## Comparison of different simulation methods for effective medium computer generated holograms |

Optics Express, Vol. 21, Issue 10, pp. 12424-12433 (2013)

http://dx.doi.org/10.1364/OE.21.012424

Acrobat PDF (2178 KB)

### Abstract

The arrangement of binary subwavelength structures is a promising alternative to the conventional multiheight level technique to generate computer generated holograms (CGHs). However, the current heuristic design approach leads to a slight mismatch between the target signal and experimental data. To evaluate this deviation, a diffractive beam splitter design is investigated rigorously using a finite-difference time-domain (FDTD) method. Since the use of a rigorous Maxwell-equation solver like FDTD requires a massive computational effort, an alternative scalar approach, a fast Fourier transform beam propagation method (FFT-BPM), is investigated with a substantial higher computing speed, showing still a good agreement with the FDTD simulation and experimental data. Therefore, an implementation of this scalar approach into the CGH design process offers the possibility to significantly increase the accuracy.

© 2013 OSA

## 1. Introduction

1. B. Goebel, L. L. Wang, and T. Tschudi, “Multilayer technology for diffractive optical elements,” Appl. Opt. **35**(22), 4490–4493 (1996). [CrossRef] [PubMed]

4. E. Noponen and J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A **11**(3), 1097–1109 (1994). [CrossRef]

9. W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt. **39**(20), 3531–3536 (2000). [CrossRef] [PubMed]

10. W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design of binary subwavelength multi-phase level computer generated holograms,” Opt. Lett. **35**(5), 676–678 (2010). [CrossRef] [PubMed]

11. W. Freese, T. Kämpfe, W. Rockstroh, E.-B. Kley, and A. Tünnermann, “Optimized electron beam writing strategy for fabricating computer-generated holograms based on an effective medium approach,” Opt. Express **19**(9), 8684–8692 (2011). [CrossRef] [PubMed]

13. W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design and fabrication of a highly off-axis binary multi-phase level computer-generated hologram based on an effective medium approach,” Proc. SPIE **7927**(792710), 792710, 792710-7 (2011). [CrossRef]

10. W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design of binary subwavelength multi-phase level computer generated holograms,” Opt. Lett. **35**(5), 676–678 (2010). [CrossRef] [PubMed]

11. W. Freese, T. Kämpfe, W. Rockstroh, E.-B. Kley, and A. Tünnermann, “Optimized electron beam writing strategy for fabricating computer-generated holograms based on an effective medium approach,” Opt. Express **19**(9), 8684–8692 (2011). [CrossRef] [PubMed]

## 2. Heuristic design approach of an effective medium computer generated hologram

16. U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. **229**(1-6), 11–21 (2004). [CrossRef]

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

19. S. Mellin and G. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express **8**(13), 705–722 (2001). [CrossRef] [PubMed]

10. W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design of binary subwavelength multi-phase level computer generated holograms,” Opt. Lett. **35**(5), 676–678 (2010). [CrossRef] [PubMed]

## 3. Rigorous simulation of an effective medium computer generated hologram

## 4. Fast Fourier transform beam propagation method

21. M. D. Feit and J. A. Fleck Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. **17**(24), 3990–3998 (1978). [CrossRef] [PubMed]

24. B. Hermansson, D. Yevick, and J. Saijonmaa, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” J. Opt. Soc. Am. A **1**(6), 663–671 (1984). [CrossRef]

25. M. Fertig and K.-H. Brenner, “Vector wave propagation method,” J. Opt. Soc. Am. A **27**(4), 709–717 (2010). [CrossRef] [PubMed]

## 5. Fast Fourier transform beam propagation method versus finite-difference time-domain method

^{4}floating point operations in this particular example, it also scales better with the number of sampling points concerning memory and computation speed. This enables the use of this method for more complicates problems where an FDTD simulation is not possible because of memory issues. Further an optimization of the designed phase function might be possible with this approach in the future.

## 6. Element fabrication and characterization

1. B. Goebel, L. L. Wang, and T. Tschudi, “Multilayer technology for diffractive optical elements,” Appl. Opt. **35**(22), 4490–4493 (1996). [CrossRef] [PubMed]

2. J. M. Miller, M. R. Taghizadeh, J. Turunen, and N. Ross, “Multilevel-grating array generators: fabrication error analysis and experiments,” Appl. Opt. **32**(14), 2519–2525 (1993). [CrossRef] [PubMed]

^{th}spot

^{th}spot and the overall transmitted optical power

## 7. Comparison of different simulation methods and experiment

## 8. Conclusion

## Acknowledgments

## References and links

1. | B. Goebel, L. L. Wang, and T. Tschudi, “Multilayer technology for diffractive optical elements,” Appl. Opt. |

2. | J. M. Miller, M. R. Taghizadeh, J. Turunen, and N. Ross, “Multilevel-grating array generators: fabrication error analysis and experiments,” Appl. Opt. |

3. | M. Banasch, L.-C. Wittig, and E.-B. Kley, “Fabrication tolerances of binary and multilevel Computer Generated Holograms (CGHs) with submicron Pixel Size,” |

4. | E. Noponen and J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A |

5. | J. Mait, D. Prather, and M. Mirotznik, “Design of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A |

6. | P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff,” J. Opt. Soc. Am. A |

7. | C. Ribot, P. Lalanne, M. S. Lee, B. Loiseaux, and J. P. Huignard, “Analysis of blazed diffractive optical elements formed with artificial dielectrics,” J. Opt. Soc. Am. A |

8. | H. J. Hyvärinen, P. Karvinen, and J. Turunen, “Polarization insensitive resonance-domain blazed binary gratings,” Opt. Express |

9. | W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt. |

10. | W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design of binary subwavelength multi-phase level computer generated holograms,” Opt. Lett. |

