Journal of Modern Physics, 2011, 2, 498-518
doi:10.4236/jmp.2011.26060 Published Online June 2011 (
Copyright © 2011 SciRes. JMP
A New Theoretical Technique for the Measurement of
High-Frequency Relic Gravitational Waves
R. Clive Woods1, Robert M. L. Baker2, Fangyu Li3, Gary V. Stephenson4,
Eric W. Davis5, Andrew W. Beckwith2,3
1Department of Electrical and Computer Engineering, Louisiana State University,
Baton Rouge, USA
2GravWave® LLC and Transportation Sciences Corporation,
Playa del Rey, USA
3Department of Physics, Chongqing University, Chongqing,
Peoples Republic of China
4Seculine Consulting, Redondo Beach, USA
5Institute for Advanced Studies at Austin, Austin, USA
Received March 3, 2011; revised April 17, 2011; accepted April 27, 2011
Under most models of the early universe evolution, high-frequency gravitational waves (HFGWs) were pro-
duced. They are referred to as “relic” high-frequency gravitational waves or HFRGWs and their detection
and measurement could provide important information on the origin and development of our Uni-
verse—information that could not otherwise be obtained. So far three instruments have been built to detect
and measure HFRGWs, but so far none of them has achieved the required sensitivity. This paper concerns
another detector, originally proposed by Baker in 2000 and patented, which is based upon a recently discov-
ered physical effect (the Li effect); this detector has accordingly been named the “Li-Baker detector”. The
detector has been a joint development effort by the P. R. China and the United States HFGW research teams.
A rigorous examination of the detector’s performance is important in the ongoing debate over the value of
attempting to construct a Li-Baker detector and, in particular, an accurate prediction of its sensitivity in the
presence of significant noise will decide whether the Li-Baker detector will be capable of detecting and
measuring HFRGWs. The potential for useful HFRGW measurement is theoretically confirmed.
Keywords: High-Frequency Gravitational Waves, High-Frequency Relic Gravitational Wave Measurement,
Primordial Gravitational Waves, Microwaves, Cosmology, General Relativity
1. Introduction
Thus far three instruments have been built to detect and
measure high-frequency gravitational waves or HFGWs
[1-6], but so far none of them has achieved the required
sensitivity to detect and measure “relic” HFGWs from
the early universe or HFRGWs. This paper concerns an-
other detector, originally proposed by Baker in 2000 and
patented [7], which is based upon a recently discovered
physical effect by Li [8]; this detector has accordingly
been named the “Li-Baker detector.” The detector has
been a joint development effort by the P. R. China and
the United States HFGW research teams.
Subsequent to the 2008 publication in EPJC, it has
been generally accepted that a perturbative photon flux
(PPF) is generated by high-frequency gravitational
waves (HFGWs) in the presence of a static magnetic
field and an electromagnetic (EM) beam having the same
frequency and a suitable phase difference (synchroreso-
nance) superimposed on the HFGW [9]. This effect is
discussed in Section 3 and its proof summarized in the
Appendix. In order to utilize this new theoretical tech-
nique for detection of cosmologically generated HFRGWs
two important questions arise: 1) What are the character-
istics and cosmological significance of the relic high-
frequency gravitational waves from the early universe
and 2) what is the adverse effect of the noise generated
by the relatively intense EM beam on the ability to detect
Copyright © 2011 SciRes. JMP
and measure the very weak PPF signal that they produce?
An objective of this paper is to answer these questions.
2. High-Frequency Relic Gravitational
2.1. Cosmological Significance of the Relic
High-Frequency Gravitational Waves
from the Early Universe
Although there is no definite evidence of the existence of
HFRGWs their measurement would allow for validation
or falsification of various cosmological theories. Most
models of early universe evolution predict that HFGWs
were produced as a result of the violent expansion of the
young universe. and their measurement and characteriza-
tion could provide important information on the origin
and development of our Universe since their properties
were uniquely determined by the most violent event in
the history of the Universe. This information is a vital
piece in the jigsaw of understanding how the young uni-
verse evolved, and that information cannot be obtained
by any other means. A number of techniques have been
proposed for measuring relic GWs at both low frequency
and high frequency, and some GW detectors have been
built, so far without any success in detection. Like the
Laser Interferometer Space Antenna (LISA) [10], the
swarm of Cosmic Microwave Background (CMB) sen-
sors [11] and the Russian gravitational-electromagnetic
resonance high-frequency gravitational wave detector
[12,13] (all proposed for sensing primordial or relic
gravitational waves), no Li-Baker detector has yet been
As is well known, Einstein [14] predicted the possibil-
ity of waves in four-dimensional spacetime, i.e., the
usual three dimensions of space plus time. These waves
are gravitational waves whose spacetime strain is h. This
spacetime strain is analogous to mechanical strain in a
beam, and is the ratio of the change in length to the
original length (without the stress of a passing gravita-
tional wave). Thus, the strain, h, has units of meters per
meter (m/m) and is dimensionless as is its amplitude A.
The importance of measuring the HFRGW strain h
and dimensionless energy density gw is that predictions
of their values produced by the “Big Bang” under infla-
tionary universe models [15-20] and cosmological string
scenarios [21-23] are available, and so direct measure-
ment will allow discrimination between the various
models. Many of these models predict maximum
HFRGW amplitude around 10GHz, with h in the ap-
proximate range 10–30 to ~ 10–34. Low-frequency gravita-
tional wave detectors such as LIGO, based upon optical
interferometers, have an optimal detection frequency
~100 Hz with upper frequency detection limit of ~2000
Hz, and accordingly cannot detect HFRGWs [24]. In
order to detect and measure high frequencies at small
amplitudes, detectors utilizing different techniques must
be employed, complementary to the low frequency de-
tectors. Krauss, Scott and Meyer [11] suggest “… pri-
mordial (relic) gravitational waves also leave indirect
signatures that might show up in CMB (Cosmic Wave
Background) maps.” They propose the use of thousands
of new detectors (possibly as many as 50,000) as well as
spacecraft-borne detectors to obtain the required sensi-
Theorized cosmological signatures (i.e., frequency
spread, polarization and phase) of the HFRGWs are im-
portant because of the uncertainty surrounding cosmo-
logical parameters leading to variations in the early uni-
verse [25]. One of the most important parameters for
analysis of the beginning of the Universe is the dimen-
sionless relic gravitational wave energy density, gw [19,
26-28]. According to these estimates, the upper limit of
gw for relic GWs should be smaller than 10–5. In fact,
recent estimates [10] show that the upper limit of gw
should be 6.9 × 10–6 at about 100 Hz. The spectra of di-
mensionless primordial relic GW strains h as a function
of frequency have been estimated in detail by Grishchuk
[19,26-28]. Detailed observational data for h and its
variation in time and direction can be used to refine the
estimated value of gw, and hence to differentiate among
the competing cosmological theories for the beginning of
the Universe.
2.2. Presently Fabricated and Proposed HFGW
As has been mentioned three high-frequency gravita-
tional wave (HFGW) detectors have been built [29] and
another has been proposed [12,13], all utilizing different
measurement techniques. These are promising for future
detection of HFRGWs having frequencies above 100
kHz (the definition of HFGWs adopted by Douglass and
Braginsky [30]), but their sensitivities are each many
orders of magnitude less than that required to detect and
measure primordial HFRGWs. These existing HFGW
detectors deal with the detection of a single plane
HFRGW, but a stochastic background needs at least two
detectors to be utilized for cross correlating the observed
data. It is intended, therefore, that the Li-Baker detectors
be utilized in pairs, which would also guard against false
alarms. The Li-Baker detector is direction dependent and
a background can be seen as a stochastic superposition of
many plane waves propagating in all directions. The di-
rectionality is discussed in Section 4.4.
The first of these HFGW detectors has been con-
Copyright © 2011 SciRes. JMP
structed at Birmingham University, England. The Bir-
mingham HFGW detector measures changes in the po-
larization state of a microwave beam (indicating the
presence of a GW) moving in a waveguide [1,2]. It is
expected to be sensitive to HFRGWs having spacetime
strains of h ~2 × 10–13.
