## Time evolution of the diffraction pattern of an ultrashort laser pulse

Optics Express, Vol. 11, Issue 10, pp. 1114-1122 (2003)

http://dx.doi.org/10.1364/OE.11.001114

Acrobat PDF (262 KB)

### Abstract

An analytical expression for the time evolution of the diffraction pattern of an ultrashort laser pulse passing through a circular aperture is obtained in the Fresnel regime. The diffraction is not constant in time as the pulse travels through the aperture. This may have implications in experiments involving fast dynamics. Examples of the evolution of the diffraction pattern are given.

© 2003 Optical Society of America

## 1. Introduction

2. Z. Ziang, 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]

## 2. Theoretical aspects

1. M. Gu and X. S. Gan, “Fresnel diffraction by a circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A **13**, 771–778 (1996). [CrossRef]

_{1}=

*r*

_{1}

*/a*and ρ=

*r/a*are the normalised radial co-ordinates with

*r*

_{1}

*)*

^{1/2}and

*r=(x*

_{2}

*+y*

_{2}

*)*

^{1/2}being the real co-ordinates in the aperture and the observation planes, respectively. Here, the Fresnel number is defined as

*N*=π

*a*

^{2}

*/(*λ

*z)*and

*z*is the distance between these two planes.

*Ẽ*

_{1}is the spectrum of the pulse in the aperture plane given by [12

12. C. F. R. Caron and R. M. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. **46**, 1881–1891 (1999). [CrossRef]

14. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes”, J. Opt. Soc. Am. A **18**, 2594–2600 (2001). [CrossRef]

_{0}is the frequency shift, which gives an estimate of the average wavelength of the spectrum, ω

_{1}is the spectrum width, and s is a positive parameter (here we are interested in the small values of

*s*, such as 0, 1 or 2, which corresponds to very short pulses in the time domain). Γ is the Gamma function, and particularly Γ(

*s*+1)=

*s*! if

*s*is a positive integer. This model for the spectrum of the pulse satisfies the physical requirement that it includes only positive-frequency components and the pulse has a simple analytical form. Note also that as s increases, the spectrum tends to a Gaussian [12

12. C. F. R. Caron and R. M. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. **46**, 1881–1891 (1999). [CrossRef]

*Ẽ*

_{1}(ρ

_{1}, ω) does not depend on ρ

_{1}. As a consequence it is straightforward to calculate the integral of Eq. (1) analytically, using the Lommel functions [11]:

*Ẽ*in

*Ẽ*

_{1}+

*Ẽ*

_{2}, each field

*Ẽ*

_{j}being associated with the Lommel function

*U*

_{j}. The temporal evolution of the electric field

*E*

_{j}is obtained by taking the Fourier transform of Eq. (5) (note that the integral starts at the point ω

_{0}because of the function

*f*

_{s}):

_{0}=0. Thus, by the change of variables ω'←ω-ω

_{0}and ω←ω' we obtain

- First, we write the Lommel function as a sum of Bessel functions. Then, by interchanging the sum sign and the integral sign (which is allowed when the Lommel function converges, i.e., for the shadow region ρ>1)[11], we obtain:

*J*

_{-ν}(

*x*)=(-1)

^{ν}

*J*

_{ν}(

*x*).

*J*

_{ν}(

*a+b*) can be rewritten as

*s*is positive, (which is always verified in our case), and provided |

*b*|<|

*a*| (either the two parameters are real or complex).

*n*) with n<0. Then the second form of the addition theorem using only positive indexes for the Bessel and Gamma functions becomes:

*E*(ρ,

*z, t*)=

*E*

_{1}(ρ,

*z, t*)+

*iE*

_{2}(ρ,

*z, t*). The instantaneous intensity (considering that the detector has a delta function response in time) is given by

*I*(ρ,

*z,t*)=|

*E*(ρ,

*z,t*)|

^{2}[1

1. M. Gu and X. S. Gan, “Fresnel diffraction by a circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A **13**, 771–778 (1996). [CrossRef]

2. Z. Ziang, 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]

*U*Lommel functions in Eq. (7), which diverge as ρ tends to 0. We verified numerically however that this divergence is very slow. If we use Eq. (14) to compute the diffraction patterns, this problem does not seem to be disturbing since the diffraction pattern extends to an area which is much bigger than ρ=1. In order to avoid the centre (ρ=0) we simply substitute ρ=0 by a small number. The problem in using the

*U*Lommel function will appear if one tries to calculate the temporal evolution of the intensity of the diffracted light exactly on the axis. Then an algorithm involving for example the

*V*Lommel function will be necessary [11]. Here we concentrate mostly in looking at the diffraction pattern on a large area, so that the calculation of the field exactly at the axis was avoided. Actually, the calculation on the axis only can be done directly and has been already analysed for example in Ref. [2

2. Z. Ziang, 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]

*t*=0, it arrives around

*t=z/c*in the observation plane. We can observe the diffraction pattern around

*t*=0 by simply substituting

*t*by

*t+z/c*in the formula, i.e., by just rewriting

## 3. Simulations

*a*is 0.5 mm and the distance between the two planes

*z*is 1 m. The spectrum width is 6.10

^{12}rad/s (about 1 THz), which corresponds to a pulse whose duration is about 1 ps, and the frequency shift is 10

^{14}rad/s (about 16 THz or a wavelength of 20 µm). With these parameters we have a Fresnel number which is equal to 0.04. The parameter s is set to 1. The non-diffracted pulse is at its maximum when

*t*=0.

