Investigation of an enhanced near-field evanescent

cooling technique for possible use in E-LIGO

Alex Alemia, David Reitzeb, David Tannerb, Rich Ottensb

July 27, 2007

a Physics Department, California Institute of Technology, Pasadena, CA

b Physics Department, University of Florida, Gainesville, FL

Abstract

Cyrogenic techniques are being considered for use in Enhanced LIGO.

The sensitivity gains to be had are very promising. Theory describes a

possible strong thermal coupling effect due to near-field evanescent waves.

An experiment has begun to investigate whether the effect can be observed

between 1 inch sapphire optical disks kept parallel for separation distances

ranging roughly 10 microns to 1 cm. The experiment is in its final build

stages, and calibration and data runs should be able to begin.

Introduction

LIGO (Laser Interferometer Gravitational-Wave Observatory) is an exciting

project that illustrates the technological prowess of the human race. The

project’s aim is to detect gravitational waves that as yet have evaded detec-

tion. The problem in observation stems from the very minute scales on which

the effect should manifest itself, even for such energy-rich sources as black hole

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– black hole spirals within a few megaparsecs. As such, the LIGO project has

required the manufacturing of some of the most precise and sensitive measuring

devices to date.

The current LIGO employs various techniques that attempt to reduce the

vibrational noise in the instrument to unheard-of levels (10−21 m/√

Hz for Ad-

vanced LIGO). At these low noise levels, thermal excitations become dominate.

In fact, the thermal noise prevents LIGO from nearing the desired quantum

limit as evidenced in Figure 1.

Figure 1: The sensitivity budget for LIGO. Internal thermal vibrations preventreaching the quantum limit. Taken from [4]

Since the internal thermal noise is proportional to the internal temperature

(1/2kBT per degree of freedom), great gains in sensitivity are to be had by moving

to cooler temperatures. Calculations show that the necessary sensitivity could

be reached with 90◦ K [4]. Unfortunately this is easier said than done, as the

LIGO mirrors must be free test masses, isolated from all mechanical contact to

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reduce vibrations, as well as existing in vacuum. Black body radiative cooling

is not enough to reach the desired temperature for the mirrors. As such, I

am investigating an active near-field evanescent cooling technique that could be

adapted for Enhanced-LIGO. The principle relies on near-field electromagnetic

effects present in all media. For distances between two bodies comparable to a

thermal wavelength, there is a large increase in the power flux heat transfer due

to the exponentially decaying evanescent waves from the surface.

My project involves cooling a 1-inch sapphire laser optic disk by bringing

another cooled disk to close proximity (i.e. on the order of a single thermal wave-

length). This will be accomplished in two stages: first, by utilizing a precise

piezo-electric squiggle motor for the coarse adjustment whose position will be

checked by a high precision magnet rotary sensor; the final approach and align-

ment will be accomplished with three piezo-electric stacks capable of roughly

3-micron advancement. The whole apparatus will be operated inside of a two

stage cryostat, consisting of an outer dewar of liquid nitrogen (∼ 70◦K) and an

inner dewar of liquid helium (∼ 4◦K). Various temperature sensors and heater

elements will allow a dynamic sampling of our device’s response at various tem-

peratures and distance separations. The hope is to gain evidence that such

near-field evanescent cooling can be utilized for larger scale objects such as the

LIGO test masses.

The experiment has been designed and built and should be ready to begin

calibration and data runs.

Background

The LIGO mirrors can be expecting to collect 2 W of power from the laser.

Traditional cooling pathways cannot compensate for this. At the required 90◦K,

far-field radiative cooling is predicted by the Stefan-Boltzman equation, Q =

3

εσT 4. This has a strong temperature dependence. As we cool the mirrors, the

effective far-field cooling diminishes drastically. At 90◦K the mirrors radiate

a paltry 300 mW. Radiative cooling is not enough to outpace the absorbed

heat. Meanwhile the only conductive heat pathways available are through the

suspension wires whose dimensions are determined by noise requirements. These

too are inadequate for proper cooling.

Near-field cooling

Evanescent cooling provides a clear alternative. First described by Polder and

Hove in 1971 [3], near-field effects can cause a significant increase in heat transfer

for bodies that are close compared to the thermal wavelength, given by Wien’s

law.

λ =2.9× 10−3 m K

T(1)

Some temperatures and their corresponding thermal wavelengths are given in

Table 1.

Table 1: Some temperatures and corresponding thermal wavelengths

T (K) λ(µm)300 9.777 384 725

The effect at these small distances can be viewed either quantum mechani-

cally as photons tunneling across the vacuum gap, or more classically as putting

a body into the region occupied by the evanescent waves associated with ther-

mally induced currents. In optics this effect is known as FTIR (Frustrated Total

Internal Reflection) and is used in such common objects as beam splitters, which

rely on evanescent coupling between two separated prisms to obtain their effect.

