## Lamellar grating optimization for miniaturized fourier transform spectrometers

Optics Express, Vol. 17, Issue 23, pp. 21289-21301 (2009)

http://dx.doi.org/10.1364/OE.17.021289

Acrobat PDF (613 KB)

### Abstract

Microfabricated Lamellar grating interferometers (LGI) require fewer components compared to Michelson interferotemeters and offer compact and broadband Fourier transform spectrometers (FTS) with good spectral resolution, high speed and high efficiency. This study presents the fundamental equations that govern the performance and limitations of LGI based FTS systems. Simulations and experiments were conducted to demonstrate and explain the periodic nature of the interferogram envelope due to Talbot image formation. Simulations reveal that the grating period should be chosen large enough to avoid Talbot phase reversal at the expense of mixing of the diffraction orders at the detector. Optimal LGI grating period selection depends on a number of system parameters and requires compromises in spectral resolution and signal-to-bias ratio (SBR) of the interferogram within the spectral range of interest. New analytical equations are derived for spectral resolution and SBR of LGI based FTS systems.

© 2009 OSA

## 1. Introduction

2. T. Sandner, C. Drabe, H. Schenk, A. Kenda, and W. Scherf, “Translatory MEMS actuators for optical path length modulation in miniaturized Fourier-transform infrared spectrometers,” MEMS MOEMS **7**(2), 021006 (
2008). [CrossRef]

3. J. Strong and G. A. Vanasse, “Lamellar grating far-infrared interferometer,” J. Opt. Soc. Am. **50**(2Issue 2), 113 (
1960). [CrossRef]

^{−1}spectral resolution.

## 2. LGI theory

*λ*

_{min}-

*λ*

_{max}] and in terms of the wavenumber

*k*= 1/

*λ*as [

*k*

_{max}–

*k*

_{min}]. The spectral resolution of the FTS depends on the distance that the moving part travels, ±

*d*, and how well the source is collimated. Assuming a point source which can be collimated perfectly, the spectral resolution is expressed with the well known equation [1]where

*Δk*is the spectral resolution, OPD stands for optical path difference between the two parts of the interferometer, and

**is the maximum deflection of the moving part of the LGI. The half divergence angle,**

*d**θ*

_{d}, is a measure of how well the source is collimated, which is expressed by the size of the source (

*D*

_{s}) or source aperture and the focal length of the first collimating mirror (

*f*

_{1}):

*θ*

_{d}) = 1 -

*θ*

_{d}

^{2}/2 brings the following criteria to the half divergence angle in order to achieve the spectral resolution depicted by Eq. (1) [1]:

*d*< T/2, which brings a new restriction to the grating period, such that:

## 3. FTS design case study

^{−1}and a spectrum of interest in the Mid-Wave Infrared (MWIR)

*λ*= [2.5um - 16um], or

*k*= [625cm

^{−1}- 4000cm

^{−1}], the following requirements can be derived:

## 4. Numerical simulations

### 4.1 Algorithm and interferogram results

*x.*Once the transmitted wave reaches the fixed grating, the diffracted pattern is multiplied by the amplitude function of the fixed grating and the reflected beam is propagated once again a distance of

*x.*Finally the resultant diffracted pattern is multiplied by the amplitude transmittance function of the movable grating and added with the initially reflected beam to form the reflected wave pattern. The detector intensity is then calculated after a Fraunhofer propagation, which is essentially a Fourier transformation followed by integration of the beam energy within the detector window. The procedure is repeated for all incidence angles in the range [-

*θ*

_{d}–

*θ*

_{d}], which are added together in incoherent (i.e., intensity) basis, and for all deflections in the range

*x*= [-

*d*– +

*d*] to form the interferogram, which is a function of interference with respect to deflection

*x*. The final spectrum is the modulus square of the Fourier transform of the interferogram.

