Find the optical gain at threshold of a laser diode having following parametric values R1 = R2 = 0.42,   α = 5cm-1 and L = 560 µm

To find the optical gain at the threshold of a laser diode, we can use the following relationship:

\[
\text{Threshold Gain, } g_{th} = \frac{1}{L} \ln\left(\frac{1}{R_1 R_2}\right) + \alpha
\]

Where:
- \( g_{th} \) is the threshold gain.
- \( L \) is the length of the laser cavity.
- \( R_1 \) and \( R_2 \) are the reflectivities of the mirrors.
- \( \alpha \) is the internal loss coefficient.

Given:
- \( R_1 = R_2 = 0.42 \)
- \( \alpha = 5 \, \text{cm}^{-1} \)
- \( L = 560 \, \mu\text{m} = 560 \times 10^{-4} \, \text{cm} \)

Now, let's calculate \( g_{th} \).

First, calculate the product \( R_1 R_2 \):

\[
R_1 R_2 = 0.42 \times 0.42 = 0.1764
\]

Next, take the natural logarithm:

\[
\ln\left(\frac{1}{R_1 R_2}\right) = \ln\left(\frac{1}{0.1764}\right) = \ln(5.67) \approx 1.734
\]

Now, apply the formula for \( g_{th} \):

\[
g_{th} = \frac{1.734}{560 \times 10^{-4}} + 5
\]

Simplifying:

\[
g_{th} \approx \frac{1.734}{0.056} + 5 \approx 30.96 + 5 = 35.96 \, \text{cm}^{-1}
\]

So, the optical gain at threshold \( g_{th} \) is approximately **35.96 cm\(^{-1}\)**.

Would you like more details or have any questions?

Here are some related questions you might be interested in:
1. What factors influence the threshold gain in a laser diode?
2. How do internal losses affect the efficiency of a laser diode?
3. What is the significance of the reflectivity values in determining the threshold gain?
4. How does the cavity length impact the threshold gain of a laser diode?
5. Can the threshold gain be minimized, and how would that improve laser performance?

**Tip:** Understanding the relationship between cavity length and threshold gain can help in designing more efficient laser diodes.

Learn how to calculate the optical gain at threshold of a laser diode using given parameters: R1 = R2 = 0.42, α = 5cm-1, L = 560 µm. Detailed step-by-step solution provided with formulas and calculations.

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