Abstract

The linearly polarized real-argument Hermite–Gauss beam is investigated by the Fourier transform method. The complex power is obtained and the reactive power of the paraxial beam is found to be zero. The complex space source required for the full-wave generalization of the real-argument Hermite–Gauss beam is deduced. The resulting basic full real-argument Hermite–Gauss wave is determined. The real and the reactive powers of the full wave are evaluated. The reactive power of the basic full real-argument Hermite–Gauss wave is infinite, and the reasons for this singularity are described. The real power depends on kw0, m, and n, where k is the wavenumber, w0 is the e-folding distance of the Gaussian part of the input distribution, and m and n are the mode numbers. The variation in the real power with respect to changes in kw0 for specified m and n as well as with respect to changes in m and n for a specified kw0 is examined.

© 2010 Optical Society of America

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  1. S. R. Seshadri, “Reactive power in the full Gaussian light wave,” J. Opt. Soc. Am. A 26, 2427–2433 (2009).
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    [CrossRef]
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    [CrossRef]
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  6. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  7. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), Chap. 6.
  8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).
  9. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1965).
  10. S. R. Seshadri, “Dynamics of the linearly polarized fundamental Gaussian light wave,” J. Opt. Soc. Am. A 24, 482–492 (2007).
    [CrossRef]
  11. S. R. Seshadri, “Full-wave generalizations of the fundamental Gaussian beam,” J. Opt. Soc. Am. A 26, 2515–2520 (2009).
    [CrossRef]
  12. S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971).

2009

2007

1995

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

1976

1972

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), Chap. 6.

1971

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

S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971).

1966

1965

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1965).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1965).

Deschamps, G. A.

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

Felsen, L. B.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

Kogelnik, H.

Li, T.

Mandel, L.

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

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), Chap. 6.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

Seshadri, S. R.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1965).

Wolf, E.

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

Appl. Opt.

Electron. Lett.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971).

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

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), Chap. 6.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, 1965).

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Figures (3)

Fig. 1
Fig. 1

Real power P r e as a function of k w 0 for 0.2 < k w 0 < 10 and (a), ( m , n ) = ( 0 , 0 ) . (b), ( m , n ) = ( 1 , 0 ) . (c), ( m , n ) = ( 2 , 0 ) . The normalization is such that the real power in the corresponding paraxial beam is 2 W.

Fig. 2
Fig. 2

Same as in Fig. 1 but for (a), ( m , n ) = ( 0 , 0 ) . (b), ( m , n ) = ( 0 , 1 ) . (c), ( m , n ) = ( 0 , 2 ) .

Fig. 3
Fig. 3

Same as in Fig. 1 but for (a), ( m , n ) = ( 1 , 0 ) . (b), ( m , n ) = ( 0 , 1 ) . (c), ( m , n ) = ( 1 , 1 ) .

Equations (79)

