Abstract

A boundary-value problem for Maxwell’s equations is formulated whose solutions represent ultrashort pulses of electromagnetic energy that travel along an axis. A paraxial approximation to the solution is introduced that, in the case of Gaussian boundary data, is expressed as a single integral over frequency. Calculations are presented for a pulse of Gaussian cross section and Gaussian time profile. A careful study is made of the error introduced by the paraxial approximation, and an error bound is derived.

© 1985 Optical Society of America

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References

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  1. R. Fork, C. Shank, R. Yen, C. Hirliman, “Femtosecond optical pulses,” IEEE J. Quantum Electron. QE-19, 500–505 (1983);S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag, New York, 1977).
    [CrossRef]
  2. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966);A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1971).
    [CrossRef]
  3. J. Cooper, “Exact and approximate pulsed beam solutions of Maxwell’s equations,” Math. Meth. Appl. Sci. (to be published).
  4. D. Kuizenga, A. Siegman, “FM and AM locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
    [CrossRef]
  5. H. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
    [CrossRef]
  6. A. Friberg, E. Wolf, “Angular spectrum representation of scattered electromagnetic fields,” J. Opt. Soc. Am. 73, 26–32 (1983);E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1969).
    [CrossRef]
  7. E. Marx, “Free-space propagation of light pulses,” National Bureau of Standards Internal Rep. 84–2835 (U.S. Department of Commerce, Washington, D.C., 1984);“Free-field propagation of localized pulses,” IEEE Trans. Antennas Propag. (to be published).
  8. L. Schwartz, Mathematics for the Physical Sciences, (Addison-Wesley, Reading, Mass., 1966).

1983 (2)

R. Fork, C. Shank, R. Yen, C. Hirliman, “Femtosecond optical pulses,” IEEE J. Quantum Electron. QE-19, 500–505 (1983);S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag, New York, 1977).
[CrossRef]

A. Friberg, E. Wolf, “Angular spectrum representation of scattered electromagnetic fields,” J. Opt. Soc. Am. 73, 26–32 (1983);E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1969).
[CrossRef]

1975 (1)

H. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

1970 (1)

D. Kuizenga, A. Siegman, “FM and AM locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966);A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1971).
[CrossRef]

Cooper, J.

J. Cooper, “Exact and approximate pulsed beam solutions of Maxwell’s equations,” Math. Meth. Appl. Sci. (to be published).

Fork, R.

R. Fork, C. Shank, R. Yen, C. Hirliman, “Femtosecond optical pulses,” IEEE J. Quantum Electron. QE-19, 500–505 (1983);S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag, New York, 1977).
[CrossRef]

Friberg, A.

Haus, H.

H. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

Hirliman, C.

R. Fork, C. Shank, R. Yen, C. Hirliman, “Femtosecond optical pulses,” IEEE J. Quantum Electron. QE-19, 500–505 (1983);S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag, New York, 1977).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966);A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1971).
[CrossRef]

Kuizenga, D.

D. Kuizenga, A. Siegman, “FM and AM locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966);A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1971).
[CrossRef]

Marx, E.

E. Marx, “Free-space propagation of light pulses,” National Bureau of Standards Internal Rep. 84–2835 (U.S. Department of Commerce, Washington, D.C., 1984);“Free-field propagation of localized pulses,” IEEE Trans. Antennas Propag. (to be published).

Schwartz, L.

L. Schwartz, Mathematics for the Physical Sciences, (Addison-Wesley, Reading, Mass., 1966).

Shank, C.

R. Fork, C. Shank, R. Yen, C. Hirliman, “Femtosecond optical pulses,” IEEE J. Quantum Electron. QE-19, 500–505 (1983);S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag, New York, 1977).
[CrossRef]

Siegman, A.

D. Kuizenga, A. Siegman, “FM and AM locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
[CrossRef]

Wolf, E.

Yen, R.

