Abstract

We study the effect of diffraction on the performance of microelectromechanical system Michelson interferometers. By using a simple Gaussian model, we calculate the degradation of the interferometer visibility due to the diffraction effect. We then use this model to estimate the optimum detector diameter to maximize the fringe visibility at the interferometer output and study its effect on the resolution of Fourier transform spectrometers based on Michelson interferometers.

© 2010 Optical Society of America

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References

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  1. P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy2nd ed. (Wiley, 2007).
    [CrossRef]
  2. M. Kraft, A. Kenda, T. Sandner, and H. Schenk, “MEMS-based compact FT-spectrometers—a platform for spectroscopic mid-infrared sensors,” IEEE Sens. J. 130–133 (2008).
    [CrossRef]
  3. D. Reyes, E. R. Schildkraut, J. Kim, R. F. Connors, P. Kotidis, and D. J. Cavicchio, “A novel method of creating a surface micromachined 3-D optical assembly for MEMS-based, miniaturized FTIR spectrometers,” Proc. SPIE 6888, 68880D(2008).
    [CrossRef]
  4. O. Manzardo, Y. Petremand, H. P. Herzig, W. Noell, and N. De Rooij, “Micro-fabricated system for wavelengths monitoring,” presented at the Conference on Optical MEMS and Their Applications, Lugano, Switzerland, 20–23 August 2002.
  5. B. Saadany, D. Khalil and T. Bourouina, “System, method and apparatus for a micromachined interferometer using optical splitting,” European patent EPO 1906159 B1 (3 February 2010).
  6. B. Saadany, H. Omran, M. Medhat, F. Marty, D. Khalil, and T. Bourouina, “MEMS tunable Michelson interferometer with robust beam splitting architecture,” presented at the IEEE LEOS Optical MEMS and Nanophotonics Conference, Clearwater Beach, Fla., USA, 17–20 August 2009.
  7. D. Khalil, H. Omran, M. Medhat, and B. Saadany, “Miniaturized tunable integrated Mach-Zehnder MEMS interferometer for spectrometer applications,” Proc. SPIE 7594, 75940T (2010)..
  8. K. Zhang and D.Li, Electromagnetic Theory for Microwaves and Optoelectronics, 2nd ed. (Springer, 2008).
  9. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  10. K. Aït-Ameur and F. Sanchez, “Transverse effects as source of error in coherence length measurements,” Opt. Commun. 233, 39–43 (2004).
    [CrossRef]

2008

M. Kraft, A. Kenda, T. Sandner, and H. Schenk, “MEMS-based compact FT-spectrometers—a platform for spectroscopic mid-infrared sensors,” IEEE Sens. J. 130–133 (2008).
[CrossRef]

D. Reyes, E. R. Schildkraut, J. Kim, R. F. Connors, P. Kotidis, and D. J. Cavicchio, “A novel method of creating a surface micromachined 3-D optical assembly for MEMS-based, miniaturized FTIR spectrometers,” Proc. SPIE 6888, 68880D(2008).
[CrossRef]

2004

K. Aït-Ameur and F. Sanchez, “Transverse effects as source of error in coherence length measurements,” Opt. Commun. 233, 39–43 (2004).
[CrossRef]

Aït-Ameur, K.

K. Aït-Ameur and F. Sanchez, “Transverse effects as source of error in coherence length measurements,” Opt. Commun. 233, 39–43 (2004).
[CrossRef]

Bourouina, T.

B. Saadany, H. Omran, M. Medhat, F. Marty, D. Khalil, and T. Bourouina, “MEMS tunable Michelson interferometer with robust beam splitting architecture,” presented at the IEEE LEOS Optical MEMS and Nanophotonics Conference, Clearwater Beach, Fla., USA, 17–20 August 2009.

B. Saadany, D. Khalil and T. Bourouina, “System, method and apparatus for a micromachined interferometer using optical splitting,” European patent EPO 1906159 B1 (3 February 2010).

Cavicchio, D. J.

D. Reyes, E. R. Schildkraut, J. Kim, R. F. Connors, P. Kotidis, and D. J. Cavicchio, “A novel method of creating a surface micromachined 3-D optical assembly for MEMS-based, miniaturized FTIR spectrometers,” Proc. SPIE 6888, 68880D(2008).
[CrossRef]

Connors, R. F.

