Abstract

The fundamental electromagnetic Gaussian beam is constructed from a single component of the electric vector potential oriented normal to the propagation direction. The potential is cylindrically symmetrical about the propagation direction. The paraxial beam and the first-order nonparaxial beam are obtained. In solving the inhomogeneous paraxial wave equation governing the evolution of the nonparaxial beam, both the particular integral and the complementary function are included. A procedure for deducing the proper asymptotic state of the nonparaxial beam is summarized. The amplitude coefficients of the cylindrically symmetric complex-argument Laguerre–Gauss beams, which constitute the complementary function are determined by requiring the potential to have the proper behavior asymptotically at infinity and near the input plane. From the potential function, the electromagnetic fields are developed and the electrodynamics of the fundamental electromagnetic Gaussian beam beyond the paraxial approximation is investigated. The role of the first-order nonparaxial beam in determining the average beam characteristics is examined.

© 2008 Optical Society of America

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References

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  1. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss and gain variation,” Appl. Opt. 4, 1562-1569 (1965).
    [CrossRef]
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1978), Chap. 6, p. 230.
  3. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Chaps. 1 and 6.
  4. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712-2719 (1998).
    [CrossRef]
  5. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 847 formula 7.421.2, p. 1037, formula 8.970.1.
  6. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation for light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826-829 (1985).
    [CrossRef]
  7. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Chap 5, pp. 108-109.
  8. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  9. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751-760 (1976).
    [CrossRef]
  10. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
    [CrossRef]
  11. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575--578 (1979).
    [CrossRef]
  12. M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355--359 (1981).
    [CrossRef]
  13. S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27, 1872-1874 (2002).
    [CrossRef]

2002 (1)

1998 (1)

1985 (1)

1981 (1)

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355--359 (1981).
[CrossRef]

1979 (1)

1976 (1)

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

1965 (1)

Agrawal, G. P.

Belanger, P. A.

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355--359 (1981).
[CrossRef]

Couture, M.

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355--359 (1981).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

Felsen, L. B.

Fukumitsu, O.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 847 formula 7.421.2, p. 1037, formula 8.970.1.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Chap 5, pp. 108-109.

Kogelnik, H.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1978), Chap. 6, p. 230.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Pattanayak, D. N.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 847 formula 7.421.2, p. 1037, formula 8.970.1.

Seshadri, S. R.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Chaps. 1 and 6.

Takenaka, T.

Yokota, M.

Appl. Opt. (1)

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Phys. Rev. A (2)

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355--359 (1981).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Other (4)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Chap 5, pp. 108-109.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 847 formula 7.421.2, p. 1037, formula 8.970.1.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1978), Chap. 6, p. 230.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Chaps. 1 and 6.

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Equations (86)

