We report on measurements that show the strength of the spontaneous Raman scattering in strongly confining silicon waveguides to depend significantly on the propagation direction of the amplified signal wave with respect to the pump wave. Furthermore, the strength of this nonreciprocity depends on the orientation of the waveguide with respect to the crystallographic axes. We find that when changing the orientation from 〈011〉 to 〈001〉, the Raman-induced nonreciprocity increases by almost a factor of 3.
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Nonreciprocal components are of great importance and commonly used in optical telecommunication systems . For example, isolators are required for sensitive laser sources that need to be protected from back-reflected light. Circulators can be utilized to separate signals in bidirectional transceivers. As compared to the rapid progress in the development of other components in silicon photonics , nonreciprocal components are still rare [3, 4]. A nonreciprocal behaviour was implemented, e.g., by bonding a magneto-optic garnet on top of a silicon-photonic waveguide. Using this method an isolation of 21dB was experimentally achieved so far .
The use of Raman scattering for nonreciprocal components promises new possibilities for the functionality of silicon photonic circuits. Up to now, Stimulated Raman Scattering (StRS) has been mainly applied for amplification purposes [5, 6]. Recently, the nonreciprocal behavior of StRS was predicted allowing the implementation of isolators  or unidirectional ring lasers . These components would be all-optical and dynamically reconfigurable and do not require any additional technological effort such as garnet bonding.
According to , the Raman nonreciprocity is caused by large longitudinal field components. It manifests itself in that the local Raman gain seen by the Stokes signal wave depends on the propagation direction with respect to the pump wave. The effect has been theoretically predicted to be present in both silicon nanowires  and nanowire silica optical fibers . In  we characterized the nonreciprocity in terms of ρ = Γ−/Γ+, the ratio of the local Raman gains for counter- (Γ−) and co-propagation (Γ+). There we have also shown that ρ strongly depends on the waveguide orientation with respect to the crystallographic structure. Assuming a silicon waveguide with a width of 500nm, a height of 220nm and a TE-polarized pump and Stokes signal, the nonreciprocity was 1.5 for the usual 〈011〉 waveguide orientation, whereas a nonreciprocity of 340 was predicted for waveguides arranged along a 〈001〉 orientation. This increase is due to the fact that the forward Stokes scattering is suppressed down to an extremely small level as compared to the backward scattering. However, an experimental verification of this nonreciprocity has not been done so far.
Here, we present the results of our experimental investigations on the behaviour of the Spontaneous Raman Scattering (SpRS) showing good quantitative agreement with our prediction of nonreciprocal Raman scattering . For this purpose, light generated by SpRS was measured from each side of a silicon waveguide. The measurements were performed on two waveguide orientations with respect to the crystallographic axes 〈011〉 and 〈001〉 showing the predicted strong dependence of the nonreciprocity on waveguide orientation.
2. Theory of nonreciprocal Raman gain in silicon waveguides
In this section we summarize the theory of nonreciprocal Raman gain in silicon waveguides . The pump and Stokes waves are assumed to propagate in waveguide modes with electric mode fields p and s, respectively. The Raman interaction with the silicon is modelled using a classical localized polarizability model of StRS; the result can be formulated in terms of a nonlinear susceptibility tensor χ (3) whose structure is determined by the Raman tensors of the triply degenerate Brillouin-zone-center optical phonon in silicon . The effect of the presence of this χ (3) on the propagating modes can be treated within full-vectorial coupled-mode theory. The power S(z) of the Stokes wave inside the waveguide is then found to evolve according to dS(z)/dz = ΓP(z)S(z), where P(z) is the local pump power, and the modal gain coefficient Γ is proportional to an effective susceptibility averaged over the waveguide core, 
where superscripts denote cartesian components along the vertical (x), lateral (y) and longitudinal (z) directions (with respect to the waveguide axis), and θ is the orientation of the waveguide on the substrate, i. e., the angle of the waveguide axis with the 〈001〉 direction (such that θ = 45° corresponds to a 〈011〉 direction). Furthermore, np,s are the refractive indices of silicon at the pump and Stokes wavelengths, Z is the vacuum impedance, and Np,s are the usual mode-normalization integrals. Finally, g is the bulk Raman-gain constant of silicon.
