## On-axis diffraction of an ultrashort light pulse by circularly symmetric hard apertures

Optics Express, Vol. 15, Issue 8, pp. 4546-4556 (2007)

http://dx.doi.org/10.1364/OE.15.004546

Acrobat PDF (394 KB)

### 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

## 1. Introduction

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]

*N*annular hard apertures is attained. Such a result can be used to investigate temporal effects associated with the diffraction of short pulses by different types of hard apertures (for instance: zone plates [9

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]

## 2. Diffracted field by a hard aperture

*U*(

*z*,

*t*), can be determined by means of an inverse Fourier transform, with respect to

*ω*frequencies, of the amplitude spectrum of the pulse,

*A*

_{0}(

*ω*), multiplied by the spectral components of the diffracted field

*U*

_{0}(

*z*,

*ω*). That is,

*N*annular hard apertures, it can be shown (see Appendix A) that,

*r*and

_{im}*r*denote the inner and the outer radii, respectively, of the

_{om}*m*annular aperture. Note that,

*r*

_{i1}= 0 when the innermost circular aperture is open at the axis. In particular, for a circular hard aperture of radius

*r*

_{0},

*r*

_{i1}= 0 and

*r*

_{o1}=

*r*

_{0}. The velocity of light in vacuum is denoted by

*c*, and

*t*reads as the physical time. The terms

*s*= (

_{im}*z*

^{2}+

*r*

^{2}

_{im})

^{1/2}and

*s*= (

_{om}*z*

^{2}+

*r*

^{2}

_{om})

^{1/2}are the distances from the edges of the

*m*annular aperture to the axial point

*z*. The total diffraction field,

*U*(

*z*,

*t*), can be thought as a coherent sum of 2

*N*boundary wave pulses [7

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

*π*between the two boundary wave pulses originated from an annular aperture. The amplitude of the boundary wave pulses vary with respect to

*z*from 0 to 1, but it tends to 1 very fast as we move away from the aperture for typical values of

*r*and

_{im}*r*. These amplitude functions differ from the ones achieved within the framework of Kirchhoff theory of diffraction [7

_{om}7. Z. L. Horváth and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E. **63**, 026601 (2001). [CrossRef]

*s*-

_{om}*s*)/

_{im}*c*between the boundary wave pulses decreases with the path length difference

*s*-

_{om}*s*. So, wherever (

_{im}*s*-

_{om}*s*)/

_{im}*c*is shorter than the temporal width of neighboring pulses, the interaction between them results in a modification of the envelope of the pulse bursts. The overall time window of the pulse burst can be estimated from (

*S*-

_{oN}*S*

_{i1})/

*C*.

*u*

_{0}(

*t*), is expressed in a quite general form as,

*P*(

*t*) > 0 is the pulse envelope and

*ω*

_{0}reads for the central frequency of the pulse. Equation (4) holds when the bandwidth is only a small fraction of the carrier frequency. With the help of the above equations and regarding only the boundary wave pulses originated from the

*m*annular aperture, we conclude that,

- The time difference from peak to peak is given by (
*s*-_{om}*s*)/_{im}*c*. - The phase difference between them changes during their propagation owing to the term Δ
*φ*=*ω*_{0}(*s*-_{om}*s*)/_{im}*c*. Therefore, at the axial points derived from the condition Δ*φ*=*π*(2*n*+1), with*n*= 0, 1, …, these waves arrive in phase. In the same way, at those axial points determined from the condition Δ*φ*= 2*π*(*n*+1), they keep the initial phase difference. - If
*P*(*t*) is an even function, at the instant*t*=*T*_{0m}= (*s*+_{om}*s*)/2_{im}*c*the total diffraction field given by Eq. (3) transforms into the expression,From Eq. (5) we infer that the diffracted field from this annular aperture at the axial positions characterized by the condition exp(*i*Δ*φ*)=*s*/_{om}*s*has no contribution to_{im}*U*(*z*,*T*_{0m}). If the ratio*s*/_{om}*s*≅ 1, this condition is fulfilled when Δ_{im}*φ*= 2*π*(*n*+1). Note that,*T*_{0m}is exactly half the time interval between the passage of the wave peaks by the considered axial point*z*.

