Threshold condition

The threshold condition states what a cavity mode needs in order to lase: after one round trip the optical field must return with the same amplitude and the same phase. The amplitude half says that the gain experienced by the mode equals all of its losses; the phase half selects the frequencies at which this can happen. For a 300 µm Fabry–Perot InP laser with cleaved facets the condition requires a modal gain of about 50 cm⁻¹, which with a quantum-well confinement factor of 0.05 means a material gain of about 1000 cm⁻¹. The broader idea of laser threshold, including the threshold current, the L–I curve and pump-power thresholds, has its own entry; this one covers the condition itself and how to evaluate it.

Gain and phase conditions

For a cavity of length $L$ with field reflectances $r_1$ and $r_2$ (power reflectances $R = |r|^2$), the round trip requires

$$r_1 r_2\, e^{(\Gamma g - \alpha_i)L}\, e^{-2j\beta L} = 1 .$$

Here $g$ is the material gain of the active region, $\Gamma$ the confinement factor, $\alpha_i$ the modal internal loss and $\beta = 2\pi n_\text{eff}/\lambda$ the propagation constant. The magnitude gives the gain condition,

$$\Gamma g_\text{th} = \alpha_i + \alpha_m,$$ $$\alpha_m = \frac{1}{2L}\ln\frac{1}{R_1 R_2},$$

with $\alpha_m$ the mirror loss spread over the cavity length. The phase gives $2\beta L = 2\pi m$ for integer $m$, which fixes the longitudinal modes. The mode closest to the gain peak reaches the gain condition first.

Worked example

Take $L$ = 300 µm, $R_1 = R_2$ = 0.3 (cleaved InP), $\alpha_i$ = 10 cm⁻¹ and $\Gamma$ = 0.05:

• mirror loss: $\alpha_m = \ln(1/0.09)/(0.06\ \text{cm})$ = 40.1 cm⁻¹
• threshold modal gain: $\Gamma g_\text{th}$ = 50.1 cm⁻¹
• threshold material gain: $g_\text{th}$ = 1003 cm⁻¹

The mirrors transmit 70% at each reflection, so the net modal gain must amplify the light 11.1 times per round trip. With $n_\text{eff}$ = 3.2 at 1550 nm the phase condition is met for $m$ near 1239, and with $n_g$ = 3.6 adjacent modes are 1.11 nm (139 GHz) apart. The same total loss gives a photon lifetime $\tau_p = 1/[v_g(\alpha_i + \alpha_m)]$ of 2.4 ps.

The condition shows directly how design choices move the threshold. A high-reflection coating on the back facet ($R_1$ = 0.9, $R_2$ = 0.3) lowers $\alpha_m$ to 21.8 cm⁻¹ and $g_\text{th}$ to 636 cm⁻¹. Doubling the length to 600 µm gives $\alpha_m$ = 20.1 cm⁻¹ and $g_\text{th}$ = 601 cm⁻¹; halving it to 150 µm gives 80.3 cm⁻¹ and 1805 cm⁻¹. Because quantum-well gain rises roughly logarithmically with carrier density, the threshold density grows exponentially with $g_\text{th}$, which is why the threshold current density rises steeply for short cavities.

Gain clamping

Above threshold, stimulated emission holds the modal gain at $\Gamma g_\text{th}$, and with it the carrier density, so extra pump goes into photons. The laser threshold entry covers this clamping and its limits, and the rate equations derive it.

Other cavity types

In a DFB laser the feedback is distributed along a grating, and the threshold gain depends on the coupling strength $\kappa L$, the facet reflectances and the facet phases; the same principle, round-trip gain equal to one, is solved for the coupled forward and backward waves. In a VCSEL the gain region is only tens of nanometers thick within a cavity about a wavelength long, so the condition is written per round trip with mirror reflectances above 99%. In a solid-state laser $\Gamma g$ becomes $\sigma \Delta N$ and the condition fixes the threshold inversion, worked for Nd:YAG in the laser threshold entry.

Pitfalls

Mixing field and power reflectances changes the mirror loss by a factor of two. Losses quoted in dB/cm must be divided by 4.34 before use in the exponential form: 10 cm⁻¹ is 43.4 dB/cm. With equal facets the mirror term is often written $(1/L)\ln(1/R)$, which is the same quantity. Measured gain from a finished device is the modal gain $\Gamma g$; dividing by an assumed $\Gamma$ to get material gain carries the uncertainty of that assumption.

Common questions

What is the threshold condition for a laser?

The round-trip gain must equal the round-trip loss and the round-trip phase must be a multiple of 2π. In loss-coefficient form, $\Gamma g_\text{th} = \alpha_i + (1/2L)\ln(1/R_1R_2)$.

What threshold gain does a semiconductor laser need?

For a 300 µm cleaved InP laser with 10 cm⁻¹ internal loss, about 50 cm⁻¹ of modal gain, or about 1000 cm⁻¹ of material gain with a confinement factor of 0.05.

Why do short cavities need more gain to reach threshold?

The mirror loss $(1/2L)\ln(1/R_1R_2)$ spreads the same facet transmission over less length: a cleaved InP laser needs 50 cm⁻¹ of modal gain at 300 µm but 90 cm⁻¹ at 150 µm.

How do dB/cm losses convert to the threshold equation?

Divide by 4.34: the equation uses natural-log coefficients in cm⁻¹, so 10 cm⁻¹ of internal loss is 43.4 dB/cm.

References: L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012). A. E. Siegman, Lasers (University Science Books, 1986). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).