1. Introduction

An octave-spanning femotosecond supercontinuum is a prerequisite to single-cycle optical pulse synthesis and isolated attosecond pulse generation that are critical in the study of attosecond science [1–3]. Supercontinuum with high pulse energy will facilitate seeding ultrafast optical parametric amplifiers [4], high-speed data acquisition in optical coherence tomography [5] as well as ultrafast, multi-dimensional molecular spectroscopy [6, 7]. It can further facilitate rapid control of the carrier-envelope offset in frequency combs [8, 9] and improve the precision in carrier-envelope phase stabilization of ultrashort optical pulses [10].

The physics of supercontinuum generation is well studied [11–14]: an octave-spanning supercontinuum is produced when a femtosecond pulse with a peak power that is slightly above the critical power P_cr for self-focusing is focused inside a medium. Intrapulse self-phase-modulation broadens the pulse spectrum. Kerr-induced nonlinear dispersion leads to steepening of the pulse envelope and the formation of an optical shock that is accompanied by a rapid phase jump. This phase jump produces a blue spectral pedestal forming the supercontinuum [11, 12, 15]. However, the supercontinuum pulse power does not scale with laser power. Multiphoton absorption, plasma formation, intensity clamping, plasma defocusing and group velocity dispersion combine to disrupt the pulse and prevent the supercontinuum power from scaling upward [16, 17]. In practice, random amplitude and phase fluctuations in the pulse field lead to modulation instability that breaks up the pulse, producing local pockets of shock waves that destroy the overall spatio-temporal coherence of the pulse [18]. Consequently, the generation of quality supercontinuum has been limited to input power of a few times P_cr. In gases P_cr is about 10 GW [19, 20] whereas in condensed matter it is a few MW [21]. It has not been possible to generate a coherent supercontinuum by conventional techniques for laser powers between 10 GW and several MW.

The difficulties mentioned above in power scaling could be resolved if supercontinuum generation takes place in an environment where the factors that disrupt the pulse are suppressed while spectral broadening remains unaffected. Supercontinuum pulses are conventionally generated in a continuous, uninterrupted medium [11]. The situation involving a medium with a distribution of interfaces and refractive index discontinuities in the longitudinal direction has not been investigated. In this paper we report on theoretical study of the novel approach of supercontinuum generation in a distributed medium consisting of multiple gas-solid interfaces located along the laser beam. We establish basic criteria to guide the setting of essential initial values of the incident laser intensity and the spacing between pairs of plates for this new multi-plate generation process. The simulated results obtained using the proposed basic criteria are in qualitative agreement with experimental observations on femtosecond superconitnuum generation performed in a medium consisting of several thin fused-silica plates [22]. In the first report on that experiment, the parameters were chosen empirically and the mechanism was not fully explained.

We note that the spatio-temporal behavior of a broadband pulse propagating through a single gas-solid-gas segment has been explored previously to explain the restoration of a self-compressed femtosecond pulse [23]. Here we analyze the case of cascaded spectral broadening of a femtosecond pulse in a distributed multi-plate arrangement.

2. Approach

For the simulation, we follow the approach commonly used in modeling filamentation and supercontinuum generation [17]. We numerically solve the generalized nonlinear Schrdinger equation simplified for forward propagation with radial symmetry [Eq. (1)] coupled with the electron plasma density equation [Eq. (2)]:

