Abstract

The analytical solution of the Rayleigh-Sommerfeld on-axis diffraction integral for an ultrashort light pulse diffracted by circularly symmetric hard apertures is derived. The particular case of a circular aperture is treated in detail. The time changes of the instantaneous intensity along the axial direction are predicted. An analysis of the standard deviation width shows a pulse broadening about the axial positions where the instantaneous intensity reaches a zero value. We show that the temporal shape of the instantaneous intensity depends on the number of oscillation cycles at the central frequency of the real electric field.

© 2007 Optical Society of America

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References

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  1. S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (2006).
    [CrossRef]
  2. 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]
  3. H. E. Hwang and G. H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (2002).
    [CrossRef]
  4. H. E. Hwang, G. H. Yang, and P. Han, "Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture," Opt. Eng. 42, 686-695 (2003).
    [CrossRef]
  5. M. Lefrançois and S. F. Pereira, "Time evolution of the diffraction pattern of an ultrashort laser pulse," Opt. Express 11, 1114-1122 (2003).
    [CrossRef] [PubMed]
  6. Z. Jiang, R. Jacquemin, and W. Eberhardt, "Time dependence of Fresnel diffraction of ultrashort laser pulses by a circular aperture," Appl. Opt. 36, 4358-4361 (1997).
    [CrossRef] [PubMed]
  7. Z. L. Horváth and Zs. Bor, "Diffraction of short pulses with boundary diffraction wave theory," Phys. Rev. E. 63, 026601 (2001).
    [CrossRef]
  8. M. Gu and X. S. Gan, "Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam," J. Opt. Soc. Am. A 13, 771-778 (1995).
    [CrossRef]
  9. C. J. Zapata-Rodríguez, "Temporal effects in ultrashort pulsed beams focused by planar diffracting elements," J. Opt. Soc. Am. A 23, 2335-2341 (2006).
    [CrossRef]
  10. H. Zhang, J. Li, D.W. Doerr, and D. R. Alexander, "Diffraction characteristics of a Fresnel zone plate illuminated by 10 fs laser pulses," Appl. Opt. 45, 8541-8546 (2006).
    [CrossRef] [PubMed]
  11. A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1926-1960, (2000).
    [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 3.
  13. R. Peng and D. Fan, "Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam," Opt. Commun. 246, 241-248 (2005).
    [CrossRef]
  14. 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]
  15. J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 1996), Chap. 1.

2006 (4)

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (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]

C. J. Zapata-Rodríguez, "Temporal effects in ultrashort pulsed beams focused by planar diffracting elements," J. Opt. Soc. Am. A 23, 2335-2341 (2006).
[CrossRef]

H. Zhang, J. Li, D.W. Doerr, and D. R. Alexander, "Diffraction characteristics of a Fresnel zone plate illuminated by 10 fs laser pulses," Appl. Opt. 45, 8541-8546 (2006).
[CrossRef] [PubMed]

2005 (1)

R. Peng and D. Fan, "Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam," Opt. Commun. 246, 241-248 (2005).
[CrossRef]

2003 (2)

H. E. Hwang, G. H. Yang, and P. Han, "Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture," Opt. Eng. 42, 686-695 (2003).
[CrossRef]

M. Lefrançois and S. F. Pereira, "Time evolution of the diffraction pattern of an ultrashort laser pulse," Opt. Express 11, 1114-1122 (2003).
[CrossRef] [PubMed]

2002 (2)

H. E. Hwang and G. H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (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)

Z. L. Horváth and Zs. Bor, "Diffraction of short pulses with boundary diffraction wave theory," Phys. Rev. E. 63, 026601 (2001).
[CrossRef]

2000 (1)

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1926-1960, (2000).
[CrossRef]

1997 (1)

1995 (1)

Alexander, D. R.

Bor, Zs.

