D. W. Prather, S. Shi, and J. S. Bergey, “Field stitching algorithm for the analysis of electrically large diffractive optical elements,” Opt. Lett. **24**, 273–275 (1999). [CrossRef]

_{1}and n

_{2}, respectively. The computation region is truncated along its outside faces by implementing perfectly matched layer (PML) absorbing boundary conditions (ABCs) [11

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

12. D. W. Prather and S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A , **16**, 1131–1142 (1999). [CrossRef]

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

## 3. Numerical design examples

_{0}=5 µm is normally incident from a silicon substrate (n

_{1}=3.4) onto the DOE profile. The exit medium is air (n

_{2}=1.0). The desired field distribution is observed in a plane 100 µm from the DOE. The aperture of the DOE is 50 µm. Since the minimum feature size is 2 µm, the DOE aperture is divided into 25 feature cells which means that the associated optimization problem has 25 variables, each of which is the etch depth of a particular feature cell. The etch depth at each cell position is coded as a real variable, which spans the range of [0, 1.2 µm].

_{analy}of the analytical micro-lens profile is used as a target function for µGA. We therefore specify the fitness function for the DOE profile as

*i*

^{th}sample point in the observation plane. The design task is to minimize the fitness function. We start from a randomly chosen DOE profile and run 500 µGA generations. The convergence curve (i.e.,

*f*as a function of the µGA generation) is shown in Fig. 3(a). It rapidly converges to a final value of the fitness function of three, which is very close to the global minimum (in this case zero). The µGA-optimized field distribution along with the field distribution for the analytical micro-lens profile is shown in Fig. 3(b), while the corresponding DOE profiles are shown in Fig. 3(c). The field distribution for the optimized DOE profile is nearly identical to the target field distribution. However, the optimized DOE profile, while close to the analytical profile, nevertheless exhibits some small differences. This illustrates a general result that we have found in our work with µGA-FDTD in which nearly the same diffraction pattern can be generated by a number of differing DOE profiles. This is also an example of the strength of genetic algorithms for global searches, while at the same time they can be weak at local searches in the neighborhood of the global solution.

*rect*() is a rectangular window function. Our design method found several profiles that can achieve the desired beamfanning function. Fig. 4 shows two resultant DOE profiles and the corresponding field distributions in the observation plane. Note that the field distributions are very similar, while the DOE profiles are quite different.

## 4. Summary and Discussion

## Acknowledgements

## References and links

1. | G. Nordin, J. Meier, P. Deguzman, and M. Jones, “Micropolazier array for infrared imaging polarimetry,” J. Opt. Soc. Am. A , |

2. | G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, “Diffractive Optical Element for Stokes Vector Measurement With a Focal Plane Array,” in |

3. | D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodectors,” Opt. Eng. , |

4. | D. A. Pommet, M. G. Moharam, and E. Gram, “Limits of scalar diffarction theory for diffractive phase elements,” J. Opt. Soc. Am. A , |

5. | 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 , |

6. | D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A , |

7. | A. Taflove, |

8. | K. Krishnakumar, “Micro-genetic algorithm for stationary and non-stationary function optimization,” SPIE |

9. | J. N. Mait., “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A , |

10. | D. W. Prather, S. Shi, and J. S. Bergey, “Field stitching algorithm for the analysis of electrically large diffractive optical elements,” Opt. Lett. |

11. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

12. | D. W. Prather and S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A , |

13. | G. S. Smith, |

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

15. | D. E. Goldberg, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.1220) Diffraction and gratings : Apertures

(050.1970) Diffraction and gratings : Diffractive optics

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 20, 2000

Published: September 11, 2000

**Citation**

Jianhua Jiang and Gregory Nordin, "A rigorous unidirectional method for designing finite aperture diffractive optical elements," Opt. Express **7**, 237-242 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-6-237

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### References

- G.Nordin, J. Meier, P. Deguzman and M. Jones, "Micropolazier array for infrared imaging polarimetry," J. Opt. Soc. Am. A, 16, 1168-1174 (1999). [CrossRef]
- G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, "Diffractive Optical Element for Stokes Vector Measurement With a Focal Plane Array," in Polarization: Measurement, Analysis, and Remote Sensing II, Dennis H. Goldstein, David B. Chenault, Editors, Proceedings of SPIE, 3754, 169-177, (1999).
- D. W. Prather, "Design and application of subwavelength diffractive lenses for integration with infrared photodectors," Opt. Eng., 38, 870-878 (1999). [CrossRef]
- D. A. Pommet,M. G. Moharam, and E. Gram, "Limits of scalar diffarction theory for diffractive phase elements," J. Opt. Soc. Am. A, 11, 1827-1834 (1995). [CrossRef]
- 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, 1599-1607 (1998). [CrossRef]
- D. W. Prather, M. S. Mirotznik and J. N. Mait, "Boundary integral methods applied to the analysis of diffractive optical elements," J. Opt. Soc. Am. A, 14, 34-43 (1997). [CrossRef]
- A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Boston, Mass.,1995).
- K. Krishnakumar, "Micro-genetic algorithm for stationary and non-stationary function optimization," SPIE 1196, 289-296 (1989).
- J. N. Mait., "Understanding diffractive optic design in the scalar domain," J. Opt. Soc. Am. A, 12, 2145-2158 (1995). [CrossRef]
- D. W. Prather, S. Shi, J. S. Bergey, "Field stitching algorithm for the analysis of electrically large diffractive optical elements," Opt. Lett. 24, 273-275 (1999). [CrossRef]
- J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
- D. W. Prather and S. Shi, "Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements," J. Opt. Soc. Am. A, 16, 1131-1142 (1999). [CrossRef]
- G. S. Smith, An Introduction to Classical Electromagnetic Radiation, (Cambridge Univ. Press, Cambridge, Mass., 1997).
- E. G. Johnson and M. A. G. Abushagur, Microgenetic-algorithm optimization methods applied to dielectric gratings," J. Opt. Soc. Am. A, 12, 1152-1160 (1995). [CrossRef]
- D. E. Goldberg, Genetic Algorithm in Search, Optimization, and Machine Learning, (Addision-Wesley, Reading, Mass., 1989).

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