The second detector, built at INFN Genoa, Italy. It is a
resonant HFRGW detector, comprising two coupled,
superconducting, spherical, resonant chambers a few
centimeters in diameter and configured as oscillators.
The oscillators are designed to have (when uncoupled)
almost equal resonant frequencies and when the fre-
quency of the HFGW is just equal to the frequency dif-
ference between the normal modes in the two coupled
spherical cavities, the EM energy conversion between
the cavities will be maximum and the HFGW sensed.
The system is expected to have a sensitivity to HFRGWs
of about h ~ 2 × 10–17 with future expectation of ~ 3 ×
10–20 [3-5] and “… system sensitivity could be increased
by several orders of magnitude.” However, there is no
further planned development of the INFN Genoa
HFRGW detector.
The third detector is the Kawamura 100 MHz
HFRGW detector that has been built by the Astronomi-
cal Observatory of Japan. It comprises two synchronous
interferometers having arms lengths of 75 cm. Minia-
turization of the detector to 10GHz wavelengths would
be 100:1.Its sensitivity is h 10–20, projected to improve
to ~10–23 for a 1000 W laser [6]. It appears that due to the
small size of a miniaturized instrument and the lack of
enough photons in the sampling period to perform accu-
rate interferometry, it will be difficult for this design to
operate satisfactorily at 10GHz with their projected sen-
Another HFGW detector, under development at the
Steinberg Astronomical Institute in Russia [12,13] de-
tects gravitational waves by their action on an electro-
magnetic wave in a closed waveguide or resonator.
2.3. New Measurement Technique
An objective of this paper is to present the rationale be-
hind a proposed and planned HFRGW detector utilizing
a new measurement technique termed the “Li effect.”
This theory was first published in 1992 [8]. Subsequently,
the Li effect has been developed further in nine later
peer-reviewed research papers [9,32-39] and is scruti-
nized by Valentine Rudenko and Nikolai Kolosnitsyn of
the Sternberg Astronomical Institute of Moscow State
University [40]. The key results are summarized in ref.
[9] and a detailed discussion of the detection mechanism
is given in ref. [39] and presented in compact form in the
Appendix. Often the Li effect is identified as a three-
dimensional synchro-resonance electromagnetic coupling
effect or 3D SR.
As mentioned in the Introduction this new detection
technique is based upon coupling between an HFGW, a
Gaussian-type microwave photon beam (having the same
frequency, direction and suitable phase as the HFRGW
being detected), and a static magnetic field. The result of
this coupling is a flux of detection photons or perturbat-
ive photon flux (PPF), and reflectors would typically be
used to direct the PPF towards sensitive microwave re-
ceivers [41]. First estimates of the Li-Baker detector’s
sensitivity in the microwave band have been similar to
those needed for detection of primordial HFRGW [9,20,
27,39]. There are, however, operational concerns such as
fundamental noise sources that must be examined.
Sources of noise in this detector include: background
photon noise from the highly energetic Gaussian micro-
wave beam (GB) including scattering, diffraction, ther-
mal noise from the detector’s containment vessel,
dark-background shot noise, Johnson noise in the mi-
crowave receivers, preamplifier noise, and quantization
In the Li-Baker-detector the key parameter is the
first-order detection photons (proportional to GW strain
amplitude A), or perturbative photon flux (PPF), and not
the second-order PPF (proportional to A2). The first-order
PPF, or the flux of detection photons produced by the
Li-effect interaction with the GWs, is therefore propor-
tional to √Ωgw and not gw. The spectra predicted by the
pre-big-bang models (Figure 2 of [10]) shows that gw
of relic GWs is almost constant at 6.9 × 10–6 in the fre-
quency range = 10 Hz to 10 GHz. Cosmic string models
predict gw ~10–8 in the range 1 Hz to 10 GHz; its peak
value is at about 10–7 to 10–6 Hz, in the low-frequency
regions—much lower than HFGW frequencies. Also, it
is shown [10] that only the Advanced LIGO may achieve
the requisite sensitivity for relic GWs predicted by the
pre-big-bang model in the frequency band around ~100
Hz; the present LIGO cannot detect relic GWs in that
region. However, the Li-Baker detector could make ob-
servations of h at around 10 GHz and, unlike the current
Low-Frequency Relic Gravitational Wave (LFRGW)
detectors, could be sensitive enough to measure relic
gravitational waves. Furthermore, with the dimensionless
cosmological Hubble parameter n = 1.0 and 1.2, there are
sharp peaks of gw at 10 GHz [42] as shown in Figure 1.
Grishchuk’s analyses that define these peaks are too
lengthy to be included here, but can be found in Refs.
[19,26-29]. A frequency scan, discussed in Section 4.5,
could reveal other HFRGW effects of interest in the early
universe at a variety of HFRGW base frequencies other
than 10 GHz.
Copyright © 2011 SciRes. JMP
Figure 1. Predicted relic gravitational wave energy density as a function of frequency (slide 6, [42]).
3. Electromagnetic and Gravitational Wave
Interaction: The Gertsenshtein and Li
The Li Effect is very different from the well-known clas-
sical (inverse) Gertsenshtein effect [43], in which a GW
travelling in a region in which there is a uniform constant
applied magnetic field will produce a coupled electro-
magnetic (EM) wave having exactly the same frequency
and wave-vector as the incoming GW. The coupled EM
wave will exhibit a flux proportional to A2. By contrast,
in the Li effect, an electromagnetic (EM) traveling wave
of a Gaussian beam (GB) in the presence of a perpen-
dicular static magnetic field is found to interact with an
incoming GW having exactly the same frequency and
wave-vector (including the direction of propagation) as
those of the EM wave and exhibit a flux proportional to
A. This is known as the “synchro-resonance condition,”
which may typically be satisfied by one Fourier compo-
nent of a continuous spectrum of incoming GWs. This
interaction produces a resultant second EM wave of the
same frequency as the EM and GW waves, but propa-
gating perpendicular to both the applied uniform mag-
netic field and to the applied EM wave, as shown in
Figure 2. It is unlike the (inverse) Gertsenshtein effect,
in which the resultant EM wave is parallel to (rather than
perpendicular to) the incoming GW, and in which there
is no applied EM wave used to synchronized to the in-
coming GW.
There are two proven features of the Li effect (proof
contained in the ten peer-reviewed references already
cited [8,9,32-39], each covering a different aspect of the
Li effect, and summarized in Appendix A). One is that
the gravitational wave although transverse interacts with
a Gaussian beam (GB) and the PPF do not travel in the
same direction as the incoming gravitational wave. The
second is that, regardless of direction, the PPF is a very
low-impedance wave of order A (e.g., for a HFGW of
frequency 2.9 GHz and amplitude A ~ 10–30, the imped-
ance of PPF is about 4.1 × 10–11 ohms). By the way, the
impedance of free space is 377 ohms and the impedance
of copper for an EM wave of frequency 30 GHz is 0.02
ohms. In other words, free space looks like a very good
“superconductor” to the PPF.
The perpendicular propagation direction of the PPF
exhibited in Figure 2 is a fundamental physical require-
ment; otherwise the EM fields will not satisfy the Helm-
holtz equation, the electrodynamics equation in curved
spacetime, the non-divergence condition in free space,
and the laws of energy-conservation as discussed in the
Appendix and in [35]. A significant feature of the
Li-effect is that the PPF move both outward away from
the GB’s axis and in ward toward the GB’s axis. Thus
reflectors in the GB itself could reflect and focus a por-
tion of the PPF to microwave receivers in regions of the
detector proper that are relatively noise free. The BPF
noise, whatever its source (except for scattering as dis-
cussed in Section 5.1), mainly propagates radially out
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Figure 2. Li effect PPF-directed to the ends of the x-axis.
from the GB’s axis and is not focused to the microwave
4. Description of Li-Baker HFGW Detector
and Its Physical Parameters
4.1. Gaussian Beam
A traveling-wave Gaussian microwave beam (GB) is
used as the applied EM wave required in the Li effect. In
the Appendix is a more complete description of the GB.
