Electromagnetic counterparts of high-frequency gravitational waves having additional polarization states: distinguishing and probing tensor-mode, vector-mode and scalar-mode gravitons

GWs from extra dimensions, very early universe, and some high-energy astrophysical process, might have at most six polarizations: plus- and cross-type (tensor-mode gravitons), x-, y-type (vector-mode), and b-, l-type (scalar-mode). Peak or partial peak regions of some of such GWs are just distributed in GHz or higher frequency band, which would be optimal band for electromagnetic(EM) response. In this paper we investigate EM response to such high-frequency GWs(HFGWs) having additional polarizations. For the first time we address:(1)concrete forms of analytic solutions for perturbed EM fields caused by HFGWs having all six possible polarizations in background stable EM fields; (2)perturbed EM signals of HFGWs with additional polarizations in three-dimensional-synchro-resonance-system(3DSR system) and in galactic-extragalactic background EM fields. These perturbative EM fields are actually EM counterparts of HFGWs, and such results provide a novel way to simultaneously distinguish and display all possible six polarizations. It is also shown: (i)In EM response, pure cross-, x-type and pure y-type polarizations can independently generate perturbative photon fluxes(PPFs, signals), while plus-, b- and l-type polarizations produce PPFs in different combination states. (ii) All such six polarizations have separability and detectability. (iii)In EM response to HFGWs from extra-dimensions, distinguishing and displaying different polarizations would be quite possible due to their very high frequencies, large energy densities and special properties of spectrum. (iv)Detection band(10^8 to 10^12 Hz or higher) of PPFs by 3DSR and observation range(7*10^7 to 3*10^9 Hz) of PPFs by FAST (Five-hundred-meter-Aperture-Spherical Telescope, China), have a certain overlapping property, so their coincidence experiments will have high complementarity.


I. INTRODUCTION
Recently, the LIGO Scientific Collaboration and the Virgo Collaboration reported multiple gravitational wave (GW) evidences,e,g. (GW150914, GW151226, GW170104, GW170608, GW170814, GW170817) [1][2][3][4][5][6] and a candidate (LVT151012) [7]etc. These GW events and candidate are mainly produced by binary black hole mergers (frequencies are distributed around 30Hz to 450Hz, and dimensionless amplitudes in region of the Earth are h ∼ 10 −21 to h ∼ 10 −22 ), and one event of them is the detection evidence of GWs produced by the for such GWs shows the rationality of weak field linear approximation and the perturbation theory of gravity. Thus it further increases the possibility and hope of searching gravitons of spin-2 in the quantization process of gravity.
3. Successful detection of the GWs by the groundbased GW detectors such as LIGO and Advanced Virgo indicates the validity of tetrad coordinates, i.e., observable quantities should be the projections of the physical quantities of the GWs as a tensor on tetrads of the observer's world-line. 4.
The information (including related energymomentum) carried by the GWs from the wave sources of the binary compact objects also provides a strong evidence of the positive definite property of the energymomentum tensor for the GW fields themselves. 5. Observation of the GWs emitted by the binary neutron star merger and related EM counterpart (the gamma-ray radiation) gives a strict limit to the propagating velocity of the GWs in the intermediate frequency band (ν g ∼ 1 Hz to 1000 Hz). Also, it provides effective constraint for the geometry of extra dimensions.
On the other hand, the results obtained by LIGO and Virgo should not mean the end of seeking the GWs. On the contrary, they are just the beginning of the GW astronomy. This is because of the following reasons: 1. With the continuous improvement of the sensitivities of LIGO, Virgo, GEO, KAGRA, AIGO etc., searchable sky area and detectable space scale will be further expanded. Therefore, it is very possible that more GW evidences will be detected and found during the next decade.
2. The GWs recently detected by LIGO and Virgo Collaboration are located in an interesting but special intermediate frequency range (ν ∼ 30Hz to ν ∼ 450Hz ), and their durations of signals in the detectors were very short. Thus, observation and detection to the continuous GWs, other kinds of GWs and the GWs in other frequency bands will be urgent affairs. 3. Except for the GWs predicted by General Relativity (GR), series of modified gravity theories and the gravity theories beyond GR also expect the GWs. These gravity theories include general metric theory of gravity [8,9], Brans-Dicke theory [10,11], scalar-tensor theories of gravity [12,13] and vector-tensor theories of gravity [14], f(R) gravity theory [15], and so on [16][17][18][19]. An important difference to the GR is that the GWs in some of such gravity theories, might have additional polarization states, which can be at most six polarization states in our 3+1 dimension spacetime, while the GWs in the GR have only two polarization states, i.e., ⊕-type and ⊗-type polarization states. Thus, further theoretical study and experimental observation of the GWs, will provide important criterions for the polarization states, the propagating speed, the waveforms, and other possible novel properties of the GWs.
