Stability and interaction of few-cycle pulses in a Kerr medium

Zhan-Jie Gao; Hui-Jun Li; Ji Lin

doi:10.1364/OE.26.009027

1. Introduction

During the last two decades, the advanced experimental techniques in nonlinear optics and laser physics have allowed the generation and control of ultrashort pulses with duration of a few optical cycles [1–5]. These few-cycle pulses (FCPs) have extensive application in nonlinear optics, including high-order harmonic generation [6], supercontinuum generation [7–10] and filamentation [11, 12]. As a result, the theoretical studies of propagation characteristics of the FCPs in the nonlinear medium have recently aroused considerable interest [13–15]. Since the slowly varying envelope approximation (SVEA) is invalid for the FCPs, various models have been proposed to describe the propagation of the FCPs in the nonlinear medium, such as the generalized nonlinear Schrödinger-type equation [16–19], the modified Korteweg-de Vries (mKdV) equation [20,21], the sine-Gordon (sG) equation [20,21] and the coupled mKdV [22,23].

In 2004, Schäfer and Wayne derived the short-pulse equation (SPE) for the propagation of FCP in silica optical fibers [24]. The SPE is proven to be completely integrable [25–27]. The pulse solutions, single-hump solutions and multiloop solitons of SPE are given [25, 27–32]. These analytical solutions give an accurate description of the propagation characteristics of the FCP. However, the SPE does not exit smooth traveling waves solution [24]. Thus, the regularized short pulse equation (RSPE) which supports smooth traveling waves solution is proposed as a higher-order regularization of the SPE [33]. The RSPE can be written in an appropriate dimensionless form as [33]

u_{ξ τ} + α u + {(u^{3})}_{τ τ} + β u_{τ τ τ τ} = 0,

where u(ξ, τ) represents the magnitude of the electric field and subscripts ξ and τ appended to u denote the partial differentiation, while α and β are real parameters. The RSPE (1) was first derived [34] and the derivation process of the RSPE is similar with that of the mKdV-sG equation although their physical interpretation are slightly different [21, 35, 36]. Moreover, the RSPE (1) can be obtained from the mKdV-sG equation by a small amplitude approximation [23]. The RSPE (1) and its vector form have been successfully used by several authors in various studies including multisolitons with different polarization states and their binary-collision dynamics in a Kerr medium [37], dispersive broadening of FCP and supercontinuum generation in transparent nonlinear optical medium [38], nonlinear propagation dynamics of circularly polarized few-cycle soliton and pulse self-compression in a Kerr medium [39].

However, the RSPE (1) is non-integrable and no exact analytical solution has been obtained yet. It is difficult to obtain directly any analytical solution of the RSPE(1) even though we applied successfully the classical Lie-group reduction method to some non-intergrable equations in nonlinear optics to derive some analytical travelling solutions [40]. We notice that the RSPE (1) is reduced to the nonlinear Schrödinger (NLS) equation and high-order NLS (HNLS) equation by the multiple scales method and only the approximate multi-cycle soliton solutions are obtained [41]. In this paper, we intend to construct the approximate single FCP solution, two- and three-FCPs solutions of the RSPE. Simultaneously, we demonstrate numerically that the FCPs’ duration has a decisive effect on the stability of the single FCP. And we also investigate the interaction of the two- and three-FCPs by the numerical method and demonstrate that their interaction modes also depend on the pulse duration.

Our article is organized as follows. In the next section, the RSPE is introduced. And we demonstrate that the RSPE provides an accurate description of the dynamics of the FCP whose duration is larger than a single optical period when the FCP’s spectrum is in the medium’s anomalous dispersion regime. In section 3, we derive the NLS equation from RSPE by the multiple scales method. Then, we construct the single FCP solutions of RSPE from one soliton solution of the NLS equation and study their stability. We calculate the transition distance of these single FCP solutions. In section 4, we study the interaction behavior of two- and three-FCPs by the numerical method. Finally, in the section 5, a short summary and discussions are given.

2. Model equation

The propagation of the optical pulse in the one dimensional medium obeys the following equation

\frac{\partial^{2} E}{\partial x^{2}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} \int_{- \infty}^{t} ϵ (t - t^{'}) E (t^{'}) d t^{'} - \frac{4 π}{c^{2}} \frac{\partial^{2} P_{n l}}{\partial t^{2}} = 0

Here, E is the electric field, P_nl is the nonlinear polarizability, ϵ is the linear permittivity and c is the speed of light in the vacuum. The approximate expression of linear permittivity in the transparency window between two resonances can be written as [33, 37–39, 42, 43]

ϵ = ϵ_{0} - a_{1} \frac{ω_{0}^{2}}{ω^{2}} - a_{2} \frac{ω^{2}}{ω_{0}^{2}} - a_{3} \frac{ω^{4}}{ω_{0}^{4}} + \dots,

Here, ω₀ is the reference frequency and a₁, a₂, a₃ are dimensionless fit parameters. In many practical situations, the linear permittivity (3) should be truncated. Since the Fourier spectrum of FCP is very broad, the effect of higher order dispersion terms is very important. For example, the generalized nonlinear Schrödinger-type equation usually contains dispersive terms up to the sixth order [44]. According to this line of thought, ϵ is truncated at ω⁴ and can be written as

ϵ_{6} = ϵ_{0} - a_{1} ω_{0}^{2} / ω^{2} - a_{2} ω^{2} / ω_{0}^{2} - a_{3} ω^{4} / ω_{0}^{4}