Z. L. Horváth and Zs. Bor, "Diffraction of short pulses with boundary diffraction wave theory," Phys. Rev. E. 63, 026601 (2001).
[CrossRef]

Doerr, D.W.

Eberhardt, W.

Fan, D.

R. Peng and D. Fan, "Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam," Opt. Commun. 246, 241-248 (2005).
[CrossRef]

Foley, J. T.

Gan, X. S.

Gu, M.

Han, P.

H. E. Hwang, G. H. Yang, and P. Han, "Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture," Opt. Eng. 42, 686-695 (2003).
[CrossRef]

Horváth, Z. L.

Z. L. Horváth and Zs. Bor, "Diffraction of short pulses with boundary diffraction wave theory," Phys. Rev. E. 63, 026601 (2001).
[CrossRef]

Hwang, H. E.

H. E. Hwang, G. H. Yang, and P. Han, "Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture," Opt. Eng. 42, 686-695 (2003).
[CrossRef]

H. E. Hwang and G. H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (2002).
[CrossRef]

Jacquemin, R.

Jiang, Z.

Lefrançois, M.

Li, J.

Peng, R.

R. Peng and D. Fan, "Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam," Opt. Commun. 246, 241-248 (2005).
[CrossRef]

Pereira, S. F.

Schimmel, H.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (2006).
[CrossRef]

Sharma, D. K.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (2006).
[CrossRef]

Veetil, S. P.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (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]

Vijayan, C.

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]

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (2006).
[CrossRef]

Viswanathan, N. K.

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]

Weiner, A. M.

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1926-1960, (2000).
[CrossRef]

Wolf, E.

Wyrowski, F.

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]

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (2006).
[CrossRef]

Yang, G. H.

H. E. Hwang, G. H. Yang, and P. Han, "Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture," Opt. Eng. 42, 686-695 (2003).
[CrossRef]

H. E. Hwang and G. H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (2002).
[CrossRef]

Zapata-Rodríguez, C. J.

Zhang, H.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

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. Mod. Opt. (1)

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (2006).
[CrossRef]

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

Opt. Commun. (1)

R. Peng and D. Fan, "Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam," Opt. Commun. 246, 241-248 (2005).
[CrossRef]

Opt. Eng. (2)

H. E. Hwang and G. H. Yang, "Far-field diffraction characteristics of a time-variant Gaussian pulsed beam propagating from a circular aperture," Opt. Eng. 41, 2719-2727 (2002).
[CrossRef]

H. E. Hwang, G. H. Yang, and P. Han, "Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture," Opt. Eng. 42, 686-695 (2003).
[CrossRef]

Opt. Express (1)

Phys. Rev. E. (1)

Z. L. Horváth and Zs. Bor, "Diffraction of short pulses with boundary diffraction wave theory," Phys. Rev. E. 63, 026601 (2001).
[CrossRef]

Rev. Sci. Instrum. (1)

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1926-1960, (2000).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 3.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 1996), Chap. 1.

Supplementary Material (1)

» Media 1: GIF (1303 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Second time derivate of I(Z,τ+ z/c)×10-29 evaluated at τ = T 0

Fig. 2.
Fig. 2.

Temporal evolution of the instantaneous intensity between two axial points of maximum destructive interference.

Fig.3.
Fig.3.

Standard deviation ratio of diffracted and non diffracted pulses.

Fig. 4.
Fig. 4.

(1.30 MB) Movie of the time evolution of real electric fields of geometric and boundary wave pulses, including their pulse envelope. [Media 1]

Fig. 5.
Fig. 5.

Second time derivate of I(Z,τ+ z/c)×10-29 evaluated at τ = T 0 for different pulse shape models.

