Abstract

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.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Sec. 10-7.
  2. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Sec. 8-2.
  3. L. D. Dickson, “Characteristics of a propagating Gaussian beam,” Appl. Opt. 9, 1854–1861 (1970). In this paper the effects of aperture truncation are considered and a condition under which they may be neglected is derived.
    [Crossref] [PubMed]
  4. C. S. Williams, “Gaussian beam formulas from diffraction theory,” Appl. Opt. 12, 871–876 (1973).
    [Crossref]
  5. R. M. Herman, J. Pardo, T. A. Wiggins, “Diffraction and focusing of Gaussian beams,” Appl. Opt. 24, 1346–1354 (1985).
    [Crossref] [PubMed]
  6. A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
    [Crossref]
  7. J. P. Campbell, L. G. DeShazer, “Near fields of truncated Gaussian apertures,”J. Opt. Soc. Am. 59, 1427–1429 (1969).
    [Crossref]
  8. G. O. Olaofe, “Diffraction by Gaussian apertures,”J. Opt. Soc. Am. 60, 1654–1657 (1970).
    [Crossref]
  9. R. G. Schell, G. Tyra, “Irradiance from an aperture with truncated-Gaussian field distribution, J. Opt. Soc. Am. 61, 31–35 (1971).
    [Crossref]
  10. V. P. Nayyar, N. K. Verma, “Diffraction by truncated-Gaussian annular apertures,”J. Opt. (Paris) 9, 307–310 (1978).
    [Crossref]
  11. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
    [Crossref] [PubMed]
  12. Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [Crossref]
  13. K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 24, 1098–1101 (1985).
    [Crossref] [PubMed]
  14. D. D. Lowenthal, “Maréchal intensity criteria modified for Gaussian beams,” Appl. Opt. 13, 2126–2133, 2774 (1974).
    [Crossref] [PubMed]
  15. D. D. Lowenthal, “Far-field diffraction patterns for Gaussian beams in the presence of small spherical aberrations,”J. Opt. Soc. Am. 65, 853–855 (1975).
    [Crossref]
  16. E. Sklar, “Effects of small rotationally symmetrical aberrations on the irradiance spread function of a system with Gaussian apodization over the pupil,”J. Opt. Soc. Am. 65, 1520–1521 (1975).
    [Crossref]
  17. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [Crossref] [PubMed]
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 416.
  19. V. N. Mahajan, “Included power for obscured circular pupils,” Appl. Opt. 17, 964–968 (1978).
    [Crossref] [PubMed]
  20. V. N. Mahajan, “Luneburg apodization problem I,” Opt. Lett. 5, 267–269 (1980).
    [Crossref] [PubMed]
  21. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 484.
  22. M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 887.
  23. V. N. Mahajan, “Aberrated point-spread function for rotationally symmetric aberrations,” Appl. Opt. 22, 3035–3041 (1983).
    [Crossref] [PubMed]
  24. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,”J. Opt. Soc. Am. 71, 75–85; erratum, 1408 (1981).
  25. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).
  26. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 9.
  27. S. Szapiel, “Aberration balancing technique for radially symmetric amplitude distributions: a generalization of the Maréchal approach,”J. Opt. Soc. Am. 72, 947–956 (1982).
    [Crossref]
  28. V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,”J. Opt. Soc. Am. 73, 860–861 (1983).
    [Crossref]
  29. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. (Academic, New York, 1965), p. 717.
  30. A. Yoshida, “Spherical aberration in beam optical systems,” Appl. Opt. 21, 1812–1816 (1982).
    [Crossref] [PubMed]
  31. R. Herloski, “Strehl ratio for untruncated aberrated Gaussian beams,” J. Opt. Soc. Am. A 2, 1027–1030 (1985).
    [Crossref]
  32. K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 24, 1098–1101 (1985).
    [Crossref] [PubMed]
  33. V. N. Mahajan, “Comparison of uniform and Gaussian beam diffraction,” Proc. Soc. Photo-Opt. Instrum. Eng.560(to be published).

