Abstract

In this paper we study the effect of material dispersion on the performance of a moving-optical-wedge Fourier transform spectrometer. The spectrum is thus evaluated numerically using a test spectrum source. The obtained numerical results show that the classical technique for numerical dispersion compensation, usually used with a Michelson interferometer, cannot be efficiently used with wedge interferometers as it is limited to the cases of weak dispersion. The error in this technique is thus evaluated in different cases and a new numerical technique is proposed to overcome this error. We also notice shrinkage in the interferogram spread in the spatial domain in contradiction with the normal dispersion effect in a Michelson interferometer.

© 2009 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 Spectroscopy, 2nd ed. (Wiley, 2007).
    [CrossRef]
  2. Q. Yang, R. Zhou, and B. Zhao, “Principle and analysis of the moving-optical-wedge interferometer,” Appl. Opt. 47, 2186-2191 (2008).
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    [CrossRef] [PubMed]
  5. T. P. Sheahen, “Use of chirping to compensate for nonlinearities in Fourier spectroscopy,” J. Opt. Soc. Am. 64, 485-493 (1974).
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  6. M. L. Forman, W. H. Steel, and G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 56, 59-63 (1966).
    [CrossRef]
  7. R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, and M. C. Abrams, “ Phase correction of emission line Fourier spectroscopy,” J. Opt. Soc. Am. 12, 2165-2171 (1995).
    [CrossRef]
  8. H. Sakai, G. A. Vanasse, and M. L. Forman, “Spectral recovery in Fourier spectroscopy,” J. Opt. Soc. Am. 58, 84-90(1968).
    [CrossRef]
  9. P. Hlubina, “White-light spectral interferometry with the uncompensated Michelson interferometer and the group refractive index dispersion in fused silica,”Opt. Commun. 193, 1-7 (2001).
    [CrossRef]

2008

2001

P. Hlubina, “White-light spectral interferometry with the uncompensated Michelson interferometer and the group refractive index dispersion in fused silica,”Opt. Commun. 193, 1-7 (2001).
[CrossRef]

1995

R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, and M. C. Abrams, “ Phase correction of emission line Fourier spectroscopy,” J. Opt. Soc. Am. 12, 2165-2171 (1995).
[CrossRef]

1975

1974

1968

1966

Abrams, M. C.

R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, and M. C. Abrams, “ Phase correction of emission line Fourier spectroscopy,” J. Opt. Soc. Am. 12, 2165-2171 (1995).
[CrossRef]

Brault, J. W.

R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, and M. C. Abrams, “ Phase correction of emission line Fourier spectroscopy,” J. Opt. Soc. Am. 12, 2165-2171 (1995).
[CrossRef]

de Haseth, J. A.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy, 2nd ed. (Wiley, 2007).
[CrossRef]

Forman, M. L.

Griffiths, P. R.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy, 2nd ed. (Wiley, 2007).
[CrossRef]

Hlubina, P.

P. Hlubina, “White-light spectral interferometry with the uncompensated Michelson interferometer and the group refractive index dispersion in fused silica,”Opt. Commun. 193, 1-7 (2001).
[CrossRef]

Learner, R. C. M.

R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, and M. C. Abrams, “ Phase correction of emission line Fourier spectroscopy,” J. Opt. Soc. Am. 12, 2165-2171 (1995).
[CrossRef]

Sakai, H.

Sheahen, T. P.

Steel, W. H.

Thorne, A. P.

R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, and M. C. Abrams, “ Phase correction of emission line Fourier spectroscopy,” J. Opt. Soc. Am. 12, 2165-2171 (1995).
[CrossRef]

Vanasse, G. A.

Wynne-Jones, I.

R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, and M. C. Abrams, “ Phase correction of emission line Fourier spectroscopy,” J. Opt. Soc. Am. 12, 2165-2171 (1995).
[CrossRef]

Yang, Q.

Zhao, B.

Zhou, R.

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Figures (10)

Fig. 1
Fig. 1

Michelson interferometer.

Fig. 2
Fig. 2

Moving-optical-wedge interferometer.

Fig. 3
Fig. 3

Refractive index as a function of wavelength using the Sellmeier equation and the proposed approximate equation.

Fig. 4
Fig. 4

Source spectrum and recovered spectrum for a specific case.

Fig. 5
Fig. 5

Interferogram as a function of lateral displacement of the wedge for different values of dispersion using single Gaussian source spectrum: (a)  G = 0 , (b)  G = 1 , (c)  G = 60 .

