# Thermal lensing and deformations from ring heaters¶

Ring heaters are used to introduce a thermal substrate lens within an optic and also actuate on the surface curvature of the mirror. Such devices allow us to compensate for self-heating from laser beam absorption. The equations implemented in finesse.thermal.ring_heater are from  which analytically provides the substrate temperature profile through a cylindrical optic when heated around its barrel over a small area.

import finesse
import numpy as np
import matplotlib.pyplot as plt
import finesse.thermal.ring_heater as ring_heater
from finesse.materials import FusedSilica
finesse.init_plotting()

a = 0.17 # mirror radius
b = 56e-3 # LIGO ring heater position
c = 76e-3 # LIGO ring heater position
h = 0.2  # mirror thickness
w = 53e-3 # spot size radius
r = np.linspace(-a, a, 100) # radial points
z = np.linspace(-h/2, h/2, 100) # longitudinal points
material = FusedSilica

T_rh_per_W = ring_heater.substrate_temperature(r, z, a, b, c, h, material)

plt.colorbar(label='T-$\mathrm{T}_{\mathrm{ext}}$ [K/W]')
plt.ylabel("Depth [m]")

Text(0, 0.5, 'Depth [m]') The total substrate thermal lens optical path difference can also be computed using the finesse.thermal.ring_heater.thermal_lens() method, again returning a distortion per Watt of power ring heater power absorbed by the mirror.

Z_rh_per_W = ring_heater.thermal_lens(r, a, b, c, h, material)
plt.plot(r, Z_rh_per_W)

Text(0, 0.5, 'OPD [m/W]') 