. With the use of ϵ₆ and the nonlinear polarizability P_nl = χ⁽³⁾E³ (χ⁽³⁾ is the cubic nonlinear susceptibility), equation (2) can be reduced to the following equation

\frac{\partial^{2} u}{\partial ξ \partial τ} + α u + β \frac{\partial^{4} u}{\partial τ^{4}} + μ \frac{\partial^{6} u}{\partial τ^{6}} + \frac{\partial^{2} u^{3}}{\partial τ^{2}} = 0

where

u = 2 \sqrt{\frac{χ^{(3)} π}{a_{1}}} E

,

ξ = \frac{a_{1} ω_{0}}{2 \sqrt{ϵ_{0} c}} x

, and

τ = ω_{0} t - \frac{ω_{0} \sqrt{ϵ_{0}}}{c} x

is the dimensionless retarded time. And the parameters α = 1, β = a₂/a₁, µ = −a₃/a₁. However, as shown in [37, 39], the stable few-cycle soliton can be excited in the medium’s anomalous dispersion regime. So we just consider the medium’s linear permittivity in the anomalous dispersion regime. In this situtation, it is a good approximation to describe the linear permittivity using the first three terms of the linear permittivity (3) which is written as

ϵ_{4} = ϵ_{0} - a_{1} ω_{0}^{2} / ω^{2} - a_{2} ω^{2} / ω_{0}^{2}

. In this case, we ignore the sixth order dispersion term of Eq. (4) and obtain the RSPE

\frac{\partial^{2} u}{\partial ξ \partial τ} + α u + β \frac{\partial^{4} u}{\partial τ^{4}} + \frac{\partial^{2} u}{\partial τ^{4}} + \frac{\partial^{2} u^{3}}{\partial τ^{2}} = 0

In order to show the validity of the approximation linear permittivity ϵ₆ and ϵ₄, as examples, we use them to compare with the linear permittivity curves of fused silica and fluoride glass [45–47]. As shown in Fig. 1(a), ϵ₄ and ϵ₆ can give a good approximation to the linear permittivity curve of fused silica from 100 THz to 200 THz and from 75 THz to 420 THz, respectively. Figure 1(b) shows the fitting range of ϵ₄ and ϵ₆ for the fluoride glass are from 240 THz to 1250 THz and from 240 THz to 2100 THz, respectively. These two examples reveal that the ϵ₄ can apply to approximate the linear permittivity curve in the anomalous dispersion regime which is in the relatively low frequency region of the medium’s transparency window. Besides, Table I and II show that the FCPs’ spectrums are in the fitting range of ϵ₄ as the FCP’s center frequency is smaller than 160 THz (in the case of fused silica) or 900 THz (in the case of fluoride glass). But when the FCP’s center frequency increase, the spectrum of FCP exceeds the fitting range of the ϵ₄ and the higher terms need to be considered.

Fig. 1 (a) The black line denotes the linear permittivity curve of fused silica. The blue line denotes ϵ₄ with ω₀ = 100 THz, ϵ₀ = 2.111336, a₁ = 0.0965 and a₂ = 0.000133. The red line denotes ϵ₆ with ω₀ = 100 THz, ϵ₀ = 2.113213, a₁ = 0.100369, a₂ = 0.0003111 and a₃ = −0.00005125. ω_c = 230THz is the zero dispersion frequency of fused silica. (b) The black line denotes the linear permittivity curve of fluoride glass. The blue line denotes ϵ₄ with ω₀ = 100 THz, ϵ₀ = 1.5507, a₁ = 1.86222694 and a₂ = 0.0000007620895. The red line denotes ϵ₆ with ω₀ = 100 THz, ϵ₀ = 1.549467, a₁ = 1.8470735, a₂ = 0.00000172049 and a₃ = −0.00000010453898. ω_c = 1615 THz is the zero dispersion frequency of fused silica.

Download Full Size | PDF

Table 1. Comparison of the FCP’s spectrum and the linear permittivity’s fitting range for the fused silica

View Table | View all tables in this article

¹CF denotes the center frequency and its unity is THz.

Table 2. Comparison of the FCP’s spectrum and the linear permittivity’s fitting range for the fluoride glass

View Table | View all tables in this article

¹CF denotes the center frequency and its unity is THz.

3. Multiple scale analysis and stability of the single FCP

In order to construct the approximate FCP solutions by applying the multiple scale method, we introduce multiple-scale variables

τ_{0} = τ, τ_{1} = δ τ, ξ_{0} = ξ, ξ_{1} = δ ξ, ξ_{2} = δ^{2} ξ,

where δ is a small parameter. And we expand the dependent variable u as

u = δ u_{1} + δ^{2} u_{2} + δ^{3} u_{3} + \dots,

Here, u_j (j = 1, 2, 3, ⋯) are functions of the multiple-scale variables. Inserting the multiple-scale variables (6) and the expansion (7) into the RSPE (5), we obtain the following linear inhomogeneous equation

(\frac{\partial^{2}}{\partial τ_{0} \partial ξ_{0}} + β \frac{\partial^{4}}{\partial τ_{0}^{4}} + α) u_{j} = Γ^{(j)},

for j = 1, 2, 3, ⋯. The concrete expressions of Γ⁽^j⁾ can be systematically and analytically obtained and are omitted here.

