## Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design

Optics Express, Vol. 8, Issue 13, pp. 705-722 (2001)

http://dx.doi.org/10.1364/OE.8.000705

Acrobat PDF (540 KB)

### Abstract

We have designed high-efficiency finite-aperture diffractive optical elements (DOE’s) with features on the order of or smaller than the wavelength of the incident illumination. The use of scalar diffraction theory is generally not considered valid for the design of DOE’s with such features. However, we have found several cases in which the use of a scalar-based design is, in fact, quite accurate. We also present a modified scalar-based iterative design method that incorporates the angular spectrum approach to design diffractive optical elements that operate in the near-field and have sub-wavelength features. We call this design method the iterative angular spectrum approach (IASA). Upon comparison with a rigorous electromagnetic analysis technique, specifically, the finite difference time-domain method (FDTD), we find that our scalar-based design method is surprisingly valid for DOE’s having sub-wavelength features.

© Optical Society of America

## 1. Introduction

8. Pierre St. Hilaire, “Phase profiles for holographic stereograms,” Opt. Eng. **34**, 83–89 (1995). [CrossRef]

12. F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. **54**, 1481–1571 (1991). [CrossRef]

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

14. I.O. Bohachevsk;y, M.E. Johnson, and M.L. Stein, “Generalized simulated annealing for function optimization,” Technometrics **28**, 209–217 (1986). [CrossRef]

18. M.R. Feldman and C.C. Guest, “High-efficiency hologram encoding for generation of spot arrays,” Opt. Lett. **14**, 479–481 (1989). [CrossRef] [PubMed]

19. J. Jiang and G. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Optical Express , **7**, 237–242 (2000), http://www.opticsexpress.org/oearchive/source/23164.htm. [CrossRef]

## 2. Heuristic design and analysis of 1–2 beamfanners

_{0}, of 5 µm is normally incident on a silicon-air interface bounded by a 50 µm (10 λ

_{0}) metallic aperture having infinite conductivity. The minimum feature size of the DOE is chosen to be 1 µm (0.2 λ

_{0}) and the observation plane is located at a distance of 100 µm in air (20 λ

_{0}). Next, we consider a heuristic method to design a DOE to perform the 1–2 beam-fanning function.

_{0}/(2Δn) for normally incident light and Δn is the difference in refractive index between air and silicon. Here, we define a zone as a region in which the DOE profile is locally continuous, e.g., for a binary grating with a 50% fill factor, a zone would constitute half the grating period. Next, a quadratic phase profile is added to the grating to focus the light in the near field observation plane.

_{0}in which d(x) is the etch depth along the DOE. The Fresnel transmission coefficient, τ, is given by τ=2n

_{1}cos(θ

_{1})/(n

_{1}cos(θ

_{1})+n

_{2}cos(θ

_{2})) in which n

_{1}and n

_{2}are the refractive indices of silicon and air, respectively, θ

_{1}is the angle of the incidence with respect to the optical axis and θ

_{2}is the refracted angle as calculated by Snell’s law. In every case in this paper, we only consider normally incident light, for which the Fresnel transmission coefficient reduces to τ=2 n

_{1}/(n

_{1}+n

_{2}). The field incident on a DOE is multiplied by the transmission function, t(x), which gives the field just past the DOE interface. This field is then propagated to a chosen observation plane using the angular spectrum approach. Once the field is known in this observation plane, all other field component can be obtained via Maxwell’s equations, and the power and the diffraction efficiency can be calculated. By using AS scalar-based theory, the goal is to determine what DOE profile, d(x), best performs its intended function. In the heuristic designs presented, we specifically design 1–2 beamfanners.

_{0}, 4 λ

_{0}, 2 λ

_{0}, 1.6 λ

_{0}, 1.4 λ

_{0}, and 1.2 λ

_{0}, and the resultant DOE profiles for these cases are shown in Fig. 2.

