Dispersion‐Engineered Super‐Gain Parametric Amplification in Nanoscale Optical Cavities

The gain factor of the optical parametric amplification (OPA) process is known to be negligible in the small scale due to low‐interaction‐medium length. Hence, in the nanoscale, OPA is deemed as infeasible. Therefore, in small‐scale‐integrated optical devices, stimulated‐emission‐based amplifiers (lasers) are employed instead of OPAs. In contrast, the major advantage of OPAs over lasers is that unlike lasers which only provide amplification over a narrow spectral band, OPAs provide high‐gain amplification over a very large, user‐controlled spectral band. In this article, it is shown that OPA can yield wideband high‐gain amplification over a nanoscale beam propagation distance through dispersion engineering. This is achieved by a proper tuning of the pump (source) wave frequency, which can maximize the effective medium nonlinearity by a few orders of magnitude, while concurrently maximizing the intracavity energy density, thereby compensating for the small co‐propagation distance for the input (signal) and pump beams. In this study, it is shown that an input wave can be amplified by a factor 108$\left(10\right)^{8}$ in a nanoscale cavity via precise dispersion engineering. Both empirical and computational formulations are used for the investigation, which display a reasonable agreement.


Introduction
Optical parametric amplifiers (OPAs) are known for their efficiency in providing wideband high-gain amplification through the use of bulk nonlinear crystals that are usually a fewcentimeters long.For this reason, OPAs are often preferred over lasers for amplification in tabletop optical devices.OPA involves amplifying a low-intensity input wave using a high-intensity pump wave, through co-propagation along a nonlinear interaction medium. [1,2]OPA efficiency is strongly dependent on the intensity of the pump wave and the length of the interaction medium.In the nanoscale, as the interaction length is extremely small, one has to rely on the intensity of the pump wave to achieve significant OPA, although even very high pump-wave intensities are not sufficient to provide a useful OPA gain in the nanoscale without causing thermal damage in the medium. [3,4]Recently, there has been a few ambitious research on attaining notable OPA performance in the microscale-nanoscale. [1,[3][4][5][6][7][8] These studies proposed that OPA is feasible using ultranonlinear artificial materials/structures through doping and structural modifications, which reduce the required pumpwave amplitude for OPA with a high-gain factor.[8][9] Hence, nanoscale OPA still remains a big challenge.In this article, it will be shown that the gain-bandwidth product of the OPA process can be drastically enhanced at the nanoscale through computational dispersion engineering.[12][13] In this article, we will refrain from using the term "nanocavity" as the term often refers to ultrasmall structures that are below 100 nm.Instead, we will use the term "nanoscale cavity" to refer to all submicron-sized cavities.The focus of this study is not on the design of a special nanoscale cavity for performing OPA, but rather on whether high-gain OPA can be attained inside a nanocavity whose optical characteristics are already given, through extensive computational analysis.In fact nanoscale cavities can be of great use for small-scale OPA as they allow for a very high-nonlinear constructive interference to be attained for relatively long optical pulses via a simultaneous maximization of the effective medium nonlinearity and the intracavity pump-wave amplitude, at a certain pump-wave frequency.Using electromagnetic field definitions, i.e., the wave equation and Lorentz dispersion equations, we will investigate the variation of the peak input wave amplitude (which is the OPA gain factor given that the excitation amplitude of the input wave is normalized to 1 V m À1 ) against the pumpwave frequency inside a nanoscale cavity with known optical characteristics, using the Gradient-Descent algorithm.Once the maximum input wave amplitude is attained for the optimal pump-wave frequency, we will choose this frequency as the pump-wave excitation frequency and use it for OPA of a lowintensity input wave inside a nanoscale cavity.Performing OPA inside a nanoscale cavity not only allows for the attainment of a high pump-wave amplitude through enhanced nonlinear constructive interference, but also compensates for the smallinteraction-medium length via providing optical feedback.The goal of OPAs, as with all optical amplifiers, is to achieve the widest amplification bandwidth, highest-gain factor, lowestpumping power, and the highest tunability.Achieving nanoscale OPA is specifically important for employing OPAs in small-scale integrated photonic devices to attain a large-gain-bandwidth product as opposed to lasers.Table 1 illustrates some of the well-known preceeding research on OPAs and their reported characteristics.
The dispersion engineering procedure that is carried out in this study is naturally theoretical/computational. We will first investigate the problem of OPA in a nanoscale cavity through the use of the empirical formula that is confirmed by both theoretical derivations and experimental measurements.Next, we will examine the feasibility of nanoscale OPA via determining the necessary pump-wave amplitude under a given interactionmedium nonlinearity.After that we will simulate the OPA process based on the finite difference time domain (FDTD) discretization of the corresponding wave equation and the associated nonlinear Lorentz dispersion equations, and perform computations to compare with the results of empirical formulations.Using our computational model, we will search for the optimum pump-wave frequency that maximizes the intracavity input wave amplitude.Followingly, under this optimum pump-wave frequency, we will compute the resulting gain spectrum for the given input wave.As the final step, a conclusion will be driven regarding the performance of nanoscale dispersion-engineered OPAs in comparison to lasers.

