## Spectral synthesis using an array of micro gratings

Optics Express, Vol. 16, Issue 12, pp. 9132-9143 (2008)

http://dx.doi.org/10.1364/OE.16.009132

Acrobat PDF (3941 KB)

### Abstract

In this paper, we present the design of a novel spectral synthesis system that is based on an array of in-plane moving micro gratings, instead of using a conventional out-of-plane micromirror array. Utilizing the unique nondispersive characteristic of the optical phase modulation mechanism based on in-plane movable micro gratings, we demonstrate that the synthetic spectra problem can be greatly simplified and effectively reduced to that of conventional phase retrieval.

© 2008 Optical Society of America

## 1. Introduction

*U*(

*u*) at a fixed diffraction angle

*θ*is described by the following equation: [8]

*u*is the wavenumber of the light,

*C*(

*u*) is a wavenumber dependent constant,

*d*is the out-of-plane displacement of the

_{m}*m*

^{th}micromirror,

*w*is the width of a micromirror element, and

*M*is the total number of elements in the array. The synthetic spectra problem is equivalent to solving the above equation for micromirror displacement

*d*for a given desired intensity spectrum of

_{m}*U*(

*u*). This is a challenging problem and typically, complicated optimization algorithms have to be employed to obtain an approximate solution [2

2. 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]

4. M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express **14**, 12590–12612 (2006). [CrossRef] [PubMed]

8. G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express **11**, 1392–1399 (2003). [CrossRef] [PubMed]

9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

## 2. Spectral synthesis using micro gratings

### 2.1. Operation principle

10. G. Zhou and F. S. Chau, “Nondispersive optical phase shifter array using microelectromechanical systems based gratings,” Opt. Express 15, 10958–10963 (2007). [CrossRef] [PubMed]

*θ*. The grating lines are aligned along the

_{i}*z*-axis. The incident light wave is then:

*U*

_{0}(

*u*) is the amplitude of the incident wave and

*u*is the wavenumber. At

*y*=0, the compensation grating bestows a spatial phase modulation exp[

*i*2

*πuh*(

*x*)] on the incident light wave, which can be expanded in a Fourier series due to its periodicity along the

*x*-axis as:

*h*(

*x*) is a function describing the grating surface profile,

*k*represents the diffraction order and

*p*is the grating period. If we select

*k*=-1order beam and direct it to the micro grating array, the light wave diffracted from the compensation grating is then:

*d*is the in-plane displacement of the grating element driven by its MEMS actuator for phase modulation. Using the Fourier expansion:

_{m}*k*=-1, we get:

*Δ*is the width of the element along the

*z’*axis, and 2

*M*+1 is the total number of grating elements. Again, working in the Fraunhofer approximation, the light wave diffracted to a predetermined direction specified by the direction cosines (-sin

*θ*, cos

_{i}*θ*, sin

_{i}*α*) in the

*x’y’z’*coordinate system as shown in Fig. 2(a) is given by [11]:

*A*is a constant that does not depend on wavenumber

*u*. Substituting Eq. (8) into Eq. (9), we obtain:

*u*within the spectral range of interest [

_{n}*u*,

_{min}*u*]:

_{max}*n*=0,±1, …, ±

*M*, and

*l*is an integer. We further define:

*DFT*stands for the operation of discrete Fourier transform. The synthetic spectra problem is now equivalent to solving the above equation for phase modulation

*ϕ*for a given desired intensity spectrum

_{m}*I*or |

_{n}*U*(

*u*)|

_{n}^{2}. The synthetic spectra problem using an array of nondispersive phase modulators has therefore now been reduced to a conventional phase retrieval problem [12–13

12. Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Comm. **30**, 304–308 (1979). [CrossRef]

*U*(

*u*)|

_{n}^{2}and 1. A number of approaches have been proposed for this latter problem, including the Gerchberg-Saxton algorithm [14–15], simulated annealing [16] and genetic algorithm [17

17. 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]

### 2.2. Spectral resolution and bandwidth

*M*+1)Δsin

*α*, which is actually the total length of the grating array along the

*z’*axis projected onto the propagation direction of the outgoing light of interest. In addition, from Eqs. (10), (11) and (13), it is clear that the amplitude spectrum |

*U*(

*u*)| is periodic with a period of 1/(Δsin

*α*). Therefore, for practical design considerations of the proposed system, one can simply determine the width of the grating element in the array based on the reciprocal of the spectral bandwidth of interest, i.e.,

*δu*,

### 2.3. Efficiency considerations

*u*

_{1}, then the efficiency of the system can be potentially very high. This is true because we separate the input beam into multiple identical channels and recombine them to form a single output. If we configure the system such that the subdivided light beams emerging from all channels are in phase at wavenumber

*u*

_{1}, then the light of this particular wavenumber can be completely transmitted through the system, thereby achieving high efficiency. However, if the desired spectrum contains a set of non-zero values at different wavenumbers, then the in-phase condition has to be compromised to achieve the desired intensity spectrum. According to Eq. (17), the values of |

*U*(

*u*)|

_{n}^{2}at these targeted wavenumbers will generally reduce since the total amount is fixed. As a result, at each wavenumber, the light is attenuated by the system.

