1. Introduction

Almost four decades ago, dispersion-free Airy waves were suggested in the pioneering work of Berry and Balazs [1]. Airy wave packets are the only non-spreading solutions of the 1D potential free quantum Schrödinger equation. The work stayed relatively unknown due to the limitation of an experimental realization of the proposed theoretical concept. Since there exists an analogy between the Schrödinger equation in quantum mechanics and the paraxial wave equation in optics, the concept has been extended to the realm of optics. In this context, Airy beams have been suggested and investigated in the seminal work of Siviloglou and Christodoulides [2,3]. However, by definition, the diffraction-free solution contains infinite energy, and therefore its amplitude must be truncated in any experimental realization. Finite energy beams were achieved by exponentially truncating the infinitely extended field profile [3]. These generated Airy beams are quasi diffraction-free and self-accelerating, i.e., the field profiles are invariant along the propagation direction and traverse along a curved trajectory. The experimental demonstration also consists of peculiar properties of Airy beams, e.g., self-acceleration even in the absence of a refractive index gradient, and self-healing of the main lobe in the presence of an obstacle [4–7]. Besides homogeneous free-space, the experimental realization of Airy beams has also been shown in various other media, including media with nonuniform index distribution [8–17], non-linear media [18–33], plasmonic metallic surfaces [34–46], photonic crystals [47–49], and 2D materials [50,51].

Since Airy beams are the only 1D non-diffracting solutions of the paraxial wave equation, they are a natural ally for flat land photonics, namely plasmonics, a branch of optics that has grown in a discipline of its own. The surface wave nature of plasmons allows a direct realization of 1D Airy beams at metal-dielectric interfaces known as Airy plasmons. Many configurations have been suggested to generate Airy plasmons [35–37] experimentally. Minovich et al. [35] demonstrated a method which is particularly interesting since a diffraction grating imprinted on the metal surface generates the Airy profile and simultaneously couples the free-space electromagnetic radiation to a metal-dielectric interface. The Airy plasmons launched by such a grating possess the characteristics of quasi non-diffracting, self-accelerating, and self-healing properties. In the previous studies, the Airy plasmons were only excited by CW radiation, and the focus was only on the effect of plasmons' diffraction during spatial propagation.

In this article, we investigate numerically the dispersion and diffraction effects of Airy plasmons excited by an optimized diffraction grating under an ultrashort pulse illumination. We show that even in the case of ultrashort pulsed Airy plasmons, the properties of Airy plasmons are retained. The spatio-temporal evolution of Airy plasmon pulses is investigated in detail. A grating parameter-dependent geometrical model is developed to incorporate the temporal broadening effects. Potential applications of Airy plasmon pulses could be found in, e.g., time-resolved biomedical imaging [52], non-disruptive and highly localized optical probing [53], and spatio-temporally resolved spectroscopy.

2. Design and optimization of Airy grating

A diffraction grating (or Airy Grating) based on the original design of Ref. [35] has been optimized for the generation of Airy plasmon pulses. The schematic of the grating is shown in Fig. 1(a). A plane wave ultrashort Gaussian laser pulse excites the grating from the top (air) side on the Gold surface to generate an Airy plasmon pulse. An analytically calculated image of the Airy plasmon pulse is shown by an image laid out after the grating edge (not to scale). Figure 1(b) depicts the Airy function's amplitude, initial phase, and the grating design. The varying width of the Airy grating is in accordance with the zeros of the amplitude of the Airy function along the transverse x-direction. The depicted grating consists of 2 full periods and one-half period (2.5 periods) with a period of $p = 745$ nm. Each full-period has two rows of slits separated by $p/2$, which imprints the Airy function's initial phase.

 

Fig. 1. (a) Schematic of the sample layout: The grating is excited from the top (air side) at normal incidence (y-axis) with a plane wave laser beam having Gaussian pulse shape. Pulsed Airy plasmons are excited by a diffraction pattern at the gold-air interface. The diffraction grating consists of 2.5 periods of the slit pattern. An image of an analytically calculated time-averaged Airy plasmon pulse is overlaid with the sample layout on the homogeneous gold surface after the grating (not to scale). The Airy plasmons propagate in the xz-plane with z-axis being the main forward propagation direction and x-axis being the transverse direction of diffraction. (b) Absolute value and phase of the airy function $A$. The varying width of the grating along the x-direction is in accordance with the zeros of the airy profile. Each consecutive column is displaced in the z-direction by $p/2$ to match the phase change of $\pi$. The half-width of the main lobe is $x_{0} = 700$ nm. Each full period of the grating is composed of two elements to imprint the initial phase profile of the Airy wave. The slits are (150 nm) thick in the z-direction and the grating period in the z-direction is $p = 745$ nm.