Equation (1) has been written assuming the convention  that the mode fields have real transverse (x,y) and imaginary longitudinal (z) field components. Now, it is a well-known property of waveguide modes that a reversal of the mode propagation direction also reverses the relative phase between transverse and longitudinal mode-field components—mathematically, the direction reversal corresponds to a complex conjugation of the mode field . When performing this conjugation in Eq. (1), we see that the second term (∣p · s∣2) under the integral in Eq. (1) changes, since in general ∣p·s∣2 ≠ ∣p* ·s∣2 due to the presence of imaginary longitudinal (z) mode-field components pz and sz. Only when these components are negligible (such as in rib waveguides with dimensions larger than a micrometer [5, 6]), then the relation ∣p·s∣2 = ∣p* ·s∣2 holds and Raman gain is reciprocal. The physical origin of the Raman-induced nonreciprocity formally derived above is that the strength of the force that drives the optical-phonon lattice vibration depends on the relative propagation directions of the pump and Stokes waves as discussed in .
In our experiments reported here, we use silicon photonic wires of such small cross-sectional dimensions (see Sect. 3.2) that the longitudinal field components lead to the large nonreciprocities observed here (see Sect. 4).
Before proceeding, we note that Saito et al. have reported nonreciprocal Raman scattering in waveguides made of gallium phosphide (GaP) [13, 14]. The dimensions of their waveguides, however, are still so large that longitudinal mode-field components are negligibly small, yet a significant Raman nonreciprocity (of more than a factor of 10) has been observed [13, 14]. Saito et al. attribute this nonreciprocity to the fact that due to the polar nature of the crystal, longitudinal-optical and transverse-optical phonons in GaP are not degenerate (as they are in silicon) but have significantly different frequencies. In GaP waveguides, this difference can manifest itself as a Raman nonreciprocity even in the absence of significant longitudinal mode-field components, in contrast to the silicon case considered in this paper.
3. Measurement Setup and Theory
3.1. Measurement Setup
As illustrated in Fig. 1, pump light at a wavelength of 1455 nm was coupled into a silicon photonic wire. This pump generates light at a wavelength of 1574nm by SpRS that propagates into both directions of the waveguide. These two signals are coupled into standard single mode fibers using grating couplers  on each side of the chip and measured by optical spectrum analyzers (OSAs). The same couplers are used to launch the pump power into the TE mode of the waveguide. Pump and Stokes signals are separated from each other by wavelength-division multiplexers (WDMs).
3.2. Waveguides under Test
Straight waveguides arranged along the usual crystal orientation 〈011〉 with three different lengths (5mm, 15mm and 35mm) have been investigated as well as zig-zag-directed 〈001〉 waveguides (sketched in Fig. 2).
All waveguides contain access structures consisting of grating couplers  followed by a tapered part where the waveguide width decreases from 10µm (grating coupler section) to the nominal width of 500nm. The waveguide height is 220nm. Transmission measurements revealed losses of 3.0 … 3.5dB/cm. The 〈001〉-oriented waveguides (as shown in Fig. 2) include several bends, which however did not perceptibly decrease the overall transmission.
The measured power levels of the Stokes signals are influenced by the unknown input and ouput coupling efficiencies as well as the insertion losses of the external elements (e.g. the WDMs). Interchanging input and ouput connections of the pump source and thereby reversing the propagation direction of the pump provides a possibility of eliminating these unknown quantities. The latter could otherwise lead to results that are falsely interpretable as a Raman nonreciprocity.