At lowest order j = 1, equation (8) is a linear equation where Γ⁽¹⁾ = 0. Solving this linear equation, the first-order approximation solution reads as

u_{1} = A (τ_{1}, ξ_{1}, ξ_{2}, ξ_{3}) e^{i (Ω τ - k ξ)} + c . c .

where c.c. denotes the complex conjugate, k and Ω satisfy the dispersion relations k = βω³ +α/Ω. At order j = 2, by substituting u₁ into Eq. (8), we get

(\frac{\partial^{2}}{\partial τ_{0} \partial ξ_{0}} + β \frac{\partial^{4}}{\partial τ_{0}^{4}} + α) u_{2} = - i Ω (\frac{\partial A}{\partial ξ_{1}} + \frac{- 3 β Ω^{4} + α}{Ω^{2}} \frac{\partial A}{\partial τ_{1}}) e^{i (Ω τ - k ξ)} + c . c .

Cancelling the secular terms from the right hand side of the Eq. (10), we obtain the following linear equation

i (\frac{\partial A}{\partial ξ_{1}} + \frac{1}{v_{g}} \frac{\partial A}{\partial τ_{1}}) = 0,

where v_g = dΩ/dk = Ω²/(−3βΩ⁴ + α) is the velocity of the wave packet. At order j = 3, the cancellation of the secular terms yields the following nonlinear equation

i \frac{\partial A}{\partial ξ_{2}} - 6 β Ω \frac{\partial^{2} A}{\partial τ_{1}^{2}} + \frac{1}{Ω} \frac{\partial^{2} A}{\partial τ_{1} \partial ξ_{1}} - 3 Ω {| A |}^{2} A = 0 .

Using a coordinate transformation, we obtain the NLS equation

i \frac{\partial U}{\partial ξ} + P \frac{\partial^{2} U}{\partial T^{2}} + Q {| U |}^{2} U = 0,

where U = δA, T = τ − ξ/v_g, P = −3Ωβ − α/Ω³ and Q = −3Ω. Omitting the higher order terms u_j(j ≥ 2), the third-order approximate solution is

u = U (T, ξ) e^{i (Ω τ - k ξ)} + c . c .

where U is the solution of the NLS equation (13). From the one soliton solution of the NLS equation (13), the concrete expression of the third-order approximate solution (14) can be written as

u = 2 b \sqrt{\frac{2 P}{Q}} sech (b τ - b ξ / v_{g}) \cos (Ω τ + (P b^{2} + k) ξ)

with the conditions PQ > 0 and parameter b is a real constant. In order to test the stability of the approximate solution, the RSPE (5) is solved by the Runge-Kutta method. We take the approximate solution with white noise as an input, with α = 1, β = 1/250, Ω = 4, and vary the pulse duration. Here, we use the number of optical cycles

n = Ω \ln (2 + \sqrt{3}) / b π

to represent the pulse duration. The numerical simulation reveals that the single FCP’s maximum amplitude has two kinds of transition during the propagation. The first kind of transitions is shown in Figs. 2(a)–2(c) with n = 1.2. The profile and the velocity of the FCP do not change during the evolution as shown in Figs. 2(a) and 2(b). In Fig. 2(c), the spatial evolution behavior of the FCP’s maximum amplitude occurs a transition at ξ = 127. The FCP’s maximum amplitude changes oscillatorily between 0.315 and 0.186 as 0 ≤ ξ ≤ 127 and changes oscillatorily between 0 315 and 0.278 as 127 < ξ ≤ 500. The second kind of transitions is shown in Figs. 2(d)–2(f) with n = 1.17. Although the FCP’s width and amplitude do not change during the evolution, figure 2(e) shows a distinct phase difference between the initial FCP and the final FCP. In Fig. 2(f), the spatial evolution of the FCP’s maximum amplitude reveals a transition occurs at ξ = 206. The FCP’s maximum amplitude changes oscillatorily between 0.31 and 0.188 as 0 ≤ ξ ≤ 206 and changes oscillatorily between 0.25 and 0.188 as 206 < ξ ≤ 500. Comparing Fig. 2(c) with Fig. 2(f), it is obvious these two kinds of transition are different.

Fig. 2 (a) The evolution of the FCP with n = 1.2. (b) The blue (solid) line denotes the input FCP at ξ = 0 and the red (dashed) line denotes the output FCP at ξ = 500. The location of the FCP at ξ = 500 is artificially reset to the initial location. (c) The variation of the maximum value of the pulse with ξ. (d) The evolution of the FCP with n = 1.17. (e) The blue (solid) line denotes the input pulse at ξ = 0 and the red (dashed) line denotes the output FCP at ξ = 500. The location of the FCP at ξ = 500 is artificially reset to the initial location. (f) The variation of the maximum value of the pulse with ξ. In (a), (b) and (c), α = 1, β = 1/250, n = 1.2. Dispersion length for these parameters is L_d = 2.3. In (d), (e) and (f), α = 1, β = 1/250, n = 1.17. Dispersion length for these parameters is L_d = 2.1.

Download Full Size | PDF

For the sake of completeness, we calculate the transition distance ξ_tr for some different optical cycles n. The curve ξ_tr versus n is plotted in Fig. 3. For comparison, a benchmark exponentially function is also plotted. We see that the transition distance increases faster than the benchmark exponentially function as n ≥ 1.28. In addition, the transition distance becomes very long for n ≥ 1.28. For instance, the transition distance is 1303 as n = 1.28 which is a very large length for the nonlinear medium. Thus, we can hold that the FCP is very stable as the number of optical cycles n ≥ 1.28.

Fig. 3 The red line (dashed) denotes the curve of transition distance ξ_tr versus the number of optical cycles n and the blue (solid) line denotes an exponential growth. Here α = 1, β = 1/250, Ω = 4. For these parameters, the dispersion length is L_d = 2.6 when the FCP’s number of optical cycles is n = 1.28.

