Читать книгу VCSEL Industry - Babu Dayal Padullaparthi - Страница 25

 L: cavity length

 d: thickness of active layer

  φ 1, φ 2: phase shift of each reflection

 r1, r2: electric field reflectance coefficients of the mirrors at both ends

 R1, R2 ,: power reflection coefficients of the mirrors at both ends

 g: gain coefficient

  α : loss coefficient

 β: propagation constant (=w/c = 2 πf /c)

 ω: angular frequency

Consider that the light wave in the resonator travels in the z direction from z = 0 and is reflected by the reflector r₂ at z = L; then it goes backward by the length of L and returns to the starting point z = 0. If the electric field is sustainable, we should have:

(1.1)

By comparing the imaginary and real parts, we have:

(1.2a)

(1.2b)

For the threshold gain g_th required for oscillation, use ln, the natural logarithm; from Eq. (1.2b),

(1.3)

The first term is absorption by the medium, and in GaAs, absorption by the free carrier has a magnitude of about 10 cm⁻¹. In the second term, the reflectance of a reflector made by cleaving the surface of a semiconductor is

(1.4)

Therefore, in the case of a GaAs edge‐emitting laser (n = 3.5) with L = d = 300 μm, it is about 39 cm⁻¹. To oscillate, a threshold gain of 10 + 39 = 49 cm⁻¹ or more is required.

The electric field E_out of the output light, with E₀: field at the end of cavity, is given by:

(1.5)