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

doi:10.1364/OE.15.004546

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

Omel Mendoza-Yero, Gladys Mínguez-Vega, Jesús Lancis, Mercedes Fernández-Alonso, and Vicent Climent »View Author Affiliations

Optics Express, Vol. 15, Issue 8, pp. 4546-4556 (2007)
http://dx.doi.org/10.1364/OE.15.004546

View Full Text Article

Acrobat PDF (394 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Full Text
Article Info
References (15)
Cited By
Figures (5)
Multimedia (1)
Metrics

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.

1. Introduction

In recent years, the time evolution of the diffraction pattern of ultrashort light pulses diffracted by hard apertures has been a subject of increasing interest [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]

]. Approximate expressions for the instantaneous intensity generated by a pulsed light with time-variant Gaussian shape, diffracted by a circular aperture were derived [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]

]. With the help of the Lommel functions, an analytical expression for the time evolution of the diffracted field in the Fresnel regime caused by the pass of an ultrashort pulse through a circular aperture was obtained [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]

]. However, due to problems in using the Lommel functions, the corresponding on-axis diffraction field cannot be obtained. Time dependences of the on-axis Fresnel diffraction of ultrashort laser pulses with spatially Gaussian distribution, diffracted by a circular aperture were reported by means of a numerical analysis [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]

]. Based on the Miyamoto-Wolf theory of the boundary diffraction wave, an exact solution for the on-axis diffraction field of short pulses diffracted by a circular aperture has been published [7

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

]. The Fresnel diffraction patterns by circular and serrated apertures, under ultrashort pulsed-laser illumination, were also investigated [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]

], achieving time-averaged intensity formulas for the on-axis case.

In this paper, the exact analytical expression, within the Rayleigh-Sommerfeld diffraction theory, for the on-axis field of an ultrashort light pulse diffracted by a set of 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

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]

] and axicons). In particular, we provide a detailed analytical study on the time changes of the instantaneous intensity originated by the on-axis diffraction of an ultrashort light pulse by a circular hard aperture. Our analysis allows us to find the axial positions where the time splitting effects occur. In addition, the positions of the flattop profiles are found. Potential applications of this work range from the time characterization of ultrashort pulses after their pass by planar diffractive elements [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]

] to the control of laser beam waveforms for ultrafast spectroscopy, nonlinear fiber optics, and high-field physics [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

Let’s consider the on-axis diffraction of an ultrashort pulse by a circularly symmetric hard aperture. The total diffraction field, 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,

U (z, t) = \frac{1}{2 π} \int_{- \infty}^{\infty} A_{0} (ω) U_{0} (z, ω) \exp (- iωt) dω .

(1)

In Eq. (1), U ₀(z,ω) is given by the first Rayleigh-Sommerfeld integral formula [12

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

] and A ₀(ω) is related to the temporal amplitude u ₀(t) of the incident pulse by the Fourier transform operation

A_{0} (ω) = \int_{- \infty}^{\infty} u_{0} (t) \exp (iωt) dt .

(2)

For a set of N annular hard apertures, it can be shown (see Appendix A) that,

U (z, t) = \sum_{m = 1}^{N} \frac{z}{\sqrt{z^{2} + r_{im}^{2}}} u_{0} (t - \frac{\sqrt{z^{2} + r_{im}^{2}}}{c}) - \sum_{m = 1}^{N} \frac{z}{\sqrt{z^{2} + r_{om}^{2}}} u_{0} (t - \frac{\sqrt{z^{2} + r_{om}^{2}}}{c}) .

(3)

In Eq. (3), r_im and r_om denote the inner and the outer radii, respectively, of the 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 2N boundary wave pulses [7

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

] emitted at the same time from the annular edges. Note that there is also a phase shift of 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_im and r_om . These amplitude functions differ from the ones achieved within the framework of Kirchhoff theory of diffraction [7

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

]. This is because of the use of different obliquity factors [12

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

]. The time delay [s_om -s_im )/c between the boundary wave pulses decreases with the path length difference s_om -s_im . So, wherever (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.

