## Modeling of the angular tolerancing of an effective medium diffractive lens using combined finite difference time domain and radiation spectrum method algorithms |

Optics Express, Vol. 18, Issue 17, pp. 17974-17982 (2010)

http://dx.doi.org/10.1364/OE.18.017974

Acrobat PDF (1636 KB)

### Abstract

A new rigorous vector-based design and analysis approach of diffractive lenses is presented. It combines the use of two methods: the Finite-Difference Time-Domain for the study in the near field, and the Radiation Spectrum Method for the propagation in the far field. This approach is proposed to design and optimize effective medium cylindrical diffractive lenses for high efficiency structured light illumination systems. These lenses are realised with binary subwavelength features that cannot be designed using the standard scalar theory. Furthermore, because of their finite and high frequencies characteristics, such devices prevent the use of coupled wave theory. The proposed approach is presented to determine the angular tolerance in the cases of binary subwavelength cylindrical lenses by calculating the diffraction efficiency as a function of the incidence angle.

© 2010 OSA

## 1. Introduction

1. M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. **31**(22), 4453–4458 (1992). [CrossRef] [PubMed]

1. M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. **31**(22), 4453–4458 (1992). [CrossRef] [PubMed]

4. 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 **16**(5), 1143–1156 (1999). [CrossRef]

9. D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodetectors,” Opt. Eng. **38**(5), 870–878 (1999). [CrossRef]

4. 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 **16**(5), 1143–1156 (1999). [CrossRef]

5. A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, B. Blasi, A. Heinzel, D. Sporn, W. Doll, and V. Wittwer, “Subwavelength-structured antireflective surfaces on glass,” Thin Solid Films **351**(1-2), 73–78 (1999). [CrossRef]

6. 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]

7. J. M. Miller, N. de Beaucoudrey, P. Chavel, E. Cambril, and H. Launois, “Synthesis of a subwavelength-pulse-width spatially modulated array illuminator for 0.633 µm,” Opt. Lett. **21**(17), 1399–1402 (1996). [CrossRef] [PubMed]

10. J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary Subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A **16**(5), 1157–1167 (1999). [CrossRef]

11. P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. **43**, 2063–2085 (1996). [CrossRef]

1. M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. **31**(22), 4453–4458 (1992). [CrossRef] [PubMed]

9. D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodetectors,” Opt. Eng. **38**(5), 870–878 (1999). [CrossRef]

4. 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 **16**(5), 1143–1156 (1999). [CrossRef]

6. 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]

12. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. **23**(14), 1081–1083 (1998). [CrossRef]

14. P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, “Towards a full vectorial and modal technique for the analysis of integrated optics structures: the Radiation Spectrum Method,” Opt. Commun. **140**(1-3), 128–145 (1997). [CrossRef]

## 2. Proposed combined FDTD-RSM numerical method principles

15. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. **31**, 2972–2974 (2006). [CrossRef] [PubMed]

14. P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, “Towards a full vectorial and modal technique for the analysis of integrated optics structures: the Radiation Spectrum Method,” Opt. Commun. **140**(1-3), 128–145 (1997). [CrossRef]

16. D. Khalil, “Extension of the radiation spectrum method for the reflection calculation at the end of a strongly guiding optical waveguide,” Opt. Quantum Electron. **35**(8), 801–809 (2003). [CrossRef]

## 3. Binary subwavelength design

^{N}phase levels surface spatial modulation in order to design the layout of a subwavelength lens which acts as a 2

^{N}levels digital lens. For a dielectric grating in a transmission configuration, the curve giving the phase difference as a function of the fill factor presents a strictly positive slope [11

11. P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. **43**, 2063–2085 (1996). [CrossRef]

*h*and the maximum phase difference

*ϕ*are linked therefore by Eq. (1):where

_{max}*n*is the refractive index of the medium at the wavelength

*λ*. The diffraction efficiency of the zeroth order is defined by the ratio between the electromagnetic energy (calculated by the flux of the Poynting vector) before the grating, and the energy in the zeroth order. For simulations presented in Fig. 1 below, the considered refractive index was 1.46, corresponding to the index of the fused silica at the 632.8 nm designed wavelength that we selected (He-Ne laser). Using the FDTD method, we calculated the main parameters of the zeroth order gratings. An eight phase levels diffractive structure simulated by the effective medium method was targeted, for which the higher phase difference was 7/8 x 2π rad.

## 4. Lens design method

### 4.1. Diffractive lens layout design

*R*, the refractive index

*n*and the focal length

*f*, is used (Eq. (3)):The profile of the superwavelength diffractive lens is estimated by “cutting” the phase function into 2π slices with a straight forward design. The resulting profile defines a simple Fresnel lens. Once the 2π modulation has been reached, the resulting profile may be quantified into N phase levels to obtain a N-levels diffractive lens. In fact, the diffractive lens introduces a phase modulation described by a relief surface to the incident wave. As we observed in the previous section, zeroth order gratings can be considered as DOEs which modulate the incident phase. Our aim is now to change the relief modulation into a new structure which contains only binary subwavelength gratings. Figure 3 summarizes the process to design the subwavelength lens.