R. Fork, C. Shank, R. Yen, C. Hirliman, “Femtosecond optical pulses,” IEEE J. Quantum Electron. QE-19, 500–505 (1983);S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag, New York, 1977).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. Kuizenga, A. Siegman, “FM and AM locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
[CrossRef]

R. Fork, C. Shank, R. Yen, C. Hirliman, “Femtosecond optical pulses,” IEEE J. Quantum Electron. QE-19, 500–505 (1983);S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag, New York, 1977).
[CrossRef]

J. Appl. Phys. (1)

H. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966);A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1971).
[CrossRef]

Other (3)

J. Cooper, “Exact and approximate pulsed beam solutions of Maxwell’s equations,” Math. Meth. Appl. Sci. (to be published).

E. Marx, “Free-space propagation of light pulses,” National Bureau of Standards Internal Rep. 84–2835 (U.S. Department of Commerce, Washington, D.C., 1984);“Free-field propagation of localized pulses,” IEEE Trans. Antennas Propag. (to be published).

L. Schwartz, Mathematics for the Physical Sciences, (Addison-Wesley, Reading, Mass., 1966).

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Figures (4)

Fig. 1
Fig. 1

Energy density of the approximate fields of a pulse propagating in the z direction at time t = 0. Although the approximation is defined for z < 0, we defined the exact solution only for z > 0.

Fig. 2
Fig. 2

Energy density of the approximate fields of a pulse propagating in the z direction with center of the pulse at z = ct = 5 m.

Fig. 3
Fig. 3

Energy density of the approximate fields of a pulse propagating in the z direction with center of the pulse at z = ct = 20 m.

Fig. 4
Fig. 4

Graphs of the x component of the approximate electric field on the z axis (r = 0) for a pulse centered at z = ct = 0 (solid curve), z = ct = 5 m (dashed curve), and z = ct = 20 m (dotted curve). The units for the vertical axis are arbitrary.

Tables (2)

Tables Icon

Table 1 Location θ* of the Minimum of A

Tables Icon

Table 2 Minimum Value of A

Equations (137)

