## Abstract

We present a theoretical investigation of dispersive wave (DW) generation in nonlinear metamaterials (MMs). The role of the anomalous self-steepening (SS) effect, which can be either positive or negative, and the negative SS parameter can have a very large value compared to an ordinary positive-index material, in DW generation is particularly identified. It is demonstrated that the SS effect exerts a great impact on the peak power while has little effect on the frequency shift of DW. For positive third-order dispersion (TOD), the negative SS broadens the pulse spectrum and weakens the DW’s peak power significantly, opposite to the case of positive SS. For negative TOD, however, the negative SS narrows the pulse spectrum and enhances the DW’s peak power, also opposite to the case of positive SS. The results suggest that the DW generation in nonlinear MMs can be manipulated by SS effect to a large extent.

© 2012 Optical Society of America

## 1. Introduction

Optical soliton is the robust balance between dispersion and nonlinearity. In particular, the conventional fundamental optical soliton is the exactly balance between self-phase modulation and anomalous group-velocity dispersion (GVD). However, the high-order dispersion and nonlinearity are ever-present under the realistic condition. These effects disrupt the balance and split the soliton into its components [1]. During the fission process, dispersive wave (DW) is generated due to the energy transfer from soliton to narrow resonance in the presence of the high-order dispersions [2–9]. The frequencies of DW are determined by the phase-matching condition between the soliton and DW [2, 6, 7]. DW is of particular importance for supercontinuum generation and broadband light sources [4, 10]. Recently, the generation of DW in the supercontinuum generation process inside the nonlinear microstructured fiber or photonic crystal fiber is explored under different operational conditions [4, 5, 10, 11]. The roles of high-order dispersions in the generation and control of DW are disclosed [6–9], and the manipulations of DW by the dispersion profile of the specific fiber [9] or frequency chirp are unfolded [12]. In this paper, we study the influences of the anomalous SS effect on generating DW based on the numerical simulations.

Metamaterials (MMs) are artificial structures that display properties beyond those available in naturally occurring materials [13–15]. MMs host a number of unusual properties, and the controllable optical magnetic responses are essential for various applications such as perfect lenses [16], superlenses [17, 18], subwavelength waveguides and antennas [19, 20], and electromagnetic cloaking devices [21, 22]. Recently, the realizability of the optical MMs [13, 14] and achievable nonlinear MMs [23, 24] have stimulated many investigations on the nonlinear optical properties of the MMs. In particular, the rich linear and nonlinear electromagnetic properties enable MMs to be potential candidates for stable soliton and other nonlinear phenomena [25–33]. A lot of work about ultrashort pulse propagating in nonlinear MMs was reported in recent years [29–33]. It was demonstrated that the dispersive magnetic permeability of MMs generates more nonlinear effects, including the anomalous self-steepening (SS) effect, second-order nonlinear dispersion, and other high-order nonlinear dispersion [29, 32, 33]. In particular, the anomalous SS effect and second-order nonlinear dispersion lead to significant changes of the conditions for the MI and soliton [32], compared with the case in an ordinary positive-index material. Hence, the controllable nonlinear MMs provide us more freedom to manipulate the ultrashort pulse and soliton. In our previous paper [33], we have presented a theoretical investigation on the controllability of the Raman soliton self-frequency shift in nonlinear MMs and identified the combined effects of the anomalous SS effect, third-order dispersion (TOD), and Raman effect on soliton self-frequency shift.

In the present paper, we show that the DW generation can be manipulated by the anomalous SS effect. The paper is organized as follows. In Sec. II, the theoretical model for ultrashort pulse propagation in nonlinear MMs with TOD and Raman delayed response is introduced. In Sec. III, we discuss the controllable DW generation in the nonlinear MMs, and reveal the roles of the anomalous SS effect in the DW generation. Finally, in Sec. IV, we summarize our results and conclusions.

## 2. Theoretical model for ultrashort pulse propagation in nonlinear MMs with TOD and Raman delayed response

To specially elucidate the role of SS effect in DW generation in MMs, we keep the linear dispersion terms to the third order, only keep the first-order time derivative of nonlinearity terms and neglect the fifth-order nonlinearity, the generalized nonlinear schr**ö**dinger equation (GNLSE) is written in the following normalized form [33],

*τ*=

*T/T*,

_{p}*ξ*=

*Z*/|

*l*

_{d}_{2}|, and

*U*=

*A/A*

_{0}, where

*T*is the duration of the input pulse,

_{p}*A*

_{0}is the amplitude of the input field, ${L}_{dm}={T}_{p}^{m}/{\beta}_{m}$ is the

*mth*-order dispersion length, and ${L}_{nl}={\left({\gamma}_{0}{A}_{0}^{2}\right)}^{-1}$ is the nonlinear length.