It is to be produced by a conventional microwave trans-
mitter with its antenna aimed along the +z-axis of Figure
2. Its frequency and direction are the same as the fre-
quency and direction of the incoming HFGW signal that
is to be detected [44] as shown in Figure 3. The GB fre-
quency is expected to be typically around 10 GHz for
GB directed along the +z-axis will allow detection of a
HFRGW also directed along the +z-axis.
In order to reduce the thermal load on the refrigeration
system the microwave transmitter and main GB micro-
wave absorber are in separate chambers sealed off from
the main detector chamber by microwave transparent
walls. A high-vacuum system able to evacuate the
chamber from 10–6 to 10–11 Torr (nominally about 7.5 ×
10–7 Torr) is needed to allow cryogenic operation and to
reduce thermal noise (see Section 5.5).
4.2. Magnetic Field and Sensitivity
A static magnetic field B (generated typically using one
or more superconductor magnets such as those found in a
conventional MRI medical body scanner) is directed
along the y-axis, as shown schematically in Figure 4.
Rather than using one pair as shown schematically in
Figure 4, it may be more cost-effective to use a number
of magnet pairs spaced equally in the z-direction. The
intersection of the magnetic field and the GB defines the
“interaction volume” where the PPF is produced and
move out in both x directions on both sides of the
y-z-plane (as in Figure 2. A of the Appendix). The in-
teraction volume in the GB for the proposed or nominal
design is roughly cylindrical in shape, about 30 cm in
length and about 9 cm cross-section diameter. In order to
estimate the detection signal, or the number of detection
photons (PPF) produced per second for a given ampli-
tude HFGW, we will utilize Equation (7) of the analyses
in [46], which is a simplification of Equation (59) in [34]
for a near-field approximation as discussed in [46],
n is the number of x-directed detection photons
Copyright © 2011 SciRes. JMP
Figure 3. Gaussian-beam transmitter compartment.
Figure 4. Schematic of the Li-Baker HFGW detector.
Copyright © 2011 SciRes. JMP
per second produced in the interaction volume, ћ =
Planck’s reduced constant,
e = angular frequency of the
EM wave (= 2πνe), νe = frequency of the EM wave, A =
the HFGW amplitude, By = y-component of the magnetic
field, ψ0 = electrical field of the EM Gaussian beam and
δs is the cross-sectional area of the interaction volume
perpendicular to the PPF. For the proposed nominal de-
sign, the minimum cross-section diameter or waist of the
GB is located about 20 cm away from the antenna; the
radius of the GB at its waist, W, is (λez/π)1/2 = 4.4 cm at
10GHz, so that its diameter is 8.8 cm (approximately
the width of the interaction volume); and the length of
the interaction volume is l = 30 cm, so δs = 2Wl = 2.58 ×
10–2 m
2. From the analysis presented in ref. [32], the
electrical field of the EM GB, ψ, is 1.26 × 104 Vm–1 for
transmitter power 1kW. For the present proposed design,
νe = 1010 s–1, ωe = 6.28 × 1010 rad/s, A = 10-30, and By =
16 T. Thus (1) gives
n = 99.2 PPF detection photons
per second. For a 103 second observation accumulation
time interval, there would be about 105 detection photons
created (the PPF). About one-fourth of them would be
focused at each receiver, since half would be directed
towards +x and half directed towards –x on each side of
the focusing reflectors in the y-z plane (only the half of
the photons directed toward the reflectors is focused to
the microwave receivers the other half is directed away
from the reflectors and unfocused and does not reach the
receivers). Table 1 provides values for an interaction
volume cross section of δs = 0.1 m × 0.05 m = 0.005 m2
(a very small detector), Table 2 is for δs = 0.30 m ×
0.088 m = 0.0258 m2 (the proposed or nominal design)
and Table 3 is for δs = 6 m × 0.5 m = 1.5 m2 (a large
detector design). Table 3 is valid under the assumption
that the near–field approximation of (1) still holds and
account is taken of the spreading property of the GB. If
the interaction volume is very large in one direction, for
example much greater than 1m, then the computation of
the total PPF could be somewhat more accurately ob-
tained by a numerical integration of Equation (59) of
[34], specifically, the numerical integration of the coeffi-
cients in equations (60) of [34]. In such a case the
evacuation pressure would also need to be somewhat
lower in order to increase the GB photon mean free path
and minimize GB photon scattering (see Section. 5.1).
Such a refinement is not judged to be necessary so the
approximation of Equation (1) was utilized in Table 3.
The proposed design or nominal case selected was a
subjective judgment of the coauthors based upon their
collective knowledge and experience with the cost, com-
plexity of the fabrication of novel laboratory equipment
and the availability of superconductor magnets and mi-
crowave transmitters and receivers.
It is again to be emphasized that unlike the Gertsen-
shtein effect, the Li effect produces a first-order PPF
whose amplitude is proportional to the incoming gravita-
tional wave (GW) amplitude A as in Equation (1) (and is
not a second-order effect proportional to A2). In the in-
verse Gertsenshtein effect, the EM wave produced is a
second-order effect; from Equation (7) in [46], the num-
ber of EM photons produced in the inverse Gertsenshtein
effect is “… proportional to the amplitude squared of the
relic HFGWs, A2,” and it would be necessary to accu-
mulate such EM photons for at least 1.4 × 1016 seconds
or 444 million years in order to achieve HFRGW detec-
tion utilizing the inverse Gertsenshtein effect as com-
puted in [34]. Since in the Li effect the number of EM
photons is proportional to the amplitude of the relic
HFGWs, which is typically A 10–30, not its square, so
that it would be necessary to accumulate such EM pho-
tons for at most about 102 to 105 seconds in order to
achieve relic HFGW detection as computed in [34]. The
JASON report [45] confuses the two effects and errone-
ously suggests that the Li-Baker HFGW detector utilizes
the inverse Gertsenshtein effect. The Li-Baker HFGW
detector does not utilize the inverse Gertsenshtein effect,
and it has a theoretical sensitivity that is about A/A2 =
1030 greater than the value incorrectly reported in the
JASON report [45] for HFRGWs.
Table 1. PPF (photons per second) for various values of By
and transmitter power for δs = 0.005 m2.
Power = 100 WPower = 1000 W Power = 10,000 W
By = 9 T3.4 10.8 34.2
By = 16 T6.1 19.2 60.8
By = 20 T7.6 24 76
Table 2. PPF (photons per second) for various values of By
and transmitter power for δs = 0.0258 m2. The design or
nominal case.
Power = 100 WPower = 1000 W Power = 10,000 W
By = 9 T17.6 55.8 176.4
By = 16 T31.4 99.2 313.7
By = 20 T39.2 124 392
Table 3. PPF (photons per second) for various values of By
and transmitter power for δs = 1.5 m2.
Power = 100 WPower = 1000 W Power = 10,000 W
By = 9 T1.023 × 103 3.2 × 103 1.026 × 104
By = 16 T1.83 × 103 5.8 × 103 1.82 × 104
By = 20 T2.3 × 103 7.2 × 103 2.3 × 104
Copyright © 2011 SciRes. JMP
For an advanced Li-Baker detector [37], also included
would be a resonance chamber (Q ~ 103) in the interac-
tion volume and more sensitive microwave receivers so
that the sensitivity could be further improved. These re-
finements will be considered elsewhere.
4.3. Microwave Reflectors
Semi-paraboloid reflectors are situated back-to-back in
the y-z plane of the GB, as shown in Figures 5 and 6, to
reflect the +x and –x propagating PPF to the microwave
receivers. The effective aperture of each reflector is 60
cm and the sagitta or depth of curvature of such a mirror
is about 2.26 cm. Since this is greater than one tenth of a
wavelength of the PPF, λe/10 = 0.3 cm, such a paraboloid
reflector is desirable rather than only a tilted plane mirror.
