Abstract

The far-field spectral anomalies of a short Gaussian pulse incident on a circular aperture with a variable wedge are theoretically studied, and some numerical examples are given to illustrate these effects. It is shown that some anomalous behaviors (such as red shift or blue shift for the spectral peak of the diffracted pulse) can be found under different conditions. Also, the important phenomenon called “spectral switches” is presented, which can be controlled by varying the angle of the wedge part in the circular aperture. Its potential application in information encoding and transmission for free-space communications is proposed, and this scheme has the benefit of easy implementation compared with other previous methods.

© 2009 Optical Society of America

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References

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  1. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.
    [CrossRef]
  2. J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, 1999).
  3. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
    [CrossRef]
  4. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
    [CrossRef]
  5. T. D. Visser and E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371-373 (2003).
    [CrossRef]
  6. J. T. Foley and E. Wolf, “Phenomenon of spectral switches as a new effect in singular optics with polychromatic light,” J. Opt. Soc. Am. A 19, 2510-2516 (2002).
    [CrossRef]
  7. P. Han, “Far-field spectral intensity characteristics of a time-dependent Gaussian pulse from two types of apodized slits,” Jpn. J. Appl. Phys. 43, 3386-3393 (2008).
    [CrossRef]
  8. P. Han, “Spectrum compression of Gaussian pulse from annular aperture in far-field,” Jpn. J. Appl. Phys. 47, 914-917 (2008).
    [CrossRef]
  9. G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901-1-013901-4 (2002).
    [CrossRef]
  10. J. Pu, C. Cai, and S. Nemoto, “Spectral anomalies in Young's double-slit interference experiment,” Opt. Express 12, 5131-5139 (2004).
    [CrossRef] [PubMed]
  11. B. Lü and L. Pan, “Spectral switching of Gaussian Schell-model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340-344 (2002).
    [CrossRef]
  12. S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211-1213 (2002).
    [CrossRef]
  13. Y. Yang and Y. Li, “Spectral shifts and spectral switches of a pulsed Bessel-Gaussian beam from a circular aperture in the far field,” Opt. Laser Technol. 39, 1478-1484 (2007).
    [CrossRef]
  14. J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 99, 57-63 (1999).
    [CrossRef]
  15. G. H. Yang, H. E. Hwang, and P. Han, “Spectral intensity distribution of a time dependent Gaussian pulsed beam from a circular aperture in the near-field diffraction,” Opt. Rev. 10, 19-23 (2003).
    [CrossRef]
  16. P. Han, “Far-field diffraction characteristics of a Gaussian pulse incident on a sinusoidal phase grating,” J. Opt. A, Pure Appl. Opt. 10, 035003-035007 (2008).
    [CrossRef]
  17. B. Qu, J. Pu, and Z. Chen, “Experimental observation of spectral switch of partially coherent light focused by a lens with chromatic aberration,” Opt. Laser Technol. 39, 1226-1230 (2007).
    [CrossRef]
  18. H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A, Pure Appl. Opt. 3, 296-299 (2001).
    [CrossRef]
  19. G. Popescu and A. Dogariu, “Spectral anomalies at wave-front dislacations,” Phys. Rev. Lett. 88, 183902 (2002).
    [CrossRef] [PubMed]
  20. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), p. 511 and p.464.
  21. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), p. 53.
  22. K. Iizuka, Elements of Photonics (Wiley, 2002), p. 18.
  23. P. Han, “Spectral anomalies for a right triangle aperture with an adjustable hypotenuse's slope,” J. Opt. A, Pure Appl. Opt. 2008 (to be published).

2008 (4)

P. Han, “Far-field spectral intensity characteristics of a time-dependent Gaussian pulse from two types of apodized slits,” Jpn. J. Appl. Phys. 43, 3386-3393 (2008).
[CrossRef]

P. Han, “Spectrum compression of Gaussian pulse from annular aperture in far-field,” Jpn. J. Appl. Phys. 47, 914-917 (2008).
[CrossRef]

P. Han, “Spectral anomalies for a right triangle aperture with an adjustable hypotenuse's slope,” J. Opt. A, Pure Appl. Opt. 2008 (to be published).