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The better known van der Waal and Casmir forces arise from similar physics.

Theory

The theoretical treatment relies heavily on the Fluctuation Dissipation Theo-

rem, first hinted at by Albert Einstein in his famous 1905 Brownian motion

paper [1]. In modern notation, the theorem dictates that a given dissipative

force D(ω) generates the fluctuating force F by

⟨F 2

⟩=

2π

∫ ∞

0

D(ω)Θ(ω, T ) dω (2)

with

Θ(ω, T ) =~ω

exp(

~ωkBT

)− 1

(3)

the Planck distribution, or the frequency distribution of an oscillator at tem-

perature T .

This theorem can be used to calculate the current density in a dielectric ma-

terial at a given temperature. Knowing the current density, a Green’s function

approach can be used to calculate the electric and magnetic fields. Knowing

these, the Poynting vector can be calculated, and from this the flux into and

out of the surface. This tells us our net heat flux.

The mathematics are complex but the result is that heat transfer between

two bodies is approximately constant in the far-field and exponential for dis-

tances short compared to the thermal wavelength [2].

Experiment

While experimental proof of evanescent cooling exists for small scale metallic

surfaces [5], evidence for larger scale non-metallic parallel surfaces is lacking.

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Research was begun at the University of Florida by Stacy Wise investigating

the thermal coupling between two 1 inch sapphire disks. This project was

inherited by Ramsey Lundock. Initial components were ordered and the cyrostat

assembled. It is from him that the current experiment was inherited.

The project aims to prove the existence of strong distance dependent ther-

mal coupling between parallel sapphire disks. Measurements will be taken over

distances ranging roughly 10 µm – 1 cm.

Setup

To observe evanescent coupling, all other sources of thermal coupling must be

damped. To reduce convection, we must operate at a moderate vacuum (roughly

10−7 torr) obtained with a dispersion pump. To reduce conduction, our plates

must be well separated and well insulated. To help ensure thermal insulation

our heated disk is separated by a ceramic disk from the body of the device. To

ensure that our disks do not contact, we have deposited a thin layer of gold on

the surface of each plate, the geometries of which are shown in Figure 2. These

capacitors give a clear signal upon contact. To reduce far-field radiative contact,

the experiment will be preformed in a cryostat maintaining temperatures of 4◦K

for the final run. This will not only damp black-body radiative effects, but will

increase the thermal wavelength, reducing our proximity requirements.

Figure 2: Geometry and location of the gold-plated capacitors on the sapphiredisks

To move the disks relative to one other, we utilize an SQ-100 series piezo-

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electric Squiggle R© motor. The motor utilizes ultrasonic vibrations in its four

piezoelectric crystals to turn a 100 threads per inch screw.

Further motion will be obtained by three piezoelectric stacks, each of which is

capable of 14 µm movement at room temperature, from which we expect roughly

3 µm movement at 4◦K. Due to the arrangement of these piezos, they will also

be utilized for tip-tilt adjustment of our plate, to help ensure parallelism.

The distance between our plates will be known both from our capacitance

readings on each of our four capacitors, as well as from a rotary sensor attached

to the motor axis.

The rotary sensor is a chip capable of measuring the alignment of a magnetic

field. Attached to the motor axis is a small perpendicular bar magnet. When the

motor is turned, the sensor chip outputs two channels of voltage. Each channel

outputs a sinusoidal voltage as the magnet is rotated, going through two full

periods for each rotation. The two channel outputs are shifted π2 relative to one

another. This sensor along with developed software will act as a feedback circuit

for motor positioning, with precision determined by the stability of sensor output

voltages for a fixed motor position, good to a part in 103. With a known thread

count per inch, this system will also give us a displacement sensor with precision

determined by the machine tolerances for the thread width and spacing.

The placement and relative sizes of our components are visible in Figure 3.

Located on the lower (hot) disk is both a silicon temperature sensor and

a strain gauge for use as a heater. On the top (cold) disk is another silicon

temperature sensor. The lower temperature sensor and heater are connected to

a PID controller. In addition, located on the inner can is another heater wire.

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Figure 3: The location and placements of the measuring and controlling devices

Procedure

The experiment is interested in measuring the heat coupling between the two

disks at various distances. This will be accomplished by using hot disk heater

controller to maintain a higher temperature than the cold disk. The controller

will stabilize the higher temperature, and the heater power required will increase

as the coupling increases between the two disks. Our controller outputs heater

power in watts.

The experiment will be slow. The time to obtain a stabilized 3 degree tem-

perature differential at room temperature is on the order of tens of minutes.

