LASER INTERFEROMETER GRAVITATIONAL WAVE …



LASER INTERFEROMETER GRAVITATIONAL WAVE OBSERVATORY

-LIGO-

CALIFORNIA INSTITUTE OF TECHNOLOGY

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

|Technical Note LIGO-T060224- 00- D 09/15/06 |

|TCS Actuator Noise Coupling |

|Phil Willems |

This is an internal working note

of the LIGO Project.

California Institute of Technology

LIGO Project – MS 51-33

Pasadena CA 91125

Phone (626) 395-2129

Fax (626) 304-9834

E-mail: info@ligo.caltech.edu

Massachusetts Institute of Technology

LIGO Project – MS 20B-145

Cambridge, MA 01239

Phone (617) 253-4824

Fax (617) 253-7014

E-mail: info@ligo.mit.edu

WWW:

File /home/blacke/documents/T9000xx.ps – printed November xx, 1999

INTRODUCTION

This document provides a detailed estimate of the coupling of TCS actuator power fluctuations into interferometer noise. It follows the analysis Stefan Ballmer used in his thesis, but with extension to the noise requirements for compensation plates, ring heaters acting on the barrel of test masses, and transmissive flexure noise.

MODEL

The basic equation for estimating TCS actuator noise injection onto a test mass is given by equation 2.39 in Stefan Ballmer’s thesis:

Here, P is the net TCS power, assumed to have the same Gaussian spot size w as the IFO beam; ρ and C are the density and heat capacity of fused silica, η is the Poisson ratio, α the thermal expansion coefficient, n and dn/dT the index of refraction and its temperature dependence, F the finesse of the arm cavity, and h the thickness of the ITM. The factor Cnum is a numerical factor describing the flexure noise coupling. This factor is a minor contribution for central heating but tends to dominate the noise for annular heating, of which barrel heating is an extreme case. Strictly speaking, this formula applies for central heating, but can be easily adapted to annular heating. The spatial overlap of the Gaussian heat profile and the Gaussian IFO beam is contained in the 1/(w2 term and to some extent in the Cnum term.

Flexure noise arises because the whole test mass will deform due to the thermoelastic expansion of any part of it, including a thin heated surface layer. Because in general the HR surface will move relative to the mirror’s center of gravity as a consequence, this couples TCS intensity noise into the arm cavity. The stress field that flexes the optic will also create stress birefringence noise, which we will also consider. The heating of the barrel of the test mass involves no overlap of the TCS beam and IFO beam, so only the flexure noise contributes.

A convenient way to calculate the flexure noise is to use the fact that the TCS heat fluctuations in the LIGO frequency band penetrate only into a layer of thickness

[pic]

where Dth is the thermal diffusivity. For fused silica and at 100 Hz, t~1 micron. This is so thin that the rest of the mirror effectively constrains the thin layer not to expand tangentially, i.e. within its surface. Although the mirror will deform, and the thin heated layer stretch under this deformation, the amount of stretch is so much less than it would be if the thin layer were free of the substrate, that the stretching force on the rest of the mirror can be well approximated by assuming the thin layer does not expand at all. The thin layer will also be bent due to the thermoelastic flexure of the mirror, but the forces and moments required to bend a thin layer are so small compared to those required to stretch it that they can be ignored. This is true even given the gradient of temperature across the layer’s thickness, so this gradient can be ignored, and the layer approximated as having uniform temperature. The expansion of the thin layer’s thickness is unconstrained.

COMPENSATION PLATE FLEXURE NOISE

We consider three types of heating of the compensation plate: central Gaussian, ideal anti-Gaussian, and uniform. The first would apply for a hot point design with underheated optics, the second for overheated optics, and the last approximately compensates for the negative lens caused by the ITM ring heater at full power.

All three heating schemes irradiate only one face of the compensation plate, so the thin heated layer causing the flexure noise lies in a plane. The free expansion of the thin layer will keep it within this plane, so the stresses required to constrain the free thermal expansion also act in the plane, which we take to be the x-y plane.

The stress field σ required to constrain the thermal expansion of a heated body is given by

[pic]

where Y is the Young’s modulus. Considering only the thermal gradients in the plane of the layer, the surface traction required to provide this stress field is

[pic]

This traction applies everywhere on the thin heated layer, except at the edges. There, the force per unit length applied to the line element defining the boundary of the heated layer is just

[pic]

This force is applied normally to the line element. It can readily be shown for any thin layer of finite dimension that the resultant force on the layer as a whole is zero.

To solve for the flexure of the compensation plate due to the thermoelastic stress of the thin heated layer, we can now simply apply to the plate the opposite of the surface tractions and line forces required to constrain the thin layer.

COMPENSATION PLATE NOISE COUPLINGS

In all the cases below, the noise coupling is proportional to the integrated heat over a power fluctuation cycle, and so is proportional to 1/f.

