Stimulated Raman amplification in GaAs/AlAs intermixed superlattices

The enhancement of stimulated Raman scattering (SRS) with a GaAs/AlAs intermixed superlattice that works as a χ(3)-quasi-phase-matched structure is studied by taking account of Kerr-induced effects, e.g., four-wave mixing (FWM), self-phase modulation (SPM), cross-phase modulation (XPM), and two-photon absorption (TPA). In particular, the efficiency of anti-Stokes generation is enhanced here, which inherently has an extremely small efficiency due to a phase mismatch in the interaction of the pump, Stokes, and anti-Stokes waves (while the efficiency of Stokes generation is sufficiently large because of no such phase mismatch). The superlattice enhances the anti-Stokes efficiency up to the order of 103 when compared with that without the superlattice, particularly at a small pump intensity. In this enhancement, it is seen that there is an efficiency boost via simultaneous FWM. In this situation, it is shown how much SPM and XPM degrade the efficiency enhancement. Furthermore, an optimal superlattice length is identified that provides the highest efficiency. The degradation of the efficiency at the optimized length due to TPA is also analyzed. Finally, to gain more anti-Stokes efficiency (or control the sizes of the Stokes and anti-Stokes efficiencies), a photonic-bandgap cavity structure is proposed. PACS numbers: 42.65.Dr, 42.65.Hw, 42.65.Wi, 71.55.Eq ∗ I.Tomita@soton.ac.uk


I. INTRODUCTION
Studies on the enhancement of stimulated Raman scattering (SRS) have been attracting a great deal of attention in recent years, aiming for its potential scientific and technological applications, e.g., probing/sensing [1,2], optical on-chip [3][4][5][6][7], and telecom [8,9] applications. Among them, the realization of SRS in silicon (Si) [3][4][5][6][7] gave a great impetus to the development of Si on-chip optical devices at telecom wavelengths, which provided a potential integrated optical source with Si photonics. However, there still remains an issue that large two-photon absorption (TPA) in Si at around 1.55 µm must be avoided for practical applications and yet cannot easily be resolved because the bandgap energy (1.1 eV) of Si is much smaller than the TPA energy (2 × 0.8 = 1.6 eV) of 1.55-µm optical signals.
Semiconductor optical amplifiers [10] made of compound semiconductors can avoid TPA via bandgap engineering with composition-ratio and/or strain controls, but their carriers, or electron-hole pairs, that give rise to radiative recombination for optical signal amplification distort the signal waveforms via plasma effects, thus preventing the operation speed from going beyond tens of gigahertz without additional systems. Fortunately, silica-glass optical fibers (SiO 2 , insulator with no carriers) have a wide bandgap that can avert TPA at around 1.55 µm, but their Raman gain coefficient g (cm/GW) is very small compared with that of semiconductors, e.g., Si, GaP, AlGaAs (see Table I), thus requiring a very long fiber on the order of kilometers.
Here, we propose a compact device design that utilizes compound-semiconductor superlattices that can avoid TPA whilst utilizing the third-order optical nonlinearity enhancement, and study the enhancement behaviors from nonlinear optical physics, where the use of instantaneous nonlinear interactions (with no TPA) provides this superlattice device with potential that goes beyond the current operation limitations of Si-photonic devices and semiconductor optical amplifiers.

A. Research target, method, and applications
In this paper, we focus on a merit of our device that can boost the output efficiency of the anti-Stokes waves that originally have a vanishingly small efficiency due to a large phase mismatch ∆k = 2k 0 − k −1 − k 1 that arises in the interacting pump, Stokes, and anti-Stokes waves of wavevectors k 0 , k −1 , and k 1 , respectively [17]. Fortunately, the Stokes waves are strongly generated in this device because of the absence of such phase mismatch requirement.
For example, the ratio of the anti-Stokes-wave power to the Stokes-wave power, e.g., in Si waveguides, is only 10 −5 -10 −6 [18,19]. But our proposed device should provide the same order of output power for both Stokes and anti-Stokes waves, as will be seen in §IV B 4. An application of the anti-Stokes-wave amplification with our device is that, since it generates a short wavelength (e.g., 1.49 µm) from an input pump beam (e.g., at 1.56 µm), it can amplify the downstream signals at 1.49 µm in gigabit-ethernet passive optical networks (GE-PONs) [20]. It can also be used for channel conversion and amplification in wavelength-division multiplexing (WDM) systems.
To attain high amplification by removal of ∆k, we employ quasi-phase matching (QPM) [21] for χ (3) -processes, in a collinear beam configuration (e.g., in a waveguide) that gives a strong field coupling among those three waves and thus achieves high output efficiency.
In this paper, instead of QPM structures proposed in SiO 2 , Ba(NO 3 ) 2 , and H 2 [12,14], which are not easily fabricated, we employ a GaAs/AlAs intermixed superlattice, which was originally developed as a χ (2) -periodic QPM structure [22]. This has a small refractiveindex change (∼ 0.01) in χ (2) -domains (i.e., only a 0.3 % change compared with the average refractive index in the periodic structure) [23], and has an advantage of avoiding significant additional optical scattering loss caused by interface roughness between the discrete χ (2)domains in orientation-patterned AlGaAs structures [24].
An important aspect in the GaAs/AlAs intermixed superlattice is that it also works as a χ (3) -periodic QPM structure because a χ (3) -change is also induced in a similar way to the χ (2) -change caused by blue-shifted resonance in nonlinear susceptibility [25]. The removal of ∆k with this χ (3) -periodic structure realizes strong anti-Stokes-wave generation.

