arXiv:1103.2389v1 [physics.optics] 11 Mar 2011
Near-field radiative heat transfer between macroscopic planar surfaces
Richard Ottens,1V. Quetschke,2Stacy Wise,1, ∗A.A. Alemi,1, †R.
Lundock,1, ‡G. Mueller,1D.H. Reitze,1D.B. Tanner,1and B.F. Whiting1
1Department of Physics, University of Florida, P.O. Box 118440, Gainesville, FL 32611-8440, USA
2Department of Physics and Astronomy, University of Texas at Brownsville, 80 Fort Brown, Brownsville, TX 78520, USA
(Dated: March 15, 2011)
Near-field radiative heat transfer allows heat to propagate across a small vacuum gap in quantities
that are several orders of magnitude greater then the heat transfer by far-field, blackbody radia-
tion. Although heat transfer via near-field effects has been discussed for many years, experimental
verification of this theory has been very limited. We have measured the heat transfer between two
macroscopic sapphire plates, finding an increase in agreement with expectations from theory. These
experiments, conducted near 300 K, have measured the heat transfer as a function of separation
over mm to µm and as a function of temperature differences between 2.5 and 30 K. The experiments
demonstrate that evanescence can be put to work to transfer heat from an object without actually
PACS numbers: 44.40.+a,78.20.Ci
Humans knew of radiative heat transfer at least as
early as the discovery of fire, and physicists have investi-
gated this process for centuries, culminating in the black-
body theory of Planck and the birth of the quantum the-
ory. Planck’s equation for black-body radiation contains
only the temperature and some fundamental constants.
When actual materials are involved, their emissivities en-
ter the discussion, but little else. For example, the heat
transfer per unit area between two semi-infinite planes
is set by their temperatures and integrated emissivities
but does not depend on their separation or other geo-
metrical quantities. When the two planes approach each
other closely the situation changes.
regime, each material interacts with exponentially de-
caying evanescent electromagnetic fields generated in and
existing outside the other material; these fields can drive
currents and generate heat.[1–3] This near-field radiative
heat transfer can be several orders of magnitude greater
than far-field blackbody radiation.
In this near-field
Much like the Casimir and van der Waals force, near-
field heat transfer deals with fluctuations that only exist
over small distances. The first in-depth theory for near-
field heat transfer between planar surfaces was derived by
Polder and Van Hove, building on the work of Rytov.
There have been several other theoretical approaches,
and in general the theory seems complete, except per-
haps at distances comparable to atomic dimensions.
Although heat transfer via near-field effects has been
discussed for many years, experimental verification of the
theory for heat transfer between two planar surfaces has
been limited. Hargreaves has presented room tem-
perature observations for two Cr surfaces at distances
as small as 1 µm. Domoto et al. reported results at
cryogenic temperatures but for relatively large (50 µm)
separations, where near-field effects were barely observ-
able. Neither study compared experiment to theory. A
comparison at a fixed spacing has been put forward, but
the plates were separated by polyethylene spacers, so the
distance could not be varied. There have also been sev-
eral recent results using a sphere-plane geometry.[4, 7–9]
There are engineering reasons for this approach, as, un-
like a parallel plane geometry, a sphere-plane geometry
needs no angular alignment. Using scanning probe and
micro-machine technologies, these experiments cover a
wide distance range.
In this paper, we report a set of measurements of heat
transfer between two planar surfaces and present a de-
tailed comparison to theory. In addition to its intrin-
sic interest, this work was motivated by possible ap-
plications in cooling objects without actually touching
them, such as the mirrors of a future laser interferometer
gravitational-wave detector. This application would
require the parallel plane geometry in order to get large
areas of close approach of the two objects and thus sig-
nificant heat transfer.
The near-field heat-transfer process can be thought
of as a form of frustrated total internal reflection.
Evanescent waves, exponentially-decaying electromag-
netic fields, exist outside, but near to, the surface of
a material medium at temperature T. These decaying
fields are a consequence of travelling waves inside the
medium experiencing total internal reflection; this phe-
nomenon occurs when there is no valid solution to Snell’s
Law, nisinθi= ntsinθt, where niand ntare the indices
of refraction and θiand θtthe angle between wave vec-
tor and surface normal on the two sides of the interface.
Although there is no energy transmission across the inter-
face, an electric field exists on the far side. Furthermore,
if another medium is brought near this exponentially de-
caying field some of the energy from the incident beam
will propagate across the gap and into the new mate-
rial. If one medium is hotter than the other, this photon
tunneling will lead to heat transfer from hot to cold.
Polder and Van Hove considered two half spaces of
identical material but different temperatures. The com-
puted heat transfer coefficient W comes from the tem-
perature derivative of the z component of the Poynting
vector (Sz) from each medium, which is evaluated just in-
side surface of the other medium. W is the temperature
derivative of this difference as can be seen in Eq. 1.