Equation (3) is the main result of this paper. It gives the exact analytical solution, within the Rayleigh-Sommerfeld formulation of diffraction, for the on-axis field of a light pulse diffracted by a circularly symmetric hard aperture. It is derived for a spatially plane wave pulse with no restrictions over the temporal amplitude of the incident pulse. At this point, u ₀(t), is expressed in a quite general form as,

u_{0} (t) = P (t) \exp (- i ω_{0} t),

(4)

where, 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 Δφ = ω ₀(s_om - s_im )/c. Therefore, at the axial points derived from the condition Δφ = π(2n+1), with n = 0, 1, …, these waves arrive in phase. In the same way, at those axial points determined from the condition Δφ = 2π(n+1), they keep the initial phase difference.
If P(t) is an even function, at the instant t = T _0m = (s_om + s_im )/2c the total diffraction field given by Eq. (3) transforms into the expression,

$U (z, T_{0 m}) = \sum_{m = 1}^{N} P (\frac{s_{om} - s_{im}}{2 c}) {\frac{z}{s_{im}} \exp (- i \frac{Δ φ}{2}) - \frac{z}{s_{om}} \exp (i \frac{Δ φ}{2})}$

(5)
From Eq. (5) we infer that the diffracted field from this annular aperture at the axial positions characterized by the condition exp(iΔφ)= s_om / s_im has no contribution to U(z,T _0m). If the ratio s_om / s_im ≅ 1, this condition is fulfilled when Δφ = 2π(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.

In the Fresnel regime (z ≫ r_im and z ≫ r_om ), Eq. (3) reduces to the form,

U (z, t) = \sum_{m = 1}^{N} u_{0} (t - \frac{z + \frac{r_{im}^{2}}{2 z}}{c}) - \sum_{m = 1}^{N} u_{0} (t - \frac{z + \frac{r_{om}^{2}}{2 z}}{c}) .

(6)

Although Eq. (6) is not an exact solution of the diffraction problem, it can be useful in many practical implementations where the paraxial approximation holds. For instance, when 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.

On the other hand, if a diverging (or converging) focused wave is diffracted by a set of N circularly symmetric hard apertures, the total diffraction field yields (see appendix B),

U (z, t) = \frac{1}{z + z_{0}} \sum_{m = 1}^{N} u_{0} (t - \frac{z + \frac{r_{im}^{2}}{2 z}}{c} - \frac{z_{0} + \frac{r_{im}^{2}}{2 z_{0}}}{c}) - \frac{1}{z + z_{0}} \sum_{m = 1}^{N} u_{0} (t - \frac{z + \frac{r_{om}^{2}}{2 z}}{c} - \frac{z_{0} + \frac{r_{om}^{2}}{2 z_{0}}}{c})

(7)

The source (or the converging point) of the spherical wave is located at the axial position 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.

In the next section, the simplest case corresponding to the diffraction of an ultrashort pulse by a circular hard aperture is considered in detail. We focus our attention in the time chances of the instantaneous intensity along the z axis. In order to compare our results with those of the previous contribution Ref. [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]

], Eq. (6) instead of Eq. (3) is used.

3. Diffracted field by a circular hard aperture

Within the paraxial approximation, we consider an incident plane-wave pulse diffracted by a circular aperture of radius r ₀. Its temporal amplitude is given by u ₀(t) = exp(-iω ₀ t)exp[-t ²/(4σ² 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/(2ω ₀σ₀)<9.30×10^-2 (σ_ω is the standard deviation of the power spectrum). Otherwise, the complex analytical signal needs to be used [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]

]. Under the above-mentioned assumptions, the total diffraction field yields,

U (z, τ + \frac{z}{c}) = \exp (- i ω_{0} τ) \exp (- \frac{τ^{2}}{4 σ_{0}^{2}}) - \exp [- i ω_{0} (τ - \frac{r_{0}^{2}}{2 cz})] \exp [- \frac{1}{4 σ_{0}^{2}} {(τ - \frac{r_{0}^{2}}{2 cz})}^{2}] .

