Abstract

Based on the Fresnel diffraction integral and by introducing a hard-aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression for the near-field spectral intensity distribution of a space–time-dependent Gaussian pulsed beam passing through an annular aperture is derived, which permits us to study the on- and off-axis spectral anomalies that are near phase singularities of the diffracted Gaussian pulsed beam in the near-field. The expressions for a circular black screen and a circular aperture are given as special cases of the general results. The relative spectral shift of a space–time-dependent Gaussian pulsed beam versus the different values of the truncation parameters and the position parameters of observation points are also studied and illustrated with numerical calculations. It is shown that the spectral switch appears near phase singularities in the near-field, and the near-field spectral behavior depends on the truncation parameters, the pulse duration τ, and the position parameter. The results of this work have potential applications in free-space information encoding and transmission.

© 2007 Optical Society of America

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References

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  1. J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
    [CrossRef]
  2. M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219-276.
    [CrossRef]
  3. 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]
  4. J. Pu and S. Nemoto, "Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture," IEEE J. Quantum Electron. 36, 1407-1411 (2000).
    [CrossRef]
  5. J. Pu and S. Nemoto, "Spectral changes and 1 × N spectral switches in diffraction of partially coherent light by an aperture," J. Opt. Soc. Am. A 19, 339-344 (2000).
    [CrossRef]
  6. G. Gbur, T. D. Visser, and E. Wolf, "Singular behavior of the spectrum in the neighborhood of focus," J. Opt. Soc. Am. A 19, 1694-1700 (2002).
    [CrossRef]
  7. 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]
  8. G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
    [PubMed]
  9. S. A. Ponomarenko and E. Wolf, "Spectral anomalies in a Fraunhofer diffraction pattern," Opt. Lett. 27, 1211-1213 (2002).
    [CrossRef]
  10. H. C. Kandpal, "Experimental observation of the phenomenon of spectral switch," J. Opt. A: Pure Appl. Opt. 3, 296-299 (2001).
    [CrossRef]
  11. H. C. Kandpal, S. Anand, and J. S. Vaishya, "Experimental observation of the phenomenon of spectral switching for a class of partially coherent light," IEEE J. Quantum Electron. 38, 336-339 (2002).
    [CrossRef]
  12. G. Popescu and A. Dogariu, "Spectral anomalies at wave-front dislocations," Phys. Rev. Lett. 88, 183902 (2002).
    [CrossRef] [PubMed]
  13. Q. Zou and B. Lü, "Anomalous spectral behaviour near phase singularities in diffraction of pulsed Laguerre-Gaussian beams," J. Opt. A: Pure Appl. Opt. 8, 531-536 (2006).
    [CrossRef]
  14. Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture," Optik 115, 447-454 (2004).
    [CrossRef]
  15. S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, "Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities," Appl. Phys. Lett. 89, 041119 (2006).
    [CrossRef]
  16. J. Li, H. Zhang, D. Doerr, and D. R. Alexander, "Optical communications with femtosecond lasers," Proc. SPIE 6399, 639908 (2006).
  17. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  18. J. J. Wen and M. A. Breazeale, "A different beam field expressed as a superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
    [CrossRef]
  19. H. Mao and D. Zhao, "Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system," J. Opt. Soc. Am. A 22, 647-653 (2005).
    [CrossRef]
  20. J. Gu and D. Zhao, "Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture," J. Mod. Opt. 52, 1065-1072 (2005).
    [CrossRef]
  21. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Elsevier, 2004), p. 698.
  22. J. Pu, C. Cai, and S. Nemoto, "Spectral anomalies in Young's double-slit interference experiment," Opt. Express 12, 5131-5139 (2004).
    [CrossRef] [PubMed]
  23. B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, "Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture," Opt. Commun. 269, 253-260 (2007).
    [CrossRef]

2007

B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, "Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture," Opt. Commun. 269, 253-260 (2007).
[CrossRef]

2006

Q. Zou and B. Lü, "Anomalous spectral behaviour near phase singularities in diffraction of pulsed Laguerre-Gaussian beams," J. Opt. A: Pure Appl. Opt. 8, 531-536 (2006).
[CrossRef]

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, "Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities," Appl. Phys. Lett. 89, 041119 (2006).
[CrossRef]

J. Li, H. Zhang, D. Doerr, and D. R. Alexander, "Optical communications with femtosecond lasers," Proc. SPIE 6399, 639908 (2006).

