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We present for the first time an all-optical wavelength conversion (AOWC) scheme supporting modulation format independency without requiring phase matching. The new scheme is named “spoof” four wave mixing (SFWM) and in contrast to the well-known FWM theory, where the induced dynamic refractive index grating modulates photons to create a wave at a new frequency, the SFWM is different in that the dynamic refractive index grating is generated in a nonlinear Bragg Grating (BG) to excite additional reflective peaks at either side of the original BG bandgap in reflection spectrum. This fundamental difference enable the SFWM to avoid the intrinsic shortcoming of stringent phase matching required in the conventional FWM, and allows AOWC with modulation format transparency and ultrabroad conversion range, which may have great potential applications for next generation of all-optical networks.
©2012 Optical Society of America
1. Introduction
Wavelength conversion is indispensiable for wavelength routing in photonic network systems such as Wavelength Division Multiplexing (WDM) and Optical Time Division Multiplexing (OTDM) [1]. Some of the critical functions it can fulfill include reconfigurable routing, contention resolution, wavelength reuse, multicasting, and traffic balancing. Compared with the conventional optical-electrical-optical (O/E/O) wavelength conversion schemes, all-optical wavelength conversion (AOWC) is especially attractive because of the potential advantages of high speed, compactness, lower-power consumption, and high integration [2–4]. Increasing demands on global delivery of high-performance network-based applications, such as cloud computing and (ultra)high definition video-on-demand streaming, calls for next generation optical networks of higher capacity and more powerful signal processing capability [5–7]. This, in turn, requires wavelength conversion techniques with broad conversion wavelength tuning range and high modulation format transparency. However, so far as we know, none of the existing AOWC techniques can meet the requirements.
The existing AOWC techniques can be mainly classified into two categories: optical gating and coherent mixing. Optical gating wavelength conversion techniques [e.g. cross gain modulation (XGM) and cross phase modulation (XPM)] exploit carrier depletion and/or carrier density-induced refractive index changes, so only intensity-modulated input signals can be converted [8–11]. Coherent mixing wavelength conversion techniques [i.e. difference frequency generation (DFG) and four wave mixing (FWM)] utilize the photon conversion function in the second- or the third-order nonlinear materials which requires phase matching for efficient conversion [12–23]. FWM-based AOWC has been investigated widely in recent years in different structures such as fibers [12–14], silicon waveguides [17–21], and semiconductor optical amplifiers [2, 22–24], and is commonly thought one of the most promising AOWC techniques because of its unique advantages of supporting format-transparent operation [2]. However, due to the need of phase-matching, FWM suffers from poor conversion wavelength tunability, which could severely limit the development of the next generation networks.
In this paper, by means of exciting additional reflective peaks (ARPs) by optically modulating a Bragg grating (BG), we propose a novel modulation-format-transparent AOWC scheme which does not require phase matching and therefore has excellent tunability of wavelength conversion. In the proposal, a dynamic refractive index grating induced by the beating of the co-propagating pump and signal is able to modulate a BG to create ARPs at either side of the unperturbed BG bandgap. When a probe wave located at the wavelength of ARP is counter-propagating, it is reflected from the induced ARPS while tracking the signal data information but at the new wavelength. The proposed scheme and the conventional FWM are similar in that a dynamic refractive index grating is induced by the beating of pump and signal light. However, instead of modulating photons to generate new photons, the dynamic refractive index grating in the proposed scheme modulates a BG to generate ARPs without generate new photons. So we name the new scheme “spoof” four wave mixing (SFWM).