P. Han, “Far-field diffraction characteristics of a Gaussian pulse incident on a sinusoidal phase grating,” J. Opt. A, Pure Appl. Opt. 10, 035003-035007 (2008).
[CrossRef]

2007 (2)

B. Qu, J. Pu, and Z. Chen, “Experimental observation of spectral switch of partially coherent light focused by a lens with chromatic aberration,” Opt. Laser Technol. 39, 1226-1230 (2007).
[CrossRef]

Y. Yang and Y. Li, “Spectral shifts and spectral switches of a pulsed Bessel-Gaussian beam from a circular aperture in the far field,” Opt. Laser Technol. 39, 1478-1484 (2007).
[CrossRef]

2004 (1)

2003 (2)

T. D. Visser and E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371-373 (2003).
[CrossRef]

G. H. Yang, H. E. Hwang, and P. Han, “Spectral intensity distribution of a time dependent Gaussian pulsed beam from a circular aperture in the near-field diffraction,” Opt. Rev. 10, 19-23 (2003).
[CrossRef]

2002 (5)

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901-1-013901-4 (2002).
[CrossRef]

G. Popescu and A. Dogariu, “Spectral anomalies at wave-front dislacations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

B. Lü and L. Pan, “Spectral switching of Gaussian Schell-model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340-344 (2002).
[CrossRef]

S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211-1213 (2002).
[CrossRef]

J. T. Foley and E. Wolf, “Phenomenon of spectral switches as a new effect in singular optics with polychromatic light,” J. Opt. Soc. Am. A 19, 2510-2516 (2002).
[CrossRef]

2001 (1)

H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A, Pure Appl. Opt. 3, 296-299 (2001).
[CrossRef]

1999 (1)

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 99, 57-63 (1999).
[CrossRef]

1996 (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Cai, C.

Chen, Z.

B. Qu, J. Pu, and Z. Chen, “Experimental observation of spectral switch of partially coherent light focused by a lens with chromatic aberration,” Opt. Laser Technol. 39, 1226-1230 (2007).
[CrossRef]

Dogariu, A.

G. Popescu and A. Dogariu, “Spectral anomalies at wave-front dislacations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Foley, J. T.

Gbur, G.

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901-1-013901-4 (2002).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), p. 53.

Han, P.

P. Han, “Spectrum compression of Gaussian pulse from annular aperture in far-field,” Jpn. J. Appl. Phys. 47, 914-917 (2008).
[CrossRef]

P. Han, “Far-field spectral intensity characteristics of a time-dependent Gaussian pulse from two types of apodized slits,” Jpn. J. Appl. Phys. 43, 3386-3393 (2008).
[CrossRef]

P. Han, “Far-field diffraction characteristics of a Gaussian pulse incident on a sinusoidal phase grating,” J. Opt. A, Pure Appl. Opt. 10, 035003-035007 (2008).
[CrossRef]

P. Han, “Spectral anomalies for a right triangle aperture with an adjustable hypotenuse's slope,” J. Opt. A, Pure Appl. Opt. 2008 (to be published).

G. H. Yang, H. E. Hwang, and P. Han, “Spectral intensity distribution of a time dependent Gaussian pulsed beam from a circular aperture in the near-field diffraction,” Opt. Rev. 10, 19-23 (2003).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), p. 511 and p.464.

Hwang, H. E.

G. H. Yang, H. E. Hwang, and P. Han, “Spectral intensity distribution of a time dependent Gaussian pulsed beam from a circular aperture in the near-field diffraction,” Opt. Rev. 10, 19-23 (2003).
[CrossRef]

Iizuka, K.

K. Iizuka, Elements of Photonics (Wiley, 2002), p. 18.

James, D. F. V.

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

Kandpal, H. C.