*r*=

_{im}*p*(

*m*-1)

^{1/2}and

*r*=

_{om}*p*(

*m*-1/2)

^{1/2}, it allows the analysis of the ultrashort pulse diffraction by a zone plate [9], being

*p*

^{2}the period of the zone plate transmittance.

*N*circularly symmetric hard apertures, the total diffraction field yields (see appendix B),

*z*

_{0}. When the sign of

*z*

_{0}is positive, a diverging focused wave impinges on the aperture, whereas a negative sign of

*z*

_{0}holds for a converging diffracted wave that is focused in toward the position

*z*

_{0}after the aperture.

*z*axis. In order to compare our results with those of the previous contribution Ref. [6

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]

## 3. Diffracted field by a circular hard aperture

*r*

_{0}. Its temporal amplitude is given by

*u*

_{0}(

*t*) = exp(-

*iω*

_{0}

*t*)exp[-

*t*

^{2}/(4σ

^{2}

_{0})]. The previous expression corresponds to the complex amplitude envelope representation of a pulsed Gaussian beam. This approximation can be adopted if σ

_{ω}/

*ω*

_{0}= 1/(2

*ω*

_{0}σ

_{0})<9.30×10

^{-2}(σ

_{ω}is the standard deviation of the power spectrum). Otherwise, the complex analytical signal needs to be used [13

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]

*τ*=

*t*-

*z*/

*c*is a time delay, and σ

_{0}is the standard deviation width of the incident Gaussian pulse. The first right hand term of Eq. (8) can be interpreted, within the Fresnel regime, as a geometric wave (a wave that propagates according with the laws of geometric optics), while the second one is due to a boundary wave pulse emitted at the edge of the circular aperture. Now, from the previous section, we can come to the following conclusions:

- The peaks of geometric and boundary wave pulses are separated by a time difference of
*r*^{2}_{0}/(2*cz*). - The phase of the boundary wave pulse changes during its propagation, with respect to the phase of the geometric wave, because of the term
*N*_{0}=*ω*_{0}*r*^{2}_{0}/(2*cz*), which is the Fresnel number corresponding to the central frequency. Hence, geometric and boundary wave pulses arrive with equal phase at the axial points*z*^{+}_{n}=*ω*_{0}*r*^{2}_{0}/[2*πc*(2*n*+1)]. In contrast, they get in with a phase shift of*π*for*z*^{-}_{n}=*ω*_{0}*r*^{2}_{0}/[4*πc*(*n*+1)]. - The total diffraction field reaches a zero value at the instant
*τ*=*T*_{0}=*r*^{2}_{0}/(4*cz*) on axial points*z*=*z*^{-}_{n}. Note that, within the paraxial approximation*T*_{0}=*T*_{01}-*z*/*c*.

## 4. Instantaneous intensity analysis

*I*

_{0}(

*z*,

*τ*+

*z*/

*c*) = exp[-

*τ*

^{2}/(2σ

^{2}

_{0})] is the on-axis instantaneous intensity of the incident pulse after free propagation (without diffraction) a distance

*z*from the circular aperture. The right hand term between curly brackets can be interpreted as an intensity modifier function of

*I*

_{0}(

*z*,

*τ*+

*z*/

*c*). From Eq. (9), we can recognize that as the pulse width increases, the intensity pattern approaches the on-axis time-average intensity of a CW illumination diffracted through a circular hard aperture, which is characterized by the well-known function 4sin

^{2}[

*ω*

_{0}

*r*

^{2}

_{0}/(4

*cz*)], see Eq. (13) in Ref. [8

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]

*z*=

*z*

^{+}

_{n}, the instantaneous intensity yields,

*π*to the axial points

*z*=

*z*

^{-}

_{n}we find that,

*τ*=

*T*

_{0}=

*r*

^{2}

_{0}/ 4

*cz*. Here, it would be advisable to recall that, at the time instant

*τ*=

*T*

_{0}, the peak of the geometric wave has passed through a given axial point

*z*

_{0}and is distant in time from it

*r*

^{2}

_{0}/ 4

*cz*, whereas the peak of the boundary wave pulse still needs a time

*r*

^{2}

_{0}/4

*cz*to arrive to

*z*

_{0}. Using Eq. (9), it can be verified that [

*dI*(

*z*,

*τ*+

*z*/

*c*)/

*d*

*τ*]

_{τ = T0}=0.

