Abstract

The full wave theory of focused waves is developed and the radiation intensity distribution is determined. In the appropriate limit, the full wave theory correctly reproduces the paraxial beams. The limitations of paraxial beam theories are discussed. The method of treatment of full waves is presented with reference to the scalar Bessel–Gauss beam and wave. The necessary theoretical formulas for other beams and waves are also given. For the scalar Bessel–Gauss wave, the beam shape parameter can be adjusted to yield a flat-topped radiation pattern. The ratio of the power in the paraxial beam to that in the full wave is used as a parameter to measure the quality of the paraxial beam approximation. Lower-order waves are found to have better paraxial beam quality than do higher-order waves. The difference in the paraxial beam quality increases as kw0 is decreased where k is the wavenumber and w0 is the waist of the paraxial beam. The radiation patterns of waves are presented for some tightly focused waves.

© 2006 Optical Society of America

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References

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  1. C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579-1587 (2001) and references cited therein, in particular Refs. 18-20, 34, and 51.
    [CrossRef]
  2. S. R. Seshadri, "Virtual source for the Bessel-Gauss beam," Opt. Lett. 27, 998-1000 (2002).
    [CrossRef]
  3. G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  4. L. B. Felsen, "Evanescent waves," J. Opt. Soc. Am. 66, 751-760 (1976).
    [CrossRef]
  5. E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. A 18, 1588-1611 (2001) and references cited therein.
    [CrossRef]
  6. An anonymous reviewer for Applied Optics suggested this parameter for the estimation of the paraxial beam quality.
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  8. C. J. R. Sheppard and T. Wilson, "Gaussian-beam theory of lenses with annular aperture," IEE J. Microwaves , Opt. Acoust. 2, 105-112 (1978).
    [CrossRef]
  9. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  10. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).
    [CrossRef]
  11. R. Borghi, M. Santarsiero, and M. A. Porras, "Nonparaxial Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 1618-1626 (2001).
    [CrossRef]
  12. S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971).
  13. S. R. Seshadri, "Partially coherent Gaussian Schell-model electromagnetic beams," J. Opt. Soc. Am. A 16, 1373-1380 (1999).
    [CrossRef]
  14. S. R. Seshadri, "Dynamics of the linearly polarized fundamental Gaussian light wave," presented on July 7, 2005 at the 2005 IEEE AP-S International Symposium and at the USNC/URSI National Radio Science Meeting, Washington, D.C., July 3-8, 2005.

2002 (1)

2001 (3)

1999 (1)

1996 (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).
[CrossRef]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

1978 (1)

C. J. R. Sheppard and T. Wilson, "Gaussian-beam theory of lenses with annular aperture," IEE J. Microwaves , Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

1976 (1)

1971 (1)

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

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).
[CrossRef]

Borghi, R.

Deschamps, G. A.

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

Felsen, L. B.

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gori, F.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Heyman, E.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Porras, M. A.

Santarsiero, M.

R. Borghi, M. Santarsiero, and M. A. Porras, "Nonparaxial Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 1618-1626 (2001).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).
[CrossRef]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).
[CrossRef]

Seshadri, S. R.

S. R. Seshadri, "Virtual source for the Bessel-Gauss beam," Opt. Lett. 27, 998-1000 (2002).
[CrossRef]

S. R. Seshadri, "Partially coherent Gaussian Schell-model electromagnetic beams," J. Opt. Soc. Am. A 16, 1373-1380 (1999).
[CrossRef]

S. R. Seshadri, "Dynamics of the linearly polarized fundamental Gaussian light wave," presented on July 7, 2005 at the 2005 IEEE AP-S International Symposium and at the USNC/URSI National Radio Science Meeting, Washington, D.C., July 3-8, 2005.

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

Sheppard, C. J. R.

C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579-1587 (2001) and references cited therein, in particular Refs. 18-20, 34, and 51.
[CrossRef]

C. J. R. Sheppard and T. Wilson, "Gaussian-beam theory of lenses with annular aperture," IEE J. Microwaves , Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

Spagnolo, G. Schirripa

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).
[CrossRef]

Wilson, T.

