Abstract

Experimental data with multiple overlapping spectral lines must often be fitted to theoretical line shape models. This paper describes an efficient program for performing least-squares fits of such data to Galatry and Voigt profiles. The algorithm and program design considerations are presented in detail, and some examples are given to demonstrate its use. The procedure described in this paper may also be used for more complex line shape profiles.

© 1989 Optical Society of America

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References

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  1. P. L. Varghese, R. K. Hanson, “Collisional Narrowing Effect on Spectral Line Shapes Measured at High Resolution,” Appl. Opt. 23, 2376 (1984).
    [CrossRef] [PubMed]
  2. L. Galatry, “Simultaneous Effect of Doppler and Foreign Gas Broadening on Spectral Lines,” Phys. Rev. 122, 1218 (1961).
    [CrossRef]
  3. P. L. Varghese, “Tunable Infrared Diode Laser Measurements of Spectral Parameters of Carbon Monoxide and Hydrogen Cyanide,” Report 6-83-T, HTGL, Stanford U., Stanford, CA (1983).
  4. F. Herbert, “Spectrum Line Profiles: a Generalized Voigt Function Including Narrowing,” J. Quant. Spectrosc. Radiat. Transfer 14, 943 (1977).
    [CrossRef]
  5. J. J. More, “The Levenberg-Marquardt Algorithm: Implementation and Theory, Numerical Analysis,” Lect. Notes Math. 630, 106 (1977).
  6. C-T. Chen, One Dimensional Digital Signal Processing (Marcel Dekker, New York, 1979).
  7. B. C. Garbow, K. E. Hillstrom, J. J. More, MINPACK Project, Argonne National Laboratory (1980).
  8. D. S. Cline, P. L. Varghese, “High Resolution Spectral Measurements in the ν5 Band of Formaldehyde Using a Tunable IR Diode Laser,” Appl. Opt. 27, 3219 (1988).
    [CrossRef] [PubMed]
  9. P. L. Varghese, R. K. Hanson, “Tunable Infrared Diode Laser Measurements of Spectral Parameters of HCN at Room Temperature,” J. Quant. Spectrosc. Radiat. Transfer 31, 545 (1984).
    [CrossRef]
  10. R. A. Toth, L. R. Brown, R. H. Hunt, “Line Positions and Strengths of Methane in the 2862 to 3000 cm−1 Region,” J. Mol. Spectrosc. 67, 1 (1977).
    [CrossRef]
  11. D. L. Gray, A. G. Robiette, A. S. Pine, “Extended Measurement and Analysis of the ν3 Infrared Band of Methane,” J. Mol. Spectrosc. 77, 440 (1979).
    [CrossRef]
  12. D. S. Cline, P. L. Varghese, “Temperature Dependent Parameters of Formaldehyde and Methane,” in preparation.

1988 (1)

1984 (2)

P. L. Varghese, R. K. Hanson, “Tunable Infrared Diode Laser Measurements of Spectral Parameters of HCN at Room Temperature,” J. Quant. Spectrosc. Radiat. Transfer 31, 545 (1984).
[CrossRef]

P. L. Varghese, R. K. Hanson, “Collisional Narrowing Effect on Spectral Line Shapes Measured at High Resolution,” Appl. Opt. 23, 2376 (1984).
[CrossRef] [PubMed]

1979 (1)

D. L. Gray, A. G. Robiette, A. S. Pine, “Extended Measurement and Analysis of the ν3 Infrared Band of Methane,” J. Mol. Spectrosc. 77, 440 (1979).
[CrossRef]

1977 (3)

R. A. Toth, L. R. Brown, R. H. Hunt, “Line Positions and Strengths of Methane in the 2862 to 3000 cm−1 Region,” J. Mol. Spectrosc. 67, 1 (1977).
[CrossRef]

F. Herbert, “Spectrum Line Profiles: a Generalized Voigt Function Including Narrowing,” J. Quant. Spectrosc. Radiat. Transfer 14, 943 (1977).
[CrossRef]

J. J. More, “The Levenberg-Marquardt Algorithm: Implementation and Theory, Numerical Analysis,” Lect. Notes Math. 630, 106 (1977).

1961 (1)

L. Galatry, “Simultaneous Effect of Doppler and Foreign Gas Broadening on Spectral Lines,” Phys. Rev. 122, 1218 (1961).
[CrossRef]

Brown, L. R.

R. A. Toth, L. R. Brown, R. H. Hunt, “Line Positions and Strengths of Methane in the 2862 to 3000 cm−1 Region,” J. Mol. Spectrosc. 67, 1 (1977).
[CrossRef]

Chen, C-T.

C-T. Chen, One Dimensional Digital Signal Processing (Marcel Dekker, New York, 1979).

Cline, D. S.

Galatry, L.

L. Galatry, “Simultaneous Effect of Doppler and Foreign Gas Broadening on Spectral Lines,” Phys. Rev. 122, 1218 (1961).
[CrossRef]

Garbow, B. C.

