1. Introduction

The gyroscope, which is used to detect the rotation, is one of the most potential applications of the ring laser. Soon after the first Raman ring laser on silicon-on-insulator was demonstrated [1], some groups study its gyro characteristics [2]. However, to make a gyro work, one of the preconditions is the coexistence of two laser beams travel in opposite directions in the ring cavity [3]. In this paper, we employed the small-signal analysis to study the bidirectional operating characteristics in a Raman ring laser. The material in discussion is not only silicon but also polar crystals, e.g. GaP, which provides different forward and backward Raman gains.

2. Modeling

In our modeling, we have four counter propagating waves inside the resonator: two quasi-TM polarizing (dominant vertical component of electric field) pump beams (“p”) and two quasi-TE polarizing (dominant horizontal component) Stokes waves (“s”). The starting point in the development of the ring laser model is the time differential intensity equations for each clockwise (CW, “−”) or counterclockwise (CCW, “+”) propagating wave [4]:

In the equations, ${\tau }_{p,s}^{cv}$is the life time of a pump or Stokes photon in the ring resonator, whose wavelengths in vacuum are${\lambda }_{p,s}$respectively. It is assumed that both the pump and Stokes waves have the same group velocity$v_g$in the ring cavity with a length L. The stimulated Raman scattering (SRS) with the gain factor$g_R$, two-photon absorption with the absorption factor β, and the free carrier absorption with the effective factor$\overline{\phi }$are taken into concern in this model. The forward and backward SRS coefficients${\kappa }_{f,b}$can be set different because the forward and backward SRS efficiencies in some polar crystals [5], e.g. GaP [6], can be different though they are all the same in most materials, e.g. silicon [2,4]. The effective free carrier number can be expressed as follows

Without Stokes light emission, the pump intensities are$I_{p,\pm }^{\left(0\right)}$. And considering the near threshold situation, both the Stokes waves are so weak that small-signal analysis works. So $I_{s,+}/I_{p,\pm }^{\left(0\right)}\ll 1$, $I_{s,-}/I_{p,\pm }^{\left(0\right)}\ll 1$, and the pump intensity can be approximately written in the form

By substituting (4) and (5) into the following equations,

And$I_{p,\pm }^{\left(0\right)}$satisfy the following equations

The coefficients can be obtained by solving the linear Eqs. (7). With the help of these coefficients, we substitute the pump intensities in the time differential equations of Stokes light, and rewrite the equations in the follow forms with second-order (third-order on the point of electric field) terms, such as the self- and cross-saturation effect.

3. Discussions

Consider the simplest situation, the injection is symmetric, that is$I_{p,\pm }^{ext}=I_p^{ext}$and$I_{s,\pm }^{ext}=I_s^{ext}$. An obvious consequence is that both pump intensities $I_{p,\pm }^{\left(0\right)}$are the same$I_p^{\left(0\right)}$ when Stokes lights do not emit. So the equations for these counter running Stokes waves become

For the first case, suppose there is no injection at the wavelength of the Stokes light, that is$I_s^{ext}=0$. At the stationary solution$I_{s,\pm }^{\left(s\right)}$ which satisfy the stationary condition

Substituting them into (13), we find

The stationary solution is stable only if both the eigenvalues of the stability matrixΘ are negative. For a second order matrix, this requires its trace is negative and its determinant is positive. This leads to

It is easy to prove that both${\alpha }_{slf}$and${\alpha }_{crs}$are positive in a normally operating laser, and

This reveals that if the backscattering between the two Stokes beams is weak, such as in a laser gyroscope, the ring laser with materials of which SRS has no directional preference, e.g. silicon, does not support stable bidirectional operating state. Because${\alpha }_{slf}$is always smaller than${\alpha }_{crs}$in this case. This can be explained in a positive feedback way. When the ring laser works at the bidirectional operating stationary point, as the forward and backward Raman gains are all the same, the only difference between the gain/loss factors of these two oppositely running Stokes beams originates from the difference between self- and cross-TPA. Once intensities of these two Stokes beams depart a little from the stationary point, the departure will be enlarged because the cross-TPA is twice as large as self-TPA. This positive feedback makes the bidirectional stationary operating point unstable.

