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Multistep cascading and fourth-harmonic generation

We apply the concept of multistep cascading to the problem of fourth-harmonic generation in a single quadratic crystal. We analyze a new model of parametric wave mixing and describe its stationary solutions for two- and three-color plane waves and spatial solitons. Some applications to the optical frequency division as well as the realization of the double-phase-matching processes in engineered QPM structures with phase reversal sequences are also discussed.

Cascading effects in optical materials with quadratic (second-order or χ (2) ) nonlinear response provide an efficient way to lower the critical power of all-optical switching devices [1]. The concept of multistep cascading [2] brings new ideas into this field, leading to the possibility of an enhanced nonlinearity-induced phase shift and generation of multicolor parametric spatial solitons. In particular, multistep cascading can be achieved by two nearly phase matched second-order nonlinear processes, second-harmonic generation (SHG) and sum-frequency mixing (SFM), involving the third-harmonic wave [3,4]. In this Letter, we extend the concept of multistep cascading to nonlinear effects of the fourth order and the fourth-harmonic generation (FHG) in a single noncentrosymmetric crystal. In particular, we analyze a new model of multistep cascading that involves the FHG process, and describe its stationary solutions for normal modes -plane waves and spatial solitons. Our study provides the first systematic analysis of the problem of FHG via a pure cascade process, observed experimentally more than 25 years ago [5] and later studied in a cascading limit only [6].
We consider the FHG via two second-order parametric processes: ω + ω = 2ω and 2ω + 2ω = 4ω, where ω is the frequency of the fundamental wave. In the approximation of slowly varying envelopes with the assumption of zero absorption of all interacting waves, we obtain where A, S, and T are the envelopes of the fundamental-frequency (FF), second-(SH) and fourthharmonic (FH) waves respectively, γ 1,2 are proportional to the elements of the second-order susceptibility tensor, and ∆k 1 = 2k 1 − k 2 and ∆k 2 = 2k 2 − k 4 are the corresponding wave-vector mismatch parameters. We introduce the normalized envelopes (u, v, w) according to the following relations: where a is the characteristic beam width, and z d = k 1 a 2 is the diffraction length of the FF component. In order to describe a family of nonlinear modes characterized by the propagation constant λ, we look for solutions in the form u(x, z) → λU (x |λ|, z|λ|)e iλz/4 , v(x, z) → λV (x |λ|, z|λ|)e iλz/2 , and w(x, z) → λW (x |λ|, z|λ|)e iλz , and obtain the normalized equations: Here s = sign(λ) = ±1, χ = γ 1 /(4γ 2 ) is a relative strength of two parametric processes, and the normalized mismatches are defined as α = 4 + β/λ and α 1 = 1/4 + β 1 /λ, where β = 8∆k 2 z d and β 1 = −∆k 1 z d . First, we analyze the plane-wave solutions of Eq. (1) which do not depend on x. In this case, the total intensity I is conserved, and we present it in terms of the unscaled variables as I = I u +I v +I w , where I u = |u| 2 /4, I v = |v| 2 , and I w = 4|w| 2 . Solutions {U 0 , V 0 , W 0 }, which do not depend on z, are the so-called normal modes. The simplest one-component FH mode {0, 0, W 0 } exists at α = 0. It has a fixed phase velocity λ = −β/4 and an arbitrary amplitude, being unstable for I w > β 2 /4 due to a parametric decay instability.
Two-mode solution {0, √ 2α, 1} describes a parametric coupling between SH and FH waves, and it exists for α > 0, bifurcating at α = 0 from the FH mode. Coupling of this two-mode plane wave to a FF wave can lead to its decay instability, provided |α 1 | < α To understand the physical meaning of this inequality, we note that the family of solutions characterized by the propagation constant λ corresponds to a straight line in the (α, α 1 )-parameter space, see Figs. 1(a,b). Moreover, all such lines include the point (4, 1/4) as the asymptotic limit for |λ| → +∞. This special point belongs to the instability region if the relative strength of the FF-SH interaction exceeds a critical value, i.e. for χ > χ (cr) = 1/(8 √ 2) ≃ 0.088. However, for χ < χ (cr) this decay instability is suppressed for highly intense waves, owing to a strong coupling with the FH field. Finally, a three-mode solution, 1 , and (iii) α < 0 and α 1 > 0. In the limit |λ| → +∞, such three-wave modes are possible only for χ > χ (cr) . In the region (i), stability properties of the three-wave solutions are determined by a simple criterion: the modes are stable if ∂I/∂|λ| > 0, and unstable, otherwise. For the parameter regions (ii) and (iii), oscillatory instabilities are possible as well. Existence and stability of all types of stationary plane-wave solutions of the model (1) are summarized in Figs. 1(a-d).