Download Full Size | PDF

4. Interaction of multi-FCPs

We try to obtain some multi-FCPs solutions of the RSPE(5) and investigate their interacting properties in this section. From the Hirota method [48], we can get the two-soliton solution of the NLS equation (13)

U = \frac{m_{1} e^{θ_{1}} + m_{2} e^{θ_{2}} + b_{11} e^{θ_{1} + θ_{2} + θ_{1}^{*}} + b_{12} e^{θ_{1} + θ_{2} + θ_{2}^{*}}}{1 + a_{11} e^{θ_{1} + θ_{1}^{*}} + a_{12} e^{θ_{1} + θ_{2}^{*}} + a_{21} e^{θ_{2} + θ_{1}^{*}} + a_{22} e^{θ_{2} + θ_{2}^{*}} + m_{11} e^{θ_{1} + t h e t a_{2} + θ_{1}^{*} + θ_{2}^{*}}}

with the conditions PQ > 0, where

θ_{j} = i P w_{j}^{2} ξ + w_{j} T

, w_j = w_j₁ + iw_j₂,

a_{j l} = Q m_{j} m_{l} / 2 P {(w_{j} + w_{l}^{*})}^{2}

,

b_{j l} = Q m_{j} m_{l}^{2} {(w_{j} - w_{l})}^{2} / 8 P w_{l 1}^{2} {(w_{j} + w_{l}^{*})}^{2}

with j, l = 1, 2. And m₁₁ = b₁₁b₁₂/m₁m₂, m₁, m₂, w₁₁, w₁₂, w₂₁, w₂₂ are arbitrary real constants. The approximate two-FCPs solution of the RSPE (5) can be obtained by substituting (16) into (14). In order to study the interaction properties of two-FCPs, we take the approximate solution with white noise as the initial data of RSPE (5) and get the revolution of FCP by the numerical method. We set parameters as α = 1, β = 1/250, Ω = 4, m₁ = m₂ = 1/10, w₁₂ = 1, w₂₂ = −1/2 and vary the number of optical cycles

n_{1} = (Ω + 1) \ln (2 + \sqrt{3}) / (w_{11} π)

,

n_{2} = (Ω - 1 / 2) \ln (2 + \sqrt{3}) / (w_{21} π)

. We find the interaction properties of the two-FCPs are affected by their pulse duration. Figure 4(a) shows an elastic collision between two-FCPs with n₁ = 1.28 and n₂ = 1.49. Figure 4(b) shows their input profile at ξ = −80 and output profile at ξ = 120. These two-FCPs move toward each other and collide. After the collision, they separate and move away without any change of the shape and velocity. Thus the interaction of two-FCPs is elastic. When one of the two-FCPs satisfies n_i ≤ 1.28 (i = 1, 2), the interaction of the two-FCPs becomes inelastic. Figure 5(a) shows an inelastic collision between two-FCPs with n₁ = 2.1, n₂ = 1.1. Figure 5(b) shows their input profile at ξ = −75 and output profile at ξ = 80. The number of optical cycles of the FCP 1 and FCP 2 are 2.1 and 1.1 respectively. In Fig. 5(a), we can see two-FCPs move toward each other and collide. After the collision, the FCP 2 become longer with a change of its velocity and the FCP 1 becomes shorter. Thus the interaction of these two-FCPs is inelastic. When two-FCPs satisfy n_i ≤ 1.28 (i = 1, 2), there is a repulsive interaction between two-FCPs. Figure 5(c) shows a repulsive interaction between two-FCPs with n₁ = 1, n₂ = 1. As shown in Fig. 5(c), they first move toward each other. In the process of collision, the propagation direction of the FCP 1 almost remains unchanged and the propagation direction of the FCP 2 changes gradually. Figure 5(d) shows their input profiles at ξ = −60 and output profiles at ξ = 700. Comparing the input and output profiles, the FCP 1 has a slight growth and the FCP 2 has a decay.

Fig. 4 (a) The elastic collision of two-FCPs. (b) The blue (solid) line is the initial profile of the FCPs at ξ = −80 and the red (dashed) line denotes the final profile of the FCPs at ξ = 120. Here w₁₁ = 5/4 and w₂₁ = 3/4.

Download Full Size | PDF

Fig. 5 (a) The inelastic collision of two-FCPs. (b) The blue (solid) line is the initial profile of the pulses at ξ = −75 and the red (dashed) line denotes the final profile of the pulses at ξ = 80. In (a) and (b), w₁₁ = 1 and w₂₁ = 133/100. (c) The repulsive interaction of two FCPs. (d) The blue (solid) line is the initial profile of the pulses at ξ = −60 and the red (dashed) line denotes the final profile of the pulses at ξ = 700. In (c) and (d), w₁₁ = 42/25 and w₂₁ = 73/50.