For the intermediate frequency GWs (ν ∼ 1Hz to ν ∼ 10 3 Hz), Nishizawa et al investigated effective methods and schemes to detect and separate the different polarization states of such GWs. In fact, such schemes are based on the correlation analysis of multiple groundbased GW detectors [20,21]. For the detection of lowfrequency GWs (ν ∼ 1Hz to ν ∼ 0.1Hz), related scheme is based on the configuration of space-based GW detectors [22]. These schemes would be promising for displaying and separating the all polarization states of the GWs. Obviously, all of the above schemes are based on the tidal action caused by these polarization states of the GWs. Moreover, L Visinalli et al studied the scheme probing extra dimensions by the intermediate frequency GW signals and related EM counterparts from the mergers of the binary neutron stars [23]. Such scheme and direct detection of the additional polarizations of the HFGWs from the extra dimensions, will be complementary in the frequency band and in the principles.
On the other hand, almost all mainstream inflationary theories and universe models predicted the primordial (relic) GWs, and the spectrum of these relic GWs would distribute in a very wide frequency region, which may be from extreme-low frequency range ( ν ∼ 10 −16 Hz to ν ∼ 10 −17 Hz) to high-frequency band ( ν ∼ 10 8 Hz to ν ∼ 10 10 Hz or higher) [24][25][26][27][28][29][30][31]. Especially, the peak region or partial peak region of the energy densities of the relic HFGWs predicted by the pre-big-bang models [26,27], the quintessential inflationary models [28][29][30], and the short-term anisotropic inflation model [31], are just distributed in the typical microwave band ( ν ∼ 10 8 Hz to ν ∼ 10 10 Hz), in which corresponding amplitudes might reach up to h ∼ 10 −26 to h ∼ 10 −30 . Moreover, the frequency of the HFGWs (KK-gravitons) expected by the braneworld scenarios [18,19] in extra-dimensions and the HFGWs predicated by interaction of astrophysical plasma with intense electromagnetic waves (EMWs) [32], have been extended to ∼ 10 9 Hz to ∼ 10 12 Hz or higher (related amplitudes of these HFGWs would be expected to be h ∼ 10 −21 to ∼ 10 −27 [18,19,32]), and some works predicted HFGWs in very high-frequency band even over 10 19 Hz from coherent oscillation of electronpositron pairs and fields [33] or from the magnetars [34]. Another important possible source of HFGWs is produced during the preheating in the early universe and it would have frequency over GHz [35,36]. Mergers [37] and evaporation [38] of primordial black holes are also possible HFGW sources, and their frequency band can be ∼ 10 9 Hz to 10 13 Hz, in which, related amplitudes might be h ∼ 10 −31 to h ∼ 10 −36 , respectively. In fact, according to the GR, and even modified gravity theories and the gravity theories beyond the GR, any energy-momentum tensor in the high-frequency oscillating states with deviation from the spherical symmetry or cylindrical symmetry, would be possible and potential HFGW sources. Thus such mechanisms and process would be very common in the high-energy astrophysics and cosmology, and these HFGWs would contain abundant astrophysical and cosmological information.
However, the frequencies of these HFGWs are far beyond the detection or observation range of the intermediate-frequency GWs (ν g ∼ 1Hz to ∼ 10 3 Hz, by ground-based GW detectors), the low-frequency GWs ( ν g ∼ 1Hz to ∼ 10 −7 Hz, by space GW detectors) and the extreme-low frequency GWs ( ν g ∼ 10 −16 Hz to ∼ 10 −17 Hz, by B-mode polarization in the CMB). Therefore, detection and observation of such HFGWs need new principle and scheme, including separation of the additional polarization states of the HFGWs. So far, the separation and distinguishing of the additional polarization states of the HFGWs in the microwave frequency band by the electromagnetic (EM) response, almost have not been reported in the past.
In this paper, based on the electrodynamics and the quantum electronics in curved spacetime, we investigate a novel way to distinguish and display the all possible six polarization states of the HFGWs. We will address the concrete forms of analytic solutions for the perturbative EM fields caused by the HFGWs having all six possible polarizations in the background stable EM fields, and study the perturbative EM signals (actually they are also the EM counterparts) of the HFGWs with additional polarizations in the 3DSR system (in laboratory scale) and in galactic-extragalactic background EM fields. Our attention will focus mainly on the EM response to the HFGWs from the extra-dimensions, from very early stage of the universe and from some high-energy astrophysical process, especially the HFGWs (KK-gravitons) from the braneworld [18,19], the relic HFGWs from the short-term anisotropic inflation [31], from the pre-big-bang [26,27], from the quintessential inflation [28][29][30], and the HFGWs from the astrophysical plasma oscillation [32], etc. This is because they involve following important scientific issues and problems: the extra-dimensions of space and the brane universes, the very early universe and inflationary epoch, the start point of time or the information from the pre-big-bang, the essence and candidates of dark energy, and the interaction mechanism of the astrophysical plasma with intense EM radiations.
