Numerical Study of Resonant Optical Parametric Amplification via Gain Factor Optimization in Dispersive Microresonators

: The achievement of wideband high ‐ gain optical parametric amplification has not been shown in micrometer ‐ scale cavities. In this paper we have computationally investigated the optical parametric amplification process in a few micrometer ‐ long dispersive microresonator. By performing a gain medium resonance frequency dependent analysis of optical parametric amplification, we have found that it is possible to achieve a wideband high ‐ gain optical amplification in a dispersive microresonator. In order to account for the effects of dispersion (modeled by the polarization damping coefficient) and the resonance frequency of the gain medium on optical parametric amplification, we have solved the wave equation in parallel with the nonlinear equation of electron cloud motion, using the finite difference time domain method. Then we have determined the resonance frequency values that yield an enhanced or a resonant case of optical parametric amplification, via gain factor optimization. It was observed that if the microresonator is more dispersive (has a lower polarization damping coefficient), then there are more resonance frequencies that yield an optical gain resonance. At these gain resonances, a very wideband, high ‐ gain optical amplification seems possible in the micron scale, which, to our knowledge, has not been previously reported in the context of nonlinear wave mixing theory.


Introduction
Parametric amplification has been studied extensively in the context of nonlinear optics. It has been well established that we can amplify a relatively low power input wave, by using another wave of high intensity, which is called the pump wave, in a nonlinear medium [1][2][3][4]. This theory has been mostly investigated experimentally rather than computationally. This is because the concept is almost always studied in the micrometer or nanometer wavelength range, and yet the required medium length to observe any significant nonlinear effect is in the millimeter or centimeter range, which requires an extremely high computational cost for obtaining meaningful results. Furthermore, based on the current concept of parametric amplification, it is impossible to achieve a wideband nonnegligible wave amplification in a few micrometer-sized gain medium or a cavity, as the required gain medium length for significant parametric amplification is mostly on the order of centimeters. For this reason, the concept of parametric amplification is not feasible to be used for optical microsystems. However, if it can be achieved in the micrometer scale, super gain optical parametric amplification can lead to much more efficient macroscale high power devices by being engineered to operate in an array form to maximize optical interference, such as laser beam welding machines and petawatt lasers that are used for particle acceleration and fusion reactions. Another potential technological advancement can be in the field of optical antennas via the generation of a supercontinuum for achieving ultra-wideband operation.
In this paper, we have carried out a computational analysis that aims to show that it is possible to amplify a low power input wave by mixing it with an intense pump wave of ultrashort duration, in a wide range of frequencies and with a very large gain coefficient, inside a low-loss microcavity of several micrometers of length. This can be achieved by using a gain medium whose resonance frequency matches with one of the gain resonances of the amplification in a microcavity. The gain resonances of an optical amplification in a microcavity can be determined by solving the wave equation in parallel with the equation of nonlinear electron motion. For this reason, we will start our analysis by investigating wave propagation in nonlinear dispersive media.

Wave Propagation in Nonlinear Dispersive Media
Nonlinearity arises when at least one of the waves that propagate in a medium has a very high intensity. Such high intensities are only possible with very short duration pulses, such as the pulses of a mode locked laser, which have durations on the scale of picoseconds or femtoseconds [5,6]. When high intensity pulses have very short durations, which is almost always the case, we must account for the wave dispersion, as the impulse response of the charge polarization density of many dielectric media last much longer in duration than the pulse durations of such high intensity pulses [7][8][9][10]. One can solve the wave equation in parallel with the nonlinear equation of electron cloud motion for a nonlinear dispersive medium. This is because the parameters such as the resonance frequency, damping coefficient, atom density, and atomic diameter have known values for most solids, which makes the simulation results much more meaningful and precise. In order to determine the time variation of the electric field in a nonlinear dispersive medium, we need to solve the following two equations [1] (see Figure 1): : Polarization density C/m , : Electron charge, m: Electron mass, : Electric field .
In Equation (2) we have made an expansion up to the third order of the nonlinear charge polarization density as higher order terms will be negligibly small. For a dielectric medium, the electrical conductivity can be assumed as negligible 0 . Some typical values for solid dielectric media are [1]. Consider the wave that has a very high amplitude, without the presence of the low amplitude wave , the pair of equations that describe the propagation of in a nonlinear dispersive medium is given as : Charge polarization density due to the electric field E ; : Charge polarization density due to the electric field E ; : Background infinite spectral band permittivity Now assume that and are propagating together in the same nonlinear dispersive medium (see Figure 2). In this case the pair of equations that represent the total electric field propagation is given as We want to determine the propagation of the low amplitude wave in the presence of the high amplitude wave ; in other words we want to determine the propagation of when there is an energy coupling from as a result of the nonlinear interaction. In order to do that we subtract Equation (3a,b) from Equation (4a,b) respectively, which gives us From Equations (3a,b) and (5a,b) we can see that and are coupled to each other. Based on Equation (5a,b), we will investigate whether it is possible to amplify the low power electric field , by drawing energy from a low loss microcavity that is energized by the high-power electric field .