B. Device physics research
In our semiconductor χ (3) -periodic structure, we carefully examine the anti-Stokes-wave generation in terms of a pump intensity and a device length from our interest in highlyefficient output with a moderate pump intensity and a reasonable device length.
Furthermore, we carefully examine other nonlinear effects (Kerr-induced effects), such as four-wave mixing (FWM), self-phase modulation (SPM), cross-phase modulation (XPM), because these are much greater than those in Si. As for two-photon absorption (TPA), since the bandgap energy in the GaAs/AlAs superlattice is comparable to the TPA energy (1.6 eV), TPA can be avoided by subtle input-wavelength adjustment or bandgap engineering, which can be done easily when compared with the Si case.
The influence of stimulated Brillouin scattering (SBS) coupled with low-frequency acoustic-phonon modes is neglected here, because the Brillouin-shifted frequency and its linewidth are only f SBS ∼ 20 GHz and ∆f SBS ∼ 170 MHz, e.g., in GaAs [26], respectively.
Thus if we use relatively short input pulses (e.g., with a width of ∼ 10 ps or a spectral width of ∼ 100 GHz used often in optical communications), then SBS nearly ceases to occur, because most of the spectral components are out of the SBS gain linewidth ∆f SBS . This SBS-stopping phenomenon is well known in optical fibers for short pump pulses [27]. On the other hand, since SRS coupled with high-frequency optical-phonon modes has a much wider gain linewidth ∆f SRS ∼ 100 GHz, SRS can keep its large gain, e.g., for the 10-ps pulses, at the Raman-shifted frequency f SRS ∼ 9 THz [16].
In this way, by including such nonlinear interactions as optical Raman and Kerr effects in the proposed periodic structure, we examine the SRS efficiency; more specifically, that of SRS for anti-Stokes waves with a far smaller efficiency than that of Stokes waves.

C. Paper structure
After this Introduction section ( §I), we show a model for the periodic GaAs/AlAs intermixed superlattice in §II. We then derive coupled nonlinear equations that deal with the interacting pump, Stoke-and anti-Stokes waves in the superlattice in §III. In §IV, we obtain analytical solutions with some approximations to the coupled equations to see the properties of those nonlinear effects on the output efficiency. Furthermore, we obtain numerical solutions that contain no such approximations, and compare them with the analytical solutions to check numerical accuracy. The numerical computations give the efficiency when there is the superlattice, which is compared with the efficiency without the superlattice, and improvements are discussed. Finally, §V is devoted to summary.

II. MODEL FOR χ (3) -PERIODIC SUPERLATTICE
We show our device model that contains a periodic GaAs/AlAs intermixed superlattice in Fig. 1, where the pump, Stokes and anti-Stokes wavelengths are larger than the halfbandgap wavelength of the superlattice, which prevents TPA. (The influence of TPA will be examined in detail in §IV B 7.) In case where one of those three wavelengths is shorter than the half-bandgap wavelength, strong TPA will emerge, and considerable free carrier absorption (FCA) will also emerge due to TPA-induced carriers. In this case, we use a reverse-biased p-n junction embedded in the device to avert FCA, as shown in Fig.1, where the superlattice waveguide is sandwiched between buffer layers of Al x Ga 1−x As with x = 0.56, 0.6 [23] with a lower refractive index than that of the waveguide. If those three wavelengths are longer than the half-bandgap wavelength, such a p-n junction is not necessary (or no bias-voltage application is necessary for the p-n junction).

As-grown region
Intermixed region

Buffer layer
FIG. 1. Device structure. A GaAs/AlAs superlattice waveguide containing periodic as-grown and intermixed regions is sandwiched between buffer layers that have a lower refractive index than that of the waveguide to confine light in it, where the thickness of the buffer layers is adjusted to strongly confine the light, and the waveguide together with the buffer layers is also sandwiched between p-GaAs and n-GaAs to form a p-n junction.
When the as-grown domains of the waveguide in Fig. 1  On the other hand, the intermixed regions have a blue-shifted absorption edge as regards the χ (3) -property, and the χ (3) value is considerably decreased when compared with that in the as-grown domains (actually it is nearly a half of χ (3) ) [28]. We set this value to be εχ (3) (0 ≤ ε ≤ 1) in our calculations, as will be shown in §IV B 4. In the next section, we derive coupled nonlinear equations to describe the evolution of the pump, Stokes and anti-Stokes waves for the above waveguide device.