The plan of this paper is organized as follows. In sec. II, we shall show the general form of the HFGWs having six polarization states. In sec. III, we address the analytic solutions for the perturbative EM fields caused by HFGWs having six polarization states in the background stable EM fields, and the separation and displaying of such six polarizations. In sec. IV, we study the EM signals caused by the six polarizations of HFGWs in the 3DSR system (in laboratory scale), and their separation and displaying effects. In sec. V, numerical estimations of the perturbative photon fluxes in the 3DSR system and in the galactic-extragalactic magnetic fields, are given. Our brief conclusion is summarized in sec. VI. 1. In general, the "monochromatic components" of the GWs having six polarization states and propagating along the z-direction can be written as where ⊕, ⊗, x, y, b and l represent ⊕-type, ⊗-type polarizations (tensor-mode gravitons), x-type, y-type polarizations (vector-mode gravitons), b-type and l-type polarizations (scalar-mode gravitons), respectively. For the coherent and non-stochastic GWs, such as the GWs from extra dimensions (e.g., the K-K GWs from braneworld [18,19]), the A ⊕ , A ⊗ , A x , A y , A b and A l are constant values of amplitudes of the GWs in the laboratory frame of reference. For the relic GWs, the A ⊕ , A ⊗ , A x , A y , A b and A l are stochastic values of the amplitudes of the relic GWs in the laboratory frame of reference, which contain the cosmology scale factor; the k g and ω g are wave number and angular frequency of the GWs, respectively. Eq.(1) can be conveniently represented as following matrix form [20] 2. Phase modification issues in the EM response to the HFGWs.
Here, the above expressions are based on following assumption, i.e., the all components of the GWs have the same propagating velocity (the speed of light). In this case, dose it mean that the above gravitons must be massless (i.e., must be gravitons predicted by the GR)? The answer to this question is that it depends on different cases. In fact, according to a series of modified and extended gravitational theories, the gravitons might have very small but non-vanishing masses, and the propagating velocities of such gravitons (e.g., massive gravitons) will be different to the speed of light. However, even if there is such difference, the modifications to the propagating velocity of the GWs (gravitons) will be extremely small values, and the phase difference between the GWs This means that smaller mass m g and higher frequency ω g will have a better coherence resonance effect between the HFGWs (gravitons) and the perturbative EM fields (signal photon fluxes). Table I shows that in most cases such phase modification would be much less than 2π ×10 −3 . Notice that even if ∆z ∼ 10 21 m, which is equivalent to the diameter of the Galaxy, the final phase difference will still be less than or much less than ∼ 10 −3 × 2π. Thus the assumption of v g = c in Eq.(1) is reasonable and valid in the typical distance discussed in this paper. Obviously, the above result is exactly valid for the GWs in the framework of GR.
3. The metric components of the HFGWs having six polarization states.
It is well known that for the weak GWs, the metric can often be expressed as a small perturbation to the background spacetime η µν , i.e., From Eqs. (1) and (2), the covariant components of the metric tensor for the HFGWs can be given by g 00 = −1, g 01 = g 02 = g 03 = g 10 = g 20 = g 30 = 0, where In the above expressions, we neglected the second-order infinite small quantity h 2 of the perturbation h ij . Obviously, the HFGWs expressed by Eqs. (8) and (9) do not satisfy the transverse and traceless gauge condition (TT-gauge condition). Different polarization states in Eq. (1) corresponds to different kinds of gravitons, where the ⊕-type and ⊗-type polarization states represent gravitons of spin-2; the x-type and y-type polarization states indicate gravitons of spin-1, and the b-type and l-type polarization states denote gravitons of spin-0, respectively.
In fact, the EM perturbation effects produced by the HFGWs and their detections had been discussed. These EM detection systems include constructed and proposed ones like the two coupled spherical cavities [45], high-frequency phonon trapping acoustic cavities [46], closed cylindrical superconducting cavity [47] and coupling system between the planar superconducting open cavity and static magnetic field [48]. However, in these previous researches, the HFGWs interacting with such EM systems are almost the GWs in the GR framework, i.e., they satisfy the TT-guage condition. So far, it is not clear what a concrete form (of the EM response to the HFGWs having additional polarization states) should be; it is also not clear whether such different polarization states of the HFGWs can be separated and displayed, and how to separate and display them. We shall show these issues in the expanded EM systems (see sect. IV), which consist of not only static magnetic fields, but also static electric fields. Moreover, they also include the coupling system between the Gaussian-type microwave photon flux (Gaussian beam) and the static high EM fields. Then, the EM perturbative effects produced by the different polarization states of the HFGWs will have different physical behaviors. Therefore, these different polarization states, in principle, can be separated and displayed. the Earth) discussed in this paper due to the possible graviton masses. For the possible HFGW sources in the braneworld [18], such distance is estimated as ∆z ∼ 10 19 m and ∼ 10 21 m. ∆φ f is the final phase difference.
According to Eqs. (10) and (11), the covariant and the contra-variant components of the background EM field tensor can be given (we use MKS units): Interaction of the HFGWs, Eq.(1), with such background EM fields, Eqs. (10) and (12), will generate the EM perturbation, and the perturbative effects can be calculated by the electrodynamics equations in curved spacetime: Γ α µν is the second kind of Christoffel connection, and J µ indicates the four-dimensional electric current density.
For the EM perturbation in the free space (vacuum), it has neither a real four-dimensional electric current nor other equivalent electric current, so J µ = 0 in Eq. (13). Moreover, for the EM perturbation generated by the weak HFGWs, we have F µν =F is the background EM field tensor, Eqs.(10) and (12), andF (1) µν is the first-order perturbation to F (0) µν ; the "∼" represents time-dependent perturbative EM fields. Here we neglected the second-order and higher-order infinite small perturbations. In this case, we have: Introducing Eqs.