The waveguides investigated here are driven in a linear regime by pumping at accordingly small power levels so that nonlinear effects such as StRS, free-carrier-absorption (FCA) or two-photon-absorption (TPA) can be neglected in our measurements. This assumption is validated experimentally in Sect. 4.1. Thus we assume that light propagation is governed by linear losses and SpRS only. We modify the corresponding differential equations [17, 18] for the case that the SpRS efficiencies are different for co- and counterpropagating Stokes signals with respect to the pump:
Here, ℓ is the waveguide length and α are the waveguide losses which are assumed equal at the pump and Stokes wavelengths. S f,b(z) and P f,b(z) represent the Stokes signal and pump powers propagating in forward (f) and backward (b) direction. Finally, q −(z) and q +(z) are the z-dependent SpRS efficiencies for counterpropagation (−) and copropagation (+) of pump and Stokes waves, respectively.
In order to solve these equations we assume the pump waves inside the waveguides to be attenuated only by the linear waveguide losses and write
Here, P 0 is the nominal pump power provided by the laser source, while ηL,RP accounts for the coupling efficiencies for the pump when launched from the left-hand (L) or the right-hand (R) side of the waveguide. In the case, that either a forward-propagating pump Pf (z) or backward-propagating pump Pb(z) is present, we obtain
The SL,RL are the out-coupled Stokes powers at the left-hand waveguide end when the pump power was launched from the left (L) or the right (R) side, respectively, while the SL,RR are the corresponding Stokes powers measured at the right-hand waveguide end. The ηL,RS account for the out-coupling efficiencies of the Stokes signals on each side of the waveguide.
We define a total SpRS imbalance as
where the unknown coupling efficiencies as well as the nominal pump power P 0 no longer appear.
3.4. Imbalance for constant waveguide orientation and cross section
For the first interpretation of our measurements we neglect any change in waveguide width (tapered section) or orientation along z. The SpRS efficiencies can then be assumed constant, q ±(z) = q ± = const., and Eq. (12) simplifies to
Here, on the one hand,
is the loss-induced imbalance that quantifies the well-known imbalance between forward and backward spontaneous Stokes powers which is caused by the presence of linear waveguide losses alone, even if the Raman process itself was completely reciprocal [17, 18]. On the other hand,
is our sought-after Raman nonreciprocity. In view of potential applications for nonreciprocal components [7, 8] it should be noted that the spontaneous Raman efficiencies q ± are directly proportional to the stimulated Raman-gain coefficients Γ± discussed in , and the SpRS nonreciprocity q −/q + is identical to the StRS nonreciprocity Γ−/Γ+.
It should be noted here, that the waveguide orientation of our nominal 〈001〉 zig-zag structure is not constant as it includes 〈011〉-oriented access waveguides as well as several bends (see Fig. 2). To begin with, however, we will neglect the corresponding change of q ± in the evaluation of our measured data (Sect. 4.2). Afterwards, we will involve the change of q ± in Sect. 4.3 and calculate the total imbalance to be expected in our mixed-oriented waveguides according to Eq. (12).
To sum up, the measurement of the four Stokes powers SL,RL,R together with the knowledge of the linear waveguide losses α from independent measurements (see Sect. 3.2) provides a method of determining the Raman-induced nonreciprocity ρ where the influence of unknown coupling efficiencies is eliminated.
4.1. Measurement technique and confirmation of linearity
Figure 3 shows measured Stokes spectra (solid curves) for three different nominal pump powers P0 for forward [Fig. 3(a)] and backward [Fig. 3(b)] scattering. To evaluate this measurement we calculated the corresponding Lorentzian fits (dashed curves). The FHWM of this fit is 102± 5.2GHz (throughout all our measurements), as expected from Raman scattering in silicon . Furthermore, Fig. 3(b) shows a pump-power-dependent noise floor that is caused by backward SpRS in the feeding silica fiber; our fitting ansatz includes such a constant offset.