III. COUPLED NONLINEAR EQUATIONS FOR THE DEVICE
In deriving the coupled nonlinear equations, we assume that stimulated Raman scattering (SRS) is greater in photon generation than spontaneous Raman scattering (i.e., beyond the SRS threshold power) [17]. In this case, since there is almost no difference in optical power obtained by classical and quantum analyses, we describe SRS in the waveguide device with classical wave equations derived from Maxwell's equations that are coupled with molecular vibrations in the employed semiconductor.
To perform the analysis for SRS, we start with the following wave equation obtained via the standard procedure [29].
E = E(z, t) is the electric field of the coupled three waves (i.e., the pump, Stokes, and anti-Stokes waves) that propagate in the z direction. P LN is the linear polarization defined by P LN = ǫ 0 χ (0) E with the vacuum permittivity ǫ 0 and the linear susceptibility χ (0) . The relative permittivity ǫ is related to χ (0) as ǫ = ǫ 0 (1 + χ (0) ) that gives the refractive index n as n = ǫ/ǫ 0 . P NL is the nonlinear polarization containing the Raman and Kerr effects.
µ 0 is the magnetic permeability. For non-magnetic semiconductors (e.g., Al x Ga 1−x As), they have the same value µ 0 in vacuum. c is the velocity of light that has the relation with ǫ 0 and µ 0 as c = 1/ √ ǫ 0 µ 0 . In the above, TE polarization has been assumed for E, P LN , and P NL , which are excited by TE-mode pump lasers.
The nonlinear polarization for the Raman effect is given by [14,30] Here, χ ′(3) is the imaginary part of the third-order nonlinear susceptibility. Q is the phononwave amplitude. ω ν is the Raman frequency. τ is the relaxation time of the molecular oscillation. γ is a constant that characterizes the coupling between electric fields and polarized molecules. In addition, the nonlinear polarization for the Kerr effect is given by where χ (3) is the real part of the third-order nonlinear susceptibility. Experimentally, the Kerr effect arises together with the Raman effect [31], and thus the total nonlinear polarization P NL is expressed as the sum of Eqs. (3) and (5).
In the above, a choice of the tensor component in the third-order nonlinear susceptibility is taken so that TE-mode input gives TE-mode output.
A. Simplification of the space and time derivatives of the wave equation The space and time derivatives in the left-hand side of the wave equation Eq. (2) are simplified when we use an optical pulse with its spatial width larger than a characteristic length of the waves (i.e., the wavelength). This is known as the slowly-varying-envelope approximation [29]. We use this approximation throughout the paper.
To express ǫE in Eq. (2) containing the three waves, we use the following sum of electric fields.
where j = −1, 0, 1 stand for the Stokes, pump, and anti-Stokes waves, respectively, ω j is the frequency of the j-th wave, and ǫ j is the ω j -component of ǫ, i.e., ǫ j = ǫ(ω j ). k j is the wavenumber of the j-th wave. Using the Raman frequency ω ν , we can write ω j as ω j = ω 0 + jω ν . The Stokes and anti-Stokes waves (j = −1, 1) that we deal with here are the first excited ones. Since the growth of higher excited waves than the first excited waves needs a much longer interaction length than that for the first exited waves [14], and since we are interested in a relatively short waveguide on the order of a centimeter, the emergence of higher excited waves is ignored.
In addition to Eq. (6), if we set ǫ = ǫ j = 1 in it, the expansion of E takes the form.
In Eq. (2), we insert Eq. (6) into ∂ 2 (ǫE)/∂t 2 and Eq. (7) into ∇ 2 E, where a relation ∂ 2 E/∂x 2 = ∂ 2 E/∂y 2 = 0 holds because of the assumption E = E(z, t). The spread of the electric fields in the x and y directions is included as a constant cross-sectional area of the waveguide.
A simplified equation in the left-hand side of Eq. (2) by substituting Eq. (7) for ∂ 2 E/∂z 2 is of the form.
Also, by substituting Eq. (6) for ∂ 2 (ǫE)/∂t 2 , we obtain Inserting Eqs. (8) and (9) into the left-hand side of Eq. (2) with the relation ǫ j = n 2 j (n j is the refractive index for the j-th wave), we obtain where the relation ω j = (c/n j )k j was used. By replacing n j (j = −1, 0, 1) with the averaged value n = n j in Eq. (10), we can further simplify the space and time derivatives of Eq. (10).
This replacement does not severely affect the results because a refractive-index deviation (∆n ≈ 0.01) from n is only 0.3 % of n = n j = 3.09, which is obtained from n j ≈ 3.08 -3.10 at 1450 -1600 nm [23]. We then use the transformations, z new = z old , t new = t old − (n/c)z old [27], and obtain This transformation corresponds to a frame change that the optical pulse in the waveguide is observed from a moving coordinate in the z direction with a velocity of c/n.

