List of Figures

View Image Figure 1:
Intensity distribution in an LG5,5 mode of width parameter w = 3.5 cm. Dashed circle: edge of a mirror of radius 17.5 cm.
View Image Figure 2:
Intensity distribution in an axisymmetric LG5,5 mode of width parameter w = 3.5 cm. Dashed circle: edge of a mirror of radius 17.5 cm.
View Image Figure 3:
Solid line: Intensity profile of a normalized mesa mode of parameters b f = 10.7 cm, w0 = 3.2 cm. Dashed line: nearest flat beam profile (b = 9.1 cm).
View Image Figure 4:
Surface of a mirror matching the mesa beam of parameters b f = 10.7 cm, w0 = 3.2 cm
View Image Figure 5:
Power distribution of a Gauss–Bessel mode of parameters 𝜃 = 54 μRd , w0 = 5.2 cm, z = 1.5 km
View Image Figure 6:
Surface of a mirror matching a Gauss–Bessel mode of parameters 𝜃 = 54 μRd , w0 = 5.2 cm, z = 1.5 km
View Image Figure 7:
Notations for a cylindrical mirror
View Image Figure 8:
Error in intensity reconstruction (50 Fourier–Bessel terms) for LG0,0, w = 2 cm (black curve) and LG5,5, w = 3.5 cm (red curve); Cut: φ = 0
View Image Figure 9:
Reconstructed intensity (FB series) for LG5,5 mode (φ = 0)
View Image Figure 10:
Temperature field in the substrate, 1 W dissipated in the coating, mode LG 0,0, w = 2 cm (φ = 0) [logarithmic scale]
View Image Figure 11:
Temperature field in the substrate, 1 W dissipated in the bulk substrate, mode LG 0,0, w = 2 cm (φ = 0) [logarithmic scale]
View Image Figure 12:
Temperature field in the substrate, 1 W dissipated in the coating, flat mode, b = 9.1 cm (φ = 0) [logarithmic scale]
View Image Figure 13:
Temperature field in the substrate, 1 W dissipated in the bulk substrate, flat mode, b = 9.1 cm (φ = 0) [logarithmic scale]
View Image Figure 14:
Temperature field in the substrate, 1 W dissipated in the coating, mode LG5,5, w = 3.5 cm (φ = 0) [logarithmic scale]
View Image Figure 15:
Temperature field in the substrate, 1 W dissipated in the bulk substrate, mode LG5,5, w = 3.5 cm (φ = 0) [logarithmic scale]
View Image Figure 16:
Nonaxisymmetric mode LG5,5, w = 3.5 cm, temperature on the coating (coating absorption) (z = − h∕2)
View Image Figure 17:
Nonaxisymmetric mode LG5,5, w = 3.5 cm. Temperature in the meridian plane (bulk absorption) (z = 0)
View Image Figure 18:
Temperature at various depths in the substrate (coating absorption) case of LG5,5 (φ = 0)
View Image Figure 19:
Temperature at various depths in the substrate (flat mode, bulk absorption) (φ = 0)
View Image Figure 20:
Coupling losses as functions of the dissipated power on the coating. Solid line: total losses, numerical integration of Equation (3.76View Equation). Dashed line: harmonic losses after Equation (3.80View Equation).
View Image Figure 21:
Thermal lens, heating by 1 W bulk absorption. The dashed line is the nearest paraboloid ˆ Z (r) (in the sense of least squares, weighted by the normalized beam intensity).
View Image Figure 22:
Thermal lens for 1 W absorbed from the mesa mode (solid line) and the flat mode (dashed line)
View Image Figure 23:
Thermal deformation of the mirror under three types of readout beams (1 W absorbed power in the coating and exaggerated by a factor of 2 × 105)
View Image Figure 24:
Deformation of the reflecting coating for three types of readout beams (coating absorption). Dashed line: best parabolic fit 2 ˆuz + δˆuz = cr + d (weighted by the intensity profile) giving the effective curvature radius
View Image Figure 25:
Thermal expansion of the mirror under three types of readout beams (heating by 1 W internal absorption of light, exaggerated by a factor of 2 × 105)
View Image Figure 26:
Thermal aberration caused by internal heating for 1 W absorbed power. Dashed lines: nearest paraboloid (weighted by the intensity distribution)
View Image Figure 27:
Normalized intensity I(r) on the mirror rear face from a ring radiator
View Image Figure 28:
Thermal lensing from a ring radiator. Red dashed curve: nearest paraboloid (weighted by the readout beam intensity). The readout beam is TEM00 with w = 2 cm. The curvature is weakly dependent on the beam width: 87 km W for w = 2 cm, 95 km W for w = 6.65 cm. Green dashed curve: Zernike expansion of the lens
View Image Figure 29:
Thermal compensation with a ring radiator: minimization of coupling losses
View Image Figure 30:
Ring radiator: correction of the thermal lensing caused by a TEM 00 beam of width w = 2 cm dissipating 100 mW on the coating. Dashed line: nearest paraboloid (flat for the optimal TCS power of 26.5 W)
View Image Figure 31:
Thermal lens profile created by an axicon system
View Image Figure 32:
Thermal compensation with an axicon: minimization of coupling losses. Solid line, short dashed, long dashed: resp. 10 mW, 20 mW, 30 mW, dissipated by the readout beam.