*d*=

*n*λ/2 for 0th order diffracted light [Fig. 4(a) ] and

*d*= λ/4 +

*n*λ/2 for the 1st order diffracted light [Fig. 4(b)] where

*n*is any integer. Divergence of the source results in a shift on the order locations as illustrated in Fig. 4(c) for an intermediate point:

*d*=

*k*λ/2 + λ/8. The light is collected through an integration window that is tailored to include all 0th orders for the calculated half divergence angle of 2.5°. Once all 0th orders are collected, some of the 1st orders are also included within the integration window, lowering the light efficiency and fringe contrast. Effect of mixing of the orders and size of integration window on spectral resolution will be discussed further.

*E(x)*is a slowly varying envelope function effected by all the system parameters and will be discussed later. Amplitude of DC components is taken as bias and the amplitude of spectral component at excitation wavelength λ

_{0}is taken as the signal. Signal to Bias ratio (SBR) can be represented as

*sinc*function centered on λ

_{o}.

^{−1}which is close to the theoretical limit of 10cm

^{−1}for a displacement of 500um.

### 4.2 Optimization results

*θ*

_{d}= 0) and a finite size source with

*θ*

_{d}limited to 2.5°. The spectral resolution worsens at large wavelengths and small grating periods, where Talbot phase reversal distance is smaller than the mechanical displacement and interferograms with cyclic constrast variation are observed. Furthermore SBR is inversely proportional with the spectral resolution since SBR decreases due to Talbot phase reversals as well. The irregularities in the lower right parts of the figure are due to the large fluctuations in the width and the peak values of a multi-lobed spectrum [e.g. see Fig. 5(c) and Fig. 5(d)]. Those regions are not of interest due to poor resolution and low SBR performance.

### 4.3 Analytical formulas

*S(k)*is the Fourier transform of Eq. (10).

*S(k)*has a total of 7 dirac-delta terms. Those 3 terms of interest that are around the center wavenumber

*k*

_{o}= 1/λ

_{o}are given as below and illustrated in Fig. 8(b):

*k*

_{ο}, when Talbot period is small in compared to the displacement. This multi-peaking behavior was observed in Fig. 5(c) and Fig. 5(d).

## 5. Experimental results

*Λ =*40um in order to demonstrate the Talbot effect using a dynamic moving reflector placed underneath the grating. The moving platform is made using FR4 material and developed in our laboratory for FTS applications [7]. The distance of the FR4 platform to the grating was varied for 10 mm and the fringe contrast was recorded. Figure 10 illustrates the simulation and the experimental data of the interferogram envelope with respect to the distance between FR4 platform and the grating. Both simulated and experimental data agree perfectly with Talbot half period of

*Τ/2*= 2.5 mm. The experimental and simulation results are sensitive to incidence angle and the Talbot period changes rapidly if the incidence angle deviates from normal incidence in the direction perpendicular to the grating fingers.

3. J. Strong and G. A. Vanasse, “Lamellar grating far-infrared interferometer,” J. Opt. Soc. Am. **50**(2Issue 2), 113 (
1960). [CrossRef]

8. R. T. Hall, D. Vrabec, and J. M. Dowling, “A High-Resolution, Far Infrared Double-Beam Lamellar Grating Interferometer,” Appl. Opt. **5**(7), (
1966). [CrossRef] [PubMed]

9. R. L. Henry and D. B. Tanner, “A Lamellar Grating Interferometer for the Far-Infrared,” Infrared Phys. **19**(2), 163–174 (
1979). [CrossRef]

4. O. Manzardo, R. Michaely, F. Schädelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. **29**(13), 1437–1439 (
2004). [CrossRef] [PubMed]

10. F. Lee, G. Zhou, H. Yu, and F. S. Chau, “A MEMS-based resonant-scanning lamellar grating Fourier transform micro-spectrometer with laser reference system,” Sens. Actuators A Phys. **149**(2), 221–228 (
2009). [CrossRef]

## 6. Conclusion

*d*and waveband in order to achieve the optimal spectral resolution and SBR.