We injected the pump at the left-hand side here and estimated an in-coupled pump power of 2mW, whereas the nominal pump power was set to P 0 = 200mW. This poor coupling efficiency was caused by the bandwidth limitation of the grating couplers . The wavelength with maximum coupling efficiency could be tuned to a desired wavelength by tilting the feeding fiber to an appropriate angle with respect to the grating coupler plane. However, we optimized this arrangement for maximum Stokes coupling efficiency, so that we can reliably detect the weak Stokes signals expected here.
From the measurements of Fig. 3 we obtain the total SpRS power emitted from the silicon waveguide by integrating only over the Lorentzian portion of the fit. The total forward- (SLR) and backward- (SLL) scattered Raman powers as a function of the nominal pump power P 0 are shown in Fig. 4 as circles and crosses, respectively.
Before proceeding, we note that Fig. 4 also shows the residual pump power detected at the right-hand waveguide end (as stars). The observed nearly linear dependence of all three powers in Fig. 4 on the nominal pump power suggests that nonlinear effects such as StRS, TPA and FCA are not significant at the power levels used here. Indeed, the estimated in-coupled pump power of at most 2mW is much lower than the power levels at which StRS has been observed in comparable waveguides before . This supports the validity of our model Eqs. (2)–(5).
Without taking into account the non-zero waveguide losses as well as the (unknown) coupling efficiencies (see Sect. 3.1) the results shown in Fig. 4 would lead to a total imbalance of I = 1.6 (that could be interpreted as a nonreciprocity).
However, this imbalance could also be induced by linear losses and differing coupling efficiencies alone as mentioned above. Therefore, an extended measurement method should be applied as shown in the next section to exclude the loss-caused imbalance Iα as well as the unknown coupling efficiencies from the measured data by applying Eqs. (12)–(15).
4.2. Measurement results
In order to extract the net Raman nonreciprocity from the measured Stokes powers [by applying Eq. (13)] we performed the four measurements corresponding to Eqs. (8)–(11) on a total of 27 waveguides with different lengths and orientations. The nominal pump power was kept at P 0 = 200mW which made sure that the recorded Stokes spectra were sufficiently above the noise floor (see Fig. 3) and that we are still in the linear regime where StRS, TPA and FCA are negligible (see Fig. 4). The evaluation was done according to the procedure explained in Sect. 3.4.
Figure 5 shows the results after taking the four measurements SL,RL,R for each of the 27 waveguides. The total imbalance I is plotted as squares in Fig. 5. It contains the Raman-induced nonreciprocity ρ as well as the linear-loss-based imbalance Iα [see Eq. (13)]. The latter is calculated assuming losses of 3.5dB/cm (see Sect. 3.2) for all measured waveguides and is indicated as crosses in Fig. 5. Correcting I from the loss-induced imbalance by means of dividing it by Iα yields the sought-after Raman nonreciprocity ρ = I/Iα = q −/q + shown as diamonds in Fig. 5.
Looking closer to the rightmost waveguides (#25 – #27 in Fig. 5), an imbalance of I =4.79±0.69 can be seen. The waveguide length here is 35mm yielding a loss-induced imbalance Iα = 2.96. Dividing I by Iα ends up in a Raman nonreciprocity of ρ = 1.62±0.23 (diamonds in Fig. 5). This shows good agreement to our theoretically predicted value of 1.5 from .
Waveguides #13 – #24 in Fig. 5 have smaller lengths of 15mm and 5mm but the same orientation (〈011〉). It can be seen that the corresponding imbalances decrease to 2.06±0.07 and 1.67±0.11, respectively. Excluding Iα (which also decreased) yields a Raman nonreciprocity of about 1.6 again.
The pure Raman nonreciprocity ρ of all these 〈011〉-oriented waveguides (#13 – #27) does therefore not depend on the waveguide length as it was expected from our previous analysis. In fact, we read a common Raman nonreciprocity of ρ = 1.63±0.12 for these waveguides.
