Optical quadratic nonlinear processes, such as sum-frequency generation (SFG), difference-frequency generation (DFG), and optical parametric amplification (OPA), provide a powerful and flexible tool for generating coherent light in the spectral regions inaccessible to laser emissions. One important merit of such quadratic three-wave nonlinear interactions lies in their outstanding designability. On the one hand, various phase-matching schemes can be applied, including quasi-phase matching (QPM)^[1–3], adiabatic phase matching^[4], and multistep cascade phase matching^[5,6]. On the other hand, nonlinear crystals can also be fabricated with distinct structures, including periodically poled grating patterns^[1–3] and sectional tiling^[2,7], and can be doped with rare-earth ions^[8,9].

One inherent characteristic of quadratic nonlinear processes is that they allow both forward and backward energy transfers among the interacting waves in the crystal^[10]. Such back conversion will be detrimental when a unidirectional energy transfer is desired. It is well known that phase mismatch among the interacting waves can cause back conversion, which can be prevented using QPM^[1–3]. Under the perfect phase-matching conditions, however, back conversion can still occur when one interacting wave is depleted. This back conversion will set an ultimate limit on the application performance of a quadratic nonlinear process^[11–14]. As of now, the physical reason for the occurrence of this back conversion is still unclear, and there is still no effective method to inhibit it.

One can usually answer the most basic and important questions about optics through the phase of a wave. A phase jump of π widely exists in linear optics, such as the half-wave loss associated with an interface reflection and the Gouy phase-shift associated with beam focusing^[15,16]. In this Letter, we numerically find an analog of this phase jump in nonlinear optics, which occurs when one wave is completely depleted in the process of three-wave interactions. This nonlinear phase jump is indeed the origin of back conversion in a phase-matched DFG process. From this new perspective, we can explore a method to inhibit such a back conversion, which has long been regarded as impossible. For a parametric down-conversion process, such as DFG or OPA, the back conversion can be inhibited by removing the phase jump of the idler wave through deliberate crystal absorption. Our studies would be beneficial for a deep understanding of optical quadratic nonlinear processes and pave the way toward the design of highly efficient DFG systems.

We start with the standard nonlinear coupled-wave equations that govern three-wave interactions (represented by W1,W2, and W3, with the angular frequencies ω1<ω2<ω3,ω1+ω2=ω3)^[10], dA1dz=−iω1deffn1cA2*A3e−iΔkz,(1)dA2dz=−iω2deffn2cA1*A3e−iΔkz,(2)dA3dz=−iω3deffn3cA1A2eiΔkz,(3)where Aj(z)=ρj(z)exp[iφj(z)] is the electric field envelope, and ρj and φj refer to the amplitude and phase, respectively. nj is the refractive index of the crystal, deff is the effective nonlinear coefficient, and c is the light speed in a vacuum. Δk=k3−k2−k1 is the wave-vector mismatch among the three interacting waves. After defining a total phase θ=Δkz−φ3(z)+φ2(z)+φ1(z), employing the transformation eiθ=cosθ+isinθ, and equating the real and imaginary parts, Eqs. (1)–(3) become dρ1dz=−ω1deffn1cρ2ρ3sinθ,(4)dρ2dz=−ω2deffn2cρ1ρ3sinθ,(5)dρ3dz=ω3deffn3cρ1ρ2sinθ,(6)dφ1dz=−ω1deffn1cρ2ρ3ρ1cosθ,(7)dφ2dz=−ω2deffn2cρ1ρ3ρ2cosθ,(8)dφ3dz=−ω3deffn3cρ1ρ2ρ3cosθ.(9)From Eqs. (4)–(6), we can see that the direction of the energy flow among the three interacting waves is determined by sinθ. When sinθ<0(dρ3/dz<0,dρ1/dz>0,dρ2/dz>0), the energy flow is from wave W3 to waves W1 and W2 (i.e., W3→W1+W2). When sinθ>0(dρ3/dz>0,dρ1/dz<0,dρ2/dz<0), the energy flow is inverse, i.e., W1+W2→W3. From the definition of θ, a phase accumulation due to Δk can induce the symbol change of sinθ, so a phase mismatch can cause the back conversion. However, here we only consider the perfect phase-matching condition (Δk=0). In this case, sinθ=sin[φ1(z)+φ2(z)−φ3(z)], that is, the direction of the energy flow depends on the phase changes of the three interacting waves. Let us consider a DFG process, in which only W2 and W3 are seeded onto the crystal. The initial phases of these two waves are assumed to be equal to zero, i.e., φ2(0)=0, and φ3(0)=0. In this case, the newly generated wave W1 accommodates its phase to φ1(0)=−π/2 in order to maximize the conversion rate from W3 to W1 and W2(sinθ=−1). As there is no phase mismatch, sinθ remains at a value of −1 until the pump W3 is completely depleted at a position z=L0. From Eq. (9), when ρ3=0,dφ3/dz→∞, so there exists a phase jump on φ3. This phase jump activates the back conversion in the SFG. From Eqs. (4)–(6), to maximize the rate of such a back conversion, it requires sinθ=1, i.e., φ1(L0)+φ2(L0)−φ3(L0)=π/2. As φ1(L0)=−π/2 and φ2(L0)=0, φ3 has to jump from 0 to −π, i.e., φ3(L0)=−π. Such a phase jump is the underlying reason for the back conversion. At the position z=2L0, the idler will be completely depleted. Similar to the above analysis, φ1(2L0) will jump from −π/2 to π/2 to change sinθ from 1 to −1 to activate the forward DFG process again. In such a way, the forward and backward conversions happen periodically. Note that the phase of W2 does not change with the distance during the whole process, because it cannot be depleted at any time.