2. ARP excitation in a refractive-index-modulated BG
To illustrate the principle of the SFWM, first we consider the optical property of a refractive-index-modulated BG. When a refractive index profile Δn(z) = nc + ndcos(2πz/Λ + ϕ) with period Λ is superimposed onto a BG whose refractive index profile is denoted as n1(z) with period d, the refractive index of the perturbed BG becomes n(z) = n1(z) + Δn(z). To investigate the photonic bandgap of the perturbed BG, we assume Λ/d to be integer to ensure that n(z) is periodic, so that the photonic dispersion theory can be readily applied. Dividing the unit cell of the perturbed BG into Q sections, the transmission matrix for single period in the perturbed BG yields [21]
where δj = (2π/λ)njΔz, γj = nj(ε0μ0)0.5, Δz = Λ/Q, and nj = Δn(jΔz). Here, Q is an integer, λ is the incident light wavelength, and ε0 and μ0 are the permittivity and permeability of light in vacuum, respectively. Based on Eq. (1) the dispersion equation for the optical wave propagating in the perturbed BG can be written as [21]Here, K is the Bloch-wave vector, and M11 and M22 are the diagonal elements of the matrix M.In our calculations, the refractive indexes of the two materials in BG are na = 1.5 and nb = 1.6, respectively, and the their lengths in each period are da = db = 220 nm. So the Bragg wavelength is λB = 2(nada + nbdb) = 1364 nm and the grating period is d = da + db = 440 nm. For the refractive index Δn(z) used to perturb BG, nc = nd = 2.5 × 10−3, Λ = 11d, and ϕ = 0. With above parameters and using Eq. (2), we calculate the dispersion diagrams both for the perturbed and unperturbed BG, as shown in Fig. 1(a) . In photonic crystals, bandgaps appear either at the center or at the border of the Brillouin zone, i.e., at either K = 0 or 1. We note from Fig. 1(a) that the unperturbed BG has a bandgap at K = 1 with central wavelength of λB. While the Δn(z) is superimposed onto it, multiple ARPs are created and located at either side of the λB. Using the transfer matrix method (TMM) [21] and assuming the total number of periods of the unperturbed BG to be N = 12000, we obtain the reflectivity spectra both for the perturbed and unperturbed BG, as shown in Fig. 1(b). It shows that dual ARPs occur at either side of the main BG reflective peak with central wavelengths of λl = 1249.7 nm and λr = 1506.5 nm, respectively, which aggress well with the dispersion diagram. The perturbed BG should have multiple ARPs according to Fig. 1(a), but only have dual ARPs. The reason will be explained latter. The insets of Fig. 1(b) show the reflectivity and phase spectra of the dual ARPs, indicating that the induced ARPs perform full-width at half-maximum (FWHM) as narrow as ~0.2 nm. Dependences of the wavelengths of the induced ARPs on the Λ (when λB = 1364 nm and da,b = 220 nm) and the λB (when Λ = 11d and da,b = λB/(4na,b)) are plotted in Fig. 1(c) and 1(d), respectively. We note that varying the Λ can effectively tune the induced ARPs to anywhere outside the BG bandgap. Meanwhile, variation in the λB (determined by na, nb, da, and db, thus given by the BG design) can linearly tune the ARPs. Finally, dependence of ARPs’ phases on ϕ (phase of the Δn(z)) are also studied by varying the ϕ while remain other parameters. Since changing the ϕ does not change the amplitude and period of the profile Δn(z), will not affect the ARPs’ wavelength but their inherent phases as shown Fig. 1(e).
Fig. 1 (a) Dispersion diagram both for the perturbed (dashed line) and unperturbed BG (solid line). The red circles mark the position of the induced bandgaps. The parameters are na = 1.5, nb = 1.6, da = db = 220 nm, nc = nd = 2.5 × 10−3, Λ = 11d, and ϕ = 0. (b) Reflectivity spectra for the perturbed (red solid line) and unperturbed BG (black circle line), respectively. The insets show the dual ARPs’ reflectivity and phase spectra in detail. (c) and (d) illustrate dependence of the induced ARPs’ wavelengths on Λ and λB, respectively. (e) Phase variations versus ϕ at the ARPs’ wavelengths λl and λr, respectively.