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t E × H = 0 , E = 0 , μ t H + × E = 0 , H = 0 .
t 2 p c 2 Δ p = 0 for z > 0 , p ( x , y , 0 , t ) = q ( t ) f ( x , y ) ,
| d m q d t m | 2 d t < , | m f x j y m j | 2 d x d y <
q ̂ ( ω ) 1 2 π q ( t ) e i ω t d t ,
f ̂ ( ξ , η ) 1 2 π f ( x , y ) exp [ i ( ξ x + η y ) ] d x d y .
| q ˆ ( ω ) | 2 ω 2 m d ω < , | f ˆ ( ξ , η ) | 2 ρ 2 m d ξ d η <
E = × ( × Z ) , H = t ( × Z ) .
E = ( z z 2 p y y 2 p , x y 2 p , x z 2 p ) , H = ( 0 , z t 2 p , y t 2 p ) .
p ( x , t ) = 1 2 π q ̂ ( ω ) u ( x , ω / c ) e i ω t d ω ,
Δ u + k 2 u = 0 , z > 0 , u ( x , y , 0 , k ) = f ( x , y ) , u / R i k u = o ( 1 / R ) as R = | x | .
u ( x , k ) = 1 2 π f ̂ ( ξ , η ) exp [ i ( ξ x + η y + ξ z ) ] d ξ d η ,
ξ 2 + η 2 + ξ 2 = k 2
ζ = ( k 2 ρ 2 ) 1 / 2 .
ζ = { k [ 1 ( ρ / k ) 2 ] 1 / 2 for k 2 > ρ 2 , i | k | [ ( ρ / k ) 2 1 ] 1 / 2 for k 2 > ρ 2 .
u ¯ ( x , k ) = u ( x , k ) ,
U in = ( E 0 × H 0 ) z d x d y d t ,
( E 0 × H 0 ) z = ( y y 2 p + z z 2 p ) z t 2 p z = 0 .
y y 2 p = 1 ( 2 π ) 3 q ̂ ( ω ) η 2 f ̂ ( ξ , η ) × exp [ i ( ξ x + η y + ζ z ω t ) ] d ξ d η d ω ,
z t 2 p = 1 ( 2 π ) 3 ω q ̂ ( ω ) ζ f ̂ ( ξ , η ) × exp [ i ( ξ x + η y + ζ z ω t ) ] d ξ d η d ω .
y y 2 p z t 2 p z = 0 d x d y d t = ω η 2 ζ ¯ | q ̂ ( ω ) | 2 | f ̂ ( ξ , η ) | 2 d ξ d η d ω = c η 2 | f ̂ ( ξ , η ) | 2 k ζ ¯ | q ̂ ( c k ) | 2 d k d ξ d η .
k | q ̂ ( c k ) | 2 ζ ¯ d k = c 2 | k | > ρ k 2 [ 1 ( ρ k ) 2 ] 1 / 2 | q ̂ ( c k ) | 2 d k .
y y 2 p z t 2 p z = 0 d x d y d t = c 2 η 2 | f ̂ ( ξ , η ) | 2 | k | > ρ k 2 × [ 1 ( ρ k ) 2 ] 1 / 2 | q ̂ ( c k ) | 2 d k d ξ d η .
z z 2 p z t 2 p z = 0 d x d y d t = c 2 | f ̂ ( ξ , η ) | 2 | k | > ρ k 4 × [ 1 ( ρ k ) 2 ] 3 / 2 | q ̂ ( c k ) | 2 d k d ξ d η .
ζ = k [ 1 ( 1 / 2 ) ( ρ / k ) 2 + ( 1 / 8 ) ( ρ / k ) 4 + ] .
ζ = k [ 1 ( 1 / 2 ) ( ρ / k ) 2 ]
ũ ( x , k ) = 1 2 π f ̂ ( ξ , η ) exp [ ( ξ x + η y + + ζ z ) ] d ξ d η .
Δ t ψ + 2 i k z ψ = 0 for z > 0 , ψ ( x , y , 0 , k ) = f ( x , y ) ,
