# Laser saturation and gain

## Homework Statement

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(a) Why does input intensity affect gain?
(b) Derive the expression
(c) Find saturation intensity in terms of input intensity

## The Attempt at a Solution

Part(a)
Consider a narrow band radiation as an input, where its bandwidth is much smaller than the spectral with of transitions, so in general
$$\frac{dN_2}{dt} = S_2 - (N_2B_{21}-N_1B_{12}) \int g_H(\omega - \omega_0) \rho(\omega) d\omega + \cdots$$
$$= S_2 - N^{*} \int B_{21} g_H(\omega - \omega_0)\rho(\omega) d\omega + \cdots$$
$$= S_2 - N^{*}\sigma_{21}(\omega_L - \omega_0) \frac{I}{\hbar \omega_L}$$

Thus, we see that rates depend on input intensity ##I_T##, which influences the gain on a laser amplifier when we solve for the steady state solutions.

Part(b)
Bookwork. Managed to derive it.
$$\alpha(\omega) = \frac{\alpha_0(\omega)}{1 + \frac{I}{I_{sat}}}$$
where saturation intensity is ## I_{sat} = \frac{\hbar \omega_L}{\sigma_{21} \tau_R}## and relaxation time is ##\tau_R = \tau_2 + \frac{g_2}{g_1} \tau_1 (1- \tau_2A_{21})##.

Part(c)
The equation for intensity is given by
$$\frac{dI}{dz} = \alpha I = \frac{\alpha_0}{1 + \frac{I}{I_{sat}}}I$$
which may be integrated to give
$$ln \left( \frac{I(z)}{I_{(0)}} \right) + \frac{I_{(z)} - I_{(0)}}{I_{sat}} = \alpha_0 z$$

At low intensity where ##I_{(z)} << I_{sat}##, the equation becomes ##I_{(z)} = I_{(0)} = e^{\alpha_0 z}##.
At high intensity where ##I_{(z)} \approx I_{(0)}##, the equation becomes ##I_{(z)} = I_{(0)} + \alpha_0 I_{sat} z##.

Thus initially the beam intensity is weak, then it becomes strong, so
$$e^{\alpha_0 z} = 100$$
$$200I_0 = \alpha_0 I_{sat}z$$

bumpp

would appreciate any input on the last bit, many thanks in advance!

bumpp

bumpp on last part

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