Optimization of Optical Parametric Amplification Gain Performance
Goal: Maximize the stimulus wave magnitude | | with respect to the resonance frequency of the interaction medium, with the high power pump wave and the low power stimulus wave time variations at the cavity input (excitations) respectively being (see Figure 3) , cos 2 : , The equations that need to be solved at each iteration of the optimization process are given as At each iteration, we use an update equation based on Newton's method.

, …
With the step size being updated at each iteration as (optimized by trial and error for fast convergence). We choose the left cavity wall as an optical isolator and the right cavity wall as an optical bandpass filter.

Finite Difference Time Domain Formulation Based Solution of the Gain Factor Optimization Problem in Optical Parametric Amplification
We can discretize Equations (3a,b) and (5a,b) using the finite difference time domain method as shown in Equations (6a,b) and (7a,b), at each iteration of the optimization problem. Our first aim is to discretize Equation (3a,b) and solve for , 1, i.e., the value of at a given point at the next time step. Since is coupled to , we first solve for , 1, and then substitute it into the equation for , 1, . We keep on solving these two equations iteratively for all time steps and for all points in the spatial domain of a given one dimensional problem. For a higher accuracy of the resulting solution, we choose ∆ and ∆ as small as possible [11]. Then we discretize Equation (5a,b) and substitute the value of , , obtained from Equation (6a,b) to solve for , 1, in Equation (7a,b). Finally, we modify the value of the resonance frequency (see Equation (8a)) based on Newton's method, and we repeat this whole procedure for each iteration of the optimization process until the desired gain factor is attained.
x: Spatial coordinate, t: Time, k: Iteration number, , , ∆ , ∆ , → , , By solving these six equations simultaneously, along with the initial condition and the boundary conditions of a given optical parametric amplification problem, we can maximize the gain factor. In order to test our computational model, we have compared our computational results with the experimentally verified theoretical results in the well-established context of nonlinear sum frequency generation, in the Appendix A. Now we move on to the discussion of the simulation results.

Finding the Optimal Resonance Frequency for High Gain Optical Parametric Amplification
Assume that a 440 THz stimulus wave and a high power 282 THz pump wave (Nd:YAG laser field) are propagating inside a low-loss cavity that has two reflecting walls. The reflecting wall on the left side can be thought as an optical isolator and has a reflection coefficient of 1, the one on the right side represents a switch controlled optical band-pass filter with a frequency dependent reflection coefficient . Both waves are generated at x = 0 μm and at the time instant t = 0 (see Figure   Problem definition: Find the optimum resonance frequency , that maximizes | | in the cavity, for 10 THz < 1000 THz (THz to UV), for 0 μm 10 μm, 0 30 ps, such that Switch controlled optical bandpass filter condition: Full reflection at 10 μm for t 30 picoseconds, frequency dependent reflection at 10 μm after t = 30 picoseconds;

