Segmented Composite Optical Parametric Amplification

Abstract

We propose a novel optical parametric amplification scheme that combines quasi-phase-matching with a composite pulse approach that involves crystal segments of specific lengths. The presented scheme highly increases the robustness of the frequency conversion against variations of the nonlinear coupling and of the pump, idler, or signal wavelengths, and has therefore the potential to enhance high amplification and broadband operation. Simulation examples applied to LiNbO${}_3$ are given.

1. Introduction

Some optical applications require short optical pulses with large peak power, which may be obtained with the help of optical parametric amplifiers (OPAs) [1,2,3,4] that are among the most useful nonlinear optical devices. Optical parametric amplification consists of the nonlinear interaction of three waves. In this process, the two waves at the longer wavelengths—the input signal wave as well as the idler wave—gain power at the expense of the pump wave being at the shortest wavelength. In OPA, the main obstacle encountered when short pulses are used is to combine a high signal amplification and a sufficiently broad amplification bandwidth. The latter is limited because material dispersion imposes that, for a given wave interaction configuration, the exact phase-matching condition can be strictly satisfied only for a single set of wavelengths of the three waves. Presently, the most common way to achieve the combination of broadband and high amplification is to use chirped quasi-phase-matching [5,6,7,8,9,10]. Even though chirped quasi-phase-matching approaches have the advantage of being broadband, they require high pump input intensity and/or very long nonlinear crystals. Very high pump intensities have the drawback of a possible damage of the nonlinear crystal when they approach its damage threshold.

In this paper, we explore an alternative method to achieve broadband amplification bandwidth together with high amplification. The technique involves a combination of quasi-phase-matching (QPM) gratings together with a segmentation of the crystal that implements the equivalent to the composite pulses approach used in Nuclear Magnetic Resonance (NMR) to prepare given quantum states in a robust way [11,12,13] (see Appendix A). The present approach leads to a highly increased robustness of the nonlinear amplification process with respect to both the phase mismatch (associated to a change of wavelength or of temperature) and the coupling strength. Note that the approach presented here does not involve the stretching and chirping of the pulses before the parametric amplification and a final recompression, as used in Optical Parametric Chirped Pulse Amplification (OPCPA) [14,15,16,17]. Such steps are generally necessary for the shortest (few cycles) pulses and the highest pulse energies to avoid excessive third order nonlinear effects and/or a damage of the samples.

The paper is organized as following. Section 2 gives the general theory, and Section 3 describes the numerical approach and the simulation results obtained with the best crystal segmentation. Some practical examples are given for the important case of MgO-doped quasi phase-matched LiNbO${}_3$. Finally, Section 4 summarizes.

2. Theory

We start with the simmetrized coupled wave equations for collinear three-wave mixing in the slowly varying envelope approximation [18,19],

$$\begin{array}{cc}\hfill i{\partial }_z{A}_1& =\tilde{\mathsf{\Omega }}{A}_2^{*}{A}_{3}exp\left[-i\Delta kz\right],\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill i{\partial }_z{A}_2& =\tilde{\mathsf{\Omega }}{A}_1^{*}{A}_{3}exp\left[-i\Delta kz\right],\hfill \end{array}$$

(1b)

$$\begin{array}{cc}\hfill i{\partial }_z{A}_3& =\tilde{\mathsf{\Omega }}{A}_1{A}_{2}exp\left[i\Delta kz\right],\hfill \end{array}$$

(1c)

where $\tilde{\mathsf{\Omega }}=-(2{\chi }^{\left(2\right)}/\pi c)\sqrt{{\omega }_1{\omega }_2{\omega }_3/n_1{n}_2{n}_3}$ is the effective nonlinear coupling coefficient for first-order QPM, z is the position along the propagation axis, ${\omega }_j$ are the frequencies of the three involved waves, and $n_j$ are their refractive indices. Here $j=1,2,3$ refer to the signal, idler and pump waves, respectively. The quantity ${\chi }^{\left(2\right)}$ in $\tilde{\mathsf{\Omega }}$ is the effective second-order susceptibility and c is the speed of light in vacuum. The amplitudes $A_j\equiv \sqrt{n_j/{\omega }_j}{E}_j$ in (1) are proportional to the amplitudes $E_j$ of the wave electric fields; $|A_j{|}^2$ is proportional to the number of photons associated to the jth wave. Note that Equation (1) is written in a form that assumes that quasi-phase-matching is implemented and that the quasi-phase matching period is sufficiently short as compared to the interaction length. Therefore, the phase mismatch parameter $\Delta k$ already contains the mismatch compensation term associated to the periodic grating, that is,

$$\Delta k=k_1+k_2-k_3+2\pi /\mathsf{\Lambda }\equiv \widetilde{\Delta k}+2\pi /\mathsf{\Lambda }.$$

