finesse.thermal package

Submodules

finesse.thermal.hello_vinet module

Hello-Vinet equations for thermal lenses in cylindrical mirrors. Higher order mode thermal effects are not implemented here. Therefore all the functions return some axially symmetric data.

Equations all based on [37]:

Jean-Yves Vinet, “On Special Optical Modes and Thermal Issues in Advanced Gravitational Wave Interferometric Detectors” Living Review 2009

finesse.thermal.hello_vinet.eval_p_n_s_numerical(result)

Evaluates a Fourier-Bessel decomposition fit performed with finesse.thermal.hello_vinet.get_p_n_s_numerical().

Returns

I_rndarray: Irradiance from fit

finesse.thermal.hello_vinet.get_p_n_s(a, w, n, m, chi, eta_n_s, eta_n_s_sq)

Returns beam intensity overlap coefficients as calculated by Eq 3.33 in [37].

Parameters

afloat: mirror radius
wfloat: spot size radius
n, mint: LG mode order
chifloat: Reduced radiation constant
eta_n_sarray_like: 2D array with dimensions (n_max, s_max) for the n-th order Bessel and the first s zeros of the Bessel equation 3.11 [37]. See finesse.thermal.hello_vinet.zeros_of_xjn_1_chi_jn() to compute these.
eta_n_s_sqarray_like: Cached (eta_n_s)^2 values

Notes

This is currently limited to computing only the LG00 mode, higher order modes are still on the todo list.

finesse.thermal.hello_vinet.get_p_n_s_numerical(I, a, s_max, material, barrel_material=None, n_max=0, T_ext=293.15, root_accuracy=1e-6, newton_cotes_order=2)

Performs a Fourier-Bessel decomposition of some axisymmetric irradiance distribution.

Parameters

Indarray: Axisymmetric irradiance distribution [Wm^-2], defined from r = 0 -> a. Radial point array is internally inferred from the size of I.
afloat: mirror radius
s_maxint: Number of zeros in each Bessel expansion
materialMaterial: Mirror substrate material, see finesse.materials
barrel_materialMaterial: Barrel coating material, see finesse.materials
n_maxint, optional: Number of bessel functions to expand with, typically 0
T_extfloat, optional: External temperature around mirror
root_accuracyfloat, optional: Absolute accuracy of root Bessel function root finding
newton_cotes_weightint, optional: Order of newton-cotes weight for integral

Returns

rndarray: Radial points [m]
chi_edgefloat: Reduced thermal constant for barrel surface
chi_facefloat: Reduced thermal constant for end faces
p_n_sndarray: Fourier-Bessel coefficients
eta_n_sndarray: Zeros of Bessel function
Jn_k_ns_r_andarray: Fourier-Bessel expansion bases

Examples

import finesse.materials import finesse.thermal.hello_vinet as hv import numpy as np import matplotlib.pyplot as plt from scipy.special import eval_hermite

finesse.init_plotting() # aLIGO like test mass substrate material = finesse.materials.FusedSilica a = 0.17 h = 0.2 w = 53e-3 r = np.linspace(0, a, 101) # 5th order hermite radial distribution E = eval_hermite(5, np.sqrt(2)*r/w) * np.exp(-(r/w)**2) I = E*E plt.plot(r, I) # perform Fourier-Bessel decomposition fit = hv.get_p_n_s_numerical(I, a, 10, material) plt.plot(r, hv.eval_p_n_s_numerical(fit), ls=’–’, lw=2, label=’s_max=10’) # perform Fourier-Bessel decomposition fit = hv.get_p_n_s_numerical(I, a, 20, material) plt.plot(r, hv.eval_p_n_s_numerical(fit), ls=’–’, lw=2, label=’s_max=20’) plt.legend() plt.xlabel(“r [m]”) plt.ylabel(“I(r) [Wm^-2]”) plt.title(“Fourier-Bessel decomposition of irradiance”)

Notes

This returns beam intensity overlap coefficients as calculated by Eq 3.15 in [37] for an arbitrary radial intensity distribution. Typically n_max=0 and s_max is chosen for the required fit. The integral is performed using composite Newton-Cotes rule, which by default is set to a Simpsons Rule (order 2) which provides better accuracy when fewer sample points in the intensity are present. Higher orders are not necessarily more accurate as over-fitting from using too high a polynomial order can introduce artifacts.

