Abstract

The basic full wave corresponding to the fundamental Gaussian beam was discovered for the outwardly propagating wave in a half-space by the introduction of a source in the complex space. There is a class of extended full waves all of which reduce to the same fundamental Gaussian beam in the appropriate limit. For the extended full Gaussian waves that include the basic full Gaussian wave as a special case, the sources are in the complex space on different planes transverse to the propagation direction. The sources are cylindrically symmetric Gaussian distributions centered at the origin of the transverse planes, the axis of symmetry being the propagation direction. For the special case of the basic full Gaussian wave, the source is a point source. The radiation intensity of the extended full Gaussian waves is determined and their characteristics are discussed and compared with those of the fundamental Gaussian beam. The extended full Gaussian waves are also obtained for the oppositely propagating outwardly directed waves in the second half-space. The radiation intensity distributions in the two half-spaces have reflection symmetry about the midplane. The radiation intensity distributions of the various extended full Gaussian waves are not significantly different. The power carried by the extended full Gaussian waves is evaluated and compared with that of the fundamental Gaussian 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, “Linearly polarized anisotropic Gaussian light wave,” J. Opt. Soc. Am. A 26, 1582-1587 (2009).
    [CrossRef]
  6. S. R. Seshadri, “Dynamics of the linearly polarized fundamental Gaussian light wave,” J. Opt. Soc. Am. A 24, 482-492 (2007).
    [CrossRef]
  7. S. R. Seshadri, “Independent waves in complex source point theory,” Opt. Lett. 32, 3218-3220 (2007).
    [CrossRef] [PubMed]
  8. S. R. Seshadri, “Quality of paraxial electromagnetic beams,” Appl. Opt. 45, 5335-5345 (2006).
    [CrossRef] [PubMed]

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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 (3)

Fig. 1
Fig. 1

Radiation intensity pattern Φ t ( θ , ϕ ) in watts per steradian ( W sr 1 ) of the extended full Gaussian wave for curves a, ϕ = 0 ° ; b, ϕ = 30 ° ; c, ϕ = 60 ° ; d, ϕ = 90 ° ; and e, cylindrically symmetric radiation intensity pattern Φ p ( θ , ϕ ) in watts per steradian ( W sr 1 ) of the corresponding paraxial Gaussian beam as functions of θ for 0 ° < θ < 90 ° . Other parameters are w 0 λ = 0.2487 , b t b = 0.8 ; the power in the paraxial Gaussian beam is 1 W , and that in the extended full Gaussian wave is 0.7226 W .

Fig. 2
Fig. 2

Radiation intensity pattern Φ t ( θ , ϕ ) in watts per steradian ( W sr 1 ) of the extended full Gaussian wave for curves a, b t b = 1 ; b, b t b = 0 as functions of θ for 0 ° < θ < 90 ° . Other parameters are w 0 λ = 0.2487 , ϕ = 0 ° ; the power in the paraxial Gaussian beam is 1 W , and that in the extended full Gaussian wave is 0.7001 W for b t b = 1 and 0.8428 W for b t b = 0 .

Fig. 3
Fig. 3

Power P t in watts in the extended full Gaussian wave as a function of b t b for 0 b t b 1 . Other parameters are w 0 λ = 0.2487 ; the power P p in the paraxial Gaussian beam is P p = 1 W .

Equations (46)

