Abstract

The object of the work reported in this paper was to find a simple and easy to calculate approximation to the Voigt function using the Padé method. To do this we calculated the multipole approximation to the complex function as the error function or as the plasma dispersion function. This generalized Lorentzian approximation can be used instead of the exact function in experiments that do not require great accuracy.

© 1981 Optical Society of America

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References

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  1. A. Mitchell, M. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. P., London, 1971), pp. 93–103.
  2. Z. Kucerovsky, E. Brannen, D. G. Rumbold, W. J. Sarjeant, Appl. Opt. 12, 226 (1973).
    [CrossRef] [PubMed]
  3. U. List, “Druckabhangigkeit der Absorption von CO-Laser-Linien durch NO,” Thesis, Institut für Angewandte Physik, Universitat Bonn, Feb.1976.
  4. M. W. Zemansky, Phys. Rev. 36, 219 (1930).
    [CrossRef]
  5. A. Reichel, J. Quant. Spectrosc. Radiat. Transfer 8, 1601 (1968).
    [CrossRef]
  6. H. A. Farach, H. Teitlbaum, Can. J. Phys. 45, 2913 (1967).
    [CrossRef]
  7. J. F. Kielkopf, J. Opt. Soc. Am. 63, 987 (1973).
    [CrossRef]
  8. G. K. Wertheim, M. A. Butler, K. W. West, D. N. Buchanan, Rev. Sci. Instrum. 45, 1369 (1974).
    [CrossRef]
  9. P. Martín, M. A. González, Phys. Fluids 22, 1413 (1979).
    [CrossRef]
  10. P. Martín, G. Donoso, J. Zamudio-Cristi, J. Math. Phys. 21, 280 (1980).
    [CrossRef]
  11. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 296–328.
  12. B. D. Fried, S. Conte, The Plasma Dispersion Function (Academic, New York, 1961).
  13. P. Martín, J. Zamudio-Cristi, G. Donoso, J. Math. Phys. 21, 1332 (May1980).
    [CrossRef]

1980 (2)

P. Martín, G. Donoso, J. Zamudio-Cristi, J. Math. Phys. 21, 280 (1980).
[CrossRef]

P. Martín, J. Zamudio-Cristi, G. Donoso, J. Math. Phys. 21, 1332 (May1980).
[CrossRef]

1979 (1)

P. Martín, M. A. González, Phys. Fluids 22, 1413 (1979).
[CrossRef]

1974 (1)

G. K. Wertheim, M. A. Butler, K. W. West, D. N. Buchanan, Rev. Sci. Instrum. 45, 1369 (1974).
[CrossRef]

1973 (2)

1968 (1)

A. Reichel, J. Quant. Spectrosc. Radiat. Transfer 8, 1601 (1968).
[CrossRef]

1967 (1)

H. A. Farach, H. Teitlbaum, Can. J. Phys. 45, 2913 (1967).
[CrossRef]

1930 (1)

M. W. Zemansky, Phys. Rev. 36, 219 (1930).
[CrossRef]

Brannen, E.

Buchanan, D. N.

G. K. Wertheim, M. A. Butler, K. W. West, D. N. Buchanan, Rev. Sci. Instrum. 45, 1369 (1974).
[CrossRef]

Butler, M. A.

G. K. Wertheim, M. A. Butler, K. W. West, D. N. Buchanan, Rev. Sci. Instrum. 45, 1369 (1974).
[CrossRef]

Conte, S.

B. D. Fried, S. Conte, The Plasma Dispersion Function (Academic, New York, 1961).

Donoso, G.

P. Martín, J. Zamudio-Cristi, G. Donoso, J. Math. Phys. 21, 1332 (May1980).
[CrossRef]

P. Martín, G. Donoso, J. Zamudio-Cristi, J. Math. Phys. 21, 280 (1980).
[CrossRef]

Farach, H. A.

H. A. Farach, H. Teitlbaum, Can. J. Phys. 45, 2913 (1967).
[CrossRef]

Fried, B. D.

B. D. Fried, S. Conte, The Plasma Dispersion Function (Academic, New York, 1961).

González, M. A.

P. Martín, M. A. González, Phys. Fluids 22, 1413 (1979).
[CrossRef]

Kielkopf, J. F.

Kucerovsky, Z.

List, U.

U. List, “Druckabhangigkeit der Absorption von CO-Laser-Linien durch NO,” Thesis, Institut für Angewandte Physik, Universitat Bonn, Feb.1976.

Martín, P.

