1. Introduction

Practical photonic devices require a low input power, high conversion efficiency and flexible functionality. One challenge in designing and building such a device is to achieve convenient and efficient frequency conversion.

Nonlinear frequency conversion in bulk a medium, especially using third-order nonlinear effect, requires high intensity and long material length because of the small nonlinear susceptibility [1]. In the case of stimulated Raman scattering (SRS) with a quasi-continuous wave as the pump, the intensity of the Stokes wave is given by [1]

Where I_p and I_s are the intensities of the pump and the Stokes waves respectively; I_SN represents the intensity of the weak Stokes seed; g_R represents Raman gain coefficient; L is the interaction length between the pump and the Stokes waves, which is related to the interaction time t by L = (c/n₀ )t, with n₀ the average refractive index. For a liquid N ₂, as an example, with the Raman gain coefficient of g_R = 2 × 10^-10 m/W [2], the threshold of interaction length to generate the Stokes wave was demonstrated in theory as L = 17 cm by Eq. (1) and observed in experiment as L = 15 cm [3], for the pump wave of 6.5 ns duration at 60 MW/cm² pump intensity. The 15 cm interaction length is correspondent to the nonlinear interaction time of the order of 1 ns. Hence if the period of the stopping time of the pulse is over 1 ns, it should be possible to achieve the same efficiency of conversion. In this Letter, we propose a new scheme that utilizes zero-velocity (ZV) optical pulse as pump to enhance nonlinear frequency conversion via SRS, and we demonstrate that a very efficient third-order nonlinear frequency conversion can be realized in a compact photonic device.

2. Theory

In a bulk medium, the temporal walk-off effects of SRS between the pump and frequency-shifted pulses can be negligible because the difference in the group velocities is small and the temporal walk-off length is tens of meters, which is much larger than the interaction length [4]. However, the difference in the group velocities can be very large in the case of SRS with ZV pulse as the pump. Hence the basic requirement to convert the energy of ZV pulse to another frequency is to reduce or even to cancel the walk-off effects. The ideal case is that the generated signal should also be a ZV soliton. If not possible, slow propagation of generated nonlinear signal would help to increase the nonlinear optical interaction hence the conversion efficiency.

A decelerated or stopped light pulse can be obtained with advanced technology at present. Experiments have demonstrated that the light pulses in a fiber Bragg grating (FBG) can be reduced to 50% of the light speed through the Kerr effect [5]. Use of a defect inside a photonic band gap (PBG) [6], creation of a Raman gap soliton (GS) [7], or GS collision [8] were proposed to generate ZV short light, although they have not yet been demonstrated experimentally due to the requirement of high input intensity. Optical pulses were also stopped by electromagnetically induced transparency (EIT) in a laser-induced grating [9, 10]. However, the intensity was too low to stimulate nonlinear processes such as SRS for the required long duration pump pulses. Based on EIT, a new all-optical mechanism was designed to stop light with linear optics [11, 12]. It was also predicted that the light can be decelerated and stopped by a PBG based on second-harmonic generation [13]. Recently, a structure of resonantly absorbing Bragg reflectors (RABR), a periodic array of thin layers of resonant two-level systems separated by half-wavelength nonabsorbing dielectric layers, was proposed and numerically demonstrated to be able to decelerate or stop short light pulse as ZV GS [14–19].

Reduction of the temporal walk-off effects between the pump and Stokes pulses can be realized with a one-dimensional (1D) doubly resonant Bragg reflector (DRBR), which is similar to a superposed grating to generate multicomponent gap solitons [20]. This structure consists of 1D periodically arranged thin atomic layers with the period at half of the pump wavelength and a passive Bragg grating with the stop band at the Stokes wavelength. The two stop bands are not overlapped in the spectra and no passive Bragg grating is at the pump wavelength because the bandwidth of each stop band is ~100 cm^-1 while the SRS has a typical frequency shift at hundreds of cm^-1. On the one hand, the pump pulse can be decelerated and stopped by the thin atomic layers [17, 18]. On the other hand, the Stokes pulse can be generated as a stopped or slowly oscillating soliton by the passive Bragg reflector [7]. By the interaction of these two kinds of stopped light pulses, the energy of the pump pulse can be efficiently shifted to the Stokes pulse. Furthermore, as described in Ref [7], the energy of the Stokes pulse will eventually leak out from both sides of the finite length of DRBR.