*ϑ*= ±1 for focusing and defocusing nonlinearity respectively;

*N*

^{2}= |

*L*

_{d}_{2}|/

*L*,

_{nl}*N*is the soliton-order;

*δ*

_{3}=

*L*

_{d}_{2}/6

*L*

_{d}_{3},

*s*

_{1}=

*S*

_{1}/

*T*,

_{p}*τ*=

_{R}*T*/

_{R}*T*.

_{p}*T*is related to the slope of the Raman gain. Clearly, as the pulse duration

_{R}*T*decreases,

_{p}*s*

_{1},

*δ*

_{3}and

*τ*increase, meaning that the SS effect, TOD parameter and Raman effects become more important.

_{R}In the GNLSE,
${\beta}_{2}={\eta}_{2}-1/{k}_{0}{v}_{g}^{2}$ is GVD, *β*_{3} = *η*_{3} − 3*β*_{2}/*k*_{0}*v _{g}* is TOD,

*S*

_{1}= 1/

*ω*

_{0}+

*γ*

_{1}/

*γ*

_{0}− (

*k*

_{0}

*v*)

_{g}^{−1}is SS coefficient, where

*v*= 2

_{g}*k*

_{0}/(

*F*

_{0}

*G*

_{1}+

*F*

_{1}

*G*

_{0}) is the group velocity,

*η*=

_{m}*m*!

*d*/2

_{m}*k*

_{0}, ${\gamma}_{m}={\omega}_{0}{\epsilon}_{0}{\chi}_{p}^{(3)}{G}_{m}/2{k}_{0}$, ${d}_{m}=\sum _{l=0}^{m}{F}_{l}{G}_{m-l}/l!\left(m-l\right)!$,

*F*=

_{m}*∂*

*[*

^{m}*ωε*(

*ω*)]/

*∂ω*|

^{m}_{ω=ω0}, and

*G*=

_{m}*∂*[

^{m}*ωμ*(

*ω*)]/

*∂ω*|

^{m}_{ω=ω0}.

*ε*(

*ω*) and

*μ*(

*ω*) are the medium permeability and permeability respectively,

*k*

_{0}is the wave number,

*ω*

_{0}is the carrier frequency, and ${\chi}_{p}^{(3)}$ is the third-order electric susceptibility.

The definitions of various variables can be found from our previous paper [33]. Obviously, the values of *β*_{2}, *δ*_{3} and *s*_{1} are determined by basic electromagnetic characteristic of MMs, especially the dispersive permittivity *ε*(*ω*) and magnetic permeability *μ*(*ω*). For the Drude dispersive model for both *ε*(*ω*) and *μ*(*ω*), it has been indicated that in the anomalous GVD region, *s*_{1} has a very large value compared to an ordinary positive-index material and can be either positive or negative determined by the structure parameters of MMs [30, 33], and *δ*_{3} is always positive [33], however the TOD can also be negative by adopting other dispersive-model or structure parameters of MMs. In the following discussions, we focus on the role of SS effect in DW generation and assume that TOD parameter can be either positive or negative.

## 3. The controllable DW generation in MMs

The dispersive magnetic permeability and tailorable linear and nonlinear effective properties by simple engineering the MM provide the favorable conditions for generating and controlling the DWs. In the MMs, in addition to the impact of the high-order dispersion, the soliton propagation is also affected by the high-order nonlinear terms, including the SS effect and Raman scattering. In the next analysis, we first discuss the combined effects of the anomalous SS and TOD on the DW generation in the MMs.

To discuss the DW generation in MMs, we employ the standard split-step Fourier method to solve the GNLSE numerically. We have adopted the normalized input pulse in the numerical simulation, *U*(0, *τ*) = *sech*(*τ*). We only consider the DW generation of the second-order soliton (*N* = 2), and the qualitative behavior of the higher-order soliton (*N* > 2) is similar. To gain a physical understanding of the effects, the loss of MMs is neglected. If no otherwise specified, only the anomalous GVD (*sgn*(*β*_{2}) = −1) and self-focusing (*ϑ* = 1) nonlinearity are considered.