*I*(

*z*,

*τ*+

*z*/

*c*) evaluated at

*T*

_{0}, we find (for a fixed set of parameters

*ω*

_{0},

*r*

_{0}and

*σ*

_{0}) that

- If the condition sin
^{2}(*ω*_{0}*r*^{2}_{0}/4*cz*)<2(*r*^{2}_{0}/8*cz**σ*_{0})^{2}holds for an axial point*z*, there is a time minimum of the instantaneous intensity at*τ*=*T*_{0}and its temporal profile is mainly made up of a single lobe with an inward central ripple. At the positions*z*=*z*^{-}_{n}, it splits in two separated lobes. Note that*I*[*z*^{-}_{n},*T*_{0}+*z*/*c*) = 0, see Eq. (11). - If the condition sin
^{2}(*ω*_{0}*r*^{2}_{0}/4*cz*)>2(*r*^{2}_{0}/8*cz**σ*_{0})^{2}holds, there is a time maximum of the instantaneous intensity at*τ*=*T*_{0}and its temporal profile is made up of a single lobe with Gaussian-like form. - If the equality sin
^{2}(*ω*_{0}*r*^{2}_{0}/4*cz*)=2(*r*^{2}_{0}/8*cz**σ*_{0})^{2}holds, the instantaneous intensity has a flattop temporal profile about*τ*=*T*_{0}. It can be shown that the number of flattop profiles is approximately given by 2Round(√2*σ*_{0}*T*_{0},*π*)-1, for*σ*_{0}*T*_{0}> √2/2.

*d*

^{2}

*I*(

*z*,

*τ*+

*z*/

*c*)/

*d*

*τ*

^{2}]

_{τ = T0}provides a useful tool to analyze the temporal evolution of the instantaneous intensity along the axial direction. It should be pointed out that the axial positions

*z*=

*z*

^{+}

_{n}can satisfy either condition I or II. In Fig. 1, this function is plotted for

*σ*

_{0}= 3 fs, starting from

*z*= 0.5 m. Note that a Gaussian pulse of

*σ*

_{0}= 3 fs has an amplitude full width at half maximum of 10 fs. The radius of the circular aperture is

*r*

_{0}=2 mm and the central wavelength is

*λ*

_{0}=800 nm (

*ω*

_{0}=3

*π*/4×10

^{15}Hz). For these physical conditions

*σ*/

_{ω}*ω*

_{0}≅ 7.07×10

^{-2}. In accordance with the previous analysis we conclude that there are three axial intervals at which the condition I and the condition II are satisfied. An instantaneous intensity with a flattop profile occurs at five axial positions, see the condition III.

*z*

_{1}=

*ω*

_{0}

*r*

^{2}

_{0}/(12

*πc*) and

*z*

_{2}=

*ω*

_{0}

*r*

^{2}

_{0}/(8

*πc*), is shown in Fig. 2. Its time changes are in complete agreement with the behavior predicted from Fig. 1. In particular, Fig. 2(

*a*) and Fig. 2(

*i*) are obtained from the condition I, in the case of a maximum destructive interference. Fig. 2(

*c*) and Fig. 2(

*g*) show instantaneous intensities with a flattop profile, see the condition III. In addition, Fig. 2(

*e*) corresponds to a maximum constructive interference satisfying the condition II.

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]

*α*=

*N*

_{0}/(4√ln2

*σ*

_{0}

*ω*

_{0}). In fact, Fig. (2) of Ref. [6

**36**, 4358–4361 (1997). [CrossRef] [PubMed]

*N*

_{0}decreases. In particular, the instantaneous intensity was plotted for

*N*

_{0}= 40 ,

*N*

_{0}= 20 , and

*N*

_{0}= 5, with pulse parameters

*ω*

_{0}= 3

*π*/4×10

^{15}Hz and

*σ*

_{0}= 3 fs. For a circular aperture of radius

*r*

_{0}= 2 mm, the above Fresnel numbers correspond to the axial positions

*z*= 0.39 m,

*z*= 0.78 m, and

*z*= 3.14 m, respectively. Now, with the help of Fig. 1, one can see that in the transition from

