Abstract

We investigate the possibility of light beams that are both narrow and long range with respect to the wavelength. On the basis of spectral electromagnetic field representations, we have studied the decay of the evanescent waves, and we have obtained some bounds for the width and range of a light beam in the near-field region. The range determines the spatial bound of the near field in the direction of propagation. For a number of representative examples we found that narrow beams have a short range. Our analysis is based on the uncertainty relations between spatial position and spatial frequency.

© 2011 Optical Society of America

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  1. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
    [CrossRef] [PubMed]
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
    [CrossRef] [PubMed]
  3. H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “The diffraction of light by narrow slits of different materials,” J. Opt. A: Pure Appl. Opt. 6, S277–S280 (2004).
    [CrossRef]
  4. H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12, 3629–3651 (2004).
    [CrossRef] [PubMed]
  5. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
    [CrossRef]
  6. L. Ronchi and M. A. Porras, “The relationship between the second order moment width and the caustic surface radius of laser beams,” Opt. Commun. 103, 201–204 (1993).
    [CrossRef]
  7. M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
    [CrossRef]
  8. M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
    [CrossRef]
  9. M. A. Alonso and G. W. Forbes, “Uncertainty products for nonparaxial wavefields,” J. Opt. Soc. Am. A 17, 2391–2401 (2000).
    [CrossRef]
  10. M. A. Alonso, R. Borghi, and M. Santarsiero, “Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case,” J. Opt. Soc. Am. A 23, 691–700 (2006).
    [CrossRef]
  11. A. Luis, “Gaussian beam and minimum diffraction,” Opt. Lett. 31, 3644–3646 (2006).
    [CrossRef] [PubMed]
  12. C. J. R. Sheppard, “High-aperture beams,” J. Opt. Soc. Am. A 18, 1579–1587 (2001).
    [CrossRef]
  13. O. E. Gawhary and S. Severin, “Degree of paraxiality for monochromatic light beams,” Opt. Lett. 33, 1360–1362 (2008).
    [CrossRef] [PubMed]
  14. O. E. Gawhary and S. Severin, “Localization and paraxiality of pseudo-nondiffracting fields,” Opt. Commun. 283, 2481–2487(2010).
    [CrossRef]
  15. P. Chen, Q. Gan, F. J. Bartoli, and L. Zhu, “Near-field-resonance-enhanced plasmonic light beaming,” IEEE Photon. J. 2, 8–17(2010).
    [CrossRef]
  16. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1966).
  17. P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).

2010 (2)

O. E. Gawhary and S. Severin, “Localization and paraxiality of pseudo-nondiffracting fields,” Opt. Commun. 283, 2481–2487(2010).
[CrossRef]

P. Chen, Q. Gan, F. J. Bartoli, and L. Zhu, “Near-field-resonance-enhanced plasmonic light beaming,” IEEE Photon. J. 2, 8–17(2010).
[CrossRef]

2008 (1)

2006 (2)

2004 (2)

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “The diffraction of light by narrow slits of different materials,” J. Opt. A: Pure Appl. Opt. 6, S277–S280 (2004).
[CrossRef]

H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12, 3629–3651 (2004).
[CrossRef] [PubMed]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

2002 (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

1996 (1)

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[CrossRef]

1994 (1)

M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

1993 (1)

L. Ronchi and M. A. Porras, “The relationship between the second order moment width and the caustic surface radius of laser beams,” Opt. Commun. 103, 201–204 (1993).
[CrossRef]

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Alonso, M. A.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Bartoli, F. J.

P. Chen, Q. Gan, F. J. Bartoli, and L. Zhu, “Near-field-resonance-enhanced plasmonic light beaming,” IEEE Photon. J. 2, 8–17(2010).
[CrossRef]

Blok, H.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “The diffraction of light by narrow slits of different materials,” J. Opt. A: Pure Appl. Opt. 6, S277–S280 (2004).
[CrossRef]

Borghi, R.

Chen, P.

P. Chen, Q. Gan, F. J. Bartoli, and L. Zhu, “Near-field-resonance-enhanced plasmonic light beaming,” IEEE Photon. J. 2, 8–17(2010).
[CrossRef]

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).

Degiron, A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Forbes, G. W.

Gan, Q.

P. Chen, Q. Gan, F. J. Bartoli, and L. Zhu, “Near-field-resonance-enhanced plasmonic light beaming,” IEEE Photon. J. 2, 8–17(2010).
[CrossRef]

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Gawhary, O. E.