Performing the measurements on partly 〈001〉-oriented waveguides leads to the results shown on the left side of Fig. 5 (waveguides #1 – #12). Here, a total imbalance > 4 could be measured again. The lengths of these waveguides are only 6 – 7mm leading to an Iα that is close to unity. The measured total imbalance I can therefore only be due to a strong Raman nonreciprocity ρ. Comparing the mixed-oriented waveguides (#6 – #12) to the almost pure 〈001〉-oriented waveguides (#1 – #5) the Raman nonreciprocity increases from 2.31±0.27 to 4.35±0.29. However, in contradiction to this result we calculated a Raman nonreciprocity of 340 for a pure 〈001〉 orientation as calculated in . For this reason, the following section will give a more detailed analysis on the nonreciprocity to be expected in our partly 〈001〉-oriented waveguides, which will resolve this remaining discrepancy.
4.3. Calculation of total imbalance for waveguides with mixed 〈001〉 and 〈011〉 orientation
Due to design-given fixed positions of the 〈011〉-oriented access waveguide structures (see Fig. 2) a pure orientation along only a 〈001〉 crystallographic axis could not be realized. As shown in Fig. 2 only the zig-zag part of the structure is 〈001〉-oriented. Grating coupler, waveguide taper section as well as the small horizontal part on the right hand side of the structure are still 〈011〉-oriented. The Raman nonreciprocity for a pure 〈011〉 orientation can be calculated as 1.5 whereas a pure 〈001〉 orientation leads to a nonreciprocity of 340 . The 〈011〉 sections in our mixed structure prevent the total forward Stokes scattering of the whole waveguide from being suppressed down to the extremely low level expected for a pure 〈001〉 waveguide . The measured nonreciprocity in our zig-zag waveguides was therefore far below the value of 340 expected for a pure 〈001〉 orientation. To examine the theoretical Raman nonreciprocity occurring in our mixed-oriented structures we calculated the expected imbalances I for forward and backward pumping. Here, we used the general result of Eq. (12), where the Raman nonreciprocity can not be extracted from the total imbalance as it was possible with Eqs. (13)–(15). We will therefore compare the theoretical imbalance to our measured imbalance here instead of evaluating an explicit Raman nonreciprocity as done in Sect. 4.2.
To obtain an accurate value for the total imbalance I we calculated the local values of q ±(z) at each position z of our waveguide. Here, we considered the change of waveguide width and orientation as in . As mentioned above, the latter differs between the access-waveguides (〈011〉) and the zigzag-directed structure (〈001〉). We also included the change of orientation within each bend in the calculation of q ±(z). The losses have been assumed to be 3.5dB/cm again. Eq. (12) can then be solved numerically and leads to an imbalance I = 4.92. This is close to our measured value of I = 4.59±0.31 (squares at waveguides #1 – #5 in Fig. 5). Note that even if our waveguide would be loss-free, the calculated imbalance would be still 4.67 which is far from unity and can only be caused by the Raman-induced nonreciprocity. We repeated this calculation for waveguides #6 – #12, where only half of the structure is 〈001〉-oriented. Here, the calculated total imbalance I = 1.97 also shows reasonable agreement with the measured value of 2.40±0.28.
We have experimentally shown the presence of a nonreciprocity in the spontaneous Raman scattering in strongly confining silicon waveguides. Evaluating measured spontaneously scattered Stokes powers from both sides of a unidirectional-pumped waveguide reveals a Raman nonreciprocity of 1.63 for waveguides oriented along a 〈011〉 crystallographic axis. Partly changing this orientation to a 〈001〉 one led to a net increase of the apparent Raman nonreciprocity to 4.35. While a much stronger increase (up to 340) for a pure 〈001〉 waveguide has previously been predicted , we have shown that the value measured here is quantitatively explained by the fact that our nominally 〈001〉-oriented waveguides actually included 〈011〉-oriented access waveguides.
This work was funded by the German Research Foundation (DFG) in the framework of Forschergruppe FOR 653. The samples characterized have been fabricated at ePIXfab.
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