Abstract

The pulse envelope of an ultrashort pulsed beam is evaluated on the focal points of a Fresnel zone plate. The description of the field dynamics is given in terms of a diffraction-induced pulse train. Within these terms we follow an analytical procedure to characterize the temporal broadening observed at the principal focus, which is significant if the number of Fresnel zones exceeds the number of cycles in the pulse. For Gaussian-type envelopes, the focal field may be accurately expressed in a simple closed form. This expression has a flat-top shape at the principal focus and other odd-order foci, and a two-peak envelope in the case of a low-integer even-order focus. Finally, extremely high orders present a time-domain evolution that emulates a train of uniform pulses with temporal characteristics equivalent to those of the incident beam.

© 2006 Optical Society of America

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References

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  1. M. A. Porras, "Ultrashort pulsed Gaussian light beams," Phys. Rev. E 58, 1086-1093 (1998).
    [CrossRef]
  2. M. A. Porras, "Diffraction effects in few-cycle optical pulses," Phys. Rev. E 65, 026606 (2002).
    [CrossRef]
  3. 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 (1996).
    [CrossRef]
  4. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158-1165 (1992).
    [CrossRef]
  5. R. Ashman and M. Gu, "Effect of ultrashort pulsed illumination on foci caused by a Fresnel zone plate," Appl. Opt. 42, 1852-1855 (2003).
    [CrossRef] [PubMed]
  6. O. Hignette, J. Santamaria, and J. Bescos, "White light diffraction patterns of amplitude and phase zone plates," J. Opt. 10, 231-238 (1979).
    [CrossRef]
  7. Z. Bor, "Distortion of femtosecond laser pulses in lenses and lens systems," J. Mod. Opt. 35, 1907-1918 (1988).
    [CrossRef]
  8. Z. Bor, "Distortion of femtosecond laser pulses in lenses," Opt. Lett. 14, 119-121 (1989).
    [CrossRef] [PubMed]
  9. J. Pearce and D. Mittleman, "Defining the Fresnel zone for broadband radiation," Phys. Rev. E 66, 056602 (2002).
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, International Editions, Singapore, 1996), pp.66-67.
  11. Z. L. Horváth and Z. Bor, "Diffraction of short pulses with boundary diffraction wave theory," Phys. Rev. E 63, 026601 (2001).
    [CrossRef]
  12. A. E. Kaplan, "Diffraction-induced transformation of near-cycle and subcycle pulses," J. Opt. Soc. Am. B 15, 951-956 (1998).
    [CrossRef]
  13. We use the standard definition erf(z)=(2/π)int0zexp(−t2)dt.
  14. T. Brabec and F. Krausz, "Nonlinear optical propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
    [CrossRef]
  15. G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901-013904 (2002).
    [CrossRef] [PubMed]
  16. C. J. Zapata-Rodríguez and J. A. Monsoriu, "Spectral anomalies in focused waves of different Fresnel numbers," J. Opt. Soc. Am. A 21, 2418-2423 (2004).
    [CrossRef]

2004 (1)

2003 (1)

2002 (3)

M. A. Porras, "Diffraction effects in few-cycle optical pulses," Phys. Rev. E 65, 026606 (2002).
[CrossRef]

J. Pearce and D. Mittleman, "Defining the Fresnel zone for broadband radiation," Phys. Rev. E 66, 056602 (2002).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901-013904 (2002).
[CrossRef] [PubMed]

2001 (1)

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

1998 (2)

1997 (1)

T. Brabec and F. Krausz, "Nonlinear optical propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

1996 (2)

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 (1996).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, International Editions, Singapore, 1996), pp.66-67.

1992 (1)

1989 (1)

1988 (1)

Z. Bor, "Distortion of femtosecond laser pulses in lenses and lens systems," J. Mod. Opt. 35, 1907-1918 (1988).
[CrossRef]

1979 (1)

O. Hignette, J. Santamaria, and J. Bescos, "White light diffraction patterns of amplitude and phase zone plates," J. Opt. 10, 231-238 (1979).
[CrossRef]

Ashman, R.

Bescos, J.

O. Hignette, J. Santamaria, and J. Bescos, "White light diffraction patterns of amplitude and phase zone plates," J. Opt. 10, 231-238 (1979).
[CrossRef]

Bor, Z.