The bandgap property of the perturbed BG can also be analyzed by Fourier transform. When Λ/d is integral, the refractive index profile of the perturbed BG has a period of Λ and thereby its bandgaps, according to Fourier transform, locate at
where m is integer, Ω = 2π/Λ, β = 4πneff/λ, and neff is the effective refractive index of the perturbed BG determined byHere, n(z) is the refractive index of the perturbed BG. Since the amplitude of the Δn(z) is far smaller than the refractive index of the unperturbed BG, so neff equals approximately to (nada + nbdb)/d. Therefore, the bandgaps of the perturbed BG have central wavelengths ofwhen m is 9, 10, 11, 12, and 13 assuming Λ = 11d, the λband is 1.22λB, 1.1λB, λB, 0.92λB, and 0.84λB, respectively, which agrees well with the results obtained by the TMM as shown in Fig. 1. According to Eq. (5), the perturbed BG should have multiple ARPs, but only two bandgaps adjacent to the original BG bandgap are excited, as plotted in Fig. 1(b). The reason is that the Δn(z) is not strong enough. When the amplitude of Δn(z) is enlarged to nc,d = 2.5 × 10−2, more ARPs are excited as depicted in Fig. 2 . It is also noted that each bandgap behaves a redshift, which can be explained by Eq. (5) since a larger neff is induced.Fig. 2 Reflectivity of the perturbed BG when nc,d is 2.5 × 10−3 (blue dashed line) and 2.5 × 10−2 (red solid line), respectively. The other parameters are same with that in Fig. 1(b).
3. SFWM of AOWC and simulation results
Based on the interesting phenomenon presented above, i.e., modulating a BG by another small-amplitude grating could excite additional ARPs outside the original bandgap, we propose a new kind of SFWM scheme to perform an AOWC capable of handling data of any format without requiring phase-matching. The schematic setup for the device is presented in Fig. 3(a) , a input signal light is combined with a pump light of the same polarization to enter a third-order Kerr nonlinear BG. Beating of the pump and the signal induces a dynamic refractive index grating to modulate the BG so that ARPs are excited, like we have seen in above section. At the same time, a probe light with wavelength λ3 located at the wavelength of one of the induced ARPs is counter-propagated, with polarization perpendicular to the pump and signal. It will be reflected by the induced ARPs while tracking the information of the input signal data, but at the new wavelength λ3. As a result, the AOWC is achieved.
Fig. 3 (a) Schematic diagram of the proposed SFWM for AOWC. The pump and signal light should locate outside of BG bandgap for a high transmission. (b) The induced refractive index Δn(z) along the nonlinear BG. The inset shows the zoomed Δn(z) in the region of 3.45 mm<z<3.47 mm. Here, signal and pump wavelengths are λ1 = 1.55 μm and λ2 = 0.98 μm, their intensities are I1 = I2 = 5 × 10−3/(4n2), and their phase difference is φ = 0. The BG parameters are the same to that used in Fig. 1(b). (c) Reflectivity spectra for the perturbed BG when λ1 = 1.55 μm while λ2 = 0.85 μm, 0.95 μm, and 1.0 μm, respectively. (d) Dependence of the ARPs’ wavelengths on λ2 when λ1 = 1.55 μm.
We employ the TMM for simulations. When the pump and signal enter the BG, they interfere and yield a beating intensity profile I(z). Due to three-order Kerr nonlinearity, the I(z) induces a refractive index of Δn(z) = n2I(z) [25–27], where n2 is the nonlinear refractive index of the nonlinear BG assuming to be 4 × 10−14 cm2/W. If consider no reflection occurs when the pump and signal light propagates in the nonlinear BG, we can get I(z) = I1 + I2 + 2cos((k1-k2)z + φ), where I1 (I2) and k1 (k2) are intensity and wave vector for the signal (pump) light, respectively, and φ is phase difference between pump and signal light. In this ideal case, assuming the signal and pump light are at wavelengths of λ1 = 1.55 μm and λ2 = 0.98 μm with I1 = I2 = 5 × 10−3/(4n2) and φ = 0, the induced refractive index becomes Δn(z) = 5 × 10−3cos2((k1-k2)z/2 + φ/2), which is a cosine function profile with a period of Λ = 2π/(K1-K2) and a maximum value of 5 × 10−3. However, due to reflection being inevitably present when pump and signal light are propagating through the nonlinear BG, the induced Δn(z) will have many ripples with the maximum value being a bit smaller (~4.8 × 10−3), as indicated in Fig. 2(b). It is of particular interest to investigate the optical properties of the BG perturbed by the induced Δn(z) in Fig. 2(b).
