Gain and Loss in Waveguides

Gain and Loss

When light propagates in a gain waveguide, its intensity changes with distance as:

$$Il = I_0 \exp(Gl)$$

Where $G$ is optical gain (different for different mode). And optical gain can be expressed as:

$$G = \Gamma g - \alpha$$

Where $\Gamma$ is optical confinement factor, $g$ is gain of material, $\alpha$ is loss of light. And expression of loss is:

$$\alpha = \Gamma \alpha_{fr} + \left(1-\Gamma \right) \alpha_c + \alpha_s +\alpha_{cp}$$

Where $\alpha_{fr}$ is the free carrier absorption coefficient in the gain medium, $\alpha_c$ is the average absorption coefficient in the waveguide layer outside the gain medium, $\alpha_s$ is the scattering loss at the hetero-junction interface, and $\alpha_{cp}$ is the coupling loss outside the optical waveguide.
We can calculate material gain:

$$g = g_0\ln(J/J_0)$$

Where $J$ is current density.

However, in P-I-N diode, not all injected electrons or photons participant in luminescence. So $\eta_{inject}$ is introduced to describe the efficiency of injection. $\eta_{inject}$ of electrons is normally reduced by junction leakage and $\eta_{inject}$ of photons is normally reduced by light transmission:

$$\eta_{inject} = \frac{J-J_{leak}}{J} = \frac{P_{s}\exp(-\Sigma\alpha_i d_i)(1-\exp(-\alpha_m d_m))}{P_s}$$

Where $J_{leak}$ is leakage of current density, $\alpha_i$ is optical absorption coefficient for layers over gain medium and $d_i$ is thickness of them, $\alpha_m$ and $d_m$ is for gain medium. Thus, effective material gain is:

$$g_{eff} = \eta_{inject}g$$

How To Measure Gain

The spectrum of optical gain $G$ in waveguides can be obtained by measuring the spontaneous emission spectrum modulated by Fabry-perot cavity.

$$G(\lambda) = (1/L) \ln(1/R) + (1/L)\ln\frac{r(\lambda)-1}{r(\lambda)+1}$$

Where $r(\lambda)$ is the ratio of coherent enhancement and weak light intensity at wavelength $\lambda$ in the spontaneous emission spectrum, $L$ is the cavity length, and $R$ is the reflectivity of the cavity end interface.

If optical gain $G$ is measured, we can calculate or simulate $\alpha$ and further calculate the spectrum of material gain $g$.

Reference

1. GaAs/AIGaAs梯度折射率分别限制单量子阱结构中的光增益谱分析