1. Introduction

The study of nonlinear optical materials has been driven in recent years by the need of optically active devices that can be monolithically integrated into optoelectronic circuits. Silicon offers the capability of optoelectronic integration via Silicon-On-Insulator (SOI) technology [1]. The major limitation of silicon, in this context, has been the lack of mechanisms for light generation, amplification, and wavelength conversion. Hence, the study of optical nonlinearities in silicon waveguides is of importance to determine the feasibility of realizing SOI-based optoelectronic circuits, with both passive and active optical functionality.

Coherent Anti-Stokes Raman Scattering (CARS), Parametric Wavelength Conversion (PWC), and Optical Parametric Amplification (OPA) are well-known nonlinear optical processes [2, 3]. PWC and OPA have been extensively investigated in optical fibers in the context of broadband signal amplification, or as wavelength-conversion mechanisms [4,5]. The theory that governs these phenomena has been fully developed in the past [6]. Considering only the χ⁽³⁾ component of the optical interaction and assuming no pump power depletion, a detailed theory describing the interaction of Stokes and anti-Stokes waves via SRS and Four Wave Mixing (FWM) has been developed [7]. The present situation involves interaction of three fields: the pump, the Stokes, and the anti-Stokes (a-Stokes) fields, at wavelengths (frequencies) λ_p(ω_p), λ_S(ω_S), λ_aS(ω_aS), respectively. These fields interact inside a SOI waveguide via the third-rank nonlinear tensor of silicon (χ⁽³⁾). In what follows, for the sake of simplicity and to accurately describe the conditions of the experiments performed, it will be assumed that the a-Stokes field, E _aS, has zero value at the front facet of the waveguide. The incoming fields will be the pump, E _P (TE₀ mode), and the Stokes, E _S (TM₀ mode). In general, the χ⁽³⁾ tensor has two contributions. The first is the “electronic”, non-resonant component, ${{\chi }^{\left(3\right)}}_{\text{NR}}$, which is nearly instantaneous in response and hence broadband, and which accounts for such effects as self-phase modulation (SPM) and FWM. The second is the Raman component, ${{\chi }^{\left(3\right)}}_{\text{R}}$(Ω=ω_p-ω_S) which accounts for SRS. The ${{\chi }^{\left(3\right)}}_{\text{R}}$(Ω) component has a strong spectral dependence, with a resonant peak at Ω=Ω₀, the frequency of oscillation of zone-center optical phonons in silicon. The element ${{\chi }^{\left(3\right)}}_{\text{R}}$(Ω) has therefore the spectral signature of the Raman-active phonon modes of vibration of the crystal lattice. This implies that there is a response time associated with this component, which corresponds to the phonon de-phasing lifetime. In the case of first order Raman scattering from silicon at room temperature, where only zone-center optical phonons are involved, the bandwidth is ~ 105 GHz. This corresponds to a response time of about 10 ps. In this paper we demonstrate, for the first time, Stokes to anti-Stokes wave-conversion in silicon waveguides. A detailed presentation of the experimental results will be given, plus a qualitative description of the physics involved. The wavelength-conversion phenomenon is reminiscent of coherent anti-Stokes generation (CARS) under quasi-phase-matching conditions. Wavelength conversion has important implications for optical communication, in particular for optical packet routing. In the following section, a brief theoretical review of SRS and Stokes/anti-Stokes wave coupling in silicon will be presented. The results that are relevant to the experimental conditions of the present work will be highlighted, followed by the description of the experimental work in Section 3.

2. Background

The spontaneous and stimulated Raman scattering in bulk silicon and in the visible region of the spectrum have been studied in the past [8, 9]. Recently, our group has reported the first observation of spontaneous [10] and stimulated [11] Raman scattering in integrated silicon waveguides. In addition, we have performed a detailed coupled-mode analysis of SRS in silicon waveguides [12]. The waveguides used in our previous measurements and in the current setup were fabricated parallel to [11¯0] direction on a silicon [001] surface, due to the favorable cleaving property of silicon in this orientation. This geometry lends itself to a coordinate system (x, y, z) rotated with respect to the crystallographic axes by 45° around the [001] axis. In this system, the waveguide is oriented along the z-axis, the TM₀ mode is linearly polarized along the Y-axis, and the TE₀ mode along the X-axis. The tensorial nature and symmetry properties of ${{\chi }^{\left(3\right)}}_{\text{R}}$(Ω) in silicon are inherited from the spontaneous Raman tensor, and is carried in the three Raman tensor components, R_i, which in the present coordinate system are written as [12],