To verify the above theoretical analysis, we numerically solved Eqs. (1)–(3) by the symmetrized split-step Fourier method. A DFG process with λ3=532 and λ2=800nm in a collinear phase-matched β-BBO crystal orientated at 22.05° was simulated. Pump wave W3 is a Gaussian pulse with a full width at half-maximum (FWHM) duration of 100 ps, and its peak intensity is 5GW/cm2, while the incident signal wave W2 is a super-Gaussian pulse with an FWHM duration of 240 ps and a peak intensity of 1GW/cm2 (Fig. 1). Dispersions, walk-off, and diffraction effects are all neglected in our calculations.

Figure 2 summarizes the calculated intensity and phase evolutions of the three interacting waves in a 10-mm-long β-BBO crystal. Note that these results are calculated at the local temporal coordinate of t=0 (i.e., point A in Fig. 1). As is clearly shown in Fig. 2(a), the direction of the energy flow depends on sinθ: when sinθ=−1, W3→W1+W2, and when sinθ=1,W1+W2→W3. The sign of sinθ changes when one wave is depleted. For example, when W3 is depleted (ρ3=0) at z=1.84mm,sinθ changes from −1 to 1; when W1 is depleted (ρ1=0) at z=3.68mm, sinθ changes from 1 to −1. Through these phase evolutions [Fig. 2(b)], we can see that the full depletion of one wave indeed causes a π-phase jump, which matches our theoretical analysis very well. The phase of W2 keeps a constant value of zero because it cannot be depleted at any time. This is the reason why the signal phase can be preserved in the DFG.

Although the back conversion process is activated by the phase jump, its occurrence time depends on the coupling intensity of the nonlinear interactions, as implied by Eqs. (4)–(9). In our simulation, because pump pulse W3 has a Gaussian temporal profile (Fig. 1), back conversion will not take place simultaneously in the temporal domain. Figure 3 shows the intensity evolutions of W2 at t=0, 50, and 100 ps, respectively. Obviously, the higher pump intensity (e.g., point A) drives an earlier and stronger back conversion. Because the back conversion happens nonuniformly in the temporal domain, the energy of pump wave W3 (i.e., the integration of all the temporal coordinates) can never be depleted completely, thus significantly degrading the conversion efficiency from W3 to W2. The effect of back conversion on the conversion efficiency is more serious in the OPA, which is typically pumped by a spatiotemporally Gaussian laser. To mitigate the back conversion effects, the pump pulses can be shaped to a flat-top profile, or the pump and signal pules can be shaped to the conformal profiles^[17,18]. However, these shaping methods are all complicated and are also not applicable to the short-pulse-pumped OPAs.