Download Full Size | PDF

We can also get the three-soliton solution of the NLS equation (13). The concrete expression of the solution is listed in the Appendix and the corresponding approximate solution of the RSPE (5) can be obtained. We study the interaction properties of the three-FCPs by the numerical method. We take the approximate solution with white noise as an input, with α = 1, β = 1/250, Ω = 4, m₁ = m₃ = 1/10, m₂ = 10, w₁₂ = 1, w₂₂ = −1/2, w₃₂ = 1/50 and vary the number of optical cycles $n_{1} = (Ω + 1) \ln (2 + \sqrt{3}) / (w_{11} π)$ , $n_{2} = (Ω - 1 / 2) \ln (2 + \sqrt{3}) / (w_{21} π)$ and $n_{3} = (Ω + 1 / 50) \ln (2 + \sqrt{3}) / (w_{31} π)$ . From the numerical simulation, we also can get the elastic collision, inelastic collision and repulsive interaction of the three-FCPs. Figure 6(a) shows an elastic collision between three-FCPs with n₁ = 2.4, n₂ = 1.96 and n₃ = 2.25. These three-FCPs attract each other and collide. After the collision, they separate and move away. Figure 6(b) shows the output profile at ξ = 300 is same as the input profiles at ξ = −150. Thus the interaction of these solitons is elastic. When one of the FCPs satisfies n_i ≤ 1.9 (i= 1, 2, 3), the collision is not elastic. Figure 7(a) shows an inelastic collision between three-FCPs with n₁ = 1.23, n₂ = 1.47 and n₃ = 1.35. These three-FCPs attract each other and collide. After the collision, they separate. In Fig. 7(b), the FCP 3 decays into chaotic oscillations and the FCP 3 transfers a part of its energy to FCP 1 and FCP 2. Figures. 7(c) and 7(d) show a repulsive interaction between three-FCPs with n₁ = 1, n₂ = 1 and n₃ = 1. On the one hand, the FCP 1 repels the FCP 2 and FCP 3. On the other hand, the FCP 2 and FCP 3 move toward each other and collide. After the collision, the FCP 2 has a distinct growth and the FCP 3 has a significant decay. Further more, these two collision FCPs exchange their propagation velocity after the collision.

Fig. 6 (a) The elastic collision of three-FCPs. (b) The blue (solid) line is the initial profile of the pulses at ξ = −150 and the red (dashed) line denotes the final profile of the pulses at ξ = 300. In (a) and (b), w₁₁ = 7/8, w₂₁ = 3/4 and w₃₁ = 3/4.

Download Full Size | PDF

Fig. 7 (a) The inelastic collision of three-FCPs. (b) The blue (solid) line is the initial profile of the pulses at ξ = −65 and the red (dashed) line denotes the final profile of the pulses at ξ = 250. (c) The repulsive interaction of the three pulses. (d) The blue (solid) line is the initial profile of the pulses at ξ = −90 and the red (dashed) line denotes the final profile of the pulses at ξ = 240. In (a) and (b), w₁₁ = 17/10, w₂₁ = 1 and w₃₁ = 5/4. In (c) and (d), w₁₁ = 42/25, w₂₁ = 73/50 and w₃₁ = 42/25.

Download Full Size | PDF

In the above numerical calculations, we have observed that the interactions of the two- and three-FCPs are complicated and strongly dependent on their pulse duration. Roughly speaking, the interaction of these multi-FCPs is elastic as n is larger than a critical number n_c and becomes inelastic as at least one of the FCPs satisfies n_i ≤ n_c (i = 1, 2, 3). For two-FCPs, the critical number is around 1.28. For three-FCPs, the critical number is around 1.9. The reason for these phenomena is that the interactions of solitons in nonintegrable system is dependent on their initial velocity [49–53]. And we find the initial propagation velocities of these multi-FCPs are related to their pulse duration which can be seen directly from their evolution.

5. Conclusions

The objective of this work is to present a detailed investigation on the dynamics of FCPs in the Kerr medium. We demonstrate that the RSPE (5) can be applied to describe the evolution of the FCP whose duration is larger than single optical cycle when the FCP’s spectrum is in the medium’s anomalous dispersion regime. We construct the single FCP solutions of the RSPE (5) from the soliton solutions of the NLS equation (13) by the multiple scales method and study their stability. We find that the pulse duration strongly affects the stability of the single FCP and two kinds of transitions occur due to the instability. The transition distance of these single FCPs are calculated. It reveals that the transition distance rises sharply as the pulse duration increases and the FCP is very robust when its duration is larger than 1.28 optical cycles. The different modes of multi-FCPs’ interaction are also illustrated with suitable pulse parameters. The numerical simulations reveal that the interactions of the two- and three-FCPs rely heavily on their pulse duration. Due to their robust nature and rich physical phenomena, we hope these FCPs will have application in the ultrashort pulse optics.