p ( x , t ) = 1 2 π | ω | > ω 0 q ̂ ( ω ) ũ ( x , ω / c ) e i ω t d ω = 1 2 π | ω | > ω 0 q ̂ ( ω ) ψ ( x , ω / c ) × exp [ i ω ( t z / c ) ] d ω .
E = ( z z 2 p y y 2 p , x y 2 p , x z 2 p ) , H = ( 0 , z t 2 p , y t 2 p ) ,
E x = 1 2 π | ω | > ω 0 q ̂ ( ω ) ( k 2 ψ 2 i k z ψ z z 2 ψ y y 2 ψ ) exp [ i ω ( t z / c ) ] d ω ,
E y = 1 2 π | ω | > ω 0 q ̂ ( ω ) x y 2 ψ exp [ i ω ( t z / c ) ] d ω ,
E z = 1 2 π | ω | > ω 0 q ̂ ( ω ) ( i k x ψ + x z 2 ψ ) × exp [ i ω ( t z / c ) ] d ω ,
H x = 0 ,
H y = ( 2 π μ ) 1 / 2 | ω | > ω 0 q ̂ ( ω ) ( k 2 ψ i k z ψ ) × exp [ i ω ( t z / c ) ] d ω ,
H z = ( 2 π μ ) 1 / 2 | ω | > ω 0 q ̂ ( ω ) i k z ψ exp [ i ω ( t z / c ) ] d ω .
E x = k m 2 ψ ( x , k m ) exp [ i k m ( z c t ) ] , E y = 0 , E z = 0 ,
H x = 0 , H y = ( μ ) 1 / 2 k m 2 ψ ( x , k m ) exp [ i k m ( z c t ) ] , H z = 0 .
U ( F , G ) I = I ( 2 | F | 2 + μ 2 | G | 2 ) d x d y d z ,
| F | 2 = | F x | 2 + | F y | 2 + | F z | 2 , | G | 2 = | G x | 2 + | G y | 2 + | G x | 2 .
p l = 1 2 π | ω | < ω 0 q ̂ ( ω ) u ( ω / c ) e i ω t d ω
p h = 1 2 π | ω | > ω 0 q ̂ ( ω ) u ( ω / c ) e i ω t d ω ,
p = 1 2 π | ω | > ω 0 q ̂ ( ω ) ũ ( ω / c ) e i ω t d ω .
( E E , H H ) = ( E l , H l ) + ( E h E , H h H )
U ( E E , H H ) I 2 U ( E l , H l ) I + 2 U ( E h E , H h H ) I .
q ( t ) = cos ( ω m t ) b ( t / τ ) ,
q ̂ ( ω ) = ½ τ b ̂ [ ( ω ω m ) τ ] + ½ τ b ̂ [ ( ω + ω m ) τ ] .
f ( x , y ) = g ( x / r 0 , y / r 0 ) ,
g ( x , y ) = m + n x m y n exp ( r 2 ) , m , n 0 .
A ( α , β , θ ) = α β θ 6 ( b * ) 2 + θ 5 β 5 ,
b * = max ( 1 θ ) α < η < α { | b ̂ ( η ) | + | b ̂ ( η + 2 α ) | } [ 0 | b ̂ ( η ) + b ̂ ( η + 2 α ) | 2 d η ] 1 / 2 .
A * ( α , β ) = min 0 < θ < 1 A ( α , β , θ )
c t r 0 | I | .
U ( E E , H H ) I K A * ( α , β ) ( | I | / r 0 ) ( c t / r 0 ) 2 U in .
K 1 5 π | Δ 2 g | 2 d x d y [ | g | 2 d x d y ] 1 .
b * b ̂ [ ( 1 θ ) α ] [ 0 b ̂ ( η ) 2 d η ] 1 / 2 .
b ( t / τ ) = exp [ ( t / τ ) 2 ] ,
b ̂ ( η ) = 1 2 exp ( 1 4 η 2 ) .
b ̂ ( η + 2 α ) exp ( 100 ) b ̂ ( η ) b ̂ ( η ) ,
( b * ) 2 = ( 2 π ) 1 / 2 exp [ 1 2 α 2 ( 1 θ ) 2 ]
