Abstract

A report is given on the extension of the theory of frequency-modulation spectroscopy (FMS) to include multimode laser sources. First, the theory is developed analytically, including the effects of the laser mode intermodulation on the FMS signal. The theory is then investigated in the limit of weak absorption. Finally, the analytical results are evaluated in a number of cases through the use of a computer model that is able to simulate the FMS spectrum expected for arbitrary molecular and atomic absorption lines. The model demonstrates that absorption features that are narrow compared with the overall laser linewidth may still be discernible through the application of multimode laser FMS.

© 1991 Optical Society of America

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References

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  1. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15 (1980).
    [CrossRef] [PubMed]
  2. G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy—theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145 (1983).
    [CrossRef]
  3. T. F. Gallagher, R. Kachru, F. Gounand, G. C. Bjorklund, W. Lenth, “Frequency-modulation spectroscopy with a pulsed dye laser,” Opt. Lett. 7, 28 (1982).
    [CrossRef] [PubMed]
  4. D. E. Cooper, T. F. Gallagher, “Frequency-modulation spectroscopy with a multimode laser,” Opt. Lett. 9, 451 (1984).
    [CrossRef] [PubMed]
  5. G. R. Janik, C. B. Carlisle, T. F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 3, 1070 (1986).
    [CrossRef]
  6. M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 2, 1510 (1985).
    [CrossRef]
  7. J. D. Simmons, J. T. Hougen, “Atlas of the I2spectrum from 19,000 to 18,000 cm−1,” J. Res. Nat. Bur. Stand. A 81, 25 (1977).
    [CrossRef]

1986 (1)

1985 (1)

M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 2, 1510 (1985).
[CrossRef]

1984 (1)

1983 (1)

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy—theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145 (1983).
[CrossRef]

1982 (1)

1980 (1)

1977 (1)

J. D. Simmons, J. T. Hougen, “Atlas of the I2spectrum from 19,000 to 18,000 cm−1,” J. Res. Nat. Bur. Stand. A 81, 25 (1977).
[CrossRef]

Bjorklund, G. C.

M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 2, 1510 (1985).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy—theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145 (1983).
[CrossRef]

T. F. Gallagher, R. Kachru, F. Gounand, G. C. Bjorklund, W. Lenth, “Frequency-modulation spectroscopy with a pulsed dye laser,” Opt. Lett. 7, 28 (1982).
[CrossRef] [PubMed]

G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15 (1980).
[CrossRef] [PubMed]

Carlisle, C. B.

Cooper, D. E.

Gallagher, T. F.

Gehrtz, M.

M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 2, 1510 (1985).
[CrossRef]

Gounand, F.

Hougen, J. T.

J. D. Simmons, J. T. Hougen, “Atlas of the I2spectrum from 19,000 to 18,000 cm−1,” J. Res. Nat. Bur. Stand. A 81, 25 (1977).
[CrossRef]

Janik, G. R.

Kachru, R.

Lenth, W.

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy—theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145 (1983).
[CrossRef]

T. F. Gallagher, R. Kachru, F. Gounand, G. C. Bjorklund, W. Lenth, “Frequency-modulation spectroscopy with a pulsed dye laser,” Opt. Lett. 7, 28 (1982).
[CrossRef] [PubMed]

Levenson, M. D.

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy—theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145 (1983).
[CrossRef]

Ortiz, C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy—theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145 (1983).
[CrossRef]

Simmons, J. D.

J. D. Simmons, J. T. Hougen, “Atlas of the I2spectrum from 19,000 to 18,000 cm−1,” J. Res. Nat. Bur. Stand. A 81, 25 (1977).
[CrossRef]

Whittaker, E. A.

M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 2, 1510 (1985).
[CrossRef]

Appl. Phys. B (1)

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy—theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145 (1983).
[CrossRef]

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

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

M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B. 2, 1510 (1985).
[CrossRef]

J. Res. Nat. Bur. Stand. A (1)

J. D. Simmons, J. T. Hougen, “Atlas of the I2spectrum from 19,000 to 18,000 cm−1,” J. Res. Nat. Bur. Stand. A 81, 25 (1977).
[CrossRef]

Opt. Lett. (3)

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Figures (3)

Fig. 1
Fig. 1

FMS scans of a single Lorentzian absorption line for decreasing values of ωrt. The values of ωrt are (a) 20, (b) 1.2, and (c) 0.6, in units where Γ = 1. In all three plots, ωm = 0.1, A = 0.01, and there are 31 modes with P±N/2 = P0/2.

Fig. 2
Fig. 2

Simulated portion of the iodine spectrum. A 1001-mode laser is used; ωm = 0.3 and ωrt = 0.2, in units where Γ = 1.

Fig. 3
Fig. 3

Signal for large ωm, with the laser linewidth dominating. A 101-mode laser is used; ωm = 10, ωrt = 0.2, Γl = 20, and σ = 8.5, all in units where Γ = 1.