Appendix

From the Hirota method, the three soliton solution of NLS (13) can be written as

U = \frac{g_{1} + g_{3} + g_{5}}{1 + f_{2} + f_{4} + f_{6}},

with the conditions PQ > 0 and

\begin{array}{l} g_{1} = m_{1} e^{θ_{1}} + m_{2} e^{θ_{2}} + m_{3} e^{θ_{3}}, f_{2} = Q_{11} e^{θ 1 + θ_{2} + θ_{3} + θ_{1}^{*} + θ_{2}^{*} + θ_{3}^{*}}, \\ g_{3} = F_{12312} e^{θ_{1} + θ_{2} + θ_{3} + θ_{1}^{*} + θ_{2}^{*}} + F_{12323} e^{θ_{1} + θ_{2} + θ_{3} + θ_{2}^{*} + θ_{3}^{*}} + F_{12313} e^{θ_{1} + θ_{2} + θ_{3} + θ_{1}^{*} + θ_{3}^{*}}, \\ g_{5} = b_{121} e^{θ_{1} + θ_{2} + θ_{1}^{*}} + b_{122} e^{θ_{1} + θ_{2} + θ_{2}^{*}} + b_{123} e^{θ_{1} + θ_{2} + θ_{3}^{*}} + b_{131} e^{θ_{1} + θ_{3} + θ_{1}^{*}} + b_{132} e^{θ_{1} + θ_{3} + θ_{2}^{*}} \\ + b_{133} e^{θ_{1} + θ_{3} + θ_{3}^{*}} + b_{231} e^{θ_{2} + θ_{3} + θ_{1}^{*}} + b_{232} e^{θ_{2} + θ_{3} + θ_{2}^{*}} + b_{233} e^{θ_{2} + θ_{3} + θ_{3}^{*}}, \\ f_{4} = a_{11} e^{θ_{1} + θ_{1}^{*}} + a_{12} e^{θ_{1} + θ_{2}^{*}} + a_{13} e^{θ_{1} + θ_{3}^{*}} + a_{21} e^{θ_{2} + θ_{1}} + a_{22} e^{θ_{2} + θ_{2}} + a_{23} e^{θ_{2} + θ_{3}^{*}} \\ + a_{31} e^{θ_{3} + θ_{1}^{*}} + a_{32} e^{θ_{3} + θ_{2}^{*}} + a_{33} e^{θ_{3} + θ_{3}^{*}}, \\ f_{6} = M_{1212} e^{θ_{1} + θ_{2} + θ_{1}^{*} + θ_{2}^{*}} + M_{1213} e^{θ_{1} + θ_{2} + θ_{1}^{*} + θ_{3} *} + M_{1223} e^{θ_{1} + θ_{2} + θ_{2}^{*} + θ_{3}^{*}} \\ + M_{1312} e^{θ_{1} + θ_{3} + θ_{1}^{*} + θ_{2}^{*}} + M_{1313} e^{θ_{1} + θ_{3} + θ_{1}^{*} + θ_{3}^{*}} + M_{1323} e^{θ_{1} + θ_{3} + θ_{2}^{*} + θ_{3}^{*}} \\ + M_{2312} e^{θ_{2} + θ_{3} + θ_{1}^{*} + θ_{2}^{*}} + M_{2313} e^{θ_{2} + θ_{3} + θ_{1}^{*} + θ_{3}^{*}} + M_{2323} e^{θ_{2} + θ_{3} + θ_{2}^{*} + θ_{3}^{*}} . \end{array}

Here m₁, m₂, m₃ are real constants. And

θ_{j} = i P w_{j}^{2} ξ + w_{j} T

, w_j = w_j₁ + iw_j₂ when w_j₁, w_j₂ are real constants and j = 1, 2, 3. Other parameters have the following form

\begin{array}{l} a_{j l} = \frac{Q m_{j} m_{l}}{2 P {(w_{j} + w_{l}^{*})}^{2}}, (j, l = 1, 2, 3) \\ b_{j k l} = \frac{Q m_{j} m_{k} m_{l} {(w_{j} - w_{k})}^{2}}{2 P {(w_{j} + w_{l}^{*})}^{2} (w_{k} + w_{l}^{*})}, (j, k, l = 1, 2, 3) \\ M_{j k l o} = \frac{Q^{2} m_{j} m_{k} m_{l} m_{0} {(w_{j} - w_{k})}^{2} {(w_{l}^{*} - w_{o}^{*})}^{2}}{4 P^{2} {(w_{j} + w_{l}^{*})}^{2} {(w_{k} + w_{l}^{*})}^{2} {(w_{k} + w_{o}^{*})}^{2} {(w_{j} + w_{o}^{*})}^{2}}, (j, k, l, o = 1, 2, 3) \\ F_{j k s l o} = \frac{Q^{2} m_{j} m_{k} m_{s} m_{l} m_{0} {(w_{j} - w_{k})}^{2} {(w_{l}^{*} - w_{o}^{*})}^{2} {(w_{j} - w_{s})}^{2} {(w_{k} - w_{s})}^{2}}{4 P^{2} {(w_{j} + w_{l}^{*})}^{2} {(w_{k} + w_{l}^{*})}^{2} {(w_{k} + w_{o}^{*})}^{2} {(w_{j} + w_{o}^{*})}^{2} {(w_{s} + w_{l}^{*})}^{2} {(w_{s} + w_{o}^{*})}^{2}}, (j, k, s, l, o = 1, 2, 3) \\ Q_{11} = \frac{m_{1}^{2} m_{2}^{2} m_{3}^{2} Q^{3} {(w_{1}^{*} - w_{2}^{*})}^{2} {(w_{1} - w_{2})}^{2} {(w_{1} - w_{3})}^{2} {(w_{1}^{*} - w_{3}^{*})}^{2} {(w_{2}^{*} - w_{3}^{*})}^{2} {(w_{2} - w_{3})}^{2}}{512 w_{11}^{2} w_{21}^{2} w_{31}^{2} P^{3} {(w_{1} + w_{2}^{*})}^{2} {(w_{1} + w_{3}^{*})}^{2} {(w_{2} + w_{2}^{*})}^{2} {(w_{1} + w_{3}^{*})}^{2} {(w_{3} + w_{1}^{*})}^{2} {(w_{3} + w_{2}^{*})}^{2}} . \end{array}

It needs to mention that the parameters that don’t exist in equation (17) are equal to zero, such as M₁₁₁₁ = M₂₂₂₂ = M₃₃₃₃ = 0 and F₁₁₁₁₁ = F₂₂₂₂₂ = F₃₃₃₃₃ = 0.

Funding

Natural Science Foundation of Zhejiang Province (LZ15A050001); National Natural Science Foundation of China (NSFC) (11675146).