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a x 0 ± ( x , y , 0 ) = A x 0 ± ( x , y , 0 ) = N m n i k H m ( 2 x w 0 ) H n ( 2 y w 0 ) exp ( x 2 + y 2 w 0 2 ) ,
H 0 ( v ) = 0.
A x 0 ± ( x , y , z ) = exp ( ± i k z ) a x 0 ± ( x , y , z ) ,
( 2 x 2 + 2 y 2 ± 2 i k z ) a x 0 ± ( x , y , z ) = 0.
[ z ± i b π 2 w 0 2 ( p x 2 + p y 2 ) ] a ¯ x 0 ± ( p x , p y , z ) = 0 ,
a ¯ x 0 ± ( p x , p y , z ) = a ¯ x 0 ± ( p x , p y , 0 ) exp [ π 2 w 0 2 ( p x 2 + p y 2 ) i | z | b ] .
I m ( p x ) = d x   exp ( i 2 π p x x ) H m ( 2 x w 0 ) exp ( x 2 w 0 2 ) .
I m ( p x ) = π w 0 i m H m ( 2 π w 0 p x ) exp ( π 2 w 0 2 p x 2 ) .
a ¯ x 0 ± ( p x , p y , 0 ) = A ¯ x 0 ± ( p x , p y , 0 ) = N m n i k π w 0 2 i m + n H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) exp [ π 2 w 0 2 ( p x 2 + p y 2 ) ] .
a ¯ x 0 ± ( p x , p y , z ) = N m n i k π w 0 2 i m + n H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) exp [ π 2 w 0 2 ( p x 2 + p y 2 ) q ± 2 ] ,
q ± = ( 1 ± i z b ) 1 / 2 = q .
a x 0 ± ( x , y , z ) = N m n i k I m ( x , z ) I n ( y , z ) ,
I m ( x , z ) = π 1 / 2 w 0 i m d p x   exp ( i 2 π p x x ) H m ( 2 π w 0 p x ) exp [ π 2 w 0 2 p x 2 q ± 2 ] ,
a x 0 ± ( x , y , z ) = N m n i k q ± m + n + 2 q ± ( m + n ) H m ( 2 q ± q ± x w 0 ) H n ( 2 q ± q ± y w 0 ) exp [ q ± 2 ( x 2 + y 2 ) w 0 2 ] .
A x 0 ± ( x , y , z ) = exp ( ± i k z ) N m n i k q ± m + n + 2 q ± ( m + n ) H m ( 2 q ± q ± x w 0 ) H n ( 2 q ± q ± y w 0 ) exp [ q ± 2 ( x 2 + y 2 ) w 0 2 ] .
A ¯ x 0 ± ( p x , p y , z ) = exp ( ± i k z ) a ¯ x 0 ± ( p x , p y , z ) .
E x 0 ± ( x , y , z ) = ± H y 0 ± ( x , y , z ) = i k A x 0 ± ( x , y , z ) .
J 0 ( x , y , z ) = z ̂ × y ̂ [ H y + ( x , y , 0 ) H y ( x , y , 0 ) ] δ ( z ) = x ̂ 2 i k A x 0 ± ( x , y , 0 ) δ ( z ) .
P C = P r e + i P i m = c 2 d x d y d z E 0 ± ( x , y , z ) J 0 ( x , y , z ) = c k 2 d x d y a x 0 ± ( x , y , 0 ) a x 0 ± ( x , y , 0 ) ,
P i m = 0.
P r e = c N m n 2 d x H m 2 ( 2 x w 0 ) exp ( 2 x 2 w 0 2 ) d y H n 2 ( 2 y w 0 ) exp ( 2 y 2 w 0 2 ) .
d x H m ( a x ) H ( a x ) exp ( a 2 x 2 ) = 2 m m ! π 1 / 2 a δ m ,
P r e = c π w 0 2 2 m + n 1 m ! n ! N m n 2 .
N m n = ( 4 c π w 0 2 2 m + n m ! n ! ) 1 / 2 ,
P r e = 2   W .
P RH + = P RH = 1 2 P r e = 1   W .
C s , m n ( x , y , z ) = N m n i k π w 0 2 i m + n d p x d p y exp [ i 2 π ( p x x + p y y ) ] H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) .
C s , m n ( x , y , z ) = S e x N m n i k π w 0 2 i m + n δ ( z ) d p x d p y exp [ i 2 π ( p x x + p y y ) ] H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) .
( 2 x 2 + 2 y 2 + 2 z 2 + k 2 ) G m , n ( x , y , z ) = S e x N m n i k π w 0 2 i m + n δ ( z ) d p x d p y exp [ i 2 π ( p x x + p y y ) ] H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) .
( 2 z 2 + ζ 2 ) G ¯ m , n ( p x , p y , z ) = S e x N m n i k π w 0 2 i m + n H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) δ ( z ) ,
ζ = [ k 2 4 π 2 ( p x 2 + p y 2 ) ] 1 / 2 .