View Image Figure 33:
Source of heat on the mirror rear face for a power mask according to Equation (4.14View Equation) with wC = 16.9 cm (normalized to 1 W)
View Image Figure 34:
Source of heat on the mirror rear face for a power mask according to Equation (4.15View Equation) with wC = 16.9 cm for 1 W dissipated by the readout beam
View Image Figure 35:
Corrected thermal lens by a power mask according to Equation (4.15View Equation) with wC = 16.9 cm for 100 mW dissipated by the readout LG00 beam (w = 2 cm)
View Image Figure 36:
Time evolution of the thermal lens from room temperature to the steady state limit. Heating from coating absorption, LG0,0 mode, w = 2 cm
View Image Figure 37:
Time evolution of the thermal lens from room temperature to steady state limit. Heating from coating absorption, LG5,5 mode, w = 3.5 cm
View Image Figure 38:
Time evolution of the thermal lens from room temperature to the steady state limit. Heating from coating absorption, Flat mode, b = 9.1 cm
View Image Figure 39:
Coating absorption: time evolution of the curvature radius of the thermal lens LG 0,0 mode, w = 2 cm
View Image Figure 40:
Coating absorption: time evolution of the curvature radius of the thermal lens, LG 5,5 mode, w = 3.5 cm
View Image Figure 41:
Coating absorption: time evolution of the curvature radius of the thermal lens, flat mode, b = 9.1 cm
View Image Figure 42:
Coating absorption: time evolution of the curvature radii of thermal lenses for three examples
View Image Figure 43:
Bulk absorption: Time evolution of the thermal lens curvature radii for three examples
View Image Figure 44:
Coating absorption: time evolution of the reflecting surface caused by thermal expansion. Mode LG0,0, w = 2 cm
View Image Figure 45:
Coating absorption: time evolution of the reflecting surface caused by thermal expansion. Flat mode, b = 9.1 cm
View Image Figure 46:
Coating absorption: time evolution of the reflecting surface caused by thermal expansion. Mode LG5,5, w = 3.5 cm
View Image Figure 47:
Coating absorption: time evolution of the curvature radius of the thermal lens caused by thermal expansion. Mode LG0,0, w = 2 cm
View Image Figure 48:
Coating absorption: time evolution of the curvature radius of the thermal lens caused by thermal expansion. Flat mode, b = 9.1 cm
View Image Figure 49:
Coating absorption: time evolution of the curvature radius of the thermal lens caused by thermal expansion. Mode LG5,5, w = 3.5 cm
View Image Figure 50:
Bulk absorption: time evolution of the reflecting surface caused by thermal expansion. Mode LG0,0, w = 2 cm
View Image Figure 51:
Bulk absorption: time evolution of the reflecting surface caused by thermal expansion. Flat mode, b = 9.1 cm
View Image Figure 52:
Bulk absorption: time evolution of the reflecting surface caused by thermal expansion. Mode LG5,5, w = 3.5 cm
View Image Figure 53:
Bulk absorption: time evolution of the curvature radius caused by thermal expansion. Mode LG0,0, w = 2 cm
View Image Figure 54:
Bulk absorption: time evolution of the curvature radius caused by thermal expansion. Flat mode, b = 9.1 cm
View Image Figure 55:
Bulk absorption: time evolution of the curvature radius caused by thermal expansion. Mode LG5,5, w = 3.5 cm
View Image Figure 56:
Coating absorption: transfer function from power to displacement. Dashed line: asymptotic regime
View Image Figure 57:
Bulk absorption: transfer function from power to displacement. Dashed line: asymptotic regime.
View Image Figure 58:
Gain in thermal noise PSD1/2 for LG modes having each a w parameter tuned for 1 ppm clipping losses, with respect to the Virgo input mirrors and beams
View Image Figure 59:
Virtually deformed mirror under beam pressure. (1 N integrated pressure) by modes having 1 ppm clipping losses. For more clarity, the displacements have been amplified by a factor of 7 × 107.
View Image Figure 60:
Distribution of strain energy in the mirror substrate (LG0,0 w = 2 cm, logarithmic scale)
View Image Figure 61:
Distribution of strain energy in the mirror substrate (Flat mode b = 9.1 cm, logarithmic scale)
View Image Figure 62:
Distribution of strain energy in the mirror substrate (LG5,5 w = 3.5 cm, logarithmic scale)
View Image Figure 63:
Distribution of ⃗ 2 (∇E ) in the case of a LG0,0 mode (w = 2 cm). (Logarithmic scale, arbitrary units)
View Image Figure 64:
Distribution of ⃗ 2 (∇E ) in the case of a flat mode (b = 9.1 cm). (Logarithmic scale, arbitrary units)
View Image Figure 65:
Distribution of ⃗ 2 (∇E ) in the case of a mesa mode (bf = 10.7 cm). (Logarithmic scale, arbitrary units)
View Image Figure 66:
Distribution of the square gradient of the trace of the strain tensor in the case of an LG5,5 mode (w = 3.5 cm). (Logarithmic scale, arbitrary units)
View Image Figure 67:
Mode LP 5,5 (solid line), Mode LG 5,5 (dashed line)