References and links

1. T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000). [CrossRef]

2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009). [CrossRef]

3. M. Wegener, Extreme Nonlinear Optics (Springer, 2006).

4. V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016). [CrossRef] [PubMed]

5. G. Mourou, S. Mironov, E. Khazanov, and A. Sergeev, “Single cycle thin film compressor opening the door to Zeptosecond-Exawatt physics,” Eur. Phys. J. Special Topics 223, 1181–1188 (2014). [CrossRef]

6. Z. Chen and A. Pukhov, “Bright high-order harmonic generation with controllable polarization from a relativistic plasma mirror,” Nat. Commun. 7, 12515 (2016). [CrossRef] [PubMed]

7. B. Spokoyny, C. J. Koh, and E. Harel, “Stable and high-power few cycle supercontinuum for 2D ultrabroadband electronic spectroscopy,” Opt. Lett. 40, 1014–1017 (2015). [CrossRef] [PubMed]

8. P. He, Y. Liu, K. Zhao, H. Teng, X. He, P. Huang, H. Huang, S. Zhong, Y. Jiang, S. Fang, X. Hou, and Z. Wei, “High-efficiency supercontinuum generation in solid thin plates at 0.1 TW level,” Opt. Lett. 42, 474–477 (2017). [CrossRef] [PubMed]

9. H. Leblond, Ph. Grelu, and D. Mihalache, “Models for supercontinuum generation beyond the slowly-varying-envelope approximation,” Phys. Rev. A 90, 053816 (2014). [CrossRef]

10. H. Leblond, Ph. Grelu, D. Mihalache, and H. Triki, “Few-cycle solitons in supercontinuum generation dynamics,” Eur. Phys. J. Special Topics 225, 2435–2451 (2016). [CrossRef]

11. D. Grossmann, M. Reininghaus, C. Kalupka, M. Kumkar, and R. Poprawe, “Transverse pump-probe microscopy of moving breakdown, filamentation and self-organized absorption in alkali aluminosilicate glass using ultrashort pulse laser,” Opt. Express. 24, 23221–23231 (2016). [CrossRef] [PubMed]

12. R. Šuminas, G. Tamošauskas, V. Jukna, A. Couairon, and A. Dubietis, “Second-order cascading-assisted filamentation and controllable supercontinuum generation in birefringent crystals,” Opt. Express. 25, 6746–6756 (2017). [CrossRef] [PubMed]

13. S. V. Sazonov, “On the nonlinear optics of few-cycle pulses,” Bull. Rus. Acad. Sci.: Physics 75, 157–160 (2011).

14. H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013). [CrossRef]

15. D. J. Frantzeskakis, H. Leblond, and D. Mihalache, “Nonlinear optics of intense few-cycle pulses: An overview of recent theoretical and experimental developments,” Rom. J. Phys. 59, 767–784 (2014).

16. T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78, 3282 (1997). [CrossRef]

17. M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069 (1999). [CrossRef]

18. A. A. Voronin and A. M. Zheltikov, “Soliton-number analysis of soliton-effect pulse compression to single-cycle pulse widths,” Phys. Rev. A 78, 063834 (2008). [CrossRef]

19. A. Kumar and V. Mishra, “Single-cycle pulse propagation in a cubic medium with delayed Raman response,” Phys. Rev. A 79, 063807 (2009). [CrossRef]

20. H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Phys. Rev. A 67, 013804 (2003). [CrossRef]

21. H. Leblond and D. Mihalache, “Few-optical-cycle solitons: Modified Korteweg–de Vries sine-Gordon equation versus other non–slowly-varying-envelope-approximation models,” Phys. Rev. A 79, 063835 (2009). [CrossRef]

22. H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Robust circularly polarized few-optical-cycle solitons in Kerr media,” Phys. Rev. A 83, 063802 (2011). [CrossRef]

23. H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Circularly polarized few-optical-cycle solitons in Kerr media: A complex modified Korteweg-de Vries model,” Opt. Commun. 285, 356–363 (2012). [CrossRef]

24. T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004). [CrossRef]

25. K. K. Victor, B. B. Thomas, and T. C. Kofane, “On exact solutions of the Schä er-Wayne short pulse equation: WKI eigenvalue problem,” J. Phys. A 39, 5585 (2007). [CrossRef]

26. J. C. Brunelli, “The bi-Hamiltonian structure of the short pulse equation,” Phys. Lett. A 353, 475–478 (2006). [CrossRef]

27. A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005). [CrossRef]

28. A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, 361–367 (2006). [CrossRef]

29. Y. Matsuno, “Periodic solutions of the short pulse model equation,” J. Math. Phys. 49, 073508 (2008). [CrossRef]

30. E. J. Parkes, “Some periodic and solitary travelling-wave solutions of the short-pulse equation,” Chaos Solitons Fractals 38, 154–159 (2008). [CrossRef]

31. Y. Matsuno, “Multiloop soliton and multibreather solutions of the short pulse model equation,” J. Phys. Soc. Jpn. 76, 084003 (2007). [CrossRef]

32. Y. Shen, F. Williams, N. Whitaker, P. G. Kevrekidis, A. Saxen, and D. J. Frantzeskakis, “On some single-hump solutions of the short-pulse equation and their periodic generalizations,” Phys. Lett. A 374, 2964–2967 (2010). [CrossRef]

33. N. Constanzino, V. Manukian, and C. K. R. T. Jones, “Solitary waves of the regularized short pulse and Ostrovsky equations,” SIAM J. Math. Anal. 41, 2088–2106 (2009). [CrossRef]