(1), (2), (8 to 10) and (12) into Eqs. (13), (14), we obtain following inhomogeneous hyperbolic-equations: (20) and, ∂Ẽ where "✷" indicates the d'Alembertian. From Eqs. (17) to (21), after lengthy calculations, we obtain the general solutions of these equations as follows: (24) and,Ẽ (1) The Eqs. (22) to (27) show that: (1) If the background stable EM fields have all spatial components (i.e., the x−, y− and z−components, here the HFGW propagates along the z−direction), then not only the EM response to the ⊕-type, ⊗-type polarization states in the GR framework can be produced, but also the EM response to the additional polarization states (the x-, y-, b-and l-type polarization states) beyond the GR can be generated.
(2) The transverse polarization components (Ẽ x ) of the perturbative EM fields have the space accumulation effect (i.e., their strengths are proportional to the propagating distance z of the HFGW in the background EM fields). This is because the HFGW (gravitons) and the perturbative EM waves (photons) have the same or almost the same propagating velocity (see section II). Thus, they can generate an optimal space accumulation effect in the propagating direction, i.e., they and the results [49,50] in the GR from the Feynman perturbative techniques and the Einstein-Maxwell equations are self-consistent. However, Eqs. (22) to (26) include new EM counterparts generated by the EM response to HFGWs by the longitudinal background EM fieldsÊ  Table I, e.g., massive gravitons], such correction to the space accumulation effect can still be neglected even in the typical astronomical scale. Therefore, the final phase difference caused by the phase velocity v P does not cause any essential impact to the space accumulation effect (see Table I).
(3) Eqs. (22) to (25) also show that the perturbative EM fields propagating along the opposite direction of the HFGW [i.e., the perturbative EM fields containing the propagating factor (k g z + ω g t)] do not have such space accumulation effect. Obviously, this is a self-consistent result to the EM perturbation, because there is no accumulation effect of energy for such EM fields in the opposite propagating direction of the HFGW. z . Moreover, since there is a phase difference of π/2 between the transverse perturbative EM fields [Ẽ (1) x ,B (1) x , Eqs. (22) to (25)] and the longitudinal perturbative electric fieldẼ  2. Electromagnetic counterparts: the perturbative photon fluxes in the EM response.
Interaction of the HFGWs with the EM fields will generate perturbative EM power fluxes (signal EM power fluxes). In order to conveniently represent the above EM signal in the background noise photon flux fluctuation (see Appendix B), we will express them in quantum language, i.e., the perturbative photon fluxes (the PPFs, or signal photon fluxes). It is interesting to study following several cases: (1) The EM response to the HFGWs in transverse background stable magnetic fieldB (26), i.e., onlyB (0) y has non-vanishing value. Then from Eqs. (22) to (25), we obtain following results immediately, , and , Here, as well as all discussions below, the perturbative EM fields propagating along the negative z-direction (i.e., the opposite propagating direction of the HFGWs) will be neglected, because they are very weak (they do not have the space accumulation effect) or absent [49,50]. Therefore, we can make the constants C 1 , C 2 , C 3 and C 4 equal to zero in Eqs. (22) to (25). The perturbative EM fields in Eqs. (28), (29) are similar to that in the refs. [49,50]. However, there is an important difference, i.e., the perturbative EM fields, Eq.(28), contain not only the contribution from the ⊕type polarization state of the HFGWs, but also the contribution from the longitudinal polarization (the l-type polarization) of the HFGWs. In other words, the contribution of the ⊕-type polarization state is always accompanied by the l-type polarization states. Moreover, they have the same symbol in the amplitudes of the perturbative EM fields, Eq.(28), i.e., this is a constructive coherence effect between the ⊕-type polarization and the l-type polarization states. Thus, the interaction of the HFGWs having additional polarization states with the background transverse magnetic fieldsB (0) y , will consume more energy [see Eq.(30)] than the HFGWs in the GR, i.e., the former will cause faster radiation damping than that of the latter for the HFGW sources in local regions.
Unlike such effect, the perturbative EM fields, Eq.(29), produced by the ⊗-type polarization state of the HFGWs, does not contain the contribution from the ltype or other additional polarization states.
From Eqs. (28), (29), the perturbative photon fluxes generated by the interaction of the HFGWs with the background magnetic fieldB where the * denotes complex conjugate; the angular brackets represent the average over time, and the superscript "2" represents second-order perturbation to the EM fields because they are proportional to the square of the HFGW amplitudes: A ⊕ , A l and A ⊗ .
(2) The EM response to the HFGW in transverse background stable magnetic fieldB x is not equal to zero. In the same way, from Eqs. (22) to (25), corresponding perturbative photon fluxes are (see Fig. 4): y , the transverse perturbative EM fields, E x can be generated, and they will produce the PPFs in the propagating direction of the HFGW. The PPF n contain contribution from both the ⊕-type and the l-type polarizations, and the PPF n Eq. (32) is very similar to Eq. (31), namely, the x , the transverse perturbative EM fields,Ẽ x , B (1) can be generated, and they will produce the perturbative photon fluxes n contribution of the ⊗-type polarization state is always independent of the additional polarization states. While, Eqs. (30), (33) show that the contribution of the ⊕-type polarization state is always accompanied by the l-type, or the b-type and l-type polarization states.