Empirical Investigation
The maximum attainable optical gain from the OPA process depended on the given wave and material parameters.[16] Experimental investigations confirmed the given empirical formula in Equation ( 1) through cross comparison with the theoretical derivation for single-pass OPA gain under perfect phase-matching over an interaction medium with an impedance of η and a second-order nonlinearity of ρ. [2,[15][16][17] c: speed of light, ε 0 : free space permittivity, L: medium length, I input : input signal intensity, ρ: second-order nonlinearity, n: refractive index, η: intrinsic impedance of the interaction medium, E pump : amplitude of pump electric field, ω 1 : input frequency (angular), ω 2 : pump frequency (angular), and G intensity : intensity gain.When the interaction medium was enclosed in a small-scale cavity, such as that was inside an optical chip (see Figure 1), an oscillator was naturally formed due to the strong reflection of the input/pump electric fields from the interface between the host optical crystal and the embedded interaction medium (which can be either a dielectric or a semiconductor).In this case, the expression for the input wave gain factor must account for the cavity losses per each roundtrip, which could be expressed Table 1.Previous studies on OPA (broadband: >50 THz, ultra-broadband: >100 THz, high-gain factor: >10 6 , medium gain factor: >10 4 , high pump power: >10 12 W m À2 , medium pump power: >10 9 W m À2 ).  in terms of the single-pass OPA intensity gain and the roundtrip loss as follows: Cavity roundtrip loss ¼ κ E pump ðiÞ ¼ E pump ð0Þ Â κ i : : : i∶ round trip number, gain (N) = cavity gain at the end of N round trips, Γ 1 and Γ 2 : reflectivities of the cavity walls, α d : dielectric loss, α c ∶ conduction loss, and α s ∶ scattering losses.The roundtrip cavity loss (κ) arose due to the dielectric, conduction, and scattering losses within the interaction medium.There was also an associated reflection loss from the cavity walls due to imperfections in reflectivity (intended for beam outcoupling).Compared to lasers, the roundtrip loss was more severe for cavity-based OPAs as the pump-wave amplitude, which was a critical determinant of the amplification performance, decreased at each roundtrip due to this loss.[20][21][22] To compensate for this strong roundtrip decay in pump-wave amplitude, optimizing the cavity buildup of the pump-wave amplitude via nonlinear constructive interference was essential.Based on Equation (2), Figure 2 illustrates the sensitivity of the OPA process to the pump-wave amplitude (V m À1 ) in a 700 nm long cavity, which has a round trip loss of 0.1%.It was clear that even a slight decrease in the pump-wave amplitude led to a drastic decrease in the amplitude gain of the input wave.The amplification was initially strong in the cavity, but due to the roundtrip loss, overtime the pump-wave amplitude decreased along with the rate of amplification.Eventually, the roundtrip loss exceeded the roundtrip gain and the input wave attenuated.
In OPA, the pump-wave amplitude remained much higher than that of the input wave amplitude throughout the whole amplification process.Hence, the roundtrip loss due to the energy transfer from the pump wave to the input wave was neglected in Equation ( 2).Since the interaction length was small in a nanoscale cavity, the required pump-wave amplitude for OPA was high as seen in Figure 2.For comparison, Figure 3a illustrates the necessary pump-wave amplitude for high-gain OPA in a microscale cavity.Based on the OPA dynamics, the required pump-wave amplitude for OPA was much lower for a microcavity.It was indeed this requirement of an ultrahigh pump-wave amplitude that made OPA quite difficult to realize at the nanoscale.The sharp reduction in the amplitude gain factor via a small decrease in the pump-wave amplitude for a nanoscale OPA process is shown in Figure 3b.Notice that the OPA gain factor was also sensitive to the nonlinearity of the interaction medium.These sensitivities in a cavity-based OPA process required the use of an optimization procedure for gain enhancement and feasibility based on the given wave and cavity parameters.