G ¯ m , n ( p x , p y , z ) = i S e x 2 N m n i k π w 0 2 i m + n H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) ζ 1   exp ( i ζ | z | ) .
A x ± ( x , y , z ) = i S e x 2 N m n i k π w 0 2 i m + n d p x d p y exp [ i 2 π ( p x x + p y y ) ] H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) ζ 1   exp [ i ζ ( | z | i b ) ] .
A x 0 ± ( x , y , z ) = exp ( ± i k z ) i S e x 2 k exp ( k b ) N m n i k π w 0 2 i m + n d p x d p y   exp [ i 2 π ( p x x + p y y ) ] H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) exp [ π 2 w 0 2 ( p x 2 + p y 2 ) q ± 2 ] .
S e x = i 2 k   exp ( k b ) .
A x ± ( x , y , z ) = k   exp ( k b ) N m n i k π w 0 2 i m + n d p x d p y   exp [ i 2 π ( p x x + p y y ) ] H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) ζ 1   exp [ i ζ ( | z | i b ) ] .
H y ± ( x , y , z ) = ± exp ( k b ) N m n π w 0 2 i m + n d p x d p y   exp [ i 2 π ( p x x + p y y ) ] H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) exp [ i ζ ( | z | i b ) ] .
J ( x , y , z ) = z ̂ × y ̂ [ H y + ( x , y , 0 ) H y ( x , y , 0 ) ] δ ( z ) = x ̂ 2   exp ( k b ) N m n π w 0 2 i m + n δ ( z ) d p ¯ x d p ¯ y   exp [ i 2 π ( p ¯ x x + p ¯ y y ) ] H m ( 2 π w 0 p ¯ x ) H n ( 2 π w 0 p ¯ y ) exp ( ζ ¯ b ) ,
P C = P r e + i P i m = c 2 d x d y d z E ± ( x , y , z ) J ( x , y , z ) .
P C = P r e + i P i m = c 2 d x d y E x ± ( x , y , 0 ) J x ( x , y , 0 ) ,
E x ± ( x , y , z ) = k   exp ( k b ) N m n π w 0 2 i m + n 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 ) H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) ζ 1   exp [ i ζ ( | z | i b ) ] .
P C = c 2 d x d y k   exp ( k b ) N m n π w 0 2 i m + n 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 ) H m ( 2 π w 0 p x ) H n ( 2 π w 0 p y ) ζ 1 exp ( ζ b ) ( 2 ) exp ( k b ) N m n π w 0 2 ( i ) m + n d p ¯ x d p ¯ y   exp [ i 2 π ( p ¯ x x + p ¯ y y ) ] H m ( 2 π w 0 p ¯ x ) H n ( 2 π w 0 p ¯ y ) exp ( ζ ¯ b ) .
P C = c   exp ( 2 k b ) N m n 2 π 2 k w 0 4 d p x d p y ( 1 4 π 2 p x 2 k 2 ) H m 2 ( 2 π w 0 p x ) H n 2 ( 2 π w 0 p y ) ζ 1   exp [ b ( ζ + ζ ) ] .
P C = c N m n 2 w 0 4 4 exp ( 2 k b ) 0 d p p 0 2 π d ϕ ( 1 p 2 cos 2 ϕ k 2 ) H m 2 ( w 0 p   cos   ϕ 2 ) H n 2 ( w 0 p   sin   ϕ 2 ) ξ 1   exp [ k b ( ξ + ξ ) ] ,
ξ = ( 1 p 2 k 2 ) 1 / 2     for   0 < p < k ,
ξ = i ( p 2 k 2 1 ) 1 / 2     for   k < p < .
P i m = w 0 2   exp ( 2 k b ) π 2 m + n m ! n ! k d p p 0 2 π d ϕ ( 1 p 2 cos 2 ϕ k 2 ) H m 2 ( w 0 p   cos   ϕ 2 ) H n 2 ( w 0 p   sin   ϕ 2 ) ( p 2 k 2 1 ) 1 / 2 .
P i m = exp ( k 2 w 0 2 ) w 0 2 ( m + n + 1 ) π m ! n ! k d p p p 2 ( m + n ) 0 2 π d ϕ cos 2 m ϕ sin 2 n ϕ ( 1 p 2 cos 2 ϕ k 2 ) ( p 2 k 2 1 ) 1 / 2 .
0 2 π d ϕ cos 2 m ϕ sin 2 n ϕ = π a 1 ,
0 2 π d ϕ cos 2 ( m + 1 ) ϕ sin 2 n ϕ = π a 2 .
P i m = exp ( k 2 w 0 2 ) w 0 2 ( m + n + 1 ) m ! n ! k d p p p 2 ( m + n ) ( a 1 p 2 k 2 a 2 ) ( p 2 k 2 1 ) 1 / 2 .
P i m = exp ( k 2 w 0 2 ) ( k w 0 ) 2 ( m + n + 1 ) m ! n ! 0 d τ ( 1 + τ 2 ) m + n [ a 1 a 2 ( 1 + τ 2 ) ] = .
