Abstract

The electric current sources that are required for the excitation of the fundamental Gaussian beam and the corresponding full Gaussian light wave are determined. The current sources are situated on the secondary source plane that forms the boundary between the two half-spaces in which the waves are launched. The electromagnetic fields and the complex power generated by the current sources are evaluated. For the fundamental Gaussian beam, the reactive power vanishes, and the normalization is chosen such that the real power is 2W. The various full Gaussian waves are identified by the length parameter bt that lies in the range 0btb, where b is the Rayleigh distance. The other parameters are the wavenumber k, the free-space wavelength λ, and the beam waist w0 at the input plane. The dependence of the real power of the full Gaussian light wave on btb and w0λ is examined. For a specified w0λ, the reactive power, which can be positive or negative, increases as btb is increased from 0 to 1 and becomes infinite for btb=1. For a specified btb, the reactive power approaches zero as kw0 is increased and reaches the limiting value of zero of the paraxial beam.

© 2009 Optical Society of America

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References

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2009 (1)

2008 (1)

2007 (1)

1998 (1)

1979 (1)

1976 (1)

1971 (1)

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

Agrawal, G. P.

Deschamps, G. A.

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

Felsen, L. B.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Pattanayak, D. N.

Seshadri, S. R.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

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Figures (3)

Fig. 1
Fig. 1

Real power P Re in watts of the extended full Gaussian wave as functions of b t b for 0 < b t b < 1 where b is the Rayleigh distance. Other parameters are (a) w 0 λ = 0.2487 and (b) w 0 λ = 0.4723 . The normalization is such that the total time-averaged power transported by the paraxial beam in the + z and the z directions is P p + + P p = 2 W .

Fig. 2
Fig. 2

Reactive power P Im in watts of the extended full Gaussian wave as a function of b t b for 0 < b t b < 1 and w 0 λ = 0.2487 . P Im = 0 occurs at b t b = 0.5905 . The normalization is the same as in Fig. 1.

Fig. 3
Fig. 3

Reactive power P Im in watts of the extended full Gaussian wave as a function of k w 0 for 0.2 < k w 0 < 1.2 and b t b = 0.5 . The normalization is the same as in Fig. 1.

Equations (41)