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× E = i k H ,
× H = i k E ,
E = i k A 1 i k ( A ) × F ,
H = × A + i k F 1 i k ( F ) ,
( 2 + k 2 ) ( A , F ) = 0 .
E = y ̂ × F y ,
H = i k y ̂ F y 1 i k F y y ,
( 2 ρ 2 + 1 ρ ρ + 2 z 2 + k 2 ) F y ( ρ , z ) = 0 .
F y ( ρ , z ) = exp ( i k z ) f y ( ρ , z ) ,
( 2 ρ 2 + 1 ρ ρ + 2 i k z + 2 z 2 ) f y ( ρ , z ) = 0 .
ρ n = ρ 2 ω 0 , z n = z l 0 .
( 2 ρ n 2 + 1 ρ n ρ n + 4 i z n + 2 f 0 2 2 z n 2 ) f y ( ρ n , z n ) = 0 ,
f y ( ρ n , z n ) = m = 0 m = f 0 2 m f y ( m ) ( ρ n , z n ) ,
( 2 ρ n 2 + 1 ρ n ρ n + 4 i z n ) f y ( 0 ) ( ρ n , z n ) = 0 ,
( 2 ρ n 2 + 1 ρ n ρ n + 4 i z n ) f y ( 1 ) ( ρ n , z n ) = 2 2 z n 2 f y ( 0 ) ( ρ n , z n ) .
f ( ρ n , z n ) = 0 d η n η n J 0 ( η n ρ n ) f ¯ ( η n , z n ) ,
f ¯ ( η n , z n ) = 0 d ρ n ρ n J 0 ( η n ρ n ) f ( ρ n , z n ) .
( 4 i z n η n 2 ) f ¯ y ( 0 ) ( η n , z n ) = 0 .
f ¯ y ( 0 ) ( η n , z n ) = f ¯ y ( 0 ) ( η n , 0 ) exp ( i 4 η n 2 z n ) .
f y ( 0 ) ( ρ n , 0 ) = N i k exp ( ρ n 2 ) ,
0 d ρ ρ J 0 ( η ρ ) 1 4 n ! p 2 ( n + 1 ) exp ( ρ 2 4 p 2 ) L n ( ρ 2 4 p 2 ) = 1 2 η 2 n exp ( p 2 η 2 ) ,
0 d η η J 0 ( η ρ ) 1 2 η 2 n exp ( p 2 η 2 ) = 1 4 n ! p 2 ( n + 1 ) exp ( ρ 2 4 p 2 ) L n ( ρ 2 4 p 2 ) ,
L n ( v ) = l = 0 l = n n ! l ! ( n l ) ! ( 1 ) l x l l ! .
f ¯ y ( 0 ) ( η n , 0 ) = N i k 1 2 exp ( 1 4 η n 2 ) .
f ¯ y ( 0 ) ( η n , z n ) = N i k 1 2 exp [ 1 4 η n 2 ( 1 + i z n ) ] .
f y ( 0 ) ( ρ n , z n ) = N i k 1 ( 1 + i z n ) exp ( ρ n 2 1 + i z n ) .
f y ( 0 ) ( ρ n , z n ) = N i k 1 i z n exp ( ρ n 2 i z n ) .
G y ( ρ , z ) = N i k ( i l 0 ) exp [ i k ( ρ 2 + z 2 ) 1 2 ] ( ρ 2 + z 2 ) 1 2 .
G y ( ρ n , z n ) = N i k 1 i z n exp [ i z n f 0 2 ( 1 + 2 f 0 2 ρ n 2 z n 2 ) 1 2 ] ( 1 + 2 f 0 2 ρ n 2 z n 2 ) 1 2 .
G y ( ρ n , z n ) = N i k 1 i z n ( 1 f 0 2 ρ n 2 z n 2 ) exp ( i k z ρ n 2 i z n f 0 2 i ρ n 4 2 z n 3 ) .
G y ( ρ n , z n ) = N i k exp ( i k z ) i z n exp ( ρ n 2 i z n ) ( 1 f 0 2 ρ n 2 z n 2 f 0 2 i ρ n 4 2 z n 3 ) .
G y ( ρ n , z n ) = exp ( i k z ) m = 0 m = 1 f 0 2 m g y ( m ) ( ρ n , z n ) ,
g y ( 0 ) ( ρ n , z n ) = N i k 1 i z n exp ( ρ n 2 i z n ) ,
g y ( 1 ) ( ρ n , z n ) = N i k 1 ( i z n ) 2 exp ( ρ n 2 i z n ) ( ρ n 2 i z n ρ n 4 2 ( i z n ) 2 ) .
g y ( 1 ) ( ρ n , z n ) = N i k 1 ( i z n ) 2 exp ( ρ n 2 i z n ) [ L 1 ( ρ n 2 i z n ) L 2 ( ρ n 2 i z n ) ] .