Here z is the propagation variable, and t is the retarded time. The input electric field is linearly-polarized and in the form of E(x, y, z, t) = A(x, y, z, t)e^{i(k₀z − ω₀t)} + c.c. where ω₀ is the angular frequency of carrier waves, k = n(ω)(ω/c), k₀ = k(ω₀), n₀ = n(ω₀) is the wave vector and c is speed of light in vacuum. I = |A|² is the intensity of the optical field and D is an operator accounting for dispersions of order higher than 2: D = ∑_n≥2(β_n/n!)(i∂/∂t)ⁿ, β_n = (∂ⁿ/∂ωⁿ|_ω0), is the corresponding dispersion coefficient. $T=1+\left(\frac{\overline{i}}{{\omega }_0}\right)(\partial /\partial t)$ is an approximated differential time operator. It plays an important role in self-steepening and spatio-temporal coupling. $R(t)=(1-f)\delta (t)+f\mathrm{\Theta }(t)\frac{1+{\omega }_R^2{\tau }_R^2}{{\omega }_R{\tau }_R^2}{e}^{-t/{\tau }_R}\text{sin}({\omega }_{R}t)$ is the combined instantaneous and Raman-delayed nonlinear response with δ(t) as the delta function and Θ(t) being the positive unity function. Fused silica is chosen as the solid material in this study since reliable optical parameters for fused silica are available and experimental data can be obtained for comparison. Linear dispersion in the visible and near infrared regime refers to the equation modified from Ericksons equation [24] for air and to the Sellmeier equation [25] for fused silica. We employ from the literature values for the fraction of delayed response f for silica (15%) and for air (50%) [26]. The ionization rate W(I) for air is taken from PPT (Perelomov, Popov, and Terentev) [27, 28]; for fused silica we use the Keldysh rate [29, 30]. The nonlinear refractive index n₂ for air of 1.0 × 10⁻¹⁹ cm²/W is determined by matching the spectral broadening in air measured with our laser in the laboratory. The n₂ of 2.4 × 10⁻¹⁶ cm²/W for fused silica comes from [21]. ρ_nt is the total atom density and ρ is the ionized electron density respectively of the medium. The ionization potential U_i used is 12.1 eV for oxygen (air) [26] and 9 eV for fused silica [30]. Values for the Raman response time constant τ_R, avalanche cross-section σ and recombination time τ_rec are those reported in [26].

We introduce several important criteria for the numerical calculation. First, an obvious criterion is that the incident intensity I₀ cannot exceed the dielectric breakdown threshold of the material. Second, we recognize that there is an optimal value for I₀ that will result in the most efficient and broadest supercontinuum. If I₀ is too small spectral expansion will be weak. If I₀ is too large there will be excessive conical emission that propagates off-axis to take energy away. Excessive conical emission becomes dominant when the maximum nonlinear phase shift, L/L_nl exceeds 2π where (L_nl)⁻¹ = k₀n₂I₀ is the nonlinear length and L is the thickness of the medium. Therefore we suggest as the second criterion that the nonlinear phase shift should be kept to near unity. Subsequent tests with different I₀ suggest that the best value for L/L_nl is ∼1.5π. The I₀ associated with this value would lead to efficient supercontinuum generation while inhibiting excessive off-axis emission. Moreover, by limiting the total nonlinear phase to the order of unity together with the large dispersion due to the broad spectrum of a 25 fs pulse, we avoid spatial beam breakup even for peak powers a few thousand times the crtical power for self-focusing and put a clamp on the rate of growth of multiple filament due to modulation instability in the pulse envelope [11, 18].

The I₀ that meets the first and second criteria that produces the most desirable spectrum of high quality by implication leads to a third criterion, which is that the location of each subsequent plate is to be at a distance downstream where the peak intensity of the pulse has restored to I₀ via spatio-temporal coupling and beam divergence during propagation outside the solid.

We note that these three criteria are the essence of the analysis of the multi-plate approach, allowing us to set the initial intensity and determine the location of the plates. Imposition of the criteria in the calculation distinguishes this work from conventional simulations of nonlinear generation in an optical medium.

For the simulation we use 100 μm thick fused silica plates and a 25 fs Gaussian input pulse at a center wavelength of 790 nm. The initial pulse energy is 140 μJ. Other plate thicknesses and pulse energies could be substituted by conforming to the criteria stated above. The initial peak power of 5.6 GW is ∼ 2100 times P_cr for fused silica but is below air’s P_cr of 9.2 GW. The second criterion applied to a 100 μm thick plate results in an effective I₀ 21.2 TW/cm² for a pulse incident at Brewster’s angle to the first plate. This condition could be achieved for the 5.6 GW pulse by focusing the pulse with a lens or mirror of 2150 mm focal length after collimating the pulse to a 8.9 mm diameter. At this intensity the multiphoton-induced plasma density in fused silica is calculated to be around 10¹⁸ ions/cm⁻³, safely below the documented avalanche plasma density of 2.0 × 10²⁰ ions/cm⁻³ [23]. Consequently, the effect of plasma on pulse propagation should be minimal. This first plate is placed at the waist position of the incident laser pulse. The plates are aligned at Brewster’s angle to minimize reflection loss. The simulation shows that there is very little influence on the spectral broadening by the choice of incident angle.