So far, it is not clear yet the related ratio among the intensities of the ⊕-type, b-type and l-type polarization states. Obviously, in the following cases: (i) then the effect of the ⊕-type polarization (effect in the GR), the b-type or the l-type polarization (effect beyond the GR) can mainly be displayed, respectively.
Notice that althoughB x are all the transverse stable magnetic fields, their EM response to the HFGW have certain difference. The n (1) z 1 ○ , Eq.(30), is a constructive coherence effect between the ⊕-type and the l-type polarization states (they have the same symbols), while the n (1) z 2 ○ , Eq.(33), is a destructive coherence effect between the ⊕-type and the b-type, l-type polarization states (the ⊕-type and the b-type, l-type polarization states have the opposite symbols). This is because the impacts of the b-type polarization to the ⊕type polarization in the xx-and the yy-components of the HFGW metric h µν are different [see Eq. (1)]. The former is "constructive superposition" (where the b-type and the ⊕-type polarizations have the same symbols), and the latter is "destructive superposition" (where the b-type and the ⊕-type polarizations have the opposite symbols). Moreover, the l-type polarization only appears in the zz-component of the metric h µν . In this case, the EM response ofB  (the xz-component F 13 of the EM field tensor) are non-symmetric. This is the physical origin of such difference.
(3) The EM response to the HFGWs in the transverse background stable electric fieldÊ has non-vanishing value. By using the same method, from Eqs. (22) to (26), we have (see Fig. 5) x , the transverse perturbative EM fields,Ẽ x , B (1) x can be generated, and they will produce the PPFs n  (4) The EM response to the HFGWs in the transverse background static electric fieldÊ (22) to (26), i.e., onlŷ E (0) y has non-vanishing value. According to the same way, we find (see Fig. 6): (5) The EM response to the HFGWs in the longitudi- x , B (1) can be generated, and they will produce the PPFs n nal stable EM fieldsB z . In fact, whether according to the electrodynamic equations in curved spacetime [49,51], or the Feynman perturbation techniques to analyze the conversion of GWs into EM waves (and vice versa) [50], the GWs (including the HFGWs) in the GR framework (i.e., the GWs having only the ⊕-type and the ⊗-type polarizations) do not generate any perturbation to the longitudinal static EM fields [49][50][51]. Unlikely, the HFGWs having the additional polarization states will generate EM perturbations to such EM fields. Thus, their physical behaviors are quite different. PuttingÊ Then Eqs. (22) to (25) are reduced tõ From Eqs. (38) to (41), the corresponding PPFs can be given by (see Fig. 7) It is interesting to note, such PPFs are only produced x can be generated, and they will produce the PPFs n by the pure additional polarization states (the x-type and the y-type polarizations, i.e., the vector mode gravitons). In other words, they are independent of the tensor mode gravitons (the ⊕-type and the ⊗-type polarizations) and the scalar mode gravitons (the b-type and the l-type polarizations).
If we consider only the EM response of the longitudinal stable magnetic fieldB (42) and (43), then the equations are reduced to In this case, the x-type and the y-type polarizations of the HFGWs can be more clearly and directly displayed. In fact, according to contemporary astronomic observation [52], it is certain that there are very widespread background galactic-extragalactic magnetic fields with strengths ∼ 10 −11 T to 10 −9 T within 1M pc in galaxies and galaxy clusters (see Fig. 8). These magnetic fields might provide a large space accumulation effect during the propagating of the HFGWs from possible sources to the Earth. This means that either the EM response to the HFGWs in the transverse background EM fields [see Eqs. (30), (31), (32), (33), (34), (35), (36) and (37)] or in the longitudinal background EM fields [see Eqs. (42) to (45)], it is all possible to detect or observe such effect provided these background EM fields are distributed in very widespread region. Since the wide distribution of the background galactic-extragalactic magnetic fields has been observed by the observational evidence [52], the EM response to the HFGWs in such background magnetic fields would have more realistic significance than that in the background electric fields, and the EM response in the background longitudinal magnetic fields, Eqs. (44) and (45), might provide observation evidence produced by the pure-additional polarization states (the x-and the ypolarizations of the HFGW).
