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Appendix C
QCL Linear Stability Analysis

The standard Maxwell-Bloch equations with a SA added can be rewritten as [128]:

∂tE = -c -- n∂zE -c -- n iμP ------- ℏl0Dth - c --- 2n(l0 -γ|E|2)E,(C.1)
∂tP = iμ --- 2ℏDE -P --- T2,(C.2)
∂tD = Dp - D -------- T1 + iμ --- ℏ(E*P - c.c.).(C.3)

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:

∂tE = -c- n∂zE -c- niP -c-- 2n(l - ¯γ |E |2) 0E,(C.4)
∂tP = -i -- 2DE -P --- T2,(C.5)
∂tD = -pfl0 T1T2 -D-- T1 + i(E *P - c.c.) .(C.6)

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 δE,δP, and δD.

The steady state solution can be found by setting the left-hand sides of the Eqs. (C.4)-(C.6) to zero. The steady state solutions has the form E = E, P = P, and D = D are constants in time and space satisfying:

D = -l0- T2 - ¯2 ¯γE-- T2,(C.7)
P = i- 2(l - ¯γ ¯E2) 0E,(C.8)
pf + 1 = ( ) ¯γ ¯E2 1 - ---- l0( 2 ) 1 + E¯ T1T2.(C.9)

The resulting equations regarding the fluctuations are

∂tδPI = 1- 2(D¯δE + δD ¯E ] R-δPI- T2,(C.10)
∂tδD = -T2DE∂ER - 2EδPI -δD- T 1,(C.11)
∂tδER = c -- n[ ( ) δE ] - ∂zδER + δPI - l0 - 3γ ¯E2 ---R- 2,(C.12)
∂tδPR = -1- 2DδEI -δPR- T 2,(C.13)
∂tδEI = c -- n[ ( ) δE ] - ∂zδEI - δPR - l0 - γE¯2 ---I 2.(C.14)

The two sets of equations, (C.10)-(C.12) and (C.13)-(C.14), are decoupled, and translationally invariant. Thus their eigenfunctions are plane waves [126]. It holds δPI(z,t) = δPI(t)eikz, and similar for relations δD and δE R. The stability of the cw solution is determined by the eigenvalues of the matrix

( ( ) ) - T- 1 -1-- l0 - γ¯¯E2 1-¯E | 2 2T2 2 | || || || || | c c ( 1 3 ) | M = || -- -- - --l0 + --¯γ ¯E2 - ik 0 || . || n n 2 2 || | | ( ) - 2E¯ ¯γE¯3 - l0E ¯ - T-2 1
(C.15)


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 =
- iΩ -l0c- 2n--(ΩT1-+-i)ΩT2----2(pf --1)-- (ΩT1 + i)(ΩT2 + i) - (pf - 1)


+ 2 γℏ-(pf---1)- μ2T1T2(ΩT1--+-i)(3ΩT2--+-2i) --4(pf --1) (ΩT1 + i)(ΩT2 + i) - (pf - 1),
(C.16)
where pf = Dp∕Dth and Ω = kc∕n. Taking the real part of Eq. C.16 one obtains Eq. 6.6.

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