Slope Efficiency and Differential Quantum Efficiency from LIV Measurements

Scope

This article describes the procedure for extracting external slope efficiency ${\eta }_s$ and external differential quantum efficiency ${\eta }_d$ of a semiconductor laser diode from light–current–voltage (LIV) measurements. Coverage is restricted to single-mode and multi-mode edge-emitting devices in continuous-wave or low-duty-cycle pulsed operation. Internal quantum efficiency separation by inverse-length method is treated separately.

Definitions

Slope efficiency ${\eta }_s$ is defined as the local derivative of optical output power $P_{\text{out}}$ with respect to drive current $I$ in the lasing region of the LIV curve:

$${\eta }_s={\frac{d{P}_{\text{out}}}{dI}\mid }_{I>I_{\text{th}}},$$

with units of W/A or mW/mA. It is determined by linear regression on the LIV curve above threshold.

External differential quantum efficiency ${\eta }_d$ is the dimensionless ratio of photons emitted from the device facet per electron injected above threshold:

$${\eta }_d=\frac{(P_{\text{out}}-P_{\text{out}}(I_{\text{th}}))\mathrm{/}h\nu }{(I-I_{\text{th}})\mathrm{/}q}=\frac{q}{h\nu }{\eta }_s=\frac{q\lambda }{hc}{\eta }_s,$$

where $q$ is the electron charge, $h\nu$ is the photon energy, $\lambda$ is the emission wavelength, $h$ is Planck's constant, and $c$ is the speed of light in vacuum.

For practical conversion at common telecom and laser wavelengths, the factor $q\lambda \mathrm{/}(hc)=\lambda \mathrm{/}(1.2398\mu \text{m}\cdot \text{V})$ gives:

For a single-junction laser, ${\eta }_d$ summed over both facets is bounded by the internal efficiency ${\eta }_i$ and therefore by unity; a reported value above unity indicates a collection or calibration error, with multi-junction (cascade) lasers the exception.

Distinction from other efficiencies

Slope efficiency must not be confused with:

• Wall-plug efficiency ${\eta }_{\text{WPE}}=P_{\text{out}}\mathrm{/}(IV)$, which is the total electrical-to-optical conversion at a specific operating point. ${\eta }_{\text{WPE}}$ depends on both the slope and the absolute operating current, voltage, and threshold; it is a different parameter and cannot be derived from ${\eta }_s$ alone.
• Internal quantum efficiency ${\eta }_i$, the fraction of injected carriers that produce stimulated photons inside the cavity. ${\eta }_i$ is extracted from inverse-length measurements ($1\mathrm{/}{\eta }_d$ vs. $L$) over a set of devices with varying cavity lengths and is not obtainable from a single-device LIV.
• Power conversion efficiency at threshold or below, which is not defined in the strict sense; slope efficiency describes only the lasing region.

Required measurements

The measurement configuration is the same as for any LIV characterization:

Two geometric configurations are common and produce different ${\eta }_s$ values that must not be compared without correction.

Single-facet butt collection. A photodiode or fiber is positioned close to one facet of the laser. Captures only one direction of emission and is sensitive to far-field divergence. For Fabry–Pérot devices with symmetric cleaved facets, the front-facet measurement captures approximately half of the total emitted power. For HR/AR-coated devices, the front-facet measurement captures the asymmetric majority of the emitted power.

Integrating sphere collection. The laser is placed at the entrance of an integrating sphere. Captures all emitted power independent of beam divergence. Required for devices with elliptical or non-Gaussian emission, for accurate ${\eta }_d$ extraction, and for any device with HR/AR coatings where front-facet collection is geometry-dependent.

The collection geometry must be reported alongside the ${\eta }_s$ value. Direct comparison of single-facet ${\eta }_s$ across devices is valid only when the facet geometry is identical.

