# What is the equation for saturation intensity in terms of input intensity?

• unscientific
In summary, the input intensity affects the gain of a laser amplifier because it influences the rates of transitions. The expression for gain is derived as 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})##. The saturation intensity can be found in terms of the input intensity by solving two different equations for the output intensity at different
unscientific

## Homework Statement

[/B]
(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

bumpp

bumpp

For part (c) build two different equations, each of which relates the output intensity, input intensity, and distance (hint: you have derived this equation). Then to each one of them, input the different conditions, one is for after traveling the first amplifier and the other after traveling both amplifiers. It's then straightforward to derive the expression for the saturation intensity in term of only input intensity.

unscientific
blue_leaf77 said:
For part (c) build two different equations, each of which relates the output intensity, input intensity, and distance (hint: you have derived this equation). Then to each one of them, input the different conditions, one is for after traveling the first amplifier and the other after traveling both amplifieTrs. It's then straightforward to derive the expression for the saturation intensity in term of only input intensity.

You absolutely right. I have been making unnecessary approximations. Solved this question.

## 1. What is laser saturation?

Laser saturation is a phenomenon that occurs when the gain medium of a laser is unable to produce any additional amplification of light, resulting in a plateau in the output power of the laser.

## 2. How does laser saturation affect laser performance?

Laser saturation can limit the maximum output power and efficiency of a laser. It can also cause changes in the laser's spectral and temporal characteristics.

## 3. What causes laser saturation?

Laser saturation is caused by the depletion of excited atoms in the gain medium, which reduces the number of photons available for amplification.

## 4. How can laser saturation be overcome?

Laser saturation can be overcome by using a larger or more efficient gain medium, increasing the input power, or reducing the cavity losses in the laser.

## 5. What is the relationship between laser gain and laser saturation?

Laser gain is directly related to laser saturation – as the gain of a laser increases, so does the likelihood of saturation occurring. However, higher gain also allows for higher output power before saturation is reached.

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