*h*(

*x*’) and

*f*(

*z*’) are the functions describing the grating surface profile. Here, we assume that the surface profiles of the grating elements are identical and the micro gratings are confined to move only along the

*x*’ axis to introduce phase shifts. Selecting the negative and positive first-order terms from the first and second Fourier expansions respectively on the right-hand side of Eq. (18) and assuming Δ/

*g*is an integer (each element has exact multiple full periods of the auxiliary grating profile), we find that instead of Eq. (10), the outgoing light beam from the grating array to the predetermined direction is now given by:

*ξ*(

*u*)=

*D*

_{1}(

*u*). Following the same procedure described in section 2.1, we define:

*ξ*(

*u*), the maximum of the sinc function in Eq. (20) is shifted. We can utilize this to minimize the loss of the system, for example, by setting the period

*g*of the auxiliary grating to fulfill the following equation:

*u*

_{0}is a chosen wavenumber, typically the center of the spectral band

*u*

_{0}=(

*u*+

_{max}*u*)/2, while at the same time, choosing a blazed profile at wavenumber

_{min}*u*

_{0}for the auxiliary grating so as to maximize the value of

*ξ*(

*u*). From Eq. (21), we see that the function of the auxiliary grating profile is to diffract the light of wavenumber

*u*

_{0}with high efficiency from normal incidence to the predetermined angle α. The increase in the sinc function values using an auxiliary grating profile is illustrated in Fig. 4(b).

### 2.4. Design method

*α*, we can determine the width of each grating element as well as the total number of required elements in the array using Eqs. (15) and (16). The period of the primary grating for phase shifting can be determined by considering the maximum stroke of the driving MEMS microactuator. The period of the auxiliary grating is determined by the required diffraction angle

*α*and a chosen wavenumber. Next, the detailed grating profiles and consequently, the efficiency terms

*η*(

*u*) and

*ξ*(

*u*), can be obtained by properly selecting blazing wavenumbers within the spectral range of interest. Based on the desired intensity spectrum of the outgoing light wave |

*U*

_{3}

^{d}(

*u*)|

^{2}, and using Eq. (20), we can then determine the targeted intensity constrain

*I*=|

^{d}_{n}*U*(

^{d}*u*)|

_{n}^{2}for the Fourier transform Eq. (14).

*I*(from the phase shifts

_{n}*ϕ*using

_{m}*DFT*) and the desired intensity

*I*:

^{d}_{n}*I*for a minimum

_{n}*E*fit to

*I*. It can be derived by setting the partial derivative of

^{d}_{n}*E*with respect to γ equal to 0, resulting in:

8. G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express **11**, 1392–1399 (2003). [CrossRef] [PubMed]

*IDFT*{2

*γ*(

*γI*-

_{k}*I*)

^{d}_{k}*U*(

*u*)}. With Eq. (25), we can compute the gradient direction of the objective function, allowing gradient-based search techniques including the Davidon-Fletcher-Powell (DFP) method [20] to be applied to solve this optimization problem.

_{k}8. G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express **11**, 1392–1399 (2003). [CrossRef] [PubMed]

### 2.5. Alternative setup

## 3. Design examples and results

**11**, 1392–1399 (2003). [CrossRef] [PubMed]

^{-1}to 4200 cm

^{-1}. The spectrum of the incident light beam is assumed to be uniform over the spectral range of interest. First, we set our spectral resolution to be around 2.3 cm

^{-1}, which corresponds to about 256 points in the spectral range of interest. In this work, instead of using 2048 micromirrors, we use only 256 MEMS-driven in-plane moving micro-grating elements. The design goal is to determine the phase shift or equivalently, the in-plane displacement of each grating element in the array so that the diffraction intensity spectrum of the light that passes through the spectral synthesis system shown in Fig. 2 and diffracts to a predetermined direction (

*α*=15°) is identical to a desired intensity spectrum. Based on the spectral range of interest and diffraction angle

*α*, we determine the width of each grating element to be 60 µm. The total length of the array is hence around 15.4 mm. The periods of the primary and auxiliary gratings are both set at 10 µm. All gratings are blazed at the center wavenumber

*u*

_{0}of the spectral band of interest, i.e. around 3922 cm

^{-1}. Under thin grating assumption, the efficiency terms in Eq. (20) are approximately given by:

*η*

^{2}(

*u*)=

*ξ*

^{2}(

*u*)=sinc

^{2}(1-

*u*/

*u*

_{0}) [21].

*ζ*is defined:

*I*(

*u*) is the intensity of the light for a specific wave number

_{p}*u*(typically the wave number corresponding to the peak of the target spectrum) passes through the grating-array-based spectral synthesis system and diffracts to the predetermined direction, and

_{p}*I*

_{0}(

*u*) is the intensity of the same incident light that passes through a double-mirror system and propagates to a direction with 0

_{p}^{0}diffraction angle.