As discussed in Section. 5, for elimination of any dif-
fracted photons emanating from the GB’s entrance to the
main detector chamber at B – B' of Figure 5, the reflec-
tor’s focus is below the x axis and “out of sight” of the
GB’s entrance. Thus the diffracted photons waves from
the GB entrance will have at least one reflection from the
absorbent detector walls prior to reaching the microwave
receivers. As will be calculated in Section. 5 other radia-
tion from the GB due to scattering and the natural fall off
of GB radiation in the radial direction is negligible, so
that the BPF is only due to diffraction from the transmit-
ter’s antenna, aperture or entrance to the main detector
chamber. This is why the paraboloid mirrors are slightly
tilted, which allows the focus to be slightly below the x-y
plane (similar to a Herschelian optical telescope) so that
there is no direct straight line between the microwave
receivers and the transmitting antenna. Since such a re-
flector would extend out 2.26 cm into the GB (on both
sides of y-z plane or 4.5 cm in total), a half or
semi-paraboloid mirror is used instead in order not to
block the Gaussian beam significantly. In the nominal
case the reflectors are about 30 cm high (along the z-axis)
and 9 cm wide (along the y-axis) and extend from z = 0
cm to z = +30 cm as shown in the figures. The reflec-
tors can be installed inside the GB in order that the dif-
fracted BPF from the GB transmitter’s entrance to the
detector chamber at B – B' and any diffraction perpen-
dicular to the GB will not be directly focused onto the
receivers. The only photons reflected or focused onto the
microwave receivers will be the ± x-directed PPF pho-
tons in the GB that are directed toward the GB’s center
(there could be several microwave receivers stacked at
each end of the x-axis to in increase the field of view and
account for any variations in the magnetic field from
uniform straight lines). The semi-paraboloid reflectors
are tilted “down” at about 12(3 cm/ 100 cm) = 0.015
radians (about 0.86˚) or more (in order to focus at re-
ceivers 100 cm distant and 3 cm below the base of the
GB) and extend from a sharp edge at point A at the
center of the GB, which is totally shielded from the re-
Figure 5. Side-view schematic of the Li-Baker HFGW detector, showing microwave-absorbent walls in the anechoic chamber
and, if not totally absorbed, also showing the paths of reflected, diffracted photons.
Copyright © 2011 SciRes. JMP
Figure 6. Plan-view schematic of the Li-Baker HFGW detector, exhibiting microwave-absorbent walls in the containment
vessel and the reflectors extending out on either side of the x-axis along y with edges completely shielded from the receivers.
ceivers, as shown in Figure 5. Thus there will be very
little blockage of the GB. The reflectors can be con-
structed of almost any material that is non-magnetic (to
avoid being affected by the intense magnetic field), re-
flects microwaves well and will not outgas in a high
vacuum. The material of the reflectors can be in the form
of fractal membranes that reflect more than 99% of the
incident microwaves (experimental data from figure 1c
of [47]). Apparently the fractal membranes (which con-
sist of printed microcircuits) produce little diffraction in
the presence of the GB and in the base frequency range
pass all the remainder radiation through the fractal mem-
branes [48]. Alternatively, microwave focusing lenses
can be placed outside of the GB on either side [41] as
shown in Figure 7.
4.4. Microwave Receivers and Directionality
High-sensitivity, shielded microwave receivers are lo-
cated at each reflector focal point. Possible receiver
technologies to use include a microwave horn plus re-
ceiver; a Rydberg Atom Cavity Detector [49]; a quantum
electronics device (QED) microwave receiver, such as
the Yale detector invented by Schoelkopf and Girvin
[50], and a single-photon detector [51]. Such single-
photon microwave receivers or detectors can be refriger-
ated sufficiently to be unaffected by thermal-photon
background or self noise. Of these receivers the micro-
wave horn plus receiver is most likely for initial trials
because of its off-the-shelf availability from many sup-
pliers. The synchro-resonant condition specifies that the
GW detected has the same frequency and propagation
direction as the GB. In order to achieve a larger field of
view and account for any curvature in the magnetic field,
an array of microwave receivers having, for example,
nine 3 cm 3 cm horns could be installed-parallel to the
y-z plane and 9 cm below the GB’s base. Their field of
view or directionality would be 9 cm/100 cm = 0.09 ra-
dians or approximately five degrees.
4.5. Bandwidth
The “detected bandwidth” (BW) is determined by two
random fluctuations in the GB transmitter output
causing BW broadening, and
the bandwidth of the microwave receivers. In general,
the narrower the frequency range or bandwidth is, the
more sensitive is the detector (the noise floor is low-
ered at smaller BW).
However, frequency scanning allows a wide band of
HFRGWs to be analyzed. As an example, in a 1 Hz
“bandwidth” and a 1000 s observation interval, then over
a year of observation about 30 kHz HFRGW frequency
band could be scanned. Essentially one would sample the
detected BW by a number of very narrow actual band-
widths, Bw. If the observation interval is 1000 s, then the
actual Bw is 0.001 Hz. Or, for 100 s observation interval,
then a 300 kHz band of HFRGWs could be scanned. For
a 1 kHz BW, then a 0.3 GHz band could be scanned us-
ing 100s intervals over a year, and this would be a sub-
stantial BW if centered on 10GHz base frequency.
5. Noise
The Li-Baker HFGW detector contains a huge number of
already existing EM quanta (~1026 photons per second)
in the intense GB. In addition to that there are noise
sources in the detector that are similar to those encoun-
tered in any microwave receiver and may be analyzed in
Copyright © 2011 SciRes. JMP
Containment vessel includes the anechoic chamber
and microwave-absorbent walls
Vacuum/Cr yog en ic
Containment Vessel
Microwave Receiver
- Detector #2
Microwave Receiver
- Detector #1
Gaussian Beam (GB)
Microwave Lens on Each Side of GB
Signal PPF
N magnetic pole
S magnetic pole
HFRGWs pole
Signal PPF
Microwave lenses can be fractal
membranes or metamaterial.
They are not in the GB
Figure 7. Schematic of the Li-Baker HFGW detector, exhibiting microwave lenses on each side of the GB focusing PPF on the
microwave receivers.
similar fashion. The sensitivity of the detector depends
critically on the noise at the microwave receivers. The
background noise discussed in [35,39] did not consider
the scattering or diffraction of the strong GB. However it
will be shown that classical analyses of such scattering
and diffraction [53-57] indicates that their effect will be
negligible at the sensitive microwave receivers. In the
Li-Baker detector the HFRGW signal manifests itself as
detection photons (PPF) created by the interaction of a
microwave beam (GB) and the GWs. The presence of the
microwave beam having the same frequency as the de-
tection photons gives rise to background photon flux
(BPF) that produces dark-background shot noise such as
scattering and diffraction, in addition to the usual mi-
crowave receiver noise. For example, Johnson noise
originates thermally in any electrical resistor, and is often
dominated by the contribution of the most significant
resistance in the receiver input stage. In order to account
for all these diverse noise sources, here they are trans-
lated through the detector to the actual microwave re-
ceiver(s) and usually termed noise equivalent power or
NEP [52]. Photon noise from the GB will be considered
in detail since it is likely to be the dominant source of
noise in the Li-Baker detector.
5.1. Noise Generated by the GB
The intensity of the GB is written (Equation (3) of [36])
and is:
~exp 2
nrW (2)
where r is the radial distance out from the GB’s axis and
W is the radius of the GB at its waist. The transverse BPF
in any longitudinal symmetrical surface of the GB must
vanish. Even if we treat a non-idealized situation, there
are always the special local regions in which the trans-
verse BPF vanish. If the transverse BPF in any longitu-
dinal symmetrical surface of the GB is not vanishing,
then the photon number at the symmetrical surface will
be continuously accumulated (increased) with time in
“the imploding wave” region of the GB and continuously
reduced (decreased) with time in the “outgoing wave”
region of the GB. Thus the “stability” of the GB would
be destroyed (see Figures 2 and 4 on p. 414 of [9]). In
the prototype Li-Baker HFRGW detector under analysis,
which has peak sensitivity (base frequency) at 10 GHz,
the energy per detection photon is ћ νe = 6.626 × 10–24 J,
while the HFRGWs or the GB both have the same fre-
quency for synchro-resonance. So a 103 W GB flux is
1.51 × 1026 photons/s. For 100-cm-distant microwave
receivers, the GB intensity in the z-direction, if the clas-
sical Equation (2) is accurate at such large attenuations,
is reduced to exp (–2 × 1002/4.4 2) (1.51 × 1026), which is
essentially zero.