Equations (23)

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U z t = 1 2 π A 0 ( ω ) U 0 ( z , ω ) exp ( iωt ) .
A 0 ( ω ) = u 0 ( t ) exp ( iωt ) dt .
U z t = m = 1 N z z 2 + r im 2 u 0 ( t z 2 + r im 2 c ) m = 1 N z z 2 + r om 2 u 0 ( t z 2 + r om 2 c ) .
u 0 ( t ) = P ( t ) exp ( i ω 0 t ) ,
U z T 0 m = m = 1 N P ( s om s im 2 c ) { z s im exp ( i Δ φ 2 ) z s om exp ( i Δ φ 2 ) }
U z t = m = 1 N u 0 ( t z + r im 2 2 z c ) m = 1 N u 0 ( t z + r om 2 2 z c ) .
U z t = 1 z + z 0 m = 1 N u 0 ( t z + r im 2 2 z c z 0 + r im 2 2 z 0 c ) 1 z + z 0 m = 1 N u 0 ( t z + r om 2 2 z c z 0 + r om 2 2 z 0 c )
U ( z , τ + z c ) = exp ( i ω 0 τ ) exp ( τ 2 4 σ 0 2 ) exp [ i ω 0 ( τ r 0 2 2 cz ) ] exp [ 1 4 σ 0 2 ( τ r 0 2 2 cz ) 2 ] .
I ( z , τ + z c ) = I 0 ( z , τ + z c ) { 1 + exp [ r 0 2 2 cz σ 0 2 ( τ r 0 2 4 cz ) ] 2 cos ( ω 0 r 0 2 2 cz ) exp [ r 0 2 4 cz σ 0 2 ( τ r 0 2 4 cz ) ] }
I ( z , τ + z c ) = I 0 ( z , τ + z c ) { 1 + exp [ r 0 2 4 cz σ 0 2 ( τ r 0 2 4 cz ) ] } 2 ,
I ( z , τ + z c ) = I 0 ( z , τ + z c ) { 1 exp [ r 0 2 4 cz σ 0 2 ( τ r 0 2 4 cz ) ] } 2 .
m 0 = 2 m 0 in { 1 cos ( N 0 ) exp [ 1 2 T 0 σ 0 2 ] } ,
m 1 = m 0 T 0 ,
m 2 = 2 m 0 in T 0 2 + m 0 σ 0 2 [ 1 + ( T 0 σ 0 ) 2 ] .
σ 2 = σ 0 2 { 1 + ( T 0 σ 0 ) 2 1 cos ( N 0 ) exp [ 1 2 ( T 0 σ 0 ) 2 ] } .
U 0 z ω = r 1 r 2 ( i ω c 1 z 2 + ρ 2 ) exp ( i ω c z 2 + ρ 2 ) z z 2 + ρ 2 ρdρ =
= z z 2 + r 1 2 exp ( i ω c z 2 + r 1 2 ) z z 2 + r 2 2 exp ( i ω c z 2 + r 2 2 ) .
Dirac ( ξ ) = 1 2 π exp ( iωξ ) ,
U z t = z z 2 + r 1 2 u 0 ( t z 2 + r 1 2 c ) z z 2 + r 2 2 u 0 ( t z 2 + r 2 2 c ) .
U z t = m = 1 N z z 2 + r im 2 u 0 ( t z 2 + r im 2 c ) m = 1 N z z 2 + r om 2 u 0 ( t z 2 + r om 2 c ) .
U 0 z ω = ω exp [ i ω c ( z + z 0 ) ] icz z 0 r 1 r 2 exp [ 2 c ( 1 z + 1 z 0 ) ρ 2 ] ρdρ =
= 1 z + z 0 exp [ ( z + z 0 c + r 1 2 ( z + z 0 ) 2 cz z 0 ) ] 1 z + z 0 exp [ ( z + z 0 c + r 2 2 ( z + z 0 ) 2 cz z 0 ) ] .
U z t = 1 z + z 0 m = 1 N u 0 ( t z + r im 2 2 z c z 0 + r im 2 2 z 0 c ) 1 z + z 0 m = 1 N u 0 ( t z + r om 2 2 z c z 0 + r om 2 2 z 0 c ) .

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