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

Z. Bor, "Distortion of femtosecond laser pulses in lenses," Opt. Lett. 14, 119-121 (1989).
[CrossRef] [PubMed]

Z. Bor, "Distortion of femtosecond laser pulses in lenses and lens systems," J. Mod. Opt. 35, 1907-1918 (1988).
[CrossRef]

Brabec, T.

T. Brabec and F. Krausz, "Nonlinear optical propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Gan, X. S.

Gbur, G.

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901-013904 (2002).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, International Editions, Singapore, 1996), pp.66-67.

Gu, M.

Hignette, O.

O. Hignette, J. Santamaria, and J. Bescos, "White light diffraction patterns of amplitude and phase zone plates," J. Opt. 10, 231-238 (1979).
[CrossRef]

Horváth, Z. L.

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

Kaplan, A. E.

Kempe, M.

Krausz, F.

T. Brabec and F. Krausz, "Nonlinear optical propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Mittleman, D.

J. Pearce and D. Mittleman, "Defining the Fresnel zone for broadband radiation," Phys. Rev. E 66, 056602 (2002).
[CrossRef]

Monsoriu, J. A.

Pearce, J.

J. Pearce and D. Mittleman, "Defining the Fresnel zone for broadband radiation," Phys. Rev. E 66, 056602 (2002).
[CrossRef]

Porras, M. A.

M. A. Porras, "Diffraction effects in few-cycle optical pulses," Phys. Rev. E 65, 026606 (2002).
[CrossRef]

M. A. Porras, "Ultrashort pulsed Gaussian light beams," Phys. Rev. E 58, 1086-1093 (1998).
[CrossRef]

Rudolph, W.

Santamaria, J.

O. Hignette, J. Santamaria, and J. Bescos, "White light diffraction patterns of amplitude and phase zone plates," J. Opt. 10, 231-238 (1979).
[CrossRef]

Stamm, U.

Visser, T. D.

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901-013904 (2002).
[CrossRef] [PubMed]

Wilhelmi, B.

Wolf, E.

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901-013904 (2002).
[CrossRef] [PubMed]

Zapata-Rodríguez, C. J.

Appl. Opt. (1)

J. Mod. Opt. (1)

Z. Bor, "Distortion of femtosecond laser pulses in lenses and lens systems," J. Mod. Opt. 35, 1907-1918 (1988).
[CrossRef]

J. Opt. (1)

O. Hignette, J. Santamaria, and J. Bescos, "White light diffraction patterns of amplitude and phase zone plates," J. Opt. 10, 231-238 (1979).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Phys. Rev. E (4)

M. A. Porras, "Ultrashort pulsed Gaussian light beams," Phys. Rev. E 58, 1086-1093 (1998).
[CrossRef]

M. A. Porras, "Diffraction effects in few-cycle optical pulses," Phys. Rev. E 65, 026606 (2002).
[CrossRef]

J. Pearce and D. Mittleman, "Defining the Fresnel zone for broadband radiation," Phys. Rev. E 66, 056602 (2002).
[CrossRef]

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

Phys. Rev. Lett. (2)

T. Brabec and F. Krausz, "Nonlinear optical propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901-013904 (2002).
[CrossRef] [PubMed]

Other (2)

We use the standard definition erf(z)=(2/π)int0zexp(−t2)dt.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, International Editions, Singapore, 1996), pp.66-67.

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Figures (7)

Fig. 1
Fig. 1

Pulsed plane wave is incident on a Fresnel zone plate. Multiple boundary pulsed beams interfere at the principal focus found at a distance f. The constant delay between different wavelets generates a diffraction-induced pulse train. However, short delays provoke the formation of a unique pulse of a broader width than the incident pulse.

Fig. 2
Fig. 2

Plot of the pulse envelope F normalized to a value 2 π N t . The temporal coordinate is also normalized to the pulse width τ. The values of the parameter γ are (a) 1 5 , (b) 1, and (c) 5.

Fig. 3
Fig. 3

Amplitude distribution of the pulse envelope provided by Eq. (6) and normalized to a value 2 π N t for γ = 10 and effective number of cycles N t = 5 (dashed curve) and N t = 50 (solid curve).