P. Martín, G. Donoso, J. Zamudio-Cristi, J. Math. Phys. 21, 280 (1980).
[CrossRef]

P. Martín, J. Zamudio-Cristi, G. Donoso, J. Math. Phys. 21, 1332 (May1980).
[CrossRef]

P. Martín, M. A. González, Phys. Fluids 22, 1413 (1979).
[CrossRef]

Mitchell, A.

A. Mitchell, M. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. P., London, 1971), pp. 93–103.

Reichel, A.

A. Reichel, J. Quant. Spectrosc. Radiat. Transfer 8, 1601 (1968).
[CrossRef]

Rumbold, D. G.

Sarjeant, W. J.

Teitlbaum, H.

H. A. Farach, H. Teitlbaum, Can. J. Phys. 45, 2913 (1967).
[CrossRef]

Wertheim, G. K.

G. K. Wertheim, M. A. Butler, K. W. West, D. N. Buchanan, Rev. Sci. Instrum. 45, 1369 (1974).
[CrossRef]

West, K. W.

G. K. Wertheim, M. A. Butler, K. W. West, D. N. Buchanan, Rev. Sci. Instrum. 45, 1369 (1974).
[CrossRef]

Zamudio-Cristi, J.

P. Martín, J. Zamudio-Cristi, G. Donoso, J. Math. Phys. 21, 1332 (May1980).
[CrossRef]

P. Martín, G. Donoso, J. Zamudio-Cristi, J. Math. Phys. 21, 280 (1980).
[CrossRef]

Zemansky, M.

A. Mitchell, M. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. P., London, 1971), pp. 93–103.

Zemansky, M. W.

M. W. Zemansky, Phys. Rev. 36, 219 (1930).
[CrossRef]

Appl. Opt. (1)

Can. J. Phys. (1)

H. A. Farach, H. Teitlbaum, Can. J. Phys. 45, 2913 (1967).
[CrossRef]

J. Math. Phys. (2)

P. Martín, G. Donoso, J. Zamudio-Cristi, J. Math. Phys. 21, 280 (1980).
[CrossRef]

P. Martín, J. Zamudio-Cristi, G. Donoso, J. Math. Phys. 21, 1332 (May1980).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

A. Reichel, J. Quant. Spectrosc. Radiat. Transfer 8, 1601 (1968).
[CrossRef]

Phys. Fluids (1)

P. Martín, M. A. González, Phys. Fluids 22, 1413 (1979).
[CrossRef]

Phys. Rev. (1)

M. W. Zemansky, Phys. Rev. 36, 219 (1930).
[CrossRef]

Rev. Sci. Instrum. (1)

G. K. Wertheim, M. A. Butler, K. W. West, D. N. Buchanan, Rev. Sci. Instrum. 45, 1369 (1974).
[CrossRef]

Other (4)

A. Mitchell, M. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. P., London, 1971), pp. 93–103.

U. List, “Druckabhangigkeit der Absorption von CO-Laser-Linien durch NO,” Thesis, Institut für Angewandte Physik, Universitat Bonn, Feb.1976.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 296–328.

B. D. Fried, S. Conte, The Plasma Dispersion Function (Academic, New York, 1961).

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Figures (3)

Fig. 1
Fig. 1

Algebraic maximum and minimum absolute errors vs p. The dashed lines denote the one generalized Lorentzian approximation. The dot and dashed lines are used for the two Lorentzian approximation. Appropriated scale factors are used.

Fig. 2
Fig. 2

Algebraic maximum and minimum absolute errors vs p and the corresponding exact Voigt of value for the two generalized Lorentzian approximation.

Fig. 3
Fig. 3

Algebraic maximum and minimum absolute errors vs d. The dashed lines denote the one Lorentzian approximation; the dot and dashed lines are used for the two Lorentzian cases.

Tables (2)

Tables Icon

Table I Absolute Error [V1(d/p) − V1(d,p)] × 100 for One Generalized Lorentzian Approximation

Tables Icon

Table II Absolute Error [V2(d,p) − V1(d,p)] × 1000 for Two Generalized Lorentzian Approximations

Equations (32)