Extraction procedure

1. Acquire LIV with adequate dynamic range

The LIV sweep must include current values extending well above $I_{\text{th}}$ (typically $I\ge 2I_{\text{th}}$ to $3I_{\text{th}}$, and ideally up to the rated operating current) so that the lasing region is well-sampled. Current step size in the lasing region should be small enough to give at least 30 data points in the linear fitting range; for a typical telecom DFB with $I_{\text{th}}=8$ mA and operation to 80 mA, a step of 0.5–1 mA is sufficient.

For high-power devices, pulsed measurement at $\le 0.1\mathrm{\%}$ duty cycle largely removes the self-heating curvature in the LIV that would otherwise depress the apparent slope.

2. Identify the lasing region

The lasing region is the range where $d^{2}P\mathrm{/}d{I}^2\approx 0$. Three behaviors define its endpoints:

Lower bound. Immediately above threshold the LIV curvature is nonzero due to the gradual transition from spontaneous to stimulated emission. The lower bound of the linear region is conservatively taken at $I\ge 1.2I_{\text{th}}$.

Upper bound. At high currents, three mechanisms reduce slope:

• Thermal rollover. Self-heating reduces gain. Visible as smooth concave-down curvature.
• Carrier leakage. Carriers escape over the heterojunction barriers. Material- and structure-dependent.
• Spatial hole burning and kinks. Mode redistribution produces step changes in slope; the origins and their signatures are covered in Why L-I curves kink.

The upper bound is taken at the current where the LIV begins to deviate from linear by more than a stated tolerance (commonly 1% of expected linear output, or where the local slope $dP\mathrm{/}dI$ departs from the fitted slope by more than 1%).

3. Fit

Perform a least-squares linear fit on $(I,P_{\text{out}})$ over the identified lasing region:

$$P_{\text{out}}(I)={\eta }_s(I-I_{\text{th}}^{\ast }),$$

where $I_{\text{th}}^{\ast }$ is the linear-extrapolated threshold from this fit (not necessarily the same as the threshold extracted from the kink or second-derivative method). The slope of the fit is ${\eta }_s$.

Report:

• ${\eta }_s$ in W/A or mW/mA
• The fitting range $[I_{\text{min}},I_{\text{max}}]$
• The number of data points $N$
• The fit residual standard deviation ${\sigma }_r$
• $R^2$ of the fit

A high-quality LIV in the lasing region should yield $R^2>0.999$ and residuals randomly distributed with no trend.

4. Convert to differential quantum efficiency

For a device with center wavelength $\lambda$:

$${\eta }_d={\eta }_s\cdot \frac{q\lambda }{hc}\mathrm{.}$$

If the emission spectrum changes significantly over the fitting range (e.g., mode-hopping), use the peak wavelength averaged over the same current range, or extract ${\eta }_d$ at fixed wavelength using a wavelength-resolved measurement.

Worked example

The 25 °C curve of the 1310 nm Fabry–Pérot InP laser from the T₀ extraction article, computed as described in Threshold Current Extraction Methods Compared and sampled every 0.25 mA, gives (every eighth point shown):

A linear least-squares fit over $I\in [12,20]$ mA, which holds 33 points of the sweep, yields:

$$\begin{array}{c}{\displaystyle {\eta }_s=0.550\text{ W/A},I_{\text{th}}^{\ast }=7.99\text{ mA},}\\ {\displaystyle R^2>\mathrm{0.9999.}}\end{array}$$

The extrapolated threshold agrees with the two-segment threshold of 8.00 mA found for the same curve, as expected when the transition is this sharp. On a curve with a rounder knee, the intercept of a fit that starts well above threshold falls below the two-segment value. Converting to differential quantum efficiency:

$$\begin{array}{rl}{\displaystyle {\eta }_d}& {\displaystyle =0.550\text{ W/A}\times 1.057\text{ A/W}}\\ {\displaystyle }& {\displaystyle =0.581=58.1\mathrm{\%}\mathrm{.}}\end{array}$$

This is the external differential quantum efficiency of the front facet alone. The device has a high-reflection coating on its rear facet, so most of its output leaves through the front. For a cleaved front facet with reflectivity $R_1\approx 0.30$ and a rear coating with $R_2\approx 0.90$, the front facet's share of the output is

$$\frac{(1-R_1)\sqrt{R_2}}{(1-R_1)\sqrt{R_2}+(1-R_2)\sqrt{R_1}}=0.92,$$

so the total ${\eta }_d$ from both facets is about 63%, within the bound set by the internal efficiency. The same 58% measured on one facet of an uncoated, symmetric device would imply a total of 116%, which is impossible and would point to a calibration or collection error.