Case 1: Central Gaussian Heating

The heating pattern is assumed the same as the self-heating of the ITM:

[pic]

We will assume in our model this heating pattern applied sinusoidally at 100Hz, calculate the noise coupling, and then determine the RIN that satisfies Advanced LIGO requirements. The integrated temperature fluctuation in the thin surface layer is

[pic]

This temperature distribution was plugged into the force formulae above and applied to a compensation plate of 340 mm diameter and 65 mm thickness in the Comsol Multiphysics FEM software. Since there is negligible heating at the radius of the optic, no line force was applied there. The plot below shows the elasto-optic optical path length profile through the CP resulting from the thermoelastic stress. The graphical rendering software obscures the colorbar- the maximum optical path variation is -1.307x10-14 m.

[pic]

Unlike the case of the test mass treated by Ballmer, there is no HR surface on the CP that defines a cavity length, so the only relevant displacement is the displacement of both sides relative to one another. The deformation of the two surfaces is shown below.

|[pic] |[pic] |

The optical path variation between these two surfaces has opposite sign, comparable magnitude (~1x10-14 m), and somewhat broader profile than the elasto-optic profile, so given the larger couplings below we can treat these as canceling to a negligibly small level. Note that the overall deformation of these surfaces is large compared to their relative separation.

The direct thermoelastic expansion of the thin surface layer gives the optical path variation

[pic]

At the center this has magnitude 2.3x10-14 m.

Finally, the direct thermorefractive phase variation in the thin surface layer is

[pic]

This has magnitude 7.5x10-13 m at the center and together with the direct thermoelastic expansion dominates the noise budget. Since the spatial profile of these two terms matches that of the IFO beam, the overlap integral used by Ballmer to get his formula applies. Modifying his equation to apply to the compensation plate gives

Assuming 5x10-22 m/(Hz at 100 Hz, this implies a RIN of 1.0x10-6/(Hz.

Case 2: Ideal Compensation for Gaussian Self-heating

We assume that the ideal compensation for a Gaussian self-heating is just a uniform heating pattern from which the Gaussian self-heating has been subtracted:

[pic]

In this case the total power incident on the CP is 7.5 W. This assumes that the barrel of the CP is insulated.

It is easy to show that the noise coupling for this heating profile is exactly the same as that for the central Gaussian spot distribution for the direct thermoelastic and direct thermorefractive effects by calculating the overlap of the heating profile with the IFO beam profile. This is true even though the total power is 15x larger.

The thermal stresses for this distribution were applied to the CP in the Comsol model, again assuming 100 Hz fluctuation. In this case, since significant heating extended out to the radius of the optic, a line force was applied around the heated face. The elasto-optic optical profile is shown below.

[pic]

Here the optical path is nearly uniform across the optic, with a ~50% drop at the center. Without detailed calculation, we can take a rough weighted average of the coupling as

-1.5x10-14m. For a uniform phase profile, the coupling factor is twice as large as for a matched Gaussian phase profile.

The deformations of the two faces of the CP are shown below.

|[pic] |[pic] |

Again the distribution is fairly broad, with a difference of ~2.5x10-14m at the center, and so roughly cancels the elasto-optic coupling to the 5x10-15m level, which is small compared to the direct thermoelastic and thermorefractive couplings, even given the larger coupling for broader phase profiles.

Therefore, the noise coupling and required RIN for the ideal compensation profile for Gaussian self-heating is the same as for the Gaussian profile.

Case 3: Uniform Compensation

In the case where a point absorber creates a significant thermal lens, the ideal compensation pattern is one that evens out the heating over the face of the compensation plate. We therefore take as representative a perfectly uniform heat distribution of 8 W distributed over the CP face:

[pic]

In this case the flexure noise coupling is essentially the same as for the ideal compensation, since it nearly the same power distributed similarly over the optic. However, the direct thermoelastic and thermorefractive path variations are

[pic]

and

[pic]

These have the same peak values as for the Gaussian profile, but are uniform over the face of the compensation plate, and so couple twice as strongly. Therefore, the Ballmer formula changes to

[pic]

Therefore, if the compensation for a point absorber requires as much power as would the idealized compensation for Gaussian self-heating, then the required RIN would be 2x lower, or 5x10-7/(Hz. While it is hard to estimate how much compensation of pointlike absorbers will be required, it is difficult to imagine they would so dominate the thermal lensing profile, so this should be taken as a quite conservative limit on the necessary RIN.

