The standard Maxwell-Bloch equations with a SA added can be rewritten as[128]: ∂_tE= -c/n∂_zE-c/niµP/ℏl_0 D_th-c/2n(l_0-γ|E|^2)E, ∂_tP= iµ/2ℏD E-P/T_2, ∂_tD= D_p-D/T_1+iµ/ℏ(E^*P-c.c.).
The dynamics of a two-level QCL gain medium with ring cavity can be described using the Maxwell-Bloch equations. After transformation of the variables, the Maxwell-Bloch equations can be simplified to: ∂_t E=-c/n∂_z E- c/niP-c/2n(l_0-γ|E |^2)E, ∂_t P=-i/2DE-P/T_2, ∂_t D=p_fl_0/T_1T_2-D/T_1+i(E^*P-c.c.).
To proceed with the linear stability analysis, we express each of the variables as the sum of the steady-state value and the small perturbations δ E, δ P, and δ D.
The steady state solution can be found by setting the left-hand sides of the Eqs. (C)-(C) to zero. The steady state solutions has the form E=Ē, P=P, and D=D are constants in time and space satisfying: D=l_0/T_2-γĒ^2/T_2, P=i/2(l_0-γĒ^2 )Ē, p_f+1=(1-γĒ^2/l_0)(1+Ē^2T_1T_2).
The resulting equations regarding the fluctuations are ∂_tδP_I=1/2(DδE_R+δDĒ]-δP_I/T_2, ∂_tδD= -T_2DĒ∂E_R-2ĒδP_I-δD/T_1, ∂_tδE_R=c/n[-∂_z δE_R+δP_I-(l_0-3γĒ^2)δE_R/2], ∂_tδP_R=-1/2DδE_I-δP_R/T_2, ∂_tδE_I=c/n[-∂_z δE_I-δP_R-(l_0-γĒ^2)δE_I/2].
The two sets of equations, (C)-(C) and (C)-(C), are decoupled, and translationally invariant. Thus their eigenfunctions are plane waves [126]. It holds δ PI(z,t)=δ PI(t)eikz, and similar for relations δ D and δ ER. The stability of the cw solution is determined by the eigenvalues of the matrix
| M= | ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ |
| ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ | . (C.1) |
If all eigenvalues have a negative real part, the cw solution is stable.
For l0=0 and γ=0, the eigenvalue with the greatest real part is λ0(K)=−ick/n. Putting λ(K)=λ0(K)+λ1(K) into the characteristic polynomial of M and equating the parts which are first order in l0, γ, and λ1(K), one arrives at
| λmax = |
| ||||||||||
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where pf = Dp/Dth and Ω=kc/n. Taking the real part of Eq. C one obtains Eq. 6.