1. Introduction

The mode properties of lasers are determined by the resonator boundary conditions and the effects of intracavity elements on amplitude and phase. In the case of polarisation modes, the analysis is simplified when the axes of birefringent elements are parallel to those of the gain. One notable exception is laser media suffering radial thermally-induced stress birefringence to produce complex output beam patterns including the characteristic Maltese cross pattern of ref [1]. Though this specific case has been solved [1, 2], there is another class of problems involving linearly anisotropic gain and birefringence but with non-collinear axes. Such a scenario has been recently identified in diamond Raman lasers (DRLs) in which the crystal contains in-grown stress birefringence [3].

Calculation of polarisation modes of such anisotropic resonators are often found using mode theory (see e.g., [4]) combined with a Jones matrix formalism (see e.g., [5]). Eigenvector solutions of the resonator round-trip matrix represent the polarisation states and the corresponding eigenvalues enable determination of the polarisation mode that experiences the highest net gain and thus establishes first in the cavity. Although some laser gain crystals, such as Nd:YVO${ }_4$, show anisotropy in their emission cross-section, the eigensolution analysis is determined by the passive elements [6], since they are either the dominating effect or the gain axes are parallel with the axes of the dominant polarising element. In systems with strong polarisation mode competition, such as intra-cavity frequency doubled solid-state lasers, an effective emission cross section, and thus threshold, can be defined for each eigensolution [7]. Effects of gain an isotropyin presence of birefringence have been studied numerically for microchip lasers [8] and VCSELs [9]. Coupled laser-level rate equations for two orthogonal cavity eigensolutions explain the polarisation dynamics of these lasers. In the case of spin injected VCSELs, their gain exhibits small circular anisotropy which has been recently shown to reveal complex dynamics in the polarisation modes [10–12].

A specific example of systems with strong gain anisotropy is crystalline Raman lasers [13]. Most of the practical Raman crystals are naturally birefringent [14], however, even isotropic Raman crystals such as diamond, silicon, and Ba(NO${ }_3$)${ }_2$, which are cubic [14], contain stress induced birefringence or the resonator consists of other anisotropic elements or components with stress induced birefringence of an arbitrary direction. An analytical model for the polarisation behaviour has not yet been presented for lasers possessing linearly anisotropic gain and non-collinear birefringence axes.

In this paper we extend the Jones formalism to include linearly anisotropic gain and derive the Jones matrix for the specific case of diamond. The theory is applied to standing wave and ring resonators containing stress induced linear and circular birefringence in the crystal. We show that the fully analytical calculations explain the observed difference in polarisation behaviour of pulsed and continuous wave (cw) DRLs.

2. General theory

Jones matrices describe how individual optical elements in the laser resonator act upon the polarisation state of the circulating beam. Jones matrices for standard optical elements can be found e.g., in [15]. The Jones matrix for a birefringent material with fast axis parallel to a local x axis is defined as

For a gain medium for which two orthogonal polarisation components of generated laser field amplify independently, we define the Jones matrix of gain as

In a laser resonator above threshold, G₀ is fixed by a lasing threshold condition. Assuming ${\gamma }_1\ge {\gamma }_2$, that condition is

A gain material containing only a pure linear birefringence can be modelled as a separate gain and an arbitrarily oriented waveplate as the birefringence and gain matrices commute. The roundtrip matrix of a resonator containing gain and linear birefringence is

3. The anisotropic Raman gain in crystals of F${ }_{2g}$ symmetry

We now derive the amplification factors ${\gamma }_{1,2}$ for crystals of cubic symmetry (specifically diamond), which has a triply degenerated Raman mode and features strongly anisotropic gain [13]. The gain maximizes for coincident pump and Stokes polarisations aligned to a 111 direction [16]. Note that the 111 directions are not orthogonal (70.6${ }^{\circ }$) as shown in Fig. 1. In arbitrary bases the power flow from pump to Stokes fields describe four coupled equations [17]

 

Fig. 1 Crystallographic directions in diamond with respect to usual propagation direction used in Raman lasers and an example of birefringence fast f and slow s axis orientation.