The resulting fit can be compared using the returned values by passing the tuple of results to finesse.thermal.hello_vinet.eval_p_n_s_numerical(). Comparing the above to the original intensity will show how accurate the fit is and will determine what s_max and weight ordering you should use. Sharp features in the intensity will not fit well.

finesse.thermal.hello_vinet.ring_radiator_intensity(r, a, b_c, D_c, P_c)

Calculates the intesity of incident on a mirror surface from an ideally thin ring radiator.

Parameters

rndarray: Radial points [m]
afloat: Mirror radius
b_cfloat: Radius of ring radiator (b_c > D_c) [m]
D_cfloat: Distance from ring radiator to mirror surface [m]
P_cfloat: Power emitted by ring [W]

Returns

I_rndarray(shape=(r.size,)): Radial variation in intensity on mirror from ring.

Notes

Solving equation 4.11 [37]

finesse.thermal.hello_vinet.substrate_temperatures(data, z, h)

Computes the 2D substrate temperature distribution per Watt of absorbed power in each of the coating and substrate for an arbitrary axisymmetric heating irradiance computed with finesse.thermal.hello_vinet.get_p_n_s_numerical().

Parameters

datatuple: Irradiance fit data from finesse.thermal.hello_vinet.get_p_n_s_numerical()
zndarray: Longitudinal points, should sample points between -h/2 and h/2. Sampling outside this region will yield incorrect results.
hfloat: Longitudinal points

Returns

T_coatndarray(shape=(z.size, r.size)): 2D array of temperature in substrate from coating absorption per watt of power absorbed in coating
T_bulkndarray(shape=(z.size, r.size)): 2D array of temperature throughout substrate from bulk absorption per watt of power absorber through the entire substrate.

Notes

This is using equation 3.20 and 3.25 in [37] for \(\phi=0\).

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.substrate_temperatures_HG00(r, z, a, h, w, material, barrel_material=None, T_ext=293.15, n_max=0, s_max=10, root_accuracy=1e-6)

Computes the 2D substrate temperature distribution per Watt of absorbed power in each of the coating and substrate from a HG00 beam.

Parameters

rndarray: Radial points
zndarray: Longitudinal points, should sample points between -h/2 and h/2. Sampling outside this region will yield incorrect results.
afloat: mirror radius
hfloat: mirror thickness
wfloat: spot size radius
materialMaterial: Mirror substrate material, see finesse.materials
barrel_materialMaterial: Barrel coating material, see finesse.materials
T_extfloat, optional: External temperature surrounding mirror
n_maxint, optional: Maximum Bessel order for expansion
s_maxint, optional: Maximum number of zeros to compute
root_accuracyfloat, optional: Accuracy of root finding

Returns

T_coatndarray(shape=(z.size, r.size)): 2D array of temperature in substrate from coating absorption per watt of power absorbed in coating
T_bulkndarray(shape=(z.size, r.size)): 2D array of temperature throughout substrate from bulk absorption per watt of power absorber through the entire substrate.

Notes

This is using equation 3.20 and 3.25 in [37] for \(\phi=0\).

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.substrate_thermal_expansion_depth(data, z, h)

Computes the depth displacements throughout the bulk of an optic due to coating absorption. Displacement is in units of m per absorbed Watts for a custom axisymmetric heating beam, see finesse.thermal.hello_vinet.get_p_n_s_numerical().