With regard to molecular scattering in the GB, we
utilize the Rayleigh scattered intensity of microwave
Copyright © 2011 SciRes. JMP
photons, I, from a molecule with incident photon inten-
sity Io as given by [53]
in which
is the atomic polarizability expressed as a
polarization volume (where the induced electric dipole
moment of the molecule is given by 4
is the
scattering angle, and R is the distance from particle to
detector. Note that the scattering is not isotropic (there is
-dependence), but in the present case,
= 90˚ so the
ratio of incident to scattered photon intensity is given by
. The polarizability is
1.1 × 10–30 m3 from [54]
so the scattering intensity ratio is 1.2 × 10–49 for each
atom in the chamber. The nominal volume of interaction
is about 2000 cm3 (30 cm long and roughly 8 cm 8 cm
in area) so at a pressure reduced to its convenient nomi-
nal value of 7.5 10–7 Torr at temperature 480 mK, the
number of molecules contained is about 3 1016, giving
a total scattering intensity ratio of 3.49 10–33. There are
1.51 1026 photons produced per second in the 103 W,
10 GHz GB nominal case. Therefore, in 103 s of obser-
vation time, the estimated number of photons received
from Rayleigh scattering in the interaction volume is
(3.49 10–33) (1.51 1026) (1000) = 5.310–4 and will be
5.2. Noise Generated by Diffraction
Diffraction can potentially produce x-directed photons
from a z-directed wave such as the GB in the absence of
any GW interactions. This is potentially a problem for
the Li-Baker detector design because the diffracted sig-
nal may either swamp the microwave receivers or else
will represent a significant extraneous source of shot
noise. Therefore, all sources of diffraction should be
eliminated or at least minimized [55,56]. For example,
the corners at B and B' of Figure 5 should have radii of
curvature in excess of two wavelengths (6 cm) and all
small obstructions and corners should have radii greater
than three wavelengths (e.g., 9 cm) and the only edge of
the focusing reflector at A will have its diffracted waves
absorbed prior to reaching the receivers. In spite of this,
there will be some microwave diffraction photon noise
that will need to be reduced before reaching the receivers.
Since there is no direct path perpendicular to the GB to
the microwave receivers in the x direction or from the
edges of the reflectors, due to the Herschelian optical
telescope design, all x-directed photons (moving perpen-
dicular to the axis of the GB as computed in [57]) and all
diffracted photons from the reflector edge will necessar-
ily encounter a wall of the detection chamber before
reaching the receivers.
The number of diffraction photons emanating radially
from the GB, including the effect of polarization align-
ment, is given by Equation (13) of [57] where diffraction
has been analyzed specifically for the Li-Baker detector.
The reader is encouraged to review this important analy-
sis since GB diffraction might be considered to be a ma-
jor source of noise. The diffraction photons per second
from the GB is
232 exp20.01
dif dGB
nkd Lkdn
where k = 2πνe/c (nominally, 209 rad/m at 10 GHz), c
being the speed of light, the diameter of the GB throat is
d (~ 0.09 m for the nominal case, essentially 2W), Ld is
the distance of a receiver from the GB (~ 1 m for the
nominal case) and nGB is the GB photon flux (nominally,
1.51 × 1026 photons per second). The ndif decreases
greatly with larger GB throat diameter. For example, for
d = 0.06 m, ndif = 5.4 1015, d = 0.09 m. ndif = 2.6 105,
but for d = 0.12 m, ndif = 5.3 10–10 diffraction photon
per second. In this same regard please see Figure 4 of
[57]. We will assume a single bounce or wave reflection
of this diffraction-noise wave from the detector walls.
The diffraction photon-path distance, prior to reaching
the receivers, is Ld (~1 m for the nominal case). The
number of diffraction photons, ndif, moving radially will
be almost evenly spread out on an area of a band of a
cylinder the width of which is the length of the GB, l
(~0.3 m for the nominal case), having a spread of πLd.
Thus the number of diffracted noise photon impinging on
each receiver per second, nrdif is given by
rdifdif rdab
where εab is the wall absorption coefficient (e.g., for the
nominal case to be discussed below, it would be 10–22)
and ar = area of the square receiver horn or receiving
surface (nominally, one HFRGW wavelength square or
9 × 10–4 m2).
5.3. Absorbing Walls
The chamber wall absorbers are of two types: metamate-
rial or MM absorbers, which have no reflection, only
transmission [58] at the base frequency and the usual
commercially available absorbers in which there is re-
flection, but no transmission. In theory, multiple layers
of metamaterials could result in a near “perfect” absorber
(two MM layers absorbs noise to 99.9972% or 45.5 dB
over their specific base frequency range 5 to10 GHz,
according to the experimental data of Landy, et al. (page
3 of [58]). An absorbent “mat” combination of MMs
Copyright © 2011 SciRes. JMP
(sketched as blue lines in Figures 3, 5 and 6) backed up
by commercially available microwave absorbers is shown
in Figure 8. As Landy, et al. [58] state in Physical Re-
view Letters: “In this study, we are interested in achiev-
ing (absorption) in a single unit cell in the propagation
direction. Thus, our MM structure was optimized to
maximize the [absorbance] with the restriction of mini-
mizing the thickness. If this constraint is relaxed, im-
pedance matching is possible, and with multiple layers, a
perfect [absorbance] can be achieved.” We analyze an
absorption mat (Patent Pending) consisting of two double
MM layers, each double layer having a 45 dB absorp-
tion. Behind the MM layers is a sheet of 10 GHz tuned
microwave pyramid absorbers, providing 40 dB ab-
sorption (guaranteed) before reflection back into the MM
layers. Thus the total absorption is 45 45 40 45 45 =
220 dB or an absorption coefficient of 10–22 for the two
double MM layers. There are several commercially
available pyramid microwave absorbers available that
offer the required low reflectivity, such as ARC Tech-
nologies, Cummings Microwave and the ETS Lindgren
Rantec microwave absorbers. The ETS Lindgren EHP-
5PCL absorbing pyramids seem like a good choice. At
normal incidence the typical reflectivity is down 45 dB
(guaranteed 40 dB). It is also important to note that the
incident ray can have almost any inclination. As Service
writes in his article published in SCIENCE [59] “…
Sandia Laboratories in Albuquerque, New Mexico are
developing a technique to produce metamaterials that
work with [electromagnetic radiation] coming from vir-
tually any direction.” A substrate of conventional elec-
tromagnetic-radiation absorbing material, such as an ar-
ray of pyramidal foam absorbers, may outgas into a high
vacuum and reduce the capability of the vacuum-pro-
ducing equipment to achieve a hard vacuum. The layers
above the substrate will also serve to seal the absorbing
and reflecting substrate from the external vacuum and,
therefore, not reduce the capability of the detector’s
vacuum-producing equipment. The surface and voids of
the pyramidal acoustical tiles substrate, upon which the
MM layers are deposited, is formed from dielectric ma-
terial to fill those voids as well as to fill the voids be-
tween the various MM layers. Thus the absorbing pyra-
mids will be sealed from evaporation, sublimation or out
gassing of the material composing them.
5.4. Parametric Analysis
In Table s 4-6 are to be found a parametric analysis of the
diffraction photons per second (based upon classical dif-
fraction equations) and noise equivalent power (NEP) for
three alternative configurations of detector-wall absor-
bent mats, for GB aperture diameters, d, of 2, 3 (nominal)
Figure 8. Schematic of typical multilayer metamaterial (two
in the nominal case shown) or MM absorbers and pyramid
absorber/r e flector. Patent Pend i ng.