C. J. R. Sheppard and T. Wilson, "Gaussian-beam theory of lenses with annular aperture," IEE J. Microwaves , Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

Electron. Lett. (1)

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

IEE J. Microwaves (1)

C. J. R. Sheppard and T. Wilson, "Gaussian-beam theory of lenses with annular aperture," IEE J. Microwaves , Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Opt. 43, 1155-1166 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Opt. Lett. (1)

Other (4)

An anonymous reviewer for Applied Optics suggested this parameter for the estimation of the paraxial beam quality.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

S. R. Seshadri, "Dynamics of the linearly polarized fundamental Gaussian light wave," presented on July 7, 2005 at the 2005 IEEE AP-S International Symposium and at the USNC/URSI National Radio Science Meeting, Washington, D.C., July 3-8, 2005.

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

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Figures (4)

Fig. 1
Fig. 1

(Color online) Φ(θ, ϕ) as a function of θ for 0 < θ < 90° and w 0∕λ = 0.4. (a) Scalar Gauss wave, β = 0, and (b) scalar Bessel–Gauss wave, βw 0 = 0.7π.

Fig. 2
Fig. 2

(Color online) Quality of the paraxial beam, Pp P, as a function of w 0∕λ in the range 0.05 < w 0∕λ < 1 for (a) scalar Gauss wave and (b) scalar Bessel–Gauss wave with βw 0 = 0.7π.

Fig. 3
Fig. 3

(Color online) Φ(θ, ϕ) as a function of θ for 0 < θ < 90° and w 0∕λ = 0.2. (a) Scalar Gauss wave, (b) linearly polarized Gaussian light wave, b1: ϕ = 0, b2: ϕ = 30°, b3: ϕ = 60°, and b4: ϕ = 90°, (c) cylindrically symmetric TE wave, and (d) azimuthally varying dipolar light wave, d1: ϕ = 0, d2: ϕ = 30° and d3: ϕ = 60°.

Fig. 4
Fig. 4

(Color online) Quality of the paraxial beam, Pp P, as a function of w 0∕λ in the range 0.05 < w 0∕λ < 0.5 for curve aa, scalar Gauss wave; curve bb, linearly polarized Gaussian light wave; curve cc, cylindrically symmetric TE wave; and curve dd, azimuthally varying dipolar light wave.

Equations (92)

