Tool · Laser characterization

T₀ Extraction Calculator

How quickly does the threshold current of this laser rise as it warms? From pasted threshold current against temperature data the calculator fits a straight line to ln(I_th) versus absolute temperature and returns the characteristic temperature T₀, the fitted threshold at 25 °C, and the R² of the fit. Background: characteristic temperature, threshold current, and Auger recombination.

Data sets

The two illustrative sets are generated from the model with T₀ = 50 K and 150 K, inside the usual ranges for InGaAsP (50 to 90 K) and GaAs (100 to 200 K) lasers, and rounded to 0.01 mA; they are not measurements.

Input data

Two columns. Comma, space, or tab separated. Lines starting with # are ignored.

Temperature unit

Fit results

T₀: 59.6 ± 0.3 K
I_th(25 °C) from fit: 8.024 ± 0.012 mA
I₀ (extrapolated to 0 K): 5.388e-2 mA
R²: 0.99992
Points used: 5

The ± values are one standard error, estimated from the scatter of the points about the fitted line (3 degrees of freedom). The 95 % interval on T₀ is about 58.6 to 60.6 K (t = 3.182). It reflects random scatter only; a systematic error, such as self-heating under CW drive, shifts T₀ without widening it.

The address holds the unit and the data rows.

Fit method: linear least-squares on ln(I_th) vs. T (K). Model: I_th(T) = I₀·exp(T/T₀), so T₀ = 1 / slope and I_th(25 °C) = I₀·exp(298.15 K / T₀). I₀ = exp(intercept) is the fit extrapolated to 0 K, far outside the measured range; it is a fitting constant with no physical meaning. Standard errors come from the residuals of the same fit and need at least three points. Runs entirely in your browser. Nothing you enter is uploaded or stored.

ln(I_th) vs. T with fit line

Measured points and the least-squares line through ln(I_th) against absolute temperature; the slope of the line is 1/T₀.

Parsed data

T (°C)	T (K)	I_th (mA)	ln(I_th)	fit residual
15.00	288.15	6.800	1.9169	2.29e-3
25.00	298.15	8.000	2.0794	-3.01e-3
35.00	308.15	9.500	2.2513	1.03e-3
45.00	318.15	11.200	2.4159	-2.17e-3
55.00	328.15	13.300	2.5878	1.87e-3

Methodology

For an explanation of the math, common pitfalls in T₀ extraction, and a full worked example, see the article Extracting T₀ from temperature-dependent LIV measurements.

The model

The threshold current is described by the empirical relation

I_{th} (T) = I_{0} e^{T / T_{0}}

with $T$ in kelvin. Taking logarithms gives a straight line,

\ln I_{th} = T / T_{0} + \ln I_{0}

and the tool fits it by ordinary least squares to all the points entered, reporting $T_{0}$ as the reciprocal of the slope and $I_{0}$ as the exponential of the intercept. Temperatures entered in °C are converted by adding 273.15. Because $I_{0}$ is an extrapolation to 0 K it has no physical meaning on its own, so the tool also gives the fitted threshold at 25 °C (298.15 K). The R² is computed on the logarithms, not on the currents.

With three or more points the tool also reports standard errors. The residual variance is $s^{2} = \sum r_{i}^{2} / (n - 2)$ , the slope has standard error $s / \sqrt{S_{T T}}$ with $S_{T T} = \sum (T_{i} - \overset{ˉ}{T})^{2}$ , and since $T_{0}$ is the reciprocal of the slope its standard error is that of the slope divided by the slope squared. The fitted threshold at 25 °C carries the standard error of the line at that temperature, $s \sqrt{1 / n + (T_{ref} - \overset{ˉ}{T})^{2} / S_{T T}}$ in the logarithm, multiplied by the threshold. The 95 % interval uses the Student t quantile for $n - 2$ degrees of freedom. These errors describe only the scatter of the points about the line; with two points there is no scatter to measure and none is given.

A larger $T_{0}$ means a threshold less sensitive to temperature. The exponential form is a description, not a derivation: in real lasers $T_{0}$ itself usually falls at higher temperature, as Auger recombination, carrier leakage over the barriers and intervalence band absorption grow, so a value fitted over one range should not be carried far outside it. Curvature in the plot of $\ln I_{th}$ against $T$ is the sign that a single value is not adequate. The thresholds should be measured under pulsed drive, so that the stage temperature is the active region temperature, and extracted by the same method at every temperature.

Worked example, from the default data (6.8 mA at 15 °C rising to 13.3 mA at 55 °C, five points). The fit gives $T_{0} = 59.6$ K, a fitted threshold of 8.024 mA at 25 °C and R² = 0.99992. The standard errors are 0.3 K on $T_{0}$ and 0.012 mA on the 25 °C threshold; with 3 degrees of freedom (t = 3.182) the 95 % interval on $T_{0}$ is 58.6 to 60.6 K. A two-point check with the end values gives 40/ln(13.3/6.8) = 40/0.6709 = 59.6 K. Extrapolating the fit to 85 °C gives 8.024 × exp(60/59.6) = 22.0 mA, an estimate that holds only if $T_{0}$ stays constant above 55 °C.

References: G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993). L. A. Coldren, S. W. Corzine and M. L. Mashanovitch, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).