D. Reyes, E. R. Schildkraut, J. Kim, R. F. Connors, P. Kotidis, and D. J. Cavicchio, “A novel method of creating a surface micromachined 3-D optical assembly for MEMS-based, miniaturized FTIR spectrometers,” Proc. SPIE 6888, 68880D(2008).
[CrossRef]

D.,

K. Zhang and D.Li, Electromagnetic Theory for Microwaves and Optoelectronics, 2nd ed. (Springer, 2008).

de Haseth, J. A.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy2nd ed. (Wiley, 2007).
[CrossRef]

De Rooij, N.

O. Manzardo, Y. Petremand, H. P. Herzig, W. Noell, and N. De Rooij, “Micro-fabricated system for wavelengths monitoring,” presented at the Conference on Optical MEMS and Their Applications, Lugano, Switzerland, 20–23 August 2002.

Griffiths, P. R.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy2nd ed. (Wiley, 2007).
[CrossRef]

Herzig, H. P.

O. Manzardo, Y. Petremand, H. P. Herzig, W. Noell, and N. De Rooij, “Micro-fabricated system for wavelengths monitoring,” presented at the Conference on Optical MEMS and Their Applications, Lugano, Switzerland, 20–23 August 2002.

Kenda, A.

M. Kraft, A. Kenda, T. Sandner, and H. Schenk, “MEMS-based compact FT-spectrometers—a platform for spectroscopic mid-infrared sensors,” IEEE Sens. J. 130–133 (2008).
[CrossRef]

Khalil, D.

B. Saadany, D. Khalil and T. Bourouina, “System, method and apparatus for a micromachined interferometer using optical splitting,” European patent EPO 1906159 B1 (3 February 2010).

D. Khalil, H. Omran, M. Medhat, and B. Saadany, “Miniaturized tunable integrated Mach-Zehnder MEMS interferometer for spectrometer applications,” Proc. SPIE 7594, 75940T (2010)..

B. Saadany, H. Omran, M. Medhat, F. Marty, D. Khalil, and T. Bourouina, “MEMS tunable Michelson interferometer with robust beam splitting architecture,” presented at the IEEE LEOS Optical MEMS and Nanophotonics Conference, Clearwater Beach, Fla., USA, 17–20 August 2009.

Kim, J.

D. Reyes, E. R. Schildkraut, J. Kim, R. F. Connors, P. Kotidis, and D. J. Cavicchio, “A novel method of creating a surface micromachined 3-D optical assembly for MEMS-based, miniaturized FTIR spectrometers,” Proc. SPIE 6888, 68880D(2008).
[CrossRef]

Kotidis, P.

D. Reyes, E. R. Schildkraut, J. Kim, R. F. Connors, P. Kotidis, and D. J. Cavicchio, “A novel method of creating a surface micromachined 3-D optical assembly for MEMS-based, miniaturized FTIR spectrometers,” Proc. SPIE 6888, 68880D(2008).
[CrossRef]

Kraft, M.

M. Kraft, A. Kenda, T. Sandner, and H. Schenk, “MEMS-based compact FT-spectrometers—a platform for spectroscopic mid-infrared sensors,” IEEE Sens. J. 130–133 (2008).
[CrossRef]

Li,

K. Zhang and D.Li, Electromagnetic Theory for Microwaves and Optoelectronics, 2nd ed. (Springer, 2008).

Manzardo, O.

O. Manzardo, Y. Petremand, H. P. Herzig, W. Noell, and N. De Rooij, “Micro-fabricated system for wavelengths monitoring,” presented at the Conference on Optical MEMS and Their Applications, Lugano, Switzerland, 20–23 August 2002.

Marty, F.

B. Saadany, H. Omran, M. Medhat, F. Marty, D. Khalil, and T. Bourouina, “MEMS tunable Michelson interferometer with robust beam splitting architecture,” presented at the IEEE LEOS Optical MEMS and Nanophotonics Conference, Clearwater Beach, Fla., USA, 17–20 August 2009.

Medhat, M.

B. Saadany, H. Omran, M. Medhat, F. Marty, D. Khalil, and T. Bourouina, “MEMS tunable Michelson interferometer with robust beam splitting architecture,” presented at the IEEE LEOS Optical MEMS and Nanophotonics Conference, Clearwater Beach, Fla., USA, 17–20 August 2009.

D. Khalil, H. Omran, M. Medhat, and B. Saadany, “Miniaturized tunable integrated Mach-Zehnder MEMS interferometer for spectrometer applications,” Proc. SPIE 7594, 75940T (2010)..

Noell, W.