H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A, Pure Appl. Opt. 3, 296-299 (2001).
[CrossRef]

Li, Y.

Y. Yang and Y. Li, “Spectral shifts and spectral switches of a pulsed Bessel-Gaussian beam from a circular aperture in the far field,” Opt. Laser Technol. 39, 1478-1484 (2007).
[CrossRef]

Lü, B.

B. Lü and L. Pan, “Spectral switching of Gaussian Schell-model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340-344 (2002).
[CrossRef]

Nemoto, S.

J. Pu, C. Cai, and S. Nemoto, “Spectral anomalies in Young's double-slit interference experiment,” Opt. Express 12, 5131-5139 (2004).
[CrossRef] [PubMed]

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 99, 57-63 (1999).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, 1999).

Pan, L.

B. Lü and L. Pan, “Spectral switching of Gaussian Schell-model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340-344 (2002).
[CrossRef]

Ponomarenko, S. A.

Popescu, G.

G. Popescu and A. Dogariu, “Spectral anomalies at wave-front dislacations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Pu, J.

B. Qu, J. Pu, and Z. Chen, “Experimental observation of spectral switch of partially coherent light focused by a lens with chromatic aberration,” Opt. Laser Technol. 39, 1226-1230 (2007).
[CrossRef]

J. Pu, C. Cai, and S. Nemoto, “Spectral anomalies in Young's double-slit interference experiment,” Opt. Express 12, 5131-5139 (2004).
[CrossRef] [PubMed]

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 99, 57-63 (1999).
[CrossRef]

Qu, B.

B. Qu, J. Pu, and Z. Chen, “Experimental observation of spectral switch of partially coherent light focused by a lens with chromatic aberration,” Opt. Laser Technol. 39, 1226-1230 (2007).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.
[CrossRef]

Visser, T. D.

T. D. Visser and E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371-373 (2003).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901-1-013901-4 (2002).
[CrossRef]

Wolf, E.

T. D. Visser and E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371-373 (2003).
[CrossRef]

S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211-1213 (2002).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901-1-013901-4 (2002).
[CrossRef]

J. T. Foley and E. Wolf, “Phenomenon of spectral switches as a new effect in singular optics with polychromatic light,” J. Opt. Soc. Am. A 19, 2510-2516 (2002).
[CrossRef]

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

Yang, G. H.

G. H. Yang, H. E. Hwang, and P. Han, “Spectral intensity distribution of a time dependent Gaussian pulsed beam from a circular aperture in the near-field diffraction,” Opt. Rev. 10, 19-23 (2003).
[CrossRef]

Yang, Y.

Y. Yang and Y. Li, “Spectral shifts and spectral switches of a pulsed Bessel-Gaussian beam from a circular aperture in the far field,” Opt. Laser Technol. 39, 1478-1484 (2007).
[CrossRef]

Zhang, H.

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 99, 57-63 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. Lü and L. Pan, “Spectral switching of Gaussian Schell-model beams passing through an aperture lens,” IEEE J. Quantum Electron. 38, 340-344 (2002).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (4)

P. Han, “Spectral anomalies for a right triangle aperture with an adjustable hypotenuse's slope,” J. Opt. A, Pure Appl. Opt. 2008 (to be published).

T. D. Visser and E. Wolf, “Spectral anomalies near phase singularities in partially coherent focused wavefields,” J. Opt. A, Pure Appl. Opt. 5, 371-373 (2003).
[CrossRef]

P. Han, “Far-field diffraction characteristics of a Gaussian pulse incident on a sinusoidal phase grating,” J. Opt. A, Pure Appl. Opt. 10, 035003-035007 (2008).
[CrossRef]

H. C. Kandpal, “Experimental observation of the phenomenon of spectral switch,” J. Opt. A, Pure Appl. Opt. 3, 296-299 (2001).
[CrossRef]

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (2)

P. Han, “Far-field spectral intensity characteristics of a time-dependent Gaussian pulse from two types of apodized slits,” Jpn. J. Appl. Phys. 43, 3386-3393 (2008).
[CrossRef]