f s ( ρ n , 0 ) = N i k ( 1 ) s s ! 2 2 s exp ( ρ n 2 ) L s ( ρ n 2 ) ,
f ¯ s ( η n , 0 ) = N i k ( 1 ) s 1 2 η 2 s exp ( 1 4 η n 2 ) .
f ¯ s ( η n , z n ) = N i k ( 1 ) s 1 2 η 2 s exp [ 1 4 η n 2 ( 1 + i z n ) ] .
f s ( ρ n , z n ) = N i k ( 1 ) s s ! 2 2 s ( 1 + i z n ) s + 1 exp ( ρ n 2 1 + i z n ) L s ( ρ n 2 1 + i z n ) .
( 4 i z n η n 2 ) f ¯ y ( 1 ) ( η n , z n ) = η n 4 8 f ¯ y ( 0 ) ( η n , z n ) .
f ¯ y ( 1 ) ( η n , z n ) = i z n η n 4 32 N i k 1 2 exp [ 1 4 η n 2 ( 1 + i z n ) ] .
[ f y ( 1 ) ( ρ n , z n ) ] P I = N i k i z n ( 1 + i z n ) 3 exp ( ρ n 2 1 + i z n ) L 2 ( ρ n 2 1 + i z n ) .
f y ( 1 ) ( ρ n , z n ) = A 1 ( 1 ) N i k 4 ( 1 + i z n ) 2 exp ( ρ n 2 1 + i z n ) L 1 ( ρ n 2 1 + i z n ) + N i k ( i z n + 32 A 2 ( 1 ) ) 1 ( 1 + i z n ) 3 exp ( ρ n 2 1 + i z n ) L 2 ( ρ n 2 1 + i z n ) + N i k s = 0 s = A s ( 1 ) ( 1 ) s s ! 2 2 s ( 1 + i z n ) s + 1 exp ( ρ n 2 1 + i z n ) L s ( ρ n 2 1 + i z n ) .
f y ( 1 ) ( ρ n , z n ) = A 1 ( 1 ) N i k 4 ( i z n ) 2 exp ( ρ n 2 i z n ) L 1 ( ρ n 2 i z n ) N i k 1 ( i z n ) 2 exp ( ρ n 2 i z n ) L 2 ( ρ n 2 i z n ) + N i k s = 0 s = A s ( 1 ) ( 1 ) s s ! 2 2 s ( i z n ) s + 1 exp ( ρ n 2 i z n ) L s ( ρ n 2 i z n ) .
A 1 ( 1 ) = 1 4 , A s ( 1 ) = 0 for s 1 , 2 .
f y ( 1 ) ( ρ n , z n ) = N i k 1 ( 1 + i z n ) 2 exp ( ρ n 2 1 + i z n ) L 1 ( ρ n 2 1 + i z n ) + N i k ( i z n + 32 A 2 ( 1 ) ) 1 ( 1 + i z n ) 3 exp ( ρ n 2 1 + i z n ) L 2 ( ρ n 2 1 + i z n ) .
A 2 ( 1 ) = 1 32 .
f y ( 1 ) ( ρ n , z n ) = N i k 1 ( 1 + i z n ) 2 exp ( ρ n 2 1 + i z n ) [ L 1 ( ρ n 2 1 + i z n ) L 2 ( ρ n 2 1 + i z n ) ] ,
= N i k exp ( ρ n 2 1 + i z n ) [ ρ n 2 ( 1 + i z n ) 3 ρ n 4 2 ( 1 + i z n ) 4 ] ,
F y ( ρ , z ) = exp ( i k z ) N i k exp ( ρ n 2 1 + i z n ) { 1 ( 1 + i z n ) + f 0 2 [ ρ n 2 ( 1 + i z n ) 3 ρ n 4 2 ( 1 + i z n ) 4 ] } .
S z ( ρ , z ) = c 2 Re [ E x ( ρ , z ) H y * ( ρ , z ) ] .
E x ( ρ , z ) = exp ( i k z ) N exp ( ρ n 2 1 + i z n ) { 1 ( 1 + i z n ) + f 0 2 [ 1 ( 1 + i z n ) 2 + 2 ρ n 2 ( 1 + i z n ) 3 ρ n 4 2 ( 1 + i z n ) 4 ] } ,
H y ( ρ , z ) = exp ( i k z ) N exp ( ρ n 2 1 + i z n ) { 1 ( 1 + i z n ) + f 0 2 [ 1 ( 1 + i z n ) 2 + ( ρ n 2 + 2 y n 2 ) ( 1 + i z n ) 3 ρ n 4 2 ( 1 + i z n ) 4 ] } ,
S z ( ρ , z ) = N 2 c 2 exp ( 2 ρ n 2 1 + z n 2 ) { 1 ( 1 + z n 2 ) + f 0 2 [ 2 ( 1 + z n 2 ) 2 + ( 3 ρ n 2 + 2 y n 2 ) ( 1 z n 2 ) ( 1 + z n 2 ) 3 ρ n 4 ( 1 3 z n 2 ) ( 1 + z n 2 ) 4 ] } ,
