Hakki-Paoli method

The Hakki-Paoli method measures the gain of a semiconductor laser chip from the spectrum of its own emission. Below threshold, the amplified spontaneous emission leaving a cleaved Fabry-Perot laser is modulated by the cavity resonances into a comb of fringes. The fringe contrast depends on how much light survives one round trip, which in turn depends on the net gain, so measuring the ratio $r$ of each fringe maximum to its neighboring minimum gives the net modal gain $\Gamma g - \alpha_i$ at that wavelength. Repeating at several currents traces out the gain spectrum as the chip approaches threshold. For a 400 µm chip with uncoated facets, fringe ratios run from about 2 at low current to 30 or more just below threshold, corresponding to net gains from about −16 to +19 cm⁻¹. The method was introduced by B. W. Hakki and T. L. Paoli in 1975 for GaAs double-heterostructure lasers.

Formula

A Fabry-Perot cavity with single-pass power gain $G = e^{(\Gamma g - \alpha_i)L}$ and facet reflectances $R_1$ and $R_2$ transmits with an Airy response whose peak-to-valley ratio is

$$r = \left(\frac{1 + G\sqrt{R_1R_2}}{1 - G\sqrt{R_1R_2}}\right)^2.$$

Solving for $G$ and taking the logarithm gives the net modal gain:

$$\Gamma g - \alpha_i = \frac{1}{L}\ln\frac{\sqrt{r}-1}{\sqrt{r}+1}$$ $$\qquad + \frac{1}{2L}\ln\frac{1}{R_1R_2}.$$

The second term is the mirror loss $\alpha_m$. As $r$ grows without limit the first term goes to zero and the net gain approaches $\alpha_m$, which is the threshold condition.

Worked example

Take $L = 400$ µm and $R_1 = R_2 = 0.32$, within the 0.27–0.32 range of cleaved III-V facets. The mirror loss is $\ln(1/0.32)/0.04\text{ cm} = 28.5$ cm⁻¹. Measured fringe ratios translate as follows:

On the long-wavelength side of the gain spectrum, where photons are below the band edge and $\Gamma g \approx 0$, the net gain flattens to $-\alpha_i$. For this chip an internal loss of 10 cm⁻¹ would appear as a fringe ratio of about 2.4 that no longer changes with current.

Measurement

The chip is driven below threshold, usually on a temperature-controlled mount, and its output is collected into a high-resolution optical spectrum analyzer or a grating spectrometer. A polarizer selects the TE emission, since the TE and TM gains differ, and the device should support a single lateral mode, otherwise fringes of several transverse modes overlap. The fringe spacing is the free spectral range, $\lambda^2/(2n_gL)$, which is 0.83 nm for the 400 µm chip at 1550 nm with a group index of 3.6. Each pair of maximum and adjacent minimum gives one gain value at the fringe wavelength; plotting these for several currents gives a family of gain spectra from which peak gain versus current, transparency current and differential gain are read. The shift of the fringe positions with current gives the linewidth enhancement factor from the same data.

Pitfalls

Spectral resolution is the main error source. Convolving the fringe pattern with the instrument response fills in the minima, lowers $r$ and makes the gain appear smaller. For the example chip at $r = 30$, a resolution of 0.06 nm lowers the measured ratio to about 25 and the gain by 0.9 cm⁻¹; at 0.1 nm the ratio falls to about 21 and the gain error reaches 2 cm⁻¹. The error grows near threshold, where the minima are deepest and narrowest. A resolution of about one fortieth of the mode spacing keeps the error below 0.2 cm⁻¹ in this case.

Other errors come from the facet reflectance, which enters directly through $\alpha_m$ and is only approximately the Fresnel value for a guided mode; from heating under CW drive, which shifts the spectrum during the scan; from back-reflection into the chip from the collection optics; and from spontaneous emission that is not coupled into the lasing mode and adds an unmodulated background, again lowering $r$. Coated facets with low reflectance reduce the contrast so far that the method loses sensitivity.

Common questions

What does the Hakki-Paoli method measure?

It measures net modal gain, $\Gamma g - \alpha_i$, as a function of wavelength and current. Separating material gain requires a calculated confinement factor, and separating internal loss requires reading the long-wavelength limit.

What are the alternatives?

The segmented-contact method measures gain from the emission of a multi-section device and works with single-pass amplified spontaneous emission, so it needs no fringes. Methods based on the mode sum or the minimum of each fringe are also used, and cavity-length series of threshold current give the gain at threshold only.

Why must the laser be below threshold?

Above threshold the gain is clamped at the threshold value and the lasing mode dominates the spectrum, so the fringes no longer contain information about the gain at other wavelengths.

References: B. W. Hakki and T. L. Paoli, "Gain spectra in GaAs double-heterostructure injection lasers," J. Appl. Phys. 46, 1299 (1975); 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).