#### 3.1. Dispersive wave generation for positive TOD

In the anomalous GVD of MMs described by the Drude dispersive model, the SS coefficient can be positive, zero or negative [32, 33]. Hence, we will discuss the role of the SS effect in the DW generation by the positive and negative SS coefficients, respectively.

Figure 1 demonstrates how the positive SS effect influences the DW generation, here only the positive TOD (*δ*_{3} = 0.04) is considered and the Raman effect is neglected. In each figure, the upper one is the contour maps of the pulse evolution, and the lower one is the input pulse (dotted line) and output pulse (solid line) spectra at *ξ* = 4.0. In the following analyses, the lowest limit relative power level of −80 dB is assumed for the formation of the DW peak in the output spectrum, hence the relative power level litter than −80 dB has been neglected.

As the propagating distance increasing, TOD and self-phase modulation induce the fission of the second-order soliton and the radiation emitted by the soliton begins to emerge in the blue-shifted band (see Fig. 1). At *ξ* = 4.0, a distinct peak generated through DW is observed in the high frequency range. As the distance increases, the peak position of DW is fixed, however the output spectrum is widened. As a consequence, the output pulse spectra are widened significantly at the joint action of generating DW and broadening soliton spectra.

In the absence of SS effect, as shown in Fig. 1(a), the peak of DW occurs near (*v* − *v _{s}*)

*T*= 2.23, and the peak power of the DW is about −21dB. Comparing Fig. 1(a)–(d), it is observed that the SS effect is important for the DW generation. With the increase of the positive SS coefficients, the evolutions of the output spectra have the following characteristics: (1) The pulse spectra become narrow. When the SS coefficient increases from zero to 0.1, the output spectra become narrow obviously, this is disadvantage to the supercontinuum generation. But the spectra broaden again due to a new sidelobe with lower power generation in the soliton fission for

_{p}*s*

_{1}= 0.2 (Fig. 1(d)); (2) The peak positions of DW move toward the blue side of pulse spectra. But the shifting is not remarkable, which indicates that the influences of the SS effect on the frequency shifts of DW are less important than TOD; (3) The peak power of DW is enhanced, and the positive SS coefficient enhances the peak power of DW for positive TOD coefficient. For

*s*

_{1}= 0.2, the peak position of DW shifts to (

*v*−

*v*)

_{s}*T*= 2.36, and the peak power is about −9dB.

_{p}In order to fully understand the roles of the SS effects in the DW generation, in Fig. 2 we have shown the frequency shifts (a) and relative peak power (b) of the DW peak plotted as a function of the positive SS coefficient *s*_{1} for *δ*_{3} = 0.04. The square symbols show the results obtained without considering the Raman effect. It can be seen that the peak position shifts slowly to higher frequency with the increasing SS coefficient *s*_{1}, and the shifts have gradually become saturated when *s*_{1} > 0.1. However, the peak power of DW increases as the SS coefficient *s*_{1} increases and does not exist saturation phenomenon for 0 < *s*_{1} < 0.2. Therefore, we can enhance the peak powers of DW as expected by manipulating the positive SS effect in MMs.

Next, we turn to the discussion about the role of the negative SS coefficient in DW generation, as shown in Fig. 3. In stark contrast to the results of the positive SS coefficient, as the increasing absolute value of negative SS coefficient, the soliton and DW generation have following characteristics: (1) The spectra of ultrashort pulse become more and more wide, which suggests that the negative SS coefficient and positive TOD coefficient in nonlinear MMs have valuable potentials for supercontinuum generation and broadband source; (2) The peak position of DWs red-shifts and the movement is significant than the positive SS coefficient; (3) The peak powers of DWs are weakened. When *s*_{1} = −0.08, the peak powers of DW have less than −30dB. Therefore, we can suppress the peak powers of DW as expected by manipulating the negative SS effect in MMs. Furthermore, it is observed that some burrs occur in the DW’s spectra for larger |*s*_{1}|. The burrs finally lead to the fission of the DW (see Fig. 3(d)).

To get a thorough understanding of the influence of the negative SS coefficient on the DW generation, Fig. 4 indicates the frequency shift (Square symbol line) and relative peak power (Circle symbol line) of the DW peak plotted as a function of negative SS coefficient. With the increasing |*s*_{1}|, the peak position of DWs shifts to lower frequency. At *s*_{1} = −0.08, the peak position has shifted to (*v* − *v _{s}*)

*T*= 1.85, however, the peak powers of DWs weaken as SS coefficient |

_{p}*s*

_{1}| increases. These results demonstrate that the peak power of DW can be suppressed by the negative SS coefficient and the peak position can be shifted slightly to red side by increasing |

*s*

_{1}|.