1985 (4)

1983 (3)

1982 (3)

1980 (1)

1978 (2)

V. N. Mahajan, “Included power for obscured circular pupils,” Appl. Opt. 17, 964–968 (1978).
[Crossref] [PubMed]

V. P. Nayyar, N. K. Verma, “Diffraction by truncated-Gaussian annular apertures,”J. Opt. (Paris) 9, 307–310 (1978).
[Crossref]

1975 (2)

1974 (1)

D. D. Lowenthal, “Maréchal intensity criteria modified for Gaussian beams,” Appl. Opt. 13, 2126–2133, 2774 (1974).
[Crossref] [PubMed]

1973 (1)

C. S. Williams, “Gaussian beam formulas from diffraction theory,” Appl. Opt. 12, 871–876 (1973).
[Crossref]

1972 (1)

1971 (1)

1970 (2)

1969 (1)

1967 (1)

A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 484.

Abromowitz, M.

M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 887.

Avizonis, P. V.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 9.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 416.

Buck, A. L.

A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
[Crossref]

Campbell, J. P.

DeShazer, L. G.

Dickson, L. D.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Sec. 10-7.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. (Academic, New York, 1965), p. 717.

Hauchi, K.

Herloski, R.

Herman, R. M.

Holmes, D. A.

Korka, J. E.

Li, Y.

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

Lowenthal, D. D.

Mahajan, V. N.

Nayyar, V. P.

V. P. Nayyar, N. K. Verma, “Diffraction by truncated-Gaussian annular apertures,”J. Opt. (Paris) 9, 307–310 (1978).
[Crossref]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).

Olaofe, G. O.

Pardo, J.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. (Academic, New York, 1965), p. 717.

Saga, N.

Schell, R. G.

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Sec. 8-2.

Sklar, E.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 484.

M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 887.

Szapiel, S.

Tanaka, K.

Tyra, G.

Verma, N. K.

V. P. Nayyar, N. K. Verma, “Diffraction by truncated-Gaussian annular apertures,”J. Opt. (Paris) 9, 307–310 (1978).
[Crossref]

Wiggins, T. A.

Williams, C. S.

C. S. Williams, “Gaussian beam formulas from diffraction theory,” Appl. Opt. 12, 871–876 (1973).
[Crossref]

Wolf, E.

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 9.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 416.

Yoshida, A.

Appl. Opt. (11)

L. D. Dickson, “Characteristics of a propagating Gaussian beam,” Appl. Opt. 9, 1854–1861 (1970). In this paper the effects of aperture truncation are considered and a condition under which they may be neglected is derived.
[Crossref] [PubMed]

C. S. Williams, “Gaussian beam formulas from diffraction theory,” Appl. Opt. 12, 871–876 (1973).
[Crossref]

R. M. Herman, J. Pardo, T. A. Wiggins, “Diffraction and focusing of Gaussian beams,” Appl. Opt. 24, 1346–1354 (1985).
[Crossref] [PubMed]

K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 24, 1098–1101 (1985).
[Crossref] [PubMed]

D. D. Lowenthal, “Maréchal intensity criteria modified for Gaussian beams,” Appl. Opt. 13, 2126–2133, 2774 (1974).
[Crossref] [PubMed]

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
[Crossref] [PubMed]

V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
[Crossref] [PubMed]

V. N. Mahajan, “Included power for obscured circular pupils,” Appl. Opt. 17, 964–968 (1978).
[Crossref] [PubMed]

V. N. Mahajan, “Aberrated point-spread function for rotationally symmetric aberrations,” Appl. Opt. 22, 3035–3041 (1983).
[Crossref] [PubMed]

A. Yoshida, “Spherical aberration in beam optical systems,” Appl. Opt. 21, 1812–1816 (1982).
[Crossref] [PubMed]

K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 24, 1098–1101 (1985).
[Crossref] [PubMed]

J. Opt. (Paris) (1)

V. P. Nayyar, N. K. Verma, “Diffraction by truncated-Gaussian annular apertures,”J. Opt. (Paris) 9, 307–310 (1978).
[Crossref]

J. Opt. Soc. Am. (8)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

Opt. Lett. (1)

Proc. IEEE (1)

A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
[Crossref]

Other (9)

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Sec. 10-7.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Sec. 8-2.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 416.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 484.

M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 887.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 9.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. (Academic, New York, 1965), p. 717.

V. N. Mahajan, “Comparison of uniform and Gaussian beam diffraction,” Proc. Soc. Photo-Opt. Instrum. Eng.560(to be published).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Diffraction geometry. A beam of outer and inner diameters D and ∊D, respectively, is focused at a distance R from the aperture plane. The beam is aberration free when a spherical wave front of radius of curvature R is centered at the intended focus and passes through the center of the aperture.