This time should decrease as the device is cooled, but stabilization time is lim-

ited by the response time of our PID filter and will probably be on the order of

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a minute or so when cooled. The PID coefficients for the heater controller will

be obtained by experiment.

The experimental procedure will require a certain calibrated distance to be

obtained, with as high a level of parallelism as possible. Then after allowing

the device to stabilize, flux can be measured according to the heater power

output from the controller. By measuring at a wide range of distances including

separations as large as a centimeter or so, thermal flux through conduction

through the device should be able to be determined directly for the bottom disk.

The top (cold) disk’s conduction pathway should be more substantial, and can

be measured directly by heating the disk, quickly backing off the bottom plate,

and observing the cool-down profile. Confidence is measured characteristics will

have to be ensured with certain calibration techniques.

Problems

Obtaining a proper distance calibration is not straight forward. While motor

displacements are known accurately through the rotary sensor, in order to trans-

form these displacements to known distances requires a known distance. While

the plates can be crashed together, typically due to tilts introduced in the de-

vice and attachment of the disk, one quadrant will touch well before the others.

Our piezo’s allow some control in this area, but they have so far proved less

reliable and less reproducible than desired. Our device is extremely sensitive to

geometrical changes introduced by handling the device, vibrations, and cooling.

The best we can hope to do is to align the disks roughly, at which point

taking a crash event as a zero distance is as inaccurate as the non-uniformities

in deposition and non-parallelism, estimated to be distances on the order of a

micron or so.

Making the problem worse our gold deposition is starting to decay. It was

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noticed during testing that occasionally a fleck of gold will loosen itself from the

rest and being attracted to the other plate, bridge the gap, registering an off-

scale capacitor measurement (seen as a touch). While the worst of these events

can be seen visually while the can if off and the device at room temperature,

their occurrence makes the case for a crash calibration error prone. These

types of events can occur at distances as large as 100 microns or so and are

unpredictable. Steps can be taken to clean the plates prior to insertion, but the

possibility for these types of events remains.

Our capacitors were originally intended to serve as an accurate and preci-

sion distance measurement, but again suffers from problems. The chip we use

to measure capacitance is itself un-calibrated. An additional annoyance is the

fact that at any given setting it can only measure a capacitance range of roughly

8 pF with a full range of 25 pF. This requires during the course of the experi-

ment changing the offset capacitance of the chip, referred to as the CAPDAC.

There exists a noticeable nonlinearity in chip readings introduced by changing

the CAPDAC, which is different for the different channels. In addition, our

wiring setup introduces a certain base constant capacitance which differs for

all four channels. Furthermore, as our gold depositions start to decay, our four

channels have different multiplicative factors. Again a straightforward capacitor

calibration would be done in terms of known distances, which are difficult to

obtain.

A detailed calibration run, measuring the capacitance on all four channels

across various CAPDACs for known displacements, combined in a crash zeroing

technique should be able to result in a fairly accurate distance determination.

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Discussion

Most of this summer has been spent ordering parts, fixing problems and building

the final apparatus. In addition, the software for the motor feedback circuit and

general experiment interface was done in LabView. At present, the device should

be in its final form so all that remains should be the detailed calibration runs

and experimental runs at various temperatures.

So far, the only data collected has been rough data of thermal coupling

at room temperature without a detailed distance calibration. However, these

results prove promising. The coupling is strong, as can be expected as far-field

radiative cooling is still strong at room temperature, however, qualitatively, a

stronger than can be expected coupling was evidenced at very close separation

distances.

Calibration and data runs should be ready to be initiated within a week’s

time or so. I hope to stay on in order to see the project through as long as

possible.

Acknowledgments

I would first like to thank Prof. Kevin Ingersent and Kristin Nichola for making

my participation in this program possible and enjoyable. Next I would like

to thank Prof. David Reitze for his sponsorship, Rich Ottens and Prof. David

Tanner for their experience and knowledge which they dutifully shared with me.

Also Mallory Gerace for her support. Finally, I owe thanks to the University of

Florida Physics Department and NSF for financial support.

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References

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electromagnetic waves thermally excited: Radiative heat transfer, coherence

properties and Casimir forces revisited in the near field. Surface Science

Reports, 57(3-4):59–112, 2005.

[3] D. Polder and M. Van Hove. Theory of Radiative Heat Transfer between

Closely Spaced Bodies. Physical Review B, 4(10):3303–3314, 1971.

[4] S. Wise. Sensitivity enhancement in future interferometric gravitational

wave detectors. PhD Thesis, Proquest Dissertations And Theses 2006. Sec-

tion 0070, Part 0606 102, 2006.

[5] J. Xu, K. Lauger, R. Moller, K. Dransfeld, and I. Wilson. Heat transfer

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