(2)

where $\mathsf{\Lambda }$ is the quasi-phase-matching period, that is, the first-order local poling period in the case of periodically poled crystals. Obviously, for the central operation wavelengths at which the device is designed, one has $\Delta k=0$. Concurrently, the true phase mismatch $\widetilde{\Delta k}=k_1+k_2-k_3$, which depends only on the wave-vectors $k_j$ of the three interacting waves, is generally quite far from vanishing. The set of Equation (1) could have been written also by using the quantities $\widetilde{\Delta k}$ instead of $\Delta k$, however in this case $\tilde{\mathsf{\Omega }}$ would need to switch its sign after each distance $\mathsf{\Lambda }/2$ and the term $2/\pi$ would need to be dropped in the nonlinear coupling coefficient.

Depending upon the initial conditions $A_j\left(z=0\right)$, different processes can arise: sum frequency generation (SFG), difference frequency generation (DFG) or OPA. Here, we consider the OPA case, we assume that ${\omega }_3={\omega }_1+{\omega }_2$ and we treat first Equation (1) in the limit of validity of the undepleted pump approximation ($A_3=$ const). In this limit the set of Equation (1) turns into

$$\begin{array}{cc}\hfill i{\partial }_z{A}_1& =\mathsf{\Omega }{A}_2^{*}exp\left[-i\Delta kz\right],\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill i{\partial }_z{A}_2& =\mathsf{\Omega }{A}_1^{*}exp\left[-i\Delta kz\right],\hfill \end{array}$$

(3b)

with $\mathsf{\Omega }=\tilde{\mathsf{\Omega }}{A}_3$ being a modified coupling coefficient. When the wave vector mismatch $\Delta k$ and the coupling coefficient $\mathsf{\Omega }$ are constant, then Equation (3) possess exact analytic solutions [20],

$$\begin{array}{cc}\hfill A_1\left(z\right)& =e^{i\Delta kz/2}\left[A_1\left(0\right)\left(cosh\left(gz\right)-\frac{i\Delta k}{2g}sinh\left(gz\right)\right)+\frac{i\mathsf{\Omega }}{g}{A}_2^{*}\left(0\right)sinh\left(gz\right)\right],\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill A_2\left(z\right)& =e^{i\Delta kz/2}\left[A_2\left(0\right)\left(cosh\left(gz\right)-\frac{i\Delta k}{2g}sinh\left(gz\right)\right)+\frac{i\mathsf{\Omega }}{g}{A}_1^{*}\left(0\right)sinh\left(gz\right)\right],\hfill \end{array}$$

(4b)

where $g=\sqrt{{\mathsf{\Omega }}^2-{\left(\frac{\Delta k}{2}\right)}^2}$ is a gain coefficient. In the case $\Delta k=0$, and for a vanishing idler wave at the input ($A_2\left(0\right)=0$), from the last equations one can easily see that both signal and idler increase initially exponentially in a parallel way as

$$\left|A_1\left(z\right)\right|\approx \left|A_2\left(z\right)\right|\approx \frac{\left|A_1\left(0\right)\right|}{2}exp\left[\mathsf{\Omega }z\right]$$

(5)

In this work, we are interested in an optimization of the signal intensity amplification factor a, defined as the ratio of the intensity of the signal wave ($A_1$) taken at distance z to its intensity at the entrance of the crystal

$$a=\frac{{\left|A_1\left(z\right)\right|}^2}{{\left|A_1\left(0\right)\right|}^2}=\frac{I_1\left(z\right)}{I_1\left(0\right)}.$$

(6)

The above argumentation, in connection with Equation (5), indicates that, when the phase-matching condition is satisfied ($\Delta k=0$), the OPA process is the most efficient. However, this is not entirely true, because the solutions (5) are derived only in the limit of the undepleted pump approximation. If one considers the depleted pump regime, the nonlinear Equation (1) has solutions in terms of Jacobi elliptic functions [21,22]. These, like trigonometric functions, are periodic functions, and thus the energy transfer oscillates back and forth between pump field and signal field. The chirped quasi-phase-matching techniques [5,6,7,8,9], in addition to the improvement of the bandwidth, eliminate the problem of back conversion and can be used even in the case of the depleted pump regime.