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( 2 ρ 2 + 1 ρ ρ + 2 z 2 + k 2 ) F ( ρ , z ) = 0 ,
F ( ρ , z ) = exp ( i k z ) F ˜ ( ρ , z ) ,
( 2 ρ 2 + 1 ρ ρ + 2 i k z + 2 z 2 ) F ˜ ( ρ , z ) = 0.
b = k w 0 2 / 2 .
( 2 ρ 2 + 1 ρ ρ + 2 i k z ) F ˜ p ( ρ , z ) = 0.
F ˜ p ( ρ , z ) = 0 d η η J 0 ( η ρ ) F ¯ p ( η , z ) ,
F ¯ p ( η , z ) = 0 d ρ ρ J 0 ( η ρ ) F ˜ p ( ρ , z ) ,
F ¯ p ( η , z ) = F ¯ p ( η , 0 ) exp ( i η 2 z / 2 k ) .
F ˜ p ( ρ , 0 ) = N exp ( γ ) J 0 ( β ρ ) exp ( ρ 2 / w 0 2 ) ,
γ = β 2 w 0 2 / 4 ,
F ¯ p ( η , 0 ) = ( N b / k ) I 0 ( η d ) exp ( η 2 w 0 2 / 4 ) ,
d = β w 0 2 / 2
F ( ρ , z ) = exp ( i k z ) ( N b / k ) 0 d η η J 0 ( η ρ ) I 0 ( η d ) × exp [ η 2 w 0 2 4 ( 1 + i z b ) ] .
S z ( ρ , z ) = ( ω k / 2 ) F * ( ρ , z ) F ( ρ , z ) ,
P p = 0 2 π 0 d ρ ρ ω k 2 ( N b k ) 2 0 d η η J 0 ( η ρ ) × J 0 ( i η d ) exp [ η 2 w 0 2 4 ( 1 i z b ) ] × 0 d η ¯ η ¯ J 0 ( η ¯ ρ ) × J 0 ( i η ¯ d ) exp [ η ¯ 2 w 0 2 4 ( 1 + i z b ) ] .
0 d ρ ρ J 0 ( η ρ ) J 0 ( η ¯ ρ ) = δ ( η η ¯ ) / η ,
P p = ( π ω k N 2 b 2 / k 2 ) 0 d η η J 0 ( i η d ) J 0 ( i η d ) × exp ( η 2 w 0 2 / 2 )
= 2 2 ω k π w 0 2 N 2 exp ( d 2 w 0 - 2 ) I 0 ( d 2 w 0 - 2 ) ,
N = [ 4 / ω k π w 0 2 exp ( γ ) I 0 ( γ ) ] 1 / 2 .
S ex = i 4 π b exp ( k b ) N ;
ρ ex = i β w 0 2 / 2 = i d ,
z = z ex = i b ,
( 2 ρ 2 + 1 ρ ρ + 2 z 2 + k 2 ) F ( ρ , z ) = S ex δ ( ρ ρ ex ) 2 π ρ δ ( z z ex ) .
( 2 z 2 + ζ 2 ) F ¯ ( η , z ) = S ex 2 π J 0 ( i η d ) δ ( z i b ) ,
ζ = ( k 2 η 2 ) 1 / 2 for   k > η and
ζ = i ( η 2 k 2 ) 1 / 2 for   k < η .
F ¯ ( η , z ) = i S ex ( 4 π ζ ) 1 J 0 ( i η d ) exp [ i ζ ( z i b ) ] .
F ( ρ , z ) = N b exp ( k b ) 0 d η η J 0 ( η ρ ) × J 0 ( i η d ) ζ 1 exp [ i ζ ( z i b ) ] .
S z ( ρ , z ) = 2 1 Re [ i ω F * ( ρ , z ) ( / z ) F ( ρ , z ) ] .
P = 0 2 π 0 ρ 2 1 ω N 2 b 2 exp ( 2 k b ) × Re { 0 d η η J 0 ( η ρ ) J 0 ( i η d ) ( ζ * ) 1 × exp [ i ζ * ( z + i b ) ] 0 d η ¯ η ¯ J 0 ( η ¯ ρ ) J 0 ( i η ¯ d ) × exp [ i ζ ¯ ( z i b ) ] } ,
P = 2 1 ω N 2 b 2 exp ( 2 k b ) 0 2 π d ϕ Re × { 0 2 π d η η J 0 ( i η d ) J 0 ( i η d ) ( ζ * ) 1 × exp [ i ζ * ( z + i b ) ] exp [ i ζ ( z i b ) ] } .
P = 2 1 ω N 2 b 2 exp ( 2 k b ) × 0 2 π d ϕ 0 k η I 0     2 ( η d ) ζ 1 exp ( 2 ζ b ) .
η = k sin θ and ζ = k cos θ .
P = 0 2 π 0 π / 2 d θ sin θ Φ ( θ , ϕ ) ,
