Gérard Gouesbet,1
James A. Lock,2
and Gérard Gréhan1
1Laboratoire d’Energétique des Systèmes et Procédés, Unité de Recherche Associée Centre National de la Recherche Scientifique No. 230, Complex de Recherche Interdisciplinaire en Aerothermochemie, Institut National des Sciences Appliquées de Rouen, B.P. 08, 76131 Mont-Saint-Aignan, France.
2Department of Physics, Cleveland State University, Cleveland, Ohio 44115.
Gérard Gouesbet, James A. Lock, and Gérard Gréhan, "Partial-wave representations of laser beams for use in light-scattering calculations," Appl. Opt. 34, 2133-2143 (1995)
In the framework of generalized Lorenz–Mie theory, laser beams are described by sets of beam-shape coefficients. The modified localized approximation to evaluate these coefficients for a focused Gaussian beam is presented. A new description of Gaussian beams, called standard beams, is introduced. A comparison is made between the values of the beam-shape coefficients in the framework of the localized approximation and the beam-shape coefficients of standard beams. This comparison leads to new insights concerning the electromagnetic description of laser beams. The relevance of our discussion is enhanced by a demonstration that the localized approximation provides a very satisfactory description of top-hat beams as well.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
BSC’s as a Function of Partial Wave for s = 0.001 for the Localized Approximation (LA); the Modified Localized Approximation (MLA); the First- (D1), Third- (D3), and Fifth-order (D5) approximations to the Standard Beam; and the Standard Beam (S)a
n
LA
MLA
D1
D3
D5
k, S
1
0.999997750
1.000000000
1.000000000
1.000000000
1.000000000
1, Same as D1
2
0.999993750
0.999996000
0.999996000
0.999996000
0.999996000
1, Same as D1
5
0.999969750
0.999972000
0.999972000
0.999972000
0.999972000
1, Same as D1
10
0.999889756
0.999892006
0.999892000
0.999892006
0.999892006
3, Same as D3
50
0.997452999
0.997455243
0.999452000
0.999455238
0.997455238
3, Same as D3
100
0.989950586
0.989952813
0.080902000
0.989952793
0.989952793
3, Same as D3
1000
0.367511653
0.367512480
<0
0.332834669
0.366292083
15, 0.367511867
2500
0.001925633
0.001925638
<0
<0
<0
31, 0.001925639
5000
0.138186 × 10−10
0.138187 × 10−10
<0
<0
<0
101, 0.138208 × 10−10
For the standard beam, the number of terms in the infinite series of Eq. (27) required for convergence to 9 significant figures (k) is also given.
Table 3
BSC’s as a Function of Partial Wave for s = 0.16 for the Localized Approximation (LA); the Modified Localized Approximation (MLA); the First- (D1), Third- (D3), and Fifth-Order (D5) Approximations to the Standard Beam; and the Standard Beam (S)a
n
LA
MLA
D1
D3
D5
k, S
1
0.944027482
1.000000000
1.000000000
1.000000000
1.000000000
1, Same as D1
2
0.852143789
0.902668412
0.897600000
0.897600000
0.897600000
1, Same as D1
4
0.595472542
0.630778820
0.539200000
0.616138215
0.618138215
3, Same as D3
6
0.339052607
0.359155441
<0
0.327063245
0.343026339
5, Same as D5
10
0.059463060
0.062988600
<0
<0
<0
9, 0.058365667
15
0.002132629
0.002259075
<0
<0
<0
15, 0.002267813
20
0.000021266
0.000022526
<0
<0
<0
19, 0.000031912
25
0.589603 × 10−7
0.624562 × 10−7
<0
<0
<0
25, 1.853835 × 10−7
For the standard beam, the number of terms in the infinite series of Eq. (27) required for convergence to 9 significant figures (k) is also given.
Tables (3)
Table 1
Coefficients α, β, and γ of Eqs. (34), (35), and (39), Respectively, as a Function of Partial Wave
BSC’s as a Function of Partial Wave for s = 0.001 for the Localized Approximation (LA); the Modified Localized Approximation (MLA); the First- (D1), Third- (D3), and Fifth-order (D5) approximations to the Standard Beam; and the Standard Beam (S)a
n
LA
MLA
D1
D3
D5
k, S
1
0.999997750
1.000000000
1.000000000
1.000000000
1.000000000
1, Same as D1
2
0.999993750
0.999996000
0.999996000
0.999996000
0.999996000
1, Same as D1
5
0.999969750
0.999972000
0.999972000
0.999972000
0.999972000
1, Same as D1
10
0.999889756
0.999892006
0.999892000
0.999892006
0.999892006
3, Same as D3
50
0.997452999
0.997455243
0.999452000
0.999455238
0.997455238
3, Same as D3
100
0.989950586
0.989952813
0.080902000
0.989952793
0.989952793
3, Same as D3
1000
0.367511653
0.367512480
<0
0.332834669
0.366292083
15, 0.367511867
2500
0.001925633
0.001925638
<0
<0
<0
31, 0.001925639
5000
0.138186 × 10−10
0.138187 × 10−10
<0
<0
<0
101, 0.138208 × 10−10
For the standard beam, the number of terms in the infinite series of Eq. (27) required for convergence to 9 significant figures (k) is also given.
Table 3
BSC’s as a Function of Partial Wave for s = 0.16 for the Localized Approximation (LA); the Modified Localized Approximation (MLA); the First- (D1), Third- (D3), and Fifth-Order (D5) Approximations to the Standard Beam; and the Standard Beam (S)a
n
LA
MLA
D1
D3
D5
k, S
1
0.944027482
1.000000000
1.000000000
1.000000000
1.000000000
1, Same as D1
2
0.852143789
0.902668412
0.897600000
0.897600000
0.897600000
1, Same as D1
4
0.595472542
0.630778820
0.539200000
0.616138215
0.618138215
3, Same as D3
6
0.339052607
0.359155441
<0
0.327063245
0.343026339
5, Same as D5
10
0.059463060
0.062988600
<0
<0
<0
9, 0.058365667
15
0.002132629
0.002259075
<0
<0
<0
15, 0.002267813
20
0.000021266
0.000022526
<0
<0
<0
19, 0.000031912
25
0.589603 × 10−7
0.624562 × 10−7
<0
<0
<0
25, 1.853835 × 10−7
For the standard beam, the number of terms in the infinite series of Eq. (27) required for convergence to 9 significant figures (k) is also given.