1 Introduction

Recently nonlinear optics research beyond the so-called slowly varying envelope approximation (SVEA) has a received tremendous boost due to various reasons, primarily for richness in physics and possible applications in many diverse areas, such as ultrafast spectroscopy, metrology, medical diagnostics and imaging, optical communications, manipulation of chemical reactions and bond formation, material processing etc. [1, 2]. Particularly, the availability of sources of light in the near-single optical cycle has opened new possibilities for physicists and scientists to explore many of the fundamental concepts and assumptions [3, 4]. The validity of SVEA is already questioned by many authors in this new domain of optical science [5–10]. Many authors have attempted to modify the SVEA so that it might be extended to the few-cycle regimes. The first widely accepted model in this regard has been developed by Brabec and Krausz [5]. Some other authors have offered non-SVEA models also [11, 12]. However, the model equation proposed by Brabec and Krausz has been used most extensively and successfully in various contexts [13–16]. Recently, following the model proposed by Brabec and Krausz, Moses and Wise have derived a coupled propagation equation for ultrashort pulses in a degenerate three-wave mixing process in quadratic (χ (2)) media [17]. In passing, it is worthwhile to mention that owing to the efficient manipulation of spectral and temporal properties of femtosecond pulses through cascaded processes in quadratic materials, both theoretical and experimental research is getting a tremendous boost in recent years [18–24]. We should also note that the Moses–Wise model is restricted to the case of strongly mismatched interaction where the conversion efficiency to second or higher harmonics is negligible. In fact, a more generalized nonlinear envelope equation is derived by Conforti et al. to describe the propagation of broadband optical pulses in second-order nonlinear materials [22, 23]. However, Moses and Wise went on to present, using cascaded-quadratic nonlinearity, theoretical and experimental evidence of a new quadratic effect, namely the controllable self-steepening (SS) effect. The controllability of the SS effect is very useful in nonlinear propagation of ultrashort pulses as it may be used to cancel the propagation effects of group velocity mismatch. After publication of the Moses–Wise model it was pointed out that the cascaded nonlinearity induces a Raman-like term into the model owing to the nonlocal nature of the cascaded nonlinearity [24]. It may be noted that, traditionally, the intensity-dependent refraction (IDR) effects in quadratic media are not expected in quadratically nonlinear media owing to the phase mismatch of the fundamental harmonic with the higher ones within the SVEA [25]. In this work, we have studied the modulation instability (MI) of the single-field equation for the fundamental field (FF) derived by Moses and Wise [17]. Our study is mainly motivated by the fact that IDR effect is closely related to MI, particularly to the existence of optical solitons in a nonlinear medium. It is well known that the modulation instability is a fundamental and ubiquitous process that appears in most nonlinear systems in nature [26–31]. It occurs as a result of interplay between the nonlinearity and dispersion in time domain or diffraction in spatial domain. It may be noted that the modulational instability in a quadratic medium was first studied by Trillo and Ferro taking into account the parametric interaction between fundamental and second harmonic field in a transparent dispersive medium [32]. However, cascading in quadratic media and solitary wave propagation were first discussed in the work of Menyuk, Schiek and Torner [33]. Later, MI in quadratic media is explored by many authors, in various contexts, such as competition between spatial and temporal break up [34], quantum-noise-initiated break up [35] etc. In this work we are specifically interested to see the role of group velocity mismatch (GVM) between the fundamental (FF) and second-harmonic (SH) field on MI as well as the role of self-steepening (SS) parameter.