In general, the system (1) is nonintegrable and its dynamics are irregular. However, we find that in some cases a quasi-periodic energy exchange between the harmonics is possible. Figure 2(a) shows one such case, when the intensities of unstable two-wave and stable three-wave stationary modes are close to each other, and an unstable two-wave mode periodically generates a FF component. Less regular dynamics are observed for other cases, such as for the generation of both SH and FH waves from an input FF wave [ Fig. 2(b)]. This example also illustrates the possibility of effective energy transfer to higher harmonics close to the double phase matching point.  Fig. 1(c)], (b) Generation of higher harmonics from a FF input [parameters correspond to Fig. 1(d)]. FF, SH, and FH components are shown by dotted, solid, and dashed curves, respectively.
Equations (1) may have a different physical meaning provided the normalized amplitude v stands for the mode of the fundamental frequency ω. Then, Eqs. (1) describe the optical frequency division by two (the field u) via parametric amplification and down-conversion (see, e.g., [7]), provided both FF (v) and SH (w) fields are launched simultaneously at the input. Such a frequency division parametric process is in fact shown in Fig. 2(a), where this time the generated u wave corresponds to the frequency ω/2, and is shown as dotted.
We now look for spatially localized solutions of Eqs. (1), quadratic solitons. First, we note that two-wave solitons consisting of the SH and FH components can be approximated as [2,8]: where all parameters are functions of α only. Bright solitons either do not exist or are unstable being in resonance with linear waves [4] outside the parameter region α > 0 and α 1 > 0, at s = +1. We find that in this region, similar to plane-wave modes, three-wave solitons exist for 0 < α 1 < α (cr) 1 , where the critical (cut-off) value α (cr) 1 corresponds to a bifurcation from the twowave solution (2). In order to find α (cr) 1 , we should solve the first equation of the system (1) with the SH profile from Eq. (2) (see also [2]). However, such a linear eigenvalue problem has no exact analytical solution for arbitrary p, and thus we introduce an approximation V 0 (x) ≃ V 0 (x) = V m sech 2 (x/q), requiring that the functions coincide at the amplitude level V m /2, and then define the scaling parameter as q = p cosh −1 (2 1/p )/cosh −1 (2 1/2 ). Such an approximation adequately describes an effective soliton waveguide, and thus should provide overall good accuracy (except for some limiting cases). After solving the eigenvalue problem with the potential χ V 0 (x), we obtain an approximate expression for the bifurcation points: α (cr) 1 ≃ ( 1 + 4V m χq 2 − 1 − 2n) 2 /(4q 2 ), where n is the order of the mode guided by the two-component parametric soliton waveguide [2]. For a single-hump mode (n = 0) the behavior of this cut-off is very similar to that of the plane waves. Indeed, in the cascading limit (α ≫ 1), we have V m ≃ 2 √ α, and α (cr) 1 ≃ 2χ √ α, which differs by √ 2 from the corresponding result for plane waves. The critical value of χ for one-hump solitons can also be found from the approximate solution, χ (cr) ≃ 0.132. We performed numerical simulations and found that the accuracy of our approximation is of the order of (and usually better than) 1% in a wide range of parameters (χ > 10 −2 and α > 10 −2 ), see Fig. 3(a). where two-component solutions exhibit parametric decay instability. An example of such an instability is presented in Figs. 3(c,d), where an unstable twowave soliton generates a stable three-wave state. Such instability-induced dynamics are very different from that of plane waves where, instead, quasi-periodic energy exchange is observed [see Fig. 2(a)]. In the case of localized beams, diffraction leads to an effective power loss and convergence to a new (stable) state.
In order to observe experimentally the multistep cascading and multi-frequency parametric effects described above, we should satisfy the double-phase matching conditions. Using the conventional quasi-phasematching (QPM) technique [9] for FHG via a pure cascade process in LiTaO 3 , we find that there exists only one wavelength (2.45µm), for which two parametric processes can be phase-matched simultaneously by the different orders m of the QPM structure with the period Λ Q = 34µm. However, for the so-called phasereversal QPM structures [10] characterized by two periods, the QPM period Λ Q and the modulation period Λ ph (Λ ph > Λ Q ), double-phase matching is possible in a broad spectral range, provided the periods are selected to satisfy the conditions: Λ Q = 4π|∆k 1 + ∆k 2 | −1 , Λ ph = 4πm|∆k 1 − ∆k 2 | −1 , where m is the grating order. Thus, the engineered QPM structures suggested in [10] are more efficient than, e.g., the Fibonacci superlattices, and they can be used to achieve double-phase matching and to support different types of multistep cascading processes.
In conclusion, we have introduced a new model of multistep cascading that describes the fourth-harmonic generation via parametric wave mixing. We have analyzed the existence and stability of the stationary solutions of this model for normal modes -plane waves and spatial solitons. We have also discussed the possibility of double-phase-matching in engineered QPM structures with phase-reversal sequences.