Linewidth enhancement factor (α)

The linewidth enhancement factor $\alpha$, also called the Henry factor or linewidth broadening factor, measures how strongly the refractive index of a semiconductor gain medium changes when its gain changes. Both depend on the carrier density $N$, and $\alpha$ is the ratio of the two changes:

$$\alpha =-\frac{4\pi }{\lambda }\frac{dn\mathrm{/}dN}{dg\mathrm{/}dN},$$

where $n$ is the refractive index, $g$ the material gain and $\lambda$ the wavelength; the minus sign makes $\alpha$ positive, because the index falls as the carrier density and the gain rise. Any change in the gain therefore also shifts the optical frequency. C. H. Henry showed in 1982 that this raises the linewidth of a semiconductor laser by a factor $1+{\alpha }^2$ above the Schawlow-Townes value, seventeen times for $\alpha =4$. The same coupling produces frequency chirp under direct modulation, makes semiconductor lasers sensitive to optical feedback, and causes filamentation in broad-area lasers. In multiple-quantum-well telecom lasers $\alpha$ is typically 2 to 5; gas and solid-state lasers, whose gain lines are symmetric and which operate near their centers, have $\alpha$ close to zero.

Where the coupling comes from

Injecting carriers changes the gain spectrum of a semiconductor, and by the Kramers-Kronig relations a change of gain at some frequencies is accompanied by a change of refractive index at others. The gain spectrum of a semiconductor is strongly asymmetric, so at the lasing wavelength the index change does not cancel; the free carriers add a further index reduction through the plasma dispersion effect. The factor rises toward longer wavelengths across the gain spectrum and falls where the differential gain is large. This is one reason DFB lasers for direct modulation are often designed with the grating wavelength on the short-wavelength side of the gain peak, and why strained quantum wells, with their higher differential gain, have lower $\alpha$ than bulk active regions. Quantum-dot and quantum-cascade active regions, with more symmetric gain spectra, can have smaller values still, although measured values vary widely with the operating conditions.

Linewidth

Each spontaneous emission event adds a small random change to the phase of the laser field, which alone gives the Schawlow-Townes linewidth. It also changes the intensity, and as the laser relaxes back to its steady state the carrier density and so the refractive index change as well, adding a further phase shift $\alpha$ times as large. The two contributions add in power:

$$\mathrm{\Delta }\nu =(1+{\alpha }^2)\mathrm{\Delta }{\nu }_{\text{ST}}\mathrm{.}$$

A laser with a Schawlow-Townes linewidth of 100 kHz and $\alpha =4$ has a linewidth of 1.7 MHz. Because $\alpha$ is hard to reduce much, narrow-linewidth semiconductor lasers lower the Schawlow-Townes term instead, with long or external cavities and high-Q feedback such as self-injection locking.

Chirp under modulation

Modulating the current modulates the carrier density, so the frequency follows the power. In the rate-equation model,

$$\mathrm{\Delta }\nu (t)=\frac{\alpha }{4\pi }[\frac{d}{dt}\ln P(t)+\kappa P(t)],$$

with a transient term during changes of power and an adiabatic term proportional to the power, whose coefficient $\kappa$ is proportional to the gain compression. Combined with fiber dispersion, this chirp broadens the pulses of directly modulated lasers and limited 1550 nm direct modulation to short links; the frequency chirp entry works through the consequences. The chirp parameter quoted for an external modulator is a related quantity that describes the modulator rather than a gain medium, and it can be zero or negative.

Feedback, injection and filamentation

Light returning into the cavity changes the carrier density, and through $\alpha$ the optical frequency, which in turn changes the phase of the returning light. This loop is why the feedback thresholds of the optical feedback regimes depend strongly on $\alpha$, and why moderate feedback can drive a laser into coherence collapse. Under external light injection, $\alpha$ makes the injection-locking range asymmetric, extending further on the long-wavelength side. In broad-area lasers and high-power amplifiers, bright regions deplete the carriers, which raises the local refractive index, focuses the light further and breaks the beam into filaments.

Measuring α

• Below threshold (Hakki-Paoli). As the current rises, the Fabry-Perot modes of the chip shift to shorter wavelengths while the modal gain rises; the ratio of the wavelength shift to the gain change, scaled by the mode spacing and cavity length, gives $\alpha$. Pulsed current keeps heating from adding its own red shift.
• FM to AM ratio. Under small-signal current modulation well above the adiabatic range, the ratio of frequency modulation to intensity modulation, $2\beta \mathrm{/}m$ in terms of the FM index $\beta$ and the AM index $m$, approaches $\alpha$.
• Fiber transfer function. After a length of dispersive fiber, the small-signal response of a directly modulated laser shows dips at frequencies set by the dispersion and by $\alpha$, which a network analyzer measures directly.
• Injection locking. The asymmetry of the locking range, measured by injecting a tunable master laser, gives $\alpha$ above threshold.

Because $\alpha$ depends on wavelength, carrier density and power, different methods applied to the same laser can give noticeably different values, and a quoted value is meaningful only with its wavelength and operating point.

Common questions

What is the Henry factor?

Another name for the linewidth enhancement factor, after C. H. Henry, who derived the $1+{\alpha }^2$ broadening of semiconductor laser linewidths in 1982.

Why do semiconductor lasers have broader linewidths than other lasers?

For two reasons: their short, lossy cavities give a large Schawlow-Townes linewidth to begin with, and the linewidth enhancement factor multiplies it by $1+{\alpha }^2$, typically between 5 and 26 for $\alpha$ of 2 to 5.

How does the linewidth enhancement factor cause chirp?

A change in drive current changes the carrier density, which changes the gain and, through $\alpha$, the refractive index and the lasing frequency. The frequency deviation is proportional to $\alpha$ and to the rate of change of the logarithm of the power, plus a term proportional to the power.

References: C. H. Henry, "Theory of the linewidth of semiconductor lasers," IEEE Journal of Quantum Electronics 18, 259 (1982). M. Osinski and J. Buus, "Linewidth broadening factor in semiconductor lasers: an overview," IEEE Journal of Quantum Electronics 23, 9 (1987). C. Harder, K. Vahala and A. Yariv, "Measurement of the linewidth enhancement factor α of semiconductor lasers," Applied Physics Letters 42, 328 (1983). T. L. Koch and J. E. Bowers, "Nature of wavelength chirping in directly modulated semiconductor lasers," Electronics Letters 20, 1038 (1984). F. Devaux, Y. Sorel and J. F. Kerdiles, "Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter," Journal of Lightwave Technology 11, 1937 (1993).