2005

H. Mao and D. Zhao, "Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system," J. Opt. Soc. Am. A 22, 647-653 (2005).
[CrossRef]

J. Gu and D. Zhao, "Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture," J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

2004

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

Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture," Optik 115, 447-454 (2004).
[CrossRef]

2003

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]

2002

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[PubMed]

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]

G. Gbur, T. D. Visser, and E. Wolf, "Singular behavior of the spectrum in the neighborhood of focus," J. Opt. Soc. Am. A 19, 1694-1700 (2002).
[CrossRef]

H. C. Kandpal, S. Anand, and J. S. Vaishya, "Experimental observation of the phenomenon of spectral switching for a class of partially coherent light," IEEE J. Quantum Electron. 38, 336-339 (2002).
[CrossRef]

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

2001

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

2000

J. Pu and S. Nemoto, "Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture," IEEE J. Quantum Electron. 36, 1407-1411 (2000).
[CrossRef]

J. Pu and S. Nemoto, "Spectral changes and 1 × N spectral switches in diffraction of partially coherent light by an aperture," J. Opt. Soc. Am. A 19, 339-344 (2000).
[CrossRef]

1999

J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
[CrossRef]

1988

J. J. Wen and M. A. Breazeale, "A different beam field expressed as a superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Appl. Phys. Lett.

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, "Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities," Appl. Phys. Lett. 89, 041119 (2006).
[CrossRef]

IEEE J. Quantum Electron.

H. C. Kandpal, S. Anand, and J. S. Vaishya, "Experimental observation of the phenomenon of spectral switching for a class of partially coherent light," IEEE J. Quantum Electron. 38, 336-339 (2002).
[CrossRef]

J. Pu and S. Nemoto, "Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture," IEEE J. Quantum Electron. 36, 1407-1411 (2000).
[CrossRef]

J. Acoust. Soc. Am.

J. J. Wen and M. A. Breazeale, "A different beam field expressed as a superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

J. Mod. Opt.

J. Gu and D. Zhao, "Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture," J. Mod. Opt. 52, 1065-1072 (2005).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

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

Q. Zou and B. Lü, "Anomalous spectral behaviour near phase singularities in diffraction of pulsed Laguerre-Gaussian beams," J. Opt. A: Pure Appl. Opt. 8, 531-536 (2006).
[CrossRef]

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]

J. Opt. Soc. Am. A

Opt. Commun.

J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
[CrossRef]

B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, "Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture," Opt. Commun. 269, 253-260 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture," Optik 115, 447-454 (2004).
[CrossRef]

Phys. Rev. Lett.

G. Popescu and A. Dogariu, "Spectral anomalies at wave-front dislocations," 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 (2002).
[PubMed]

Proc. SPIE

J. Li, H. Zhang, D. Doerr, and D. R. Alexander, "Optical communications with femtosecond lasers," Proc. SPIE 6399, 639908 (2006).

Other

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219-276.
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Elsevier, 2004), p. 698.

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Figures (9)

Fig. 1
Fig. 1

Normalized spectral modifier K ( ρ , z ; ω ) versus relative propagation distance z / z 0 and the relative transverse distance ρ / w 0 for a fixed annular aperture, respectively, where ω 0 = 2.36 fs 1 and τ = 3   fs . The solid curves denote the case using Eq. (14), and the dotted curves denote the case using the diffraction integral formula directly: (a) δ a = 1 , δ b = 0.5 , and ρ = 0 ; (b) δ a = 1 , δ b = 0.8 , and z / z 0 = 1 .