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The optimization of the grating for the pulsed excitation has been performed by calculating the Airy plasmon's generation efficiency by gratings with different numbers of periods and slit thicknesses. The purpose of optimization is to obtain a grating with both a large generation bandwidth and a high generation efficiency. The grating has been extensively optimized for a different number of periods and slit widths. Here the optimization method is based on the idea that Airy plasmons are formed by the scattering from the grating, where the scattered field may be assumed to be made of the Airy plasmon mode and non-Airy plasmon modes. An overlap integral [54] is utilized to quantify the Airy plasmons mode generation from the diffraction grating, which is given by the following equation:

Here, subscript ‘a’ means analytical solution, ‘$n$’ stands for numerical solution, and ‘$d\mathbf {S}$’ is the line integral along the transverse x-direction. The overlap is calculated over the analytically evolved fields and the numerical field at a distance in the z-direction of $10\,\mu$m after the excitation grating. The analytical solution can be thought of as a target Airy profile, which would like to be achieved by an ideal diffraction grating.

The analytical electric and magnetic fields for CW excitation were developed using a simple formulation in reference [54]. Here we introduce it once again for the sake of consistency of our discussion. We assume a TM polarized plane surface wave bound to the gold-air interface in the xz-plane at y=0 and decaying exponentially away from the interface along the y-direction. The material above the interface (air) has the permittivity $\varepsilon _{1}$ and the material below the interface (gold) has the permittivity $\varepsilon _{2}$. The entire distribution of the TM-polarized surface plasmons in real space is then described as

Figure 2(a) shows the Airy plasmon's generation efficiency (Eq. (1)) for the excitation wavelength range $\lambda = 600$ to 1100 nm for 1, 3, 5, and 7 periods of the diffraction grating comparing CW response of numerics with the monochromatic analytical field. It can be noticed that for 3 periods, both the generation bandwidth and the generation efficiency is large. In a further optimization of the slit thickness, it was observed that a smaller slit thickness (in the z-direction) of 150 nm as compared to 200 nm produces a slightly larger generation efficiency with similar generation bandwidth of Airy plasmons for 3 periods. Therefore, we have chosen to further optimize the grating around 3 periods with a slit thickness of 150 nm. Figure 2(b) depicts these further optimization results. The number of grating periods is chosen to be 2, 2.5, 3, and 3.5. The generation bandwidth is the largest for 2.5 periods, as can be noticed by its high excitation efficiency even for longer (900 to 1000 nm) wavelengths. For this reason, we have chosen our grating configuration with 2.5 periods. This configuration is not only a 77% reduction in size from the previously reported design [54] but also provides a larger bandwidth for the generation of Airy plasmon pulses.

 

Fig. 2. Design optimization for broadband Airy plasmon generation. Airy plasmon generation efficiency is calculated using Eq. (1) for different numbers of periods. (a) for 1, 3, 5, and 7 periods the optimum generation efficiency is for 3 periods. (b) Optimizing the grating further around 3 periods. The generation efficiency is calculated for 2, 2.5, 3, and 3.5 periods. The bandwidth is highest for 2.5 periods, which is chosen for the further spatio-temporal analysis.

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3. Analytical, semi-analytical and numerical models

Before proceeding to numerically investigating the spatio-temporal behavior of Airy plasmon pulses, we assist our numerically calculated results with an analytical and semi-analytical model.

The analytical solution for pulsed excitation is based on the non-paraxial solutions presented in Eqs. (2)–(4) and is obtained by superimposing the spectral components of a 6 fs Gaussian pulse. The analytically calculated time-integrated intensity distribution is shown in Fig. 3(a). The half-width of the main lobe and the exponential apodization parameters are $x_{0} = 0.7\,\mu$m and $a = 0.04$. The analytical model offers a quick and efficient way of simulating the effects of different parameters such as $x_{0}$ and $a$ on the Airy plasmon pulse dynamics.