The anisotropic scattering efficiency, S(Ω), defined as the percentage of scattered radiation per unit of solid angle, per unit length, is related to the Raman tensor as follows

where ê_i , ê_s are the polarizations of the incident and scattered radiation, respectively. S_o(Ω) is a constant of proportionality that contains the material parameters. The value of S(Ω) at the peak Raman shift (Ω=15.6 THz) was measured to be 4.1×10 ^-7 cm ^-1Sr ^-1 [10]. For the stimulated Raman effect in bulk material, the gain coefficient g_s, is obtained from the spontaneous efficiency following reference [9].

The relevant parameters for SRS and FWM in silicon can be summarized as,

Where N is the Bose occupation factor (0.1 at room temperature), n is the refractive index, and Δω is the FWHM of the spontaneous Raman line-shape. Equation (3.2) defines the Raman susceptibility, ${{\chi }^3}_{\text{R}}$. In Eq. (3.3), the value of ${{\chi }^{\left(3\right)}}_{\text{NR}}$ is the χ₁₂₂₁ component of the nonlinear susceptibility tensor [13, 14]. For a waveguide along the Z-axis, the amplitude of the Y-axis component (TM₀ mode) of the anti-Stokes field, E_aS(z), is related to the initial amplitude of the Y-axis component (TM₀ mode) of the Stokes field, E_So, by [7]

Where I_p is the effective pump intensity and Δk is the phase mismatch between pump, Stokes and anti-Stokes waves, defined as,

Note that in Eq. (5), the phase difference takes into account the two different polarization modes in the waveguide (TE₀ and TM₀) and therefore, waveguide-induced birefringence will play a crucial role in the efficiency of Stokes to anti-Stokes conversion. The coefficient A, in Eq. (4), is given by [7]

Note that A is in general a complex number. Its real part is responsible for an exponential gain in the a-Stokes wave [7]. The imaginary part of A results in an oscillatory behavior of the a-Stokes conversion efficiency, along the waveguide direction, z. As will be shown in Figure 1, it also manifests itself as a characteristic sinc² dependence of the a-Stokes power on the phase mismatch. From Eqs. (4) and (6), two regimes can be clearly identified,

The first case corresponds to an SRS dominated regime in which the large phase mismatch diminishes the FWM efficiency. In this regime, the Stokes signal is exponentially amplified with pump power and waveguides length, and the generated a-Stokes wave is negligible. The second case corresponds to a nearly phase-matched condition, in which the anti-Stokes signal is linearly amplified via the FWM effect. This process depletes the pump wave, effectively suppressing the exponential SRS gain. With the values given in Eq. (3) for the nonlinear coefficients in silicon, and assuming an effective pump power of 0.7 W inside the waveguide, the value of the threshold that defines these two regimes is

Therefore, to operate the waveguide as an amplifier, the amplitude of Δk must be larger than 2.1 cm^-1, whereas to realize efficient wavelength conversion, the phase mismatch must have much smaller amplitude than 2.1 cm^-1. Figure 1 shows the Stokes to anti-Stokes conversion efficiency vs. Δk, calculated using Eq. (4). The efficiency has a sinc² dependence on Δk, for all values larger than the threshold defined in Eq. (8). At the threshold value, Δk=2.1 cm^-1, the conversion efficiency is approximately 0.5%. For |Δk| smaller than 10 cm^-1, the maximum efficiency drops by at least two orders of magnitude, compared to the phase-matched condition. From this figure, it is evident that in the SRS regime the anti-Stokes conversion efficiency is not a single-valued function of Δk. Furthermore, the lower the conversion efficiency is, the wider the range of values of Δk that can fit it.

 

Fig. 1 Conversion efficiency, calculated from Eq. (4) using an effective pump power of 0.7 W at the input facet of the waveguide. The measured efficiency in the experiment is 1×10^-5.