Fig. 2
Fig. 2

Normalized spectral intensity I ( ρ , z ; ω ) of a space–time-dependent Gaussian pulsed beam from an annular aperture with different position parameters, where ω 0 = 2.36 fs 1 , τ = 3   fs , δ a = 1.0 , δ b = 0.8 , and z / z 0 = 1.0 :(a) ρ / w 0 = 0 , 0.214, and 0.3; (b) ρ / w 0 = 0.4 , 0.4225, and 0.45; (c) ρ / w 0 = 0.4225 , 0.971, and 1.533.

Fig. 3
Fig. 3

Normalized spectral intensity I ( ρ , z ; ω ) within the region 0.35 ρ / w 0 0.5 versus ( ω ω 0 ) / ω 0 and relative transverse distance ρ / w 0 . The other calculation parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Relative spectral shift ( ω max ω 0 ) / ω 0 of a space–time-dependent Gaussian pulsed beam passing through an annular aperture versus relative transverse distance ρ / w 0 . Other calculation parameters are the same as in Fig. 2 except that τ = 6   fs is used in the dotted curve.

Fig. 5
Fig. 5

Real time-dependent waveform u ( t ) = exp [ ( t / τ ) 2 / 2 ] cos ( ω 0 t ) corresponding to Eq. (3) for ω 0 = 2.36 fs 1 and τ = 3   fs and 6 fs.

Fig. 6
Fig. 6

Relative spectral shift ( ω max ω 0 ) / ω 0 of a space–time-dependent Gaussian pulsed beam passing through a circular aperture versus relative propagation distance z / z 0 , where ω 0 = 2.36 fs 1 , ρ = 0 , δ a = 1.0 , and δ b = 0 .

Fig. 7
Fig. 7

Normalized spectral intensity I ( ρ , z ; ω ) of a space–time-dependent Gaussian pulsed beam from a circular aperture for different values of τ = 3   fs and 6 fs, where z / z 0 = 0.252 and the other calculation parameters are the same as in Fig. 6.

Fig. 8
Fig. 8

Relative spectral shift ( ω max ω 0 ) / ω 0 of a space–time-dependent Gaussian pulsed beam versus truncation parameter δ a within the region 0.1 δ a 3 , where τ = 3   fs , ω 0 = 2.36 fs 1 , ρ / w 0 = 1 , z / z 0 = 0.5 , and δ b = 0 .

Fig. 9
Fig. 9

Relative mean frequency variation ( ω ¯ ω 0 ) / ω 0 of a space–time-dependent Gaussian pulsed beam versus the aperture truncation parameter δ a , where ω 0 = 2.36 fs 1 , τ = 3   fs , δ b = 0.8 , ρ / w 0 = 0.4 , and z / z 0 = 1 .

Equations (19)