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k ( ν ) k 0 = p π - exp ( - y 2 ) p 2 + ( d - y 2 ) d y V ( d , p ) ,
d = D b = ν - ν 0 b ; p = α / b ,
b = Δ ν D ( 2 ln 2 ) - 1 / 2 ,
k ( ν ) = k 0 exp ( - d ) 2 .
k ( ν ) = S π · α α 2 + D 2 = k 0 p π 1 p 2 + d 2 ,
z = d + i p = D b + i α b ,
V ( d , p ) = Re V ( z ) = Re [ w ( i z ) ] = 2 π Re ( exp z 2 erfcz ) ,
V ( z ) = w ( i z ) = m = 0 ( - 1 ) m z m Γ ( m 2 + 1 ) = 1 - 2 π z + z 2 - 4 3 π z 3 + z 4 2 ,
V ( z ) = w ( i z ) = 1 π [ 1 z + 1 z m = 1 ( - 1 ) m 1 × 3 × ( 2 m - 1 ) ( 2 z 2 ) m ] = 1 π z - 1 2 z 3 ± ,
V n ( z ) = i = 0 n - 1 p i z k 1 + l = 1 n q l z l
V n ( z ) = j = 1 n c j z - a j .
V 1 ( z ) = 1 1 + π .
V ( z ) = 1 i π Z ( i z ) .
V 2 ( z ) = ( π - 2 ) π + ( 4 - π ) z ( π - 2 ) π + π z + ( 1 - π ) π z 2 = c 2 z - a 2 + c 2 * z - a 2 * ,
a 2 = - π 2 ( 4 - π ) + i ( 23 π - 32 - 4 π 2 ) 1 / 2 2 ( 4 - π ) ,
c 2 = 1 2 π - i 2 π - 5 2 ( 23 π - 32 - 4 π 2 ) 1 / 2 .
a j = α j + i β j ,
c j = γ j + i δ j ,
V n ( d , p ) = Re V n ( z ) = j = 1 n γ j ( p - α j ) + δ j ( d - β j ) ( p - α j ) 2 + ( d - β j ) 2 .
V 1 ( d , p ) = 1 π ( p + 1 π ) ( p + 1 π ) 2 + d 2 .
V 1 ( D b , α b ) = 1 π b + 1 π b 2 ( α - 1 π b ) 2 + D 2 .
lim b 0 V 1 ( D b , α b ) = b π α α 2 + D 2 = 1 π P p 2 + d 2 .
V 1 ( d ) = lim α 0 V 1 ( D b , α b ) = 1 π 1 ( D b ) 2 + 1 π = 1 π d 2 + 1 ,
V 2 ( d , p ) = γ 2 ( p - α 2 ) + δ 2 ( p - β 2 ) ( p - α 2 ) 2 + ( d - β 2 ) 2 + γ 2 ( p - α 2 ) - δ 2 ( d + β 2 ) ( p - α 2 ) 2 + ( d + β 2 ) 2 ,
α 2 = - π 2 ( 4 - π ) = - 1.0324 ,
β 2 = ( 23 π - 32 - 4 π 2 ) 1 / 2 2 ( 4 - π ) = 0.5138 ,
γ 2 = 1 2 π = 2.8210 ,
δ 2 = - 2 π - 5 2 ( 23 π - 32 - 4 π 2 ) 1 / 2 = - 0.7273.
V 2 ( D b , α b ) = b γ 2 ( α - α 2 b ) + δ 2 ( D - β 2 b ) ( α - α 2 b ) 2 + ( D - β 2 b ) 2 + b γ 2 ( α - α 2 b ) - δ 2 ( D + β 2 b ) ( α - α 2 b ) 2 + ( D + β 2 b ) 2 .
lim b 0 V 2 ( D b , α b ) = b 2 γ 2 α α 2 + D 2 = 1 π p p 2 + d 2 ,
lim α 0 V 2 ( D b , α b ) = b δ 2 ( D - β 2 b ) - α 2 γ 2 b 2 ( D - β 2 b ) 2 + α 2 2 b 2 - b δ 2 ( D + β 2 b ) + α 2 γ 2 b 2 ( D + β 2 b ) 2 + α 2 b 2 = 2 d 2 ( β 2 δ 2 - α 2 γ 2 ) - 2 ( α 2 2 + β 2 2 ) ( β 2 δ 2 + α 2 γ 2 ) ( d 2 + α 2 2 - β 2 2 ) 2 + 4 α 2 2 β 2 2 .
V 2 ( d ) = 4 ( 4 - π ) 2 · ( 3 - π ) ( 4 - π ) d 2 + ( π - 2 ) 2 [ 2 ( 4 - π ) 2 d 2 + ( 2 π 2 + 16 - 11 π ) ] 2 + π ( 23 π - 32 - 4 π 2 ) = - 0.1649 d 2 + 1.7686 ( d 2 + 0.8018 ) 2 + 1.1257 .

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