Application of criterion 2 that avoids assymetric beam behavior validates the use of radial symmetry ( ${\nabla }_{\perp }^2=r^{-1}{\partial }_{r}r{\partial }_r$) and use 0.33 fs, 3 THz, and 12.5 μm respectively for the time, spectral and radial step increments in the simulation. We present here the on-axis results. Results obtained by integrating over the transverse central zone that contains the supercontinuum exhibit similar behaviors.

3. Results and discussion

We first examine the spatio-temporal evolution of the pulse in its propagation through the multiplate medium. Results of the simulation show that most changes to the spatial and temporal properties of the pulse occur in the space between the plates. As it passes through the first plate, the pulse acquires a temporal chirp that lengthens the FWHM of the pulse duration by ∼11% with a smaller increase in the transverse beam size. While these changes are small, the acquired nonlinear phase inside the plate of 1.55π, close to the design value of 1.5π, is sufficient to cause the pulse, upon exiting the plate, to come to a focus outside the plate. In the present case this introduces a sharp rise in the pulse intensity to more than 100 TW/cm² at a distance less than one cm outside the first plate as shown in Fig. 1(a). Meanwhile the acquired nonlinear phase triggers the pulse to compress and shift its peak to the rear part of the pulse in time. Since this happens outside the solid medium, the self-focused beam does not induce damage. The pulse intensity recedes back to the initial value due to diffraction. At the location that this happens we insert the next plate in accordance with our third criterion.

 

Fig. 1 3-D plot of the on-axis (r = 0) spatio-temporal (top trace) and the spectral (bottom trace) evolution of the pulse as it propagates through the multi-plate system: (a) The vertical axis displays the retarded time relative to the moving frame of the pulse. x = 0 of the horizontal axis represents entrance to the first plate. Instantaneous intensity is given by the color bar. The white arrows indicate the location of each plate. Evolution inside the plates is not shown due to insufficient spatial resolution. (b) Normailized pectral intensity shown color-coded in dB. The spectrum is normalized to the highest point in the entire plot. Substantial increase to the overall width of the spectrum appears after passage through each plate.

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As illustrated in Fig. 1, this focusing behavior repeats after each passage through the plates. Eventually, beginning with passage through the second plate, the pulse peak is shifted close to its rear edge to create a sharp trailing edge for the pulse. Very distinctive spectral expansion from each plate can be seen in Fig. 1(b). It begins symmetric expansion that is characteristic of self-phase modulation in the first plate. Extension to the high frequency side begins in the second plate, growing to become a plateau with a sharp drop-off in subsequent plates. The effect is clearly evident after the third plate. Detail of the spectral expansion inside the plates is shown in Fig. 2. This spectral signature is the consequence of the pulse shifting to form a sharp increase in the slope dI/dt of the E field leading up to the entrance to a plate, and the nonlinear interaction of this sharp field gradient with the solid medium inside each plate to give rise to the large blue shift Δω = −dϕ/dt = −(ω₀/c)d(n₂Il)/dt by taking advantage of the orders-of-magnitude larger nonlinearity of the solid medium. This phenomenon distinguishes itself when compared to the traditional supercontinuum in filaments in that the steepening of the pulse and the spectral expansion happen sequentially in the multi-plate system instead of simultaneously as in a gas. Figure 2 also shows the result of adding a fifth plate. The addition mainly redistributes the spectral energy and does little to broaden the spectrum.

 

Fig. 2 Evolution of on-axis spectral broadening inside each plate. Pulse propagates towards the right. The spectral intensity is normalized to the highest point in the plots to highlight expansion of the spectrum, especially growth of a plateau toward shorter wavelength (upward direction) while preserving total spectral intensity (preserve energy).