Φ ( θ , ϕ ) = I 0 2 ( k d sin θ ) 2 π f 0 2 exp ( γ ) I 0 ( γ ) exp [ k 2 w 0 2 ( 1 cos θ ) ] .
Φ ( θ , ϕ ) = ( 2 π f 0 2 ) 1 exp [ k 2 w 0 2 ( 1 cos θ ) ] , for β = 0.
P = 1 exp ( k 2 w 0 2 ) for β = 0 .
H x = E y = exp ( i k z ) N ( 1 + i z / b ) 2 [ 1 2 x 2 w 0 2 ( 1 + i z / b ) ] × exp [ x 2 + y 2 w 0 2 ( 1 + i z / b ) ] ,
H y = E x = exp ( i k z ) N ( 1 + i z / b ) 3 2 x y w 0 2 exp [ x 2 + y 2 w 0 2 ( 1 + i z / b ) ] .
S z = c 2 Re ( E x H y * E y H x * ) = c 2 N 2 ( 1 + z 2 / b 2 ) 2 × [ 1 4 x 2 w 0 2 ( 1 + z 2 / b 2 ) + 4 x 2 ( x 2 + y 2 ) w 0 4 ( 1 + z 2 / b 2 ) ] × exp [ 2 ( x 2 + y 2 ) w 0 2 ( 1 + z 2 / b 2 ) ] .
P 1 + 2 = d x d y c 2 N 2 ( 1 + z 2 / b 2 ) 2 × [ 1 4 x 2 w 0 2 ( 1 + z 2 / b 2 ) ] × exp [ 2 ( x 2 + y 2 ) w 0 2 ( 1 + z 2 / b 2 ) ] = 0.
P p = c 2 N 2 4 ( 1 + z 2 / b 2 ) 3 w 0 4 d x d y ( x 4 + x 2 y 2 ) × exp [ 2 ( x 2 + y 2 ) w 0 2 ( 1 + z 2 / b 2 ) ] = c 4 π w 0 2 N 2 .
x 2 1 + 2 = 1 P p 2 π w 0 2 ( 1 + z 2 / b 2 ) 2 d x d y x 2 × [ 1 4 x 2 w 0 2 ( 1 + z 2 / b 2 ) ] × exp [ 2 ( x 2 + y 2 ) w 0 2 ( 1 + z 2 / b 2 ) ] = w 0 2 2 0.
ρ 2 = ( w 0 2 / 2 ) ( a 0 + a 2 z 2 / b 2 ) .
a 0 = m = 0 m = M m ! f 0 2 m , a 2 = m = 0 m = M ( m + 2 ) ! 2 f 0 2 m .
F ( ρ , z ) = exp ( i k z ) N b k 0 d η η J 0 ( η ρ ) × exp [ η 2 w 0 2 ( 1 + i z / b ) 4 ] .
S z ( ρ , z ) = ( ω k / 2 ) F 0 * ( ρ , z ) F 0 ( ρ , z ) .
P p = 0 2 π d ϕ 0 d ρ ρ S z ( ρ , z ) = 1 ,
N = ( 4 / πω k w 0 2 ) 1 / 2 .
F ( ρ , z ) = N b exp ( k b ) 0 d η η J 0 ( η ρ ) ζ - 1 × exp [ i ζ ( z i b ) ] ,
S z ( ρ , z ) = 1 2 Re [ i ω F 0 * ( ρ , z ) z F 0 ( ρ , z ) ] ,
P = 0 2 π 0 dρρ S z ( ρ , z ) = 0 2 π 0 π / 2   sin   θΦ ( θ , ϕ ) .
Φ ( θ , ϕ ) = ( 2 π f 0     2 ) 1 exp [ k 2 w 0 2 ( 1 cos θ ) ] ,
f 0 = 1 / k w 0 .
P = 1 exp ( k 2 w 0 2 ) .
For     k w 0 0 , P = k 2 w 0 2 .
E ϕ ( ρ , z ) = H ρ ( ρ , z ) = exp ( i k z ) N b k 0 d η η 2 J 1 ( η ρ ) × exp [ η 2 w 0 2 ( 1 + i z / b ) 4 ] ,
S z ( ρ , z ) = ( c / 2 ) Re [ E ϕ ( ρ , z ) H ρ * ( ρ , z ) ] ,
P p = 0 2 π d ϕ 0 d ρ ρ S z ( ρ , z ) = 1 ,
N = ( 2 / π c ) 1 / 2 .
E ϕ ( ρ , z ) = N b exp ( k b ) 0 d η η 2 J 1 ( η ρ ) ζ - 1 × exp [ i ζ ( z i b ) ] ,
H ρ ( ρ , z ) = ( N b / k ) exp ( k b ) 0 d η η 2 J 1 ( η ρ ) × exp [ i ζ ( z i b ) ] ,
P = 0 2 π d ϕ 0 π / 2 d θ sin θ Φ ( θ , ϕ ) ,
Φ ( θ , ϕ ) = ( 4 π f 0 4 ) 1 sin 2 θ exp [ k 2 w 0 2 ( 1 cos θ ) ] ,