Table 4. Diffraction photons s–1 and NEP W for a mat
composed of absorbent microwave pyramids only, exhibit-
ing an absorption of –40 dB.
GB aperture
diameter Ld = 0.5 m Ld = 1.0 m
(nominal) Ld = 2.0 m
d = 6 cm 3.4 × 106 s–1,
2.3 × 10–17 W
4.3 × 105 s–1,
3 × 10–18 W
5.3 × 104 s–1,
4 × 10–19 W
d = 9 cm
3.3 × 10–4 s–1,
2.2 × 10–27 W
4.2 × 10–5 s–1,
2.8 × 10–28 W
5.2 × 10–6 s–1,
3.5 × 10–29 W
d = 12 cm 1 × 10–18 s–1,
7 × 10–42 W
1.3 × 10–19 s–1,
8.7 × 10–43 W
1.6 × 10–20 s–1,
1 × 10–43 W
Table 5. Diffraction photons s–1 and NEP W for absorbent
microwave pyramids and one MM layer (one layer of two
MMs), exhibiting an absorption of –130 dB.
GB aperture
diameter Ld = 0.5 m Ld = 1.0 m
(nominal) Ld = 2.0 m
d = 6cm 3.4 × 10-3 s–1,
2.3 × 10–26 W
4.3 × 10–4 s–1,
3 × 10–27 W
5.3 × 10–5 s–1,
4 × 10–28 W
d = 9 cm
3.3 × 10–13 s–1,
2.2 × 10–36 W
4.2 × 10–14 s–1,
2.8 × 10–37 W
5.2 × 10–15 s–1,
3.5 × 10–38 W
d = 12 cm 1 × 10–27 s–1,
7 × 10–51 W
1.3 × 10–28 s–1,
8.7 × 10–52 W
1.6 × 10–29 s–1,
1 × 10–52 W
Table 6. Diffraction photons s–1 and NEP W for absorbent
microwave pyramids and four (two layers of two) MM lay-
ers (nominal), exhibiting an absorption of –220 dB.
GB aperture
diameter Ld = 0.5 m Ld = 1.0 m
(nominal) Ld = 2.0 m
d = 6 cm 3.4 × 10–12 s–1,
2.3 × 10–35 W
4.3 × 10–13 s–1,
3 × 10–36 W
5.3 × 10–14 s–1,
4 × 10–37 W
d = 9 cm
3.3 × 10–22 s–1,
2.2 × 10–45 W
4.2 × 10–23 s–1,
2.8 × 10–46 W
5.2 × 10–24 s–1,
3.5 × 10–47 W
d =12 cm 1 × 10–36 s–1,
7 × 10–60 W
1.3× 10–37 s–1,
8.7 × 10–61 W
1.6 × 10–38 s–1,
1 × 10–61 W
Copyright © 2011 SciRes. JMP
and 4 GB microwave wavelengths (6, 9, and 12 cm) and
for single-reflection diffraction path lengths, Ld, of 0.5,
1.0 (nominal) and 2.0 m from the GB throat to the re-
ceivers. These distances are approximately the distance
along the x-axis of the microwave receivers from the axis
of the GB. The nominal design choice is discussed in
Section 4.2.
Note that if during prototype-detector tests it became
apparent that diffraction rays reached a microwave re-
ceiver without being intercepted by an absorbent wall,
then one would increase the diameter of the nominal or
design GB from 9 cm to 12 cm resulting in diffraction
flux at a receiver of 1.31 × 10–15 s
–1, which would be
negligible. Such a design change would also increase
detector size and cost, so this alternative design would
not be pursued unless needed.
5.5. Noise Generated by Thermal Photons
In addition, isolation from background noise is further
improved by cooling the detector chamber and proper
choice microwave receiver apparatus [49-51] to reduce
thermal noise background to a negligible amount. A
cooling system is selected so that the temperature T sat-
isfies kBT ћ
, where kB is Boltzmann’s constant and
T ћ
/kB 480 mK for detection at 10 GHz and for
the detector’s narrow bandwidth. This condition is satis-
fied by the temperature for the detector enclosure T <
480 mK, which can be conveniently obtained using a
common helium-dilution refrigerator so that virtually no
thermal photons from the chamber walls will be radiated
at 10 GHz. According a study accomplished at the Uni-
versity of Western Australia “It is shown that this tech-
nology (new low noise microwave technology and ul-
tra-cryogenic techniques) could measure the standard
quantum limit of a macroscopic resonant-mass dis-
placement detector.” So that experimental data supports
the reduction in thermal photons, through the use of
modern refrigeration methods utilized in available mi-
crowave equipment [60], to allow high-sensitivity mi-
crowave detection.
5.6. Comprehensive Noise Summary
A standard sensor design method, already mentioned, for
aggregating noise sources is to translate all noise terms
through the system, or “refer them” from the location at
which they occur to the equivalent noise or NEP detec-
tion photon microwave receiver (s) [52]. Such an expres-
sion of noise is equivalent to the amount of power that
this amount of noise would represent at the detector. ll
the uncorrelated noise components can be root-sum-
squared together, so that:
  
ndns nj npa nqa
where the equivalent-power noise components are de-
fined as follows and the values for noise shown are based
upon experimental data:
The dark-background shot noise is proportional to the
square root of the number of photons present in a sample
and is mitigated by using the absorption layers on the
detector walls, larger GB throat diameter and wall ge-
ometry (Herschelian telescope geometry) to keep the
microwave receivers “below” and “out of sight” of the
GB entry-aperture source of diffraction and all x-directed
diffraction from the GB kept “above” and not directed to
the receivers as shown in Figure 5. (The x-directed dif-
fraction from the GB move in planes parallel to the x-y
plane). Ta bles 4 -6 present the calculated diffraction with
the nominal design given in Tabl e 5 . Stray BPF spillover
and diffraction that still manages to get reflected onto the
detectors will create the shot noise, but such noise could
be filtered out by pulse-modulating the magnetic field
and a baffle arrangement shown in Figure 8.
The signal shot noise is Pns = ћν(Ns)/Δt where Ns is
the signal-photon count, and Δt is the sample or accu-
mulation time. This “noise” is part of the useful data and
should not to be subject to elimination.
The Johnson noise (due to the thermal agitation of
electrons when they are acting as charge carriers in a
power amplifier) is Pnj = 4kBTRLBW, [61] where RL is the
equivalent resistance of the front-end amplifier and BW is
the bandwidth. Mitigation of this noise source is accom-
plished by reducing bandwidth or reducing load resis-
tance. However, in practice the bandwidth is often fixed
by the application, in this case by the detection band-
width. And the load resistance is required to generate a
large voltage from a very small current. Hence there is in
practice an optimum selection of load resistance that will
optimize the signal to noise output during the initial tests
of the Li-Baker detector, and the selection of this load
resistance is the essence of impedance matching in its
most basic form. Johnson noise is generally reduced or
eliminated by refrigeration to 0.48 K. At a Bw of 0.001
Hz and a sample interval of Δt =1000 seconds the noise
is 3.37 × 10–28 W or 5 × 10–5 noise photons per second
[62].The preamplifier noise is Pnpa = Bw/f1, [61] which is
essentially 1/f noise, where the crossover frequency f is
related to stray capacitance and load resistance; in which
f1 = 1/(2πRLCjn), [61] where Cjn = detection capacitance
plus FET (field effect transistor) input capacitance plus
stray capacitance. This noise source is mitigated by re-
ducing bandwidth, reducing load resistance, or reducing
stray capacitance. From [63] at a Bw of 0.001 Hz and a
sample interval of Δt =1000 seconds the noise is 7.57 ×
Copyright © 2011 SciRes. JMP
10–30 W or 1.13 × 10–6 noise photons per second [64].
The quantization noise is Pnqa = QSE12 , where
QSE is the quantization step equivalent or the value of
one LSB (Least Significant Bit , the smallest value that is
quantized by an ADC, or Analog to Digital Converter).
