Tuning Fabry-Perot cavity modal frequencies using controlled thermoelastic deformations on mirror surface

Goal

To correct for the modal frequency shifts in the FP arm cavity. We are not looking to change the overall RoC of the mirror or suppress the higher order modes; but creating phase shifts that would result in desired modal frequency shifts. This will be done by imaging heat patterns on the ETM surface. The thermoelastic deformations created on the mirror surface introduces phase shifts to the cavity modes. The green ALS system will be used to mode scan the cavity continuously. A feedback control system will actively correct for the frequency shifts based on the cavity mode scan information obtained from the green probe laser mode scans.

The lessons learned from this exercise will help in designing/implementing a similar kind of system for the folding mirrors in the signal recycling cavity.

Overview

CTD.pdf

Description

The PSL laser is locked to the arm cavity using the IR PDH error signal. The green laser is injected from the ETM side of the arm. The relative phase between the two lasers is kept constant using a phase-locked-loop (PLL) servo. The transmitted end-green from the arm interferes with the frequency-doubled PSL and produces a beatnote.

Taking into account the following:

1. The amplitude of the beatnote depends on the intensity of the transmitted green.
2. Changing the frequency of the end-green laser (using the PLL local oscillator) will affect the resonance conditions in the arm cavity and excite the various transverse modes.
3. The frequencies at which the various transverse mode resonances occur depend on the spatial inhomogeneities on the mirror surface.

we can conclude that the amplitude of the beatnote as a function of the end-green laser frequency (or the local oscillator frequency) will hold the information about the cavity resonance frequencies and hence the effect of the mirror distortions created by various heat patterns can be mapped.

We expect to correct for frequency shifts in the order of a few KHz (deformations/length change of the order of a few nm --> dl/l = df/f).

Simulation/Modelling

• SIS modelling of the arm cavity using the phase map measurements made for the ITM and ETM.

This will give an estimate of the modal content of the arm cavity which we will use as reference to start with. The cavity field can be expressed in terms of Zernike polynomials of order 'm'. Comparing this with desired cavity field, we can obtain the correction in terms of Zernike polynomials. This should give an idea about the range of frequency shifts we are looking at for each higher order mode, the highest mode order correction we are going to limit ourselves to,

• COMSOL model:

Model thermal deformations on the ETM. Use the generated correction maps and estimate deformation depths and points on the ETM. Based on this, we can decide the limitations on the system if we consider only positive corrections (thermo-elastic expansion) and if it is essential to design a heating system that can generate positive and negative corrections to the mirror.

Estimate the heating power that can bring about these deformations on the HR surface of the mirror. Check for limitations posed on the number of heating elements (heat pixels in the heater array) and their size.

Design and construction

• Heater array:

Customised heater array. Array size, individual heating element size and pixelation of heater will be decided based on the required heating power, desired actuation area on the ETM, space limitations in the vacuum chamber...

• Heater electronics:

Control over heat power generated by individual heat elements is mandatory. The current driver heating the elements of the array will be the actuator driven by the CTD feedback servo.

• Telescope components: Based on the pros and cons, we must decide if

  --> we will reflect or refract heat patterns on the ETM. Layout as to how each of these will look like, is shown in the figure.

  --> the heater will be installed in-vacuum or out-of vacuum (For out-of-vacuum heater installation, we will have to think about compatible vacuum windows).

  --> the heater and the reflecting/refracting elements will be installed on fixed or movable stages. Telescope specs will be decided based on actuation area and the material properties (absorption spectrum) of the mirror.

• Apertures: ???

• PLL for green laser: Estimate the LO frequency and amplitude in the PLL. Design analog PLL filters.

• Inversion matrix:

• Front ends:

Tasks/Timeline

1. Initial modelling to decide on our requirements: Steps under simulation/modelling take priority. This will help us decide on what are our requirements of the system and if our requisites are plausible in the first place.

2. Hardware design/installation:

Drawbacks