So far, we just consider the DW generation and manipulation by the SS effect for the fixed and positive TOD coefficient, next we discuss the frequency shift and peak power as the function of the TOD coefficient, as shown in Fig. 5. We see that the peak position of DW shifts to lower frequency and the peak power enhances as *δ*_{3} increases. Moreover, both the peak position and the peak power of DW become saturated gradually. In Fig. 5, we also show the influence of SS coefficient on DWs. For *δ*_{3} < 0.05, the influence of the positive SS coefficient (*s*_{1} = 0.2) on the peak position of DWs is significant; however, in the range 0.05 < *δ*_{3} < 0.1, the influence of the positive SS coefficient on the peak position of DWs can be neglected. When the negative SS effect (*s*_{1} = −0.06) is considered, it is obvious that frequency shifts are firstly increased and then decreased. The peak power of DWs is influenced by the SS effect distinctly, especially when the TOD coefficient is small. With these analysis results, we stress that the positive SS effect enhances the DWs generation and the negative SS effect weakens the DWs generation for positive TOD. When *δ*_{3} = 0.01, the peak power is about −45dB if SS effect is neglected; however, it can be enhanced by the positive SS coefficient to −10dB at *s*_{1} = 0.2 and it can be weakened by the negative SS coefficient to −94dB at *s*_{1} = −0.06.

#### 3.2. Dispersive wave generation for negative TOD

The former section just considers the role of the positive TOD parameter in the DW generation. Actually, we can also obtain the negative value of TOD parameter if we control the composite materials and the structure parameters of the each unit in MMs. Next, we consider the influence of the negative TOD parameter and abnormal SS effect together on the DW generation. Fig. 6(a)–(c) show the contour and output spectra of a second-order soliton (*N* = 2) at the different SS coefficients for *δ*_{3} = −0.04, and Fig. 6(d) indicates the frequency shift and relative peak power of the DW peak plotted as a function of SS coefficient. Obviously, the DWs occur in the red side of the spectra (low frequency) for negative TOD, and the negative SS coefficient narrows the pulse spectra. But for *s*_{1} = 0.06, the positive SS coefficient play a leading role in the fission of the DW’s spectrum. With the increase of SS coefficient from negative to positive, the peak position of DW has a blue shift to higher frequency and the peak power weakens continuously. Moreover, it is found that the frequency shift of DW is sharply for the positive SS coefficient and it becomes slow for *s*_{1} < −0.1. Hence, we can enhance or suppress the DW generation in MMs by controlling the SS effect for the negative TOD parameter.

#### 3.3. Influence of Raman effect on DW generation in MMs

The Raman scattering effect has been neglected in the above researches. Now we investigate the influence of the Raman scattering effect on the DW generation, as shown in Fig. 2, here the circle symbols show the results obtained when the Raman effect is considered (*τ _{R}* = 0.05). Compared with the results without considering the Raman effect (

*τ*= 0), we observe that the peak positions of DW’s spectra have a tiny red shift, and the frequency shift difference induced by Raman effect may exceed 0.1 at

_{R}*s*

_{1}= 0.2. Moreover, the peak powers of DW’s spectra are suppressed greatly by the Raman effect, and the reduced peak powers may exceed a factor of 8dB at

*s*

_{1}= 0.

## 4. Conclusion

We identified the combined effect of anomalous SS effect and TOD on DW generation in nonlinear MMs. Our results show that DW generation can be controlled by the anomalous SS effect in nonlinear MMs depending on the sign of SS coefficient and TOD parameter. For the positive value of TOD parameter, the amplitude of the DW peak increases rapidly with positive SS coefficient, and decreases with negative SS coefficient. However, the results are just opposite to the case of negative value of TOD parameter: the amplitude of the DW peak decreases rapidly with positive SS coefficient, and increases with negative SS coefficient. These results suggest that the DW generation can be manipulated as required in nonlinear MMs and nonlinear MMs are valuable candidates for applications in which a supercontinuum source is required.

## Acknowledgments

This work is supported by the National Basic Research Program (973 Program) of China (Grant No. 2012CB315701), the National Natural Science Foundation of China (Grant No. 61025024, 11004053, and 10974049), the China Postdoctoral Science Foundation (Grant No. 2012M511365), the Hunan Provincial Natural Science Foundation of China (Grant No. 11JJB001 and 12JJ7005), and the Young Teacher Development Plan of Hunan University.

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