Reflectivity spectra of the perturbed BG are depicted in Fig. 2(c), for a signal wavelength of 1.55 μm and different pump wavelengths. Strikingly, ARPs with narrow FWHM are observed on both sides of the original BG bandgap, even though the induced Δn(z) is not perfect cosine function profile like the one considered in the ideal calculations of Fig. 1. Figure 2(d) depicts the dependence of the induced ARPs’ wavelengths on the pump light when the signal light is at λ1 = 1.55 μm. We noted that varying λ2 could change the period of the induced Δn(z), which thereby flexibly tune the ARPs. The larger the difference between λ1 and λ2, the further the ARPs (and thus the converted output λ3) locates from the original BG bandgap.
As shown from Fig. 3(c), when λ2 is varied from 1.0 μm to 0.85 μm, the ARPs are and widely tuned while always remaining high efficiency (unity reflectivity). Therefore, compared with traditional FWM-based AWOC technique which has significant drop of conversion efficiency as the frequency difference between the signal and converted wave increases, the SFWM-based AOWC has very excellent wavelength conversion ability. While the change of λ2 provides a control mechanism to obtain different λ3 for any given λ1, the other parameter that has to be considered is the design on the BG. Variations in the BG’s period or refractive index (generally by changing temperature or strain) would define the BG bandgap, defining the same time the wavelength region where it will induce ARPs (and thus the location of our output λ3). Therefore, the output λ3 in our SFWM can be almost anywhere, providing enormous wavelength converting capability. This ultrabroad conversion wavelength tuning range will prove to be very valuable for the next generation of all-optical networks.
A description on how the ARPs behave (in terms of reflectivity and phase) for an input signal (with fixed pump) is presented in Fig. 4 . Peak value of the right ARP against signal intensity is shown in Fig. 4(a). Since intensity of output light λ3 depends directly on the reflectivity of the induced ARPs and this is tuned by varying signal intensity, the proposed SFWM works well for intensity-modulated signal formats. In Fig. 4(b), we further plot the phase spectra of the ARPs when the phase difference between pump and signal light is φ = -π/2 and π/2, respectively. It is interesting to find that when φ is changed (like it will be in differential phase-shift keying and coherent systems), the inherent phase of the ARPs also change accordingly while their reflectivity remain unchanged. This effect enables the proposed AOWC to work for phase-modulated signal. Therefore, in contrast to the XGM and XPM converters that only allow of conversion of intensity-modulated signal, the SFWM-based AOWC has advantage of modulation format independency. We also find from Fig. 4(a) that the induced ARPs are saturated with unity reflectivity at larger input intensity. It is quite helpful since high immunity to intensity noise can be achieved when this technique is used for converting phase-modulated signals.
Fig. 4 (a) Peak value of the right ARP (i.e, at wavelength λr = 1584.2nm) versus the signal intensity I1 when the pump intensity I2 = 5 × 10−3/(4n2), λ1 = 1.55 μm, λ2 = 0.98 μm, and φ = 0. (b) Reflectivity and phase spectra for the right ARP when I1 = I2 = 5 × 10−3/(4n2), λ1 = 1.55 μm and λ2 = 0.98 μm. Here, Red and blue graphs relate to the ARP when φ is -π/2 and π/2, respectively.
4. Conclusions
To conclude, we have investigated the optical properties of an optically refractive-index-modulated BG, and found that tunable and ultranarrow ARPs can be excited at either side of the original BG bandgap. Taking advantage of this interesting phenomenon, we have proposed a SFWM theory that allows performing AOWC with advantages over existing methods for wavelength conversion. The SFWM enables the AOWC with modulation format independency and excellent ultrabroad conversion range due to the removal of phase matching requirement, which is believed to have great potential applications in the next generation of all optical networks.
In the present paper, we assumed that both the materials in the BG had nonlinear response, but the ARPs can also be excited when one single material has nonlinear behavior. So different kinds of nonlinear grating structures, such as fiber Bragg grating, distributed feedfack structure, and two-dimensional surface relief gratings, may be potentially used to build the SFWM-based AOWC device. Also, ARPs excitation is not limited to the third-order Kerr nonlinear effect as used in this paper. Any other effects that can provide an effective refractive index change may also be to excite ARPs. Further steps in the development of this technique can be given in considering nonlinear material dispersion and loss, and especially reducing the required input power.
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