A ( α , β , θ ) = ( 2 π ) 1 / 2 α β θ 6 exp [ 1 2 α 2 ( 1 θ ) 2 ] + θ 5 β 5 .
θ * ( α , β ) β h ( α ) ,
A * ( α , β ) 2 β 5 [ 1 h ( α ) ] .
b ( t / τ ) = exp [ ( t / τ ) 2 ] ,
g ( x / r 0 , y / r 0 ) = exp ( r 2 / r 0 2 ) .
ψ ( r , z , k ) = r 0 w ( z ) exp { ( r w ( z ) ) 2 + i [ k r 2 2 R ( z ) s ( z ) ] } ,
w 2 ( z ) = r 0 2 [ 1 + ( 2 z r 0 2 k ) 2 ] ,
R ( z ) = z [ 1 + ( r 0 2 k 2 z ) 2 ] ,
s ( z ) = arctan ( 2 z r 0 2 k ) .
r 0 = 10 3 m , z 0 = 10 6 m , k m = 10 7 m 1 .
E x = 1 2 π | ω | > ω 0 q ̂ ( ω ) k 2 ψ exp [ i ω ( t z / c ) ] d ω ,
E y = 0 ,
E z = 1 2 π | ω | > ω 0 q ̂ ( ω ) i k x ψ exp [ i ω ( t z / c ) ] d ω ,
H x = 0 ,
H y = ( 2 π μ ) 1 / 2 | ω | > ω 0 q ̂ ( ω ) k 2 ψ exp [ i ω ( t z / c ) ] d ω = ( μ ) 1 / 2 E x ,
H z = ( 2 π μ ) 1 / 2 | ω | > ω 0 q ̂ ( ω ) i k y ψ exp [ i ω ( t z / c ) ] d ω .
Γ = ( 2 π μ ) 1 / 2 | ω | > ω 0 q ̂ ( ω ) i k ψ exp [ i ω ( t z / c ) ] d ω .
½ ( | E x | 2 + | E z | 2 ) + ½ μ ( | H y | 2 + | H z | 2 ) = ( | E x | 2 + ½ | x Γ | 2 + ½ | y Γ | 2 ) = ( | E x | 2 + ½ | r Γ | 2 )
q ̂ ( ω ) = ( τ / 2 2 ) { exp [ ¼ τ 2 ( ω ω m ) 2 ] + exp [ ¼ τ 2 ( ω + ω m ) 2 ] } ,
r 2 r 0 2 z 2 ( r 0 2 k m 2 ) 2 = 1 ,
p = 1 2 π q ̂ ( ω ) ψ ( r , z , ω / c ) exp [ i ω ( t z / c ) ] d ω .
φ = k ( z c t ) arctan [ 2 z / ( r 0 2 k ) ] .
U ( E E , H H ) I 6 z 0 ( c t ) 2 r 0 3 K ( θ * β ) A * ( α , β ) U in 3.57 × 10 11 ( c t ) 2 U in .
G ( γ ) = 1 2 π 0 2 π | g ̂ ( γ cos ϕ , γ sin ϕ ) | 2 d ϕ
L j = 0 G ( γ ) γ j d γ , j = 1 , 3 , 5 , .
| f ̂ ( ξ , η ) | 2 υ ( ρ ) d ξ d η = 2 π r 0 2 0 G ( γ ) υ ( γ / r 0 ) γ d γ .
h ( z ) 2 2 = | h ( x , y , z ) | 2 d x d y
h I 2 = I | h ( x , y , z ) | 2 d x d y d z = I h ( z ) 2 2 d z .
U ( F , G ) I = 2 ( F x I 2 + F y I 2 + F z I 2 ) + μ 2 ( G x I 2 + G y I 2 + G z I 2 ) .
z z 2 ( u ũ ) ( k ) = 1 2 π f ̂ ( ξ , η ) × exp [ i ( ξ x + η y ) ] z z 2 ( e i ζ z e i ζ z ) d ξ d η ,
z z 2 ( u ũ ) ( z , k ) 2 2 = | f ̂ ( ξ , η ) | 2 | z z 2 ( e i ζ z e i ζ z ) | 2 d ξ d η .
| ζ ξ | ½ ( ρ / k ) 4 for 0 ρ | k |