Fig. 4
Fig. 4

Pulse envelope F n , normalized to a value 2 π N t , of a focused Gaussian beam of N t = 50 for different values of the orders n and a factor (a) γ = 10 and (b) γ = 1 .

Fig. 5
Fig. 5

Envelope pattern of a Gaussian quasi-monochromatic beam of N t = 500 focused by a Fresnel diffractive lens of M = 50 zones ( γ = 1 10 ) at the focal points of lowest odd orders n.

Fig. 6
Fig. 6

Pulse envelope F n ( t ) of a focused Gaussian beam of N t = 50 at the second focus n = 2 , for values of (a) γ = 1 5 , (b) γ = 1 , and (c) γ = 5 .

Fig. 7
Fig. 7

Pulse envelope F n ( t ) of a focused Gaussian beam of N t = 5 and γ = 1 at the extremely high-order foci (a) n = 51 and (b) n = 52 .

Equations (21)

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F ( t ) = F ̃ 0 ( Ω ) exp ( i Ω t ) U ( ω 0 + Ω , z ) d Ω ,
U ( ω , z ) = ω i c z 0 P ( r ) exp ( i ω 2 c z r 2 ) r d r
P ( r ) = m = 0 M 1 rect [ r 2 p 2 ( m + 1 4 ) 1 2 ] ,
U ( ω , z ) = [ 1 exp ( i ω p 2 4 c z ) ] m = 0 M 1 exp ( i ω p 2 2 c z m ) .
U ( ω 0 + Ω , f ) = [ 1 + exp ( i Ω T 0 2 ) ] m = 0 M 1 exp ( i Ω T 0 m ) ,
F ( t ) = m = 0 M 1 [ F 0 ( t m T 0 ) + F 0 ( t ( m + 1 2 ) T 0 ) ] .
m = 0 M 1 exp ( i m x ) = 1 exp ( i M x ) 1 exp ( i x ) ,
U ( ω 0 + Ω , f ) = [ 1 + exp ( i π Ω ω 0 ) ] 1 exp ( i 2 π M Ω ω 0 ) 1 exp ( i 2 π Ω ω 0 ) = 1 exp ( i 2 π M Ω ω 0 ) 1 exp ( i π Ω ω 0 ) .
F ( t ) F ̃ 0 ( Ω ) exp ( i Ω t ) 1 exp ( i 2 π M Ω ω 0 ) i π Ω ω 0 d Ω .
t F ( t ) = ω 0 π F ̃ 0 ( Ω ) exp ( i Ω t ) [ 1 exp ( i 2 π M Ω ω 0 ) ] d Ω = 2 T 0 [ F 0 ( t ) F 0 ( t M T 0 ) ] .
F ( t ) = 2 T 0 t [ F 0 ( t 0 ) F 0 ( t 0 M T 0 ) ] d t 0 = 2 M { 1 M T 0 t M T 0 t F 0 ( t 0 ) d t 0 } .
F ( t ) 2 M t d F 0 ( t 0 ) d t 0 d t 0 = 2 M F 0 ( t ) .
F ( t ) 2 T 0 F 0 ( t 0 ) d t 0 = F max
G ( t ) = 2 T 0 t F 0 ( t 0 ) d t 0 .
F ( t ) = π ( τ T 0 ) [ erf ( t τ ) erf ( t τ M τ T 0 ) ] .
F n ( t ) = 2 M { 1 M n T 0 t M n T 0 t F 0 ( t 0 ) d t 0 } .
F n ( t ) = m = 0 M 1 [ F 0 ( t m n T 0 ) + F 0 ( t ( m + 1 2 ) n T 0 ) ] .
F n ( t ) = F ̃ 0 ( Ω ) exp ( i Ω t ) 1 exp ( i Ω n T 0 2 ) 1 exp ( i Ω n T 0 ) [ 1 exp ( i M Ω n T 0 ) ] d Ω 1 2 F ̃ 0 ( Ω ) exp ( i Ω t ) [ 1 exp ( i M Ω n T 0 ) ] d Ω .
F n ( t ) 1 2 [ F 0 ( t ) F 0 ( t M n T 0 ) ] .
F n ( t ) M n T 0 2 d F 0 d t .
F n ( t ) = m = 0 M 1 [ F 0 ( t m n T 0 ) F 0 ( t ( m + 1 2 ) n T 0 ) ] .

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