Parameters

datatuple: Irradiance fit data from finesse.thermal.hello_vinet.get_p_n_s_numerical()
zndarray: Longitudinal points, should sample points between -h/2 and h/2. Sampling outside this region will yield incorrect results.
hfloat: mirror thickness

Returns

U_z_coat_per_Wndarray(shape=(z.size, r.size)): D Array of z displacements throughout the substrate per absorbed Watts of HG00 beam in coating

Notes

Solving equation 3.117 and 3.118 in [37]

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.substrate_thermal_expansion_depth_HG00(r, z, a, h, w, material, barrel_material=None, T_ext=293.15, n_max=0, s_max=20, root_accuracy=1e-6)

Computes the depth displacements throughout the bulk of an optic due to coating absorption. Displacement is in units of m per absorbed Watts for a HG00 heating beam.

Parameters

rndarray: Radial points
zndarray: Longitudinal points, should sample points between -h/2 and h/2. Sampling outside this region will yield incorrect results.
afloat: mirror radius
hfloat: mirror thickness
wfloat: spot size radius
materialMaterial: Mirror substrate material, see finesse.materials
barrel_materialMaterial: Barrel cotaing material, see finesse.materials
T_extfloat, optional: External temperature surrounding mirror
n_maxint, optional: Maximum Bessel order for expansion
s_maxint, optional: Maximum number of zeros to compute
root_accuracyfloat, optional: Accuracy of root finding

Returns

U_z_coat_per_Wndarray(shape=(z.size, r.size)): D Array of z displacements throughout the substrate per absorbed Watts of HG00 beam in coating

Notes

Solving equation 3.117 and 3.118 in [37]

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.surface_deformation_coating_heating(data, h)

Computes the depth displacement change of the surface of an optic due to coating absorption. Displacement is in units of m per absorbed Watts for a custom axisymmetric heating profile, see finesse.thermal.hello_vinet.get_p_n_s_numerical() on how to generate the data for this.

Parameters

datatuple: Irradiance fit data from finesse.thermal.hello_vinet.get_p_n_s_numerical()
zndarray: Longitudinal points, should sample points between -h/2 and h/2. Sampling outside this region will yield incorrect results.
hfloat: mirror thickness

Returns

U_z_coat_per_Wndarray(shape=(r.size.)): Array of z displacements

Notes

Solving equation 3.121 and 3.122 in [37]

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.surface_deformation_coating_heating_HG00(r, a, h, w, material, barrel_material=None, T_ext=293.15, n_max=0, s_max=20, root_accuracy=1e-6)

Computes the depth displacement change of the surface of an optic due to coating absorption. Displacement is in units of m per absorbed Watts for a HG00 heating beam.

Parameters

rndarray: Radial points
afloat: mirror radius
hfloat: mirror thickness
wfloat: spot size radius
materialMaterial: Mirror substrate material, see finesse.materials
barrel_materialMaterial: Barrel cotaing material, see finesse.materials
T_extfloat, optional: External temperature surrounding mirror
n_maxint, optional: Maximum Bessel order for expansion
s_maxint, optional: Maximum number of zeros to compute
root_accuracyfloat, optional: Accuracy of root finding per Watt of power absorbed in coating

Returns

U_z_coat_per_Wndarray(shape=(r.size.)): Array of z displacements

Notes

Solving equation 3.121 and 3.122 in [37]

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.surface_deformation_substrate_heating(data, h)

Computes the depth displacement change of the surface of an optic due to bulk absorption from a generic axisymmetric heating profile. Displacement returned is in units of m per absorbed Watts through entire substrate.

The accuracy of this computation decreases as h/a > 1. The substrate modelled must be disk like. The error is more pronounced towards the edge of the substrate.

Parameters

datatuple: Irradiance fit data from finesse.thermal.hello_vinet.get_p_n_s_numerical()
hfloat: mirror thickness

Returns

U_z_bulk_per_Wndarray(shape=(r.size.)): Array of z displacements per watt of absorbed power through entire substrate [1W/h]

Notes

Solving equation 3.165 [37]

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.surface_deformation_substrate_heating_HG00(r, a, h, w, material, barrel_material=None, T_ext=293.15, n_max=0, s_max=20, root_accuracy=1e-6)

Computes the depth displacement change of the surface of an optic due to bulk absorption from a HG00 beam. Displacement returned is in units of m per absorbed Watts through entire substrate.

