High Sensitive Position Sensor based on PDH technique

Team members

Angela Anna Baiju and Chenyue Gu

Idea

This project is aiming of using Pound–Drever–Hall (PDH) technique to sense the vibration and the movement of the mirror driven by the noise from the environment (acoustic, thermal, etc.).

Pound–Drever–Hall laser frequency stabilisation is a powerful technique providing a means to stabilise the frequency of a laser to a specific reference frequency. It is commonly employed in precision measurements and experimental setups where precise control over the laser frequency is required, such as interferometric gravitational-wave detector. Alternatively, if a stable laser is available, the PDH technique can be used to stabilise and/or measure the instabilities in an optical cavity length via the error signal readout from the cavity.

Pound-Drever-Hall Method

The idea behind the PDH method is simple: A laser’s frequency is measured with a Fabry–Perot cavity, either transmission or reflection, and then is fed back to the input laser to suppress frequency fluctuations. The process begins with modulating the laser light. This is typically achieved by imposing a high-frequency (radio frequency or microwave) signal onto the laser beam (Local Oscillator). This modulation creates sidebands around the carrier frequency of the laser light. These sidebands are symmetrically spaced around the carrier frequency. The modulated laser light then passes through an electro-optic modulator (EOM). The EOM applies phase modulation to the light, causing the sidebands to shift in frequency relative to the carrier frequency. The amount of phase modulation applied determines the frequency separation between the carrier and the sidebands. The phase-modulated laser light is directed onto a frequency reference, which can be a stable Fabry–Perot cavity. The reflected beam is then picked off and compared with the local oscillator's signal via a mixer whose output is the product of its inputs. From the output of the mixer, by using a low-pass filter, the dc (or very low frequency) part which is of interest remains and the ac (or high frequency) part that is usually the twice the modulation frequency is filtered out. The remaining dc signal, so called error signal, then goes through the feedback loop. The feedback loop typically involves a control mechanism, such as adjusting the current supplied to the laser diode or controlling the frequency of the modulation signal. The feedback loop aims to minimise the error signal by stabilising the laser frequency to match the reference frequency. A basic setup of PDH method is shown in Fig. 1 .

Fig. 1 A basic diagram of the setup implementing Pound-Drever-Hall method. In this digram, the laser frequency is modulated with a Pockels cell, driven by some local oscillator. The reflected beam is picked off with an optical isolator (a polarising beamsplitter and a quarter-wave plate makes a good isolator) and sent into a photodetector, whose output is compared with the local oscillator’s signal via a mixer. A low-pass filter on the output of the mixer isolates this low frequency signal, which then goes through a servo am- plifier and into the tuning port on the laser, locking the laser to the cavity. The Faraday isolator keeps the reflected beam from getting back into the laser and destabilizing it.

Math Model

To implement the method, we need to know well about how to process the error signal, which is the heart of the technique. And we follow the derive from reference <>To start the math behind it, we pick a point outside the cavity and measure the electric field over time. Then the magnitude of the electric field of the incident and the reflected beams can be written as

$E_{in}=E_{0}e^{i\omega t}$, $E_{ref}=E_{1}e^{i\omega t}$,

where $E_{0}$ and $E_{1}$ are complex, and we account of the relative phase between the two waves in via them.

Modification to the Project

In this project, we will lock the cavity by reading the error signal of reflection/transmission of the cavity and then feed it back to the input mirror, moving the mirror with a PZT. And then we can read out the cavity length change from the error signal. This sensing technique is commonly using in LVK (LIGO-Virgo-KAGRA) interferometric gravitational wave detection community, as well as quantum optomechnics.

Setup

We plan for a setup.... (the whole setup and explain how we applied the PDH locking)

Reference

• Black, E. D. (2001). An introduction to Pound–Drever–Hall laser frequency stabilization. American journal of physics, 69(1), 79-87.