P. Han, “Spectrum compression of Gaussian pulse from annular aperture in far-field,” Jpn. J. Appl. Phys. 47, 914-917 (2008).
[CrossRef]

Opt. Commun. (1)

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 99, 57-63 (1999).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (2)

Y. Yang and Y. Li, “Spectral shifts and spectral switches of a pulsed Bessel-Gaussian beam from a circular aperture in the far field,” Opt. Laser Technol. 39, 1478-1484 (2007).
[CrossRef]

B. Qu, J. Pu, and Z. Chen, “Experimental observation of spectral switch of partially coherent light focused by a lens with chromatic aberration,” Opt. Laser Technol. 39, 1226-1230 (2007).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

G. H. Yang, H. E. Hwang, and P. Han, “Spectral intensity distribution of a time dependent Gaussian pulsed beam from a circular aperture in the near-field diffraction,” Opt. Rev. 10, 19-23 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

G. Popescu and A. Dogariu, “Spectral anomalies at wave-front dislacations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901-1-013901-4 (2002).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Rep. Prog. Phys. (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

Other (5)

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.
[CrossRef]

J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, 1999).

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), p. 511 and p.464.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), p. 53.

K. Iizuka, Elements of Photonics (Wiley, 2002), p. 18.

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Figures (7)

Fig. 1
Fig. 1

(a) Geometry and notation for an incoming Gaussian pulse incident on a wedged circular aperture. (b) Structure of the aperture with radius a and wedge angle ξ.

Fig. 2
Fig. 2

Normalized spectral intensity of I y ( 0 , ω ) ω 2 G ( ω ) for two different bandwidths, γ = 1 ( ω 0 τ ) = 0.3 and γ = 0.5 . The spectrum is always blue-shifted. As the bandwidth γ increases, the amount of the peak’s shift increases. (For all the spectral curves in the following figures, each curve is normalized to its maximum value.)

Fig. 3
Fig. 3

Normalized spectra for G ( ω ) (dotted curve), M ( θ , ω ) (dashed curve), I y ( θ , ω ) (solid curve) at different angles. (a) sin ( θ ) = 1.0 × 10 4 . (b) sin ( θ ) = 1.67 × 10 4 . (c) sin ( θ ) = 2.5 × 10 4 . (The same curve types are used consistently in all the following figures.)

Fig. 4
Fig. 4

Normalized spectra for G ( ω ) (dotted curve), M ( θ , ω ) (dashed curve), I y ( θ , ω ) (solid curve) at different angles. (a) sin ( θ ) = 3.06 × 10 4 . (b) sin ( θ ) = 3.16 × 10 4 . (c) sin ( θ ) = 3.26 × 10 4 . The small solid dot(s) in the plots indicates the position of the maximum of the spectrum and the capital letter L (R) indicates the left (right) peak. The small circles on the horizontal axis indicate the frequency where M ( θ , ω ) and I ( θ , ω ) have zero amplitude.

Fig. 5
Fig. 5

Plot of the normalized frequency shift Ω as a function of sin ( θ ) for the parameter values ω 0 = 3 × 10 15 rad s , a = 1 mm , ξ = π 3 , and γ = 0.5 . The six angles indicated on the x axis from θ 1 to θ 6 marked with “×” correspond to the picked angles for Figs 3a, 3b, 3c and Figs. 4a, 4b, 4c, respectively. It is found that at θ 2 there is no shift for spectrum’s peak [ Ω = 0 as in Fig. 3b], and at θ 5 there is a discontinuous jump (spectral switch) as in Fig. 4b.

Fig. 6
Fig. 6

Normalized spectra for G ( ω ) (dotted curve) and I y ( θ , ω ) (solid curve) under different wedge angles at sin ( θ ) = 3.16 × 10 4 . (a) ξ = π 2 , (b) ξ = π 3 , (c) ξ = π 4 .