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A ¯ x 0 ± ( p x , p y , z ) = exp ( ± i k z ) A ¯ x 0 ± ( p x , p y , 0 ) × exp [ π 2 w 0 2 ( p x 2 + p y 2 ) i | z | b ] ,
A x 0 ± ( x , y , z ) = exp ( ± i k z ) N i k q ± 2 exp [ q ± 2 ( x 2 + y 2 ) w 0 2 ] ,
q ± = ( 1 ± i z b ) 1 2 ,
N = ( 4 c π w 0 2 ) 1 2 ,
J 0 ( x , y , z ) = z ̂ × y ̂ [ H y 0 + ( x , y , 0 ) H y 0 ( x , y , 0 ) ] δ ( z ) = x ̂ 2 N exp [ ( x 2 + y 2 ) w 0 2 ] δ ( z ) .
P C 0 = c 2 d x d y d z E ( x , y , z ) J * ( x , y , z ) = c N 2 d x d y exp [ 2 ( x 2 + y 2 ) w 0 2 ] = c N 2 π w 0 2 2 = 2 W .
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 ) ] ,
ζ = [ k 2 4 π 2 ( p x 2 + p y 2 ) ] 1 2 .
E x ( x , y , z ) = i k ( 1 + 1 k 2 2 x 2 ) A x ( x , y , z ) ,
H y ( x , y , z ) = z A x ( x , y , z ) .
E x ± ( x , y , z ) = N 2 π b exp ( k b t ) d p x d p y × exp [ i 2 π ( p x x + p y y ) ] ( 1 4 π 2 p x 2 k 2 ) × 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 ) ] ,
H y ± ( x , y , z ) = ± N π w 0 2 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 ) ] × exp [ i ζ ( | z | i b t ) ] .
J ( x , y , z ) = z ̂ × y ̂ [ H y + ( x , y , 0 ) H y ( x , y , 0 ) ] δ ( z ) = x ̂ 2 H y + ( x , y , 0 ) δ ( z ) = δ ( z ) x ̂ 2 N π w 0 2 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 ) ] exp ( ζ ¯ b t ) ,
P C = c 2 d x d y d z E ( x , y , z ) J * ( x , y , z ) = c 2 d x d y { N 2 π b exp ( k b t ) d p x d p y × exp [ i 2 π ( p x x + p y y ) ] ( 1 4 π 2 p x 2 k 2 ) × exp [ π 2 w 0 2 ( p x 2 + p y 2 ) ( 1 b t b ) ] ζ 1 exp ( ζ b t ) × ( 2 ) N π w 0 2 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 ) ] exp ( ζ ¯ * b t ) } .
P C = c N 2 π w 0 2 2 π b exp ( 2 k b t ) d p x d p y × ( 1 4 π 2 p x 2 k 2 ) exp [ 2 π 2 w 0 2 ( p x 2 + p y 2 ) ( 1 b t b ) ] × ζ 1 exp [ ( ζ + ζ * ) b t ] .
2 π p x = p cos ϕ , 2 π p y = p sin ϕ .
P C = P Re + i P Im = w 0 2 π exp ( 2 k b t ) 0 d p p 0 2 π d ϕ ( 1 p 2 cos 2 ϕ k 2 ) × exp [ w 0 2 2 ( 1 b t b ) p 2 ] ξ 1 exp [ k b t ( ξ + ξ * ) ] ,
ξ = ( 1 p 2 k 2 ) 1 2 .
P Re = w 0 2 π exp ( 2 k b t ) 0 k d p p 0 2 π d ϕ ( 1 p 2 cos 2 ϕ k 2 ) × exp [ w 0 2 2 ( 1 b t b ) p 2 ] ξ 1 exp [ 2 k b t ξ ] .
p = k sin θ ,
P Re = 2 P t ,
P t = 0 π 2 d θ sin θ 0 2 π d ϕ 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 ) ] ,
P Im = w 0 2 π exp ( 2 k b t ) k d p p 0 2 π d ϕ ( 1 p 2 cos 2 ϕ k 2 ) × exp [ w 0 2 2 ( 1 b t b ) p 2 ] ξ Im 1 ,
ξ Im = ( p 2 k 2 1 ) 1 2 .
p 2 k 2 = 1 + τ 2 .
P Im = k 2 w 0 2 exp [ k 2 w 0 2 2 ( 1 + b t b ) ] × 0 d τ ( 1 τ 2 ) exp [ k 2 w 0 2 2 ( 1 b t b ) τ 2 ] .
P Im = ( π 2 ) 1 2 k w 0 ( 1 b t b ) 1 2 exp [ k 2 w 0 2 2 ( 1 + b t b ) ] × [ 1 1 k 2 w 0 2 ( 1 b t b ) ] .
F ¯ 0 ( p x , p y , z ) = exp ( ± i k z ) f ¯ s ( p x , p y ) × exp [ π 2 w 0 2 ( p x 2 + p y 2 ) i | z | b ] ,
F ¯ ( p x , p y , z ) = f ¯ s ( p x , p y ) exp ( i ζ | z | ) ,
J ¯ 0 ( p x , p y , z ) = 2 i k f ¯ s ( p x , p y ) δ ( z ) ,
J ¯ ( p x , p y , z ) = 2 i ζ f ¯ s ( p x , p y ) δ ( z ) .
F ¯ ( p x , p y , z ) = f ¯ s ( p x , p y ) ( k ζ ) exp ( i ζ | z | )
F ¯ ( p x , p y , 0 ) = f ¯ s ( p x , p y ) ( k ζ ) ,
× E = i k H
× H = i k E + x ̂ J x ,
E = i k ( x ̂ + 1 k 2 x ) A x ,
H = x ̂ × A x ,
( 2 + k 2 ) A x = J x .
E 0 ± = x ̂ i k A x 0 ± ,
H 0 ± = ± y ̂ i k A x 0 ± .

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