Abstract

Frequency-modulation (FM) spectroscopy is known to be a sensitive spectroscopic technique capable of accurately measuring the frequency dependence of the absorption and index of refraction of narrow spectral features. The absorption and index of refraction are coupled by a form of the Kramers–Kronig (K-K) relations, and both components provide information about the spectral feature. In this Letter, we propose a processing technique based on fitting the data to a complex signal model derived from the K-K relation. By using this complex constraint and only processing a single quadrature, our model predicts significant improvement in the minimum detectable absorption compared with conventional FM spectroscopy.

© 2011 Optical Society of America

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References

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  1. G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, Appl. Phys. B 32, 145 (1983).
    [CrossRef]
  2. F. Devaux, Y. Sorel, and J. Kerdiles, J. Lightwave Technol. 11, 1937 (1993).
    [CrossRef]
  3. J. Goodman, Statistical Optics (Wiley, 2000).
  4. M. Gehrtz, G. C. Bjorklund, and E. A. Whittaker, J. Opt. Soc. Am. B 2, 1510 (1985).
    [CrossRef]

1993 (1)

F. Devaux, Y. Sorel, and J. Kerdiles, J. Lightwave Technol. 11, 1937 (1993).
[CrossRef]

1985 (1)

1983 (1)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, Appl. Phys. B 32, 145 (1983).
[CrossRef]

Bjorklund, G. C.

M. Gehrtz, G. C. Bjorklund, and E. A. Whittaker, J. Opt. Soc. Am. B 2, 1510 (1985).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, Appl. Phys. B 32, 145 (1983).
[CrossRef]

Devaux, F.

F. Devaux, Y. Sorel, and J. Kerdiles, J. Lightwave Technol. 11, 1937 (1993).
[CrossRef]

Gehrtz, M.

Goodman, J.

J. Goodman, Statistical Optics (Wiley, 2000).

Kerdiles, J.

F. Devaux, Y. Sorel, and J. Kerdiles, J. Lightwave Technol. 11, 1937 (1993).
[CrossRef]

Lenth, W.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, Appl. Phys. B 32, 145 (1983).
[CrossRef]

Levenson, M. D.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, Appl. Phys. B 32, 145 (1983).
[CrossRef]

Ortiz, C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, Appl. Phys. B 32, 145 (1983).
[CrossRef]

Sorel, Y.

F. Devaux, Y. Sorel, and J. Kerdiles, J. Lightwave Technol. 11, 1937 (1993).
[CrossRef]

Whittaker, E. A.

Appl. Phys. B (1)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, Appl. Phys. B 32, 145 (1983).
[CrossRef]

J. Lightwave Technol. (1)

F. Devaux, Y. Sorel, and J. Kerdiles, J. Lightwave Technol. 11, 1937 (1993).
[CrossRef]

J. Opt. Soc. Am. B (1)

Other (1)

J. Goodman, Statistical Optics (Wiley, 2000).

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Figures (4)

Fig. 1
Fig. 1

Experimental arrangement for measuring the spectral feature of a sample (thick lines represent the optical path, and thin lines are the electrical connections).

Fig. 2
Fig. 2

Frequency domain illustration of the proposed spectroscopy technique.

Fig. 3
Fig. 3

(a) Plot of the constraint line showing system parameters. (b) Pictorial representation of the phase-shift standard deviation before projection, σ Δ ϕ , and after projection, σ Δ ϕ c .

Fig. 4
Fig. 4

Scaled minimum detectable phase shift as a function of the scaled modulation frequency. Δ Ω is the HWHM of the absorption line. (The dashed line is an estimate of the minimum practical detectability floor for a given Δ Ω .)

Equations (11)

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I 1 = I 0 [ 1 + M cos ( ω m t ) ] ,
E 1 ( t ) = I 0 exp ( j ω c t ) [ M 4 exp ( j ω m t ) + 1 + M 4 exp ( j ω m t ) ] ,
E 2 ( t ) = I 0 exp ( j ω c t ) [ T 1 M 4 exp ( j ω m t ) + T 0 + T 1 M 4 exp ( j ω m t ) ] .
S ( t ) = [ S I + n I ( t ) ] cos ( ω m t ) + [ S Q + n Q ( t ) ] sin ( ω m t ) ,
S I = A cos ( Δ ϕ ) , S Q = A sin ( Δ ϕ ) ,
SNR = Δ ϕ 2 ¯ σ Δ ϕ 2 = A 2 Δ ϕ 2 ¯ σ 2 = e 2 ( δ 1 + δ 0 ) R 2 P 0 2 M 2 Δ ϕ 2 ¯ 2 e R P 0 Δ f + ( 4 k T R ) Δ f ,
Δ ϕ min = [ 2 e Δ f e 2 ( δ 1 + δ 0 ) R P 0 M 2 ] 1 / 2 .
σ Δ ϕ c 2 = σ 2 A 4 b 2 1 + a 2 .
σ Δ ϕ c 2 σ Δ ϕ 2 sin 2 ( ψ ) .
Δ ϕ min c = | sin ( ψ ) | Δ ϕ min .
ψ ( ω m ) = tan 1 [ ϕ ( ω c + ω m ) ϕ ( ω c ) δ ( ω c + ω m ) + δ ( ω c ) ] .

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