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We limit the discussion to cases near lasing threshold, in which the effects caused by uneven depletion of P₁ and P₂ and resulting pump polarisation rotation can be neglected. If the pump basis orientation is aligned with the input linear polarisation direction, the P₂ is zero and Eqs (7) simplify to

As the polarisation bases for the pump and the Stokes fields are not required to coincide, primes were added to the pump basis labels in Eqs. (8) to discern it from the Stokes basis. As shown below, it is useful to choose a different Stokes basis. We calculate the ${\chi }^{\left(3\right)}$ terms as a function of the pump and Stokes basis angles α and β, respectively, using the tensor for diamond [13]. The ‘direct’ susceptibility components ${\chi }_{11^{\prime }11^{\prime }}^{\left(3\right)}$ and ${\chi }_{21^{\prime }21^{\prime }}^{\left(3\right)}$, where the pump interacts with S₁ and S₂ independently, are [17]

Equations (8) simplify when choosing a basis for which the ${\chi }_{\text{cross}}$ terms are zero (so that the Stokes field components are independently amplified). Figure 2 shows the calculation of χ and ${\chi }_{\text{cross}}$ for all combinations of pump and Stokes bases orientations α and β. It is seen that, for every pump polarisation, there exists a Stokes basis in which the cross terms vanish (see Fig. 2(b)) and are 90${ }^{\circ }$ apart. For example, for a pump polarisation angle of $\alpha ={90}^{\circ }$, the ${\chi }_{\text{cross}}$ terms are zero for the Stokes basis angle $\beta =0^{\circ }$. For this choice, the polarisations amplify independently with gains proportional to maxima and minima values of χ (shown as black and white full lines in Fig. 2(a)). With such directions chosen as a Stokes basis Eqs. (8) reduce to

 

Fig. 2 (a) χ as a function of pump and Stokes polarisation angles. Full black and white lines show maxima γ₁ and minima γ₂, respectively. (b) χ_cross as a function of pump and Stokes polarisation angles.

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The maxima γ₁ and minima γ₂ of χ are found by parametrizing Eq. (9) and finding its extremes. The solutions for γ₁ and γ₂ are

Eqs. (12) describe the black and white lines in Fig. 2(a). Since the coupling coefficients are normalised to the value of pump polarisations parallel to $\left[110\right]$ direction, γ₁ grows from 1 to 4/3 and falls back to 1, and γ₂ decreases from 1 through 1/3 to 0 as the pump polarisation is rotated from 0${ }^{\circ }$ through 35.3${ }^{\circ }$ to 90${ }^{\circ }$, respectively. With these simplifications, the gain matrix G takes the form of Eq. (2) allowing the solution procedure of Sec. 2. G is a function of only the combined loss and reflectivity of the mirrors R and the angle of the pump polarisation α, which in turn defines the Stokes basis and amplification coefficients ${\gamma }_{1,2}$.

4. Model results

The eigenvectors and eigenvalues of the roundtrip matrix M (Eq. (6)) predict the output Stokes polarisation dependencies. M is a function of the four parameters; the birefringence phase difference ξ, the angle between the birefringence fast axis and Stokes bases τ, the angle of the pump polarisation α which defines the coupling parameters γ₁ and γ₂, and the effective reflectivity of the mirrors R that determines steady-state gain strength.

The impact of ξ is the strongest when the Stokes bases is 45${ }^{\circ }$ to birefringence axes and is zero if they coincide. The effect of Γ increases from zero to $\ln (R^{-1/2})$ as the pump polarisation direction changes from 0${ }^{\circ }$ to 90°. The relative strength of Γ and ξ distinguishes several cases. When the gain term dominates, (i.e., $\mathrm{\Gamma }\gg \xi$), the cavity eigenvectors are linear polarisations aligned with the Stokes basis. When birefringence dominates ($\mathrm{\Gamma }\ll \xi$), the eigenvectors are linear polarisations in the directions of the birefringence fast and slow axis and for $\mathrm{\Gamma }\approx \xi$ the output polarisations are elliptical. The following sections discuss model results for the three cases and a caseinvolving the addition of circular birefringence.

Note that for $\mathrm{\Gamma }=0$, the solutions are eigenvectors along the birefringence axes directions with identical eigenvalue and the laser can operate on either of them. If ξ is also zero, the solution of the roundtrip matrix is the identity for whichall eigenvectors are solutions with identical eigenvalue. Experiments have shown that linear polarisations are produced under such conditions, but with direction that changed randomly from pulse to pulse [16].