11. | W. Freese, T. Kämpfe, W. Rockstroh, E.-B. Kley, and A. Tünnermann, “Optimized electron beam writing strategy for fabricating computer-generated holograms based on an effective medium approach,” Opt. Express |

12. | W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Multi-phase-level diffractive elements realized by binary effective medium patterns,” Proc. SPIE |

13. | W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design and fabrication of a highly off-axis binary multi-phase level computer-generated hologram based on an effective medium approach,” Proc. SPIE |

14. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) |

15. | J. W. Goodman, |

16. | U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun. |

17. | A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, and S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. |

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

19. | S. Mellin and G. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express |

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

21. | M. D. Feit and J. A. Fleck Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. |

22. | S. Kumar, T. Srinivas, and A. Selvarjan, “Beam propagation method and its application to integrated optic structures and optical fibers,” J. Phys. |

23. | M. D. Feit and J. A. Fleck Jr., “Calculation of dispersion in graded-index multimode fibers by a propagating-beam method,” Appl. Opt. |

24. | B. Hermansson, D. Yevick, and J. Saijonmaa, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” J. Opt. Soc. Am. A |

25. | M. Fertig and K.-H. Brenner, “Vector wave propagation method,” J. Opt. Soc. Am. A |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.1380) Diffraction and gratings : Binary optics

(090.1970) Holography : Diffractive optics

(090.2890) Holography : Holographic optical elements

(220.2560) Optical design and fabrication : Propagating methods

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Holography

**History**

Original Manuscript: February 14, 2013

Revised Manuscript: April 10, 2013

Manuscript Accepted: April 17, 2013

Published: May 14, 2013

**Citation**

Wiebke Eckstein, Ernst-Bernhard Kley, and Andreas Tünnermann, "Comparison of different simulation methods for effective medium computer generated holograms," Opt. Express **21**, 12424-12433 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12424

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### References

- B. Goebel, L. L. Wang, and T. Tschudi, “Multilayer technology for diffractive optical elements,” Appl. Opt.35(22), 4490–4493 (1996). [CrossRef] [PubMed]
- J. M. Miller, M. R. Taghizadeh, J. Turunen, and N. Ross, “Multilevel-grating array generators: fabrication error analysis and experiments,” Appl. Opt.32(14), 2519–2525 (1993). [CrossRef] [PubMed]
- M. Banasch, L.-C. Wittig, and E.-B. Kley, “Fabrication tolerances of binary and multilevel Computer Generated Holograms (CGHs) with submicron Pixel Size,” MOC´04–10th Microoptics Conference, Germany (2004).
- E. Noponen and J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A11(3), 1097–1109 (1994). [CrossRef]
- J. Mait, D. Prather, and M. Mirotznik, “Design of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A16(5), 1157–1167 (1999). [CrossRef]
- P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff,” J. Opt. Soc. Am. A16(5), 1143–1156 (1999). [CrossRef]
- C. Ribot, P. Lalanne, M. S. Lee, B. Loiseaux, and J. P. Huignard, “Analysis of blazed diffractive optical elements formed with artificial dielectrics,” J. Opt. Soc. Am. A24(12), 3819–3826 (2007). [CrossRef] [PubMed]
- H. J. Hyvärinen, P. Karvinen, and J. Turunen, “Polarization insensitive resonance-domain blazed binary gratings,” Opt. Express18(13), 13444–13450 (2010). [CrossRef] [PubMed]
- W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt.39(20), 3531–3536 (2000). [CrossRef] [PubMed]
- W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design of binary subwavelength multi-phase level computer generated holograms,” Opt. Lett.35(5), 676–678 (2010). [CrossRef] [PubMed]
- W. Freese, T. Kämpfe, W. Rockstroh, E.-B. Kley, and A. Tünnermann, “Optimized electron beam writing strategy for fabricating computer-generated holograms based on an effective medium approach,” Opt. Express19(9), 8684–8692 (2011). [CrossRef] [PubMed]
- W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Multi-phase-level diffractive elements realized by binary effective medium patterns,” Proc. SPIE7591(75910Z), 75910Z-1–75910Z-7 (2010). [CrossRef]
- W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design and fabrication of a highly off-axis binary multi-phase level computer-generated hologram based on an effective medium approach,” Proc. SPIE7927(792710), 792710, 792710-7 (2011). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.)35, 227–246 (1972).
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
- U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,” Opt. Commun.229(1-6), 11–21 (2004). [CrossRef]
- A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, and S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt.31(17), 3320–3336 (1992). [CrossRef] [PubMed]
- D. Pommet, M. Moharam, and E. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A11(6), 1827–1834 (1994). [CrossRef]
- S. Mellin and G. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express8(13), 705–722 (2001). [CrossRef] [PubMed]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Boston: Artech House, 2000).
- M. D. Feit and J. A. Fleck., “Light propagation in graded-index optical fibers,” Appl. Opt.17(24), 3990–3998 (1978). [CrossRef] [PubMed]
- S. Kumar, T. Srinivas, and A. Selvarjan, “Beam propagation method and its application to integrated optic structures and optical fibers,” J. Phys.34, 347–358 (1989).
- M. D. Feit and J. A. Fleck., “Calculation of dispersion in graded-index multimode fibers by a propagating-beam method,” Appl. Opt.18(16), 2843–2851 (1979). [CrossRef] [PubMed]
- B. Hermansson, D. Yevick, and J. Saijonmaa, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” J. Opt. Soc. Am. A1(6), 663–671 (1984). [CrossRef]
- M. Fertig and K.-H. Brenner, “Vector wave propagation method,” J. Opt. Soc. Am. A27(4), 709–717 (2010). [CrossRef] [PubMed]

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