## Design of the diffractive optical elements for synthetic spectra

Optics Express, Vol. 11, Issue 12, pp. 1392-1399 (2003)

http://dx.doi.org/10.1364/OE.11.001392

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### Abstract

We present a novel efficient gradient-based optimization algorithm for the design of diffractive optical elements (DOE’s) for synthetic spectra applications. Two design examples are given. The results demonstrate that the DOE’s obtained by the proposed algorithm can accurately produce the desired intensity spectra at a predetermined diffraction angle.

© 2003 Optical Society of America

## 1. Introduction

7. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. **A12**, 2145–2158 (1995). [CrossRef]

4. F. Wyrowski and O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography” J. Opt. Soc. Am. **A5**, 1058–1065 (1988). [CrossRef]

6. G. Zhou, Y. Chen, Z. Wang, and H. Song, “Genetic local search algorithm for optimization design of diffractive optical elements,” Appl. Opt. **38**, 4281–4290 (1999). [CrossRef]

*et al.*reported that DOE’s can also be used to synthesize the infrared spectra of real compounds [8

8. M. B. Sinclair, M. A. Butler, S. H. Kravitz, W. J. Zubrzycki, and A. J. Ricco, “Synthetic infrared spectra,” Opt. Lett. **22**, 1036–1038 (1997). [CrossRef] [PubMed]

9. M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, “Synthetic spectra: a tool for correlation spectroscopy,” Appl. Opt. **36**, 3342–3348 (1997). [CrossRef] [PubMed]

*et al.*also developed a modified Gerchberg-Saxton algorithm for the design of DOE’s that reproduce the infrared spectra of important compounds [9

9. M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, “Synthetic spectra: a tool for correlation spectroscopy,” Appl. Opt. **36**, 3342–3348 (1997). [CrossRef] [PubMed]

8. M. B. Sinclair, M. A. Butler, S. H. Kravitz, W. J. Zubrzycki, and A. J. Ricco, “Synthetic infrared spectra,” Opt. Lett. **22**, 1036–1038 (1997). [CrossRef] [PubMed]

9. M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, “Synthetic spectra: a tool for correlation spectroscopy,” Appl. Opt. **36**, 3342–3348 (1997). [CrossRef] [PubMed]

## 2. Design method

*M*micromirror elements each of width Δ, for a total length of

*L*(=

*M*Δ). The position of each elastically supported micromirror element in the array is controlled precisely by applying a driving voltage between the mirror and the electrode underneath. This programmable, variable-element DOE can be fabricated using microfabrication technology. For simplicity, here we assume that the amplitude of the incident wave does not vary with position or wavelength. The diffracted field just leaving the DOE can be expressed by:

*A*is the amplitude of the incident wave, λ is the wavelength, and

*d*

_{m}is the displacement of the

*m*th micromirror element. Working in the Fraunhofer approximation, the diffracted field at a fixed diffraction angle

*θ*is described by the Fourier transform [10]:

*u*

_{n}(

*u*

_{n}=1/λ

_{n},

*n*=1, 2, …,

*N*) within the spectral range of interest [

*u*

_{min},

*u*

_{max}] is given by:

*E*fit to the target spectrum. It can be derived by setting the partial derivative of

*E*with respect to γ equal to 0, resulting in:

*λ*

_{max}=1/

*u*

_{min}corresponds to the maximum wavelength of the spectral range of interest.

*E*with respect to

*d*

_{m}is given by:

*δ*

_{nk}is the discrete Kronecker delta function, defined as:

*F*(

**Φ**) with respect to

**Φ**, where

**Φ**is the vector of design variables and

*F*(

**Φ**) is the objective function, the basic recursion formula for the DFP method is:

*s*

_{k}is determined by a line search in the direction

**d**

_{k}:

**d**

_{k}is determined iteratively by following formulas:

*F*(

**Φ**

_{k}) is gradient vector of the objective function, and

**I**is the identity matrix. The correction matrix

**A**

_{k}is derived from information collected during the last iteration, i.e., from the change in the variable vector:

## 3. Design examples and results

^{-1}to 4200 cm

^{-1}, is 256 chosen at equal intervals. The coefficients

*G*

_{nm}in Eq. (5) do not change from iteration to iteration. Therefore, to minimize the processing time, they are generated and stored at the beginning of the run. In DFP search procedures, if a new parameter generated is outside the specified boundary, which is defined by the Inequality (10), it will be set to the parameter close to the nearest boundary. The calculations were performed on a 2GHz Pentium computer with 256MB memory. The design started from a randomly chosen initial point and terminated when the error defined by Eq. (7) converged or a predetermined number of iterations (300 iterations was chosen here) was reached. Under these conditions, the design required approximately 15 minutes. Figure 3 shows a part of the surface profile of the resulting DOE with the displacement of each micromirror set to the designed value. Only the first 64 of the 2048 micromirror elements are shown. Figure 4 shows the diffracted intensity spectrum generated by the programmable grating-like DOE shown in Fig. 3. In order to investigate the efficiency of the DOE, the following parameter