P = 1 f 0 2 ( 0.5 k 2 w 0 2 f 0     2 ) exp ( k 2 w 0 2 ) .
For     k w 0 0 , P = ( 1 / 3 ) k 4 w 0     4 .
H x ( x , y , z ) = E y ( x , y , z )
= exp ( i k z ) 2 π 3 w 0 4 N d p x d p y × exp [ i 2 π ( p x x + p y y ) ] p x 2 × exp [ π 2 w 0 2 ( p x 2 + p y 2 ) ( 1 + i z / b ) ] ,
H y ( x , y , z ) = E x ( x , y , z )
= exp ( i k z ) 2 π 3 w 0 4 N d p x d p y × exp [ i 2 π ( p x x + p y y ) ] p x p y × exp [ π 2 w 0 2 ( p x 2 + p y 2 ) ( 1 + i z / b ) ] ,
S z ( x , y , z ) = c 2 Re [ E x ( x , y , z ) H y * ( x , y , z ) E y ( x , y , z ) H x * ( x , y , z ) ] ,
P p = S z ( x , y , z ) d x d y = 1 ,
N = ( 4 / π c w 0     2 ) 1 / 2 .
E x ( x , y , z ) = 2 π 3 k w 0 4 exp ( k b ) N d p x d p y × exp [ i 2 π ( p x x + p y y ) ] × p x p y ζ 1 exp [ i ζ ( z i b ) ] ,
E y ( x , y , z ) = 2 π 3 k w 0 4 exp ( k b ) N d p x d p y × exp [ i 2 π ( p x x + p y y ) ] × p x 2 ζ 1 exp [ i ζ ( z i b ) ] ,
H x ( x , y , z ) = 2 π 3 w 0 4 exp ( k b ) N d p x d p y × exp [ i 2 π ( p x x + p y y ) ] × p x 2 exp [ i ζ ( z i b ) ] ,
H y ( x , y , z ) = 2 π 3 w 0 4 exp ( k b ) N d p x d p y × exp [ i 2 π ( p x x + p y y ) ] × p x p y exp [ i ζ ( z i b ) ] .
ζ = [ k 2 4 π 2 ( p x 2 + p y 2 ) ] 1 / 2 , for   k 2 < 4 π 2 ( p x 2 + p y 2 ) ,
ζ = i [ 4 π 2 ( p x 2 + p y 2 ) k 2 ] 1 / 2 , for   k 2 > 4 π 2 ( p x 2 + p y 2 ) ,
P = 0 π / 2 d ϕ 0 π / 2 sin θ Φ ( θ , ϕ ) ,
Φ ( θ , ϕ ) = ( 8 π f 0 6 ) 1 cos 2 ϕ sin 4 θ × exp [ k 2 w 0 2 ( 1 cos θ ) ] ,
P = 1 3 f 0 2 + 3 f 0 4 ( 0.125 k 4 w 0 4 0.5 + 3 f 0 4 ) exp ( k 2 w 0 2 ) .
For   k w 0 0 , P = ( 1 / 15 ) k 6 w 0 6 .
F ( ρ , z ) = exp ( i k z ) N b k 0 d η η J 0 ( η ρ ) I 0 ( ηd ) × exp [ η 2 w 0 2 ( 1 + i z / b ) 4 ] ,
d = β w 0 2 / 2 ; β = 0   leads   to   the fundamental   scalar Gaussian   beam .
S z ( ρ , z ) = ( ω k / 2 ) F 0 * ( ρ , z ) F 0 ( ρ , z ) ,
P p = 0 2 π d ϕ 0 d ρ ρ S z ( ρ , z ) = 1 ,
N = [ 4 / ω k π w 0 2 γ I 0 ( γ ) ] 1 / 2 , γ = β 2 w 0 2 / 4 .
F ( ρ , z ) = N b exp ( k b ) 0 d η η J 0 ( η ρ ) I 0 ( η d ) ζ - 1 × exp [ i ζ ( z i b ) ] ,
S z ( ρ , z ) = 1 2 Re [ i ω F 0 * ( ρ , z ) z F 0 ( ρ , z ) ] ,
P = 0 2 π d ϕ 0 d ρ ρ S z ( ρ , z ) = 0 2 π d ϕ 0 π / 2 d θ sin θ Φ ( θ , Φ ) ,
Φ ( θ , ϕ ) = I 0 2 ( k d   sin   θ ) exp [ k 2 w 0 2 ( 1 cos θ ) ] 2 π f 0 2 exp ( γ ) I ( γ ) .

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