By contrast, the generation of Raman GS [7] without ZV GS as the pump was difficult in experiment because the temporal walk-off between the pump and Stokes pulses was significant. In that case, the pump pulse had to be traveling waves while the Raman GS was a slowly oscillating field inside the sample [7]. Thus the pump pulse width had to be very large so as to treat it as a continuous wave (cw) to make sure that the Stokes pulse did not separate with the pump pulse until the SRS threshold was achieved. Therefore, a very high pulse energy density (200 J/cm²) with high intensity (400 GW/cm²) and long pulse duration (510 ps, corresponding to the length of 3 cm FBG) was needed. As a result, conversion efficiency would be very low and optical damage was likely to occur before SRS was observed.

The dynamics of a doubly resonant SRS process can be described by the Maxwell-Bloch (M-B) equations. Higher-order Stokes signals, which can be efficiently generated in a bulk medium, can be neglected in DRBR because their wavelengths are different from any Bragg resonance, and they would walk off quickly from the pump and the first Stokes component. As we shall see in the following discussion, the interaction time of SRS is determined by the period of the stopped pulses, which is in the order of ns and much longer than the Raman transverse relaxation time (orders of ten of ps).Thus the effect of Raman transverse relaxation can be neglected. Consulting the two-wave Maxwell-Bloch equations (TWMB) in Ref. [15] and referring to the Raman and Kerr effects in Refs. [4] and [7], we have M-B equations for the forward and backward pump amplitudes (${E^{\pm }}_p$) and Stokes amplitudes (${E^{\pm }}_s$) given by

Where ${{\sum }^{\pm }}_{p\mathit{,}s}$ = (2μτ_c /ħ)${{\sum }^{\pm }}_{p\mathit{,}s}$; τ_c = (2ħn ₀ /μ ₀ c ² ω_p μ ² ρ)^1/2 is the cooperative resonant absorption time; n ₀ is the average refractive index of the DRBR; μ ₀ is the vacuum permeability; c is the speed of light in vacuum; ω_p and ω_s are the frequencies of the pump and the Stokes waves; μ is the dipole matrix element; ρ is the density of two-level systems; P and n are the polarization and density of inverse population; ζ = zn ₀ /cτ_c and τ = t /τ_c are the dimensionless spatial coordinate and time, respectively; Δ = δcτ_c /n ₀ and K = κcτ_c /n ₀ are the dimensionless detuning and coupling constant corresponding to the detuning and coupling constant of the passive Bragg grating, respectively; f_R is the fraction of the nonlinearity arising from molecular vibrations, and a typical value is 0.18 [21]; Γ_p,s = μ_p,s n ₂ ε ₀ cħ ²/8,τ ² τ_c are the dimensionless nonlinear coefficients for the pump and the Stokes waves, with n ₂ the nonlinear index coefficient; ε ₀ is the vacuum permittivity; G_s = (ε ₀ c ² ħ ²/8μ ² τ_c )g_R and G_p = (ω_p /ω_s )G_s are the dimensionless gain coefficients, with g_R the Raman gain coefficient.

On the right-hand side of M-B Eqs. (2a) and (2b), the first term represents self- and crossed-phase modulation by Kerr and Raman effects, which modified both the pulses shapes and spectra, the same effect as in the bulk Raman active medium [4]; the second term represents the Raman gain; the third term of Eq. (2a) describes the polarization caused by the periodic atomic layers; the last terms of Eq. (2b) are the effects of coupling coefficient and grating detuning of the passive Bragg reflect. The Bloch equations for the polarization P and inversion n induced by the pump in the periodic atomic layers are given in Eqs. (2c) and (2d).