Here, in analogy with the technique of composite pulses from quantum physics [11,12,13,23,24,25,26,27,28], we propose to use segmented composite crystals for OPA. We note that the composite pulse analogy was already used in nonlinear optics, but for the sum frequency generation or second harmonic generation (SHG) [29,30,31]. In these cases, in the undepleted pump approximation the differential equations governing the spatial dynamics have a SU(2) symmetry [32,33], which is exactly the same symmetry possessed by quantum systems with two states [27,28]. Therefore, the mapping between two-state quantum systems and SFG in the undepleted pump regime is complete and one can use the known analytic solutions from quantum physics to find robust solutions in nonlinear optics [29,30,31]. In the case of OPA, there is no SU(2) symmetry and it is thus not possible to exploit known composite pulses analytic solutions. Instead, we are going to derive numerically solutions that achieve broadband amplification bandwidth together with high amplification in the depleted pump regime.

3. General Numerical Approach and LiNbO${}_3$ Crystal Simulations

The procedure that we track is the following. The period of the flip sign of the nonlinear susceptibility ${\chi }^{\left(2\right)}$ is such that the phase mismatch for the OPA process will be zero in Equation (2) due to QPM, resulting in a local modulation period $\mathsf{\Lambda }$ (Figure 1a). However, in contrast to the periodic design, we induce additional sign flips of the coupling coefficient at specific boundaries. In practice, at each segment boundary, there are two domains with the same orientation that merge in a single double-as-long domain, as shown in Figure 1b. The periodic sign switch of ${\chi }^{\left(2\right)}$ ensures the phase matching for OPA ($\Delta k=0$) and the additional sign switches of ${\chi }^{\left(2\right)}$ at the segment boundaries will change the sign of $\tilde{\mathsf{\Omega }}$ in the whole crystal segment in a similar fashion as in Shaka–Pines pulses from NMR [12,13] discussed in the Appendix A. We denote the intervals between two double length domains as $l_1,l_2,l_3\dots l_N$, as shown in Figure 1b. The determination of the optimum segment lengths $l_k$ is done using Monte Carlo simulations in the depleted pump regime. In practice, it consists in a maximization of the integral Q of the (normalized) amplification a over a surface of interest in the $(\Delta k,\tilde{\mathsf{\Omega }})$-space, where the values of $\Delta k$ and $\tilde{\mathsf{\Omega }}$ are in units of the reciprocal crystal length $1/L$. The Figure-of-Merit integral Q is bounded by 1 and is given as

$$Q\equiv \frac{1}{r}\frac{1}{2{\tilde{\mathsf{\Omega }}}_{max}\Delta k_{max}}{\int }_{-\Delta k_{max}}^{\Delta k_{max}}{\int }_0^{{\tilde{\mathsf{\Omega }}}_{max}}a\left(\tilde{\mathsf{\Omega }},\Delta k\right) d(\Delta k) d\tilde{\mathsf{\Omega }},$$

(7)

where r is the initial pump-to-signal photon-intensity ratio, $r\equiv I_3\left(0\right)/I_1\left(0\right)$. In our case, we have chosen $\Delta k_{max}=15/L$ and ${\tilde{\mathsf{\Omega }}}_{max}=30/L$, the optimization of the integral Q is done over 10${}^5$ random sets of the segment lengths $l_1,l_2,l_3\dots l_N$. Those corresponding to the best solutions are listed in

Table 1. Numerically found segment lengths $l_i$ (in units of total crystal length L) for composite segmented periodically poled design with N segments. The given values for $l_i$ are such as to optimize the robustness of the OPA process against variations of the nonlinear coupling coefficient and of the phase mismatch $\Delta k$.