 

Fig. 3. Numeric and semi-analytical model of Airy plasmon pulse excited by 6 fs Gaussian pulse centered at 800 nm. (a) Logarithmic plot of analytically calculated time-integrated intensity under a paraxial approximation. (b) The time-domain response is calculated semi-analytically by using the impulse response of the grating. (c) Logarithmic plot of time-integrated intensity calculated using Finite-Difference Time-Domain method (FDTD). (d) The trajectory of the main lobe is compared for analytical, semi-analytical, and numerical models. The three models agree well with similar main lobe propagation trajectories.

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Next, we explain our semi-analytical model methodically, which can be used to calculate the time-domain response of the system to an arbitrary signal. The methodology to calculate the response is as follows:

This is a standard method to efficiently calculate the time-domain response to an arbitrary signal in a linear system. In Fig. 3(b), our semi-analytical model is used to calculate the spatio-temporal response of the optimized Airy grating configuration under 6 fs Gaussian input pulse. The transfer function H($\nu$) is obtained by calculating frequency domain fields from Lumerical FDTD simulations for wavelengths ranging from 600 to 1100 nm under normal incidence.

Figure 3(c) shows the logarithm of the time-integrated intensity for an Airy plasmon pulse obtained for normal incidence 6 fs Gaussian pulse excitation using the rigorous Lumerical FDTD solutions. In Fig. 3, the results show reasonably good agreement between the analytical, semi-analytical, and numerical models. It captures the accelerating and quasi non-diffracting properties of the Airy plasmon pulse. Figure 3(d) compares the main lobe trajectories for the analytic, semi-analytic, and numeric models. The main lobes propagate along a curved trajectory with similar curvature for these three models. The propagation length is over a distance of at least $15\,\mu$m. It is noteworthy that even under ultra-short pulse excitation, spatio-temporal plasmonic pulsed beams can maintain the standard Airy plasmons properties. In the next sections, we will advance our discussions in the direction of spatio-temporal evolution of Airy plasmon pulses based on the rigorous numerical results obtained in this section.

4. Spatio-temporal evolution of Airy plasmon pulses

There are many numerical and experimental studies in the literature on the spatial properties of Airy plasmons (e.g., see [10,35,38,41–43]) whereas a detailed study of the pulse nature of these spatially shaped plasmon fields is lacking. The fact that ultrashort pulsed Airy plasmons are accelerating and quasi non-diffracting makes them exciting for investigating their spatio-temporal evolution as targeted in our current work. The temporal evolution of such Airy plasmon pulses can be controlled by the dispersion and the diffraction of the excitation grating and the metal properties. The grating has been optimized for the largest plateau of excitation bandwidth with a considerable generation efficiency of the Airy plasmons. However the maximum generation efficiency is only 52%. The moderate generation efficiency means that the diffracting plasmon waves from the grating potentially contain significant non-Airy contributions. So, it is important to comprehend the effects of such diffraction on the temporal evolution of Airy plasmon pulses. Figure 4(a) shows the time-averaged intensity evolution of the Airy plasmon pulse on a linear intensity scale. The plasmons' profile is generated by the complex-shaped grating; therefore, it is better to compare it against a simple benchmark design. The aim here is to observe and evaluate the effects of many columns in the diffraction grating relative to a single column benchmark design. For this purpose, we choose the separated first column of the diffraction grating (only the rightmost column of slits in Fig. 1(b)), as our benchmark design. The excitation structure therefore consists of two slits, each of width $1.4\,\mu$m with a center to center separation of 745 nm. The time-averaged intensity generated by such a grating is shown in Fig. 4(b). It can be noticed that the pulsed plasmons generated from this simple structure, diffracts quickly and the intensity becomes significantly lower already after $5\,\mu$m.

 

Fig. 4. Spatial and temporal dynamics of plasmons excited by the Airy grating and by a reference structure consisting of a single-column of slits: (a) Time integrated Airy plasmon pulse evolution. The white band follows the main lobe propagation and has a width of $0.7\,\mu$m. (b) Time integrated intensity of single column slits (having 2 periods with slit width $w = 1.4\,\mu$m, thickness 150 nm and period $p = 745$ nm). A White rectangular region with a width of $0.7\,\mu$m is used for spatial averaging. (c) Temporal intensity profile at 8 spatial locations starting from $z = 0.5\,\mu$m at each $2\,\mu$m distance in a co-moving frame over time. Temporal intensity is averaged over a spatial width of $0.7\,\mu$m (shown by white bars in (a)). In the inset, an enlarged view of the pulse profile for the temporal window 40 fs to 70 fs is also shown. (d) Temporal profile for Gaussian reference pulse spatial averaged over white bars at 8 subsequent propagation distances as shown in Fig. 4(b).