B. C. Garbow, K. E. Hillstrom, J. J. More, MINPACK Project, Argonne National Laboratory (1980).

Gray, D. L.

D. L. Gray, A. G. Robiette, A. S. Pine, “Extended Measurement and Analysis of the ν3 Infrared Band of Methane,” J. Mol. Spectrosc. 77, 440 (1979).
[CrossRef]

Hanson, R. K.

P. L. Varghese, R. K. Hanson, “Tunable Infrared Diode Laser Measurements of Spectral Parameters of HCN at Room Temperature,” J. Quant. Spectrosc. Radiat. Transfer 31, 545 (1984).
[CrossRef]

P. L. Varghese, R. K. Hanson, “Collisional Narrowing Effect on Spectral Line Shapes Measured at High Resolution,” Appl. Opt. 23, 2376 (1984).
[CrossRef] [PubMed]

Herbert, F.

F. Herbert, “Spectrum Line Profiles: a Generalized Voigt Function Including Narrowing,” J. Quant. Spectrosc. Radiat. Transfer 14, 943 (1977).
[CrossRef]

Hillstrom, K. E.

B. C. Garbow, K. E. Hillstrom, J. J. More, MINPACK Project, Argonne National Laboratory (1980).

Hunt, R. H.

R. A. Toth, L. R. Brown, R. H. Hunt, “Line Positions and Strengths of Methane in the 2862 to 3000 cm−1 Region,” J. Mol. Spectrosc. 67, 1 (1977).
[CrossRef]

More, J. J.

J. J. More, “The Levenberg-Marquardt Algorithm: Implementation and Theory, Numerical Analysis,” Lect. Notes Math. 630, 106 (1977).

B. C. Garbow, K. E. Hillstrom, J. J. More, MINPACK Project, Argonne National Laboratory (1980).

Pine, A. S.

D. L. Gray, A. G. Robiette, A. S. Pine, “Extended Measurement and Analysis of the ν3 Infrared Band of Methane,” J. Mol. Spectrosc. 77, 440 (1979).
[CrossRef]

Robiette, A. G.

D. L. Gray, A. G. Robiette, A. S. Pine, “Extended Measurement and Analysis of the ν3 Infrared Band of Methane,” J. Mol. Spectrosc. 77, 440 (1979).
[CrossRef]

Toth, R. A.

R. A. Toth, L. R. Brown, R. H. Hunt, “Line Positions and Strengths of Methane in the 2862 to 3000 cm−1 Region,” J. Mol. Spectrosc. 67, 1 (1977).
[CrossRef]

Varghese, P. L.

D. S. Cline, P. L. Varghese, “High Resolution Spectral Measurements in the ν5 Band of Formaldehyde Using a Tunable IR Diode Laser,” Appl. Opt. 27, 3219 (1988).
[CrossRef] [PubMed]

P. L. Varghese, R. K. Hanson, “Collisional Narrowing Effect on Spectral Line Shapes Measured at High Resolution,” Appl. Opt. 23, 2376 (1984).
[CrossRef] [PubMed]

P. L. Varghese, R. K. Hanson, “Tunable Infrared Diode Laser Measurements of Spectral Parameters of HCN at Room Temperature,” J. Quant. Spectrosc. Radiat. Transfer 31, 545 (1984).
[CrossRef]

P. L. Varghese, “Tunable Infrared Diode Laser Measurements of Spectral Parameters of Carbon Monoxide and Hydrogen Cyanide,” Report 6-83-T, HTGL, Stanford U., Stanford, CA (1983).

D. S. Cline, P. L. Varghese, “Temperature Dependent Parameters of Formaldehyde and Methane,” in preparation.

Appl. Opt. (2)

J. Mol. Spectrosc. (2)

R. A. Toth, L. R. Brown, R. H. Hunt, “Line Positions and Strengths of Methane in the 2862 to 3000 cm−1 Region,” J. Mol. Spectrosc. 67, 1 (1977).
[CrossRef]

D. L. Gray, A. G. Robiette, A. S. Pine, “Extended Measurement and Analysis of the ν3 Infrared Band of Methane,” J. Mol. Spectrosc. 77, 440 (1979).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

F. Herbert, “Spectrum Line Profiles: a Generalized Voigt Function Including Narrowing,” J. Quant. Spectrosc. Radiat. Transfer 14, 943 (1977).
[CrossRef]

P. L. Varghese, R. K. Hanson, “Tunable Infrared Diode Laser Measurements of Spectral Parameters of HCN at Room Temperature,” J. Quant. Spectrosc. Radiat. Transfer 31, 545 (1984).
[CrossRef]

Lect. Notes Math. (1)

J. J. More, “The Levenberg-Marquardt Algorithm: Implementation and Theory, Numerical Analysis,” Lect. Notes Math. 630, 106 (1977).