Computational Investigation
The empirical results suggested that OPA in a nanoscale cavity required an extensive amount of pump energy that corresponded to a pump-wave amplitude on the order of 10 10 V m À1 for ultrashort pulses.This was equivalent to an optical intensity of 10 18 W m À2 , which was practically hard to generate and would induce optical damage in the interaction medium.However, the empirical formulation assumed that the pump-wave amplitude was not depleted throughout the input wave amplification in the interaction medium, and that perfect phase-matching occurred between the input and pump waves.Furthermore, the frequency dependence of the electric susceptibility was not considered in the empirical formula.In this section, we investigated the OPA process through an elaborate wave-optics formulation, such that frequency dependence, pump-depletion, and phase-mismatch were all accounted.Most importantly, we analyzed the frequency dependence of the nonlinearity coefficient via computational implementation of the electric field wave equation and the corresponding nonlinear Lorentz dispersion equation for modeling the electric polarization term that was associated with energy transfer from the pump-wave-energized cavity to the input wave.25] We would focus on the numerical analysis of nanoscale semiconductor cavities (Figure 4) with resonance frequencies in the near-infrared region, where optical nonlinearity often had a spectral peak.
For computational analysis, the input and pump waves (E in and E p ) were excited from the left cavity edge x ¼ x exc and the pump-wave frequency f p ðkÞ (at search iteration k) was varied within the range 10 THz < f p ðkÞ < 1000 THz.
A in and A p : input and pump wave amplitudes; f in and f p : input and pump wave frequencies.
The electric field wave equation that models the co-propagation of both waves (total wave) is given as whereas the time variation of the polarization density ðP in þ P p Þ of the total wave (E in þ E p ) is expressed by the nonlinear Lorentz dispersion equation: P in/p : polarization density induced by the input/pump field, N: bound charge density, ω r : angular resonance frequency, ε 0 : free space permittivity, ε ∞ : background permittivity, Γ: polarization damping rate, E in : input wave electric field, E p : pump wave electric field, σ: conductivity, d: atom diameter, e: unit charge, x: spatial coordinate, t: time, m: electron mass, and μ 0 : free space permeability.
To attain the input wave amplitude as a function of time, one first needed to solve for the pump-wave amplitude under the assumption that the input wave initially did not exist within the cavity, and then subtracted the resulting equations from Equation ( 5) and ( 6), which represent the propagation of the total wave.Hence, if one removed the input wave from the cavity, the wave dynamics within the cavity was expressed solely in terms of the pump wave as follows: Subtracting Equation ( 7) and (8) from Equation ( 5) and ( 6) yielded the set of equations that modeled the propagation of the input wave while it was co-propagating with the high-intensity pump wave: Then came the most important problem of identifying the optimal pump frequency f p, opt .This required a search algorithm that adjusted the pump frequency within the range 10 THz < f p < 1000 THz such that the input wave amplitude was maximized at a desired excitation frequency.Therefore, our objective function F was the spectral area that was confined around the quasi-monochromatic input wave excitation frequency f exc within a bandwidth Δf , where Δf was the bandwidth of the input wave at the excitation point x ¼ x exc .Here, we would use the Gradient-Descent algorithm to search for the optimal pump frequency within the intended frequency range, which is outlined as follows: Fðf p ðkÞ Þ∶ objective function, μ k : step size, x 0 ∶ computation point in space, and k: iteration number.Ω∶ dummy frequency variable, ΔT: simulation duration, t: time, f exc : excitation frequency, and ΔT Â Δf ≈ 0.5 (simulation duration À bandwidth product).
The step size μ k was chosen at every iteration based on the sufficient increase criteria, which states Notice that we introduced penalty coefficients to the objective function to keep the pump frequency in the desired range of investigation.
Hence, to model the OPA of an input wave for a super-gain amplification inside a nanoscale cavity, one needed to solve Equation ( 7)-( 11) simultaneously as these equations were coupled to each other.The easiest way to solve these equations was to discretize them based on the FDTD method. [23]The resulting discretized equations are expressed at each search-step k as follows: k∶ Iteration counter, x: spatial coordinate, and t: time.Eðx, tÞ ¼ EðiΔx, jΔt, Þ !Eði, jÞ, i ¼ 1, 2, : : : N, j ¼ 1, 2, : : : The values of the penalty coefficients δ and ζ were set according to the frequency-range precision of the problem.In our computations, the spatial discretization was fixed at Δx ¼ 20 nm and the temporal discretization was fixed at Δt ¼ 0.05f s.One should be careful while discretizing Equation ( 7)-( 10) as the Courant stability criterion restricted the value of the time step Δt.According to the criterion, Δt < Δx=v p had to be satisfied (where v p is the phase velocity of the wave).The violation of the criterion led to the accumulation of the discretization error in time which led to instabilities and inaccurate results.Detailed mathematical description on FDTD discretization is available in ref. [23].
Once the optimum pump frequency that maximized the input wave amplitude was found, we performed a parallel investigation on the associated electric energy density W e and the secondorder nonlinearity ρ 2 to understand the dynamics behind high-gain amplification.These quantities were evaluated using the following formulations: Total wave equation ( 21) -order nonlinear polarization densityÞ ( 25) Here, the computation of the third-order nonlinear polarization density was omitted as its contribution to the overall polarization density was much smaller than the second-order polarization density.