Fig. 2
Fig. 2

Aperture irradiance distribution for uniform and Gaussian (γ = 1) beams of a given total power P0. (a) Circular aperture. (b) Annular aperture with = 0.5. (c) Ratio Ig()/Iu() of peak values of aperture irradiance as a function of for several values of γ. The units of aperture irradiance are P0/A, where A is the area of the circular aperture.

Fig. 3
Fig. 3

Focal-plane irradiance and encircled-power distributions for uniform and Gaussian (γ = 1) beams of a given total power P0. The irradiance and encircled power are in units of P0A2R2 and P0, respectively. The radial distance r or r0 in the focal plane is in units of λR/D. The focal point is at r = 0.

Fig. 4
Fig. 4

Focal-point irradiance ratio η for uniform and Gaussian beams as a function of γ and . Note that γ = a / ω.

Fig. 5
Fig. 5

Axial irradiance of a beam focused at a fixed distance R with a Fresnel number a2R = N = 1, 10, 100. The irradiance is in units of the focal-point irradiance P0A2R2 for a uniform circular beam. For a uniform beam, the minima of axial irradiance are located at z/R = 1/3, 1/5, 1/7,…, when = 0 and at z/R = 3/11, 3/19, 3/27,…, when = 0.5. The Gaussian beam results in this figure are for γ = 1.

Fig. 6
Fig. 6

Central irradiance on a target at a fixed distance z from the aperture plane when a beam is focused at various distances R. The quantity Nz = a2z represents the Fresnel number of the aperture as observed from the target. The irradiance is in units of P0A2z2. The Gaussian beam results in this figure are for γ = 1.

Fig. 7
Fig. 7

Defocused irradiance and encircled-power distributions for uniform and Gaussian (γ = 1) beams. The amount of defocus aberration Φ0 = 2.783 rad (0.443λ) is such that it gives a central irradiance of 0.5 for a uniform circular beam of large Fresnel number N. The units of r, r0, I(r), and P(r0) are the same as in Fig. 3.

Fig. 8
Fig. 8

Axial irradiance of collimated uniform and Gaussian beams. The irradiance is in units of the uniform irradiance P0/A at the aperture of a uniform circular beam. The distance z is in units of the far-field distance D2/λ for a uniform circular aperture.

Fig. 9
Fig. 9

The focal-plane irradiance and encircled-power distributions for a Gaussian beam with γ = 2. The irradiance and encircled power are in units of P0A2R2 and P0, respectively. The radial distance r or r0 in the focal plane is in units of λR/D. The focal point is at r = 0. The solid curves have been obtained using Eqs. (17) and (18), and the dashed curves represent their corresponding approximations given by Eqs. (81a) and (88a), respectively.

Fig. 10
Fig. 10

Axial irradiance of a Gaussian beam with γ = 2 focused at a distance R with a Fresnel number N = 1, 10, 100. The irradiance is in units of P0A2R2. The solid curves have been obtained by using Eq. (24), and the dashed Curves represent their corresponding approximations given by Eq. (83a).

Tables (6)

Tables Icon

Table 1 Maxima and Minima of Focal-Plane Irradiance Distribution and Corresponding Encircled Powers for a Circular Beama

Tables Icon

Table 2 Radial Polynominals for Balanced Primary Aberrations for Uniform and Gaussian Beams

Tables Icon

Table 3 Standard Deviation of Primary Aberrations and of the Corresponding Balanced Aberrations for Uniform and Gaussian (γ = 1) Circular Beamsa

Tables Icon

Table 4 Standard Deviation of Primary Aberrations in Weakly Truncated ( γ 3) Gaussian Circular Beamsa

Tables Icon

Table 5 Factor by Which the Standard Deviation of a Classical Aberration across a Circular Aperture Is Reduced When It Is Optimally Balanced with Other Aberrations

Tables Icon

Table 6 Standard Deviation Factor for Primary Aberrations for a Gaussian Circular Beam With Various Values of γa

Equations (120)

Equations on this page are rendered with MathJax. Learn more.