(8)

In Eq. (8), τ = t-z/c is a time delay, and σ₀ 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/(2cz).
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/(2cz), 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/[2πc(2n+1)]. In contrast, they get in with a phase shift of π for z ^- _n = ω ₀ r ² ₀/[4πc(n+1)].
The total diffraction field reaches a zero value at the instant τ = T ₀ = r ² ₀/(4cz) on axial points z = z ^- _n. Note that, within the paraxial approximation T ₀ = T ₀₁ - z/c.

4. Instantaneous intensity analysis

With the help of Eq. (8), the instantaneous intensity I(z,τ+z/c)=|U(z,τ+z/c)| is calculated

I (z, τ + \frac{z}{c}) = I_{0} (z, τ + \frac{z}{c}) {1 + \exp [\frac{r_{0}^{2}}{2 cz σ_{0}^{2}} (τ - \frac{r_{0}^{2}}{4 cz})] - 2 \cos (\frac{ω_{0} r_{0}^{2}}{2 cz}) \exp [\frac{r_{0}^{2}}{4 cz σ_{0}^{2}} (τ - \frac{r_{0}^{2}}{4 cz})]}

(9)

where I ₀(z,τ+z/c) = exp[-τ ²/(2σ² 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 4sin²[ω ₀ r ² ₀/(4cz)], see Eq. (13) in Ref. [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]

When geometric and boundary wave pulses arrive in phase to the axial points z = z ⁺ _n, the instantaneous intensity yields,

I (z, τ + \frac{z}{c}) = I_{0} (z, τ + \frac{z}{c}) {1 + \exp [\frac{r_{0}^{2}}{4 cz σ_{0}^{2}} (τ - \frac{r_{0}^{2}}{4 cz})]}^{2},

(10)

whereas, if they arrive with a phase shift of π to the axial points z = z ^- _n we find that,

I (z, τ + \frac{z}{c}) = I_{0} (z, τ + \frac{z}{c}) {1 - \exp [\frac{r_{0}^{2}}{4 cz σ_{0}^{2}} (τ - \frac{r_{0}^{2}}{4 cz})]}^{2} .

(11)

In order to study the time changes of the instantaneous intensity, we analyze its behavior at the time instant τ = T ₀ = r ² ₀/ 4cz. 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/ 4cz, whereas the peak of the boundary wave pulse still needs a time r ² ₀/4cz to arrive to z ₀. Using Eq. (9), it can be verified that [dI(z,τ+z/c)/d τ]_{τ = T
₀} =0.

From the calculus of the second time derivate of 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 sin²(ω ₀ r ² ₀/4cz)<2(r ² ₀/8cz 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 sin²(ω ₀ r ² ₀/4cz)>2(r ² ₀/8cz 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 sin²(ω ₀ r ² ₀/4cz)=2(r ² ₀/8cz 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(√2σ ₀ T ₀,π)-1, for σ ₀ T ₀ > √2/2.

Therefore, the function [d ² I(z,τ+z/c)/d τ ²]_{τ = T
₀} 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 =3π/4×10¹⁵ 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.

Fig. 1. Second time derivate of I(Z,τ+ z/c)×10^-29 evaluated at τ = T ₀

The instantaneous intensity, for nine axial positions between z ₁ = ω ₀ r ² ₀/(12πc) and z ₂ = ω ₀ r ² ₀/(8πc), 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.

Fig. 2. Temporal evolution of the instantaneous intensity between two axial points of maximum destructive interference.

In Ref. [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]

] the effects of the circular aperture on the temporal shape of the on-axis instantaneous intensity were estimated by using the parameter α = N ₀/(4√ln2σ ₀ ω ₀). In fact, Fig. (2) of Ref. [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]

] suggests that the temporal profile of the instantaneous intensity only pass from having two lobes to a single one as the Fresnel number 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 = 3π/4×10¹⁵ 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

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]

] is correct but is not complete, giving rise to a possible misunderstanding about the time evolution of the instantaneous intensity.

