Partial-wave representations of laser beams for use in light-scattering calculations

Gérard Gouesbet, James A. Lock, and Gérard Gréhan

Gérard Gouesbet,^{1} James A. Lock,^{2} and Gérard Gréhan^{1}

^{1}Laboratoire 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.

^{2}Department 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.

Gérard Gouesbet and James A. Lock Appl. Opt. 52(5) 897-916 (2013)

References

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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.