B. Derivation of terms for the Raman effect
To derive the coupled equations of the three waves with j = −1, 0, 1, we first need to extract nonlinear polarization terms for the Raman effect that satisfy ω 0 − ω −1 = ω ν and ω 1 − ω 0 = ω ν , which we call resonant terms. Here, ω −1 , ω 0 , and ω 1 stand for the frequencies of the Stokes, pump, and anti-Stokes waves, respectively. The relation of ω −1 , ω 0 , ω 1 , and ω ν is illustrated in Fig. 2. To calculate the nonlinear polarization in Eq. (3), we insert the electric field E of Eq. (7) and the following phonon-wave amplitude Q into Eq. (3), where k ν is the wavenumber of the phonon wave with a frequency of ω ν . By this insertion, we have Next, we perform the second-order time derivative of P R NL that is necessary for the calculation of the right-hand side of Eq. (2), and we have where the slowly-varying-envelope approximations, |ω j q| ≫ |∂q/∂t|, |ω j E j | ≫ |∂E j /∂t|, were used.
To extract resonant terms from Eq. (14) for ω j (j = −1, 0, 1), we write down the terms The q and q * in Eqs. (15), (16), and (17) are related to E j via Eqs. (4) and (12). To clarify the relation between them, we calculate Eq. (4) by inserting Eq. (12) into it. This calculation needs the first-and the second-order time derivatives of Q, which are of the form.
where the slowly-varying-envelope approximation was used again.
To obtain the relation between q (or q * ) and E j , we substitute Eqs. (18) and (19) for Eq. (4) and obtain where the approximation ω ν ≫ 1/τ was used, which arises from the relaxation time τ that is much greater than the time scale (∼ 1/ω ν ) of high molecular eigenfrequencies.
Next, we calculate the right-hand side of Eq. (21). Inserting Eq. (7) into it, we obtain Extracting resonant terms from Eq.

C. Derivation of terms for the Kerr effect
In this section, we calculate the nonlinear polarization Eq. (5) for the Kerr effect. To do this, we substitute Eq. (7) for Eq. (5) and obtain By performing the second-order time derivative of P K NL necessary to calculate the right-hand side of Eq. (2), we get where the slowly-varying-envelope approximation, |ω j E j | ≫ |∂E j /∂t|, was used.
To extract resonant terms from Eq. (27), we here write down the terms resonant to ω j in In addition, because of the relation are resonant to ω −1 , ω 0 , and ω 1 , respectively, and their second-order time derivatives are of the form with the addition of µ 0 .
where the relation Eq. (32) was used.