Equations (35)

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E = Re ( n = - N / 2 N / 2 E n exp { i [ ( ω c + n ω rt ) t + ϕ n ] } ) ,
E pm = Re [ E exp ( i M sin ω m t ) ] = Re [ E j = - j = J j ( M ) exp ( i j ω m t ) ] ,
E pm = Re ( n = - N / 2 n = n / 2 j = - E n J j ( M ) × exp { i [ ω c + n ω rt + j ω m ) t + ϕ n ] } ) .
T n , j = exp ( i ϕ n , j - δ n , j ) ,
ω n , j = ω c + n ω rt + j ω m .
E FMS = Re { n , j E n J j ( M ) T n , j exp [ i ( ω n , j t + ϕ n ) ] } .
I = c 8 π E FMS 2 = c 8 π n , j n , j E n E n J j ( M ) J j ( M ) T n , j T n , j * × exp { i [ ( ω n , j - ω n , j ) t + ( ϕ n - ϕ n ) ] } .
exp [ i ( ϕ n - ϕ n ) ] t = 1 T 0 T exp { i [ ϕ n ( t ) - ϕ n ( t ) ] } d t = δ n , n ,
I mm = c 8 π n , j , j E n 2 J j ( M ) J j ( M ) T n , j T n , j * × exp [ i ( ω n , j - ω n , j ' ) t ] .
I ml = c 8 π j , j , n , n { E n E n exp [ i ( ϕ n - ϕ n ) ] } × J j ( M ) J j ( M ) T n , j T n , j * exp [ i ( ω n , j - ω n , j ' ) t ] .
ω n , j - ω n , j = ( n - n ) ω rt + ( j - j ) ω m ;
I = c 8 π n E n 2 [ j = j ± 1 J j ( M ) J j ( M ) T n , j T n , j * exp ( ± i ω m t ) ] .
T n , j T n , j * = exp ( i ϕ n , j - ϕ n , j ) exp ( - i ϕ n , j - δ n , j ) 1 + i ( ϕ n , j - ϕ n , j ) - ( δ n , j + δ n , j ) .
I = c 8 π n E n 2 j J j ( M ) J j + 1 ( M ) [ 1 + ( ϕ n , j - ϕ n , j + 1 ) × sin ω m t - ( δ n , j + δ n , j + 1 ) cos ω m t ] .
I ˜ ( ω c ) = M n c E n 2 8 π ( δ n , - 1 - δ n , 1 ) .
Γ l = N ω rt .
P n M ( c E n 2 / 8 π ) = exp [ - ( ω n - ω c ) 2 / 2 σ 2 ] .
δ ( ω ) = k = 1 K A k Γ k 2 ( ω - Ω k ) 2 + Γ k 2 .
I ˜ sm = P 0 [ δ ( ω - ω m ) - δ ( ω + ω m ) ] .
I ˜ sm = 2 P 0 ω m { [ δ ( ω c - ω m ) - δ ( ω c + ω m ) ] / 2 ω m } - 2 P 0 ω m d δ d ω .
I ˜ sm - 2 P 0 ω m d d ω c [ A Γ 2 ( ω c - Ω 1 ) 2 + Γ 2 ] = 4 P 0 ω m A Γ 2 ( ω c - Ω 1 ) [ ( ω c - Ω 1 ) 2 + Γ 2 ] 2 .
ω rt Γ ,
I ˜ max , ω r t Γ = 3 3 4 A P 0 ω m Γ ,
I ˜ mm = modes I ˜ n .
I ˜ mm = modes ( - 2 P n ω m ) d δ ( ω n ) d ω .
I ˜ mm = - 2 ω m P ± N / 2 modes d δ ( ω n ) d ω ,
I ˜ mm ( ω c ) - 2 ω m P ± N / 2 ω r t mode - N / 2 mode + N / 2 d δ d ω d ω = 2 ω m P ± N / 2 ω rt [ δ ( ω c - Γ l 2 ) - δ ( ω c + Γ l 2 ) ] .
I mm ( ω c ) 2 ω m P ± N / 2 A Γ 2 ω rt × [ 1 ( ω c - Γ l / 2 ) 2 + Γ 2 - 1 ( ω c + Γ l / 2 ) 2 + Γ 2 ] .
I ˜ max , ω rt < Γ 2 ω m P ± N / 2 A / ω rt ,
I ˜ ( ω c ) = n P n ( δ n , - 1 - δ n , 1 ) = n P n [ δ ( ω c + n ω rt - ω m ) - δ ( ω c + n ω rt + ω m ) ] .
I ˜ ( ω c ) = 1 ω rt P ( ω ) [ δ ( ω c + ω - ω m ) - δ ( ω c + ω + ω m ) ] d ω .
P ( ω ) = { exp [ - ( ω - ω c ) 2 / 2 σ 2 ] - Γ l / 2 < ω < Γ l / 2 0 otherwise .
I ˜ ( ω c ) = ( π A Γ / ω rt ) [ P ( ω m - ω c ) - P ( - ω m - ω c ) ] .
F small ω m = 8 3 3 P N / 2 P 0 Γ ω rt ,
F large ω m = π P 0 P 0 Γ ω rt .

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