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A x p ± ( x , y , 0 ) = N i k exp ( x 2 + y 2 w 0 2 ) ,
N = ( 4 c π w 0 2 ) 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 2 exp [ π 2 w 0 2 ( p x 2 + p y 2 ) ] .
A x ± ( x , y , z ) = exp ( ± i k z ) A x p ± ( x , y , z ) ,
( 2 x 2 + 2 y 2 ± 2 i k z ) A x p ± ( x , y , z ) = 0 .
A ¯ x p ± ( p x , p y , z ) = A ¯ x p ± ( p x , p y , 0 ) exp [ π 2 w 0 2 ( p x 2 + p y 2 ) i | z | b ] ,
A x p ± ( x , y , z ) = N i k π w 0 2 d p x d p y exp [ i 2 π ( p x x + p y y ) ] exp [ π 2 w 0 2 ( p x 2 + p y 2 ) q ± 2 ]
= N i k q ± 2 exp [ q ± 2 ( x 2 + y 2 ) w 0 2 ] ,
q ± = ( 1 ± i z b ) 1 2 , and q = q ± * .
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 2 4 = 1 .
A x p ± ( x , y , ± i b ) = N i k π w 0 2 δ ( x ) δ ( y ) .
G p ± ( x , y , 0 ) = A x p ± ( x , y , ± i b ) ,
( 2 x 2 + 2 y 2 + 2 z 2 + k 2 ) G ( x , y , z ) = S ex N i k π w 0 2 δ ( x ) δ ( y ) δ ( z ) ,
S ex = i 2 k exp ( k b ) .
( 2 x 2 + 2 y 2 + 2 z 2 + k 2 ) G fs ( x , y , z ) = δ ( x ) δ ( y ) δ ( z )
G fs ( x , y , z ) = exp [ i k ( x 2 + y 2 + z 2 ) 1 2 ] 4 π ( x 2 + y 2 + z 2 ) 1 2 .
G ( x , y , z ) = N i k exp ( k b ) ( i b ) exp [ i k ( x 2 + y 2 + z 2 ) 1 2 ] ( x 2 + y 2 + z 2 ) 1 2 .
A x + ( x , y , z ) = N i k exp ( k b ) ( i b ) exp { i k [ x 2 + y 2 + ( | z | i b ) 2 ] 1 2 } [ x 2 + y 2 + ( | z | i b ) 2 ] 1 2 .
A x ± ( x , y , z ) = exp ( ± i k z ) N i k q ± 2 exp [ q ± 2 ( x 2 + y 2 ) w 0 2 ] .
( 2 z 2 + ζ 2 ) G ¯ ( p x , p y , z ) = N i k i 4 π b exp ( k b ) δ ( z ) ,
ζ = [ k 2 4 π 2 ( p x 2 + p y 2 ) ] 1 2 .
G ¯ ( p x , p y , z ) = N i k 2 π b exp ( k b ) ζ 1 exp ( i ζ | z | ) .
A ¯ x ± ( p x , p y , z ) = N i k 2 π b exp ( k b ) ζ 1 exp [ i ζ ( | z | i b ) ] .
A x + ( x , y , z ) = N i k 2 π b exp ( k b ) d p x d p y exp i 2 π ( p x x + p y y ) ζ 1 exp [ i ζ ( | z | i b ) ]
A x p ± ( x , y , ± i b t ) = N i k 1 ( 1 b t b ) exp [ ( x 2 + y 2 ) w 0 2 ( 1 b t b ) ] .
G p ± ( x , y , 0 ) = A x p ± ( x , y , ± i b t )
( 2 x 2 + 2 y 2 + 2 z 2 + k 2 ) G ( x , y , z ) = S ex N i k 1 ( 1 b t b ) exp [ ( x 2 + y 2 ) w 0 2 ( 1 b t b ) ] δ ( z ) ,
S ex = i 2 k exp ( k b t ) .
( 2 z 2 + ζ 2 ) G ¯ ( p x , p y , z ) = N i k i 4 π b exp ( k b t ) × exp [ π 2 w 0 2 ( p x 2 + p y 2 ) ( 1 b t b ) ] δ ( z ) ,
G ¯ ( p x , p y , z ) = N i k 2 π b exp ( k b t ) × exp [ π 2 w 0 2 ( p x 2 + p y 2 ) ( 1 b t b ) ] ζ 1 exp ( i ζ | z | ) .
A ¯ x ± ( p x , p y , z ) = N i k 2 π b exp ( k b t ) × exp [ π 2 w 0 2 ( p x 2 + p y 2 ) ( 1 b t b ) ] ζ 1 exp [ i ζ ( | z | i b t ) ] .
ζ = k 2 π 2 k ( p x 2 + p y 2 )
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 2 exp [ π 2 w 0 2 ( p x 2 + p y 2 ) q ± 2 ] ,
A x ± ( x , y , z ) = N i k 2 π b exp ( k b t ) d p x d p y × exp [ i 2 π ( p x x + p y y ) ] × exp [ π 2 w 0 2 ( p x 2 + p y 2 ) ( 1 b t b ) ] × ζ 1 exp [ i ζ ( | z | i b t ) ] .
Φ p ( θ , ϕ ) = ( 2 π f 0 2 cos θ ) 1 exp ( 1 2 k 2 w 0 2 tan 2 θ ) ,
f 0 = 1 k w 0 .
Φ ( θ , ϕ ) = ( 1 sin 2 θ cos 2 ϕ ) exp [ k 2 w 0 2 ( 1 cos θ ) ] 2 π f 0 2 .
E x ( x , y , z ) = i k ( 1 k 2 2 x 2 + 1 ) A x ( x , y , z ) ,
H y ( x , y , z ) = z A x ( x , y , z ) .
P t = 0 2 π d ϕ 0 π 2 d θ sin θ Φ t ( θ , ϕ ) ,
Φ t ( θ , ϕ ) = ( 1 sin 2 θ cos 2 ϕ ) 2 π f 0 2 exp [ 1 2 k 2 w 0 2 ( 1 b t b ) sin 2 θ ] × exp [ k 2 w 0 2 b t b ( 1 cos θ ) ] .

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