D. Full coupled equations with the Raman and Kerr terms
To complete building full coupled equations containing the Raman and Kerr effects, we insert the Raman terms Eqs. (15)- (17) and the Kerr terms Eqs. (28)-(30),(33)-(35) into the right-hand side of Eq.
where g in Eq. (42) is the Raman gain, n 2 in Eq. (44) is the nonlinear refractive index, and At a steady state, vibrating molecules with a frequency of ω ν in the employed semiconductor have a constant amplitude for q in the motion of Q = (qe i(kν z−ων t) + c.c.)/2. Thus, the relation ∂q/∂t = 0 holds, and Eq. (39) gives In this case, substituting Eq. (47) for Eqs. (36)- (38), we obtain If we include linear-loss terms with a loss coefficient α j (j = −1, 0, 1) and nonlinear-loss terms with a TPA coefficient β TPA , the coupled equations take the form.
where α TPA j is defined by [27,32] α TPA and β TPA is defined by Note that in the above equations, FCA is neglected because the p-n junction in Fig. 1  (cm/GW) [16] and κ j ≈ 7.1 (cm/GW) [33]. For a comparison, we show g and κ j of Si at around 1.55 µm: g ≈ 20 (cm/GW) [4] and κ j ≈ 1.8 (cm/GW) [34]. For Si, the effect from the SRS terms is much stronger than that from the Kerr terms, which is completely different from the Al x Ga 1−x As case.
In the next section, we solve Eqs.
holds. Note that the Stokes and anti-Stokes waves can grow from small seed light and do not necessarily require high-power seed beams at z = 0.
(b) Since the anti-Stokes wave with a phase mismatch ∆k grows much more weakly than the Stokes wave with no such phase mismatch, an additional approximation holds.
(c) We ignore the wavelength dependence of the loss coefficients, α −1 , α 1 , α 0 , of the Stokes, anti-Stokes, and pump waves, respectively, because their differences are small at the wavelengths far from the bandgap wavelength (actually they are near the half-bandgap wavelength). In this case, we can set where α is a constant.
Employing the above approximations Eqs. (56)-(58), we obtain the following simplified equations for Eqs. (51)- (53): In the approximations used above, the effects of SPM and XPM were greatly reduced. (These effects will be examined in detail in §IV B 4.) In Eqs. (59) and (60), the linear-loss terms are easily removed by setting We then obtain To solve Eqs. (63) and (64), we integrate Eq. (63) from 0 to z and get Next, inserting Eq. (65) into Eq. (64) and integrating Eq. (64) from 0 to z, we obtain where E 1 (0) denotes the input electric field for the anti-Stokes wave. In what follows, we set E 1 (0) ≈ 0 as E 1 (0) being a very small quantity.
(iii) The output efficiency of the anti-Stokes wave in Eq. (70) oscillates with a period of Thus the period has an |E 0 | 2dependence.
As regards (i), since g and β TPA are proportional to χ ′(3) , as seen in Eqs. (42) and (55), and since n 2 is proportional to χ (3) , as seen in Eq. (44), these coefficients in the intermixed domains are smaller than those in the as-grown domains because of blue-shifted resonance for χ (3) and χ ′(3) . Thus, periodic g and n 2 are produced in the GaAs/AlAs-intermixedsuperlattice waveguide. These periodic g and n 2 form the QPM structure that can remove ∆k and enhance the output efficiency.
In connection with (ii), if the photon energies of the Stokes and anti-Stokes waves are smaller than the half-bandgap energy, TPA that degrades SRS can be greatly reduced. Even in this region, n 2 has a finite value (and becomes large near the half-bandgap energy) [28].
For this reason, we use a semiconductor device with the photon energies of the pump, Stokes, and anti-Stokes waves near and below the half-bandgap energy for highly-efficient output (or with their wavelengths near and larger than the half-bandgap wavelength).
In relation to (iii), the setting of the QPM period Λ to Λ ′ = 2π/|∆k − 2κ −1 |E 0 | 2 | is not practical in device applications, because |E 0 | 2 actually decreases as the pump wave propagates through the QPM device, as will be shown in §IV B, which requires an aperiodic QPM structure. In addition, |E 0 | 2 at z = 0 is sometimes varied, and a corresponding aperiodic QPM structure is necessary in this case. This means that there is no availability for a wide change in input pump intensity. For this reason, we set Λ = 2π/|∆k| when performing QPM.
With respect to (iv), because of κ 2 1 ≈ 50.4 and (g 2 /4)(ω 2 1 /ω 2 −1 ) ≈ 29.9 in our case, we can see that the FWM-enhanced efficiency is about 2.7 times greater than that without FWM. For a comparison, we here mention that Si has κ 2 1 ≈ 3.3 and (g 2 /4)(ω 2 1 /ω 2 −1 ) ≈ 119.6, which means that Si has a little efficiency enhancement by FWM. On the other hand, such an enhancement has been reported in an optical fiber (SiO 2 ) with no QPM structure [35].
However, our device has a QPM structure, and we need to investigate the effect of SRS with FWM on QPM. At the same time, we need to take account of SPM and XPM, which are much stronger than those in SiO 2 as well as in Si. The numerical analysis containing all those effects on the efficiency is given in the next section.

B. Numerical solutions
The solutions in §IV A with approximations for nonlinear optical terms did not contain the effects of pump depletion and very large growth of the Stokes wave under the assumption that they were at the initial stage. To investigate the evolution of the three waves more precisely, we obtain numerical solutions (i.e., without such approximations) for the case without QPM at first by the fourth-order Runge-Kutta method [36]. At the initial wave evolution, since those two effects are not serious, a comparison between the analytical and numerical solutions can be made to check numerical accuracy, which will be described in §IV B 4. After this check, we proceed to investigate the solutions for the case with QPM, and show improved output characteristics in comparison with no QPM case.

Determination of the half-bandgap, Stokes, pump, and anti-Stokes wavelengths
We have used Al x Ga 1−x As (x = 0.18), where the χ (3) -property of the as-grown GaAs/AlAs superlattice can be well approximated. These two have quite a similar χ (3) -property and almost the same bandgap energy [28]. We do not have to focus on TPA in the intermixed regions because they have a blue-shifted absorption edge and TPA in the as-grown regions emerges at first. In this superlattice, we have assumed that the Stokes, pump, and anti-Stokes wavelengths, λ i (i = −1, 0, 1), respectively, are near and larger than the half-bandgap wavelength λ half , thus keeping large n 2 and very small (almost zero) β TPA . (Here, a large figure of merit, 4πn 2 /(β TPA λ i ) ≫ 1, is kept even at λ i = λ half [37,38].) To show available λ −1 , λ 0 , λ 1 , we need to calculate λ half (µm) that is defined by where h is Planck's constant, c is the velocity of light, and E g (eV) is the bandgap energy at a temperature of T (K). E g in Eq. (71) can be determined from Varshni's equation [39] for Al x Ga 1−x As.