Sources of extraction error

The following dominate the spread of ${\eta }_s$ values reported in the literature for nominally identical devices.

Inconsistent fitting range. ${\eta }_s$ extracted on $I\in [1.05I_{\text{th}},1.5I_{\text{th}}]$ differs from ${\eta }_s$ extracted on $I\in [1.5I_{\text{th}},3I_{\text{th}}]$ by 5–15% for typical devices, due to the gradual transition region near threshold and to thermal rollover at high $I$. The fitting range must be reported with the value.

Self-heating in CW measurements. For high-power devices in CW mode, junction temperature rise during the sweep reduces gain at the end of the sweep relative to the start. The apparent ${\eta }_s$ is biased low. Magnitude depends on device thermal impedance, mounting, and sweep direction; for typical high-power devices, the bias is 5–20% relative.

Photodetector saturation or non-linearity. Power meters and photodiodes have linear ranges. Exceeding the linear range produces apparent slope reduction not related to the device. Verify the detector linear range covers the LIV power range, including margin for the high-current end.

Spectral mismatch in detector calibration. Power meter responsivity varies with wavelength. A meter calibrated at 1550 nm used at 1310 nm without recalibration can introduce 5–20% systematic error. Detectors must be calibrated at the actual measurement wavelength.

Far-field truncation. Single-facet butt-coupled photodiodes capture only the central portion of the laser far-field. For lasers with broad far fields or non-Gaussian emission, the captured fraction varies with operating conditions, biasing ${\eta }_s$ low and producing artificial current-dependence in the slope. Use an integrating sphere or a beam-expanded lens-coupled detector to eliminate.

Cleave or facet quality variation. ${\eta }_d$ depends strongly on facet reflectivity. Devices with poor or uneven cleaves show reduced and variable ${\eta }_d$ that does not reflect bulk device properties. Visual inspection of facets under high magnification before measurement identifies most facet issues.

Wavelength drift across fit range. For devices that mode-hop or have appreciable wavelength shift over the fitting range, the conversion factor $q\lambda \mathrm{/}(hc)$ changes slightly. For typical edge emitters, whose wavelength shifts by 0.3–0.5 nm/°C, and modest self-heating, the error is well under 1% (a 4 nm shift at 1310 nm changes $q\lambda \mathrm{/}(hc)$ by 0.3%) and is usually neglected.

Validation

Two checks confirm extraction quality.

The fit residuals should be randomly distributed about zero with no systematic trend versus current. Visible curvature in the residual plot indicates either an inappropriate fitting range or a real device non-linearity.

The extracted ${\eta }_d$ must respect the physical bound. Summed over both facets it cannot exceed the internal efficiency ${\eta }_i$, and so cannot exceed 1 for a single-junction laser. For a symmetric, uncoated Fabry–Pérot laser each facet emits half the power, so a single-facet ${\eta }_d$ above 50% indicates an error. For a coated device, divide the single-facet value by that facet's share of the output, as in the worked example, before comparing it with the bound.

References

For the textbook treatment of slope efficiency, differential quantum efficiency, and the inverse-length method for separating internal efficiency from mirror loss, see Coldren, Corzine, and Mašanović (2012), chapter 2. For the influence of self-heating on extracted slope and methods for thermal correction, see Piprek (2003), chapter 7.

The wall-plug efficiency calculator computes efficiency point by point from an LIV sweep, and the wall-plug efficiency article works through the same laser.