# Theory - External-Cavity Diode Lasers (PID Loops / Electronics)

## Frequency stabilisation of the diode

We have briefly discussed that we need to use a laser with a frequency resonant with the energy difference between the $5^2 S_{1/2}$level to the $5^2 P_{3/2}$ level in Rb. In practice, this is not as straightforward as it seems. At a frequency of ~384.230406 THz (our 780.5 nm of last chapter) and a linewidth of ~6 MHz, we will need to stabilize the frequency to $10^{-8}$ precision.

We will be using a semiconductor diode laser for the experiment, and its frequency stability is strongly dependent on both of

the temperature of the diode

the current running through the diode in operation

We will need to stabilize both the temperature and current of the diode, which we will do using two separate Proportional-Integral-Derivative (PID) loops.

In addition, laser diodes often have maximum current ratings, and we will be building a maximum voltage/current ceiling to prevent burning out the fragile (and expensive!) hardware.

## PID Loops

A PID loop is a control mechanism that manipulates an input variable to the experiment based on the error signal: A difference between a target value (set point) and the measured variable (process variable). We use PID loops extensively in everyday life, such as in air-conditioning units to control the temperature of a room.

The term PID stands for Proportional, Integral, and Derivative. These correspond to three different mathematical computations applied to the error signal to drive the controller's output. The proportional term produces an output value proportional to the current error value. The integral term is proportional to both the magnitude of the error and the duration of the error, making it crucial for long-term accuracy. The derivative term is computed based on the rate of change of the error, providing a prediction of future error based on the current rate of change. We can optimize the 3 terms to shape the control and response of the system so that we minimize the error signal as quickly and stably as possible.

To provide an example on how this will work in practice:

we fix a temperature set point $\tau (t)$, where the actual temperature measured by a temperature sensor (AD590) is $y(t)$.

The difference of the two is the error signal $e(t)=\tau(t)-y(t)$ that is fed into the PID controller.

The PID controller transforms the error signal based on the 3 terms mentioned above by the following equation,

where $K_p, K_i, K_d$ are parameters that we control.

$u(t)$ is feed into the feedback mechanism, in our case a peltier cooler, that is able to either remove or add heat to the system based on the voltage/current sent to it

We adjust$K_p, K_i, K_d$ such that we are able to reach the set point $e(t')=0$ stably as fast as possible.

### Question: How does this work for the current stabilisation?

<Feedback via either current or the piezoactuator> - Not done right now, include note that it can also be done for future experiments

## References

AD590 link

Peltier Cooler link

Selection rules

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