O. E. Gawhary and S. Severin, “Localization and paraxiality of pseudo-nondiffracting fields,” Opt. Commun. 283, 2481–2487(2010).
[CrossRef]

O. E. Gawhary and S. Severin, “Degree of paraxiality for monochromatic light beams,” Opt. Lett. 33, 1360–1362 (2008).
[CrossRef] [PubMed]

Gbur, G.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “The diffraction of light by narrow slits of different materials,” J. Opt. A: Pure Appl. Opt. 6, S277–S280 (2004).
[CrossRef]

Lenstra, D.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “The diffraction of light by narrow slits of different materials,” J. Opt. A: Pure Appl. Opt. 6, S277–S280 (2004).
[CrossRef]

Lezec, H. J.

H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12, 3629–3651 (2004).
[CrossRef] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Luis, A.

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1966).

Porras, M. A.

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[CrossRef]

M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

L. Ronchi and M. A. Porras, “The relationship between the second order moment width and the caustic surface radius of laser beams,” Opt. Commun. 103, 201–204 (1993).
[CrossRef]

Ronchi, L.

L. Ronchi and M. A. Porras, “The relationship between the second order moment width and the caustic surface radius of laser beams,” Opt. Commun. 103, 201–204 (1993).
[CrossRef]

Santarsiero, M.

Schouten, H. F.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “The diffraction of light by narrow slits of different materials,” J. Opt. A: Pure Appl. Opt. 6, S277–S280 (2004).
[CrossRef]

Severin, S.

O. E. Gawhary and S. Severin, “Localization and paraxiality of pseudo-nondiffracting fields,” Opt. Commun. 283, 2481–2487(2010).
[CrossRef]

O. E. Gawhary and S. Severin, “Degree of paraxiality for monochromatic light beams,” Opt. Lett. 33, 1360–1362 (2008).
[CrossRef] [PubMed]

Sheppard, C. J. R.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Thio, T.

Visser, T. D.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “The diffraction of light by narrow slits of different materials,” J. Opt. A: Pure Appl. Opt. 6, S277–S280 (2004).
[CrossRef]

Zhu, L.

P. Chen, Q. Gan, F. J. Bartoli, and L. Zhu, “Near-field-resonance-enhanced plasmonic light beaming,” IEEE Photon. J. 2, 8–17(2010).
[CrossRef]

IEEE Photon. J. (1)

P. Chen, Q. Gan, F. J. Bartoli, and L. Zhu, “Near-field-resonance-enhanced plasmonic light beaming,” IEEE Photon. J. 2, 8–17(2010).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “The diffraction of light by narrow slits of different materials,” J. Opt. A: Pure Appl. Opt. 6, S277–S280 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Opt. Commun. (4)

L. Ronchi and M. A. Porras, “The relationship between the second order moment width and the caustic surface radius of laser beams,” Opt. Commun. 103, 201–204 (1993).
[CrossRef]

M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[CrossRef]

O. E. Gawhary and S. Severin, “Localization and paraxiality of pseudo-nondiffracting fields,” Opt. Commun. 283, 2481–2487(2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Other (2)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1966).

P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).

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Figures (10)

Fig. 1
Fig. 1

Gaussian field distribution: the spatial field amplitudes as a function of x at y = 0 (left figures) and the spectral field amplitudes as a function of k x at k y = 0 (right figures), for z = 0 and z = λ / 2 , respectively. We consider σ = λ / 4 ( σ k = π / 2 ) (top figures) and σ = λ / 20 ( σ k = π / 10 ) (bottom figures).

Fig. 2
Fig. 2

Distribution of the twice-differentiated Gaussian field distribution: the spatial field amplitudes as a function of x at y = 0 (left figures) and the spectral field amplitudes as a function of k x at k y = 0 (right figures), for z = 0 and z = λ / 2 , respectively. We consider σ = λ / 4 ( σ k = π / 2 ) (top figures) and σ = λ / 20 ( σ k = π / 10 ) (bottom figures).

Fig. 3
Fig. 3

Band-limited field distribution: the spatial field amplitudes as a function of x at y = 0 (left figures) and the spectral field amplitudes as a function of k x at k y = 0 (right figures), for z = 0 and z = λ / 2 , respectively. We consider a = 2 k (top figures) and a = 5 k (bottom figures).

