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().

The HR surface is at z = +h/2 and the AR surface is at z = -h/2.

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 3.18 and 3.25 in [37] for \(\phi=0\) with the signs of the axial coordinate switched as noted above.

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.

The HR surface is at z = +h/2 and the AR surface is at z = -h/2.

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 3.18 and 3.25 in [37] for \(\phi=0\) with the signs of the axial coordinate switched as noted above.

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().

The HR surface is at z = +h/2 and the AR surface is at z = -h/2 and the surface normal to the HR surface is positive.

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] with the signs of the axial coordinate switched as noted above.

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.

The HR surface is at z = +h/2 and the AR surface is at z = -h/2 and the surface normal to the HR surface is positive.

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] with the signs of the axial coordinate switched as noted above.

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.

The HR surface is at z = +h/2 and the AR surface is at z = -h/2 and the surface normal to the HR surface is positive.

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] with the signs of the axial coordinate switched as noted above.

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.

The HR surface is at z = +h/2 and the AR surface is at z = -h/2 and the surface normal to the HR surface is positive.

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] with the signs of the axial coordinate switched as noted above.

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 HR surface is at z = +h/2 and the AR surface is at z = -h/2 and the surface normal to the HR surface is positive.

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] with the signs of the axial coordinate switched as noted above.

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 HR surface is at z = +h/2 and the AR surface is at z = -h/2 and the surface normal to the HR surface is positive.

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] with the signs of the axial coordinate switched as noted above.

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 3.41 and 3.42 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 3.18 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.