It should be pointed out that the coupling between FIG. 8: The spatial scale of distribution for the background magnetic fields in the Milky Way reaches up to ∼ thousand light-years long [52][53][54], and the magnetic fields have a stable strength and direction in the scale. Thus any HFGWs passing through the background magnetic field would generate a significant space accumulation effect in the Earth region. This figure is made based on some materials of the Ref. [54].
the longitudinal perturbative electric fieldsẼ (1) z , Eq.(26), and the transverse perturbative magnetic fields, Eqs. (23) and (25), do not generate any transverse PPFs, i.e., This is because theẼ y , Eqs. (23), (25), have a phase difference of π/2. Therefore, the average values of the transverse perturbative EM power fluxes, Eq. (46), are vanishing. In fact, such results ensure the total momentum conservation in the interaction of the HFGWs (gravitons) with the background EM fields. However, the longitudinal perturbative electric fieldẼ (1) z will play an important role in the 3DSR system (see below, sect. IV). We will show that the transverse PPFs generated by the coupling between theẼ 1 and 2). The 3DSR was discussed in Ref. [51,55], so we shall not repeat it in detail here. In this article, the 3DSR system is different to that in previous studies, and we update it into a new system for probing the HFGWs having additional polarizations to display related novel effects, i.e: (1) Unlike previous EM detection schemes, here the 3DSR system contains not only the static magnetic field, but also the static electric field, and their directions can be adjusted. In this special coupling between the static EM fields and the Gaussian-type photon flux, the perturbative EM signals generated by the different polarization states (the tensor, the vector and the scalar mode gravitons in the high-frequency band) can be effectively distinguished and displayed.
(2) Since the EM signals generated by the interaction of the HFGWs with the background static EM fields, will have the same frequencies with the HFGWs. Thus once the GB is adjusted to the resonance frequency band for the HFGWs, then the first-order perturbative EM power fluxes [the perturbative photon fluxes (PPFs), i.e., the signal photon fluxes] also have such frequency band. This means that the 3DSR can be a detection system of broad frequency band.
In order to make the system having a good sensitivity to distinguish the PPFs generated by the different polarization states of the HFGWs, we select a new group of wave beam solutions for the GB in the framework of the quantum electronic (also see Appendix A): , z are the electric and magnetic components in Cartesian coordinate system for the GB, respectively, and only x-component E (0) x of the electric field has a standard form of circular mode of the fundamental frequency GB [56]. The concrete expressions of functions F 1 , F 2 , F 3 and F 4 can be found in the Appendix A.
In fact, there are different solutions of the wave beam for the Helmholtz equation, and they can be the Gaussian-type wave beams or the quasi-Gaussian-type wave beams. One of reasons of selecting such wave beam solutions, Eqs. (47) and (48), is that it will be an optimal coupling between the Gaussian-type photon flux and the background static EM fields, and will make the PPFs (the signal photon fluxes) and the background noise photon fluxes having very different physical behaviors in the special local region. These physical behaviors include the propagating direction, strength distribution, decay rate, wave impedance, etc (see below and Appendix B). Thus, such results will greatly improve the distinguishability between the signal photon fluxes and the background noise photons. Also, they will greatly increase the separability among the tensor mode, the vector mode and the scalar mode gravitons. This is the physical origin of the very low standard quantum limit of the 3DSR system (i.e., the high sensitivity of the 3DSR system, e.g., see Ref [57]).
By using Eqs. (47) and (48), the average values of the transverse background photon flux (the Gaussian-type photon flux) with respect to time in cylindrical polar co-ordinates can be given by where (0) T 02 are 01-and 02-components of the energy-momentum tensor for the background EM wave (the GB), and [56] i.e., the transverse background photon flux at the longitudinal symmetry surface (the yz-plane and the xz-plane) of the GB is equal to zero (see Fig. 9). In fact, this is the necessary condition for the stability of GB. From Eqs. (47), (48) and (22) to (26), and under the resonance condition (ω e = ω g ), the transverse perturbative photon fluxes (PPFs) in cylindrical polar coordinates can be given by where (1) T 01 ωe=ωg and (1) T 02 ωe=ωg are average values with respect to time of 01-and 02-components of energymomentum tensor for the first-order perturbative EM fields.
In the following, we shall study the EM response to the HFGWs with the additional polarization states in some of typical cases.
1. EM response to the HFGWs in the coupling system between the transverse background static magnetic field B (0) y and the GB. ThenÊ x =B (0) z = 0 and onlŷ B (0) y = 0. In fact, this is a coupling system between the transverse static magnetic fieldsB (0) y and the Gaussiantype photon flux. In this case, from Eqs. (28) to (29) and (47), (48) and (51), the concrete forms of the transverse PPFs can be obtained: where ∆z is the spatial scale of the transverse static magnetic fieldB (0) y in the 3DSR system. Eq. (52) shows that the transverse PPF n (1) φ−⊗ is produced by the pure ⊗-type polarization state of the HFGWs, while Eq. (53) represents that the transverse PPF n (1) φ−⊕,l is generated by the combination state of the ⊕-type and the l-type polarizations of the HFGWs.
2. EM response to the HFGWs in the coupling system between the transverse background static magnetic field B (0) x and the GB. ThenÊ x = 0. In this case, from Eqs. (48) and (51), we have, Eq. (54) shows that the PPF n   47), (48) and (51), the concrete forms of the transverse PPFs can be given by: Eqs. (56) and (57) show that the transverse PPFs n (1) φ−x and n (1) φ−y are generated by the pure x-type and the pure y-type polarizations of the HFGWs, respectively.
