Abstract

The effect of sampling jitter induced by frequency fluctuations of the reference laser is analyzed theoretically. It is shown that the spectral broadening of the lines is small enough to permit the use of single-mode laser diodes in medium-to-high-resolution spaceborne instruments. The same mathematical formalism is used to give a new insight into the analysis of the spectral noise induced by random sampling jitter caused by detector and electronic noise.

© 1992 Optical Society of America

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References

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  1. H. Sakai, “Consideration of the signal-to-noise-ratio in Fourier spectroscopy,” in Proceedings of Aspen International Conference on Fourier Spectroscopy (Air Force Cambridge Research Laboratories, Bedford, Mass., 1971), pp. 19–41.
  2. E. E. Bell, R. B. Sanderson, “Spectral errors resulting from random sampling-position errors in Fourier transform spectroscopy,” Appl. Opt. 11, 688–689 (1972).
    [CrossRef] [PubMed]
  3. A. Zachor, “Drive nonlinearities: their effects in Fourier spectroscopy,” Appl. Opt. 16, 1412–1424 (1977).
    [CrossRef] [PubMed]
  4. B. Carli, F. Mencaraglia, A. Bonetti, “Sub-millimeter high-resolution FT spectrometer for atmospheric studies,” Appl. Opt. 23, 2594–2603 (1984).
    [CrossRef] [PubMed]
  5. G. Guelachvili, in Spectrometric Techniques, G. Vanasse, ed. (Academic, New York, 1981), Vol. 2, Chap. 1.
  6. W. Posselt, “Michelson interferometer for passive atmospheric sounding,” in Future European and Japanese Remote Sensing Sensors and Programs, P. N. Slater, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1490, 114–125 (1991).
  7. K. Petermann, Laser Diode Modulation and Noise (Kluwer, Dordrecht, The Netherlands, 1988).
    [CrossRef]
  8. Note that the calculation of Eq. (13) with the power spectral density of Eq. (20) leads to a result that is slightly different from Eq. (7.143) of Ref. 7.
  9. A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  10. It is assumed that the sampling pulses are generated at zero crossings of the laser interferogram. As a result, the SNR is defined by SNR = −20 log[sin(2πδx/λr)].

1984 (1)

1977 (1)

1972 (1)

Bell, E. E.

Bonetti, A.

Carli, B.

Guelachvili, G.

G. Guelachvili, in Spectrometric Techniques, G. Vanasse, ed. (Academic, New York, 1981), Vol. 2, Chap. 1.

Mencaraglia, F.

Oppenheim, A.

A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Petermann, K.

K. Petermann, Laser Diode Modulation and Noise (Kluwer, Dordrecht, The Netherlands, 1988).
[CrossRef]

Posselt, W.

W. Posselt, “Michelson interferometer for passive atmospheric sounding,” in Future European and Japanese Remote Sensing Sensors and Programs, P. N. Slater, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1490, 114–125 (1991).

Sakai, H.

H. Sakai, “Consideration of the signal-to-noise-ratio in Fourier spectroscopy,” in Proceedings of Aspen International Conference on Fourier Spectroscopy (Air Force Cambridge Research Laboratories, Bedford, Mass., 1971), pp. 19–41.

Sanderson, R. B.

Schafer, R.

A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Zachor, A.

Appl. Opt. (3)

Other (7)

G. Guelachvili, in Spectrometric Techniques, G. Vanasse, ed. (Academic, New York, 1981), Vol. 2, Chap. 1.

W. Posselt, “Michelson interferometer for passive atmospheric sounding,” in Future European and Japanese Remote Sensing Sensors and Programs, P. N. Slater, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1490, 114–125 (1991).

K. Petermann, Laser Diode Modulation and Noise (Kluwer, Dordrecht, The Netherlands, 1988).
[CrossRef]

Note that the calculation of Eq. (13) with the power spectral density of Eq. (20) leads to a result that is slightly different from Eq. (7.143) of Ref. 7.

A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

It is assumed that the sampling pulses are generated at zero crossings of the laser interferogram. As a result, the SNR is defined by SNR = −20 log[sin(2πδx/λr)].

H. Sakai, “Consideration of the signal-to-noise-ratio in Fourier spectroscopy,” in Proceedings of Aspen International Conference on Fourier Spectroscopy (Air Force Cambridge Research Laboratories, Bedford, Mass., 1971), pp. 19–41.