Determination of the phase mismatch ∆k
It is also necessary to determine the phase mismatch ∆k in Eqs. (51)-(53) when performing the numerical integration of them. This ∆k is defined by where n(λ i ) is the refractive index for λ i (i = −1, 0, 1). The form of n(λ i ) is determined by Sellmeier's equation for Al x Ga 1−x As at room temperature [42,43], which is given by where for x ≤ 0. 36 73) and (74), we obtain Λ = 189.1µm.

Determination of the linear loss coefficient α
Since the Stokes, pump, and anti-Stokes wavelengths are larger than the half-bandgap wavelength, the TPA loss does not emerge, but the linear loss emerges at all times. However, recent technological development has achieved a small linear loss, e.g., a loss of 1.2 dB/cm in Al x Ga 1−x As (x = 0.18) at around 1.55 µm for TE-mode beams [44]. In our calculations, we employ this value. In addition, since the wavelengths larger than the half-bandgap wavelength are far from the bandgap wavelength, we can justify the omission of the wavelength dependence of the linear-loss coefficients α j (j = −1, 0, 1), as given in Eq. (58), and we set α j = α = 0.276 cm −1 . This was calculated from the loss of 1.2 dB/cm so as to fit e −αz .

Output efficiencies for the cases with and without QPM
Using the above phase mismatch ∆k, the refractive indices n(λ i ) of the above wavelengths In numerical calculations with QPM, a small difference between the refractive indices, n intermixed and n as−grown , in the intermixed and as-grown domains, respectively, is ignored because the difference n as−grown − n intermixed = 0.01 is only 0.3 % of the average refractive index of the QPM structure, as already mentioned in §I. On the other hand, a large change in χ (3) of the intermixed domains is set as εχ (3) (0 ≤ ε ≤ 1) using the χ (3) of the as-grown domains, as depicted in Fig. 3. This setting means that when ε = 1, there is no QPM structure and that when 0 ≤ ε < 1, there is a QPM structure. In particular, ε = 0 gives a maximum QPM effect.
To calculate the output efficiencies, we use a waveguide with a cross-sectional area of (0.5 µm) 2 and with a pump power of 10 W [45] that can be achieved with fiber lasers. In this case, the pump intensity at z = 0 is |E 0 (0)| 2 = 4.0 (GW/cm 2 ). As seed-light intensities at The output efficiencies η S , η AS for ε = 1 (i.e., without QPM) as a function of the device length z are shown in Fig. 4, indicated by the red and blue curves, respectively. The pump intensity |E 0 (z)| 2 divided by |E 0 (0)| 2 was also added to Fig. 4, indicated by the green curve, which shows a rapid decrease with increasing z. The inset of Fig. 4 is a magnified figure of η AS at around z = 0.15 cm, which shows that η AS oscillates and does not grow sufficiently due to a non-zero phase mismatch ∆k and pump depletion. In this case, most of the pump power is transferred to the Stokes wave that has no such phase mismatch.
From the inset of Fig. 4, we can roughly estimate the ratio, |E 1 | 2 /|E −1 | 2 , between the anti-Stokes-wave and Stokes-wave intensities to be 10 −3 , which is rather larger than 10 −5 -10 −6 in Si waveguides [18,19]. This comes from |E 1 | 2 /|E −1 | 2 proportional to the pump intensity |E 0 | 2 [17] with the use of a larger |E 0 | 2 in our case.  (70), respectively, and the thick red, blue, and green curves are the same as in Fig. 4. The inset displays the same figure with the log 10 plot for the vertical axis. We observe that at the initial stage of growth of the Stokes and anti-Stokes waves, the numerical solutions well fit the analytical results but that as z increases, they deviate from the analytical results. The main cause of this is pump depletion, and there is also another effect from SPM and XPM that are greatly reduced in the approximations for the analytical solutions. The latter effect will be described below (see Fig. 7).  Figure 6 shows the output efficiencies η S , η AS for ε = 0 (i.e., with QPM that provides the maximum efficiency), where we observe a great increase in η AS due to QPM, which reaches the same order of η S (but with a factor of about 1/7). In addition, we can see that there is an optimal device length that provides the highest efficiency in η AS at z peak = 0.69 cm that is determined from a balance between optical gain and loss. This gives us useful information on determining a necessary device length when we fabricate actual devices. In this case, η S of the Stokes wave somewhat decreases because more pump power is transferred to the anti-Stokes power. We also checked the influence of an initial phase shift (i.e., at z = 0) of the QPM period on η AS , and observed very little effect on it.