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Before comparing the temporal characteristics, such as temporal widths and spectral widths, for the Airy grating with the benchmark design, it is interesting to look at the pulse profile of the main lobe of Airy plasmon pulses at various propagation distances. Figure 4(c) provides the time traces at 8 different locations by taking the spatial average over a width $0.7\,\mu$m on the main lobe of the Airy plasmon pulse as indicated by the white bars in Fig. 4(a). The intensity has been normalized, and the pulse traces are plotted in a co-moving frame of reference. Here, the group velocity of the infinitely extended surface plasmon polariton at the central pulse wavelength of 800 nm is taken as the reference velocity of the co-moving frame. A careful inspection of the temporal intensity profile reveals that for some spatial locations, there are weak secondary pulses along with the primary pulse. At position $z = 0.5\,\mu$m, we observe mainly the primary pulse. The pulse starts spreading at subsequent propagation distances due to dispersion effects. For propagation distances from $z = 4.5\,\mu$m to $10.5\,\mu$m significant secondary pulse profiles appear, as shown in the inset Fig. 4(c). For locations from $z = 12.5\,\mu$m to $14.5\,\mu$m, the primary pulse profile spreads mainly due to dispersion, and the secondary pulse profiles are clearly absent. Figure 4(d) shows the pulse profile for the benchmark design by taking a spatial average over white bars (width $0.7\,\mu$m, shown in Fig. 4(b)). Pulse spreading is negligible over the propagation distance of $15\,\mu$m for this simple reference configuration, which indicates that the observed pulse dynamics of the Airy plasmons is induced mainly by dispersive spatial diffraction effects.

Next, a comparison of the temporal width (Fig. 5(a)) and the spectral width (Fig. 5(b)) for the main lobe of the Airy plasmon pulse and the pulsed plasmon from the slit configuration (benchmark design) is considered. The temporal and spectral widths are calculated as the second-order moment of the temporal intensity and the spectral intensity. This definition of the pulse width was chosen to consider also the influence of the change of pulse shape beyond a single lobed shape, which is due to the occurrence of side lobes as illustrated in Fig. 4(c). The widths are calculated by first taking the spatial average of the intensity over a width of $0.7\,\mu$m for the main lobe of the Airy plasmon pulse's profile. Similarly, for comparison purposes, the pulsed plasmons generated by a single column grating are spatially averaged over $0.7\,\mu$m widths (width of the white box). Figure 5(a) depicts the temporal width change over the propagation distance. The temporal width remains almost constant for the reference case, while for the main lobe of the Airy plasmon pulse, the width undergoes appreciable changes. At a distance $\approx 5\,\mu$m there is a sharp increase in the temporal width, which reaches a peak value, and then decreases again. This bump in the temporal width may be attributed to the contribution of the non-Airy plasmonic beams propagating from the diffraction grating and crossing the main lobe region. Finally, the temporal width increases monotonically, which is due to planar metallic dispersion effects. In Fig. 5(b), the spectral width for the main lobe of the Airy plasmon pulse decreases monotonically. A transform-limited temporal width of the pulse is calculated corresponding to Airy grating's spectral width. This is shown by a red dotted line in Fig. 5(a). From Fig. 5(a) and Fig. 5(b) and the above discussions, it can be inferred that the hump in the temporal width variation is mainly due to diffraction grating geometry.

 

Fig. 5. Comparison of the temporal and spectral width for the main lobe of the Airy grating and a reference structure consisting of a single-column of slits: (a) Temporal width over the propagation distance. The width changes abruptly in a certain region. The dotted red curve shows the transform-limited temporal width. (b) Spectral width evolution over propagation direction. The spectral bandwidth decreases significantly over the propagation distance. Corresponding temporal broadening for the transform-limited case is larger than first column slits; however, the temporal width change for the Airy grating has a spatial dependence.