Results
For the computational investigation, we have considered a 700 nm long cavity with a typical semiconductor interaction medium whose resonance frequency is in the near-IR region.The following wave and material parameters are used for the computations: As the parameters indicate, a near-infrared input wave is amplified at 200 THz using a high-amplitude pump wave whose frequency is to be selected for gain maximization.The material parameters are chosen as the typical values for well-known semiconductors, and the wave propagation takes presence inside a cavity whose interaction-medium length is measured in nanometers.The pump-wave frequency is adjusted using Equation (11).Notice that the pump-wave amplitude is selected to be two orders of magnitude lower than the required amplitude for significant OPA based on Figure 2. Hence, without dispersion engineering, no optical amplification would take place here.Figure 5 shows the maximum input wave amplitude (gain factor) that has been reached in the cavity within a duration of 33 ps, with respect to the pump-wave frequency.As illustrated in Figure 5, only for the pump-wave frequencies of 530 and 540 THz, we observed a very strong amplification (the resolution of the frequency sweep is 10 THz).This indicates that nanoscale OPA is indeed possible for relatively low-pump-wave amplitudes under certain pump-wave frequencies.The reason for this is that at such frequencies, a super-nonlinear response is induced via a simultaneous enhancement of the accumulated electric energy density and the second-order nonlinearity.This occurrence is explained in Table 2 where energy density and nonlinearity are computed based on Equation ( 20)- (25).For f p ¼ 540 THz, the energy density and the second-order nonlinearity are concurrently high, which maximizes the rate of energy transfer to the input wave and enables strong OPA even under such a short-interaction length.Therefore, high-gain input wave amplification can take place around this pump-wave frequency.Figure 6 shows the time variation of the input wave amplitude within the nanoscale cavity under the optimum pump-wave frequency of 540 THz.As expected, the input wave amplitude originally increases due to the initially strong intracavity pump-wave amplitude.Though, overtime the pump-wave amplitude decreases beyond the threshold level for input wave amplification as a result of roundtrip loss due to dielectric, conduction, and reflection losses.
In Figure 6, the equivalent second-order nonlinearity (ρ), which is necessary to be detected for comparing computational and empirical results, is found to be ρ ¼ ð3.2 Â 10 À18 Þ C V À2 at the optimal pump frequency of 540 THz (and also for all the other pump frequencies in Table 2) using Equation ( 21)-( 25), considering a peak input wave amplitude gain of 2 Â 10 8 within an input wave bandwidth of 0.5 THz via the parameters given earlier.Consequently, the amplitude variation of the input wave is computed over time for both the computational and the empirical formulations (via Equation ( 2), ( 3), ( 7)-( 10)).The results mostly agree for the entire duration in the cavity.
Table 3 shows the resulting gain spectrum of the input wave under the given cavity parameters based on the optimum pumpwave frequency of 540 THz.Evidently, the input wave amplitude (gain) remains high within the entire range of investigation (100-900 THz) due to the concurrent maximization of the energy and the optical nonlinearity.
For a better understanding and verification of the attained results, the outlined gain enhancement procedure is reinvestigated based on an actual semiconductor.Hence, the computations are repeated for a gallium-arsenide-based nanoscale cavity with the following parameters: Spatial and temporal discretization range for the simulation: 0 ≤ x ≤ 700 nm, 0 ≤ t ≤ 10ps, relative permittivity: (ε r ) = 13 (μ r = 1), atom diameter d = 0.3 nanometers, interaction medium range: 0 nm < x < 700 nm, polarization decay rate: Γ = 10 12 Hz, transition (bandgap) frequency = 345 THz, conductivity = 10 À4 Sm À1 , left cavity wall reflectivity: R Left = 1, right cavity wall reflectivity: R Right = 0.95, left cavity wall location: x = 0 nm, right cavity wall location: x = 700 nm, and penalty coefficients: Upon examining Figure 7 and Table 4, one can see the same pattern for high-gain OPA concerning energy density and second-order nonlinearity.In this case, high-gain OPA can be realized around f p ¼ 330 THz where the energy density and the second-order nonlinearity are simultaneously maximized.
Note that for this case, due to the high roundtrip loss in the nanoscale cavity, the input wave starts to attenuate early.Table 2. Gain factor of the input wave under the given cavity parameters for a pump-wave frequency, total wave-induced energy density, and second-order nonlinearity computed at different iterations of Equation (11).The corresponding gain spectrum for the input wave based on the optimal pump frequency is shown in Table 5.As before, the gain factor remains high within the entire investigation bandwidth 100 THz < f pump < 900 THz.