Case 3: Compensation of a Ring-heated ITM

The optical path profile of an ITM whose ROC has been fully compensated by its own ring heater is shown in the left-hand figure below. It is apparent that the phase variation is characterized roughly as a negative lens, with steep variation of ~1.5x10-6 m over the outer 7 cm of radius. The optimized compensation pattern for this profile has not yet been calculated, but for the purposes of estimating noise couplings it is possible to use the heating profile that yielded the thermorefractive phase profile show below at right, which has the same overall phase variations in the opposite sense.

|[pic] |[pic] |

The heating profile used to get the right hand profile was a uniform 18W of power distributed over the inner 13 cm radius on the CP face, or 339 W/m2. For comparison, the uniform heating considered above for a point absorber is only 88 W/m2. Therefore, the noise coupling in this case will be 3.9x worse, and require an RIN of 1.3x10-7/(Hz. This may be just barely possible using the very best HgCdTe detectors but would be very difficult to achieve reliably in practice.

TEST MASS FLEXURE NOISE

In the conceptual design for TCS, the test masses will receive direct thermal compensation only on their barrel surfaces, nearer their AR surface. For this reason, only the flexure noise is relevant for calculating test mass TCS actuation noise coupling. In deference to the cylindrical symmetry of the problem we now work in a cylindrical system with coordinates [pic].

The concept of the laterally constrained surface layer remains useful in this case. However, flexure noise caused by fluctuations on the barrel of the optic is qualitatively different from that caused by fluctuations on the surface, because the constrained thin surface layer does not lie in a plane. Instead, it has the form of a hoop, which would increase its radius under free thermal expansion. It is easy to show, by considering a short curved segment of the coating around part of the barrel, that compressive forces at the edges of the coated segment will have a small resultant away from the barrel surface; the condition of no net force on the short curved segment will therefore require a balancing pressure pulling the segment into the surface. When the thin layer extends all the way around the barrel the result is the well-known formula for the hoop stress:

[pic]

The substrate must apply this radial pressure to the thin layer to constrain its expansion. (The thickness of the layer is still free to expand, and will.) The surface traction within the local plane of the thin layer (along z and () is the same- modulo the coordinate system- as given for the case of the flat surface:

[pic]

We assumed uniform 1W, 100 Hz heating of a 1 cm wide stripe around the barrel of a test mass, 2 cm from the AR surface, and modeled it using an axisymmetric 2-D model in Comsol. The resulting deformation plot for r, z > 0 is shown below.

[pic]

As expected, the thermoelastic deformation is very large in the immediate vicinity of the added heat (lower right). However, the color contour plot shows that the flexure can be seen throughout the mirror- the color represents displacement in the upward, z-direction. (The color chart is blank near the applied heat because its scale has been set to emphasize deformation in the rest of the optic. The scale of the colorbar is 10-15 m.) The motion of the HR surface is estimated by taking the z-displacement there (-3.9x10-16 m) and subtracting from it the motion of the center of mass of the optic, calculated as the average z-displacement over the rest of the volume (-1.8x10-16 m), for a net motion of 2.1x10-16 m.

We require the TCS injected displacement noise to be less than 5x10-22 m/(Hz at 100 Hz, so this 1 W fluctuation must be scaled down to 2.4x10-6 W/(Hz. In order to fully compensate the arm cavity at high IFO power, it will be necessary to provide 22W of heat to the ETM, in which case we require 1.1x10-7/(Hz RIN. This should be simple to provide with a ring heater, whose thermal inertia will passively smooth power fluctuations on its input supply.

CONCLUSIONS AND IMPLICATIONS FOR TCS

The analyses above show a convenient way to calculate the effect of TCS fluctuations on interferometer optics by means of a constrained surface layer approximation. The advantage of this technique is that by treating the surface layer as a set of boundary forces and tractions on the substrate, it is possible to model the system in a FEM without having to mesh finely enough to sample the thin layer, or to treat the thin layer as a shell element. In addition, the constrained surface layer assumption allows these boundary forces and tractions to be simply calculated from the fluctuating heating profile.

For the case of the compensation plate, these analyses have shown that for the types of heating pattern we expect to use in Advanced LIGO, the flexure noise is never a significant contribution to the overall noise coupling. This is in contrast to the case of TCS on the ITMs in initial LIGO, where for annular heating the flexure noise in the dominant contribution. The difference is that for the ITM it is the motion of the HR surface relative to the center of gravity that determines the coupling, while for the CP it is the separation of the two surfaces, and the interior elasto-optic effect, that contribute, and these latter motions are much smaller and cancel each other.

These calculations also show that for a wide range of heating profiles, the required RIN of the CO2 laser is at or above 1x10-6/(Hz, which is quite practical to reach using HgCdTe detectors. However, the likely heating profile required to compensate for an ITM that itself has been compensated using its own ring heater will require nearly heroic CO2 laser intensity stabilization.

It has already been noted that the CP ring heater cannot provide compensation for an ITM that has been curvature compensated at high power. Given the results presented here, we recommend that the ITM ring heater not be used for high power curvature compensation. The ITM ring heater might still prove useful for small amounts of static curvature compensation or for controlling parametric instability if its power can be kept below 1 W.

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