*η*is defined:

*I*(

*u*

_{p}) is the diffraction intensity produced by the DOE for a specific wave number

*u*

_{p}(typically the wave number corresponding to the peak of the target spectrum) at the given diffraction angle, and

*I*

_{0}(

*u*

_{p}) is the diffraction intensity for the same wave number

*u*

_{p}observed at 0° diffraction angle, when the displacements of the micromirror elements are all set to 0 (a flat surface profile, for this setting, the DOE is actually a reflective mirror). The value of

*η*in this case is 0.7%.

^{-1}. We also use the same programmable DOE to generate the desired intensity spectrum shown in Fig. 5 at a diffraction angle of 15°. After the optimization, the displacements of the micromirrors in the DOE are set to the resulting design values, and the diffraction intensity spectrum at the specific diffraction angle is calculated. A part of the surface profile of the resulting DOE (first 64 of the 2048 micromirror elements) is shown in Fig. 6, and the calculated spectrum is shown in Fig. 7. The value of

*η*obtained in this case is 13%.

## 4. Conclusions

## References and links

1. | G. J. Swanson and W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. |

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

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

4. | F. Wyrowski and O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography” J. Opt. Soc. Am. |

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

6. | G. Zhou, Y. Chen, Z. Wang, and H. Song, “Genetic local search algorithm for optimization design of diffractive optical elements,” Appl. Opt. |

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

8. | M. B. Sinclair, M. A. Butler, S. H. Kravitz, W. J. Zubrzycki, and A. J. Ricco, “Synthetic infrared spectra,” Opt. Lett. |

9. | M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, “Synthetic spectra: a tool for correlation spectroscopy,” Appl. Opt. |

10. | J. W. Goodman, |

11. | M. A. Wolfe, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1970) Diffraction and gratings : Diffractive optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 13, 2003

Revised Manuscript: June 4, 2003

Published: June 16, 2003

**Citation**

Guangya Zhou, Francis Tay, and Fook Chau, "Design of the diffractive optical elements for synthetic spectra," Opt. Express **11**, 1392-1399 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-12-1392

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### References

- G. J. Swanson and W. B. Weldkamp, ???Diffractive optical elements for use in infrared systems,??? Opt. Eng. 28, 605???608 (1989).
- R. W. Gerchberg and W. O. Saxton, ???A practical algorithm for the determination of phase from image and diffraction plane pictures,??? Optik 35, 237???246 (1972).
- J. R. Fienup, ???Iterative method applied to image reconstruction and to computer-generated holograms,??? Opt. Eng. 19, 297???305 (1980).
- F. Wyrowski and O. Bryngdahl, ???Iterative Fourier transform algorithm applied to computer holography??? J. Opt. Soc. Am. A 5, 1058???1065 (1988). [CrossRef]
- J. Turunen, A. Vasara, and J. Westerholm, ???Kinoform phase relief synthesis: a stochastic method,??? Opt. Eng. 28, 1162???1176 (1989).
- G. Zhou, Y. Chen, Z. Wang, and H. Song, ???Genetic local search algorithm for optimization design of diffractive optical elements,??? Appl. Opt. 38, 4281???4290 (1999). [CrossRef]
- J. N. Mait, ???Understanding diffractive optic design in the scalar domain,??? J. Opt. Soc. Am. A 12, 2145???2158 (1995). [CrossRef]
- M. B. Sinclair, M. A. Butler, S. H. Kravitz, W. J. Zubrzycki, and A. J. Ricco, ???Synthetic infrared spectra,??? Opt. Lett. 22, 1036???1038 (1997). [CrossRef] [PubMed]
- M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, ???Synthetic spectra: a tool for correlation spectroscopy,??? Appl. Opt. 36, 3342???3348 (1997). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics, 2nd Edition, (McGraw-Hill, New York, 1996), Chap. 4, 63???90.
- M. A. Wolfe, Numerical Methods for Unconstrained Optimization: an Introduction, (Van Nostrand Reinhold Company Ltd., New York, 1978), Chap. 6, 161???167.

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