Discussion
The fluctuations in the numerical input wave amplitude and the discrepancy between the computational and experimental results in Figure 6 and 7 are significant.This occurrence, we believe, is mainly because our numerical model has limitations regarding our available computational resources.First, we only use the polarization terms up to the third-order and neglect higher-order terms.The addition of higher-order terms would potentially decrease the mismatch between the experimental and the numerical results.There is also the issue of limited spatial resolution which could be another contribution to the discrepancy.Another important reason is the assumed excitation synchronicity in the numerical model.Both the input wave and the pump wave are excited at t = 0s in our simulations.Within the cavity, the pump wave exhibits amplitude fluctuations as it builds up in the cavity due to interference.Therefore, the input wave draws energy from the pump wave while the pump wave is fluctuating in amplitude during its buildup period in the cavity.We believe this is the reason for fluctuations in the numerical input wave amplitude.Whereas the experimental results assume that the pump-wave amplitude is already built up and reached to a steady state (hence the input wave would not fluctuate during the energy transfer), as it is impossible to excite the input and the pump waves at the exact same instant in such temporal scale.Table 3 and 5 indicate that once the optimum pump-wave frequency is selected, the input wave gain factor remains high over the entire infrared-ultraviolet spectral range.This is a huge Table 4. Gain factor of the input wave under the given cavity parameters for a pump-wave frequency, total wave-induced energy density, and second-order nonlinearity computed at different iterations of Equation (11).advantage of OPAs over lasers, as for lasers the amplification bandwidth is often quite narrow.Furthermore, the achieved OPA gain factor is on the order of 10 8 for pump-wave amplitudes of the same order, which is much higher than that of lasers in the nanoscale under similar pump intensities.In our computations, the required amplitude for high-gain OPA in a nanoscale cavity is found as 5 Â 10 8 , which corresponds to an optical intensity of 2.8 Â 10 14 W m À2 .Given that the intensities of sharply focused short pulses are extremely high due to spatiotemporal condensation, such optical intensities are quite practically attainable under precise focusing of ultrashort (up to 100 ns) laser beams down to a cross-sectional area on the μm 2 scale, which is the typical crosssectional area scale for nanoscale optical cavities.It is important to note that in this study we have assumed the cavity loss to be small, otherwise the OPA process could not have been achieved for the same pump-wave amplitude as OPAs are much more sensitive to cavity losses compared to lasers, [26][27][28][29] which is another drawback of OPAs apart from the requirement of high optical pump intensities.Also, the whole investigation has been carried out in 1D, which allowed us to investigate only the amplitude but not the beam cross section.A 2D or 3D FDTD analysis can allow one to investigate the beam characteristics alongside the amplitude, but naturally at a much higher computational cost.Table 2 and 4 show that the main condition for high-gain OPA in a nanoscale cavity is the concurrent maximization of the intracavity energy and the second-order nonlinearity.The upper left side of Figure 8 shows that when the pump wave does not energize (interact through strong linear polarization) the cavity, the input wave remains unamplified even if the second-order polarization (nonlinearity) is strong.[32][33] Finally, the lower portion of Figure 8 shows that the simultaneous presence of a highnonlinear polarization density and a high intracavity energy density enables a high-gain amplification of the input wave.It is worth mentioning that although the pump power is immense, the cavity material remains undamaged due to the pump-pulse being ultrashort.Optical breakdown usually occurs in the intensity scale of 10 15 W m À2 for pulses below 100 ps. [34]Therefore, while longer pulses with high intensity often lead to optical breakdown, this is rarely the case for ultrashort pulses.To summarize the discussion, the pump wave transfers its energy to the cavity, and the cavity transfers its stored energy to the input wave under strong nonlinear coupling, which enables nanoscale OPA inside a cavity, supporting previous observations on small-scale high-performance OPAs. [35,36]

