VCSEL Industry. Babu Dayal Padullaparthi

       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)

      1.1.5.2 Resonant Wavelength

Now, if λ denotes the wavelength and n the equivalent refractive index, from Eq. (1.2a) we have:

      (1.6)

      When the phase shift is 0 and the reflector is at the fixed end, for example, the standing wave is as shown in Figure 1.8b. The total length L is an integer q times the half wavelength λ /(2n) in the medium:

      (1.7)

      Now, in a laser cavity with a resonator length L longer than the wavelength, waves of many wavelengths with slightly different lengths can resonate. These modes are called longitudinal modes. On the other hand, the modes in the perpendicular direction are called the transverse modes.

      Considering a normal semiconductor laser, if λ is 1.3 μm, n = 3.5, and L = 3 μm, then q = 16.

      Therefore, even if q differs by 1, the resonance wavelength changes only slightly as Δλ. With |Δ λ | ≪̸ λ in mind, if λ → λ ₀ + Δλ , q → q + 1, then we obtain,

      (1.8)

      This |∆ λ | is called free spectral range (FSR) and is inversely proportional to cavity length, L.

      Here, n_eff is the effective index considering the dispersion of the medium and is given by the following expression:

      (1.9)

      Since ∂n/∂ λ < 0 in ordinary semiconductors, n_eff is usually larger than n. In the above example, n_eff = 4.0 and |∆ λ | = 70 nm.

      1.1.5.3 Cavity Formation

To achieve laser oscillation, a resonator that provides optical feedback to the gain medium is required. The laser resonator is formed by a pair of mirrors; a so‐called Fabry‐Pérot (FP) resonator is shown in Figure 1.8(a). In an edge‐emitting laser, the gain width is w, the cavity length is equal to L, and the mirror is usually made by simply cleaving the semiconductor crystal. In this case the refractive index of about 3.5 is higher than outside air, and the resonator edges look like open termination. (Here φ = 2 π . φ is defined in Eq. (1.1)).

      In Table 1.1 we have touched on DFB and DBR structures for single‐mode operation of edge‐emitting lasers [20–23]. In both cases, we utilize a pair of Bragg mirrors having an electric field reflectivity expressed by that sandwich some space