4.1. Output polarisation determined by gain: Case $\Gamma \gg \mathrm{\xi }$

In this case the solutions are linear polarisations along the Stokes basis where output Stokes is linearly polarised along the direction β of the highest normalised gain component γ₁, with $\beta =1/2\arctan (2\cot (\alpha ))$. Figures 3(a)-(d) indicate the polarisation, ellipticity and angle for all pump polarisation angles and for birefringence phase shift increasing up to 100 mrad. The gain dominated behaviour is shown in the bottom row in Figs. 3(a)-(d) (the 0${ }^{\circ }$ pump case was excluded). For pump polarisation rotated from 0${ }^{\circ }$ to 90${ }^{\circ }$ the output polarisation rotates in opposite direction from 45${ }^{\circ }$ to 0${ }^{\circ }$. The pump and output polarisations coincide for 111 directions, at 35.3${ }^{\circ }$ and 144.7${ }^{\circ }$. Such behaviour is typical for pulsed DRLs using relatively low R [16]. Note that output polarisations beyond 45${ }^{\circ }$ are not obtained for any pump polarisation.

 

Fig. 3 Output polarisations shown as ellipses as a function of linear pump polarisations angles and birefringence phase shift. Top axis indicates corresponding Γ. The colours red/blue indicate left/right handedness. R = 99% and τ = 0°, 20°, 45°, and 70° in (a), (b), (c), and (d), respectively.

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Fig. 4 Output polarisations shown as ellipses as a function of induced rotation angles and birefringence phase shift. (a) and (b) show the polarisation behaviour for R = 99% Γ = 3.8 × 10⁻³, τ = 20°, and α = 35° for a linear and ring cavity, respectively.

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4.2. Output polarisation determined by linear birefringence: Case $\Gamma \ll \mathrm{\xi }$

Under this condition the cavity eigenvectors are linear polarisations aligned with the birefringence fast and slow axes as shown in top rows of Fig. 3(a)-(d). The gains associated with each axis direction, and therefore which polarisation lases, is dependent on the pump polarisation and fast axis direction. In Fig. 3(a) the output polarisation is always horizontal, in 3(b) and 3(d) the polarisations flip at 120° and60°, respectively and in 3(c) the flip occurs at 90°. There is also a flip when the pump polarisations cross 0°/180°. The switching between polarisations is well produced in experiments as shown in Sec. 4.5. Vertical output polarisations are not obtained even for a vertical fast axis, as in the high gain case; however, all other orientations are possible.

The pump polarisation at which the output polarisation flips occurs at the crossing points where the gain for one axis surpasses the other. Figure 5(a) shows the normalised gain as a function of pump polarisation angle and birefringence fast axis direction. The region where the gain is highest for the slow axis direction is highlighted by white stripes. For any pump polarisation, maximum gain (indicated by dashed black lines) is achieved when birefringence axes coincide with the Stokes bases.

 

Fig. 5 (a) Normalised gain as a function of pump and birefringence fast axis angle τ for strong birefringence ξ = 15 mrad and low gain R = 99%. Black dashed lines indicate gain maxima. White stripes indicate the region where the output polarisation is parallel to the slow birefringence axis. (b) Gain as a function of birefringence fast axis angle for pumping polarisations parallel to the [110], [111], [001], and $\left[\overline{1}\overline{1}0\right]$. The lineschange from full to dashed for the Stokes polarisation parallel to fast and slow axis, respectively.

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Figure 5(b) shows vertical cuts through Fig. 5(a) for pump polarisations aligned to crystal directions of high symmetry. For any birefringence orientation (except 0° and 90°) the highest gain is obtained for the pump polarisation parallel to the ⟨111⟩ axis that is closest to a fast or slow birefringence axis. The maximum difference in gain, and thus threshold, between the two ⟨111⟩ directions is 26% for birefringence fast axis oriented at 10° and 80°. At 10°, for example (as in Fig. 1), the birefringence axes are 45° from $\left[\overline{1}\overline{1}1\right]$ and 25° from [111] direction and therefore for $\left[\overline{1}\overline{1}1\right]$ pumping the Stokes polarisation direction is pulled the furthest from its original direction unperturbed by birefringence.

4.3. Output polarisation determined by balance between linear birefringence and gain: Case $\Gamma \approx \mathrm{\xi }$

For approximately balanced gain and birefringence, the output is elliptical as shown in Figs. 3(a)-(d) as red (left-handed) and blue (right-handed) ellipses. For intermediate values of ξ, the ellipticity strongly depends on pump polarisation due to the resultant effect on Γ and the angle between the birefringence and Stokes bases. The output becomes linear when the Stokes high gain axis coincides with one of the birefringence axes. In that case the output polarisation remains unchanged for any phase shift value as it does not interact with the birefringence. Thus elliptical states are observed in the transition region where $\mathrm{\Gamma }\approx \xi$ and when the maximum gain direction is away from the birefringent axes.

