4.3 ELECTROOPTIC EFFECT AND POLARIZATION BIREFRINGENCE IN VCSELs

For VCSELs operating above threshold, a small amount of birefringence may be experienced by the polarized fields. In fact, birefringence is produced by an inevitable internal field (due to the applied voltage) arising from the electro-optic effect. This effect causes the in-plane gain anisotropy as well as the split of frequency between the two orthogonal polarizations in VCSELs. If a VCSEL is grown on a (001) substrate and an internal field is , pointed in the [001] direction, this orientation of the applied field will lead to a change in refractive index, which can be written as [31]

where the + and – signs apply to light polarized along the [110] and [110] directions and r₄₁ and R₁₂ are the linear and quadratic electrooptic effects, respectively. Hence, the corresponding difference in refractive index between the two orthogonal polarized modes is given by

where the subscripts h and v denote the polarized modes along the [110] and directions, respectively, and n(z) is the refractive index in zero electric field. The variable z indicates the position dependence of n and r₄₁ along the [001] direction so that (4.47) evaluates the local birefringence of the laser cavity.

In order to calculate the overall birefringence of the laser cavity, (4.47) has to be modified by the nonuniform distribution of optical field profile E. The optical field distribution inside the VCSEL can be obtained from [29]

where Δn is the change in refractive index. If Δn is interpreted as the birefringence (i.e., Δn =n_h– n_υ), the corresponding frequency splitting, Δω(= ω_υ – ω¾), between the two orthogonal polarizations can be estimated from (4.48). By substituting ω = ω_s + Δω and E = E_s + ΔE into (4.48), where the subscript ('s') stands for steady state, and using the first-order perturbation method, the approximated solution to Δω is given by [32]

where n_g is the group refractive index. Substituting (4.47) into (4.49) gives

The brackets denote the normalized spatial averaging over the longitudinal direction, and is defined as

As a result, the local refractive index changes induced by the internal electric field have to be weighted by the optical intensity E_s(z)² in order to evaluate their influence on the overall birefringence and cavity resonance frequency. E_s(z)² can be calculated using the transfer matrix method as discussed in Chapter 2 if the laser structure is known. The approximated value of E can also be obtained by a simple approach shown below.

It is noted that is dependent on the configuration of lasers such as material composition and doping levels as well as the external bias voltage. Figure 4.7 sketches a typical VCSEL structure with multilayered mirrors. An active layer, comprising three QWs (of material type 1), is centered in a lλ spacer layer (of material type 2), which is surrounded by multilayered mirrors composed of alternating λ/4 semiconductor layers (of material 2 and material 3). Bandgaps and refractive indices of materials 1, 2, and 3 are denoted by E_g1 < E_g2 < E_g3 and n₁ > n₂ > n₃, respectively. It is assumed that series resistance is uniformly distributed as indicated by the slope of Fermi levels shown in Figure 4.7.

There are basically three important contributions to. These contributions arise from (1) the built-in potential across the active layer, , (2) the series resistance in the device , and (3) the localized fields at various hetero-junctions of the multilayered mirrors . is due to the electrostatic potential Vactive across the active layer. If the laser is biased at the situation that the transparency is reached and population inversion is obtained, the electric field can be assumed uniformly along the active layer [i.e., V_active ~ (E_g2 – E_g1)/q] so that can be written as

Figure 4.7 Sketch of the position dependence of some important quantities in a typical VCSEL, comprising three quantum wells of material type 1 centered in a lλ cavity of material type 2, which is surrounded by multilayered mirrors composed of materials 2 and 3. The various curves show (a) the energy level of the conduction band, (b) the energy level of the upper valence band, and (c) the internal electric field. The dotted lines in (a) and (b) denote the electron and hole Fermi levels, respectively.

where the factor 2 in (4.52) is due to resonant FP mode as discussed in Section 2.3.1 and L_eff is the effective cavity length resulting from the spatial integral in the denominator of (4.52). L_eff can be expressed as L_eff ≈ L_spacer + 2L_pen, where L_pen is the penetration length of the optical field inside the multilayered mirror and can be calculated from (2.45).

Furthermore, is due to the series resistance of the VCSEL. If the series resistance is assumed to be uniformly distributed over the multilayered mirrors, can be expressed as

where V_pair is the voltage drops across a pair of dielectric layers (i.e., low and high dielectric pairs), f(≈ L_pen/(λ_R/2n)) is the effective number of pairs felt by the penetration optical intensity (i.e., λ_R/2n is the length of a dielectric pair). The negative sign is due to the fact that the electric field is pointing toward negative z (from p to n mirror).

The last term contribution to, namely, , rises from the localized fields at various heterojunctions of the mirrors. These internal fields are similar to those appearing inside a p-n homojunction except for a difference in electron affinity between the two semiconductor layers. This can be accounted for by introducing conduction band offset, η_CBO (i.e., the fraction of the bandgap difference), into the Fermi level of the composing materials. Hence, if the doping levels of the semiconductor multilayered mirrors are not too low, the electrostatic potential across each 2–3 heteroj unction, V₃₂, can be approximated by (E_g3 – E_g2)/q multiplied by η_CBO for the n-type mirror and 1 – η_CBO for the p-type mirror. As a result, can be deduced for (4.51), which is given by

where the multiplication factor 2 ƒ + 1 is due to the constructive interference of the positive polarity of the electric fields with the antinode of the optical standing wave but the negative polarity of the electric fields is eliminated by the node of the standing wave (see Fig. 4.8). Therefore, combining the three contributions gives

Assumed that material 1 is GaAs, material 2 is Al_0.18Ga_0.82As, and material 3 is AlAs. The corresponding bandgap energies and refractive indices (at room temperature and at operating wavelength λ_R, 850 nm) are E_g1 = 1.42 eV, E_g2 = 1.67 eV, E_g1 = 2.17 eV, and n₁ = 3.64, n₂ = 3.46, n₃ = 2.99, respectively. Furthermore, if L_space = 0.24 µm and the total number of dielectric pairs of the

Figure 4.8 The longitudinal distribution of optical field E_s(z) inside a typical VCSEL. Lspace is the spacer length between the two multilayered mirrors, and L_pen is the penetration depth of the optical field inside the multilayered mirrors.

n-type and p-type mirrors is (20 ± 25), it can be shown that ƒ = 3.4 and L_pen = 0.45 µm. If the device is biased at 2.5 V, it can also be shown that V_active ≈ 0.2 V, V₃₂ ≈ 2.17 V, and V_pair ≈ 0.024 V. Using these parameters, the split of frequency between h and v modes is given by

This addition clearly shows that the internal fields at the heterojunctions of multilayered mirrors are the domination factor for the determination of birefringence through the electrooptic effect. The positive value of (4.56) indicates light polarized along the [110] axis having a higher resonance frequency than light polarized along .

These calculations have been compared with the experimental measurement. It is found that the lasing mode along the [110] direction has a resonance frequency of about +10 GHz more than the nonlasing one along the direction for VCSELs from the 1 x 16 arrays but only +4 GHz more than that from the 1 x 8 arrays [33]. Hence, it has been proved that the theoretical prediction gives the right direction of detuning but the corresponding magnitude of detuning is overestimated. This implies that birefringence has significant influence on the split off frequency between the two orthogonal polarizations, but other intrinsic optical anisotropies in the laser cavity can also alternate the polarization properties of VCSELs and are further investigated in the following sections.