Optical gain

Optical gain is the exponential growth coefficient describing how a light wave amplifies as it propagates through an inverted gain medium. Gain is the optical analog of an electronic transistor's transconductance: it converts pump energy into amplification of an optical signal.

Definition. As light propagates through a gain medium, its intensity grows as:

$$I(z)=I(0)e^{(g-\alpha )z},$$

where $g$ is the gain coefficient (units of inverse length: cm⁻¹ or m⁻¹), $\alpha$ is the loss coefficient, and $z$ is propagation distance. Net amplification occurs when $g>\alpha$.

Material gain vs modal gain. Two related quantities:

• Material gain $g$: gain per unit length of the active material if a plane wave occupied the entire active volume
• Modal gain $\mathrm{\Gamma }g$: gain per unit length experienced by the actual guided mode; $\mathrm{\Gamma }$ is the confinement factor

For typical edge-emitting semiconductor lasers, $\mathrm{\Gamma }$ is 1 – 10%, so modal gain is 10 – 100× smaller than material gain.

Standard gain values.

The vast range (from ${10}^{-3}$ cm⁻¹ in HeNe to $5\times {10}^3$ cm⁻¹ in semiconductors) reflects the different oscillator strengths of the underlying transitions and the different active-material densities.

Gain in dB. Optical gain is often expressed in dB per unit length:

$$g_{\text{dB}}=4.343\cdot g{\text{(if }g\text{ in cm}}^{-1}\text{)}\mathrm{.}$$

Total gain in dB after propagating through length $L$:

$$G=10{\log }_{10}\text{\hspace{0.17em}⁣}\left(\frac{I(L)}{I(0)}\right)=4.343\cdot g\cdot L\mathrm{.}$$

So 1 mm of semiconductor waveguide with a net modal gain of 50 cm⁻¹ gives 21.7 dB of small-signal gain.

Gain coefficient vs current. For semiconductor gain media, the material gain depends on the injected carrier density $N$. Two standard models:

• Logarithmic gain:

$$g(N)=g_0\ln \text{\hspace{0.17em}⁣}\left(\frac{N}{N_{\text{tr}}}\right),$$

with $N_{\text{tr}}$ the transparency carrier density (see transparency current) and $g_0\approx 1000$ – 3000 cm⁻¹.

• Linear gain (low-density approximation):

$$g(N)=a(N-N_{\text{tr}}),$$

with differential gain $a=dg\mathrm{/}dN\approx 5\times {10}^{-16}$ cm² for InGaAsP MQW.

These two models agree at low carrier density above transparency.

Gain peak wavelength. The peak gain occurs at the wavelength of the dominant electronic transition. For semiconductors, this is set by:

• Bandgap: lowest-energy transition
• Quantum confinement: blue-shifts the peak relative to bulk
• Carrier density: increasing density blue-shifts the gain peak (band-filling)
• Temperature: increasing T red-shifts (bandgap narrowing)

A typical edge-emitting InGaAsP laser at 1550 nm has gain peak that shifts ~0.4 nm/K with temperature, several times faster than the ~0.1 nm/K index-driven drift of a DFB or DBR emission wavelength, so the grating-to-gain-peak detuning changes with temperature.

Gain saturation. As intensity increases, the inversion is depleted, reducing gain:

$$g(I)=\frac{g_0}{1+I\mathrm{/}{I}_{\text{sat}}},$$

where $I_{\text{sat}}$ is the saturation intensity. In a semiconductor, $I_{\text{sat}}=h\nu \mathrm{/}(a\tau )$, with $a$ the differential gain and $\tau$ the carrier lifetime, is of order 1 MW/cm² (0.85 MW/cm² for $a=5\times {10}^{-16}$ cm² and $\tau =0.3$ ns at 1550 nm). The saturation power is $I_{\text{sat}}$ times the active-region cross-section divided by the confinement factor, about 1 μm² in a typical amplifier waveguide (0.2 μm² of active region at $\mathrm{\Gamma }=0.2$), which gives about 8.5 mW. An SOA therefore saturates at output powers of order 10 mW, designs for higher output lower $\mathrm{\Gamma }$ to raise it, and inside a laser above threshold saturation holds the gain at its threshold value. For fiber amplifiers, $I_{\text{sat}}$ is much smaller (0.5 – 5 kW/cm²), so they run deep in saturation, and their output is set by the pump power they convert.

Net gain vs threshold gain. A laser oscillates when round-trip gain equals round-trip loss:

$$g_{\text{th}}=\alpha +\frac{1}{2L}\ln \text{\hspace{0.17em}⁣}\left(\frac{1}{R_1{R}_2}\right),$$

where $\alpha$ is internal loss, $L$ is cavity length, and $R_1,R_2$ are mirror reflectivities. At threshold, the net gain (material gain minus losses) equals the mirror loss; below threshold, the system is amplification-only; above, the system oscillates.

For typical Fabry-Perot diode lasers:

• Cavity length 300 μm, $R_1=R_2=0.32$ (cleaved facets)
• Mirror loss: $\sim$ 38 cm⁻¹
• Internal loss: 5 – 15 cm⁻¹
• Threshold modal gain: ~50 cm⁻¹

Achievable with carrier densities $\sim 2\times {10}^{18}$ cm⁻³ in MQW active regions.

Cross-section and gain. The material gain is the cross-section times the inversion density:

$$g=(N_2-N_1)\cdot {\sigma }_{\text{emission}}(\lambda ),$$

where $\sigma$ is the emission cross-section (units of area, typically ${10}^{-21}$ – ${10}^{-16}$ cm²) and $(N_2-N_1)$ is the inversion density. Materials with large $\sigma$ (Ti:sapphire, $\sigma \sim 3\times {10}^{-19}$ cm²) achieve high gain at modest inversion; materials with small $\sigma$, such as rare-earth ions in glass, with $\sigma$ of order ${10}^{-21}$ to ${10}^{-20}$ cm², require higher inversion or longer interaction length.

Polarization-dependent gain. In compressive- or tensile-strained MQW active regions, the gain is different for TE and TM polarizations (the heavy-hole and light-hole bands have different oscillator strengths for the two polarizations). Most edge-emitting lasers are TE-dominant; strained MQWs can be engineered for any desired TE/TM ratio. VCSELs may be unpolarized due to symmetric circular aperture.

Asymmetric gain shape. Real semiconductor gain spectra are asymmetric (broader on the high-energy side) due to the conduction-band density of states. Asymmetric gain shape affects:

• Mode hop direction in tunable lasers
• Multi-mode lasing patterns
• ASE spectral shape

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 14 (laser amplifiers); Coldren, Corzine & Mašanović, Diode Lasers and PICs (2nd ed., 2012), Ch. 4 (the comprehensive treatment for semiconductor gain); Agrawal, Fiber-Optic Communication Systems (4th ed., 2010), Ch. 7 for fiber-amplifier gain.