N	Name	Segment Lengths ${\mathit{l}}_1;{\mathit{l}}_2;\mathbf{\dots };{\mathit{l}}_{\mathit{N}}$ in Units of L
3	3	0.373; 0.594; 0.033
4	4	0.303; 0.522; 0.124; 0.051
6	6a	0.293; 0.258; 0.003; 0.255; 0.124; 0.067
6	6b	0.168; 0.035; 0.345; 0.023; 0.222; 0.207
6	6c	0.223; 0.005; 0.404; 0.175; 0.113; 0.080
8	8	0.022; 0.064; 0.046; 0.205; 0.270; 0.096; 0.222; 0.075

. We have found that the use of a small number of composite segments (two, three, and partly four) do not lead to any strong improvement with respect to the standard QPM case. In contrast, already, for a moderate number of segments between six and eight, we find a significant improvement of the robustness of the amplification process. In this case, our analysis shows that there are different solutions for the optimum segment configurations (given in Table 1), which works better depending on the initial amplitude of the signal wave $A_1\left(0\right)$ (as compared to the pump wave amplitude taken as $A_3\left(0\right)=1$).

Figure 2 illustrates the performance of the composite crystals with six segments (6a from Table 1) compared to standard periodic design. The amplification values are calculated numerically from Equation (1) in the cases when $A_1\left(0\right)=0.1$, $A_2\left(0\right)=0$, and $A_3\left(0\right)=1$, and thus a pump-to-signal photon-intensity ratio $r=100$. Figure 2 shows clearly that the region of high signal intensity amplification expands strongly for the segmented composite crystal compared to standard periodic design. In other words, the composite crystals exhibit much broader acceptance bandwidths compared to a standard quasi-phase-matching. The working principle of the composite concept can be recognized directly with the help of Figure 3, which depicts the evolutions of the signal wave intensity for the conditions associated to the three positions P1, P2, and P3 given in Figure 2. Clearly, each segment boundary gives rise to a “kick” for such evolutions. By choosing the z-positions of the frontiers appropriately, one can achieve that the spatial evolutions corresponding to points in the big red area in Figure 2a get very close (nearly “in phase”) near the end of the device at distance L with a high final signal wave amplification, as seen in Figure 3a. In contrast, for pure QPM without composite segments such a “re-phasing” cannot occur, as can be recognized in Figure 3b.

The simulations of Figure 2 and Figure 3 illustrate the general approach for finding the best composite sequence as compared to standard QPM. Next, we will prove the concept further by applying these results to specific practical examples and we make the numerics for a real crystal: 5 mol. % Magnesium Oxide doped Lithium Niobate (MgO:LiNbO${}_3$). This ferroelectric nonlinear crystal possesses higher damage threshold compared to undoped LiNbO${}_3$, high nonlinear optical coefficient, broad transparency range, and is suitable for domain poling [34]. We compare the standard quasi phase matching with the composite approach for OPA when the three interacting beams share the same extraordinary polarisation (Type 0 configuration, all beams polarized parallel to crystal c-axis) associated to the largest element of the nonlinear tensor $d_{333}={\chi }^{\left(2\right)}=27$ pm/V.

The color plots in Figure 4 and Figure 5 compare the signal intensity amplification a for MgO:LiNbO${}_3$ for standard QPM ($\mathsf{\Lambda }=29.71$$\mathsf{\mu }$m) and for a composite crystal made of six segments. Figure 4 is for intermediate pump-to-signal intensity ratio r while Figure 5 is for large r. Figure 4 also shows the less optimum case where there are only three composite segments. The nonlinear susceptibility is fixed and the plots are represented for varying input pump intensity (at the fixed wavelength of 1064 nm) and for varying signal wavelength (with center at ${\lambda }_1$ = 1550 nm). Note that here, to keep the ratio r constant for each plot, the input signal intensity changes in the same way as the input pump intensity. Note also that the pump intensity (y-axis) takes the role of the coupling in Figure 2. Similarly, as a variation of the signal wavelength with respect to the central one gives rise to a phase mismatch $\Delta k$; here, the signal wavelength (x-axis) takes the role of $\Delta k$ in Figure 2. The total crystal length is $L=5$ mm. The amplification values are calculated numerically from Equation (1) in the cases when $A_1\left(0\right)=0.1$, $A_2\left(0\right)=0$, and $A_3\left(0\right)=1$ for Figure 4 and $A_1\left(0\right)=0.01$, $A_2\left(0\right)=0$, and $A_3\left(0\right)=1$ for Figure 5. Clearly, a greatly enhanced robustness and frequency bandwidth of the six-segment composite OPA compared to the standard QPM OPA can be recognized. The plots in Figure 5c compare directly the signal intensity amplification spectrum for the optimal range for composite crystal (red line) and optimal range for standard periodic design (blue dash line). These are slices at the pump intensity values of 3.75 GW/cm${}^2$ and 0.8 GW/cm${}^2$ in Figure 5a,b, respectively.

