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
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, 2]. Approximate expressions for the instantaneous intensity generated by a pulsed light with time-variant Gaussian shape, diffracted by a circular aperture were derived [3, 4]. 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 . 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 . 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 . The Fresnel diffraction patterns by circular and serrated apertures, under ultrashort pulsed-laser illumination, were also investigated , 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  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  to the control of laser beam waveforms for ultrafast spectroscopy, nonlinear fiber optics, and high-field physics .
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,
For a set of N annular hard apertures, it can be shown (see Appendix A) that,
In Eq. (3), rim and rom 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 sim = (z 2 + r 2 im)1/2 and som = (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  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 rim and rom. These amplitude functions differ from the ones achieved within the framework of Kirchhoff theory of diffraction . This is because of the use of different obliquity factors . The time delay [som-sim)/c between the boundary wave pulses decreases with the path length difference som-sim. So, wherever (som-sim)/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 (SoN-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,
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 (som - sim)/c.
- The phase difference between them changes during their propagation owing to the term Δφ = ω 0(som - sim)/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 = (som + sim)/2c 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(iΔφ)= som / sim has no contribution to U(z,T 0m). If the ratio som / sim ≅ 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 ≫ rim and z ≫ rom), Eq. (3) reduces to the form,
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 rim = p(m-1)1/2 and rom = p(m-1/2)1/2, it allows the analysis of the ultrashort pulse diffraction by a zone plate , 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),
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. , 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 σω/ω 0 = 1/(2ω 0σ0)<9.30×10-2 (σω is the standard deviation of the power spectrum). Otherwise, the complex analytical signal needs to be used . Under the above-mentioned assumptions, the total diffraction field yields,
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
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. .
When geometric and boundary wave pulses arrive in phase to the axial points z = z + n, the instantaneous intensity yields,
whereas, if they arrive with a phase shift of π to the axial points z = z - n we find that,
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 = T0 =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 = 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 =3π/4×1015 Hz). For these physical conditions σω/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.
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.
In Ref.  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.  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.  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 ms (with s = 0, 1, 2.) are defined as . It is straightforward to show that
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. . From Eqs. (12)–(14), the squared standard deviation yields,
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.
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.
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 . 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 . 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.
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.
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
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.
it can be shown that
The generalization of Eq. (18) to the case of N annular hard apertures is straightforward
In Eq. (19), rim and rom represent the inner and outer edges radii of the m annular aperture.
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,
By using a procedure similar to that on the Appendix A, we can get to the following expression,
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.
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