ps
Using the formulation given in Equation (8a,b), the maximum stimulus wave amplitude that has been reached in the cavity (for 0 < t < 30 ps) is determined as | | 3.3 10 V/m, which corresponds to 519.79 THz 520 THz (see Table 1). However, there might be other gain resonances, especially around the major gain resonance 520 THz. Therefore, finding the optimum gain factor and the optimum resonance frequency is not enough by itself. To determine all the other gain resonances, we also need to sweep the resonance frequency from 10 THz to 1000 THz in 10 THz increments, and investigate the reason behind the high gain at this resonance frequency by analyzing the stored electric energy density and the intracavity charge polarization density of the pump wave. As we can see from Table 2, there are several resonance frequencies that yield a high gain for the stimulus wave, for a 1.064 μm (282 THz) Nd-YAG pump wave. The amplification peaks occur between the resonance frequencies 350 THz and 600 THz . This suggests that we can achieve strong optical parametric amplification in a micrometer sized cavity using a 282 THz pump wave, if we use a gain medium that has the correct or optimal resonance frequency for amplification. In this case the optimal resonance frequencies are as shown in Table 2. The corresponding optimal resonance frequency values of the amplification peaks are especially determined by the intracavity charge polarization density (and to a lesser extent on the electric energy density) created by the pump wave. If we look at Figure 5, the charge polarization density that is created by the pump wave is very high at the gain resonances, and as the charge polarization density decreases at higher frequencies, only less significant (minor) gain peaks are observed. This makes perfect sense because the charge polarization density created by the pump wave couples to the stimulus wave electric field intensity (see Equation (5b)) and acts as a source for the stimulus wave thereby amplifying it. Although the maximum gain factor of the stimulus wave also depends on the stored electric energy density in the cavity, if there is not enough charge polarization density in the cavity to couple this stored electric energy to the stimulus wave, amplification does not occur. This can be seen from Figures 5 and 6. There are many high electric energy density peaks beyond f 600 THz, however, since there is not enough charge polarization density that can transfer this energy density to the stimulus wave, amplification is not significant.
Since the resonance frequency values that yield a high stimulus wave gain factor is between f 350 THz and 600 THz, using any dielectric material that has a resonance frequency or an emission peak in this range (especially in the vicinity of the boldly indicated frequencies at Table 2), is suitable for achieving high gain amplification of a stimulus wave of any frequency, under a 282 THz Nd:YAG pump wave, such as the Bithiophene based red fluorescent light emitting material called BTCN [12], which has a resonance frequency value of 470 THz with a FWHM bandwidth of 80 THz. BTCN is recently suggested as a good red-light emitting material for organic light emitting applications.

Effect of the Damping Coefficient on Stimulus Wave Amplification
As the damping coefficient (polarization decay rate) increases, the amplification of the stimulus wave gets weaker (Figures 7-11). This is because the amount of stored electric energy density in the cavity depends on the damping coefficient. As the charge polarization density decays quicker due to high , the stored electric energy will have less time to accumulate in the cavity and the stimulus wave will have less time to get amplified. If we look at Figures 7-11, the gain resonances eventually disappear as the damping coefficient is increased. For 10 Hz, the amplification of the stimulus wave becomes insignificant.    The amplification for 10 Hz is almost insignificant as shown in Figure 11. At this damping rate, the magnitude of the stored electric energy density is not enough to amplify the stimulus wave significantly, as the pump wave magnitude is quickly damped to a level at which nonlinearity is lost and energy coupling does not occur. Figure 12 illustrates the sharp decrease of the gain factor for 10 Hz.