This noise source is easily mitigated and eliminated by
increasing the number of bits used in an ADC so that the
LSB is a smaller portion of the overall signal. In practice
the QSE is selected so that it does not cause lower SNR.
The noise is 1.33 × 10–26 W or 2 × 10–3 noise photons per
The mechanical thermal noise is caused by the
Brownian motion of sensor components. Mitigation is to
refrigerate the sensing apparatus to reduce thermal inputs.
The 0.48 K cooling should be sufficient, but if not an
even lower temperature can be achieved [60,65]. Also, as
mentioned earlier, there are specialty devices that could
be made readily available internationally that meet the
0.48 K temperature considered for the nominal case [60].
The phase or frequency noise (of the EM-GB) is due
to the fluctuations in the frequency of the microwave
source for the GB. Steps will need to be taken during the
Li-Baker detector tests to keep the GB source tuned pre-
cisely to the interaction volume resonance, thus reducing
phase noise and maximizing the resonant magnification
effect required from the interaction volume cavity. A
cavity-lock loop or alternatively a phase-compensating
feedback loop will be selected during post-fabrication
trials to mitigate this noise source.
The noise or noise equivalent power at the receiver(s)
or NEP as summarized in Table 7, is not a constant, but
exhibits a stochastic or random component. In order to
obtain the best estimate of the detection photons, one
would need to utilize a conventional signal-processing
filter [67].
The total NEP from Equation (5) of 1.02 × 10–26 W
(noise flux is 1.54 × 10–3 photons per second) is Quanti-
zation and thermal noise limited at roughly 1 × 10–26 to
2 × 10–27 W for a detector temperature of 0.48 K. If need
be the receivers could be further cooled and shielded
from noise by baffles [55] as shown in Figure 9 in which
the spherical BPF wave front, if significant, can be re-
duced by baffle diffraction and the PPF focused by the
reflectors passed through the baffle openings with less
interaction with baffle edges and less diffraction. Given a
signal that exhibits the nominal value given in Table 2 of
99.2 s–1 photons, one quarter of which is focused on each
of the microwave receivers, which is 24.8 s–1 photons or
1.6 × 10–22 W, the signal-to-noise ratio for each receiver
is better than 1500:1.
5.7. Noise Mitigation by Magnetic-Field
As noted, a unique feature of the Li-Baker HFGW de-
tector is that some of the noise sources are present when
the magnetic field is “off” and there is no signal or de-
tection photons present. With the magnetic field “on”
there is noise plus the signal. Thus, one can distinguish
between HFRGW generated photons and the background
generated photons from the GB. In principle one could
use coincidence gating to subtract the “noise” (with the
magnet “off”) from the “signal plus noise” with the
magnet “on” and obtain the signal alone. However, there
will still be stochastic noise sources that form a noise
spectrum that can be reduced by filtering but cannot be
completely removed. Consider a simplified case of a
uniform, low-frequency (compared with the 10 GHz
Table 7. Summary of Li-Baker detector noise based upon experimental data concerning its components.
Noise Contributor Brief Description of Noise sourceMitigation/Elimination Means Nominal Computed Value
photons s–1, NEP W
Dark-background shot noise GB noise especially diffraction Wall geometry and absorbing wall
materials 4.2 × 10–23 s–1, 2.8 × 10–46 W
Signal shot noise Noise in the signal itself Part of useful data and not to be
eliminated --
Johnson noise Thermal agitation in a power
amplifier resistance Refrigeration to low temperature 5 × 10–5 s–1, 3 × 10–28 W
Preamplifier kTC noise Stray capacitance and load
Reducing bandwidth, load
resistance and/or stray capacitance. 1 × 10–6 s–1, 8 × 10–30 W
Quantization noise Analog to Digital Converter Increasing the number of bits used 2 × 10–3 s–1, 1 × 10–26 W
Mechanical thermal noise Brownian motion of sensor
components. Refrigeration to low temperature 3 × 10–4 s–1, 2 × 10–27 W
Phase or Frequency noise Fluctuations in the frequency of the
microwave source for the GB.
Cavity-lock loop or a
phase-compensating feedback loop 5 × 10–15 s–1, 3 × 10–38 W
Copyright © 2011 SciRes. JMP
Figure 9. Schematic of microwave receiver shielded by MM absorbers and pyramid absorber/reflectors.
signal) square-wave chopper frequency energizing the
magnet, with the magnet alternatively “off” and “on”. It
could be utilized to remove some of the background
photons from the GB.
5.8. Standard Quantum Limit (SQL)
There is another possible concern here: Stephenson [68]
concluded that a HFRGW intensity of hdet = 10–30 to
10–32 m/m (time-varying strain in the fabric of space-time
whose amplitude is A) represent the lowest possible GW
strain variations detectable by each RF receiver in the
Li-Baker HFGW detector. There is a limit to this sensi-
tivity that is called “quantum back-action” or standard
quantum limit (SQL) and is a result of the Heisenberg
uncertainty principle [69]. An additional (1/2) factor
increase in maximum sensitivity applies if the separate
outputs from the two RF receivers are averaged, rather
than used independently for false alarm reduction, re-
sulting in a minimum hdet = 1.2 10–37. Because the pre-
dicted best sensitivity of the Li-Baker detector in its cur-
rently proposed configuration is A = 10–30 m/m, these
results confirm that the Li-Baker detector is pho-
ton-signal-limited, not quantum-noise-limited; that is, the
SQL is so low that a properly designed Li-Baker detector
can have sufficient sensitivity to observe HFRGW of
amplitude A 10–30 m/m or less. In theory the Li-Baker
detector is about seven orders of magnitude less sensitive
than the standard quantum sensitivity limit.
5.9. Sensitivity Increase
It may be desirable to increase the sensitivity of the pro-
totype Li-Baker detector through use of more sensitive
microwave receivers, a stronger magnetic field, a more
powerful microwave GB transmitter, a larger interaction
volume, the introduction of resonance chambers, etc. In
fact if the accumulation time, Δt, of the PPF at the re-
ceivers is increased to 1000 s, then for A = 10–33 the de-
tection photons per second .would still be 99.2 from
Equation (1). Of course the noise would also increase to
1.54 photons per second, but the signal to noise ratio
would still exhibit a respectable value of 64. Even if one
cannot greatly increase sensitivity immediately, a null
experimental result at a larger value of A would still be
valuable, since it can provide the indirect means to de-
termine whether or not some theories and scenarios
should be corrected or eliminated. For example, the data
analysis of low-frequency, laser-interferometer gravita-
tional-wave detectors, such as LIGO and Virgo [70],
have so far had null results, but have been the basis for
cosmological theory improvements and have had impor-
tant significance for further study.
6. Conclusions
Three HFGW detectors have previously been fabricated
and two others theoretically proposed, but analyses of
their sensitivity and the results provided herein suggest
that for meaningful relic gravitational wave (HFRGW)
detection, greater sensitivity than those fabricated in-
struments currently provide is necessary. The theoretical
sensitivity of the Li-Baker HFGW detector studied
herein, and based upon a different measurement tech-
nique than the other detectors, is predicted to be A =
10–30 m/m at base frequencies near to 10 GHz. This de-
tector design is not quantum-limited and theoretically
exhibits sensitivity sufficient for useful relic gravitational
wave detection. Utilization of magnetic-field pulsed
modulation allows for reduction in some types of noise.
Other noise effects, based upon classical equations or
experimental data, are found theoretically to be minimal;
but they can only be accurately determined based on the
Li-Baker prototype detector tests and some of the design
and adjustments can only be finalized during prototype
fabrication and testing. The detector can be built from
Copyright © 2011 SciRes. JMP
off-the-shelf, readily available components and its re-
search results would be complementary to the proposed
low-frequency gravitational wave (LFGW) detectors,
such as the Advanced LIGO and the proposed Laser In-
terferometer Space Antenna or LISA.