U ( r ; z ) = exp [ i k ( z + r 2 / 2 z ) ] i λ z × U ( ρ ) exp ( i k ρ 2 / 2 z ) exp ( - i k ρ · r / z ) d ρ ,
I ( r ; z ) = U ( r ; z ) 2 .
P ( r 0 ; z ) = r r 0 I ( r ; z ) d r .
I u ( ρ ) = ( P 0 / A ) / ( 1 - 2 )
I g ( ρ ) = ( P 0 / A ) f ( γ ; ) exp ( - 2 γ ρ 2 ) ,
A = π a 2
f ( γ ; ) = 2 γ / [ exp ( - 2 γ 2 ) - e - 2 γ ] ,
γ = ( a / ω ) 2 ,
I g ( ) / I u ( ) = ( 1 - 2 ) f ( γ ; ) exp ( - 2 γ 2 ) = 2 γ ( 1 - 2 ) / { 1 - exp [ - 2 γ ( 1 - 2 ) ] } .
U ( ρ ) = I ( ρ ) exp ( - i A ρ 2 / λ R ) .
I ( r ; z ) = 4 ( R / z ) 2 | 1 I ( ρ ) exp [ i Φ 2 ( ρ ) ] J 0 ( π r ρ ) ρ d ρ | 2 ,
Φ 2 ( ρ ) = A λ ( 1 z - 1 R ) ρ 2
I u ( r ; R ) = ( 1 - 2 ) - 1 { [ I c ( r ) ] 1 / 2 - 2 [ I c ( r ) ] 1 / 2 } 2 ,
I c ( r ) = [ 2 J 1 ( π r ) / π r ] 2
P u ( r 0 ; R ) = ( 1 - 2 ) - 1 [ P c ( r 0 ) + 2 P c ( r 0 ) - 4 0 0 1 J 1 ( π r r 0 ) J 1 ( π r r 0 ) d r / r ] ,
P c ( r 0 ) = 1 - J 0 2 ( π r 0 ) - J 1 2 ( π r 0 )
I g ( r ; R ) = f ( γ ; ) [ 1 exp ( - γ ρ 2 ) J 0 ( π r ρ ) ρ d ρ ] 2
P g ( r 0 ; R ) = ( π 2 / 2 ) 0 r 0 I g ( r ; R ) r d r .
I u ( 0 ; R ) = 1 - 2
I g ( 0 ; R ) = ( 2 / γ ) tanh [ ( 1 - 2 ) γ / 2 ] ,
η = I g ( 0 ; R ) / I u ( 0 ; R ) = tanh [ ( 1 - 2 ) γ / 2 ] ( 1 - 2 ) γ / 2 .
η 2 / γ ( 1 - 2 ) .
I u ( 0 ; z ) = ( R / z ) 2 ( 1 - 2 ) { sin [ ( 1 - 2 ) Φ 0 / 2 ] / [ ( 1 - 2 ) Φ 0 / 2 ] } 2
I g ( 0 ; z ) = ( R / z ) 2 [ 2 γ / ( Φ 0 2 + γ 2 ) ] { cosh [ ( 1 - 2 ) γ ] - cos [ ( 1 - 2 ) Φ 0 ] / sinh [ ( 1 - 2 ) γ ] } ,
Φ 0 = ( A λ ) ( 1 z - 1 R )
= π N ( R z - 1 ) ,
N = a 2 / λ R
Φ 0 = 2 π n / ( 1 - 2 ) ,             n = ± 1 , ± 2 ,
z / R = { 1 + [ 2 n / N ( 1 - 2 ) ] } - 1 .
tan [ ( 1 - 2 ) Φ 0 / 2 ] = ( R / z ) ( 1 - 2 ) Φ 0 / 2 ,             z R
2 ( λ z A - Φ 0 Φ 0 2 + γ 2 ) { cosh [ ( 1 - 2 ) γ ] - cos [ ( 1 - 2 ) Φ 0 ] } = - ( 1 - 2 ) sin [ ( 1 - 2 ) Φ 0 ] .
I ( r ; z ) = 4 ( R / z ) 2 1 1 ρ s [ I ( ρ ) I ( s ) ] 1 / 2 × cos [ Φ 0 ( ρ 2 - s 2 ) ] J 0 ( π r ρ ) J 0 ( π r s ) d ρ d s .
I ( 0 ; z ) = 4 ( R / z ) 2 1 1 ρ s [ I ( ρ ) I ( s ) ] 1 / 2 cos [ Φ 0 ( ρ 2 - s 2 ) ] d ρ d s .