34. S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP. 84, 221–228 (1997). [CrossRef]

35. H. Leblond, S. V. Sazonov, I. V. Mel’nikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006). [CrossRef]

36. H. Leblond, I. V. Mel’nikov, and D. Mihalache, “Interaction of few-optical-cycle solitons,” Phys. Rev. A 78, 043802 (2008). [CrossRef]

37. A. V. Kim and S. A. Skobelev, “Few-cycle vector solitons of light,” Phys. Rev. A 83, 063832 (2011). [CrossRef]

38. V. G. Bespalov, S. A. Kozlov, Y. A. Shpolyansky, and I. A. Walmsley, “Simplified field wave equations for the nonlinear propagation of extremely short light pulses,” Phys. Rev. A 66, 013811 (2002). [CrossRef]

39. S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007). [CrossRef]

40. J. Jia and J. Lin, “Solitons in nonlocal nonlinear kerr media with exponential response function,” Opt. Express 20, 7469–7479 (2012). [CrossRef] [PubMed]

41. Y. Shen, T. P. Horikis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Traveling waves of the regularized short pulse equation,” J. Phys. A 47, 315204 (2014). [CrossRef]

42. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

43. M. Born and E. Wolf, Principles of Optics (Pergamon, 1968).

44. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

45. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642 (1997). [CrossRef]

46. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965). [CrossRef]

47. C. Z. Tan, “Determination of refractive index of silica glass for infrared wavelengths by IR spectroscopy,” J. Non-Cryst. Solids 223, 158–163 (1998). [CrossRef]

48. R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys. 14, 805 (1973). [CrossRef]

49. M. J. Ablowitz, M. D. Kruskal, and J. R. Ladik, “Solitary wave collisions,” SIAM J. Appl. Math 36, 428–437 (1979). [CrossRef]

50. J. Yang and Y. Tan, “Fractal structure in the collision of vector solitons,” Phys. Rev. Lett. 85, 3624 (2000). [CrossRef] [PubMed]

51. Y. Tan and J. K. Yang, “Complexity and regularity of vector-soliton collisions,” Phys. Rev. E 64, 056616 (2001). [CrossRef]

52. P. Dorey, K. Mersh, T. Romanczukiewicz, and Y. Shnir, “Kink-Antikink Collisions in the ϕ⁶ Model,” Phys. Rev. Lett. 107, 091602 (2011). [CrossRef]

53. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010). [CrossRef]

Few cycle pulse				Fitting range of ϵ₄ overlaps the spectrum	Fitting range of ϵ₆ overlaps the spectrum
FCP	CF¹	Pulse duration	Spectrum (THz)	Fitting range of ϵ₄ overlaps the spectrum	Fitting range of ϵ₆ overlaps the spectrum
FCP1	142	11 fs (1.5 cycles)	100 – 184	Yes	Yes
FCP2	150	10 fs (1.5 cycles)	106 – 194	Yes	Yes
FCP3	150	6.6 fs (1 cycle)	84 – 216	No	Yes
FCP4	160	9.4 fs (1.5 cycle)	113 – 207	No	Yes

Few cycle pulse				Fitting range of ϵ₄ overlaps the spectrum	Fitting range of ϵ₆ overlaps the spectrum
FCP	CF¹	Pulse duration	Spectrum (THz)	Fitting range of ϵ₄ overlaps the spectrum	Fitting range of ϵ₆ overlaps the spectrum
FCP5	450	1.98 fs (1 cycle)	252 – 648	Yes	Yes
FCP6	745	1.34 fs (1 cycle)	417 – 1072	Yes	Yes
FCP7	745	0.9 fs (0.7 cycle)	277 – 1213	Yes	Yes
FCP8	900	1.11 fs (1 cycle)	504 – 1296	No	Yes

Few cycle pulse				Fitting range of ϵ₄ overlaps the spectrum	Fitting range of ϵ₆ overlaps the spectrum
FCP	CF¹	Pulse duration	Spectrum (THz)	Fitting range of ϵ₄ overlaps the spectrum	Fitting range of ϵ₆ overlaps the spectrum
FCP1	142	11 fs (1.5 cycles)	100 – 184	Yes	Yes
FCP2	150	10 fs (1.5 cycles)	106 – 194	Yes	Yes
FCP3	150	6.6 fs (1 cycle)	84 – 216	No	Yes
FCP4	160	9.4 fs (1.5 cycle)	113 – 207	No	Yes

Few cycle pulse				Fitting range of ϵ₄ overlaps the spectrum	Fitting range of ϵ₆ overlaps the spectrum
FCP	CF¹	Pulse duration	Spectrum (THz)	Fitting range of ϵ₄ overlaps the spectrum	Fitting range of ϵ₆ overlaps the spectrum
FCP5	450	1.98 fs (1 cycle)	252 – 648	Yes	Yes
FCP6	745	1.34 fs (1 cycle)	417 – 1072	Yes	Yes
FCP7	745	0.9 fs (0.7 cycle)	277 – 1213	Yes	Yes
FCP8	900	1.11 fs (1 cycle)	504 – 1296	No	Yes

Stability and interaction of few-cycle pulses in a Kerr medium

Abstract

1. Introduction

2. Model equation

3. Multiple scale analysis and stability of the single FCP

4. Interaction of multi-FCPs

5. Conclusions

Appendix

Funding

References and links

Cited By

Figures (7)

Tables (2)

Equations (19)

Optics Express