Abstract

A method is developed for deducing the required source in the complex space for the full-wave generalization of a specified paraxial beam. This method is applicable to all paraxial beams. However, the details of the method are presented for the anisotropic Gaussian beam. The required source in the complex space for the anisotropic Gaussian beam is a line source with a Gaussian distribution. An expression is derived for the anisotropic Gaussian full wave, which in the appropriate limit reproduces the anisotropic Gaussian paraxial beam. The radiation intensity of the anisotropic Gaussian light wave is determined; its characteristics are analyzed and compared with those of the limiting case of the anisotropic Gaussian paraxial beam.

© 2009 Optical Society of America

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References

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  1. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  2. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751-760 (1976).
    [CrossRef]
  3. Y. Li and E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256-258 (1982).
    [CrossRef] [PubMed]
  4. A. Yariv, Quantum Electronics, 2nd. ed. (Wiley, 1967), pp. 123-127.
  5. S. R. Seshadri, “Dynamics of the linearly polarized fundamental Gaussian light wave,” J. Opt. Soc. Am. A 24, 482-492 (2007).
    [CrossRef]
  6. S. R. Seshadri, “Independent waves in complex source point theory,” Opt. Lett. 32, 3218-3220 (2007).
    [CrossRef] [PubMed]

2007 (2)

1982 (1)

1976 (1)

1971 (1)

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

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Figures (4)

Fig. 1
Fig. 1

Radiation intensity Φ ( θ , ϕ ) in W sr 1 as functions of the polar angle θ in degrees for 0 ° < θ < 90 ° and the azimuthal angle ϕ = 0 ° . (a) Anisotropic Gaussian paraxial beam and (b) anisotropic Gaussian full wave. Other parameters are w 0 y λ = 0.2487 and w 0 x = 2 w 0 y ; the total time-averaged power in the paraxial beam is 1 W .

Fig. 2
Fig. 2

Same as for Fig. 1 but for ϕ = 30 ° .

Fig. 3
Fig. 3

Same as for Fig. 1 but for ϕ = 60 ° .

Fig. 4
Fig. 4

Same as for Fig. 1 but for ϕ = 90 ° .

Equations (33)

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A x p ( x , y , 0 ) = N i k exp ( x 2 w 0 x 2 y 2 w 0 y 2 ) ,
N = ( 4 c π w 0 x w 0 y ) 1 2 ,
f ( x , y , z ) = d p x d p y exp [ i 2 π ( p x x + p y y ) ] f ¯ ( p x , p y , z ) ,
f ¯ ( p x , p y , z ) = d x d y exp [ i 2 π ( p x x + p y y ) ] f ( x , y , z ) .
A ¯ x p ( p x , p y , 0 ) = N i k π w 0 x exp ( π 2 w 0 x 2 p x 2 ) × π w 0 y exp ( π 2 w 0 y 2 p y 2 ) .
A ¯ x p ( p x , p y , z ) = A ¯ x p ( p x , p y , 0 ) × exp [ π 2 w 0 x 2 p x 2 ( i z b x ) π 2 w 0 y 2 p y 2 ( i z b y ) ] ,
b u = 1 2 k w 0 u 2 for u = x , y .
A x p ( x , y , z ) = N i k d p x d p y exp [ i 2 π ( p x x + p y y ) ] × π w 0 x w 0 y exp ( π 2 w 0 x 2 p x 2 q x 2 π 2 w 0 y 2 p y 2 q y 2 )
= N i k q x exp ( q x 2 x 2 w 0 x 2 ) q y exp ( q y 2 y 2 w 0 y 2 ) ,
q u = ( 1 + i z b u ) 1 2 for u = x , y
E x p ( x , y , z ) = H y p ( x , y , z ) = i k A x p ( x , y , z ) .
S z ( x , y , z ) = ( c 2 ) Re [ E x ( x , y , z ) H y * ( x , y , z ) ] ,
P p = N 2 c π w 0 x w 0 y 4 = 1 .
A x p ( x , y , i b y ) = N i k ( 1 b y b x ) 1 2 exp [ x 2 w 0 x 2 ( 1 b y b x ) ] × π w 0 y δ ( y )
G p ( x , y , 0 ) = A x p ( x , y , i b y ) ,
( 2 x 2 + 2 y 2 + 2 z 2 + k 2 ) G ( x , y , z ) = S e x N i k ( 1 b y b x ) 1 2 exp [ x 2 w 0 x 2 ( 1 b y b x ) ] × π w 0 y δ ( y ) δ ( z ) ,
S e x = i 2 k exp ( k b y ) .
( 1 b y b x ) 1 2 exp [ x 2 w 0 x 2 ( 1 b y b x ) ] = d p x exp ( i 2 π p x x ) π w 0 x exp [ π 2 w 0 x 2 p x 2 ( 1 b y b x ) ] ,
δ ( y ) = d p y exp ( i 2 π p y y ) .
( 2 z 2 + ζ 2 ) G ¯ ( p x , p y , z ) = S e x N i k π w 0 x w 0 y × exp [ π 2 w 0 x 2 p x 2 ( 1 b y b x ) ] δ ( z ) ,
ζ = [ k 2 4 π 2 ( p x 2 + p y 2 ) ] 1 2 .
G ¯ ( p x , p y , z ) = i 2 ζ S e x N i k π w 0 x w 0 y × exp [ π 2 w 0 x 2 p x 2 ( 1 b y b x ) ] exp ( i ζ z ) .
A ¯ x ( p x , p y , z ) = k ζ exp ( k b y ) N i k π w 0 x w 0 y × exp [ π 2 w 0 x 2 p x 2 ( 1 b y b x ) ] exp [ i ζ ( z i b y ) ] ,
ζ = k 2 π 2 k ( p x 2 + p y 2 ) = k π 2 w 0 x 2 p x 2 b x π 2 w 0 y 2 p y 2 b y
A ¯ x ( p x , p y , z ) = exp ( i k z ) A ¯ x p ( p x , p y , z ) ,
A ¯ x p ( p x , p y , z ) = N i k π w 0 x w 0 y exp ( π 2 w 0 x 2 p x 2 q x 2 π 2 w 0 y 2 p y 2 q y 2 ) ,
A x ( x , y , z ) = N i k π k w 0 x w 0 y exp ( k b y ) d p x d p y × exp [ i 2 π ( p x x + p y y ) ] exp [ π 2 w 0 x 2 p x 2 ( 1 b y b x ) ] × ζ 1 exp [ i ζ ( z i b y ) ]
Φ p ( θ , ϕ ) = exp [ 1 2 tan 2 θ ( k 2 w 0 x 2 cos 2 ϕ + k 2 w 0 y 2 sin 2 ϕ ) ] 2 π f 0 x f 0 y cos θ ,
f 0 u = 1 k w 0 u for u = x , y .
E x ( x , y , z ) = 1 i k ( 2 x 2 + k 2 ) A x ( x , y , z ) ,
H y ( x , y , z ) = z A x ( x , y , z ) .
P = 0 2 π d ϕ 0 π 2 d θ sin θ Φ ( θ , ϕ ) ,
Φ ( θ , ϕ ) = ( 1 sin 2 θ cos 2 ϕ ) 2 π f 0 x f 0 y exp [ k 2 w 0 y 2 ( 1 cos θ ) ] × exp [ 1 2 ( k 2 w 0 x 2 k 2 w 0 y 2 ) sin 2 θ cos 2 ϕ ] .

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