P ( r 0 ; z ) = 2 π 2 1 1 ρ s [ I ( ρ ) I ( s ) ] 1 / 2 × cos [ Φ 0 ( ρ 2 - s 2 ) ] Q ( ρ , s ; r 0 ) d ρ d s ,
Q ( ρ , s ; r 0 ) = 0 r 0 J 0 ( π r ρ ) J 0 ( π r s ) r d r = ( r 0 2 / 2 ) [ J 0 2 ( π r 0 ρ ) + J 1 2 ( π r 0 s ) ]             if ρ = s ,
= [ r 0 / π ( ρ 2 - s 2 ) ] [ ρ J 1 ( π r 0 ρ ) J 0 ( π r 0 s ) - s J 1 ( π r 0 s ) J 0 ( π r 0 ρ ) ]             if ρ s .
1 1 f ( ρ , s ) d ρ d s = [ ( 1 - ) / 2 ] 2 [ i = 1 M ω i 2 f ( ρ i , s j ) + 2 i = 2 M j = 1 i = 1 ω i ω j f ( ρ i , s j ) ] ,
ρ i = s i = [ 1 + + ( 1 - ) x i ] / 2 ,
I ( r ; z ) = 4 Φ 0 2 | 1 I ( ρ ) exp [ i Φ 0 ρ 2 ] J 0 ( π r ρ ) ρ d ρ | 2 ,
Φ 0 = A / λ z
I u ( 0 ; z ) = [ 4 / ( 1 - 2 ) ] sin 2 [ π ( 1 - 2 ) / 8 z ]
I g ( 0 ; z ) = { 2 γ / [ 1 + ( 4 γ z / π ) 2 ] } { coth [ γ ( 1 - 2 ) ] - cos [ π ( 1 - 2 ) / 4 z ] / sinh [ γ ( 1 - 2 ) ] } ,
z = ( 1 - 2 ) / 4 ( 2 n + 1 ) ,             n = 0 , 1 , 2 , .
z = ( 1 - 2 ) / 8 n ,             n = 1 , 2 , .
{ 2 ( 4 z / π ) 3 γ 2 / [ 1 + ( 4 γ z / π ) 2 ] } { cosh [ γ ( 1 - 2 ) ] - cos [ π ( 1 - 2 ) / 4 z ] } = - ( 1 - 2 ) sin [ π ( 1 - 2 ) / 4 z ] .
Φ ( ρ , θ ; ) = c n m m [ ( n + 1 ) ] 1 / 2 R n m ( ρ ; ) cos m θ ,
R n m ( ρ ; ) = a n m ρ n + b n m ρ n - 2 + + d n m ρ m .
m = 1 2 , m = 0 , = 1 , m 0.
c n m = σ Φ ,
σ Φ 2 = Φ 2 - Φ 2
Φ n = 1 0 2 π I ( ρ ) Φ n ( ρ , θ ) ρ d ρ d θ / 1 0 2 π I ( ρ ) ρ d ρ d θ .
Φ s ( ρ ) = A s ρ 4 ,
σ s u = [ ( 4 - 2 - 6 4 - 6 + 4 8 ) 1 / 2 / 3 5 ] A s .
Φ b s ( ρ ) = A s ρ 4 + A d ρ 2 .
A d u = - ( 1 + 2 ) A s .
σ b s u = [ ( 1 - 2 ) 2 / 6 5 ] A s .
σ s g = [ ( 20 e 2 - 69 e + 40 ) 1 / 2 / ( e - 1 ) ] A s
= A s / 3.668.
A d g = ( b 4 0 / a 4 0 ) A s
= - 0.933 A s
σ b s g = A s / 5 a 4 0
= A s / 13.705.
σ s g / σ s u = 0.91.
σ b s g / σ b s u = 0.98.
A d g / A d u = 0.933 ,
Φ 0 = A d .
z = R - 8 λ F 2 A d ,
Φ c ( ρ , θ ) = A c ρ 3 cos θ ,
σ c u = [ ( 1 + 2 + 4 + 6 ) / 8 ] 1 / 2 A c .
Φ b c ( ρ , θ ) = ( A c ρ 3 + A t ρ ) cos θ .
A t u = - ( 2 / 3 ) [ ( 1 + 2 + 4 ) / ( 1 + 2 ) ] A c .
σ b c u = ( 1 - 2 ) ( 1 + 4 2 + 4 ) 1 / 2 6 2 ( 1 + 2 ) 1 / 2 A c .