As a specific example we consider the situation for a birefringence fast axis at $\tau ={20}^{\circ }$ (Fig. 3(b)). For pump polarisation close to 0°, the Stokes is elliptical for ξ between 0.2 mrad and 4 mrad. As the pump angle is increased, Γ also increases and, correspondingly, elliptical outputs are obtained only for larger values of ξ. For fixed ξ, the major effect of increasing the pump polarisation angle is to shift the Stokes basis towards τ. The polarisation is purely linear for the pump angle of 65°, for which the Stokes basis coincides with τ. Further increasing the pump angle ultimately leads again to elliptical output but with opposite handedness and a flip of the major axis from fast to slow birefringence axis at a pump angle of 120°.

The transition from low to high birefringence is shown in more detail by plotting ellipticity as a function of R and the birefringence phase shift. Figure 6 shows ellipticity for two cases: Fig. 6(a) for pump polarisation close (within 5°) to the [110] direction for which ${\gamma }_1\approx {\gamma }_2\approx 1$ and Fig. 6(b) for pump polarisation aligned with [001] where ${\gamma }_1=1$ and ${\gamma }_2=0$. For both plots, the birefringence fast axis is at 45° with respect to high gain axis γ₁ (i.e., 0° and 45°, respectively) to induce the highest ellipticity. Note, however, that the maximum ellipticity never reaches 1. Below the region of high ellipticity (gain dominated regime) the output polarisation tends to linear along the Stokes basis high gain axis. Conversely, above the region (birefringence dominated regime), the polarisation tends to the birefringence axis with higher gain.

The Γ value as a function of R is plotted on Figs. 6(a)-(b) (dashed line), confirming that $\mathrm{\Gamma }\approx \xi$ is the condition needed to produce strongly elliptical output. Γ deviates from this condition for small values of R and large gain difference. For increasing R (decreasing Γ) the phase shift required to induce ellipticity and ultimately flip the polarisation from the high gain to the birefringence axis is smaller. The slope of the peak ellipticity line is smaller in Fig. 6(a) than 6(b) due to the smaller gain difference and thus the birefringence dominates for smaller ξ.

 

Fig. 6 Ellipticity of the Stokes mode as a function of R and birefringence phase shift ξ for pump polarisation angles of (a) 5° and (b) 90°. The dashed black line indicates the magnitude of the gain difference Γ (also shown asthe top axis) as a function of R on the ξ axis. The insets show magnified regions for typical values observed in diamond and used in cw DRLs.

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The insets of Fig. 6(a) and 6(b) show the ellipticity for typical values used in cw DRLs. The reflectivities range from 99.5% to about 90% depending on pump power [18, 19] and birefringences at 1 µm wavelength are about 0.05 rad (but can be as high as 0.5 rad) for materials typically used in lasers [3]. These results indicate that elliptical output is predicted for experimental parameters of R = 99% and ξ = 5 mrad, for example, when using a pump polarisation of 90°.

4.4. Optical rotation

Some CVD-grown single crystal diamonds show evidence of optical rotation as a result of the compound effect of crystal stress induced birefringence with varying orientation in the propagation direction [3]. Such an element, represented as asequence of randomly oriented wave-plates, is equivalent to a single wave-plate and a rotator [20]. However, since the rotator matrix is non-commutative with the gain and birefringence matrices, a single matrix describing all effects is required to enable calculation of the cavity eigenmodes. Such a matrix can be obtained, following the derivation in [21], by dividing the medium into thin layers of separate birefringence, gain and optical rotation effects. Since the change induced in each layer is small, the exponential functions in gain and birefringence matrices and trigonometric functions in the rotation matrix can be replaced by low order Taylor expansions. Such matrices commute and, in the limit of zero thickness, combine into the overarching matrix J as

In a standing wave resonator, the rotation angle in matrix J changes sign on the second pass. However, because the gain, linear birefringence, and optical rotation act upon the field simultaneously, the rotation is not completely cancelled per round trip. The outcome is that any Stokes polarisation is rotated exactly by half of the optical rotation angle, as shown in Fig. 4(a) for the example parameters of 35° and 20° for the pump and the birefringence fast axis angles, respectively.

In contrast, in a ring resonator, the rotation accumulates for each round-trip and thus even a small amount of rotation results in an elliptical output polarisation. Referring to Fig. 4(b), for $\mathrm{\Gamma }\gg \xi$ (bottom rows), the polarisation is highly elliptical even for small optical rotation angles and with its major axis rotating towards 90°. For $\mathrm{\Gamma }\ll \xi$ (top rows), the ellipticity is induced only for much larger rotation values and the major axis remains oriented along the birefringence axis. These results indicate that the ellipticity of output for ring resonators will be highly susceptible to circular birefringence. For such conditions the normalised gain changes from 1 to 0.5 as the pump polarisation angle changes from horizontal to vertical. The output polarisation handedness depends on the direction of optical rotation and also the strength and orientation of the linear birefringence. The normalised gain of the rotated polarisations in a linear resonator follows identical rules to Fig. 5(a).