To  η AS containing both SPM and XPM effects, which is the same as that in Fig. 6.
In the black curve, we can see that QPM is more effective because there is no period deviation from Λ = 2π/|∆k| by a phase shift due to SPM and XPM. Owing to this effective QPM, η AS grows more rapidly than that with SPM and XPM as the anti-Stokes wave propagates in the z direction; in this case, about 1.6 times larger η AS is obtained at z = 0.4 cm.
Furthermore, in order to see only the SPM effect on η AS , we intentionally delete the XPM effect, as depicted in Fig. 7 by the red curve. We can see that this also causes a period deviation from Λ = 2π/|∆k|, thus decreasing η AS . But, in our semiconductor device, since we cannot remove only SPM and/or XPM effects in an artificial manner, the device cannot get rid of the efficiency reduction. Next, we check the dependence of η AS on the parameter ε (0 ≤ ε ≤ 1) while taking account of all the above nonlinear effects. Figure 8 shows the peak value of η AS at z = z peak , denoted by η AS,peak , as a function of ε. In Fig. 8, we can see that as ε approaches 0 from 1, η AS, peak increases rapidly. To obtain an enhancement factor e (or figure of merit) due to QPM when compared with that without QPM (i.e., ε = 1), we calculate the enhancement factor e defined by e = η AS, peak, ε η AS, peak, ε=1 = |E 1 | 2 peak, ε This is depicted in the inset of in Fig. 8, which shows that e is about 76 times greater at ε = 0 (with the maximum QPM effect) than that at ε = 1 (with no QPM effect). Since the actually-achievable value of ε is ∼ 0.5 [28], the enhancement factor e in this case is ∼ 11.5. FIG. 9. Dependence of peak output intensity |E 1 | 2 peak, ε on the input pump intensity |E 0 (0)| 2 for ε = 0, 0.5, 1. The inset depicts the log 10 plot of the enhancement factor e that depends on |E 0 (0)| 2 when ε = 0, 0.5.
The enhancement factor e in the inset of Fig. 8 was obtained when the initial pump intensity |E 0 (0)| 2 was fixed at 4.0 (GW/cm 2 ). As we can easily see, an increase in the initial pump intensity |E 0 (0)| 2 increases the anti-Stokes-wave intensity |E 1 | 2 peak, ε , which is shown in Fig. 9. Thus our interest now goes to the dependence of e on |E 0 (0)| 2 .
The dependence of e on |E 0 (0)| 2 is depicted in the inset of Fig. 9, where the vertical axis is set on a log 10 scale. This shows that e takes a very large value between 10 3 and 10 3.5 for 0 ≤ ε ≤ 0.5 at very small |E 0 (0)| 2 , where |E 0 (0)| 2 should be larger than the SRS threshold pump intensity δ ≈ 10 −2 (GW/cm 2 ) [46]. The increase in e originates from the cause that as |E 0 (0)| 2 approaches δ, the intensity-dependent period Λ ′ = 2π/|∆k − 2κ −1 |E 0 | 2 | in (iii) of §IV A comes nearer to Λ = 2π/|∆k|, thereby achieving highly-efficient QPM. Note that the above Λ ′ will actually be affected by not only SPM but also XPM at the region beyond the approximations of §IV A. But actually, both SPM and XPM simultaneously diminish as |E 0 (0)| 2 approaches δ, and thus the above simple interpretation holds in explaining the e-enhancement.

Dependence of peak position on the pump intensity
In actual device fabrication, it is important to determine an optimal (i.e., the smallest) device length that gives the highest output intensity. Figure 10 depicts the dependence of the optimal device length z peak on the pump intensity |E 0 (0)| 2 , which shows that z peak decreases as |E 0 (0)| 2 increases. This is because as |E 0 (0)| 2 increases, the Stokes and anti-Stokes waves grow rapidly, thus causing rapid pump depletion. After complete pump depletion, the Stokes and anti-Stokes waves do not grow and are attenuated by the linear optical loss.
Thus z = z peak is given approximately from the complete-pump-depletion point.
The important information obtained from Fig. 10 is that we can determine the lower limit of an input pump intensity |E 0 (0)| 2 for the optimal waveguide length; For instance, as shown in the inset of Fig. 10 by the dashed line for ε = 0.5, |E 0 (0)| 2 = 1.2 (GW/cm 2 ) is necessary such that z peak = 1 (cm) that is the length aiming to actual device fabrication. (GW/cm 2 ). The dashed lines indicate the necessary |E 0 (0)| 2 for z peak = 1 (cm) when ε = 0, 0.5.