P = d x d y S z ( ρ , z ) = N 2 π 2 c ω 0 2 ( 1 1 2 f 0 2 ) .
( u ) a v = 1 P d x d y u S z ( ρ , z ) = 0 for u = x , y .
( x 2 ) a v = 1 P d x x 2 d y S z ( ρ , z ) ,
= 1 2 ω 0 2 [ 1 + 1 2 f 0 2 + z n 2 ( 1 + 3 2 f 0 2 ) ] ,
( y 2 ) a v = 1 2 ω 0 2 [ 1 + 3 2 f 0 2 + z n 2 ( 1 + 1 2 f 0 2 ) ] .
( ρ 2 ) a v = ω 0 2 ( 1 + f 0 2 ) ( 1 + z n 2 ) .
[ ( ρ 2 ) a v ] 1 2 ω 0 1 = 1 2 f 0 2 .
E z ( ρ , z ) = f 0 exp ( i k z ) N i 2 x n ( 1 + i z n ) 2 exp ( ρ n 2 1 + i z n ) ,
H x ( ρ , z ) = f 0 2 exp ( i k z ) N 2 x n y n ( 1 + i z n ) 3 exp ( ρ n 2 1 + i z n ) ,
H z ( ρ , z ) = f 0 exp ( i k z ) N i 2 y n ( 1 + i z n ) 2 exp ( ρ n 2 1 + i z n ) .
( 2 ρ n 2 + 1 ρ n ρ n + 2 f 0 2 2 z n 2 + 2 f 0 2 ) F y ( ρ n , z n ) = 0 .
F y ( ρ n , z n ) = 0 d η n η n J 0 ( η n ρ n ) F ¯ y ( η n , z n ) ,
( 2 z n 2 + ζ n 2 ) F ¯ y ( η n , z n ) = 0 ,
ζ n = 1 2 f 0 ( 2 f 0 2 η n 2 ) 1 2 .
F ¯ y ( η n , z n ) = F ¯ y ( η n , 0 ) exp ( i ζ n z n ) ,
F y ( ρ n , z n ) = 0 d η n η n J 0 ( η n ρ n ) F ¯ y ( η n , 0 ) exp ( i ζ n z n ) .
F y ( ρ n , z n ) = exp ( i k z ) [ f y ( 0 ) ( ρ n , z n ) + f 0 2 f y ( 1 ) ( ρ n , z n ) ] ,
ζ n = f 0 2 η n 2 4 f 0 2 η n 4 32 .
exp ( i ζ n z n ) = exp ( i k z ) exp ( i z n η n 2 4 ) exp ( i z n f 0 2 η n 4 32 ) ,
exp ( i ζ n z n ) = exp ( i k z ) exp ( i z n η n 2 4 ) ( 1 i z n f 0 2 η n 4 32 ) .
F y ( ρ n , 0 ) = F y ( 0 ) ( ρ n , 0 ) + f 0 2 F y ( 1 ) ( ρ n , 0 ) .
F y ( 0 ) ( ρ n , 0 ) = f y ( 0 ) ( ρ n , 0 ) = N i k exp ( ρ n 2 ) ,
F ¯ y ( 0 ) ( η n , 0 ) = N i k 1 2 exp ( η n 2 4 ) .
F ¯ y ( η n , 0 ) = F ¯ y ( 0 ) ( η n , 0 ) ,
f y ( 0 ) ( ρ n , z n ) = N i k 1 2 0 d η n η n J 0 ( η n ρ n ) exp [ η n 2 ( 1 + i z n ) 4 ] ,
= N i k 1 ( 1 + i z n ) exp [ ρ n 2 ( 1 + i z n ) ] ,
f y ( 1 ) ( ρ n , z n ) = N i k 1 2 ( i z n 32 ) 0 d η n η n 5 J 0 ( η n ρ n ) exp [ η n 2 ( 1 + i z n ) 4 ] ,
= N i k i z n ( 1 + i z n ) 3 exp [ ρ n 2 ( 1 + i z n ) ] L 2 [ ρ n 2 ( 1 + i z n ) ] .
g y ( 0 ) ( ρ n , z n ) = f y ( 0 ) ( ρ n , z n ) = N i k 1 i z n exp ( ρ n 2 i z n ) ,
g y ( 1 ) ( ρ n , z n ) = f y ( 1 ) ( ρ n , z n ) = N i k 1 ( i z n ) 2 exp ( ρ n 2 i z n ) L 2 ( ρ n 2 i z n ) .
f y ( 0 ) ( ρ n , 0 ) = N i k exp ( ρ n 2 ) ,
f y ( 1 ) ( ρ n , 0 ) = 0 ,

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