Download Full Size | PDF

Figure 2 is a plot of the real part of the coefficient A, relative to the value of the Raman gain, g_s , as a function of the phase mismatch parameter, Δk, according to Eq. (6). The plot depicts the suppression of the exponential SRS gain as the phase-matched condition (Δk=0 cm^-1) is reached. It is also clear that the gain suppression is negligible for absolute values of the phase mismatch lower than 10 cm^-1. Also noticeable is the regime for small, positive values of Δk (~1 cm^-1) in which the gain coefficient is larger than that of pure SRS. This behavior has been predicted in silica fibers and has been attributed to Modulation Instability [7].

 

Fig. 2 Ratio of effective gain relative to SRS Raman gain, calculated from Eq. (6).

Download Full Size | PDF

3. Experimental setup

The experimental setup depicted in Fig. 3 is similar to the setup used to measure spontaneous Raman emission and SRS in silicon waveguides [10, 11]. The pump used is a CW, cascaded-Raman-cavity (CRC) fiber laser (Spectra Physics), operating at a wavelength of 1427 nm. The signal laser is an External-Cavity-Diode-Laser (ECDL, New Focus), tuned in the Stokes domain from 1535 nm to 1550 nm.

 

Fig. 3 Experimental setup used for observing CARS in silicon. F.- Pump laser filter. PBS.- Polarizing beam splitter. P(TM).- Polarizer to collect the TM₀ mode from SOI waveguide. OSA.- Optical Spectrum Analyzer. ECDL.- External Cavity Diode Laser, to use as a signal, with a scan range from 1535 to 1550 nm. VOA.- Variable Optical Attenuator.

Download Full Size | PDF

The Polarizing-Beam-Splitter (PBS) is used to couple the pump and signal lasers into the waveguide, in a cross-polarization scheme. The pump laser is coupled into the TE₀ mode and the signal laser is coupled into the TM₀ mode. In order to measure the conversion efficiency between the Stokes and the a-Stokes signal, a Variable Optical Attenuator (VOA) is placed after the ECDL. Only the TM₀ mode is collected at the waveguide output. No filter for the residual pump and signal beams is necessary, since the a-Stokes signal is collected by an Optical Spectrum Analyzer (OSA). The total pump-to-waveguide coupling loss is 6.1 dB consisting of 3dB loss due to the PBS, 1dB due to the pump filter, and 2.1dB of coupling loss into the waveguide. Fabry-Perot fringes from the cavity formed between the two facets of the waveguide (which are not AR coated), are small in amplitude compared to the observed spectral signature of the CARS effect. Therefore, these fringes are irrelevant for the results presented in the following section. The waveguide had a cross section of 5.4 µm² and was fabricated along the e [11¯0] direction of silicon [11].

4. Results

Figure 4 shows the anti-Stokes spectra measured as a function of the Stokes laser wavelength. There is a clear peak at 1328.8nm of anti-Stokes emission when the Stokes laser is tuned to 1542.3nm. The latter corresponds to the peak of the spontaneous Raman emission in silicon. The nature of the weaker features in Fig. 4 most likely corresponds to the wavelength-dependent character of the phase-matching parameter, Δk, in the SOI waveguide. Further analysis will be carried out in the future, considering other polarization configurations of the experiment, to determine the symmetry properties of these resonances.

 

Fig. 4 Plot of the a-Stokes spectra collected for a given value of the signal (Stokes wavelength). The z-axis represents the Stokes/a-Stokes conversion efficiency, normalized to unity. Note the clear appearance of two “satellite” resonances, as mentioned in the text.

Download Full Size | PDF

Figure 5 shows the a-Stokes power, integrated across the anti-Stokes spectrum (1323–1333 nm), as a function of the pump power effectively coupled into the waveguide (red triangles). The blue diamonds correspond to counter-propagating pump and Stokes signal in the waveguide. The relative lack of a-Stokes emission in the counter propagating condition underscores the importance of phase matching. In this case (Δk=2.8×10⁴ cm^-1), any a-Stokes signal would be due to down-converted spontaneous Stokes emission. Also shown in the figure is the measured power when the Stokes laser is off (crosses). The relative absence of power in this case is an unambiguous indication that the measured a-Stokes signal is that of down-converted Stokes signal. The figure also shows the result for the a-Stokes power obtained from Eq. (4) (solid line) using Δk~-27 cm^-1. The fact that the theoretical fit spans several values of a-Stokes power reduces the range of Δk values that match the measured data. Nonetheless, it should be stressed that the above procedure will not be construed as an accurate measurement of Δk. The latter can be achieved using procedures similar to those outlined in reference [14].