T1−T2→0|Sz(d+) − Sz(0−)
It is convenient to break W into two components
W = Wsin+ Wexp.(2)
Wsinis the part where the wave number kx of the field
is in the range 0 < kx< ω/c, where the field is propa-
gating, and which gives the ordinary (Stefan-Boltzmann)
far-field radiation. Wexpis the component where the field
is exponentially decaying away from the surface; it is the
larger contributor to the near-field limit:
where the energy transmission coefficients are
1 − cos(2χP)
cosh[2κ(d − δP)] − cos(2χP),(4)
with P = ? or ⊥ for the two polarizations and χ?and
χ⊥being the phase shifts on reflection
χ?= arg[(−iκǫ + kz)(−iκǫ − kz)∗], (5)
χ⊥= arg[(−iκ + kz)(−iκ − kz)∗], (6)
with ǫ the material dielectric function. δ?and δ⊥can be
iκǫ + kz
iκǫ − kz
iκ + kz
iκ − kz
and kz and κ are the z component of k for the medium
and the vacuum, respectively
(ǫ − 1)ω2/c2− κ2, (9)
κ = ikzv=
Note that kx> ω/c, so that kzv is imaginary above the
surface and the energy transmission coefficients Texp
are not derived from the energy reflection coeffi-
cients R?and R⊥. We have used this theory to calcu-
late the heat transfer coefficient between sapphire plates,
using Barker’s sapphire dielectric function.
FIG. 1. Heat transfer coefficient vs. distance for z-cut sap-
phire. The temperatures of the two media are Thot = 310
K and Tcold = 300 K, respectively. The violet curve shows
terms dominate at close distances. The blue curve is the total
heat transfer coefficient W.
sinwhich dominate at far distances. The red curve
exp. The green curve shows W⊥
exp. The evanescent
Figure 1 shows the prediction of the model for z-
cut sapphire; the temperatures of the two media are
Thot= 310 K and Tcold= 300 K. The Wsin term dom-
inates at large separation and is nearly constant with
separation. The Wexpterm dominates in the near-field
regime, so that the total heat transfer changes from be-
ing independent of separation at large distances towards
Wexp∝ 1/d2at short distances. The turning point be-
tween Wsinand Wexpoccurs when the separation is ap-
proximately equal to the peak wavelength of the black-
body curve of the hot half space. Wien’s displacement
law predicts that λmax≈ 9 µm when the temperature is
A sketch of our apparatus is shown in Fig. 2. It is de-
signed around two 50×50×5 mm3sapphire plates. These
have a specified flatness of λ/8 @ 633 nm per inch on the
largest surfaces and are cut such that the c axis is per-
pendicular to these surfaces (z-cut). Sapphire was used
because it has good thermal conductivity. It is also a
candidate for the test masses of future gravitational-wave
One of the plates, henceforth called the cold plate, is
attached to the thermal bath that is the vacuum cham-
ber. The other plate, the hot plate, is thermally isolated
from the bath by a Macor spacer attached to the back
side of the hot plate. The hot plate also has a heater
wound on a copper ring which is itself attached to the
back of the plate. The heater current and voltage, after
correcting for lead resistance, give the power required to
maintain a given temperature difference between the hot
and cold plates. Both plates have a Si-diode thermome-
ter fastened to their backs to read the temperature (and
for the hot plate to control it). Both plates have all four
corners coated with an approximately 200 nm thick layer
of sputtered copper. These coatings have areas about
FIG. 2. Experimental apparatus. Stepper motors allow ad-
justment of the spacing, tip, and tilt (read capacitively) of two
sapphire plates. The temperature of the hot plate is controlled
by a feedback circuit, and the power required to maintain a
temperature difference gives the heat transfer from the hot
plate to the cold plate and to the thermal bath.
1 mm2and serve as capacitor plates that are read by
four 24-bit capacitance-to-digital converter circuits to
measure the separation and angular misalignments of the
plates. The metal film is wrapped around to the sides of
the sapphire to allow electrical contact to the electrodes.
The cold plate is glued to a copper disk, which in turn
is attached to the experimental structure. The Macor
spacer on the back of the hot plate is attached to a modi-
fied kinematic mirror mount which allows for z-axis linear
movement and tip and tilt angular adjustment by turn-
ing the three adjustment screws in the back. Three step-
per motors turn screws on the kinematic mount via gear
reduction boxes; each motor step translates to a linear
movement of the hot plate by 35 nm. The components
are held together by an “L” shaped backbone (not shown)
to give rigidity. The assembly is located in a UHV cham-
ber, with a base pressure below 2 × 10−7Torr, making
gas conduction negligible. Signals to the stepper motors,
capacitance readouts, temperature readouts, and current
and voltages to the heater are all controlled and/or read
by a LabVIEW computer program.
Each pair of capacitor plates is calibrated by taking ca-
pacitance readings as the plates are driven together one
step of the stepper motors at a time. A fit is made to
C = ǫ0a/d + Cstraywhere ǫ0is the dielectric constant of
the vacuum, a is the capacitor area, and Cstrayis a par-
allel contribution independent of separation. The data
fits the equation above very well, with R2values greater
than 0.999. The fitted value of a equals the metalized
area within our knowledge of this area; Cstray≈ 0.4 pF.