| z z 2 ( e i ζ z e i ζ z ) | 2 ¼ ρ 8 k 4 ( | k | z + 1 ) 2 for ρ | k |
| z z 2 ( e i ζ z e i ζ z ) | 2 1 16 ρ 8 k 4 for ρ > | k | .
z z 2 ( u ũ ) ( z , k ) 2 2 π 2 r 0 6 k 4 [ ( | k | z + 1 ) 2 + 1 4 ] L 9 .
z z 2 ( u ũ ) ( k ) I 2 π M r 0 6 k 2 | I | z ̂ 2 ,
M ½ L 9 [ 1 + 2 ( r 0 k 0 ) 1 + 5 4 ( r 0 k 0 ) 2 ] .
z z 2 ( p h p ) ( t ) I 2 1 2 π { | ω | > ω 0 | q ̂ ( ω ) | z z 2 [ u ( ω / c ) ũ ( ω / c ) ] I d ω } 2 1 2 π c 3 | ω | > ω 0 ω 4 | q ̂ ( ω ) | 2 d ω × | k | > k 0 z z 2 [ u ( k ) ũ ( k ) ] I 2 k 4 d k
z z 2 ( p h p ) ( t ) I 2 M c 3 [ | ω | > ω 0 ω 4 | q ̂ ( ω ) | 2 d ω ] [ r 0 6 | I | z ̂ 2 k 0 k 6 d k ] 1 5 M ( | I | / r 0 ) ( r 0 k 0 ) 5 z ̂ 2 c 3 | ω | > ω 0 ω 4 | q ̂ ( ω ) | 2 d ω .
U ( E h E , H h H ) I K h ( | I | / r 0 ) z ̂ 2 ( r 0 k 0 ) 5 c 3 × | ω | > ω 0 ω 4 | q ̂ ( ω ) | 2 d ω ,
K h 1 5 L 9 [ 1 + ( r 0 k 0 ) 1 ] + ( 13 40 L 9 + 3 20 L 11 ) ( r 0 k 0 ) 2 + ( L 11 + 3 10 L 13 ) ( r 0 k 0 ) 4 + 4 5 L 13 ( r 0 k 0 ) 6 .
z z 2 u ( z , k ) 2 2 | f ̂ ( ξ , η ) | 2 | ζ | 4 d ξ d η ρ < | k | k 4 | f ̂ | 2 d ξ d η k 4 | f ̂ | 2 d ξ d η + ρ > | k | | f ̂ | 2 ( ρ 2 k 2 ) 2 d ξ d η | f ̂ | 2 ( ρ 2 k 2 ) 2 d ξ d η ,
z z 2 u ( z , k ) 2 2 2 π r 0 2 [ ( r 0 k ) 4 0 G ( γ ) γ d γ + 0 G ( γ ) γ 5 d γ ] .
z z 2 u ( k ) I 2 I z z 2 u ( z , k ) 2 2 d z 2 π r 0 2 | I | [ ( r 0 k ) 4 L 1 + L 5 ] .
z z 2 p l = 1 2 π | ω | < ω 0 q ̂ ( ω ) z z 2 u ( ω / c ) e i ω t d ω ,
z z 2 p l ( t ) I 2 1 2 π | ω | < ω 0 | q ̂ ( ω ) | 2 d ω | ω | < ω 0 z z 2 u ( ω / c ) 2 d ω 4 ω 0 2 max | ω | < ω 0 | q ̂ ( ω ) | 2 r 0 2 | I | [ ( r 0 k 0 ) 4 L 1 + L 5 ] ,
U ( E l , H l ) I 4 | I | r 0 2 ω 0 2 k 0 4 max | ω | < ω 0 | q ̂ ( ω ) | 2 K l ,
K l = 16 15 L 1 + 7 3 L 3 ( r 0 k 0 ) 2 + 6 L 5 ( r 0 k 0 ) 4 .
U in c 2 k 4 | q ̂ ( c k ) | 2 ρ < | k | | f ̂ ( ξ , η ) | 2 [ 1 ( ρ k ) 2 ] 3 / 2 d ξ d η d k .
ρ < | k | | f ̂ ( ξ , η ) | 2 [ 1 ( ρ k ) 2 ] 3 / 2 d ξ d η ρ < k 0 | f ̂ ( ξ , η ) | 2 [ 1 ( ρ k 0 ) 2 ] 3 / 2 d ξ d η 2 π r 0 2 L ( r 0 k 0 ) for | k | > k 0 ,
L ( γ ) = 0 γ G ( γ ) [ 1 ( γ γ ) 2 ] 3 / 2 γ d γ .