5. Pulse width estimation

The time width of the diffracted pulse varies along the axial direction. We determine the pulse width from the well-known relation σ ² = m ₂/m ₀-(m ₁/m ₀)², where the instantaneous intensity moments m_s (with s = 0, 1, 2.) are defined as

m_{s} = \int_{- \infty}^{\infty} I (z, t) t^{s} dt

. It is straightforward to show that

m_{0} = {2 m}_{0 in} {1 - \cos (N_{0}) \exp [- \frac{1}{2} {\frac{T_{0}}{σ_{0}}}^{2}]},

(12)

m_{1} = m_{0} T_{0},

(13)

m_{2} = 2 m_{0 in} T_{0}^{2} + m_{0} σ_{0}^{2} [1 + {(\frac{T_{0}}{σ_{0}})}^{2}] .

(14)

The term m _0in = √2σ 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

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]

]. From Eqs. (12)–(14), the squared standard deviation yields,

σ^{2} = σ_{0}^{2} {1 + \frac{{(\frac{T_{0}}{σ_{0}})}^{2}}{1 - \cos (N_{0}) \exp [- \frac{1}{2} {(\frac{T_{0}}{σ_{0}})}^{2}]}} .

(15)

Fig. (3) shows that the ratio σ/σ ₀ increases as we move in toward the circular aperture. This result is a consequence of the time delay r ² ₀/(2cz) 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.

Fig.3. Standard deviation ratio of diffracted and non diffracted pulses.

6. Analysis of results

The time changes of the instantaneous intensity are due to the interaction of geometric and boundary wave pulses. As the peaks of geometric and boundary wave pulses come closer, the front part of the boundary wave pulse approaches the rear part of geometric wave pulse. The ratio 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 4σ ₀ of the pulse envelope. Note that, a time width of 4σ 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 4σ ₀, of the real electric field. Therefore, if we consider Gaussian pulses with a finite time extension of 4σ 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).

Figure 4 shows an animation of the time evolution of real electric fields of geometric and boundary wave pulses with their corresponding amplitude envelope. A resulting field envelope of zero value at 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.

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]

The axial positions characterized by a zero value for the diffracted field at the instant τ = T ₀ are determined from the equality N ₀ = 2π(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/(4σ ² ₀)] is modified after its diffraction by a circular aperture to yield 4S ₀(ω)sin²(N/2), being N = ωr ² ₀/(2cz) the Fresnel number [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]

]. Then, at these positions the central frequency 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.

A Gaussian temporal dependence for the field is commonly assumed to represent the shape of a short pulse. In addition, temporal amplitude profiles such that P(t) = sech[πt/(2√3σ ₀)] or P(t) = [1 + (t/σ ₀)²]^-1, are also used to model real pulses [15

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

]. 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 = T
₀} 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.

Fig. 5. Second time derivate of I(Z,τ+ z/c)×10^-29 evaluated at τ = T ₀ for different pulse shape models.

7. Conclusions

Within the Rayleigh-Sommerfeld formulation of diffraction, the exact analytical solution for the total on-axis field of an ultrashort light pulse diffracted by a circularly symmetric hard aperture is derived. The light pulse before diffracting is assumed to have a spatially uniform distribution but an arbitrary temporal shape. The particular case of an incident short pulse with temporal amplitude u ₀(t) = exp(-Iω ₀ t)exp[-t ²/(4σ 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/[4πc(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 ₀/4cz, 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.