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Figures (1)

Fig. 1
Fig. 1

Apparent line shape of a monochromatic source at 2410 cm−1 in the presence of 1/f frequency noise (solid curve). A Lorentzian line shape of the same width (dashed curve) is shown for comparison. Spectral sampling interval: 1.57 MHz. A.U., arbitrary units.

Equations (36)

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L c = c π Δ ν > MPD             or             Δ ν < c π MPD ,
E ( t ) = S ( t ) exp { j [ ω t + δ Φ ( t ) ] } ,
I ( t ) = 2 S 0 cos [ v c ω t + Δ Φ ( τ ) ] ,
Δ Φ ( τ ) = δ Φ ( t ) - δ Φ ( t - τ )
t k = v λ r k 2
δ t k = - λ r 2 π v Δ Φ r ( τ k + δ τ k ) = - c ω r v Δ Φ r ( τ k + δ τ k ) ,
I k = cos [ v c ω s t k - ω s ω r Δ Φ r ( τ k + δ τ k ) ] .
δ Φ s ( τ ) = - ω s ω r δ Φ r ( τ ) .
δ Φ r ( t ) = 0 t ω ˙ r t d t = ω ˙ r t 2 2 ,
Δ Φ r ( τ ) = ω ˙ r τ 2 2 ( 2 c v - 1 )
Δ Φ s ( t , τ ) = - ω ˙ r ω s ω r τ 2 2 ( 2 c v - 1 ) .
ω ˙ s = - ω s ω r ω ˙ r .
Δ ω s = ω s ω r Δ ω r ,
Δ Φ ( τ ) 2 = τ 2 π 0 + W δ ω ( ω m ) sin 2 ( ω m τ / 2 ) ( ω m τ / 2 ) 2 d ω m ,
E ( t ) E * ( t - τ ) = S 0 exp [ - ½ Δ Φ ( τ ) 2 ] exp ( j ω τ ) ,
Δ ω r = W δ ω r ,
Δ Φ r ( τ ) 2 = W δ ω r τ .
Δ Φ s ( τ ) 2 = ( ω s ω r ) 2 W δ ω r τ .
Δ ω s = ( ω s ω r ) 2 W δ ω r = ( ω s ω r ) 2 Δ ω r
Δ ω s ω s = ω s ω r Δ ω r ω r .
W δ ω r ( ω m ) = ω N r 2 / ω m .
Δ Φ r ( τ ) 2 ( ω N r τ ) 2 π ln 2.52 ω c τ ,
Δ Φ s ( τ ) 2 = ( ω s ω r ) 2 ( ω N r τ ) 2 π ln 2.52 ω c τ ,
ω N s = ω s ω r ω N r .
[ Δ Φ r ( 0.67 × 10 - 9 s ) 2 ] 1 / 2 = 0.17 rad at 1300 nm ,
[ Δ Φ r ( τ ) 2 ] = Δ Φ .
E ( t ) E * ( t - τ ) = S 0 exp ( - ½ Δ Φ 2 ) .
X k = 1 N n = 0 N - 1 x n exp ( - j 2 π n k N ) , x n = k = 0 N - 1 X k exp ( + j 2 π n k N ) .
X k = S o ( δ k - k 0 + δ k - N + k 0 ) ,             0 < k < N - 1
I n = S 0 exp ( j 2 π n k 0 ) exp j Δ Φ n = S 0 exp ( j 2 π n k 0 ) exp ( - ½ Δ Φ 2 ) ,
γ n = I i I i - n * - I i I i - n * .
γ n = S 0 2 [ 1 - exp ( - Δ Φ 2 ) ] δ n ,
P k = 1 N S 0 2 [ 1 - exp ( - Δ Φ 2 ) ] .
k = 0 N - 1 X k 2 + k = 0 N - 1 P k = S 0 2 exp ( - Δ Φ 2 ) + S 0 2 [ 1 - exp ( - Δ Φ 2 ) ] = S 0 2 .
SNR = N exp ( - ½ Δ Φ 2 ) { [ 1 - exp ( - Δ Φ 2 ) ] } 1 / 2 .
( i = 1 M 1 N S 0 i 2 { 1 - exp [ - ( 2 π δ x / λ i ) 2 ] } ) 1 / 2 ( M N ) 1 / 2 ( 2 π δ x / λ ) S rms < 10 - 3 S rms .

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