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5. Geometrical model for spatio-temporal dynamics of Airy plasmon pulses

As discussed in the last section on the temporal dynamics of Airy plasmons, it was found that abrupt changes (humps) in the temporal pulse width have an interdependence with the grating parameters. In this section, we study systematically the evolution dynamics of the temporal pulse width for the spatial main lobe of the Airy plasmon pulse and its first 10 side lobes. Figure 6(a) shows again, the time-integrated intensity of the Airy plasmon pulse, but this time we investigate and compare the temporal evolution of the pulse length for a number of lobes. The temporal evolution of the pulses within each lobe is evaluated up to a propagation distance where the time-averaged intensity of that lobe has decayed down to an intensity just 25 times the background intensity. This propagation distance is indicated in Fig. 6(a) by the position of the white bars along the propagation distance $z$. The temporal pulse length is evaluated by spatially averaging the lobes over a width which is indicated by the white bars in Fig. 6(a) and which is equal to the respective grating column being responsible for the lobe’s excitation. In Fig. 6(a) the first six lobes are denoted by labels 1 to 6, where $n=1$ is the main lobe. The corresponding evolution of the pulse lengths for the lobes 1, 3, and 5 is shown in Fig. 6(b). The typical hump, where the pulse length abruptly increases and then decays, is found for all lobes (only lobes 1, 3, and 5 are shown in the Fig. 6(b)). As discussed before and shown in Fig. 4(c), the humps in the temporal pulse length are a characteristics of the appearance and decay of secondary pulses, which are delayed with respect to the main pulse. The position of each hump, i.e. the center position of the hump along the propagation distance $z$, is determined by applying a Gaussian pulse fit and taking the center of the Gaussian. This position will be called hump distance $Z_{n}$ for hump $n$ and is denoted by a vertical dashed line for each lobe's pulse length evolution in Fig. 6(b).

 

Fig. 6. Space-dependent temporal dynamics and its interdependence on side lobes: (a) Time-integrated intensity distribution of the Airy plasmon, where the numbers indicate the main lobe (1) and the side lobes (2, 3, $\ldots$, 6) of the Airy plasmon. The length of the white bar at each lobe corresponds to the size of the grating slit the lobe originates from and it illustrate the width over which spatial averaging is performed for calculating the temporal pulse length for each lobe. (b) Evolution of the temporal pulse length over the propagation distance for lobes 1, 3, and 5. A Gaussian fit has been performed for the humps in the traces of the temporal pulse length for each lobe. The distances, which are indicated by the dotted vertical lines, are the first-order moments of the fitted Gaussians.

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To explain the observed temporal dynamics and particularly the characteristic positions of the humps in the consecutive lobes, we developed a simple geometrical model of the lobe's interaction, which we compare to our already presented findings from the rigorous numerical simulations. We start by having a look at the hump distance $Z_{n}$, which is plotted for the individual lobes $n=1 \ldots 11$ in Fig. 7(a). It clearly shows the trend, that for larger lobe number $n$ the humps occur already for shorter propagation distances $z$. Our model to explain this observation is based on the assumption that the used binary excitation grating gives rise to a significant non-Airy plasmon mode contribution. These non-Airy parts of the excitation evolve on a different trajectory than the Airy plasmon mode. Hence, both parts interfere, which is observed as a perturbation of the Airy lobe's pulse dynamics. Our physical model describes this interference according to the scheme depicted in Fig. 7(b), where we assume that the trajectories of both parts of the excitation can be modelled approximately by Gaussian beams. The sketch shows the grating elements, from which Airy and non-Airy plasmon pulses are excited. For the quantitative description, each grating element is characterized by two parameters: its transverse size $s_{n}$ and its distance to the next grating element $\Delta d_{n}$, where ‘$n$’ stands for $n$th Airy lobe originating from the $n$th grating element. Our model assumes that the secondary pulses, which had been shown in Fig. 4(c), appear when the non-Airy part of the excitation crosses a lobe of the Airy-part of the excitation. Assuming that the non-Airy part of the excitation can be modelled by a Gaussian beam, a secondary pulse should appear at a propagation distance, when this non-Airy part of the excitation of grating element $n+1$ overlaps spatially with lobe $n$, which originated from grating element $n$.