4. The EM response to the HFGWs in the coupling system between the transverse background static electric fieldÊ (0) x and the GB. ThenÊ x = 0. In this case from Eqs. (26), (48) and (51), in the same way, under the resonance condition (ω e = ω g ), the transverse PPFs can be given by Eqs. (58) and (59) show that the transverse PPFs n φ−⊕,l is generated by the combination state of the ⊕-type and the l-type polarizations, and n (1) φ−x is produced by the pure x-type polarization state. 5. The EM response to the HFGWs in the coupling system between the transverse background static electric field E Eq. (60) shows that the transverse PPF n (1) φ−⊗ is generated by the pure ⊗-type polarization state; Eq. (61) shows that the transverse PPF n (1) φ−⊕,l is generated by the combination state of the ⊕-type and the l-type polarizations, and Eq. (62) shows that n (1) φ−y is produced by the pure y-type polarization state.
6. The EM response to the HFGWs in the coupling system between the longitudinal background static electric fieldÊ (0) z and the GB. ThenÊ In the same way, under the resonance condition (ω e = ω g ), the transverse PPFs, can be given by Eqs. (63) and (64) show that the transverse PPFs, n (1) φ−x and n (1) φ−y are generated by the pure x-type polarization and the pure y-type polarization of the HFGWs, respectively. The Eq. (65) shows that the PPF n (1) φ−b,l is produced by the combination state of the b-type and the l-type polarizations.
In all of the above discussions, the ratio [of the electric component (Ẽ z )] of the PPFs is much less than the ratio of the background noise photon flux. This means that the PPFs expressed by the Eqs. (52) to (55), (56) to (62), (63) to (65) have very low wave impedance [58,59], which is much less than the wave impedance to the BPFs (see below). Then the PPFs (i.e., the signal photon fluxes) would be easier to pass through the transmission way of the 3DSR system than the BPFs due to very small Ohm losses of the PPFs.
According to the same way, and from Eqs. (47), the PPF in the EM response to the HFGWs in the coupling system between the transverse background static magnetic fieldB (0) x and the GB can be given by Eq. (66) shows that the PPF n φ−⊕,b,l , is generated by the combination state among the ⊕-type, the b-type and the l-type polarizations of the HFGWs. Obviously the ratio of the electric component to the magnetic component of the PPF is larger than that of the PPF n [Eq. (54)], i.e., the PPF expressed by Eq. (66) has larger wave impedance. However, its angular distribution factor cos 2 φ (see Fig. 10d) is quite different to the sin 2φ of the BPF, Eq. (49), Fig. 9. Thus, it is always possible to distinguish them.
7. Angular distributions of the strengths and the "rotation directions" of typical PPFs in the cylindrical polar coordinates.
Based on the above discussion, the angular distributions of the strengths and the "rotation directions" for some of typical transverse PPFs are listed in the following figures [Figs. (10a) to (10e)].
In Fig. 10a, the n (1) φ ∝ sin 2 φ, and it includes following five cases: (i) n (1) φ−⊗ , Eq. (52). This is the transverse PPF displaying the pure ⊗-type polarization state (the tensor mode gravitons) of the HFGWs. The PPF is from the EM response to the HFGWs in the coupling between the transverse static magnetic fieldB  Best detection position of all of such PPFs should be the receiving surfaces at φ = π/2 and φ = 3π/2 (see Fig.  10a), where the PPFs have their maximum, while the BPF (the background noise photon flux) vanishes at the surface (see Fig. 9). Because the BPF from the GB will be the dominant source of the noise photon fluxes, i.e., other noise photon fluxes [e.g., shot noise, Johnson noise, quantization noise, thermal noise (if operation temperature T < 1K), preamplifier noise, etc.] are all much less than the BPF [60], in order to detect the PPFs generated by the HFGWs (ν ∼ 10 9 to 10 12 Hz, h ∼ 10 −21 to 10 −23 ) in the braneworld [18], the requisite minimal accumulation time of the signals can be less or much less than φ−y ∝ sin φ cos 2 φ, respectively.
rection" expressed by Fig. 10a is completely "left-handed circular" or completely "right-handed circular", and the "left-handed circular" or "right-handed circular" property depends on the phase factors in Eqs. (52), (54), (56), (61) and (64). In Fig. 10b, the n (1) φ ∝ sin 2φ, and it includes following five cases: (i) n (1) φ−⊕,l , Eq. (53). This is the transverse PPF displaying the combination state of the ⊕-type and the ltype polarizations (the tensor-mode and the scalar-mode gravitons) of the HFGWs. The PPF is from the EM response to the HFGWs in the coupling between the transverse static magnetic fieldB    57). This is the transverse PPF displaying the pure y-type polarization state (the vector-mode gravitons) of the HFGWs. The PPF is from the EM response to the HFGWs in the coupling between the longitudinal static magnetic fieldB (0) z and the GB in the 3DSR. (iv) n (1) φ−⊕,l , Eq. (58). This is the transverse PPF displaying the combination state of the ⊕-type and the ltype polarizations (the tensor-mode and the scalar-mode gravitons) of the HFGWs. The PPF is from the EM response to the HFGWs in the coupling between the transverse static electric fieldÊ (0) x and the GB in the 3DSR.