2 Theoretical model and MI analysis

The Moses–Wise model, with the third-order dispersion (TOD) and the fourth-order dispersion (FOD) effects, for ultrashort pulse propagation in a cascaded-quadratic medium could be written as follows [17]:

(1)

Here A is the complex envelope of the fundamental field traveling along z, β 2 is the group velocity dispersion (GVD) parameter, β 3 is the third-order dispersion (TOD) parameter, β 4 is the fourth-order dispersion (FOD) parameter, ω 0 is the carrier-wave frequency, γ is the cubic nonlinear co-efficient, Δk=2k 1−k 2 is the wave-vector mismatch between the FF and SH fields and \(\varGamma ^{2} =32\pi^{2}\omega_{0}^{4}\chi_{2\omega _{0} - \omega _{0}}^{( 2)}\chi_{\omega _{0} + \omega _{0}}^{( 2 )}/k_{1}k_{2}c^{4}\). k 1 and k 2 are, respectively, the wave vectors associated with FF and SH fields. δ measures the group velocity mismatch between FF and SH fields. It may be worthwhile to mention a few words about the sixth term of (1). Here the first term inside the brackets refers to SS arising from χ (2), while the second one refers to SS arising from χ (3) [17]. GVM-induced SS term is included in the last term through the parameter δ. It may be noted that the fifth and the sixth term of (1) vanishes if Γ 2/Δk=−γ, while the last term could vanish for some particular values of Δk and γ+Γ 2/Δk≠0 as discussed in Ref. [17]. For a systematic derivation of the Moses–Wise model readers are referred to Ref. [36] along with Ref. [17]. Now we rewrite (1) in the so-called soliton units [26] as follows:

(2)

where u is the normalized amplitude, α= sgn(β 2) and we have

(3)

Here ξ and τ are the normalized propagation distance and time, respectively, P 0 is the peak power of the incident pulse, L D is the dispersion length and N is the so-called soliton order. We would like to emphasize that in this work, s is termed as the normalized self-steepening (SS) parameter in analogy with the one found in conventional nonlinear fiber optics [26]. On the basis of (2) we would now investigate the MI of few-cycle pulses. Equation (2) has a steady-state solution given by \(u = u_{0}\exp[ i\beta u_{0}^{2} \xi]\), where u 0 is the constant amplitude of the incident plane wave. We now introduce the perturbation a(ξ,τ) together with the steady-state solution to (2) and linearize in a(ξ,τ) to obtain

(4)

Separating the perturbation to real and imaginary parts, according to a=a 1+ia 2, and assuming a 1,a 2∝exp[i(Kξ−Ωτ)], where K and Ω are the wave number and the frequency of perturbation, respectively, from (4) we obtain the following dispersion relation:

(5)

From (5), we observe that the modulation instability exists only if the quantity inside the bracket is <0 and TOD plays no role in MI. The fact that third-order dispersion contributes nothing to modulational instability is a well-known result [28, 37]. Another interesting fact is that, owing to the absence of ρ, the seventh term of (4), which is actually the seventh term of (1), in real units, plays no role in MI. It may be worthwhile to mention that, physically speaking, the net effect of the absence of ρ basically states the fact that the GVM-induced SS term, i.e. δ, has no influence on MI in cascaded-quadratic media. In view of these facts, we note that the Moses–Wise model is basically an NLSE with self-steepening (SS) effect. Cascading merely gives a tunable Kerr effect. This enables us to relate the results of this article with the one with the standard NLSE with the SS term.

To start with the MI analysis, we begin with the case of anomalous dispersion regime, for which α=−1. The expression for the so-called gain spectrum g(Ω) could be put in the following form:

$$ g( \varOmega ) = 2{\mathop{\mathrm{Im}}\nolimits} ( K ) = 2\bigl[ \varOmega _{B}^{2}- \varOmega _{A}^{2} \bigr]^{ \frac{1}{2}},$$
(6)

where

(7)

We note that for the occurrence of MI, one must have \(\varOmega _{B}^{2} >\varOmega _{A}^{2}\). To simplify our calculations, in the following analysis, we are taking δ 4=0. We discuss the role of the FOD on MI briefly at the end of this article. The maximum of the gain occurs, in the absence of FOD, at two frequencies given by

$$ \varOmega _{\max} = \pm\sqrt{2 u_{ 0}^{ 2} - 2 \eta \bigl( 2 s^{2} u_{ 0}^{4} - u_{ 0}^{ 2} \bigr) - 2\bigl( 1 + \eta ^{2} \bigr) s^{2}u_{0}^{4}}$$
(8)

Now, we would try to see the role of various controllable parameters such as the self-steepening (SS) s and wave-vector mismatch Δk on MI. Figure 1 depicts the modulation instability gain spectrum g(Ω) vs. Ω for different values of η for s=.01 and u 0=1.