 

Fig. 5. Integrated a-Stokes power signal vs. pump power, with the Stokes signal power fixed. The integration is carried out from 1323 to 1333 nm in the a-Stokes spectra (Fig. 4). The red triangles show clearly the a-Stokes amplification due to Stokes parametric down-conversion. The blue diamonds correspond to counter-propagating pump and Stokes signal in the waveguide. The crosses are the down-converted, spontaneous a-Stokes signal. The Stokes signal was set at 1542.3 nm and 300 µW of input power. The fit with Eq. (4) was performed using $A=\sqrt{-179-2.3i}$ . This corresponds to Δk ~ -27 cm^-1.

Download Full Size | PDF

Figure 6 shows the integrated power in the a-Stokes domain as a function of the Stokes wavelength, λ_s. The peak shown at the center of the Raman frequency shift (1542.3 nm) indicates that the observed parametric down conversion has the spectral characteristic of Stimulated Raman emission in silicon, as expected from Eq. (4). Note that at high pump powers, the data appears to show a double-peak spectrum. The reason for this is that at high powers, the pump laser exhibits a double-peak spectral line-shape. This is a well known artifact of CRC fiber lasers, and is well documented by the manufacturer [15]. To obtain the down-conversion efficiency, the a-Stokes integrated power was measured as a function of the Stokes signal power, for different values of pump power. The result is shown in Fig. 7.

 

Fig. 6. Spectral dependence of the a-Stokes integrated power (as in Fig. 4), against Stokes wavelength, for different values of pump power. The solid lines are lorentzian fits to the data. Data fluctuation at high pump power in the region close to the Raman resonance (1542.3 nm) is most likely due to Fabry-Perot effects from the waveguide facets [11].

Download Full Size | PDF

 

Fig. 7. Stokes to a-Stokes conversion efficiency. The Stokes power was varied using the VOA shown in Fig. 3. The efficiency, defined as the slope of the plots shown, increases with pump power, as expected from Eq. (4).

Download Full Size | PDF

From the slopes in Fig. 7, the maximum Stokes/anti-Stokes power conversion efficiency measured for this particular waveguide is found to be 1×10^-5, for an effective pump power of 0.7 W. Referring to Fig. 1, the horizontal line shows the measured conversion efficiency relative to the phase mismatch parameter. Because of the oscillatory nature of the curve, an accurate estimation of the phase-mismatch, based on the measured CARS efficiency, is not possible when the efficiency is low. Specifically, the values of Δk that match the measured conversion efficiency are in the range -55 to -10 cm^-1.

Next, an Erbium-Doped-Fiber-Amplifier was introduced in the setup, to amplify the Stokes signal up to 20 mW. The result is an effective signal power of about 5 mW inside the waveguide, leading to a maximum a-Stokes signal observed of 55 nW, as shown in Fig. 8.

 

Fig. 8. Maximum amount of a-Stokes signal obtained, at 1328.8 nm. It was obtained by increasing the Stokes signal power up to ~ 5 mW effectively coupled into the waveguide.

Download Full Size | PDF

4. Conclusions

This paper has reported the first observation of coherent anti-Stokes Raman scattering in silicon. Specifically, we have demonstrated coherent parametric wavelength conversion from 1543.3nm to 1328.8 nm. This was achieved using a 1427nm CW pump with ~2nm linewidth. The bandwidth over which the conversion occurs corresponds to that of the Raman process. Including the pump broadening effect, this bandwidth is 350GHz in the present experiment. The bandwidth may be increased using multiple pumps, suitably spaced in wavelengths. The measured Stokes / anti-Stokes power conversion efficiency is 1×10^-5. The relatively low conversion efficiency can be attributed to the non-phase-matched condition in the present waveguide. With proper design of waveguide dispersion, conversion efficiency can be drastically increased, leading to silicon based, all-optical wavelength converters, suitable for optical packet switched networks.