O. Manzardo, Y. Petremand, H. P. Herzig, W. Noell, and N. De Rooij, “Micro-fabricated system for wavelengths monitoring,” presented at the Conference on Optical MEMS and Their Applications, Lugano, Switzerland, 20–23 August 2002.

Omran, H.

B. Saadany, H. Omran, M. Medhat, F. Marty, D. Khalil, and T. Bourouina, “MEMS tunable Michelson interferometer with robust beam splitting architecture,” presented at the IEEE LEOS Optical MEMS and Nanophotonics Conference, Clearwater Beach, Fla., USA, 17–20 August 2009.

D. Khalil, H. Omran, M. Medhat, and B. Saadany, “Miniaturized tunable integrated Mach-Zehnder MEMS interferometer for spectrometer applications,” Proc. SPIE 7594, 75940T (2010)..

Petremand, Y.

O. Manzardo, Y. Petremand, H. P. Herzig, W. Noell, and N. De Rooij, “Micro-fabricated system for wavelengths monitoring,” presented at the Conference on Optical MEMS and Their Applications, Lugano, Switzerland, 20–23 August 2002.

Reyes, D.

D. Reyes, E. R. Schildkraut, J. Kim, R. F. Connors, P. Kotidis, and D. J. Cavicchio, “A novel method of creating a surface micromachined 3-D optical assembly for MEMS-based, miniaturized FTIR spectrometers,” Proc. SPIE 6888, 68880D(2008).
[CrossRef]

Saadany, B.

B. Saadany, H. Omran, M. Medhat, F. Marty, D. Khalil, and T. Bourouina, “MEMS tunable Michelson interferometer with robust beam splitting architecture,” presented at the IEEE LEOS Optical MEMS and Nanophotonics Conference, Clearwater Beach, Fla., USA, 17–20 August 2009.

D. Khalil, H. Omran, M. Medhat, and B. Saadany, “Miniaturized tunable integrated Mach-Zehnder MEMS interferometer for spectrometer applications,” Proc. SPIE 7594, 75940T (2010)..

B. Saadany, D. Khalil and T. Bourouina, “System, method and apparatus for a micromachined interferometer using optical splitting,” European patent EPO 1906159 B1 (3 February 2010).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Sanchez, F.

K. Aït-Ameur and F. Sanchez, “Transverse effects as source of error in coherence length measurements,” Opt. Commun. 233, 39–43 (2004).
[CrossRef]

Sandner, T.

M. Kraft, A. Kenda, T. Sandner, and H. Schenk, “MEMS-based compact FT-spectrometers—a platform for spectroscopic mid-infrared sensors,” IEEE Sens. J. 130–133 (2008).
[CrossRef]

Schenk, H.

M. Kraft, A. Kenda, T. Sandner, and H. Schenk, “MEMS-based compact FT-spectrometers—a platform for spectroscopic mid-infrared sensors,” IEEE Sens. J. 130–133 (2008).
[CrossRef]

Schildkraut, E. R.

D. Reyes, E. R. Schildkraut, J. Kim, R. F. Connors, P. Kotidis, and D. J. Cavicchio, “A novel method of creating a surface micromachined 3-D optical assembly for MEMS-based, miniaturized FTIR spectrometers,” Proc. SPIE 6888, 68880D(2008).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Zhang, K.

K. Zhang and D.Li, Electromagnetic Theory for Microwaves and Optoelectronics, 2nd ed. (Springer, 2008).

IEEE Sens. J.

M. Kraft, A. Kenda, T. Sandner, and H. Schenk, “MEMS-based compact FT-spectrometers—a platform for spectroscopic mid-infrared sensors,” IEEE Sens. J. 130–133 (2008).
[CrossRef]

Opt. Commun.

K. Aït-Ameur and F. Sanchez, “Transverse effects as source of error in coherence length measurements,” Opt. Commun. 233, 39–43 (2004).
[CrossRef]

Proc. SPIE

D. Reyes, E. R. Schildkraut, J. Kim, R. F. Connors, P. Kotidis, and D. J. Cavicchio, “A novel method of creating a surface micromachined 3-D optical assembly for MEMS-based, miniaturized FTIR spectrometers,” Proc. SPIE 6888, 68880D(2008).
[CrossRef]

Other

O. Manzardo, Y. Petremand, H. P. Herzig, W. Noell, and N. De Rooij, “Micro-fabricated system for wavelengths monitoring,” presented at the Conference on Optical MEMS and Their Applications, Lugano, Switzerland, 20–23 August 2002.