The average capacitance gives the distance while the in-
dividual readings are used to correct the alignment by
sending steps to the motors controlling tip and tilt.
To measure the heat transfer coefficient we compute
FIG. 3. Heat transfer coefficient vs. distance. The curves
are each offset vertically by 2 W/m2·K from the one below.
The points are the data, with error bars, determined from the
scatter in the heat transfer measurements and the uncertainty
in the distance calibration. The solid lines are the theoretical
predictions for flat plates while the dashed lines are the theo-
retical predictions for slightly curved plates (see text). Each
measured curve has a reproducible addendum due to other
heat leaks which are not included in the model and which has
been subtracted from the data. The temperatures are (top to
bottom): Thot = 327.0 K, Tcold = 308.0 K; Thot = 322.0
K, Tcold = 307.0 K; Thot = 317.0 K, Tcold = 305.8 K;
Thot= 312.0 K, Tcold= 305.2 K.
W = P/[A(Thot− Tcold)], where P is the power dissi-
pated in the heater, A is the plate surface area, and Thot
and Tcoldare the temperatures of the hot and cold plates,
respectively. The data are a sum of parallel heat path-
ways, including thermal conduction through the Macor
spacer and the other parts of the hot-plate holder, radi-
ation to the thermal bath, and the contributions of the
near- and far-field radiation between the two plates. We
can observe the near-field effect because it is the only
one of these that will change with plate separation. The
thermal conduction and radiation-to-the-bath paths just
add a constant offset to the radiative heat transfer that
the model predicts.
Data for the heat transfer coefficient versus distance
were collected for four temperature differences.
Fig. 3.) Each one shows that near-field heat transfer ex-
ists and that the data and model agree reasonably well.
Each run covers a separation range of about 2-100 µm.
The only freedom in the fit is an offset to the model. All
our measurements have an offset of 0.0435±0.0004W/K,
completely consistent with thermal conduction through
FIG. 4. Near-field heat flux vs. distance for multiple temper-
ature differences and multiple runs. (One run is shown with
open symbols and other runs with filled symbols.) The inset
shows the near-field heat flux vs. temperature difference for
several specific separation distances.
the Macor spacer and the rest of the support for the hot
plate, in parallel with radiation from the rear surface of
the plate. The chamber pressure for these runs ranged
between 5 × 10−7Torr and 2 × 10−7Torr. The vac-
uum chamber is held at constant temperature of 30.0◦C
(303.2 K). The hot plate is brought close to the cold
plate step by step. Each datum is an average of a set of
500 values, in both heat transfer coefficient and distance,
taken after the system has reached thermal equilibrium.
It takes 30 to 45 minutes after moving to a new position
to reach thermal equilibrium. We calculate the average
and its standard deviation and plot these in Fig. 3.
Theoretical curves match the experimental data well.
However, although the agreement is within experimen-
tal errors, there does appear to be a systematic offset:
the theory predicts a slightly lower heat transfer coeffi-
cient at each separation than we measure. Alternatively,
the plates could be slightly closer than measured by the
capacitive readout. We believe that the latter explana-
tion is correct. Simulations of the heat transfer between
two convex plates (shown as dashed lines) eliminate the
systematic error when the radius of curvature in each
plate is ∼1 km,corresponding to deviations from flatness
of 500 nm. Subsequently, we measured the plate cur-
vatures using optical flats and green mercury light; the
Newton’s rings interference pattern gave a central dis-
placement of 170±30 nm with respect to the perimeter,
a little smaller but of the same order of magnitude as the
Figure 4 shows the heat flux caused by near-field ef-
fects. Both far-field heat transfer and the offset due to
heat leaks to the thermal bath have been subtracted.
For each temperature difference, data are shown for sev-
eral distinct data runs; these agree very well. The inset
shows the dependence on the temperature difference of
the plates. For distance values where the near-field ef-
fects dominate, the heat flux is linear in the temperature
difference. Both data and model were fitted to
φ(∆T,d) = G(d)(∆T)α(d)
where φ(∆T,d) is the total near-field heat flux, G(d)
is a multiplicative factor, and α(d) is the exponent for
∆T. Each curve follows a power law in ∆T, with the
exponent varying from 0.70 at small distances to 0.91 at
larger distances. The differential heat transfer, Eq. 1, has
α(d) = 1; the finite temperature differences used in the
experiment bring in higher-order terms.
In summary we have measured near-field heat trans-
fer across a small gap for a parallel-plane geometry. The
data agree quite well with the theory of Polder and Van
Hove. The experiments demonstrate that significant
amounts of heat can be transferred via evanescent radi-
ation in the near-field regime.
Our research is supported by the National Science
Foundation through Grant No. PHY-0855313. A.A.A.
was supported by the University of Florida Physics REU
Site through NSF grant DMR-0552726.
mospheric Science, Dalhousie University, Halifax, NS
†Current address: Department of Physics, Cornell Univer-
sity, Ithaca, NY 14853-2501, USA
‡Current address: Astronomical Institute, Tohoku Uni-
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