The accuracy of this computation decreases as h/a > 1. The substrate modelled must be disk like. The error is more pronounced towards the edge of the substrate.

Parameters

rndarray: Radial points
afloat: mirror radius
hfloat: mirror thickness
wfloat: spot size radius
materialMaterial: Mirror substrate material, see finesse.materials
barrel_materialMaterial: Barrel cotaing material, see finesse.materials
T_extfloat, optional: External temperature surrounding mirror
n_maxint, optional: Maximum Bessel order for expansion
s_maxint, optional: Maximum number of zeros to compute
root_accuracyfloat, optional: Accuracy of root finding

Returns

U_z_bulk_per_Wndarray(shape=(r.size.)): Array of z displacements per watt of absorbed power through entire substrate [1W/h]

Notes

Solving equation 3.165 [37]

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.thermal_lenses(data, h)

Computes the substrate thermal lens per Watt of absorbed power in each of the coating and substrate for an arbitrary axisymmetric heating irradiance computed with finesse.thermal.hello_vinet.get_p_n_s_numerical().

Parameters

datatuple: Irradiance fit data from finesse.thermal.hello_vinet.get_p_n_s_numerical()
hfloat: mirror thickness

Returns

Z_coatndarray(shape=(r.size,)): Array of optical path difference in bulk from coating absorption per watt of power absorbed in coating
Z_bulkndarray(shape=(r.size,)): Array of optical path difference in bulk from bulk absorption per watt absorbed through entire substrate [1W/h]

Notes

This is using equation 3.20 and 3.25 in [37] for \(\phi=0\).

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.thermal_lenses_HG00(r, a, h, w, material, barrel_material=None, T_ext=293.15, n_max=0, s_max=20, root_accuracy=1e-6)

Computes the substrate thermal lens per Watt of absorbed power in each of the coating and substrate from a HG00 beam.

Parameters

rndarray: Radial points
afloat: mirror radius
hfloat: mirror thickness
wfloat: spot size radius
materialMaterial: Mirror substrate material, see finesse.materials
barrel_materialMaterial: Barrel cotaing material, see finesse.materials
T_extfloat, optional: External temperature surrounding mirror
n_maxint, optional: Maximum Bessel order for expansion
s_maxint, optional: Maximum number of zeros to compute
root_accuracyfloat, optional: Accuracy of root finding

Returns

Z_coatndarray(shape=(r.size,)): Array of optical path difference in bulk from coating absorption per watt of power absorbed in coating
Z_bulkndarray(shape=(r.size,)): Array of optical path difference in bulk from bulk absorption per watt absorbed through entire substrate [1W/h]

Notes

This is using equation 3.20 and 3.25 in [37] for \(\phi=0\).

Currently only works for n_max == 0.

finesse.thermal.hello_vinet.zeros_of_xjn_1_chi_jn(double chi, int n_max, int s_max, double accuracy)

Compute the roots of the equation

\[x J_{n+1}(x) - \chi J_{n}(x) = 0 \]

which is used throughout the Hello-Vinet thermal equations.

Parameters

chifloat: Reduced radiation constant
n_maxint: Max bessel order to compute up to, must be > 0
s_maxint: Max number of zeros to compute, must be >= 2
accuracydouble: absolute error on zero finding brentq algorithm

Returns

eta_n_sarray_like: Array of s_max zeros for the n_max bessel functions

Notes

This is based on the calculations in [37]. The zeros of this function are used in multiple calculations throughout the text.

This algorithm finds the first two zeros of the 0th order bessel function then from there assumes the next zero is approximately the difference between the last two further away.

For higher order bessels, the zeros are always between the zeros of n-1 zeros which can be used as bounds for root finding.

finesse.thermal.reciprocity module

Compute thermo-elastic deformations in optical substrates using reciprocity theorem [42]. This provides a more accurate method to model thermo-elastic deformation of surfaces when the temperature distribution is known throughout a substrate. Reciprocity used here requires finite-element model results for volumetric strain due to a Zernike based force applied to the surface of the substrate. Substrate temperature distributions can then be converted into displacements and overlaps with the volumetric strain calculated. The coefficients of these overlaps then describes the zernike surface decomposition.