_{0}/n), Λ is the grating period, and θ is the angle of the first diffracted order with respect to the optical axis, i.e., θ=tan

^{-1}(s/2z), s is the separation between the diffracted orders in the observation plane, and z is the distance to that plane. In Fig. 3, note that for larger grating periods the AS scalar results agree very well with those predicted by FDTD, whereas for smaller grating periods the validity of AS scalar theory appears to break down.

_{0}. This assessment appears to indicate that perhaps the validity and accuracy of AS scalar diffraction theory is slightly broader than other results reported in the literature [5]. In terms of zones, AS scalar diffraction theory is accurate, i.e., less than 10% error, for zone sizes greater than or equal to the free space wavelength of the incident light. Also note that the sub-wavelength minimum feature size is the same for each beamfanner in the analysis. Therefore, one may conclude that the validity of AS scalar diffraction theory does not necessarily depend on DOE minimum feature size but rather the size of the DOE zones.

_{0}. The field amplitudes calculated via FDTD, however, differ more dramatically from the AS scalar result, which is simply a rectangle function scaled by the Fresnel transmission coefficient. This apparent disagreement in field magnitudes can be reconciled somewhat by examining only those field components that actually propagate to the observation plane, i.e., which are not evanescent, as will be shown next.

_{0}, 4 λ

_{0}, and 2 λ

_{0}, the magnitudes of the angular spectra calculated from rigorous analysis agree well with what was calculated from the scalar analysis while the cases 1.6 λ

_{0}, 1.4 λ

_{0}, and 1.2 λ

_{0}, tend to be somewhat different. Also note that the spatial frequency components outside the range of [-1/λ

_{0},1/λ

_{0}] are evanescent. These limits are denoted by vertical red lines.

_{0}, 4 λ

_{0}, and 2 λ

_{0}. However, AS scalar results tend to differ significantly from rigorous results when the grating period is less than 2 λ

_{0}, which is consistent with what we observe in the near-field diffraction patterns. From this, we conclude that the amplitude and the phase of the

*propagating*field just past the DOE is the most dominant factor in determining whether AS scalar diffraction theory is valid in our test cases. This also suggests that any DOE profile that generates the desired propagating field just past the DOE, regardless of the evanescent field content of the total field, will yield the same diffracted beam (with the possible exception that the overall diffraction efficiency may be different). Hence the evanescent field components represent a general design freedom for creating a diffractive optical element.

## 3. Iterative angular spectrum algorithm

### 3.1 Design method

8. Pierre St. Hilaire, “Phase profiles for holographic stereograms,” Opt. Eng. **34**, 83–89 (1995). [CrossRef]

12. F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. **54**, 1481–1571 (1991). [CrossRef]

### 3.2 1–2 beamfanner design

*ε*≪1. We also found that we had to impose an additional constraint to prevent the algorithm from converging to two adjacent microlenses that focus light independently of one another. To prevent this outcome (which was important for our eventual application), only one half (either left or right) of the DOE is illuminated and the energy is equalized about the central position (x=0) in the observation plane. All of the constraints are applied in each iteration until an acceptable irradiance pattern is obtained. For all cases considered, IASA produced an acceptable irradiance pattern within 1000 iterations.

### 3.3 1–3 and 1–4 beamfanner designs

## 4. Summary

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | J. Goodman, |

3. | G.S. Smith. |

4. | D. A. Gremaux and N. C. Gallagher, “Limits of scalar diffraction theory for conducting gratings,” Appl. Opt. |

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

6. | M. G. Moharam and T. K. Gaylord, “Diffraction analysis of surface-relief gratings,” J. Opt. Soc. Am. A |

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

8. | Pierre St. Hilaire, “Phase profiles for holographic stereograms,” Opt. Eng. |

9. | N.C. Gallagher and B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. |

10. | J.R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. |

11. | F. Wyrowski, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A |

12. | F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. |

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

14. | I.O. Bohachevsk;y, M.E. Johnson, and M.L. Stein, “Generalized simulated annealing for function optimization,” Technometrics |