U in 2 π r 0 2 L ( r 0 k 0 ) c 3 | ω | > ω 0 ω 4 | q ̂ ( ω ) | 2 d ω .
U ( E h E , H h H ) I K h 2 π L | I | r 0 ( z ̂ r 0 ) 2 ( r 0 k 0 ) 5 U in .
q ̂ ( ω ) = 1 2 τ b ̂ [ ( ω ω m ) τ ] + 1 2 τ b ̂ [ ( ω + ω m ) τ ] .
| ω | > ω 0 ω 4 | q ̂ ( ω ) | 2 d ω 2 ω m ω 4 | q ̂ ( ω ) | 2 d ω = 1 2 τ 2 ω m ω 4 | b ̂ [ ( ω ω m ) τ ] + b ̂ [ ( ω + ω m ) τ ] | 2 d ω = τ 2 0 | b ̂ ( η ) + b ̂ ( η + 2 α ) | 2 ( ω m + η / τ ) 4 d η 1 2 τ ω m 4 0 | b ̂ ( η ) + b ̂ ( η + 2 α ) | 2 d η .
U in π r 0 2 ( / c 3 ) L ω m 4 τ 0 | b ̂ ( η ) + b ̂ ( η + 2 α ) | 2 d η .
max | ω | < ω 0 | q ̂ ( ω ) | = max 0 ω ω 0 | q ̂ ( ω ) | = max ( 1 θ ) α η α ( τ / 2 ) | b ̂ ( η ) + b ̂ ( η + 2 α ) | .
b * = max ( 1 θ ) α η α | b ̂ ( η ) + b ̂ ( η + 2 α ) | × [ 0 | b ̂ ( η ) + b ̂ ( η + 2 α ) | 2 d η ] 1 / 2 .
U ( E l , H l ) I K l π L | I | r 0 α β θ 6 ( b * ) 2 U in .
U ( E E , H H ) I 2 U ( E l , H l ) I + 2 U ( E h E , H h H ) I K ( θ β ) ( | I | / r 0 ) [ α β θ 6 ( b * ) 2 + ( z ̂ / r 0 ) 2 θ 5 β 5 ] U in ,
K = max ( 2 K l , K h ) / ( π L ) .
L j ( j 1 ) L j 2 , j 3 ,
K = K h / ( π L ) .
U ( E E , H H ) I K ( θ β ) ( | I | / r 0 ) ( c t / r 0 ) 2 A ( α , β , θ ) U in
A ( α , β , θ ) = α β θ 6 ( b * ) 2 + θ 5 β 5 .
A * ( α , β ) = min 0 < θ < 1 A ( α , β , θ )
min 0 < θ < 1 K ( θ β ) A ( α , β , θ ) min θ * θ 1 K ( θ β ) A ( α , β , θ ) K ( θ * β ) A * ( α , β ) .
U ( E E , H H ) I K ( θ * β ) A * ( α , β ) ( | I | / r 0 ) ( c t / r 0 ) 2 U in .
g = m + n x m y n exp ( r 2 ) ,
G ( γ ) = C m . n γ 2 ( m + n ) exp ( γ 2 / 2 ) ,
L j = [ j 1 + 2 ( m + n ) ] L j 2 , j 3 ,
L 13 16 L 11 16 14 L 9 for m + n 2 .
K h 1 5 L 9 for m + n 2 .
L 1 L ( γ ) = 0 γ G ( γ ) [ 1 ( γ γ ) 2 ] 3 / 2 γ d γ 0 γ G ( γ ) [ 1 ( γ γ ) 2 ] 2 γ d γ 0 γ G ( γ ) γ d γ 2 ( γ ) 2 × 0 γ G ( γ ) γ 3 d γ L 1 3 γ 2 L 3 .
L ( γ ) ( 1 18 γ 2 ) L 1 for m + n 2 ,
K L 9 / ( 5 π L 1 ) for m + n 2 .
L 9 = 0 G ( γ ) γ 9 d γ = 1 2 π | Δ 2 g | 2 d x d y ,
L 1 = 0 G ( γ ) γ d γ = 1 2 π | g | 2 d x d y .

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