A New Theoretical Technique for the Measurement of High-Frequency Relic Gravitational Waves

doi:10.4236/jmp.2011.26060

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Journal of Modern Physics, 2011, 2, 498-518

doi:10.4236/jmp.2011.26060 Published Online June 2011 (http://www.SciRP.org/journal/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,

People’s Republic of China

4Seculine Consulting, Redondo Beach, USA

5Institute for Advanced Studies at Austin, Austin, USA

E-mail: DrRobertBaker@GravWave.com

Received March 3, 2011; revised April 17, 2011; accepted April 27, 2011

Abstract

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

R. C. WOODS ET AL.

499

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

Waves

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

constructed.

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 Ωgw 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-

tivity.

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, Ωgw [19,

26-28]. According to these estimates, the upper limit of

Ωgw for relic GWs should be smaller than 10–5. In fact,

recent estimates [10] show that the upper limit of Ωgw

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 Ωgw, and hence to differentiate among

the competing cosmological theories for the beginning of

the Universe.

2.2. Presently Fabricated and Proposed HFGW

Detectors

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-

R. C. WOODS ET AL.

500

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-

sitivity.

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

noise.

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 Ωgw. The spectra predicted by the

pre-big-bang models (Figure 2 of [10]) shows that Ωgw

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 Ωgw ~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 Ωgw 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.

R. C. WOODS ET AL.

501

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

Effects

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

R. C. WOODS ET AL.

502

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

receivers.

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],



xey

nABs









 (1)

where



n is the number of x-directed detection photons

R. C. WOODS ET AL.

503

Figure 3. Gaussian-beam transmitter compartment.

Figure 4. Schematic of the Li-Baker HFGW detector.

R. C. WOODS ET AL.

504

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

R. C. WOODS ET AL.

505

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.

R. C. WOODS ET AL.

506

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

fapctors:

 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

R. C. WOODS ET AL.

507

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:







022

~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

R. C. WOODS ET AL.

508

photons, I, from a molecule with incident photon inten-

sity Io as given by [53]



8π1cos

II R









(3)

in which



is the atomic polarizability expressed as a

polarization volume (where the induced electric dipole

moment of the molecule is given by 4





E),



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

8π





. 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) = 5.310–4 and will be

negligible.

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















222

232 exp20.01

dif dGB

nkd Lkdn

(4a)

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

nnalL











 (4b)

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

R. C. WOODS ET AL.

509

(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

(nominal)

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

(nominal)

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

(nominal)

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

R. C. WOODS ET AL.

510

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:



  

222

EP W

ndns nj npa nqa

PPPP P













,

(5)

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 ×

R. C. WOODS ET AL.

511

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

second.

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

Modulation

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

resistance

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

R. C. WOODS ET AL.

512

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

R. C. WOODS ET AL.

513

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

manuscript.

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516

Appendix

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

GB.

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]



exp

exptan 2

ikz tfR

























 















(A1)

where 222

rxy



, 2π





, 2

πe



,



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

[1,39]

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.

R. C. WOODS ET AL.

517

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)

exp ,

exp

nf W

























(A2)

where (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)

(1)

n and (1)

n (signal) have different decay forms:

(1) (1)

exp ,

exp

nf W

























(A3)

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

(0)

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

R. C. WOODS ET AL.

518

(1) (1)

0maxxx x

. Thus, (1)

nt (accumulated signal) can

be effectively larger than the background noise photon

fluctuation



1/2

(0)

xtot

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

(1)

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].

Acknowledgements

The work on this Appendix is supported in part by the

National Nature Foundation of China, grant No.

11075224.