Fig. 4
Fig. 4

Distribution of the twice-differentiated band-limited field distribution: the spatial field amplitudes as a function of x at y = 0 (left figures) and the spectral field amplitudes as a function of k x at k y = 0 (right figures), for z = 0 and z = λ / 2 , respectively. We consider a = 2 k (top figures) and a = 5 k (bottom figures).

Fig. 5
Fig. 5

Range functions f ( ζ ) and g ( ζ ) as a function of ζ = z / σ .

Fig. 6
Fig. 6

Gaussian beam range quantities β ( z ) , I ( 0 ) / I ( z ) , and α ( z ) , for z = Γ 2 , as a function of the normalized width w ( 0 ) / 2 λ = σ / λ . Solid lines, Gaussian; dashed lines, twice-differentiated Gaussian.

Fig. 7
Fig. 7

Beam narrowness factor 1 2 k w lb ( z ) = [ 1 + α ( z ) ] 1 2 as a function of σ / λ for the Gaussian field distribution (top figure) and its twice-differentiated one (bottom figure).

Fig. 8
Fig. 8

Normalized range Γ 2 / λ as a function of k / a , pertaining to the nondifferentiated band-limited field distribution.

Fig. 9
Fig. 9

Range quantities β ( z ) , I ( 0 ) / I ( z ) , and α ( z ) , for z = Γ 2 , as a function of k / a , pertaining to the band-limited field distribution (solid lines) and its twice-differentiated one (dashed lines).

Fig. 10
Fig. 10

Beam narrowness factor 1 2 k w lb ( z ) = [ 1 + α ( z ) ] 1 2 as a function of k / a for the band-limited field distribution (top figure) and its twice-differentiated one (bottom figure), where z 1 = Γ 2 and z 2 = 2 Γ 2 .

Tables (1)

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Table 1 Numerical Values of the Integral h ( n )

Equations (61)