15. | Y. Lin, T.J. Kessler, and G.N. Lawrence, “Design of continuous surface-relief phase plates by surface-based simulated annealing to achieve control of focal-plane irradiance,” Opt. Lett. |

16. | B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. |

17. | J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. |

18. | M.R. Feldman and C.C. Guest, “High-efficiency hologram encoding for generation of spot arrays,” Opt. Lett. |

19. | J. Jiang and G. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Optical Express , |

20. | A. Taflove, |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(230.1950) Optical devices : Diffraction gratings

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 16, 2001

Published: June 18, 2001

**Citation**

Stephen Mellin and Gregory Nordin, "Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design," Opt. Express **8**, 705-722 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-13-705

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### References

- M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, New York, 1965).
- J. Goodman, Introduction to Fourier Optics (McGraw-Hill., New York, 1968).
- G.S. Smith. An Introduction to Classical Electromagnetic Radiation (Cambridge University Press, Cambridge, 1997).
- D. A. Gremaux and N. C. Gallagher, "Limits of scalar diffraction theory for conducting gratings," Appl. Opt. 32, 1948-1953 (1993). [CrossRef] [PubMed]
- D.A. Pommet, M.G. Moharam, and E.B. Grann, "Limits of scalar diffraction theory for diffractive phase elements", Opt. Lett. 11, 1827-1834 (1995).
- M. G. Moharam and T. K. Gaylord, "Diffraction analysis of surface-relief gratings," J. Opt. Soc. Am. A 72, 1385-1392 (1982). [CrossRef]
- R.W. Gerchberg, W.O. Saxton. "A practical algorithm for the determination of phase from image and diffraction plane pictures" Optik 35, 237-246 (1971).
- Pierre St. Hilaire, "Phase profiles for holographic stereograms," Opt. Eng. 34, 83-89 (1995). [CrossRef]
- N.C. Gallagher and B. Liu, "Method for computing kinoforms that reduces image reconstruction error," Appl. Opt. 12, 2328-2335 (1973). [CrossRef] [PubMed]
- J.R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-306 (1980).
- F. Wyrowski, "Iterative Fourier-transform algorithm applied to computer holography," J. Opt. Soc. Am. A 5, 1058-1065 (1988). [CrossRef]
- F. Wyrowski and O. Bryngdahl, "Digital holography as part of diffractive optics," Rep. Prog. Phys. 54, 1481-1571 (1991). [CrossRef]
- J. N. Mait, "Understanding diffractive optic design in the scalar domain," J. Opt. Soc. Am. A 12, 2145- 2158 (1995). [CrossRef]
- I.O. Bohachevsky, M.E. Johnson, M.L. Stein, "Generalized simulated annealing for function optimization," Technometrics 28, 209-217 (1986). [CrossRef]
- Y. Lin, T.J. Kessler, G.N. Lawrence, "Design of continuous surface-relief phase plates by surface-based simulated annealing to achieve control of focal-plane irradiance," Opt. Lett. 21, 1703-1705 (1996). [CrossRef] [PubMed]
- B. K. Jennison, J. P. Allebach, and D. W. Sweeney, "Iterative approaches to computer-generated holog-raphy," Opt. Eng. 28, 629-637 (1989).
- J. Turunen, A. Vasara, and J. Westerholm, "Kinoform phase relief synthesis: a stochastic method," Opt. Eng. 28, 1162-1167 (1989).
- M.R. Feldman and C.C. Guest, "High-efficiency hologram encoding for generation of spot arrays," Opt. Lett. 14, 479-481 (1989). [CrossRef] [PubMed]
- J. Jiang and G. Nordin, "A rigorous unidirectional method for designing finite aperture diffractive optical elements," Opt. Express 7, 237-242 (2000), http://www.opticsexpress.org/oearchive/source/23164.htm. [CrossRef]
- A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).

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