P r e = w 0 2 π 2 m + n m ! n ! 0 k d p p 0 2 π d ϕ ( 1 p 2 cos 2 ϕ k 2 ) H m 2 ( w 0 p   cos   ϕ 2 ) H n 2 ( w 0 p   sin   ϕ 2 ) ξ 1   exp [ k 2 w 0 2 ( 1 ξ ) ] .
P r e = k 2 w 0 2 π 2 m + n m ! n ! 0 π / 2 d θ   sin   θ 0 2 π d ϕ ( 1 cos 2 ϕ sin 2 ϕ ) H m 2 ( w 0 k   sin   θ   cos   ϕ 2 ) H n 2 ( w 0 k   sin   θ   sin   ϕ 2 ) exp [ k 2 w 0 2 ( 1 cos   θ ) ] .
P r e = 2 k 2 w 0 2 0 π / 2 d θ   sin   θ ( 1 1 2 sin 2 θ ) exp [ k 2 w 0 2 ( 1 cos   θ ) ]     for   ( m , n ) = ( 0 , 0 ) ,
P r e = k 4 w 0 4 0 π / 2 d θ   sin   θ sin 2 θ ( 1 3 4 sin 2 θ ) exp [ k 2 w 0 2 ( 1 cos   θ ) ]     for   ( m , n ) = ( 1 , 0 ) ,
P r e = k 4 w 0 4 0 π / 2 d θ   sin   θ sin 2 θ ( 1 1 4 sin 2 θ ) exp [ k 2 w 0 2 ( 1 cos   θ ) ]     for   ( m , n ) = ( 0 , 1 ) ,
P r e = k 6 w 0 6 4 0 π / 2 d θ   sin   θ sin 4 θ ( 1 1 2 sin 2 θ ) exp [ k 2 w 0 2 ( 1 cos   θ ) ]     for   ( m , n ) = ( 1 , 1 ) ,
P r e = k 2 w 0 2 2 0 π / 2 d θ   sin   θ [ k 4 w 0 4 ( 3 4 sin 4 θ 5 8 sin 6 θ ) k 2 w 0 2 ( 2 sin 2 θ 3 2 sin 4 θ ) + 2 sin 2 θ ] exp [ k 2 w 0 2 ( 1 cos   θ ) ]     for   ( m , n ) = ( 2 , 0 ) ,
P r e = k 2 w 0 2 2 0 π / 2 d θ   sin   θ [ k 4 w 0 4 ( 3 4 sin 4 θ 1 8 sin 6 θ ) k 2 w 0 2 ( 2 sin 2 θ 1 2 sin 4 θ ) + 2 sin 2 θ ] exp [ k 2 w 0 2 ( 1 cos   θ ) ]     for   ( m , n ) = ( 0 , 2 ) .
H 2 2 ( w 0 k   sin   θ   sin   ϕ 2 ) = 4 ( w 0 4 k 4 sin 4 θ sin 4 ϕ 2 w 0 2 k 2 sin 2 θ sin 2 ϕ + 1 ) .
P r e = k 2 w 0 2 2 π 0 π / 2 d θ   sin   θ 0 2 π d ϕ [ w 0 4 k 4 ( sin 4 θ sin 4 ϕ sin 6 θ sin 4 ϕ cos 2 ϕ ) 2 w 0 2 k 2 ( sin 2 θ sin 2 ϕ sin 4 θ sin 2 ϕ cos 2 ϕ ) + 1 sin 2 θ cos 2 ϕ ] .
I m ( x , z ) = π 1 / 2 w 0 i m lim t = 0 m t m d p x   exp ( i 2 π p x x ) exp [ π 2 w 0 2 p x 2 q ± 2 ] exp ( t 2 + 2 t 2 π w 0 p x ) .
π 1 / 2 w 0 i m d p x   exp [ i 2 π p x ( x i 2 t w 0 ) ] exp [ π 2 w 0 2 p x 2 q ± 2 ] = q ± i m   exp [ q ± 2 ( x + i 2 t w 0 ) 2 w 0 2 ] .
I m ( x , z ) = q ± i m   exp ( q ± 2 x 2 w 0 2 ) lim t = 0 m t m exp [ t 2 α 2 2 t i 2 x q ± 2 w 0 ] ,
α 2 = 1 + 2 q ± 2 .
s = i α t .
I m ( x , z ) = q ± α m   exp ( q ± 2 x 2 w 0 2 ) lim s = 0 m s m exp [ s 2 + 2 s 2 q ± 2 x α w 0 ] .
I m ( x , z ) = q ± α m   exp ( q ± 2 x 2 w 0 2 ) H m ( 2 q ± 2 x α w 0 ) .
α = q ± q ± = q ± q .
I m ( x , z ) = q ± m + 1 q ± m H m ( 2 q ± q ± x w 0 ) exp ( q ± 2 x 2 w 0 2 ) .
C s , m ( x ) = i m d p x   exp ( i 2 π p x x ) H m ( 2 π w 0 p x ) .
C s , m ( x ) = = 0 = m i m m ! ( 1 ) 2 m 2 ! ( m 2 ) ! ( 2 π w 0 ) m 2 d p x exp ( i 2 π p x x ) p x m 2 .
C s , m ( x ) = = 0 = m a m w 0 m 2 m 2 x m 2 δ ( x ) ,
a m = m ! 2 ( m 2 ) / 2 ! ( m 2 ) ! ( 1 ) m     for   0 m ,
C s , n ( y ) = i n d p y   exp ( i 2 π p y y ) H n ( 2 π w 0 p y ) .
C s , n ( y ) = p = 0 p = p n a p n w 0 n 2 p n 2 p y n 2 p δ ( y ) ,
a p n = n ! 2 ( n 2 p ) / 2 p ! ( n 2 p ) ! ( 1 ) n     for   0 p p n ,
C s , m n ( x , y , z ) = N m n i k π w 0 2 = 0 = m a m w 0 m 2 m 2 x m 2 δ ( x ) p = 0 p = p n a p n w 0 n 2 p n 2 p y n 2 p δ ( y ) .

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