4.5. Experimental comparison ($\Gamma \gg \mathrm{\xi }$, $\Gamma \ll \mathrm{\xi }$)

We verify the validity of the model against the polarisation measurements of a DRL similar to that reported in [16] but operated close to threshold (case $\mathrm{\Gamma }\gg \xi$) and the DRL in [3] (case $\mathrm{\Gamma }\ll \xi$). Measuredand modelled polarisation angles are shown in Fig. 7 for (i) $\mathrm{\Gamma }\gg \xi$ and for $\mathrm{\Gamma }\ll \xi$ for three birefringence axis directions ((ii),(iii), and (iv)). The modelled polarisations closely match the measurements, except for one point in (iii), where the experiment switches direction later then predicted, most likely due to inaccurate determination of the birefringence axis direction or magnitude of optical rotation.

The role of birefringence magnitude and direction is analysed in more detail by plotting the observed output polarisations on the gain map of Fig. 5 as displayed in Fig. 8. For (i) the Stokes output is polarised along the high-gain directions as expected. The measurements closely follow the maximum of the gain γ₁ even for pump angles approaching 0° where a transition to unstable or random polarisation directions is expected. The absence of such behaviour may have been caused by a coincidence of the particular orientation of the birefringence axis close to 45°.

 

Fig. 7 Measured Stokes polarisation angles (full thick lines) compared to modelled Stokes polarisations (dashed). (i) this work, R = 0.3, ξ = 0.3 rad, τ unknown; (ii), (iii), (iv) from [3], R = 0.99 (R used was 99.5% plus 0.5% additional roundtrip loss), ξ = 0.06 rad, 0.2 rad, 0.27 rad, τ = 45°, 71°, 1.7°, and θ = 1.25°, 9.15°, 6.6°, respectively, measured by Mueller polarimetry.

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In the birefringence dominated regime the model correctly predicts the starkly contrasting behaviour in which the polarisation remains aligned to the birefringent axis (dashed lines) and flips at points for which the gain is surpassed by that of the other birefringence axis. The offset from the fast and slow axes caused by the presence of small amount of optical rotation is also correctly predicted. The small discrepancies between the modelled and measured polarisations (< 3°) are attributed to errors in measuring the value of circular birefringence and fast axis orientation.

Since all polarisation measurements reported to date have corresponded to the extreme cases producing linearly polarised output, our work predicts conditions that are expected to yield elliptical output. It is noted that the region of high ellipticity places tight constrains on the exact values of R and ξ as shown in Fig. 6.

 

Fig. 8 Stokes output polarisation angle as a function of pump polarisation angle. Dashed lines show fast f and slow s axis of the linear birefringence determined by Mueller polarimetry. Blue circles and full line show measuredand modelled Stokes polarisation angles for given pump polarisation angles for Γ ≫ ξ. The triangle, square and diamond symbols show measurements (ii), (iii), (iv) corresponding to spots A, C, D in [3] respectively and show the opposite case of Γ ≪ ξ for varying values of birefringence and its orientation. Full lines of the same colour are corresponding modelled results and dashed lines indicate the orientation of the linear birefringence.

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5. Conclusion

We have used a Jones formalism to calculate polarisation modes in lasers containing non-collinear anisotropic gain and birefringent axes, a scenario that occurs in diamond Raman lasers. The model describes 3 main regimes: A gain dominated regime in which linear output polarisations are observed along the high gain axis. A birefringence dominant regime in which linear polarisations are produced in directions along the fast and slow axis depending upon which is more closely aligned to the high gain axis. And a transition region producing elliptical outputs when the effects of gain and birefringence are in near balance. The model also considers the role of optical rotation in the gain medium, which is found to occur in some diamond samples. In standing wave resonators the rotation causes an angular offset in the polarisation behaviour; however, for unidirectional ring lasers, even small amounts of optical rotation lead to elliptical output.

The theory also applies to other anisotropic systems such as silicon Raman lasers (which share crystal class with diamond) and potentially inversion lasers that have a combination of strongly anisotropic gain and birefringence. Extension to other systems is straightforward provided the gain has an orthogonal basis where the polarisation components amplify independently.