Using the parameters above, the energy density of the pump and the Stokes pulses can be expressed as [14]

where |${{\sum }^{+}}_{p\mathit{,}s}$|²+|${{\sum }^{-}}_{p\mathit{,}s}$|² is the dimensionless energy density. The intensity of the forward and backward pulses can be expressed as

And Eq. (1) can be rewritten as

where |${{\sum }^{\pm }}_{SN}$|²represents the dimensionless intensity of the weak Stokes seed.

We assume no initial polarization and population inversion in DRBR, and a sech-shaped pulse with pulse width τ ₀ and peak intensity ∑₀ entering the system from the left side. The forward and backward Stokes waves evolve from a weak cw Stokes seed with intensity ${{\sum }^{\pm }}_{SN}$. The equations are solved numerically with the 4th order Rugge-Kutta numerical method [22].

3. Results

First we assume that there is no Raman gain, i.e., G_s = G_p = 0, but with Kerr effect in DRBR in order to simulate how the pump pulse would evolve. In this case, the influence of the Raman wave propagation to the pump field can be ignored, i.e., ${{\sum }^{\pm }}_S$ = 0, and the Eqs. (2) can be reduced to the two-wave Maxwell-Bloch equations (TWMB) [15] with Kerr effect. Other parameters are as follows: the dimensionless nonlinear coefficients are Γ_p= 1×10^-3 and Γ_s= 9.5× 10^-4, corresponding to the ratio of the Stokes frequency to the pump frequency at ω_s /ω_p = 0.95; the intensity of the weak Stokes seed is ${{\sum }^{\pm }}_{SN}$ = 10^-5, the incident laser field’s pulse width ∑₀ = 1.5τ_c and peak intensity ∑₀ = 2.1507, similar to the pulse width and intensity used to generate a ZV GS in an RABR [16–19]; and the dimensionless spatial coordinate, coupling constant and detuning are ζ = 6, K = 4.5, and Δ = 0, respectively.

Figure 1 shows the evolution of the pump pulse into a stable stopped short light pulse. The trace of the stopped pulse is expressed with the population density n (Fig. 1(a)) and energy density |${{\sum }^{+}}_p$|²+|${{\sum }^{-}}_p$|² (Fig. 1(b)). This stopped pulse consists of forward and backward components of equal energy density |${{\sum }^{\pm }}_p$|² = 1.4, which are coupled by the thin atomic layers [18]. The instability of the pump soliton is caused by the nonlinear optical Kerr effect. Note that our simulations (not shown here) show that the transverse and longitudinal relaxation times of the atomic layers affect the population density [18] but not the field distribution, thus the basic character of pulse localization will not change, the effect of relaxation times can be neglected.

 

Fig. 1. Evolution of (a) the population inversion n and (b) the energy density (|${{\sum }^{+}}_p$|²+|${{\sum }^{-}}_p$|²) of the pump pulse in DRBR without SRS but with optical Kerr effect. Here τ ₀ = 1.5τ_c , ∑₀ = 2.1507, K = 4.5 and Δ = 0.

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We further consider the situation with Raman gain. By substituting |${{\sum }^{\pm }}_p$|² = 1.4, |${{\sum }^{\pm }}_{\mathit{SN}}$|² = 10^-10, and a given interaction time at τ = 1500 into Eq. (5), we obtain the required gain coefficient of the Stokes wave with G_s = 9.5 × 10^-3.

Using G_p = 1 × 10^-2, G_s = 9.5 × 10^-3 and other parameters in Fig. 1, the evolution of the pump pulse and the generation of the Stokes pulse in DRBR are shown in Fig. 2. The energy density of the pump wave (|${{\sum }^{+}}_p$|²+|${{\sum }^{-}}_p$|²) and the Stokes wave (|${{\sum }^{+}}_s$|²+|${{\sum }^{-}}_s$|²) are shown in Figs. 2(a) and 2(b). In this case, the pump pulse is first stopped as a solitary wave, as shown in Fig. 1. When the interaction time between the stopped soliton and the Raman active medium is greater than τ = 1500, the Stokes pulse starts to increase its intensity at the expense of the pump fields. During this process, both group velocities of the pump and the Stokes pulses are equal to zero, as the Stokes pulse is also trapped inside the grating as a stopped GS [7].