In most DFG and OPA processes, only one wave (e.g., W2, the signal) is desirable, with the other wave (e.g., W1, the idler) as a byproduct. The fundamental method to inhibit the back conversion is to control the phase of W1. From the above analysis, if the phase of W1 does not change with the distance, i.e., φ1(z)=−π/2, the back conversion will no longer happen. This requires that the intensity of W1 should remain unusually smaller than that predicted by the Manley–Rowe relation^[9]. Therefore, the idler wave W1 must be consumed by an additional process. Linear crystal absorption is one convenient way to deplete W1. We assume that the absorption coefficient of W1 is α and rewrite Eq. (1) as dA1dz=−iω1deffn1cA2*A3e−iΔkz−12αA1.(10)By numerically solving Eqs. (1)–(3) and (10), the intensity and phase evolutions of the three waves under two absorption levels of αLnl=2 and 8 are summarized in Fig. 4, where the nonlinear length Lnl, as defined in Ref. [18], is fixed at 1.5 mm. By inducing an absorption of αLnl=8 on W1, the back conversion indeed no longer happens as expected, as shown in Fig. 4(b). At each position of the crystal, it can be viewed as a DFG process without the incidence of the idler. Therefore, phase φ1 keeps at −π/2 across the entire crystal, while φ2 and φ3 remain at a constant value of zero [Fig. 4(d)]. In this case, the conversion from W3 to W2 becomes unidirectional without back conversion [Fig. 4(b)]. The inhibition of the back conversion process ensures a complete depletion of W3 and a maximum amplification of W2.

In the case of αLnl=2, although the back conversion can still occur initially, it will diminish gradually, as shown in Fig. 4(a). Compared to the case of no absorption in W1, the depletion of W1 is enhanced by the absorption, so its phase jump takes place earlier than usual [Fig. 4(c)]. This earlier phase jump activates the forward conversion process before the end of a complete back conversion. In such a way, W2 still obtains a net gain in one forward–backward conversion process. As a result, W2 keeps rising with the decreasing oscillation amplitude [Fig. 4(a)]. Finally, W2 will approach to the quantum-limited intensity with negligible back conversion, and W3 is almost completely depleted. In practice, the choice of α depends on the initial pump and signal intensities and also on the crystal length. On the one hand, to effectively suppress the back conversion, the value of α should be larger than a certain value, αLnl>m (m=0.3 in the case of Ref. [9]). On the other hand, to ensure a maximum conversion efficiency within a limited crystal length L, the value of α should be smaller than a certain value, αL<n.

Due to the suppression of the back conversion, each temporal coordinate on W2 can obtain a maximum gain at the output end of the crystal (Fig. 5), which ensures the maximum conversion efficiency regardless of the profile of W3. Figure 5 also implies that such a DFG process with absorption in W1 is robust against the variation of the pump intensity. A more stable output can be expected from such a DFG process without back conversion.

In conclusion, we reveal that the phase jump, accompanied by one depleted interacting wave, is responsible for the intrinsic back conversion in quadratic nonlinear processes. Based on this understanding, we find an effective method to inhabit the back conversion in a parametric down-conversion process. By using crystal absorption to enhance the depletion of the undesirable wave, the back conversion can be terminated, and the conversion efficiency from the pump to the desirable wave could reach the quantum limit. This method for suppressing the back conversion by absorbing the idler wave is compatible with the technique of noncollinear phase matching, so it applies to broadband nonlinear amplification. In the case of broadband amplification, absorption on a specific wavelength (often corresponding to the magic wavelength) can be obtained by doping specific rare-earth ions in an yttrium calcium oxyborate crystal, as we did in Ref. [9]. In addition to the linear absorption method, one could also use a nonlinear method to deplete the undesirable wave, e.g., designing a second-harmonic generation process for this wave in the same crystal^[19], which, however, is hard to realize in conventional bulk crystals. The results presented in this study further our understanding of quadratic nonlinear processes and pave the way toward the design of highly efficient parametric down-converters.