B. Saadany, D. Khalil and T. Bourouina, “System, method and apparatus for a micromachined interferometer using optical splitting,” European patent EPO 1906159 B1 (3 February 2010).

B. Saadany, H. Omran, M. Medhat, F. Marty, D. Khalil, and T. Bourouina, “MEMS tunable Michelson interferometer with robust beam splitting architecture,” presented at the IEEE LEOS Optical MEMS and Nanophotonics Conference, Clearwater Beach, Fla., USA, 17–20 August 2009.

D. Khalil, H. Omran, M. Medhat, and B. Saadany, “Miniaturized tunable integrated Mach-Zehnder MEMS interferometer for spectrometer applications,” Proc. SPIE 7594, 75940T (2010)..

K. Zhang and D.Li, Electromagnetic Theory for Microwaves and Optoelectronics, 2nd ed. (Springer, 2008).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy2nd ed. (Wiley, 2007).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Michelson interferometer.

Fig. 2
Fig. 2

Radii of the first fringe, the moving arm beam, and the reference arm beam as functions of OPD. (a) Initial spot size = 30 μm and (b) initial spot size = 75 μm .

Fig. 3
Fig. 3

Radii of the first fringe, the moving arm beam, and the reference arm beam as functions of initial spot size for different values of OPD. (a) OPD = 500 μm and (b) OPD = 2000 μm .

Fig. 4
Fig. 4

Plot of the intensity of light received at the detector as a function of transverse dimensions. Dimensions are in micrometers. (a) Initial spot size = 10 μm , OPD = 1000 μm ; (b) initial spot size = 15 μm , OPD = 1000 μm ; (c) initial spot size = 30 μm , OPD = 1000 μm ; and (d) initial spot size = 30 μm , OPD = 500 μm .

Fig. 5
Fig. 5

Visibility as a function of OPD for different spot sizes. Spot sizes are 30, 50, 75, and 100 μm .

Fig. 6
Fig. 6

Visibility as a function of OPD for different detector radii. Detector radii are 30, 70, 100, and 200 μm .

Fig. 7
Fig. 7

Visibility as a function of detector radius for different initial spot sizes. Initial spot sizes are 30, 50, 75, and 100 μm .

Fig. 8
Fig. 8

Visibility as a function of detector radius for different values of OPD: OPD = 500 , 1000, 2000, 3000 μm .

Fig. 9
Fig. 9

Visibility as a function of wavelength for different initial spot sizes. Initial spot sizes are 30, 50, 75, and 100 μm . Detector radius = 400 μm and OPD = 2000 μm .

Fig. 10
Fig. 10

Visibility as a function of wavelength for different detector radii. Detector radii are 50, 100, and 300 μm . Spot size = 30 μm and OPD = 2000 μm .

Fig. 11
Fig. 11

Plot showing the variation of deduced spectral width δ ν versus angular divergence θ (in milliradians) of the Gaussian beam. Three curves are based on our analysis with detector radii of 70, 100, and 200 μm . The fourth curve (drawn as small squares) is based on the analysis of [10], which assumes infinite detector radius.

Equations (20)

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u = A exp { j [ p ( z ) + k ρ 2 2 q ( z ) ] } ,
q = z + q 0 ,
p = j ln ( z + q 0 ) .
q 0 = z 0 + j s ,
2 π u u * ρ d ρ = 1 ,
A = j k s π .
. ψ = k s π ( j z + j s ) exp { j k ρ 2 2 ( z + j s ) } e j k z .
z r = 2 z R + z D ,
z m = z r + Δ z = 2 z R + z D + Δ z .
ψ m = k s π j z m + j s exp { j k ρ 2 2 ( z m + j s ) } e j k z m ,
ψ r = k s π j z r + j s exp { j k ρ 2 2 ( z r + j s ) } e j k z r ,
ϕ m = k z m + k ρ 2 2 z m z m 2 + s 2 tan 1 z m s ,
ϕ r = k z r + k ρ 2 2 z r z r 2 + s 2 tan 1 z r s .
Δ ϕ = k Δ z + k ρ 2 2 { z m z m 2 + s 2 z r z r 2 + s 2 } + tan 1 z r s tan 1 z m s .
Δ ϕ = ± n π ,
z m z m 2 + s 2 = z r z r 2 + s 2 ,
s = z m z r .
V = I M a x I M i n I M a x + I M i n ,
δ υ = c ln 2 π Δ z 0.5 ,
θ = λ π W 0 ,

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