Analyzing the Gain Spectrum of the Stimulus Wave at the Optimal Resonance Frequency
Now let us compute the gain spectrum of the stimulus wave at the optimal resonance frequency of 520THz (see Figure 7 and Table 2), for 1 10 Hz. The configuration is shown in Figure 13: Our problem: For the gain maximizing resonance frequency of f 520 THz, find the maximum stimulus wave amplitude (gain factor) , in the cavity for each stimulus wave frequency , in the range 10 THz < 1000 THz (THz to UV), for 1 10 Hz , 0 30 ps , such that The stimulus wave is supplied to the cavity at t = 0 as a quasi-monochromatic wave. For a given initial (at t = 0 s) stimulus wave frequency, the center frequency of the band-pass filter is adjusted to be at the same frequency with the initial stimulus wave frequency. By doing so, we can observe how much gain can be obtained from the cavity for each initial stimulus wave frequency. Since the resonance frequency of f 520 THz yields the maximum stimulus wave gain, we choose this frequency to compute the gain spectrum of the stimulus wave. Then we sweep the stimulus wave frequency from 10 THz to 1000 THz in 10 THz increments. The amplification (gain) spectrum of the stimulus wave for = 520 THz is plotted below in Figure 14. As evident from Figure 14, the gain spectrum of the stimulus wave extends from the THz region to the UV region of the electromagnetic spectrum. This means that we can obtain strong output radiation at any frequency in the given band. Note that there is relatively low gain around the resonance frequency = 520 THz as the dielectric absorption is strong around the resonance region. (1) Based on our computations, changing the length of the cavity, or the length of the gain medium, does not change the spectral gain resonance locations. (2) The amplitude (3 10 V/m of the pump wave is the typical electric field amplitude of a milimeter focused 1.064 μm wavelength ultrashort Nd:YAG laser beam, which corresponds to an optical intensity of 10 GW/cm . Dielectric breakdown in dispersive media occurs at an optical intensity level of TW/cm and beyond. Therefore, at this amplitude the pump wave would not induce dielectric breakdown and would not pose an optical hazard to the medium of interaction [13][14][15]. (3) The temporal walk-off between the interacting waves that stems from the dispersion of the medium limits the effective interaction length. To counter-act this effect, we should ensure that the reflection losses at the cavity walls are minimized. The optical isolator that is used as a cavity wall should be of high quality to maximize intracavity reflections ( 1 . The bandpass transmission filter reflection coefficient must be maximized at the optical transmission stopband, so that the stored electric energy is retained in the cavity to support the high-gain amplification of the stimulus wave. In this paper we have assumed that the optical isolator (left cavity wall) reflection coefficient is 1 from the inside of the cavity (ideal optical isolator), and the reflection coefficient of the filter (right cavity wall) is close to 1 at the transmission stopband. When the reflection losses are minimized, the number of round trips in the cavity for which the nonlinearity is retained, is maximized and the reduced effective interaction length due to temporal walk-off between the interacting waves is compensated.
Another precaution to counter-act the temporal walk-off effect is to retain the high pump wave amplitude to keep the nonlinearity and nonlinear coupling as high as possible. This can be achieved by choosing an interaction medium with a low polarization damping coefficient. When the polarization damping coefficient is low, the stored electric energy is preserved for longer durations, which increases the number of round trips in the cavity for which the nonlinear energy coupling process is sustained. This also compensates for the reduced effective interaction length. The effect of the polarization damping coefficient on optical parametric amplification will be discussed in more detail in the next section.

Effect of the Damping Coefficient and the Mean Cavity Wall Reflection Coefficient on Optical Parametric Amplification
Consider the following resonant optical parametric amplification case, in which 800 THz is the spectral location of a gain resonance for a 100 THz pump wave. The input wave and the pump wave are propagating inside a cavity that has two reflecting walls on the left and right side. Both waves are generated at x = 0 μm at time t = 0 s (see Figure 15).  The variation of the maximum input wave amplitude that has been reached between 0 < t < 50 ps is plotted with respect to the damping coefficient and the mean cavity wall reflectivity in Figures 16 and 17. If we look at these figures, it is obvious that as the damping coefficient decreases and/or as the mean cavity wall reflectivity increases, the amplification efficiency improves. In Figure 16, we can clearly see that the maximum amplitude of the input (or stimulus) wave decreases sharply for 2 10 Hz . This means that the gain factor of the input wave to be amplified can be significantly enhanced by choosing an interaction medium with a low damping rate. Moreover, it is interesting to observe that the damping coefficient has a threshold value for high-gain amplification, in this case 2 10 Hz is the threshold value for high-gain amplification. Around 10 Hz, we can see that the amplification of the input wave has reduced by several orders of magnitude. In Figure 17, we can see that the mean reflection coefficient of the cavity walls plays a major role in achieving high-gain optical parametric amplification. Below a mean reflection coefficient of = 0.79, the amplification becomes insignificantly small as compared to the amplification for > 0.9, and the gain factor (maximum reached amplitude for 0 < t < 50 ps) of the input wave gradually increases as the mean reflection coefficient of the cavity walls increases. As previously explained, this is because the stored electric energy is better preserved in the cavity for a larger mean reflection coefficient, which enables the amplitude of the pump wave and hence the nonlinearity, to be retained for a longer duration, and a higher gain factor of the input wave to be obtained [15][16][17][18]. The critical (threshold) values of the damping coefficient and the mean reflection coefficient of the cavity walls, for high-gain amplification, are investigated for three different values of the resonance frequency of the interaction medium, the input/stimulus wave frequency, and the pump wave frequency, as shown in Table 3. Based on these critical/threshold values, we suggest the mean reflection coefficient of the cavity walls to be greater than 0.85 ( > 0.85) and the damping coefficient of the dielectric interaction medium to be lower than 10 Hz 10 Hz .