Fig. 1
figure 1

Modulational instability gain as a function of normalized frequency for four different values of η with u 0=1 and s=0.01

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It can be clearly seen that the gain spectrum is symmetric with respect to Ω=0. We observe from Fig. 1 that, for the given input power and a fixed self-steepening parameter, the modulation instability gain increases with increase in η. Physically speaking, MI gain increases with a decrease in the wave-vector mismatch Δk, as the parameter η is directly related to it through (7). If a probe wave at a frequency ω 0+Ω were to propagate with the CW beam at ω 0, it would experience a net power gain given by (6) as long as \({\mathop{\mathrm{Im}}\nolimits} (K) > 0\). Eventually, due to MI gain, the CW beam would break up spontaneously into a periodic pulse train known as solitons. These soliton-like pulses exist whenever the conditions \(\varOmega _{B}^{2} > \varOmega _{A}^{2}\) and Δk≠0 are satisfied. The appearance of the sidebands located around Ω=0 is clear evidence of the modulation instability.

Figure 2 shows, quite expectedly, that with increase of the amplitude, the MI gain also increases. On the other hand we find that MI gain decreases with increase in the SS parameter, s, as could be observed from Fig. 3. The SS parameter is inversely related to both the carrier-wave frequency, ω 0, and pulse width, T 0. So, we may conclude that through a judicious choice of parameters one may control the modulational instability gain, in a cascaded-quadratic medium like the one considered in this work. It may be possible to have MI in the normal dispersion regime, for which α=+1, if Δk<0, and the other parameters are chosen judiciously. This would necessitate tuning the phase mismatch to make the cascaded nonlinearity stronger than the Kerr one. To get an idea about the occurrence of the modulation instability in the normal dispersion regime, for which α=+1, in Fig. 4 we depict MI gain as a function of the normalized frequency for three different values of η with s=0.01 and u 0=1.

Fig. 2
figure 2

Modulation instability gain as a function of normalized frequency for three different values of u 0 with η=2 and s=0.1

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Fig. 3
figure 3

Modulation instability gain as a function of normalized frequency for four different values of s with η=2 and u 0=1

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Fig. 4
figure 4

Modulational instability gain as a function of normalized frequency for three different values of η with u 0=1 and s=0.01 (Normal dispersion regime)

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We note that the parameter η- and thereby the β-parameter is now negative. As stated earlier, this would require one to tune the phase mismatch so that the cascaded nonlinearity becomes stronger than the Kerr one. MI gain increases with increase in |β|. Similar analysis for other parameters may be carried out as was done for the case of the anomalous dispersion regime.

Finally, to investigate the role of the fourth-order dispersion (FOD) on MI, in Fig. 5, we plot the MI gain vs. normalized frequency, for various values of the FOD parameters, for both the normal and the anomalous dispersion regimes. We find that in the anomalous dispersion regime with increase in the FOD parameter, along with the usual peaks two side bands, appears at higher frequencies. However, we note that these side bands move toward the center as the FOD parameter increases. These side bands disappear if the FOD parameter is increased beyond δ 4=0.01. On the other hand, in the normal dispersion regime, MI gain is almost unaffected due to the presence of FOD.

Fig. 5
figure 5

Modulational instability gain as a function of normalized frequency for different values of δ 4 with u 0=1, η=2 and s=0.01. (a) Anomalous dispersion regime. (b) Normal dispersion regime

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3 Conclusion

To conclude, we have studied the modulation instability of the Moses–Wise model, with the TOD and the FOD, terms for ultrashort pulse propagation in a cascaded-quadratic medium. A nonlinear dispersion relation is worked out using standard methods. We find that subject to the fulfillment of the MI criterion and judicious choice of the parameters, MI could be generated in a cascaded-quadratic-cubic medium in both normal and anomalous dispersion regimes. We find that the group velocity mismatch-induced self-steepening effect plays no role in the modulation instability.