The slope efficiency of a laser diode is \(0.5\) W/A, and the output optical power at a current of \(100\) mA is \(30\) mW. Assuming piece-wise linear characteristics of the laser diode, the threshold current of the laser is ________ mA.
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Write the power-current line as \(P=\eta_s(I-I_{th})\) using the given slope efficiency, plug in the one known operating point, and solve for the current-axis intercept.
Step 1: Treat the P-I curve above threshold as a straight line.
Above the threshold current, a laser diode's power-current graph is a line with slope equal to the slope efficiency, $\eta_s = 0.5$ mW/mA, passing through the known point $(I, P) = (100 \text{ mA}, 30 \text{ mW})$.
Step 2: Write the line in point-slope form.
\[ P - 30 = 0.5 (I - 100) \]
Step 3: The threshold current is where this line crosses $P=0$.
By definition, $I_{th}$ is the current axis intercept of the P-I line (the current below which the laser gives no output). Setting $P=0$:
\[ 0 - 30 = 0.5(I_{th} - 100) \]
\[ -30 = 0.5\, I_{th} - 50 \]
\[ 0.5\, I_{th} = 50 - 30 = 20 \]
\[ I_{th} = \frac{20}{0.5} = 40 \text{ mA} \]
Step 4: Confirm with the slope.
From $I_{th}=40$ mA to $I=100$ mA is a current step of $60$ mA, and $0.5 \text{ mW/mA} \times 60 \text{ mA} = 30$ mW, which is exactly the given output power, so the value is confirmed.
\[ \boxed{I_{th} = 40 \text{ mA}} \]