Fig. 7
Fig. 7

Illustration for the data encoding and information transmission by controlling the wedge angle. The blue shift (B) is associated with a bit of information such as “1” and the red shift (R) is associated with a bit of “0.”

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

U ( p , ω ) = 1 j λ Σ U ( p , ω ) exp ( j ω r c ) r χ ( θ ) d σ ,
FH ( g ( ρ , ϕ ) ) = n = ( j ) n exp ( j n f ϕ ) 2 π 0 ρ g n ( ρ ) J n ( 2 π f ρ ρ ) d ρ ,
g n ( ρ ) = 1 2 π π π g ( ρ , ϕ ) exp ( j n ϕ ) d ϕ .
FH ( circ ( ρ a ) ) = a f ρ J 1 ( 2 π a f ρ ) ,
g n ( ρ ) = circ ( ρ a ) [ 1 2 π ξ 2 ξ 2 exp ( j n ϕ ) d ϕ ] = { circ ( ρ a ) 1 n π sin ( n ξ 2 ) , n 0 circ ( ρ a ) ξ 2 π , n = 0 } .
FH ( wedge portion ) = n = except n = 0 ( j ) n exp ( j n f ϕ ) 2 n sin ( n ξ 2 ) 0 a ρ J n ( 2 π f ρ ρ ) d ρ + ξ 2 π a f ρ J 1 ( 2 π a f ρ ) .
FH ( g ( ρ , ϕ ) ) = ( 1 ξ 2 π ) a f ρ J 1 ( 2 π a f ρ ) 2 n = except n = 0 ( j ) n exp ( j n f ϕ ) 1 n sin ( n ξ 2 ) 0 a ρ J n ( 2 π f ρ ρ ) d ρ .
u ( p , t ) = exp [ 1 2 ( t τ ) 2 + j ω 0 t ] ,
U ( p , ω ) = 2 π τ exp { 1 2 [ ( ω ω 0 ) τ ] 2 } .
U ( p , ω ) = 1 j λ R exp ( j k R ) U ( p , ω ) FH ( g ( ρ , ϕ ) ) ,
U ( f ρ , ϕ , ω ) = 1 j R ( ω 2 π c ) e j ( k R ) ( 2 π τ ) exp { 1 2 [ ( ω ω 0 ) τ ] 2 } { ( 1 ξ 2 π ) }
a f ρ J 1 ( 2 π a f ρ ) 2 { n = except n = 0 ( j ) n exp ( j n f ϕ ) 1 n sin ( n ξ 2 ) 0 a ρ J n ( 2 π f ρ ρ ) d ρ } .
U y ( θ , ω ) = 2 π τ a 2 2 π c j R e j ( k R ) ( ω ) exp { 1 2 [ ( ω ω 0 ) τ ] 2 } ,
{ ( 2 π ξ ) [ J 1 ( a sin ( θ ) ω c ) a sin ( θ ) ω c ] 2 n = except n = 0 1 n sin ( n ξ 2 ) 0 a ρ J n ( a sin ( θ ) ω ρ c ) d ρ } ,
I y ( θ , ω ) = A exp { [ ( ω ω 0 ) τ ] 2 } ω 2 { ( 2 π ξ ) [ J 1 ( a sin ( θ ) ω c ) a sin ( θ ) ω c ] 2 n = except n = 0 1 n sin ( n ξ 2 ) 0 a ρ J n ( a sin ( θ ) ω ρ c ) d ρ } 2 A G ( ω ) M ( θ , ω ) ,
M ( θ , ω ) = ω 2 { ( 2 π ξ ) [ J 1 ( a sin ( θ ) ω c ) a sin ( θ ) ω c ] 2 n = except n = 0 1 n sin ( n ξ 2 ) 0 a ρ J n ( a sin ( θ ) ω ρ c ) d ρ } 2
I y ( 0 , ω ) = A G ( ω ) ω 2 ,
Ω = ( ω p ω 0 ) ω 0 ,

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