*t*=- 0.5 ps is shown. This pattern is approximately the same up to

*t*almost equal to 0 (time in the observation plane minus the propagation time between aperture and observation planes). When

*t*=0, the peak widens at the top and narrows a little bit at the bottom, as shown in Fig. 1(b). At t=0.1 ps, two peaks (in fact a ring in the whole surface of the plane) start to appear from both sides of the centre whose relative value is decreasing (Fig. 1(c)). When the pulse is vanishing, the value in the centre decreases faster than the two lateral peaks as shown in Fig. 1(d).

*t*=0.8 ps, the first two peaks vanishes and there is a big flat centre. The secondary peaks also grow compared to the centre. At

*t*=1 ps, the peaks which appeared at

*t*=0.2 ps have totally disappeared; it remains only the secondary peaks whose relative value is still increasing. Finally, at

*t*=1.5 ps the centre is vanishing, but not the side peaks. The latter will still spread away from the centre while their value decreases to zero.

## 4. Conclusions

*U*Lommel function), and second, is the problem of the integral given by Eq. (12) which does not converge for a certain choice of parameters. We also would like to note that although the analytical result given by Eq. (14) seems to be very complicated, the convergence of the sums is very rapid (only a couple of terms are necessary) which makes the evaluation quite efficient. Alternatively one may also try to compute Eq. (6) directly numerically, but the main reason that we have chosen for the analytical solution is that this gives more insight on the time dependence and the shape of the diffracted pattern. With the analytical solution it is also possible to explore certain limiting parameters which may be intersting.

## Acknowledgments

## References and links

1. | M. Gu and X. S. Gan, “Fresnel diffraction by a circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A |

2. | Z. Ziang, R. Jacquemin, and W. Eberhardt, “Time dependence of Fresnel diffraction of ultrashort laser pulses by a circular aperture,” Appl. Opt. |

3. | H. Quan-Sheng and Zhu-Zhen-he, “Diffraction of ultrashort pulses,” Acta Phys. Sin. |

4. | M. Kempe and W. Rudolph, “Analysis of confocal microscopy under ultrashort light pulse illumination,” J. Opt Soc. Am. A |

5. | J. Anderson and C. Roychoudhuri, “Diffraction of an extremely short optical pulse,” J. Opt. Soc. Am. A |

6. | M. Kempe and U. Stamm. “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992). [CrossRef] |

7. | H. Ichikawa, “Analysis of femtosecond-order optical pulses diffracted by periodic structure”, J. Opt. Soc. Am. A |

8. | J. Cooper and E. Marx, “Free propagation of ultrashort pulses” J. Opt. Soc. Am. A |

9. | R. W. Ziolkowski and J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A |

10. | W. Goodman, |

11. | M. Born and E. Wolf, |

12. | C. F. R. Caron and R. M. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. |

13. | S. Feng, H. G. Winful, and R. Hellwarth, “Gouy shift and temporal reshaping of focussed single-cycled electromagnetic pulses,” Opt. Lett. |

14. | C. J. R. Sheppard, “Bessel pulse beams and focus wave modes”, J. Opt. Soc. Am. A |

15. | G. N. Watson, |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(140.0140) Lasers and laser optics : Lasers and laser optics

(320.0320) Ultrafast optics : Ultrafast optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 24, 2003

Revised Manuscript: April 12, 2003

Published: May 19, 2003

**Citation**

M. Lefrancois and S. Pereira, "Time evolution of the diffraction pattern of an ultrashort laser pulse," Opt. Express **11**, 1114-1122 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-10-1114

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### References

- M. Gu and X. S. Gan, �??Fresnel diffraction by a circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,�?? J. Opt. Soc. Am. A 13, 771-778 (1996). [CrossRef]
- Z. Ziang, 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]
- H. Quan-Sheng and Zhu-Zhen-he,�??Diffraction of ultrashort pulses,�?? Acta Phys. Sin. 37, 1432-1437 (1988).
- M. Kempe and W. Rudolph, ,�??Analysis of confocal microscopy under ultrashort light pulse illumination,�??J. Opt Soc. Am. A 10, 240-245 (1993). [CrossRef]
- J. Anderson and C. Roychoudhuri, �??Diffraction of an extremely short optical pulse,�?? J. Opt. Soc. Am. A 15, 456-463 (1998). [CrossRef]
- M. Kempe, U. Stamm. �??Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,�?? J. Opt. Soc. Am. B 9, 1158-1165 (1992). [CrossRef]
- H. Ichikawa, "Analysis of femtosecond-order optical pulses diffracted by periodic structure", J. Opt. Soc. Am. A 16, 299-304 (1999). [CrossRef]
- J. Cooper and E. Marx, "Free propagation of ultrashort pulses" J. Opt. Soc. Am. A 2, 1711-1720 (1985). [CrossRef]
- R. W. Ziolkowski and J. B. Judkins, "Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams," J. Opt. Soc. Am. A 9, 2021-2030 (1992). [CrossRef]
- W. Goodman, Introduction to Fourier optics (Mc Graw-Hill, NY, 1968).
- M. Born and E. Wolf, Principles of Optics (Pergamon, NY, 1980)(p.437-439, chap. 8.8.1).
- C. F. R. Caron and R. M. Potvliege, "Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams," J. Mod. Opt. 46, 1881-1891 (1999). [CrossRef]
- S. Feng, H. G. Winful and R. Hellwarth, "Gouy shift and temporal reshaping of focussed single-cycled electromagnetic pulses," Opt. Lett. 23, 385-387 (1998). [CrossRef]
- C. J. R. Sheppard, "Bessel pulse beams and focus wave modes," J. Opt. Soc. Am. A 18, 2594-2600 (2001). [CrossRef]
- G. N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).

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