Appendices

Appendix A

We provide the determination of the on-axis total diffraction field due to the pass of an ultrashort pulse by a set of 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

U_{0} (z, ω) = - \int_{r_{1}}^{r_{2}} (i \frac{ω}{c} - \frac{1}{\sqrt{z^{2} + ρ^{2}}}) \exp (i \frac{ω}{c} \sqrt{z^{2} + ρ^{2}}) \frac{z}{z^{2} + ρ^{2}} ρdρ =

= \frac{z}{\sqrt{z^{2} + r_{1}^{2}}} \exp (i \frac{ω}{c} \sqrt{z^{2} + r_{1}^{2}}) - \frac{z}{\sqrt{z^{2} + r_{2}^{2}}} \exp (i \frac{ω}{c} \sqrt{z^{2} + r_{2}^{2}}) .

(16)

In Eq. (16), inner and outer edge radii of the annular aperture are denoted by r ₁ and r ₂, respectively, ω = 2πc/λ is the frequency of light with wavelength λ and propagation velocity in vacuum c.

After substituting Eq. (16) into Eq. (1), using Eq. (2

) and the following integral definition of the Dirac function

Dirac (ξ) = \frac{1}{2 π} \int_{- \infty}^{\infty} \exp (iωξ) dω,

(17)

it can be shown that

U (z, t) = \frac{z}{\sqrt{z^{2} + r_{1}^{2}}} u_{0} (t - \frac{\sqrt{z^{2} + r_{1}^{2}}}{c}) - \frac{z}{\sqrt{z^{2} + r_{2}^{2}}} u_{0} (t - \frac{\sqrt{z^{2} + r_{2}^{2}}}{c}) .

(18)

The generalization of Eq. (18) to the case of N annular hard apertures is straightforward

U (z, t) = \sum_{m = 1}^{N} \frac{z}{\sqrt{z^{2} + r_{im}^{2}}} u_{0} (t - \frac{\sqrt{z^{2} + r_{im}^{2}}}{c}) - \sum_{m = 1}^{N} \frac{z}{\sqrt{z^{2} + r_{om}^{2}}} u_{0} (t - \frac{\sqrt{z^{2} + r_{om}^{2}}}{c}) .

(19)

In Eq. (19), r_im and r_om represent the inner and outer edges radii of the m annular aperture.

Appendix B

Here, the calculus of the on-axis total diffraction field caused by the pass of a diverging (or converging) wave pulse, by a set of N annular hard apertures is shown. In the paraxial approximation, the incident wave pulse is given by exp[Iω/c(z ₀+ρ ²/2z 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,

U_{0} (z, ω) = ω \frac{\exp [i \frac{ω}{c} (z + z_{0})]}{icz z_{0}} \int_{r_{1}}^{r_{2}} \exp [\frac{iω}{2 c} (\frac{1}{z} + \frac{1}{z_{0}}) ρ^{2}] ρdρ =

= \frac{1}{z + z_{0}} \exp [iω (\frac{z + z_{0}}{c} + \frac{r_{1}^{2} (z + z_{0})}{2 cz z_{0}})] - \frac{1}{z + z_{0}} \exp [iω (\frac{z + z_{0}}{c} + \frac{r_{2}^{2} (z + z_{0})}{2 cz z_{0}})] .

(20)

By using a procedure similar to that on the Appendix A, we can get to the following expression,

U (z, t) = \frac{1}{z + z_{0}} \sum_{m = 1}^{N} u_{0} (t - \frac{z + \frac{r_{im}^{2}}{2 z}}{c} - \frac{z_{0} + \frac{r_{im}^{2}}{2 z_{0}}}{c}) - \frac{1}{z + z_{0}} \sum_{m = 1}^{N} u_{0} (t - \frac{z + \frac{r_{om}^{2}}{2 z}}{c} - \frac{z_{0} + \frac{r_{om}^{2}}{2 z_{0}}}{c}) .

(21)

Acknowledgments

This research was funded by the Dirección General de Investigación Científica y Técnica, Spain, and FEDER, under the project F2004-02404. Omel Mendoza-Yero gratefully acknowledges the financial support from “Convenio UJI-Bancaixa” (Grant 06i005.17). The authors would like to thank the reviewers for their very helpful and constructive comments.

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. 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.

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: Author | 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.

Click here to see a list of articles that cite this paper