Finally, Figure 6 gives the color plots for the signal intensity amplification a as in Figure 4 and Figure 5, but for the case where the initial signal intensity is as strong as the pump ($r=1$, $A_1\left(0\right)=1$, $A_2\left(0\right)=0$ and $A_3\left(0\right)=1$). This is a highly depleted regime, and the real advantages of the segmented composite approach over the standard QPM in the depleted pump regime can be recognized by the very large red area in the two right panels. In general, a laser beam spot has an intensity distribution with strong intensity in the center of the spot and smaller intensity at the wings (Gaussian beam for example). Therefore, averaging the amplification value for the big red island of the central panel (b) in Figure 6 will lead to a significantly higher average amplification than averaging the amplification values for the oscillation islands seen for standard QPM in Figure 6a. In Figure 6, we also show that the composite approach is very tolerant with respect to errors in the segment lengths, which is typical also of composite pulse techniques in quantum physics. The color plot in Figure 6c is obtained with the sequence 8 of Table 1) like Figure 6b after adding a large random error on each segment length within $\pm 5\%$ of the segment length itself. After this procedure each segment length was renormalized by a common factor in order to maintain the same total length of 5 mm like for Figure 6b. It can be seen that the large robust red area is maintained nearly unchanged despite for the rather large allowed errors. More realistic random errors of the order of 1–2% give landscapes almost indistinguishable from the one in Figure 6b.

We can conclude from Figure 4, Figure 5 and Figure 6 that the present composite OPA approach works very well in the depleted pump case, because, in all these cases, one has a significant pump depletion, and thus a signal amplification approaching the maximum theoretically possible.

4. Summary and Conclusions

In summary, we used the similarity between the three wave mixing equations and the time-dependent Schrödinger equation to transfer concepts from quantum physics to nonlinear optics. Specifically, we have suggested to use segmented composite crystals for optical parametric amplification in analogy with the composite pulses in NMR and quantum optics. The approach used here is based on sign-alternating dual-compensating composite pulse sequences similar to those of Shaka and Pines [12,13]. These are particularly suited for optical parametric amplification because besides the standard quasi-phase-matching they require only additional sign flips of the nonlinear optical susceptibility at specific locations corresponding to the segment frontiers. We have demonstrated numerically that this technique is especially powerful for broadband OPA. The present approach does not require very long crystals and, for the given example of MgO:LiNbO${}_3$, is compatible with pump intensities significantly below the damage threshold for ps or sub-ps illumination [35,36]. Note that the derivation of the optimum segment lengths was done in the general dimensionless case and is therefore independent of the material being selected for the implementation. Therefore, the technique presented here can be applied in principle to any nonlinear material permitting phase matching at the central wavelengths of the envisaged frequency conversion process. In combination with QPM, the technique can thus be applied to any material that can be structured by periodic poling (such as LiNbO${}_3$, LiTaO${}_3$, or KTiOPO4) or by orientation patterning during the growth process such as GaAs. However, QPM is not always necessary and the phase matching at the central wavelengths may be realized also by birefringence phase matching in the cases where this is possible. In such cases, it is sufficient to stack a small number of nonlinear crystals of the appropriate thicknesses with mutually reversed axes orientations, as discussed earlier in connection to SFG [30]. In contrast to periodically poling or orientation patterning that generally lead to a limited input aperture, the crystal stacking approach may therefore permit also to realize devices being addressable by very large area pump beams.