## Acknowledgment

## References and links

1. | V. Saptari, “Fourier-Transform Spectroscopy Instrumentation Engineering”, SPIE International Society for Optical Engineering, 2003. |

2. | T. Sandner, C. Drabe, H. Schenk, A. Kenda, and W. Scherf, “Translatory MEMS actuators for optical path length modulation in miniaturized Fourier-transform infrared spectrometers,” MEMS MOEMS |

3. | J. Strong and G. A. Vanasse, “Lamellar grating far-infrared interferometer,” J. Opt. Soc. Am. |

4. | O. Manzardo, R. Michaely, F. Schädelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. |

5. | C. Ataman, H. Urey, and A. Wolter, “MEMS-based Fourier Transform Spectrometer,” J. Micromechanics and Microengineering |

6. | J. W. Goodman, Introduction to Fourier Optics, Roberts & Company Publishers, 2005. |

7. | C. Ataman, H. Urey, “Compact Fourier Transform Spectrometers using FR4 Platform” SNA: A. Physical, A 151 (2009) 9–16. |

8. | R. T. Hall, D. Vrabec, and J. M. Dowling, “A High-Resolution, Far Infrared Double-Beam Lamellar Grating Interferometer,” Appl. Opt. |

9. | R. L. Henry and D. B. Tanner, “A Lamellar Grating Interferometer for the Far-Infrared,” Infrared Phys. |

10. | F. Lee, G. Zhou, H. Yu, and F. S. Chau, “A MEMS-based resonant-scanning lamellar grating Fourier transform micro-spectrometer with laser reference system,” Sens. Actuators A Phys. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(300.6290) Spectroscopy : Spectroscopy, four-wave mixing

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: August 14, 2009

Revised Manuscript: October 23, 2009

Manuscript Accepted: November 1, 2009

Published: November 6, 2009

**Citation**

Onur Ferhanoglu, Hüseyin R. Seren, Stephan Lüttjohann, and Hakan Urey, "Lamellar grating optimization for miniaturized fourier transform spectrometers," Opt. Express **17**, 21289-21301 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-23-21289

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### References

- V. Saptari, “Fourier-Transform Spectroscopy Instrumentation Engineering”, SPIE International Society for Optical Engineering, 2003.
- T. Sandner, C. Drabe, H. Schenk, A. Kenda, and W. Scherf, “Translatory MEMS actuators for optical path length modulation in miniaturized Fourier-transform infrared spectrometers,” MEMS MOEMS 7(2), 021006 (2008). [CrossRef]
- J. Strong and G. A. Vanasse, “Lamellar grating far-infrared interferometer,” J. Opt. Soc. Am. 50(2Issue 2), 113 (1960). [CrossRef]
- O. Manzardo, R. Michaely, F. Schädelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. 29(13), 1437–1439 (2004). [CrossRef] [PubMed]
- C. Ataman, H. Urey, and A. Wolter, “MEMS-based Fourier Transform Spectrometer,” J. Micromechanics and Microengineering 16, 2516–2523 (2006).
- J. W. Goodman, Introduction to Fourier Optics, Roberts & Company Publishers, 2005.
- C. Ataman, H. Urey, “Compact Fourier Transform Spectrometers using FR4 Platform” SNA: A. Physical, A 151 (2009) 9–16.
- R. T. Hall, D. Vrabec, and J. M. Dowling, “A High-Resolution, Far Infrared Double-Beam Lamellar Grating Interferometer,” Appl. Opt. 5(7), (1966). [CrossRef] [PubMed]
- R. L. Henry and D. B. Tanner, “A Lamellar Grating Interferometer for the Far-Infrared,” Infrared Phys. 19(2), 163–174 (1979). [CrossRef]
- F. Lee, G. Zhou, H. Yu, and F. S. Chau, “A MEMS-based resonant-scanning lamellar grating Fourier transform micro-spectrometer with laser reference system,” Sens. Actuators A Phys. 149(2), 221–228 (2009). [CrossRef]

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