A t g = ( b 3 1 / a 3 1 ) A c
= - 0.608 A c
σ b c g = A c / 2 2 a 3 1
= A c / 8.802.
σ b c g / σ b c u = 0.96.
A t g / A t u = 0.91 ,
Φ a ( ρ , θ ) = A a ρ 2 cos 2 θ ,
σ a u = ( 1 / 4 ) ( 1 + 4 ) 1 / 2 A a .
Φ b a ( ρ , θ ) = A a ρ 2 cos 2 θ + A d ρ 2 .
A d = - ( 1 / 2 ) A a ,
Φ b a ( ρ , θ ) = ( 1 / 2 ) A a ρ 2 cos 2 θ .
σ b a u = ( 1 / 2 6 ) ( 1 + 2 + 4 ) 1 / 2 A a .
σ b a g = A a / 2 6 a 2 2
= A a / 5.609.
σ b a g / σ b a u = 0.87.
I ( ρ ) = 2 γ exp ( - 2 γ ρ 2 )
I ( ρ ) = ( 2 P 0 / π ω 2 ) exp [ - 2 ( ρ / ω ) 2 ] .
0 exp ( - α ρ 2 ) J 0 ( β ρ ) ρ d ρ ,
( 1 / 2 ) exp ( - β 2 / 4 α ) .
I ( r ; z ) = ( R / z ) 2 [ 2 γ / ( Φ 0 2 + γ 2 ) ] exp [ - γ π 2 r 2 / 2 ( Φ 0 2 + γ 2 ) ]
I ( r ; z ) = ( 2 P 0 / π ω z 2 ) exp ( - 2 r 2 / ω z 2 ) ,
ω z 2 = ( λ z / π ω ) 2 + ω 2 ( 1 - z / R ) 2 .
I ( 0 ; z ) = ( R / z ) 2 [ 2 γ / ( Φ 0 2 + γ 2 ) ]
I ( 0 ; z ) = 2 P 0 / π ω z 2 .
I ( r ; R ) = ( 2 / γ ) exp ( - π 2 r 2 / 2 γ )
I ( r ; R ) = ( 2 P 0 / π ω R 2 ) exp ( - 2 r 2 / ω R 2 ) ,
ω R = λ R / π ω
I ( 0 ; R ) = 2 / γ ,
I ( 0 ; R ) = 2 π P 0 ω 2 / λ 2 R 2 ,
z p / R = [ 1 + ( γ / π N ) 2 ] - 1
z p / R = [ 1 + ( λ R / π ω 2 ) 2 ] - 1 .
I ( 0 ; z p ) = ( 2 / γ ) + ( 2 γ / π 2 N 2 )
I ( 0 ; z p ) = 2 P 0 / π ω 2 z p ,
ω z p 2 = ω 2 / [ 1 + ( π ω 2 / λ R ) 2 ] .
z p / R = [ 1 + ( π N g ) - 2 ] - 1
ω z p 2 = ω 2 / [ 1 + ( π N g ) 2 ] ,
P ( r 0 ; z ) = 1 - exp [ - γ π 2 r 0 2 / 2 ( Φ 0 2 + γ 2 ) ]
P ( r 0 ; z ) = P 0 { 1 - exp [ - 2 r 0 2 / ω z 2 ] } ,
z / R z = ( A / λ z ) Im ( 1 / α ) - 1 = [ A Φ 0 / λ z ( Φ 0 2 + γ 2 ) ] - 1
= 1 - z / R ( 1 - z / R ) 2 + ( λ z / π ω 2 ) 2 - 1.
I ( r ; z ) = { 2 γ / [ 1 + ( 4 γ z / π 2 ) ] } exp { - 8 γ z 2 r 2 / [ 1 + ( 4 γ z / π ) 2 ] }
I ( r ; z ) = ( 2 P 0 / π ω z 2 ) exp ( - 2 r 2 / ω z 2 ) ,
ω z 2 = ω 2 [ 1 + ( λ z / π ω 2 ) 2 ]
R z = - z [ 1 + ( π ω 2 / λ z ) 2 ] .
p s = ρ s = ( s / 2 γ ) p s - 2 = ( s / 2 ) ! γ - s / 2 .
Φ 0 = - ( 4 / γ ) A s = - 4 γ A s ,
A t = - ( 2 / γ ) A c = - 2 γ A c ,
Φ 0 = - ( 1 / 2 ) A a = - ( γ / 2 ) A a

Metrics