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Outside the plates the spectral width is practically unchanged. The small reduction in width of the on-axis spectrum outside each plate is a consequence of the conical nature of the emission. In addition, the simulation shows that airs contribution to spectral expansion is negligible due to its three orders of magnitude smaller nonlinearity compared to fused silica.

Ultimately, accumulated nonlinear chirp in the pulse splits the pulse into sub-pulses and lengthens the FWHM of the main pulse to 45–55 fs. This, in association with energy loss due to multiphoton absorption and ionization in the plates collectively lowers the pulse intensity. The forces that form the shock front wane away. Nonlinear interaction weakens (after plate #4 in the present case). Beyond this point there is very minor increase to the spectral width upon adding more plates. The final spectrum spans more than one octave.

The location of each plate in the beam path is calculated based on the third criterion stated above. Figure 3 shows the waist size of the beam as the pulse propagates.The beam size changes mainly as a counterplay between nonlinear self-focusing and diffraction. A plate is inserted whenever the beam size is such that the instantaneous intensity in the pulse restores to I₀. Two factors lead to lower peak power so that smaller waists at the positions shown in the figure are needed to maintain the same incident pulse intensity. The first is that pulse chirp lowers the peak power of the pulse as has already been discussed in conjunction with Fig. 1. The second is multiphoton absorption and ionization in the plates that account for a ∼3% energy loss through each plate. The small amount of ionization implies that ionization has little impact on the physics of the generation here.

 

Fig. 3 Calculated Gaussian beam 1/e waist size along the beam path for 140μJ input pulse energy with four plates inserted. Arrows indicate the location where the calculated intensity equaled I₀ and a plate was inserted. Dotted lines indicate evolution of the beam size if a plate had not been inserted. The red line is the intensity evolution, which is the peak intensity in the Fig. 1(a).

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The calculated spacing between every pair of plates is shown in Table 1. They compare very well with experimental values. In the experiment, the location of each plate was determined empirically by maximizing the width of the pulse spectrum while avoiding formation of multiple filaments and optical damage in the medium. The good agreement gives credence to the three criteria introduced for this simulation and can be used to obtain the working parameters for the multi-plate approach in supercontinuum generation. By fixing the plate thickness at 100 μm we calculated the spacing between pairs of plates for different input pulse energies. The results for 35 μJ and 8.75 μJ are shown in Table 1. As expected, the spacing gets smaller and the number of plates needed increases at lower input energies. Interestingly, even at 8.75 μJ just one additional plate is needed in order to reach a similar total spectral width.

Table 1. Spacing between each pair of adjacent plates for different energies of a 25 fs pulse.

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Different degree of Kerr phase accumulation in a Gaussian transverse profile gives rise to concentric zones in the radial direction with the supercontinuum filling the central zone. The spectrum of the outer zones is dominated by the spectrum of the pump whereas the spectrum of the central zone constitutes the supercontinuum. The integrated intensity of the spectrum in the central zone is recorded and compared with the calculated integrated intensity. Details of the experimental procedure have been reported previously [22]. The spectra obtained after each plate insertion is presented in Fig. 4. The figure shows reasonable agreement between simulated result and experimental measurement. The total calculated pulse energy of the supercontinuum is 51.6% of the incident energy, agreeing with the measured value of 54%.

 

Fig. 4 Self-normalized measured and calculated spectral intensity in the central zone of the laser beam at 10 cm downstream from the plate location after the insertion of each fused silica plate into the beam. The signature plateau and blue-end drop off of a supercontinuum is clearly evident with insertion of the third plate.

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4. Summary

In summary, we showed that the standard filamentation model coupled with the three criteria we proposed serves well to visualize the physics of the multi-plate supercontinuum generation scheme. It could therefore be used as an efficient platform for further investigation of the process. A good example is to evaluate the possibility of scaling up in power with this multi-plate technique. Preliminary calculation indicates that when the input peak power reaches above the filamentation threshold for air, self-focusing leads to filamentation and spectral broadening in air before the pulse enters the first plate. This negates the good beam control afforded by the solid medium. This therefore calls for eliminating the air between the plates. Investigations of varying the plate thicknesses and using chirped input pulses would also be interesting.