*z*= 0.78 m to

*z*= 3.14 m there are additional temporal changes of the instantaneous intensity. Most significant ones are shown in Fig. 2. Therefore, the information given in Fig. 2 of Ref. [6

**36**, 4358–4361 (1997). [CrossRef] [PubMed]

## 5. Pulse width estimation

*σ*

^{2}=

*m*

_{2}/

*m*

_{0}-(

*m*

_{1}/

*m*

_{0})

^{2}, where the instantaneous intensity moments

*m*(with

_{s}*s*= 0, 1, 2.) are defined as

*m*

_{0in}= √2

*σ*

*σ*

_{0}reads for the time integral intensity of the incident pulse. We note that the ratio

*m*

_{0}/

*m*

_{0in}in Eq. (12) coincides with the on-axis time-average intensity of an ultrashort pulse diffracted by a circular aperture, see Eq. (16) in Ref. [8

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]

*σ*/

*σ*

_{0}increases as we move in toward the circular aperture. This result is a consequence of the time delay

*r*

^{2}

_{0}/(2

*cz*) between boundary and geometric wave pulses. After that, the standard deviation ratio oscillates mainly within the values 2 and 1, accordingly with the time shape variations of the instantaneous intensity. Finally, the oscillations vanish and the curve tends to unity. So, in the Fraunhofer region the time width of the diffracted pulse coalesces with that of the incident pulse. Peaks in Fig. 3 are reached close to the positions

*z*=

*z*

^{-}

_{n}.

## 6. Analysis of results

*η*=

*T*

_{0}/

*σ*

_{0}can be used as a way to determine the degree of wave mixture. The interference process depends on the relative phase of the pulses. In addition, only several effective oscillation cycles of the carrier wave exist within the time width 4

*σ*

_{0}of the pulse envelope. Note that, a time width of 4

*σ*

_{0}coincides with the 1/

*e*

^{2}width of the instantaneous intensity. Being the time changes of the instantaneous intensity caused by the interference of geometric and boundary wave pulses, we can expect these changes not to happen in a rather arbitrary way but to depend on the number of oscillation cycles, within the time width 4

*σ*

_{0}, of the real electric field. Therefore, if we consider Gaussian pulses with a finite time extension of 4

*σ*

_{0}, the conditions of maximum destructive interference at the instant

*τ*=

*T*

_{0}take place at a finite number of axial positions. Consequently, the number of on-axis zeros of the instantaneous intensity are approximately equal to the number of oscillation cycles at frequency

*ω*

_{0}of the real electric field (see Fig. 4).

*τ*=

*T*

_{0}always corresponds to the sum of real electric fields in phase opposition with maximum values of the carrier wave. Consequently, the corresponding instantaneous intensity envelope is composed of two temporal lobes. In the same way, a maximum constructive interference can yield a maximum field envelope value at

*τ*=

*T*

_{0}, if the condition II is satisfied. This is because of the sum of real electric fields with equal phase. In addition, a suitable relative time-varying phase shift is needed to obtain a flattop profile for the instantaneous intensity envelope.

*τ*=

*T*

_{0}are determined from the equality

*N*

_{0}= 2

*π*(

*n*+1). The frequency spectrum of an incident ultrashort pulse

*S*

_{0}(

*ω*) with temporal amplitude

*u*

_{0}(

*t*) = exp(-

*iω*

_{0}

*t*)exp[-

*t*

^{2}/(4

*σ*

^{2}

_{0})] is modified after its diffraction by a circular aperture to yield 4

*S*

_{0}(

*ω*)sin

^{2}(

*N*/2), being

*N*=

*ωr*

^{2}

_{0}/(2

*cz*) the Fresnel number [14

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]

*ω*

_{0}is also omitted from the spectrum of the pulse. Note that the lack of the central frequency

*ω*

_{0}physically implies a pulse broadening as Fig. 3 shows.