Equations all based on [42]:

Modeling thermoelastic distortion of optics using elastodynamic reciprocity Eleanor King, Yuri Levin, David Ottaway, and Peter Veitch Phys. Rev. D 92, 022005 - Published 20 July 2015 https://doi.org/10.1103/PhysRevD.92.022005

The methods here have been adapted from code and work done by Huy Tuong Cao <huy-tuong.cao@LIGO.org>.

class finesse.thermal.reciprocity.AxisymmetricFEAData(r=None, z=None, V=None, material=None, filepath=None)

Bases: object

An object that contains the finite element analysis data required to perform Axisymmetric thermal reciprocity calculations. This data can either be loaded from a numpy npz file that contains the r, z, V, and material data or can be provided directly to the the constructor.

Attributes

afloat: Radius of optic in meter
hfloat: Thickness of optic in meter
rarray_like: 1D radial point vector ranging from [0, a] with Nr elements
zarray_like: 1D thickness point vector ranging from [0, h] with Nz elements
R, Zarray_like: 2D meshgrid arrays of self.r, self.z
Varray_like: An (NV, Nz, Nr) shaped array representing the 2D basis functions for this reciprocity.
materialfinesse.materials.Material: An object containing various thermal and mechanical properties for the optic being modelled. Must match the values being used in the finite element model generating self.V.

property NV

property Nr: Number of points in the radial vector.

property Nz

property dr

property dz

finesse.thermal.reciprocity.zernike_coefficients_axisymmetric(data: AxisymmetricFEAData, T)

Compute the Zernike coefficient from temperature substrate profile and volumetric strain. This is for axially symmetric heating profiles and distortions.

Parameters

dataAxisymmetricFEAData: Finite element model data
Tarray_like: Array of shape (data.Nz, data.Nr) of temperature over the finite element model domain.

Returns

Z_coeffsarray_like: Array of shape data.NV of Zernike coefficients of the surface

finesse.thermal.reciprocity.zernike_surface_axisymmetric(data: AxisymmetricFEAData, Z_coeffs)

Reconstruct the surface deformation from Zernike coefficients.

Parameters

dataAxisymmetricFEAData: Finite element model data
Z_coeffsarray_like: Zernike coefficients in an array of shape data.NV

Returns

Warray_like: Surface displacement in metres

finesse.thermal.ring_heater module

Ring heater thermal equations for cylindrical mirrors

Equations all based on [38] and [39]:

finesse.thermal.ring_heater.substrate_deformation_depth(r, z, a, b, c, h, material, barrel_material=None, T_ext=293.15, m_max=10, root_accuracy=1e-12)

Computes the depth displacements throughout the bulk of an optic due to 1W absorption of ring heater. Displacement is in units of m per absorbed Watts.

Parameters

rarray_like: Radial vector [m]
z :array_like: Longitudinal points, should sample points between -h/2 and h/2.
afloat: Mirror radius [m]
bfloat: Ring heater lower boundary relative to center [m]
cfloat: Ring heater upper boundary relative to center [m]
hfloat: Mirror thickness [m]
materialfinesse.materials.Material: Mirror substrate material, see finesse.materials
barrel_materialfinesse.materials.Material: Barrel coating material, see finesse.materials
T_extfloat, optional: External temperature surrounding mirror
m_maxint, optional: Number of zeros to find, i.e. analytic order
root_accuracy: float, optional: Accuracy of root finding

Returns

U_z_rh_per_Wndarray(shape=(z.size, r.size)): Array of z displacements throughout the substrate per absorbed Watts of ring heater

Notes

See [39] for derivation of these equations.

finesse.thermal.ring_heater.substrate_temperature(r, z, a, b, c, h, material, barrel_material=None, T_ext=293.15, m_max=10, root_accuracy=1e-12)

Computes the substrate temperature distribution per Watt absorbed ring heater power.