7. Acknowledgements
This work is supported by the National Nature Science
Foundation of China under Grant No. 11075224, the
Foundation of China Academy of Engineering Physics
under Grant Nos. 2008T0401 and 2008T0402, Chong-
qing University Postgraduates Science and Innovation
Fund No. 200811B1A0100299, GravWave® LLC, Trans-
portation Sciences Corporation and Seculine Consulting.
All authors reviewed, edited and approved the manu-
script. Some material by RCW was based on work sup-
ported by the US National Science Foundation, while
working at the Foundation. Any opinion, finding, and
conclusions or recommendations expressed in this mate-
rial are those of the authors and do not necessarily reflect
the views of the US National Science Foundation. FYL
initially developed the theory; RMLBjr designed the mi-
crowave-absorbing baffles and the reflectors including
the off-axis Herschelian telescope and the exterior to GB
focusing lenses to reduce diffraction noise, suggested the
use of the vacuum and the utilization of two microwave
detectors, AWB developed the astrophysical applications,
RCW analyzed the magnet and Gaussian beam scattering,
GVS developed the standard quantum limit and the noise
equivalent power approach and RCW and GVS devel-
oped the concept for the high-sensitivity microwave re-
ceivers and signal processing. Christine S. Black pro-
vided the figures and Amara D. Angelica edited the
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Brief Summary of the Li-Effect Proof
The Li-Baker detector is based upon the Li-effect and is
a coupling system among a traveling-wave Gaussian-
type microwave photon beam, a static magnetic field,
HFGWs and fractal membranes (or other equivalent mi-
crowave optics to focus detection photons on microwave
receivers) [9,33,34,39]. This synchro-resonance electro-
magnetic coupling scheme or 3D SR or Li effect is based
upon a particular type of synchro-resonant microwave
EM beam, which is a special beam exhibiting the direc-
tion, frequency and appropriate phase as the HFGW to
be detected. That beam is a classical microwave Gaus-
sian beam or GB in which the intensity falls off radially
from the beam’s axis according to a Gaussian curve,
hence the name. The GB is a microwave beam having an
hour-glass shape that is focused down to a narrow waist.
We take the axis of the GB to be the z-axis and the axis
of the orthogonal static magnetic field to be the y-axis
with the origin at the GB’s waist. Below the origin (z < 0)
the GB is converging and above the origin (z > 0) it is
diverging. The characteristics of the GB, for example the
orthogonal E and B vectors comprising the Poynting
vectors of the EM-GB, are different in the x-y plane
cross sections of the GB. That is they are different in the
four quadrants: x > 0, y > 0; x > 0, y < 0; x < 0, y > 0; x
< 0, y < 0 of the GB. Several descriptions of a classical
GB are available in the literature [44,70-73]. In the vol-
ume defined by the GB and magnetic field intersection
there are eight octants to be considered: four for z > 0
and four for z < 0. The GB is different in each of these
octants and the HFGWs passing through these octants
interact differently with the combined GB and magnetic
field in each of these octants. Note that we choose the
GB of transverse polarized electric modes (which has
non-vanishing magnetic component) and not the GB of
transverse polarized magnetic models. These diverse
interactions between HFGWs and the GB in the various
octants are the essence of 3D SR or Li effect and the
Li-Baker detector that makes use of the 3D SR or Li ef-
fect. What has been proven theoretically in the
peer-reviewed literature [8,9,32-39](each covering a dif-
ferent aspect of the Li-effect) is that the HFGWs passing
through these octants generate microwave EM radiation
or detection photons that move out in the +x and –x di-
rections. In practice, due to the variety of directions of
the actual HFGWs, they move out in a cone whose axis
is the x-axis and vertex is at the origin of coordinates. In
order to concentrate or focus the microwave detection
photons, whose presents signal the detection of HFGWs,
at sensitive microwave receivers at both ends of the
x-axis, it is useful to introduce microwave optical de-
vices. These devices can be in the form of concave metal
reflectors or parabolic “mirrors,”fractal membranes,
metamaterial microwave lenses, etc. Depending upon the
Li-Baker detector design these optical microwave focus-
ing devices can be positioned inside or outside of the
Unlike the pure-inverse Gertsenshtein effect (G-effect),
here under the synchro-resonance condition, coherence
modulation of the HFGWs to the preexisting transverse
photon flux of the Gaussian beam (GB) is predicted to
produce the transverse (radial) first-order perturbative
photon flux (PPF) or signal due to the presence of GWs
as shown in Figure 1A, and the PPF has a maximum at a
longitudinal symmetrical surface of the GB where the
transverse background photon flux (BPF) or GB noise
vanishes. Moreover, the PPF and the BPF have obvi-
ously different decay rates in the transverse direction,
and the PPF reflected, for example by the fractal mem-
branes, exhibits a very small decay to be compared with
a very large decay of the much stronger BPF. Thus, such
properties might provide a new way to distinguish the
BPF (noise) and display the PPF (signal). The general
form of the GB of a fundamental frequency mode is [44]
exptan 2
ikz tfR
 
where 222
, 2π
, 2
01WW zf , 2
Rzfz , 0
is the ampli-
tude of the electric (or magnetic) field of the GB, W0 is
the minimum spot radius, R is the curvature radius of the
wave front of the GB. From Equation (A1) one finds
Figure 1A. First-order longitudinal PPF ((1)
n signal) and
BPF ((0)
n noise) in the Li-Baker detector in the z direction
as measured radially, r.
Copyright © 2011 SciRes. JMP
Figure 2A. Schematic diagram of strength distribution of the transverse BPF (0)
n and PPF (1)
n in the outgoing wave re-
gion of the GB [34,35].
(0) (0)
(0) (0)
(0) (0)
exp ,
exp ,
nf W
nf W
nf W
where (0)
n, (0)
n, (0)
n represent the average values in
the x-, y- and z- directions of the BPF (noise) and
 
xx yy
 
0maxzxy zz
 . Because
of the non-vanishing (0)
n and (0)
n, the GB will be
asymptotically spread as |z| increases.
Unlike (0)
n, (0)
n and (0)
n (noise), the PPF (1)
n and (1)
n (signal) have different decay forms:
(1) (1)
(1) (1)
(1) (1)
exp ,
exp ,
nf W
nf W
nf W
where (1)
, (1)
and (1)
are the functions of posi-
tion x, y, z. Therefore, the decay rate of (1)
n (signal) is
slower than that of (0)
n (noise).
In the Li-Baker detector the first-order longitudinal
PPF (1)
n and the BPF (0)
n have the same propagating
direction, and (0)
n is much lager then (1)
n in most of
the nearby regions. Thus, (1)
n will be swamped by the
n in such regions. However, the (1)
n and (0)
n will
exhibit a comparable order of magnitude in the “far-axis
region” (r > 30 cm about the distance of six radii of the
GB as shown in Figure 2A) due to the different decay
Figure 3A. The interaction of the GB, magnetic field and
HFRGWs in the Li-effect produces (1)
n photons.
rates and the (1)
n will become larger than (0)
n further
out in the radial, r, direction. Therefore, as discussed in
Section 5.8, the Li-Baker detector is photon-signal-limited,
not quantum noise limited.
Unlike Figure 1A, here (0)
n while
Copyright © 2011 SciRes. JMP
(1) (1)
0maxxx x
. Thus, (1)
nt (accumulated signal) can
be effectively larger than the background noise photon
nt at the yz-plane and at the parallel
surfaces near the yz-plane, provided that the total noise
photon flux passing through the surface can be effec-
tively suppressed as discussed in Section 5. Because
n propagates along apposite directions in the regions
of y > 0 and y < 0 in the GB, there is conservation of
total momentum in the coherent resonance interaction
and there is also an ability to focus half of the (1)
which are directed to the center of the GB, at the two
microwave receivers at opposite ends of the x-axis. The
reverse mirror image of Figure 2A in the xy-plane (z <
0), shown in Figure 3A as b-b, insures that there is con-
servation of angular momentum and the differrentiation
of the interaction volume into octants about the origin
(center or intersection of the axes of the GB and the
static magnetic field) is established by [1,39].
The work on this Appendix is supported in part by the
National Nature Foundation of China, grant No.