OSA's Digital Library

Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 7, Iss. 6 — Sep. 11, 2000
  • pp: 237–242
« Show journal navigation

A rigorous unidirectional method for designing finite aperture diffractive optical elements

Jianhua Jiang and Gregory P. Nordin  »View Author Affiliations


Optics Express, Vol. 7, Issue 6, pp. 237-242 (2000)
http://dx.doi.org/10.1364/OE.7.000237


View Full Text Article

Acrobat PDF (323 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

2. Implementation of the µGA-FDTD design method

Due to its flexibility and relative computational efficiency [10

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]

], we selected the FDTD method as the rigorous diffraction computational kernel. The geometry of the grid region in our FDTD calculations is shown schematically in Fig. 1, in which the finite aperture DOE is totally embedded in the FDTD grid. The refractive indices of the incident and exit media are n1 and n2, 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]

] bounded by a perfect electric conductor (PEC). Because of the negligible reflection error caused by the PML ABC, this boundary can be placed only a few cells from the DOE boundaries, which significantly reduces the size of the computation region and hence the computational load [12

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]

]. To excite the entire FDTD grid, a total/scattered field algorithm [7

7. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Boston, Mass.,1995).

] is employed to generate a normally incident TE plane wave. Once steady state has been reached, the electromagnetic plane wave spectrum approach [13

13. G. S. Smith, An Introduction to Classical Electromagnetic Radiation, (Cambridge Univ. Press, Cambridge, Mass., 1997).

] is used to propagate the electromagnetic field from the near field output plane to the observation plane.

Fig. 1. Schematic diagram of the 2-D FDTD geometry showing TE polarization definition

3. Numerical design examples

To illustrate the utility of our µGA-FDTD design method, we examine several DOE design examples. The physical geometry for all cases is shown in Fig. 2. The DOE profiles are assumed to be etched into a Si substrate. A monochromatic plane wave with a free space wavelength of λ0=5 µm is normally incident from a silicon substrate (n1=3.4) onto the DOE profile. The exit medium is air (n2=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].

In our first example we use a simple analytical, quadratic profile, micro-lens to test our µGA-FDTD design method to determine whether it can reproduce the analytical micro-lens profile. The electrical field distribution Eanaly 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

f=i=1LE(xi)2Eanaly(xi)2,
(1)

in which xi refers to the position of the 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.

Fig. 2. Design geometry for the numerical design examples
Fig. 3. Microlens test case for µGA-FDTD Design tool with field distribution of analytical profile as target function (a) µGA convergence curve, (b) field distributions of both analytical and optimized profiles, and (c) the analytical and optimized microlens profiles.
Fig. 4. Two µGA-FDTD optimized DOE profiles for 1-to-2 beamfanner with 25µm peak separation (a) Optimized DOE profiles and (b) their corresponding field distribution.

The second design example is a 1-to-2 beamfanner for an imaging polarimetry system, in which the objective is to design a DOE profile that creates two focused beams separated by 25 µm in the observation plane. Since the desired electric field profile is not easily specified, a Gaussian weighting function is used for µGA-FDTD to control the field distribution at the desired peak positions. We also use a constant penalty function (i.e., negative weighting) to control the side-lobes. The fitness function (which in this case must be maximized by µGA) is defined as

f=i=1LE(xi)·[exp((xi12.5)22σ2)+exp((xi+12.5)22σ2)5(1rect(xi40))]
(2)

in which σ is a peak width parameter and 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.

Fig. 5. Optimized wide-angle 1-to-3 beamfanner with 50um peak separartions (a) Optimized DOE profile and (b) its field distribution

4. Summary and Discussion

We have developed a completely rigorous design method for finite-aperture diffractive optical elements that have minimum feature sizes on the order of or less than a wavelength. The method combines a micro-genetic algorithm (µGA) for global optimization and a 2-D FDTD method for rigorous electromagnetic computation. Based on the presented design examples, µGA-FDTD appears to be an attractive approach for rigorous DOE design. We implemented µGA-FDTD in Fortran 90 running on a PC with a 500 MHz CPU and 256 Mbyte RAM. This is sufficient for the design examples considered herein, in which the width of the DOE’s is small (10λ0). Computation of a single µGA generation takes only 20 seconds. However, for electrically large DOE’s, significantly greater computational resources are needed. Currently, we are working to parallelize our code and port it to an 8-processor system.

Acknowledgements

G. P. Nordin acknowledges support by National Science Foundation CAREER Award ECS-9625040 and grant EPS-9720653.

References and links

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]

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 Polarization: Measurement, Analysis, and Remote Sensing II,Dennis H. Goldstein and David B. Chenault, Editors, Proceedings of SPIE, 3754, 169–177, (1999).

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]

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]

7.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Boston, Mass.,1995).

8.

K. Krishnakumar, “Micro-genetic algorithm for stationary and non-stationary function optimization,” SPIE 1196, 289–296 (1989).

9.

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

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]

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]

13.

G. S. Smith, An Introduction to Classical Electromagnetic Radiation, (Cambridge Univ. Press, Cambridge, Mass., 1997).

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]

15.

D. E. Goldberg, Genetic Algorithm in Search, Optimization, and Machine Learning, (Addision-Wesley, Reading, Mass., 1989).

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


Sort:  Journal  |  Reset  

References

  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]
  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 Polarization: Measurement, Analysis, and Remote Sensing II, Dennis H. Goldstein, David B. Chenault, Editors, Proceedings of SPIE, 3754, 169-177, (1999).
  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]
  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]
  7. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Boston, Mass.,1995).
  8. K. Krishnakumar, "Micro-genetic algorithm for stationary and non-stationary function optimization," SPIE 1196, 289-296 (1989).
  9. J. N. Mait., "Understanding diffractive optic design in the scalar domain," J. Opt. Soc. Am. A, 12, 2145-2158 (1995). [CrossRef]
  10. 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]
  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]
  13. G. S. Smith, An Introduction to Classical Electromagnetic Radiation, (Cambridge Univ. Press, Cambridge, Mass., 1997).
  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]
  15. D. E. Goldberg, Genetic Algorithm in Search, Optimization, and Machine Learning, (Addision-Wesley, Reading, Mass., 1989).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article

OSA is a member of CrossRef.

CrossCheck Deposited