_{l}### 4.2. Electromagnetic wave propagation simulation validating the proposed approach

*η*of the lens is then estimated as the flux of the Poynting vector through a surface that corresponds to the diameter of the Airy disk at the focal point. For a lens diameter

*D*, a focal length

*f*, a focal point located at the

*y*coordinate, at a working wavelength

_{f}*λ*and at final step of the simulation

*t*, the diffraction efficiency is calculated by Eq. (4): where ×denotes the cross product of the electric

_{f}*x*the u

_{u}^{th}spatial discrete coordinate x, at the focal plane, and

*x*the v

_{v}^{th}spatial discrete coordinate x, at the incident plane, Δ

*x*and Δ

_{FDTD}*x*stand respectively for the sampling steps for FDTD and RSM computation. Figure 4 shows the principle of the “combined” simulation. Left on the Fig. 4, we present amplitude of electric field

_{RSM}## 5. Angular tolerance analysis in a specific case

## 6. Conclusion

## References and links

1. | M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. |

2. | B. Kress, and P. Meyrueis, |

3. | S. Sinzinger, and J. Jahns, “Diffractive Optics,” in |

4. | 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 |

5. | A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, B. Blasi, A. Heinzel, D. Sporn, W. Doll, and V. Wittwer, “Subwavelength-structured antireflective surfaces on glass,” Thin Solid Films |

6. | 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. |

7. | J. M. Miller, N. de Beaucoudrey, P. Chavel, E. Cambril, and H. Launois, “Synthesis of a subwavelength-pulse-width spatially modulated array illuminator for 0.633 µm,” Opt. Lett. |

8. | F. T. Chen and H. G. Craighead, “Diffractive lens fabricated with mostly zeroth-order gratings,” Opt. Lett. |

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

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

11. | P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. |

12. | P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. |

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

14. | P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, “Towards a full vectorial and modal technique for the analysis of integrated optics structures: the Radiation Spectrum Method,” Opt. Commun. |

15. | A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. |

16. | D. Khalil, “Extension of the radiation spectrum method for the reflection calculation at the end of a strongly guiding optical waveguide,” Opt. Quantum Electron. |

17. | M. Born, and E. Wolf, “Optics of crystals,” in |

**OCIS Codes**

(050.1965) Diffraction and gratings : Diffractive lenses

(050.2065) Diffraction and gratings : Effective medium theory

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: December 1, 2009

Revised Manuscript: January 19, 2010

Manuscript Accepted: January 19, 2010

Published: August 6, 2010

**Citation**

Victorien Raulot, Philippe Gérard, Bruno Serio, Manuel Flury, Bernard Kress, and Patrick Meyrueis, "Modeling of the angular tolerancing of an effective medium diffractive lens using combined finite difference time domain and radiation spectrum method algorithms," Opt. Express **18**, 17974-17982 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17974

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### References

- M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. 31(22), 4453–4458 (1992). [CrossRef] [PubMed]
- B. Kress, and P. Meyrueis, Digital Diffractive Optics: An introduction to planar diffractive optics and related technologies (John Wiley and Sons, 2000).
- S. Sinzinger, and J. Jahns, “Diffractive Optics,” in Microoptics, (Wiley-VCH, 2nd edition, 2005), pp. 138–141.
- 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 16(5), 1143–1156 (1999). [CrossRef]
- A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, B. Blasi, A. Heinzel, D. Sporn, W. Doll, and V. Wittwer, “Subwavelength-structured antireflective surfaces on glass,” Thin Solid Films 351(1-2), 73–78 (1999). [CrossRef]
- 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]
- J. M. Miller, N. de Beaucoudrey, P. Chavel, E. Cambril, and H. Launois, “Synthesis of a subwavelength-pulse-width spatially modulated array illuminator for 0.633 µm,” Opt. Lett. 21(17), 1399–1402 (1996). [CrossRef] [PubMed]
- F. T. Chen and H. G. Craighead, “Diffractive lens fabricated with mostly zeroth-order gratings,” Opt. Lett. 23, 1081–1083 (1996).
- D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodetectors,” Opt. Eng. 38(5), 870–878 (1999). [CrossRef]
- J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary Subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A 16(5), 1157–1167 (1999). [CrossRef]
- P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085 (1996). [CrossRef]
- P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23(14), 1081–1083 (1998). [CrossRef]
- A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech, 3rd Edition, 2005).
- P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, “Towards a full vectorial and modal technique for the analysis of integrated optics structures: the Radiation Spectrum Method,” Opt. Commun. 140(1-3), 128–145 (1997). [CrossRef]
- A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006). [CrossRef] [PubMed]
- D. Khalil, “Extension of the radiation spectrum method for the reflection calculation at the end of a strongly guiding optical waveguide,” Opt. Quantum Electron. 35(8), 801–809 (2003). [CrossRef]
- M. Born, and E. Wolf, “Optics of crystals,” in Principles of Optics, (Cambridge University Press, Cambridge, 7th ed., 2005), 837–840.

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