Conclusion
Although OPA is quite insignificant in the nanoscale based on the empirical results, our computational results have shown that high-gain OPA can be realized in a nanoscale cavity at a feasible pump-wave amplitude for certain pump-wave frequencies where the energy density and the second-order nonlinearity are simultaneously enhanced.Once the optimal pump-wave frequency is selected for co-excitation with the input wave based on the dispersion characteristics of the underlying intracavity medium, the input wave can be amplified over the entire infrared-visibleultraviolet spectral range with a super-high-gain factor.This finding indicates that nanoscale cavity-based OPA is a highly programmable process concerning the maximization of the nonlinear constructive interference, which enables strong OPA for much lower pump-wave excitation amplitudes that are otherwise not sufficient for input wave amplification.Hence, once the cavity-based OPA process is optimized in the aforementioned way, OPA can be a better alternative to lasers for nanoscale optical amplification.

Figure 1 .
Figure 1.Small-scale on-chip cavity under input and pump-wave excitations for OPA inside a nonlinear medium.

Figure 2 .
Figure 2. Input wave (120 THz) amplitude gain versus roundtrip in a 700 nm long cavity for various pump-wave (180 THz) amplitudes (V m À1 ) in gallium-arsenide.The threshold pump amplitude for high-gain OPA is very high.

Figure 3 .
Figure 3. a) Input wave (120 THz) amplitude gain versus number of roundtrips in a 300 μm long microcavity under an intense pump wave (180 THz), with κ ¼ 0.001%.b) Input wave amplitude gain versus pump-wave amplitude for different interaction-medium nonlinearities (C V À2 ) within a 700 nm long cavity under the same parameters.A microscale cavity requires much less pump amplitude (V m À1 ) than a nanoscale cavity for significant OPA.

Figure 4 .
Figure 4. Numerical discretization of a nanoscale cavity under input and pump-wave excitations for OPA.

Figure 5 .
Figure 5. Input wave peak amplitude (computed within an input wave bandwidth of 1 THz) versus pump-wave frequency.

Figure 6 .
Figure 6.Input wave amplitude variation inside the cavity versus time based on computational and empirical formulations.

Figure 7 .
Figure 7. a) Input wave peak amplitude (computed within an input wave bandwidth of 1 THz) versus pump-wave frequency.b) Comparison of empirical and computational results based on the time variation of the input wave amplitude within the cavity.

Figure 8 .
Figure 8. Conditions for high-gain OPA in a nanoscale cavity.High-gain OPA requires simultaneous maximization of the intracavity energy density and the nonlinear coupling rate.

Table 3 .
Gain spectrum of the input wave under the given cavity parameters for a pump-wave frequency of 540 THz, pump-wave amplitude of 5 Â 10 8 V m À1 , and a cavity length of 700 nm.

Table 5 .
Gain spectrum of the input wave under the GaAs-based cavity parameters for a pump-wave frequency of 330 THz, pump-wave amplitude of 6 Â 10 8 V m À1 , and a cavity length of 700 nm.f input[THz]f pump[THz]