Conclusions
Recent studies on achieving high-gain optical parametric amplification in microresonators have focused on materials that display a strong nonlinearity under high intensity excitation. For instance, the recent studies mentioned in [19][20][21] have investigated lithium niobate (LN) for that matter. The study mentioned in [19] has focused on the strong nonlinearity of lithium niobate in a microcavity with a high quality factor, and the study mentioned in [20] has focused on optimal mode matching in a periodically poled waveguide. These studies have reported high harmonic generation and parametric amplification efficiencies. However, none of the recent studies have investigated the possible existence of optimum resonance frequencies for a given pump wave excitation, that can simultaneously maximize the stored electric energy density and the nonlinear coupling coefficient (pump wave polarization density) and may enable ultra high-gain amplification in microresonators with a cavity length of as small as a few micrometers.
Through rigorous computational analysis that involves the maximization of the nonlinear constructive interference and the nonlinear coupling coefficient, we have shown that a very wideband, super-gain optical parametric amplification is possible in a 10 µm long microcavity. This interaction medium length is even much smaller than the recently suggested parametric amplification waveguide of 300 μm length mentioned in [21], as our analysis directly focuses on maximizing both the nonlinear energy coupling coefficient and the stored electric energy density in the cavity, rather than the stored electric energy (quality factor) alone, which are both critical in achieving a super-gain amplification.
As a summary for resonant optical parametric amplification by nonlinear wave mixing in a microresonator under a 282 THz Nd:YAG pump wave excitation, the following requirements must be met: • The cavity should be low loss. This is satisfied in our computations by choosing a dielectric gain medium 0 , and by adjusting the reflectivities of the cavity walls to be very high.

•
The damping coefficient (polarization decay rate) of the gain medium should be low ( 10 Hz).

•
The resonance frequency 0 of the gain medium must match with one of the resonance frequencies that are found to maximize the stimulus wave gain inside the cavity. In the case of a 282 THz Nd:YAG pump wave excitation, the resonance frequency of the interaction material should reside between 350 THz and 600 THz (350 THz < 600 THz . • Bithiophene based red fluorescent light emitting material called BTCN, which has a resonance frequency value of 470 THz with a FWHM bandwidth of 80 THz, is a suitable interaction medium for resonant optical parametric amplification using a 282 THz Nd:YAG pump wave excitation. Once these requirements are satisfied, it is possible to amplify a low power input (stimulus) wave with a very large gain coefficient, in a wide range of frequencies, inside a low loss microresonator of several micrometers of length.

Appendix A. Verification of Our Computational Model
We want to test the accuracy of our computational model by using the experimentally verified theoretical formula in the well-established context of sum frequency generation via nonlinear wave mixing, which is the basis of optical parametric amplification [1,4], in the following example.

Example A1. Nonlinear sum frequency generation (frequency up-conversion).
This example is about the generation of a higher frequency component , by mixing two monochromatic waves with frequencies and , such that . In order to achieve this, at least one of the waves must have a high intensity, so that nonlinearity arises and wave mixing occurs.
The high amplitude pump wave is generated at x = 2.5 μm. It has an amplitude of V/m and a frequency of 180 THz.

μm,
sin 2 1.8 10 V/m The input wave is generated at x = 2.5 μm. It has an amplitude of V/m and a frequency of 120 THz. = Input wave amplitude = /100 ( ≪ . Figure A2. Comparison of the frequency up-conversion efficiencies for = 300 THz and d = 6.3 10 , versus the pump wave amplitude.