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U ( ρ , z ; ω ) = i ω c z exp ( i ω z c i ω ρ 2 2 c z ) 0 T ( ρ ) U ( ρ , 0 ; ω ) × exp ( i ω ρ 2 2 c z ) J 0 ( ω ρ ρ c z ) ρ d ρ ,
T ( ρ ) = { 1 b ρ a 0 otherwise ,
u ( ρ , 0 ; t ) = exp ( ρ 2 w 0 2 ) exp [ 1 2 ( t τ ) 2 + i ω 0 t ] ,
U ( ρ , 0 ; ω ) = 1 2 π u ( ρ , 0 ; t ) exp ( i ω t ) d t .
U ( ρ , 0 ; ω ) = τ 2 π exp ( ρ 2 w 0 2 ) exp { 1 2 [ ( ω ω 0 ) τ ] 2 } .
T ( ρ ) = T a ( ρ ) T b ( ρ ) ,
T a ( ρ ) = j = 1 N A j exp ( B j a 2 ρ 2 ) ,
T b ( ρ ) = j = 1 N A j exp ( B j b 2 ρ 2 ) ,
0 exp ( m x 2 ) J ν ( n x ) x ν + 1 d x = n ν ( 2 m ) ν + 1 exp ( n 2 4 m ) , Re [ m ] > 0 , Re [ ν ] > 1 ,
U ( ρ , z ; ω ) = i ω c z τ 2 π exp ( i ω z c i ω ρ 2 2 c z ) × exp { 1 2 [ ( ω ω 0 ) τ ] 2 } × { j = 1 N A j 0 exp [ ( 1 w 0 2 + B j a 2 + i ω 2 c z ) ρ 2 ] × J 0 ( ω ρ ρ c z ) ρ d ρ j = 1 N A j 0 exp [ ( 1 w 0 2 + B j b 2 + i ω 2 c z ) ρ 2 ] × J 0 ( ω ρ ρ c z ) ρ d ρ } = i ω c z τ 2 π exp ( i ω z c i ω ρ 2 2 c z ) × exp { 1 2 [ ( ω ω 0 ) τ ] 2 } × [ j = 1 N A j 2 ξ j exp ( ρ 2 4 c 2 z 2 ξ j ω 2 ) j = 1 N A j 2 χ j exp ( ρ 2 4 c 2 z 2 χ j ω 2 ) ] ,
ξ j = 1 w 0 2 + B j a 2 + i ω 2 c z ,     χ j = 1 w 0 2 + B j b 2 + i ω 2 c z .
I ( ρ , z ; ω ) = K ( ρ , z ; ω ) E ( 0 ) ( ω ) ,
E ( 0 ) ( ω ) = exp { [ ( ω ω 0 ) τ ] 2 } ,
K ( ρ , z ; ω ) = ω 2 τ 2 2 π c 2 z 2 | j = 1 N A j 2 ξ j exp ( ρ 2 4 c 2 z 2 ξ j ω 2 ) j = 1 N A j 2 χ j exp ( ρ 2 4 c 2 z 2 χ j ω 2 ) | 2 .
U 1 ( ρ , z ; ω ) = i ω c z τ 2 π exp ( i ω z c i ω ρ 2 2 c z ) × exp { 1 2 [ ( ω ω 0 ) τ ] 2 } × j = 1 N A j 2 ξ j exp ( ρ 2 4 c 2 z 2 ξ j ω 2 ) ,
I 1 ( ρ , z ; ω ) = ω 2 τ 2 2 π c 2 z 2 | j = 1 N A j 2 ξ j exp ( ρ 2 4 c 2 z 2 ξ j ω 2 ) | 2 E ( 0 ) ( ω ) .
U 2 ( ρ , z ; ω ) = i ω c z τ 2 π exp ( i ω z c i ω ρ 2 2 c z ) × exp { 1 2 [ ( ω ω 0 ) τ ] 2 } { c z w 0 2 2 c z + i ω w 0 2 × exp [ c z w 0 2 ρ 2 2 c 2 z 2 ( 2 c z + i ω w 0 2 ) ω 2 ] j = 1 N A j 2 χ j exp ( ρ 2 4 c 2 z 2 χ j ω 2 ) } ,
I 2 ( ρ , z ; ω ) = ω 2 τ 2 2 π c 2 z 2 | c z w 0 2 2 c z + i ω w 0 2 × exp [ c z w 0 2 ρ 2 2 c 2 z 2 ( 2 c z + i ω w 0 2 ) ω 2 ] j = 1 N A j 2 χ j exp ( ρ 2 4 c 2 z 2 χ j ω 2 ) | 2 E ( 0 ) ( ω ) .  
ω ¯ ( x , y , z ) = 0 ω | U ( x , y , z ; ω ) | 2 d ω 0 | U ( x , y , z ; ω ) | 2 d ω

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