Mirror loss (α_m)

Mirror loss is the fraction of the circulating light that escapes through the cavity mirrors on each round trip, expressed as if it were spread evenly along the cavity. Writing it per unit length lets it be added directly to the internal loss $\alpha_i$ and compared with the modal gain. For a cavity of length $L$ with power reflectances $R_1$ and $R_2$,

$$\alpha_m = \frac{1}{2L}\ln\frac{1}{R_1 R_2}.$$

In semiconductor lasers it ranges from about 10 to 60 cm⁻¹: a 300 µm chip with two cleaved facets of $R = 0.3$ has $\alpha_m = 40$ cm⁻¹, and a 1000 µm chip has 12 cm⁻¹. In solid-state and gas lasers with centimeter to meter cavities and an output coupler of a few percent, the same formula gives values below 0.1 cm⁻¹, and the loss is usually quoted per round trip instead.

Where the formula comes from

On one round trip the power is multiplied by $R_1 R_2 \, e^{(g - \alpha_i)2L}$ for a net modal gain $g$. At threshold this product equals 1, which gives $g_\text{th} = \alpha_i + \alpha_m$ with $\alpha_m$ as above; the condition is developed under laser threshold. The logarithm matters: with $R = 0.3$ on both facets only 9 % of the power survives a round trip at the mirrors, far from the small-loss regime in which $\ln(1/R) \approx 1 - R$.

Worked values

With a group index of 3.6, $\alpha_i = 10$ cm⁻¹ and an internal quantum efficiency $\eta_i = 0.8$, the same values as the differential-quantum-efficiency entry, cleaved facets ($R_1 = R_2 = 0.3$) give:

Here $\eta_d$ is the differential quantum efficiency, summed over both facets, and $\tau_p$ the photon lifetime. A high-reflection rear coating ($R_1 = 0.9$) with an anti-reflection front ($R_2 = 0.05$) gives $\alpha_m = 51.7$ cm⁻¹ at 300 µm and 15.5 cm⁻¹ at 1000 µm, and sends about 98 % of the output through the front facet.

Efficiency and photon lifetime

Above threshold the generated photons are shared between the two loss channels in proportion to their rates, so the fraction that leaves as output is the optical efficiency

$$\eta_\text{opt} = \frac{\alpha_m}{\alpha_m + \alpha_i}, \qquad \eta_d = \eta_i\,\eta_\text{opt}.$$

For the 300 µm cleaved chip, $\eta_\text{opt} = 0.80$. Mirror loss and internal loss together fix the photon lifetime, $1/\tau_p = v_g(\alpha_i + \alpha_m)$, and through it the cavity linewidth and part of the modulation response. A higher $\alpha_m$ shortens $\tau_p$ and raises the efficiency, at the price of a higher threshold gain, which raises the threshold current and pushes the operating carrier density up the gain curve where the differential gain is lower and the laser is more temperature sensitive. Cavity length and facet coatings are chosen to balance these: short cavities with high $\alpha_m$ suit fast directly modulated lasers, and long cavities with low $\alpha_m$ suit high power and narrow linewidth.

Distributed feedback lasers

A DFB laser has no discrete mirrors; feedback comes from a grating along the whole cavity, described by the coupling coefficient $\kappa$. The threshold gain of each mode then depends on the product $\kappa L$ and on the phase and reflectance of the facets, and plays the role of $\alpha_m$ in the threshold and efficiency relations. Practical DFBs use $\kappa L$ of order 1–2: lower values give weak feedback and a high effective mirror loss, while higher values trap the light, lower the efficiency and cause spatial hole burning. The same idea applies to a VCSEL, whose mirror reflectances above 0.99 compensate for a cavity only about a micrometer long.

Pitfalls

The reflectance in the formula is the modal reflectance of the guided mode, which differs from the plane-wave Fresnel value by a few percent for a cleaved facet and by more for a coated one. Coating reflectances quoted on a datasheet are often design values at one wavelength. Both facets count in $\eta_d$ unless the facet split is calculated, which is easy to forget when only the front-facet power is measured.

Common questions

Is mirror loss really a loss?

It is a loss to the cavity, since the light no longer circulates, but it is the laser's output. Of the photons in the lasing mode, only the share taken by internal loss is wasted.

How do you reduce the threshold of a diode laser through the mirror loss?

Make the cavity longer or raise the facet reflectance with coatings. Both lower $\alpha_m$ and the threshold gain, while also lowering the differential efficiency.

What is a typical mirror loss for a cleaved laser diode?

About 20–50 cm⁻¹ for cavities of 250–600 µm with uncoated facets; 40 cm⁻¹ at 300 µm.

References: L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012); G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993); A. E. Siegman, Lasers (University Science Books, 1986).