Parameters

rarray_like: Radial vector [m]
zarray_like: Longitudinal vector [m]
afloat: Mirror radius [m]
bfloat: Ring heater lower boundary relative to center [m]
cfloat: Ring heater upper boundary relative to center [m]
hfloat: Mirror thickness [m]
materialfinesse.materials.Material: Mirror substrate material, see finesse.materials
barrel material: finesse.materials.Material: Barrel coating material, see finesse.materials
T_extfloat, optional: External temperature surrounding mirror
m_maxint, optional: Number of zeros to find, i.e. analytic order
root_accuracy: float, optional: Accuracy of root finding

Returns

T_rh_per_Warray_like: 2D Array of temperature distribution over [r, z]

Notes

Typo for A_m terms in manuscript, sin terms are functions of u_m, not v_m. See [38].

finesse.thermal.ring_heater.surface_deformation(r, a, b, c, h, material, barrel_material=None, T_ext=293.15, m_max=10, root_accuracy=1e-12)

“Computes the surface thermo-elastic deformation of the HR surface per Watt absorbed ring heater power.

Parameters

rarray_like: Radial vector [m]
afloat: Mirror radius [m]
bfloat: Ring heater lower boundary relative to center [m]
cfloat: Ring heater upper boundary relative to center [m]
hfloat: Mirror thickness [m]
materialMaterial: Mirror substrate material, see finesse.materials
barrel_materialMaterial: Barrel coating material, see finesse.materials
T_extfloat, optional: External temperature surrounding mirror
m_maxint, optional: Number of zeros to find, i.e. analytic order
root_accuracy: float, optional: Accuracy of root finding

Returns

U_z_rh_per_Warray_like: 1D radial vector of z displacement per Watt of ring heater power

Notes

See [39] for derivation of these equations.

finesse.thermal.ring_heater.thermal_lens(r, a, b, c, h, material, barrel_material=None, T_ext=293.15, m_max=10, root_accuracy=1e-12)

Computes the substrate thermal lens per Watt absorbed ring heater power.

Parameters

rarray_like: Radial vector [m]
afloat: Mirror radius [m]
bfloat: Ring heater lower boundary relative to center [m]
cfloat: Ring heater upper boundary relative to center [m]
hfloat: Mirror thickness [m]
materialfinesse.materials.Material: Mirror substrate material, see finesse.materials
barrel_materialfinesse.materials.Material: Barrel coating material, see finesse.materials
T_extfloat, optional: External temperature surrounding mirror
m_maxint, optional: Number of zeros to find, i.e. analytic order
root_accuracy: float, optional: Accuracy of root finding

Returns

Z_rh_per_Warray_like: 1D radial vector of optical path difference from propagating through substrate.

Notes

Typo for A_m terms in manuscript, sin terms are functions of u_m, not v_m. See [38].

finesse.thermal.ring_heater.zeros_u_m(double a, double h, double tau, int m_max, double accuracy)

Compute the roots of the equation

\[u_m - au \cot(u_m h/(2a)) = 0 \]

Parameters

afloat: Mirror radius [m]
hfloat: Mirror thickness [m]
taufloat: Reduced radiation constant
n_maxint: Max bessel order to compute up to
accuracydouble: absolute error on zero finding brentq algorithm

Returns

u_marray_like: Array of m_max zeros

finesse.thermal.ring_heater.zeros_v_m(double a, double h, double tau, int m_max, double accuracy)

Compute the roots of the equation

\[v_m + au tan(v_m h/(2a)) = 0 \]

Parameters

afloat: Mirror radius [m]
hfloat: Mirror thickness [m]
taufloat: Reduced radiation constant
n_maxint: Max bessel order to compute up to
accuracydouble: absolute error on zero finding brentq algorithm

Returns

v_marray_like: Array of m_max zeros

finesse.thermal package

Submodules

finesse.thermal.hello_vinet module

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finesse.thermal.reciprocity module

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finesse.thermal.ring_heater module

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Module contents