*P*(

*t*) = sech[

*πt*/(2√3

*σ*

_{0})] or

*P*(

*t*) = [1 + (

*t*/

*σ*

_{0})

^{2}]

^{-1}, are also used to model real pulses [15]. The analysis given in the section 2 of this work applies also to the above-mentioned pulse profiles and in general, to the case where the pulse envelope

*P*(

*t*) is an even function. In particular, Eq. (3) holds for the temporal amplitude profile of real pulses. We find that for a few-cycles pulse the positions of the flattop profiles are affected by the selection of a pulse shape model. To show this, the function [

*d*

^{2}

*I*(

*z*,

*τ*+

*z*/

*c*)/

*d*

*τ*

^{2}]

_{τ = T0}is depicted in the Fig. (5) by using three different models for the pulse shape, but the same standard deviation

*σ*

_{0}= 3 fs. The main discrepancies appear for the position of the first flattop profile. The differences between the full widths at half maximum of the pulse amplitude, obtained from each model, should determine these position changes.

## 7. Conclusions

*u*

_{0}(

*t*) = exp(-

*Iω*

_{0}

*t*)exp[-

*t*

^{2}/(4

*σ*

^{2}

_{0})] was considered. We show analytically that the instantaneous intensity reaches a zero at the axial positions given by

*z*

^{-}

_{n}=

*ω*

_{0}

*r*

^{2}

_{0}/[4

*πc*(

*n*+1)]. The standard deviation of the instantaneous intensity shows a pulse broadening about the positions

*z*=

*z*

^{-}

_{n}. By analyzing the phase shift between geometric and boundary wave pulses at the center instant

*τ*=

*T*

_{0}=

*r*

^{2}

_{0}/4

*cz*, the time changes of the instantaneous intensity along the axial direction were predicted. Temporal profiles are mainly made up of a lobe with a central inward ripple or of one with a Gaussian-like form. However, for

*z*=

*z*

^{-}

_{n}the instantaneous intensity splits in two temporal lobes about

*τ*=

*T*

_{0}. In addition, the positions of the flattop profiles can also be found. It was shown for a few-cycles pulse that the time changes of the instantaneous intensity depend on the number of oscillation cycles at central frequency

*ω*

_{0}of the real electric field.

## Appendix A

*N*annular hard apertures. Using the first Rayleigh-Sommerfeld formula, the on-axis diffraction field, after a plane-wave propagation through an annular hard aperture yields

*r*

_{1}and

*r*

_{2}, respectively,

*ω*= 2

*πc*/

*λ*is the frequency of light with wavelength

*λ*and propagation velocity in vacuum

*c*.

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]

*N*annular hard apertures is straightforward

## Appendix B

*N*annular hard apertures is shown. In the paraxial approximation, the incident wave pulse is given by exp[

*Iω*/

*c*(

*z*

_{0}+

*ρ*

^{2}/2

*z*

_{0})]/

*z*

_{0}, being

*z*

_{0}the position of the pulse source. If the Kirchhoff boundary conditions and

*z*≫

*λ*hold, the Fresnel diffraction integral yields,

## Acknowledgments

## References and links

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. |

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. |

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. |

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. |

5. | M. Lefrançois and S. F. Pereira, “Time evolution of the diffraction pattern of an ultrashort laser pulse,” Opt. Express |

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

7. | Z. L. Horváth and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E. |

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 |

9. | C. J. Zapata-Rodríguez, “Temporal effects in ultrashort pulsed beams focused by planar diffracting elements,” J. Opt. Soc. Am. A |

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. |

11. | A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. |

12. | J. W. Goodman, |

13. | R. Peng and D. Fan, “Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam,” Opt. Commun. |

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 |

15. | J. C. Diels and W. Rudolph, |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(050.1940) Diffraction and gratings : Diffraction

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 17, 2006

Revised Manuscript: January 25, 2007

Manuscript Accepted: February 12, 2007

Published: April 3, 2007

**Citation**

Omel Mendoza-Yero, Gladys Mínguez-Vega, Jesús Lancis, Mercedes Fernández-Alonso, and Vicent Climent, "On-axis diffraction of an ultrashort light pulse by circularly symmetric hard apertures," Opt. Express **15**, 4546-4556 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4546

Sort: Year | Journal | Reset

### References

- 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]
- 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]
- 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]
- 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]
- Z. L. Horváth and Zs. Bor, "Diffraction of short pulses with boundary diffraction wave theory," Phys. Rev. E. 63, 026601 (2001). [CrossRef]
- 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]
- 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]
- A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1926-1960, (2000). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 3.
- 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]
- 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]
- J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 1996), Chap. 1.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.