4.4.4 Stability Analysis of Polarizations in Birefringence

For the case γ_bf ≠ 0 but γ_a = 0, four types of steady-state solutions can be obtained (see Fig. 4.10). Two of them have orthogonal linear polarization, which are defined as the h and v modes. For each of these modes, the circular polarized components have equal amplitudes. The other two types of solutions are elliptically polarized for which the circular polarized components have unequal amplitude [43]:

1. The solutions of the two orthogonal linear polarizations are derived (i.e., Figs. 4.10a and 4.10b). Letting E_h = 0, and setting the time derivative terms of

Figure 4.10 Steady-state solutions of (4.72)–(4.74): (a) v mode; (b) h mode; and (c,d) elliptically polarized (after Ref. 34).

(4.73) and (4.74) to zero, the υ-mode solution can be expressed as

and the corresponding two orthogonal polarizations are given by

Again letting E_υ = 0 and setting the time derivative terms of (4.73) and (4.74) to zero, the h-mode solution can be expressed as

and the corresponding orthogonal polarization are given by

The corresponding steady-state values of the total carrier population and the population difference between the sublevels with opposite value of the spin for both linearly polarized solutions are

Hence, it is shown that γ_bf leads to a frequency difference of 2γ_bf between the h and v modes.

2. The solutions of the two elliptical polarizations, which is similar to that of the derivation of orthogonal polarizations, are deduced (i.e., Figs 4.10c and 4.l0d). This is done by setting the time derivative terms of (4.72)—(4.74) to zero and assuming that E₊ ≠ E_- ≠ 0. The two elliptically polarized solutions are given by

These two solutions are distinguished by the two values for the normalized population difference n_s, which are given by

The corresponding value for N_s is obtained from

but is restricted to the condition that N_s > l, which implies the pumping level should satisfy

The stability of the four types of steady-state solutions given above can be examined by writing their solutions as [43]

where a_± is a complex perturbation of the field amplitude, and ∆N and δn are real perturbations of N and n, respectively.After substituting the perturbed solutions (4.94) into (4.72)—(4.74) and linearizing to the first-order in the perturbation, one obtains the following set of linear coupled differential equations for a_±, ∆N, and δn.

To simplify the notation, (4.95) is written in vectorial form

where A = (a₊, a*₊, a_-, a*_-, ∆N‚ δn)^T and M is a 6 x 6 matrix whose coefficients can be easilv derived from (4.95) The eigenvalues of M are determined by a sixth-order polynomial that has to be solved. The stability of a steady-state solution is given by the real parts of the eigenvalues λ [i.e., stable if Re(λ) < 0 for all λ], while the imaginary part of λ gives a frequency characteristic of the evolution of the perturbation.

The stability of the two orthogonal polarizations can be analyzed by substituting the steady-state v mode solutions, (4.83) and (4.87) or the steady-state h-mode solutions, (4.85) and (4.87), in (4.96). The set of equations given by (4.96) can be decoupled into two independent subsets if the equations are rewritten for the variables S = a₊ + α_ and R = a₊ — a_ [34]. The first subset is

where Q = Q_s± is assumed. These equations determine the stability of a polarization (i.e., h or v mode) with respect to the perturbation of same polarization. This subset of equations is independent of γ_bf and τ_J. The general solutions of (4.97) are

It can be shown that one of the three eigenvalues of (4.98) has zero value and the other two are of complex expressions but with negative real part. This means that each steady-state solution of the two orthogonal polarizations is unconditionally stable with respect to amplitude perturbation of the same polarization.

The second subset of equations is

where the upper and lower signs denote the stabilities of h and v modes, respectively, in steady state. This subset determines the stability of a polarized solution with respect to perturbations of the "orthogonal polarization." To determine the eigenvalues of (4.99), it is assumed that

and the resulting eigenequation (i.e., third-order polynomial) for λ is

where the upper and lower signs denote the stabilities of υ and h modes, respectively, in steady state. If the coefficient of λ⁰ of (4.101) is zero (i.e., equivalent to, say, γ_bf = 0), there is a zero eigenvalue associated with the arbitrariness of the polarization direction. It can also be shown that the other two complex eigenvalues of (4.101) always have a negative real part, which implies that the system is unconditionally stable. However, if the coefficient of λ⁰ of (4.101) is nonzero, the zero eigenvalue becomes nonzero, which determines the stability of the system. On the other hand, if α_H = 0 (i.e., if there is no coupling between amplitude and phase fluctuations), both solutions of the two orthogonal polarizations are always stable [i.e., the coefficients of the polynomial (4.101) are all positive as µ > 1]. Hence, there exists a regime of bistability for any value of µ or γ_bf and no polarization switching occurs as the injection current is changed, but the nonvanishing value of α_H, together with the phase anisotropy, causes polarization switching.

Now, the stability of a particular solution for a general value of α_H in terms of two control parameters, the injection current µ and the birefringence parameter γ_bfτ_d is analyzed. This can be done by selecting a critical value of µ so that any variation from that value of µ can lead to purely exponential growth or decay of the polarized solutions. The condition for the h-polarized solution can be obtained from (4.101) by setting the coefficient of λ⁰ to zero. The critical value of µ at which the stability of this solution change is given by

Using the same augment (i.e., the coefficient of λ⁰ should be less than or equal to zero for the requirement that all λ real parts be negative), it can be shown that the h-mode solution is always stable for any µ < µ_h. For the υ-polarized solution, the condition for the υ-polarized solution can be obtained from (4.101) by equating the product of coefficients of λ² and λ¹ to the coefficient of λ⁰. Hence, the critical value of µ at which the stability of this solution change is given by

For the υ-polarized solution to be stable, all real parts of λ values must be negative, which implies that µ > µ_υ. Figure 4.11 shows the four different regions

Figure 4.11 Stability diagram for the steady-state solutions. The h mode is stable below the solid curve, while the v mode is stable to the left of the dashed line. These two curves divide four zones of different stability for the two linearly polarized solutions. In I, both solutions are stable; in II, neither solution is stable; in III, only the h mode is stable; and in IV, only the v mode is stable. Elliptically polarized solutions are stable within the narrow region between the solid and dashed–dotted curves. (After Ref. 43).

with different stabilities for the linearly polarized solutions obtained from (4.102) and (4.103): region I—both orthogonal polarizations are stable; region II—both are unstable; region III—only the h-mode solution is stable; region IV—only the υ-mode solution is stable. The stability diagram is a consequence of the combined effect of saturable dispersion associated with the α_H factor and spin dynamics associated with a finite value of τ_J-¡. In the calculation above, the following parameters have been used : τ_p = 300 ns, τ_d = 1 ns, τ_J = 50 ns, and α_H = 3.0.

The linear stability of elliptically polarized solutions can also be examined by solving (4.96) numerically for the sixth-order polynomial eigenvalues. To find the stability of a particular elliptical polarization, the coefficients of the polynomial of the eigenequation have to be obtained numerically in order to deduce the corresponding eigenvalues. The stability is determined by looking at the real part of the eigenvalue as described previously. The condition of a stable elliptically polarized solution is also shown in Figure 4.11. The elliptically polarized solution is stable in a narrow domain of parameters bounded by the dashed and solid lines in which µ is close to but larger then µ_h.