Fig. 6
Fig. 6

Interferogram as a function of lateral displacement of the wedge for different values of dispersion using double Gaussian source spectrum: (a)  G = 0 , (b)  G = 1 , (c)  G = 60 . (d) Envelope of the interferogram versus the optical path difference.

Fig. 7
Fig. 7

Error and overlap integral as a function of dispersion strength: (a) error, (b) overlap.

Fig. 8
Fig. 8

Error and overlap integral as a function of height of second Gaussian: (a) error, (b) overlap. (c) Recovered spectrum versus the source spectrum in the case of h = 0.5 , where the first method is used to recover the spectrum.

Fig. 9
Fig. 9

Error and overlap integral as a function of the separation between Gaussians: (a) error, (b) overlap. (c) Recovered spectrum versus the source spectrum in the special case when σ 1 σ 0 = 800 cm 1 . The first method is used to recover the spectrum. Note that first method refers to the application of the conventional dispersion treatment when applied on the wedge spectrometer while the second method refers to the new proposed algorithm detailed in Section 4.

Fig. 10
Fig. 10

Flow chart for the proposed second method for dispersion compensation.

Equations (29)

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σ = 0.5 Step in OPD ,
I ( x ) = 2 0 B ( σ ) cos 2 π x σ d σ ,
I ( x ) = B ( σ ) cos 2 π x σ d σ .
B ( σ ) = 2 0 I ( x ) cos 2 π σ x d x .
I ( x ) = B ( σ ) cos [ 2 π x σ + φ ( σ ) ] d σ .
B ( σ ) e j φ ( σ ) = J ( σ ) = I ( x ) e 2 j π σ x d x ,
B ( σ ) = | J ( σ ) | = [ J r 2 ( σ ) + J i 2 ( σ ) ] 1 / 2 ,
x = 2 L sin α ( n 2 sin 2 α cos α ) .
f = x L = 2 sin α ( n 2 sin 2 α cos α ) .
f 2 sin α ( n sin 2 α 2 n cos α ) ,
n ( λ ) = n 0 + Δ ( λ ) ,
f = f 0 + f 1 ,
f 0 = 2 sin α { n 0 sin 2 α 2 n 0 cos α } ,
f 1 = 2 sin α { 1 + sin 2 α 2 n 0 2 } Δ ( λ ) ,
x = f L ( f 0 + f 1 ) L = x 0 + x 1 .
n ( λ ) = 1 + i = 1 3 A i λ 2 λ 2 B i 2 ,
n ( λ ) = n 0 + C 1 λ λ 0 δ + C 2 ( λ λ 0 ) 2 δ 2 = n 0 + Δ ( λ ) ,
φ ( σ ) = 2 π σ x 1 ( σ ) = 2 π σ f 1 ( σ ) L .
σ k = k span of x 0 = k Nb f 0 ,
B ( σ ) = e ( σ σ 0 ) 2 s 0 2 + he ( σ σ 1 ) 2 s 1 2 ,
Overlap = ( | S ( σ ) B ( σ ) | ) 2 ( | S ( σ ) | 2 ) ( | B ( σ ) | 2 ) ,
Error = | B S | 2 | B | 2 ,
I ( L ) = B ( σ ) cos [ 2 π σ f ( σ ) L ] d σ .
I ( l ) e 2 j π u L d L = B ( σ ) d σ cos [ 2 π σ f ( σ ) L ] e 2 j π u L d L = 0.5 B ( σ ) d σ [ e 2 π σ f ( σ ) L + e 2 π σ f ( σ ) L ] e 2 j π u L d L = 0.5 B ( σ ) d σ { δ [ u σ f ( σ ) ] + δ [ u + σ f ( σ ) ] } .
I ( l ) e 2 j π u L d L = 0.5 B [ g ( ε ) ] { δ ( u ε ) + δ ( u + ε ) } g ( ε ) d ε = 0.5 { B [ g ( u ) ] g ( u ) + B [ g ( u ) ] g ( u ) } = B [ g ( u ) ] g ( u ) .
B ( σ ) = I ( L ) e 2 j π u L d L g ( u ) .
k Span of lateral displacement ε < k + 1 Span of lateral displacement .
F k = n = 0 N 1 I n e 2 j π n k N .
B ( σ ) = | F ε d ε d σ | .

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