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{ E , H } = 1 4 π 2 R 2 { E ˜ , H ˜ } exp ( i k x x + i k y y ) d k x d k y .
( 2 z 2 + k 2 k x 2 k y 2 ) { E ˜ , H ˜ } = 0 , z > 0 ,
{ E ˜ ( k x , k y , z ) , H ˜ ( k x , k y , z ) } = { E ˜ ( k x , k y , 0 ) , H ˜ ( k x , k y , 0 ) } exp ( i k z z ) , z > 0 ,
k z = ( k 2 k x 2 k y 2 ) 1 2 , ( k x , k y ) D pr .
H ˜ = ( ω μ ) 1 k × E ˜ ,
{ E ˜ ( k x , k y , z ) , H ˜ ( k x , k y , z ) } = { E ˜ ( k x , k y , 0 ) , H ˜ ( k x , k y , 0 ) } exp ( γ z z ) , z > 0 ,
γ z = ( k x 2 + k y 2 k 2 ) 1 2 , ( k x , k y ) D ev .
E ( x , y , 0 ) = E 0 exp ( r 2 2 σ 2 ) , r 2 = x 2 + y 2 ,
E ˜ ( k x , k y , 0 ) = E 0 2 π σ 2 exp ( 1 2 σ 2 κ 2 ) , κ 2 = k x 2 + k y 2 .
E ( x , y , 0 ) = E 0 σ 2 ( 2 x 2 + 2 y 2 ) exp ( r 2 2 σ 2 ) .
E ˜ ( k x , k y , 0 ) = E 0 2 π σ 4 κ 2 exp ( 1 2 σ 2 κ 2 ) .
E ˜ ( k x , k y , 0 ) = E 0 ( a 2 κ 2 ) Π a ( κ ) , Π a ( κ ) = { 1 , κ < a 0 , κ > a .
E ( x , y , 0 ) = E 0 a 2 π r 2 J 2 ( a r ) ,
E ˜ ( k x , k y , 0 ) = E 0 κ 2 a 2 ( a 2 κ 2 ) Π a ( κ )
E ( x , y , 0 ) = E 0 ( 2 x 2 + 2 y 2 ) 1 π r 2 J 2 ( a r ) .
I ( z ) = R 2 | E ( x , y , z ) | 2 d x d y = E ( z ) R 2 2 ,
I ( z ) = 1 4 π 2 R 2 | E ( k x , k y , z ) | 2 d k x d k y = 1 4 π 2 E ˜ ( z ) R 2 2 = 1 4 π 2 [ E ˜ ( z ) D pr 2 + E ˜ ( z ) D ev 2 ] ,
w 2 ( z ) = 4 R 2 ( x 2 + y 2 ) | E ( x , y , z ) | 2 d x d y I ( z ) = 4 r E ( z ) R 2 2 I ( z ) ,
W 2 ( z ) = 4 1 4 π 2 R 2 ( k x 2 + k y 2 ) | E ˜ ( k x , k y , z ) | 2 d k x d k y I ( z ) = 4 1 4 π 2 κ E ˜ ( z ) R 2 2 I ( z ) ,
1 2 w ( z ) 1 2 W ( z ) 1.
2 z 2 | E ˜ ( k x , k y , z ) | 2 = 0 , for     ( k x , k y ) D pr .
2 z 2 | E ˜ ( k x , k y , z ) | 2 = 4 ( k x 2 + k y 2 k 2 ) | E ˜ ( k x , k y , z | 2 , for     ( k x , k y ) D ev .
κ E ˜ ( z ) D ev 2 = ( k 2 + 1 4 2 z 2 ) E ˜ ( z ) D ev 2 .
κ E ˜ ( z ) D pr 2 + κ E ˜ ( z ) D ev 2 = κ E ˜ ( z ) D pr 2 + ( k 2 + 1 4 2 z 2 ) E ˜ ( z ) D ev 2 .
4 π 2 I ( z ) W 2 ( z ) 4 ( k 2 + 1 4 2 z 2 ) [ E ˜ ( z ) D pr 2 + E ˜ ( z ) D ev 2 ] .
W ( z ) [ 1 + 1 4 k 2 2 I ( z ) / z 2 I ( z ) ] 1 2 2 k .
W ( z ) [ 1 + α ( z ) ] 1 2 2 k ,
α ( z ) = 2 I ( z ) / z 2 4 k 2 I ( z ) ,
w ( z ) ( 2 / k ) [ 1 + α ( z ) ] 1 2 w lb ( z ) .
β ( z ) = α ( z ) I ( z ) I ( 0 ) = 2 I ( z ) / z 2 4 k 2 I ( 0 ) .
1 2 Re { R 2 [ E × H * ] z d x d y } = 1 2 Re { R 2 [ E ˜ × H ˜ * ] z d k x d k y } = 1 2 Re { R 2 k z ω μ | E ˜ | 2 exp ( i k z z i k z * z ) d k x d k y } = 1 2 D pr k z ω μ | E ˜ | 2 d k x d k y ,
[ 16 π 2 k 2 I ( 0 ) ] 0 z 2 β ( z ) d z = D ev 4 γ z 2 | E ˜ ( k x , k y , 0 ) | 2 [ 0 z 2 exp ( 2 γ z z ) d z ] d k x d k y = γ z 1 2 E ˜ ( 0 ) D ev 2 ,