*I*

_{0}(

*u*) can be estimated using Eq. (10) by setting

_{p}*η*(

*u*) to 1, diffraction angle

*α*to 0, and at the same time all displacements

*d*to 0. The value of

_{m}*ζ*in this case is about 1.1%.

^{-1}. We used the same spectral synthesis system to produce this spectrum and again, the design again took only a few seconds to be completed. The results shown in Fig. 8(b) give a

*ζ*value of 33.5%. From these two design examples, we noted that, as compared with the conventional micromirror-based spectral synthesis systems, the proposed system using an array of in-plane moving gratings leads to improved results with fewer elements needed in the array. In addition, with reduced memory requirements and shorter computational time, the optimization algorithm can easily handle a large number of sampling points within the spectral range of interest, hence making high-resolution spectral synthesis possible.

*α*are kept the same, the system parameters including the width of grating element, grating periods, and blazing wavelength remain unchanged. We simply increased the number of grating elements to 1024, resulting in a spectral resolution of about 0.6 cm

^{-1}. The optimization process took less than a minute and results are shown in Fig. 9. The value of

*ζ*obtained in this case is around 40%.

## 4. Conclusion

## Acknowledgments

## References and links

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

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

3. | T. A. Leskova and A. A. Maradudin, “Synthetic spectra from rough-surface scattering,” Opt. Lett. |

4. | M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express |

5. | H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. |

6. | M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. |

7. | R. Belikov and O. Solgaard, “Optical wavelength filtering by diffraction from a surface relief,” Opt. Lett. |

8. | G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express |

9. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

10. | G. Zhou and F. S. Chau, “Nondispersive optical phase shifter array using microelectromechanical systems based gratings,” Opt. Express 15, 10958–10963 (2007). [CrossRef] [PubMed] |

11. | J. W. Goodman, |

12. | Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Comm. |

13. | M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. |

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

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

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

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

18. | T. J. Suleski and D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication: I. Commercial slide imagers,” Appl. Opt. |

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

20. | M. A. Wolfe, |

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

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1970) Diffraction and gratings : Diffractive optics

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: January 23, 2008

Revised Manuscript: April 4, 2008

Manuscript Accepted: April 10, 2008

Published: June 5, 2008

**Citation**

Guangya Zhou and Fook Siong Chau, "Spectral synthesis using an array of micro
gratings," Opt. Express **16**, 9132-9143 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-9132

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### References

- 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]
- T. A. Leskova and A. A. Maradudin, "Synthetic spectra from rough-surface scattering," Opt. Lett. 30, 2784-2786 (2005). [CrossRef] [PubMed]
- M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, "Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors," Opt. Express 14, 12590-12612 (2006). [CrossRef] [PubMed]
- H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, "Micromechanical gratings for visible and near-infrared spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004). [CrossRef]
- M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, "Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens," IEEE Photon. Technol. Lett. 17, 2622-2624 (2005). [CrossRef]
- R. Belikov and O. Solgaard, "Optical wavelength filtering by diffraction from a surface relief," Opt. Lett. 28, 447-449 (2003). [CrossRef] [PubMed]
- G. Zhou, F. E. H. Tay, and F. S. Chau, "Design of the diffractive optical elements for synthetic spectra," Opt. Express 11, 1392-1399 (2003). [CrossRef] [PubMed]
- J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- G. Zhou and F. S. Chau, "Nondispersive optical phase shifter array using microelectromechanical systems based gratings," Opt. Express 15, 10958-10963 (2007). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics, 2nd Edition, (McGraw-Hill, New York, 1996), Chap. 4, 63-90.
- Y. M. Bruck and L. G. Sodin, "On the ambiguity of the image reconstruction problem," Opt. Comm. 30, 304-308 (1979). [CrossRef]
- M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, "Adaptive phase-distortion correction based on parallel gradient-descent optimization," Opt. Lett. 22, 907-909 (1997). [CrossRef] [PubMed]
- 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).
- 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]
- T. J. Suleski and D. C. O'Shea, "Gray-scale masks for diffractive-optics fabrication: I. Commercial slide imagers," Appl. Opt. 34, 7507-7517 (1995). [CrossRef] [PubMed]
- G. J. Swanson and W. B. Veldkamp, "Diffractive optical elements for use in infrared systems," Opt. Eng. 28, 605-608 (1989).
- M. A. Wolfe, Numerical Methods for Unconstrained Optimization: an Introduction, (Van Nostrand Reinhold Company Ltd., New York, 1978), Chap. 6, pp. 161-167.
- B. Kress and P. Meyrueis, Digital Diffractive Optics: An introduction to Planar Diffractive Optics and related technology, (John Wiley & Sons. Ltd, New York, 2000), Chap. 1, pp. 35- 42.

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