 

Fig. 2. Evolution of the energy density of (a) the pump pulse (|${{\sum }^{+}}_{p\mathit{,}s}$|²+|${{\sum }^{-}}_p$|²) and (b) the Stokes pulse (|${{\sum }^{+}}_s$|²+|${{\sum }^{-}}_s$|²) in DRBR with SRS. Here G_p = 1 × 10^-2, G_s = 9.5 × 10^-3 and other parameters are the same as in Fig. 1.

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After the efficient power exchange between the pump and the Stokes pulses, the energy of the Stokes pulse will leak out from both sides of the DRBR and the Stokes waves extend as nanosecond pulses. Figure 3 shows this process on the right-hand side of the DRBR. The same output pulse can be obtained from the left-hand side of the sample. The efficiency of the Raman shift can be estimated by comparing the output energy of the Stokes pulses with the energy of the stopped pump pulse. Remarkably, this efficiency can be greater than 80%, i.e., nearly 85% of photons can shift from the pump pulse to the Stokes pulse.

Practically, this efficient conversion can be achieved by using periodically arranged multiquantum wells structure as an example [23]. In In_xGa_1-xAs quantum wells embedded in GaAs the average refraction index is n ₀ = 3.6; the excitation resonant wavelength is 833 nm, corresponding to the angular frequency at ω_p = 2.26 × 10¹⁵ Hz; the density of two-level systems is ρ = 10²³ m^-3; and the dipole matrix element is μ = 2.5 × 10^-30 Cm. Then the cooperative resonant absorption time is τ_c = 4 ps. If a passive Bragg grating is written on the sample with the coupling constant of κ = 140 cm^-1, the Raman shift at 600 cm^-1, and the Raman gain and nonlinear index coefficients at g_R = 2.4 × 10^-10 m/W and n ₂ = 2.4 × 10^-10 m²/W, it will be possible to observe the enhanced frequency conversion. For the incident laser pulse of 750 μJ/cm², with 6 ps duration at 60 MW/cm², and the Stokes seed of 1 mW/cm² (level of quantum noise), the required length of the DRBR is 2 mm, and the stopping time of the pump pulse exceeds 6 ns. Actually, the pump pulse can be stopped with a different shape, duration, and peak intensity, which depends on the pulse area and τ_c [18]. The pulse stopping is also possible with the transverse distribution of the pulse intensity taken into account [19]. In comparison, the required power density for the generation of Raman GS in a Bragg grating is 200 J/cm², which is about five orders of magnitude greater than that in the DRBR. By substituting the Raman gain coefficient and light intensity parameters above into Eq. (1), the threshold length to generate the Stokes pulse in a bulk medium is 140 mm, which is 70 times greater than the length of DRBR (2 mm). This threshold length in a bulk medium will be greater for a smaller Raman gain with the same pump intensity. However, the required length of DRBR is not directly related to the Raman gain, as the pump pulse exists inside the sample as a stopped short light field. Furthermore, it is difficult to achieve the conversion efficiency at 85% by SRS from one frequency to another in the bulk medium because higher-order Stokes signals would be generated [1]. By contrast, the 85% conversion efficiency is easily obtainable in DRBR as no higher-order Stokes signals will be produced as a result of the pump and signal walk-off.

 

Fig. 3. The energy density of the Stokes pulse output from right-hand side (|${{\sum }^{+}}_s$(ζ = 6)|²) of DRBR. The same Stokes pulse output from left-hand side (|${{\sum }^{-}}_s$(ζ = 0)|²) is not shown. The parameters are the same as in Fig. 2.

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4. Conclusions

In conclusion, a new scheme of using stopped short light pulses to generate SRS in a DRBR structure is proposed and numerically demonstrated. The nonlinear optical conversion is shown to be considerably more efficient than those in a bulk medium, and the required low input pulse energy is easily obtainable and so avoids nonreversible material damage by the incident laser. Although the theory in this work is applied specifically to enhance nonlinear Raman conversion, it is believed that the stopped short light pulses with doubly resonant structure should be a generally applicable technique for enhancing different kinds of nonlinear optical conversion.