But for those polar crystals whose Raman scattering has directional preference, e.g. longitude phonon scattering in GaP only occurs in the backward direction, stable bidirectional operating can be realized if

Combining with the lasing condition: the zero order net gain $\rho >0$, we have

So a difference between the forward and backward gains is needed to realize stable bidirectional operation. The left diagram in Fig. 1 gives an example of the dependence of the operating regime to the forward and backward scattering gains. The dashed line is the threshold. Only in the region above it where $\rho >0$, laser light emits. This region is divided into three areas by the solid curves, which are obtained by setting the two expressions in (22) to be equal. In the middle area where the stability condition, the right part of (22), is not satisfied, the ring laser works in unidirectional operating regime. Only in the two side areas where there is a distinct difference between the forward and backward gains, stable bidirectional operation can be obtained. To justify this conclusion, we simulate A~D four situations by the side of the threshold line. The numerical simulation is based on the discretisation of (1) by simply replacing the time derivatives with finite differences. The simulation results agree with the distribution of these work points. In the cases A and B, though the laser cavity is symmetric pumped, there is always one suppressed Stokes mode, CW or CCW, depending on the initial condition. It should be noticed that case A is the situation that the material have the same forward and backward scattering gain. But in the case C and D, both CW and CCW Stokes modes can coexist and be stable in the ring cavity after sufficiently long simulating time.

 

Fig. 1 Theory and numerical simulation results illustrating the occurrence of the two different ring laser regimes (unidirectional and bidirectional). The left diagram is the small-signal analysis results from (22). Figures A, B, C, and D are, respectively, the simulation results corresponding to the points A, B, C, and D in the left diagram. The numerical simulation is in the case of symmetric pumped. Values of other parameters, common to these diagrams, are: ${\lambda }_s$ = 1550nm, ${\lambda }_p$ = 1646nm, $\overline{\phi }$ = 6×10⁻¹⁰, ${\tau }_{eff}$ = 0.1ns, β = 0.6cm²/GW, $v_g{\tau }_p$ = 2.35cm, $v_g{\tau }_s$ = 8cm, $I_s^{ext}$ = 0, $I_p^{ext}/v_g{\tau }_p^{cp}$ = 0.15GW/cm³, and the corresponding $I_p^{\left(0\right)}$ = 0.1648GW/cm².

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As discussed above, when there is no Stokes light injected, those materials of which SRS has no directional preference do not support stable bidirectional emission. Now we add such injections. By setting${\kappa }_f={\kappa }_b=1$, we have

Consider the stationary solution$I_{s,+}^{\left(s\right)}=I_{s,-}^{\left(s\right)}=I_s^{\left(s\right)}$, which satisfy the stationary condition

Appling the above vibrations analysis, we have the stability matrix

Again, the stationary solution is stable only if the trace is negative and the determinant is positive. Because all these variables are positive, it only requires that

Combining with the stationary condition, we finally have

Even in a laser with the same forward and backward Raman gain and no backscattering, bidirectional operating is possible if the injection is strong enough. The conclusion is illustrated in Fig. 2 . As show in the left figure, when the extent injections are not strong enough, the competition between these two opposite propagating Stokes modes induces asymmetric intensities of them. Once the injections are strong enough, as shown in the right, both CW and CCW modes have the same intensity in the end.

 

Fig. 2 Simulation results ring laser regimes with injections at Stokes wavelength: $I_s^{ext}/v_g{\tau }_s^{cp}$ = 2kW/cm³ (left) and 3kW/cm³ (right), ${\kappa }_f{g}_R/\beta ={\kappa }_b{g}_R/\beta$ = 5 and values of other parameters are the same with those in Fig. 1.

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All the discussion above is based on the small-signal analysis, which requires the Raman ring laser work in the near threshold region. Once the intensity of the Stokes emission is no longer much smaller than that of the pump, all of these conclusions are invalid. In such region, further study is still needed.

4. Conclusions

Employing the small-signal analysis, we studied the bidirectional operating characteristics in a Raman ring laser. If without injections, bidirectional operating only occurs in the ring made up of polar crystals, and it requires that the difference between forward and backward stimulated Raman gains is large enough. Further, with strong enough external injections, bidirectional operating can also be obtained in a ring laser made up by general crystal, e.g. silicon, which has the same forward and backward gains.