Dependence of output efficiency on TPA
In §IV B 1, since the Stokes, pump, and anti-Stokes wavelengths, λ i (i = −1, 0, 1), respectively, were near and larger than the half-bandgap wavelength, λ half , sizable n 2 and negligibly small β TPA were obtained (i.e., high nonlinear refraction and very small TPA). But, in case of smaller λ i (i = −1, 0, 1) than λ half , TPA will emerge strongly. We will examine this effect on the output efficiency.
TPA excites carriers, which give rise to FCA, i.e., an additional optical absorption. But we can remove this additional absorption by use of the reversed-biased p-n junction in Fig. 1.
A detailed calculation containing FCA, e.g., in a Si waveguide, shows a remarkable increase in the optical losses of propagating beams with increasing input beam power, that is, the influence of FCA is far greater than that of only TPA [47]. Thus, the removal of . Input pump intensity |E 0 (0)| 2 -dependence of the peak output intensity |E 1 | 2 peak, ε of the anti-Stokes wave for ε = 0, 0.5, 1 with β TPA = 0.1 (cm/GW), indicated by the black curves, and with β TPA = 0 (cm/GW), indicated by the blue curves. The inset shows the |E 0 (0)| 2 -dependence of the efficiency η AS, peak, ε = |E 1 | 2 peak, ε /|E 0 (0)| 2 when β TPA is varied from 0 to 1 (cm/GW).

TPA-induced carriers is indispensable for device applications.
Here, without including FCA with the help of the p-n junction, we can discuss TPA on the peak value |E 1 | 2 peak, ε at z = z peak for ε = 0, 0.5, 1. This is shown in Fig. 11 when using β TPA = 0.1 (cm/GW) measured for TE-modes [48], where the black curves depict |E 1 | 2 peak, ε with β TPA = 0.1 (cm/GW) and the blue curves show that with β TPA = 0 (cm/GW) for a comparison. In Fig. 11, we can see that a decrease in |E 1 | 2 peak, ε due to non-zero β TPA becomes larger for smaller ε, where at smaller ε, the anti-Stokes-wave intensity is larger and thus there is stronger TPA.
In the above, a great decrease in the output efficiency η AS due to TPA has been shown quantitatively, and we recognize the significance of keeping λ i (i = −1, 0, 1) near and larger than λ half to avoid TPA and obtain highly-efficient output for device applications. In order to obtain much larger η AS , we propose a cavity structure with dielectric multilayer mirrors (DMMs) or photonic crystals (PhCs) [49][50][51][52][53] attached to both ends of the device, which can also shorten the device. Here, the DMMs and PhCs have a function that they reflect the Stokes waves back to the device with their bandgap with optical reflectance R to enhance the Stokes-wave intensity |E −1 | 2 in the device, and let the pump and anti-Stokes waves pass through them, as illustrated in Fig. 12. This enhancement in |E −1 | 2 increases η AS (with no necessity of boosting the pump intensity |E 0 | 2 ) owing to the relation η AS ∝ |E −1 | 2 in Eq. (70). If we use bandgap structures with R < 1, it is possible to adjust the sizes of η AS and η S (and to equalize them, if necessary). Quantitative analysis of such efficiency adjustments will be provided in a forthcoming paper.

V. SUMMARY
We have proposed a superlattice SRS device design utilizing a GaAs/AlAs intermixed superlattice with χ (3) -nonlinearity that can boost the anti-Stokes-wave efficiency η AS by QPM at 1.49 µm to the same order of the Stokes-wave efficiency η S at 1.63 µm with the input of a 1.56-µm pump beam, which could be used for channel amplification and/or conversion in GE-PONs and WDM systems.
In particular, having focused on the anti-Stokes-wave amplification, we have computed where the influence of SPM and XPM was greatly reduced in the approximations. The numerical calculations without such approximations have given a result that a difference between the analytical and numerical solutions in behavior out of the initial stage stems from pump depletion and the SPM and XPM effects. To obtain η AS precisely, we have numerically computed it for the cases with and without QPM containing FWM, SPM, and XPM effects. A comparison between the two cases has exhibited that there is a three orders of magnitude greater efficiency with QPM than that without QPM for ε ≤ 0.5 at a small pump intensity. The physical mechanism of these has been clearly shown in terms of nonlinear-interaction analysis. A TPA-effect on η AS at an optimal length has also been examined, which indicates the significance of TPA-reduction for device applications. In addition, we have proposed the use of a photonic-bandgap cavity to obtain more η AS or control the sizes of η AS and η S .
The proposed device has potential towards ultrahigh-speed data processing/routing applications using channel conversion and amplification via instantaneous optical nonlinear