Virendra N. Mahajan, "Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations," J. Opt. Soc. Am. A 3, 470-485 (1986)

Much is said in the literature about Gaussian beams. However, there is little in terms of a quantitative comparison between the propagation of uniform and Gaussian beams. Even when results for both types of beam are given, they appear in a normalized form in such a way that some of the quantitative difference between them is lost. In this paper we first consider an aberration-free beam and investigate the effect of Gaussian amplitude across the aperture on the focal-plane irradiance and encircled-power distributions. The axial irradiance of focused uniform and Gaussian beams is calculated, and the problem of optimum focusing is discussed. The results for a collimated beam are obtained as a limiting case of a focused beam. Next, we consider the problem of aberration balancing and compare the effects of primary aberrations on the two types of beam. Finally, the limiting case of weakly truncated Gaussian beams is discussed, and simple results are obtained for the irradiance distribution and the balanced aberrations.

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Factor by Which the Standard Deviation of a Classical Aberration across a Circular Aperture Is Reduced When It Is Optimally Balanced with Other Aberrations

Reduction Factor

Aberration

Uniform (γ = 0)

Gaussian (γ = 1)

Weakly Truncated Gaussian (
$\sqrt{\gamma}\ge 3$)

Spherical

4

3.74

2.24

Coma

3

2.64

1.73

Astigmatism

1.22

1.66

1.41

Table 6

Standard Deviation Factor for Primary Aberrations for a Gaussian Circular Beam With Various Values of γ^{a}

$\sqrt{\gamma}$

Balanced Spherical

Balanced Coma

Balanced Astigmatism

0

13.42

8.49

4.90

0.5

13.69

8.53

5.06

1.0

13.71

8.80

5.61

1.5

14.90

9.74

6.81

2.0

18.29

12.21

9.08

2.5

26.33

17.62

12.82

3.0

43.52

27.57

18.06

3.5

75.78

42.96

24.51

4.0

128.09

64.01

32.00

The numbers given in this table represent the factor by which the peak aberration coefficient A_{i} must be divided by in order to obtain the standard deviation.

Tables (6)

Table 1

Maxima and Minima of Focal-Plane Irradiance Distribution and Corresponding Encircled Powers for a Circular Beam^{a}

Max/Min

r, r_{0}

I(r)

P(r_{0})

Max

0

1

0

(0)

(0.924)

(0)

Min

1.22

0

0.838

(1.43)

0

(0.955)

Max

1.64

0.0175

0.867

(1.79)

(0.0044)

(0.962)

Min

2.23

0

0.9I0

(2.33)

(0)

(0.973)

Max

2.68

0.0042

0.922

(2.76)

(0.0012)

(0.976)

Min

3.24

0

0.938

(3.30)

(0)

(0.981)

Max

3.70

0.0016

0.944

(3.76)

(0.0005)

(0.983)

Min

4.24

0

0.952

(4.29)

(0)

(0.985)

Max

4.71

0.0008

0.957

(4.75)

0.0002

(0.986)

The numbers without parentheses are for a uniform beam, and those with parentheses are for a Gaussian (γ = 1) beam.

Table 2

Radial Polynominals for Balanced Primary Aberrations for Uniform and Gaussian Beams

Factor by Which the Standard Deviation of a Classical Aberration across a Circular Aperture Is Reduced When It Is Optimally Balanced with Other Aberrations

Reduction Factor

Aberration

Uniform (γ = 0)

Gaussian (γ = 1)

Weakly Truncated Gaussian (
$\sqrt{\gamma}\ge 3$)

Spherical

4

3.74

2.24

Coma

3

2.64

1.73

Astigmatism

1.22

1.66

1.41

Table 6

Standard Deviation Factor for Primary Aberrations for a Gaussian Circular Beam With Various Values of γ^{a}

$\sqrt{\gamma}$

Balanced Spherical

Balanced Coma

Balanced Astigmatism

0

13.42

8.49

4.90

0.5

13.69

8.53

5.06

1.0

13.71

8.80

5.61

1.5

14.90

9.74

6.81

2.0

18.29

12.21

9.08

2.5

26.33

17.62

12.82

3.0

43.52

27.57

18.06

3.5

75.78

42.96

24.51

4.0

128.09

64.01

32.00

The numbers given in this table represent the factor by which the peak aberration coefficient A_{i} must be divided by in order to obtain the standard deviation.