 

Fig. 7. A grating parameter-dependent geometrical model for spatio-temporal dynamics: (a) red: Hump distance $Z_{n}$ for the lobes $n=1 \ldots 11$, which illustrates at which propagation distance $z$ the secondary pulse appears on the lobe's trajectory leading to a sudden increase in the overall pulse length. blue: Error value $\epsilon _{n}$ of Eq. (5), which indicates the discrepancy of where our simple physical model predicts the appearance of the secondary pulse from where it really appears in the rigorous numerical simulations. (b) Sketch of a simple geometrical model to explain the occurrence of the secondary pulses in the Airy lobe's trajectory.

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To check this assumption by an analytical expression, we model both, the trajectory of the non-Airy part as well as the trajectory of the Airy-lobe by a simple Gaussian model. Hence, we take from any optics textbook [56] the analytical solution for the evolution of the half-width $w_{n}(z)=w_{n}(0)\sqrt {1+z^2/z^2_0}$ of a Gaussian beam, which is excited by slit $n$, assuming that the slit's size $s_{n}$ determines the initial half-width of the Gaussian as $w_{n}(0)=s_{n}/2$ and $z_0=\pi w^2_n(0)/\lambda _0$. If the half-width $w_{n}(z)$ characterizes the spreading of the non-Airy part as well as the trajectory of the Airy lobe $n$, the interference of the two should occur at a propagation distance $z^{*}$ for which

This condition considers that the non-Airy part originating from slit $n+1$ spreads faster than the Airy-lobe from slit $n$ moves to the side, since the size $s_{n+1}$ is smaller than $s_{n}$, which gives rise to a more rapid Gaussian beam diffraction. The condition is therefore fulfilled if the spreading of a Gaussian beams caught up with its neighbor, which is at a distance $\Delta d_{n}$, minus half the width of the lobe $s_{n}/2$ to consider the interference at the lob's center. Figure 7(a) shows the error function $\epsilon _{n}$ of Eq. (5), which illustrates that our model is quantitatively sufficiently close to the rigorous numerical simulations to support the stated general ideas of the model. However, it is clear that our model is just a very simple geometric representation of a complex scattering problem, which can’t explain all details. It does not consider accurately the Airy lobe's caustics and is based on the paraxial approximation of Gaussian beam propagation, which is increasingly wrong for smaller slit sizes as can be seen from the growing absolute value of the error function $\epsilon _{n}$ for larger lobe numbers $n$ in Fig. 7(a). Despite these limitations, the model could be utilized for tailoring the non-Airy plasmon part to shape the Airy plasmon's pulse profile, e.g. by tuning the geometrical parameters of the grating. Using the simple geometrical model offers tailoring of the grating parameters for shaping the pulse, which might be used in further experimental studies.

One could use a curved grating for changing the slope and achieve temporal focusing for minimal spot size in space and time. Furthermore, the input pulse shape can be used as an additional degree of freedom to control and engineer the pulsed plasmonic beams on the metal-dielectric surface. An example of this pulsed plasmonic beam can be Airy-Airy plasmonic pulsed beams, which use Airy pulses as an incident source and can exploit their non-dispersive properties for non-varying pulse shapes. The carefully engineered spatiotemporal responses can have application in generating intense 2D plasmonic light bullets. These engineering methods provide only passive control of the pulsed beam on the metal-dielectric interface. Active control of the shape and spatio-temporal properties may be achieved by utilizing the tunable properties of 2d materials such as Graphene.

6. Conclusion

In summary, we reported a numerical study of the spatio-temporal evolution of Airy plasmon pulses. The diffraction grating used for the excitation has been optimized for a maximum bandwidth of the Airy plasmon generation, resulting in a broad generation bandwidth from $\lambda =$ 600 to 1100 nm. The optimization has also reduced the grating size to just 2.5 periods, much shorter than previously reported diffraction gratings. Airy plasmon pulses are excited by an ultrashort 6 fs Gaussian pulse using the optimized diffraction grating. The generated pulsed beam maintains quasi non-diffracting and accelerating properties. We have also assisted our numerical model with an analytical and semi-analytical model and have found that results show good agreement. Furthermore, a detailed investigation of the pulse dynamics suggests that non-Airy plasmons from the diffraction grating cause significant pulse broadening. We have developed a geometrical model that considers the grating parameters to predict the distance at which the temporal width increases abruptly. This geometrical model may be used to tune the grating parameters for temporal focusing to provide spatio-temporal pulsed beams with highly localized spot size.