[ 16 π 2 k 2 I ( 0 ) ] 0 z β ( z ) d z = C 4 γ z 2 | E ˜ ( k x , k y , 0 ) | 2 [ 0 z exp ( 2 γ z z ) d z ] d k x d k y = E ˜ ( 0 ) D ev 2 ,
[ 16 π 2 k 2 I ( 0 ) ] 0 β ( z ) d z = D ev 4 γ z 2 | E ˜ ( k x , k y , 0 ) | 2 [ 0 exp ( 2 γ z z ) d z ] d k x d k y = 2 γ z 1 2 E ˜ ( 0 ) D ev 2 .
Γ 2 2 = 0 z 2 β ( z ) d z 0 β ( z ) d z = γ z 1 2 E ˜ ( 0 ) D ev 2 2 γ z 1 2 E ˜ ( 0 ) D ev 2
Γ 1 = 0 z β ( z ) d z 0 β ( z ) d z = E ˜ ( 0 ) D ev 2 2 γ z 1 2 E ˜ ( 0 ) D ev 2 .
W ( Γ ) [ 1 + α ( Γ ) ] 1 2 2 k ,
α ( Γ ) = γ z k E ˜ ( Γ ) D ev 2 E ˜ ( 0 ) R 2 2 .
w ( Γ ) 2 k [ 1 + α ( Γ ) ] 1 2
E ˜ av ( κ , z ) = [ 1 2 π 0 2 π | E ˜ ( κ , ϑ , z ) | 2 d ϑ ] 1 2 .
w ( 0 ) = 2 σ , W ( 0 ) = 2 / σ ,
Γ 1 = π 1 2 σ , Γ 2 = σ .
β ( σ ζ ) = exp ( σ 2 k 2 ) σ 2 k 2 f ( ζ ) ,
I ( σ ζ ) / I ( 0 ) = 1 exp ( σ 2 k 2 ) g ( ζ ) ,
f ( ζ ) = 1 + ζ 2 1 2 ( 3 + 2 ζ 2 ) ζ π 1 2 erfcx ( ζ ) ,
g ( ζ ) = ζ π 1 2 erfcx ( ζ ) .
w ( 0 ) = 2 σ , W ( 0 ) = 2 3 σ ,
Γ 1 = π 1 2 σ σ 4 k 4 + 2 σ 2 k 2 + 2 σ 4 k 4 + 3 σ 2 k 2 + 15 4 , Γ 2 = σ ( σ 4 k 4 + σ 2 k 2 + 3 4 σ 4 k 4 + 3 σ 2 k 2 + 15 4 ) 1 2 .
β ( σ ζ ) = exp ( σ 2 k 2 ) σ 2 k 2 { σ 4 k 4 ( 1 2 + 1 2 ζ 2 ) + σ 2 k 2 ( 2 + 9 2 ζ 2 + ζ 4 ) + 3 + 87 8 ζ 2 + 5 ζ 4 + 1 2 ζ 6 [ σ 4 k 4 ( 3 4 + 1 2 ζ 2 ) + σ 2 k 2 ( 15 4 + 5 ζ 2 + ζ 4 ) + 105 16 + 105 8 ζ 2 + 21 4 ζ 4 + 1 2 ζ 6 ] ζ π 1 2 erfcx ( ζ ) } ,
I ( σ ζ ) / I ( 0 ) = 1 + 1 2 exp ( σ 2 k 2 ) { [ 2 σ 2 k 2 ζ 2 + 9 2 ζ 2 + ζ 4 ] [ σ 4 k 4 + σ 2 k 2 ( 3 + 2 ζ 2 ) + 15 4 + 5 ζ 2 + ζ 4 ] ζ π 1 2 erfcx ( ζ ) } ,
w ( 0 ) = 2 6 / a , W ( 0 ) = a .
Γ 1 = ( 35 / 32 ) ( a 2 k 2 ) 1 2 , Γ 2 = 7 / 2 ( a 2 k 2 ) 1 2 .
β ( ζ ( a 2 k 2 ) 1 2 ) = 6 h ( 1 ) ( ζ ) ( a 2 / k 2 1 ) 4 ( a 2 / k 2 ) 3 ,
I ( ζ ( a 2 k 2 ) 1 2 ) / I ( 0 ) = 1 [ 1 6 h ( 0 ) ( ζ ) ] ( a 2 / k 2 1 ) 3 ( a 2 / k 2 ) 3 ,
α ( ζ ( a 2 k 2 ) 1 2 ) = 6 h ( 1 ) ( ζ ) ( a 2 / k 2 1 ) 4 ( a 2 / k 2 ) 3 [ 1 6 h ( 0 ) ( ζ ) ] ( a 2 / k 2 1 ) 3 ,
h ( n ) ( ζ ) = 0 1 ( 1 v 2 ) 2 exp ( 2 ζ v ) v 2 n + 1 d v .
w ( 0 ) = 4 5 / a , W ( 0 ) = 2 a .
Γ 1 = 231 128 a 4 + 3 a 2 k 2 + 6 k 4 5 a 4 + 12 a 2 k 2 + 16 k 4 ( a 2 k 2 ) 1 2 , Γ 2 = 11 2 ( a 4 + 4 a 2 k 2 + 16 k 4 5 a 4 + 12 a 2 k 2 + 16 k 4 ) 1 2 ( a 2 k 2 ) 1 2 .
β ( ζ ( a 2 k 2 ) 1 2 ) = 60 h ( 3 ) ( ζ ) ( a 2 / k 2 1 ) 6 ( a 2 / k 2 ) 5 ,
I ( ζ ( a 2 k 2 ) 1 2 ) I ( 0 ) = 6 ( a 2 / k 2 ) 5 15 ( a 2 / k 2 ) 4 + 10 ( a 2 / k 2 ) 3 + 60 h ( 2 ) ( ζ ) ( a 2 / k 2 1 ) 5 ( a 2 / k 2 ) 5 ,
α ( ζ ( a 2 k 2 ) 1 2 ) = 60 h ( 3 ) ( ζ ) ( a 2 / k 2 1 ) 6 6 15 a 2 / k 2 + 10 ( a 2 / k 2 ) 2 + 60 h ( 2 ) ( ζ ) ( a 2 / k 2 1 ) 5 ,

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