Phys. Rev. (1)

L. Galatry, “Simultaneous Effect of Doppler and Foreign Gas Broadening on Spectral Lines,” Phys. Rev. 122, 1218 (1961).
[CrossRef]

Other (4)

P. L. Varghese, “Tunable Infrared Diode Laser Measurements of Spectral Parameters of Carbon Monoxide and Hydrogen Cyanide,” Report 6-83-T, HTGL, Stanford U., Stanford, CA (1983).

D. S. Cline, P. L. Varghese, “Temperature Dependent Parameters of Formaldehyde and Methane,” in preparation.

C-T. Chen, One Dimensional Digital Signal Processing (Marcel Dekker, New York, 1979).

B. C. Garbow, K. E. Hillstrom, J. J. More, MINPACK Project, Argonne National Laboratory (1980).

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Figures (2)

Fig. 1
Fig. 1

(a) Galatry fit to synthetic data on ten spectral lines. The crosses represent the data, the solid line is computed from the best-fit parameters. Input line parameters are listed in Table IV, parameters extracted from the fit are listed in Table V. (b) Residual error from the curve fit showing random noise and no detectable systematic error.

Fig. 2
Fig. 2

(a) Galatry fit to high resolution spectroscopic data on methane between 2863.00 and 2863.18 cm−1 obtained with a tunable diode laser. The four prominent lines are part of the P(15) manifold of the ν3 fundamental band and were identified in Ref. 10; the weak line was not cataloged. The data were obtained using pure methane at 25.1 Torr and 296 K in a 20-cm absorption cell. (b) Residual error from the curve fit. No systematic residual error is detectable.

Tables (6)

Tables Icon

Table I Error Bound of galfft

Tables Icon

Table III CPU Time on a CDC Dual Cyber 170/750 Computer for Execution of galfft

Tables Icon

Table IV Synthetic Data with Ten Spectral Lines; 500 Data Points

Tables Icon

Table V Summary of Fitted Parameters from galfft with Synthetic Data of Ten Spectral Lines with ±2.5% Noise (Fig. 1)

Tables Icon

Table VI Galatry Fit of Absorption Data on the P+(15) Manifold of CH4 Shown in Fig. 2; 153 Data Points

Equations (20)

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F = i = 1 N L C i ϕ i ,
G ( x , y , z ) = 1 π Re { 0 Φ ( y , z , τ ) exp ( i x τ ) d τ } = 1 π Re { F [ Φ ( y , z , τ ) ] } .
Φ ( y , z , τ ) = exp { y τ + 1 2 z 2 [ 1 z τ exp ( z τ ) ] } ,
Φ V ( y , τ ) = lim z 0 Φ ( y , z , τ ) = exp ( y τ τ 2 4 ) .
F [ Φ ( y , z , τ ) ] 0 T Φ ( y , z , τ ) exp ( i x τ ) d τ n = 0 N 1 f n exp ( i x n Δ τ ) = F ( x ) ,
Δ τ T N 1 ,
f 0 = 0 0 . 5 Δ τ Φ ( y , z , τ ) d τ Δ τ 6 [ 2 Φ ( 0 ) + Φ ( 0 . 5 Δ τ ) ] ,
f n = ( n 0 . 5 ) Δ τ ( n + 0 . 5 ) Δ τ Φ ( y , z , τ ) d τ Δ τ 6 { Φ [ ( n 0 . 5 ) Δ τ ] + 4 Φ ( n Δ τ ) + Φ [ ( n + 0 . 5 ) Δ τ ] } .
F ( x k ) = n 0 N 1 f n exp ( i 2 π k n / N ) , k = 0 , 1 , 2 , , N 1 ,
( n ) G ( x , y , z ) x n = 1 π Re { F [ Φ x n ( y , z , τ ) ] } , with Φ x n ( y , z , τ ) ( i τ ) n Φ ( y , z , τ ) ,
( n ) G ( x , y , z ) y n = 1 π Re { F [ Φ y n ( y , z , τ ) ] } , with Φ y n ( y , z , τ ) ( τ ) n Φ ( y , z , τ ) ,
( n ) G ( x , y , z ) z n = 1 π Re { F [ Φ z n ( y , z , τ ) ] } , with Φ z n ( y , z , τ ) ( n ) Φ ( y , z , τ ) z n ,
( n ) G ( x , y , z ) x n = { 1 π Im { F [ Φ y n ( y , z , τ ) ] } n = 4 m + 1 ( n ) G ( x , y , z ) y n n = 4 m + 2 1 π Im { F [ Φ y n ( y , z , τ ) ] } n = 4 m + 3 ( n ) G ( x , y , z ) y n n = 4 m + 4 ,
T = { C 1 y + C 2 y if y > 1 ( C 1 + C 2 ) y 1 / 3 if Y c < y 1 Y c 1 / 3 if y Y c
x 10 + 6 y = X T max .
T limit = { ln ( ε y ) + 2 y if y < 1 ln ε + 2 y if y 1
| x i x o | > X T max ,
x k < x max + 2 π T .
F = i = 1 N L C i G ( x x 0 i , y i , z )
F υ = i = 1 N L C i V ( x x 0 i , y i )

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