(v) n (1) φ−x , Eq. (63). This is the transverse PPF displaying the pure x-type polarization state (the vector-mode gravitons) of the HFGWs. The PPF is from the EM response to the HFGWs in the coupling between the longitudinal static electric fieldÊ (0) y and the GB in the 3DSR. Unlike Fig. 10a, here the "rotation direction" of the PPFs are not completely "left-handed circular" or not completely "right-handed circular", and it and the trans-verse BPF have the same angular distribution [see Eq. (49)]. Thus the displaying condition in Fig. 10b will be worse than that in Fig. 10a. However, because the transverse PPFs in Fig. 10b and the BPF have other different physical behaviors, such as the different wave impedance, decay rate, and even different propagating directions in the local region, it is always possible to display and distinguish the PPFs from the BPF.
In Fig. 10c, the n (1) φ ∝ sin φ, and it includes following three cases: (i) n (1) φ−x , Eq. (59c). This is the transverse PPF displaying the pure x-type polarization state (the vector-mode gravitons) of the HFGWs. The PPF is from the EM response to the HFGWs in the coupling between the transverse static electric fieldÊ (0) x and the GB in the 3DSR. (ii) n (1) φ−y , Eq. (62b). This is the transverse PPF displaying the pure y-type polarization state (the vector-mode gravitons) of the HFGWs. The PPF is from the EM response to the HFGWs in the coupling between the transverse static electric fieldÊ (0) y and the GB in the 3DSR. (iii) n (1) φ−b,l , Eq. (65b). This is the transverse PPF displaying the combination state of the b-type and the l-type polarizations (the scalar-mode gravitons) of the HFGWs. The PPF is from the EM response to the HFGWs in the coupling between the longitudinal static electric fieldÊ (0) z and the GB in the 3DSR.
Unlike Fig. 10a, here the "rotation direction" of the PPFs is not completely "left-handed circular" or not completely "right-handed circular". However, the best detection position of the PPFs is also the receiving surfaces at φ = π/2 and 3π/2 [see Fig. 10a and 10c], where the PPFs have their peak values while the BPF vanishes.
In Fig. 10d, the n (1) φ−⊕,b,l ∝ cos 2 φ, Eq. (66), and it is the transverse PPF displaying the combination state of the ⊕-type (the tensor-mode gravitons) and the b-type, l-type polarizations (the scalar-mode gravitons) of the HFGWs. The PPF is from the EM response to the HFGWs in the coupling between the transverse static magnetic fieldB φ−⊕,b,l , Eq. (66) are surfaces at φ = 0, π. Especially, the peak value areas of the PPF (the signal photon fluxes) are just the zero value area of the BPF (the background noise photon flux), Eq. (49) and Fig. 9.
The PPFs, n φ−⊕,b,l , Eq. (54) and Eq. (66), are both from the same EM response to the HFGWs in the coupling between the transverse static magnetic fieldB (0) x and the GB. This means that displaying the PPFs at such areas would have very strong complementarity. Moreover, like the PPFs expressed in Fig. 10a, here, the PPF n (1) φ−⊕,b,l is also completely "left-handed circular" or completely "right-handed circular" (see Fig. 10d).
In Fig. 10e, the n φ−y ∝ sin φ cos 2 φ, Eqs. (59b), (59d), (62a) and (62c), and it is the transverse PPF displaying the pure x-type polarization (the vector-mode gravitons) of the HFGWs and the pure y-type polarizations (the vector-mode gravitons) of the HFGWs, respectively. For the former, the PPF is from the EM response to the HFGWs in the coupling between the transverse static electric fieldÊ (0) x and the GB. For the latter, the PPF is from the EM response to the HFGWs in the coupling between the transverse static electric fieldÊ φ−y , in Fig. 10e is worse than the PPFs in Figs. 10a to 10d. Nevertheless, considering obvious difference of other physical behaviors (e.g., the wave impedance, the decay rate, the propagating direction, etc.) between the PPFs in Fig. 10e and the BPF in Fig. 9, their distinguishing is still possible.
The above discussions show that the three polarization states (the ⊗-type, the x-type and the y-type polarizations, i.e., the tensor-mode and the vector-mode gravitons) of the HFGWs can be clearly separated and distinguished. On the other hand, ⊕-type polarization (the tensor-mode gravitons), the b-type and the l-type polarizations (the scalar-mode gravitons) of the HFGWs are often expressed as their combination states to generate the PPFs. However, from the PPFs produced by these combination states, it is easy to calculate the PPFs generated by the pure ⊕-type, the pure b-type and the pure l-type polarizations, and thus we can completely determine these polarizations, respectively. From Eqs. (53) to (54) and (65b), we have where n φ−⊕,l , Eq. (53), and n (1) φ−b,l , Eq. (65b) are the PPFs generated by the combination state of the ⊕-type, the b-type, the l-type polarizations, by the combination state of the ⊕-type, the l-type polarizations, and by the combination state of the b-type, the l-type polarizations, respectively.
Clearly, n φ−b and n (1) φ−l in Eq. (67) are the PPFs generated by the pure ⊕-type, the pure b-type and the pure l-type polarizations of the HFGWs, respectively. By using Eq. (67), it is easy to calculate and find: Notice that the each term n φ−⊕,l and n φ−l can be completely confirmed.
So far the PPFs produced by the six polarization states (the ⊗-type, the x-type, the y-type, the ⊕-type, the b-type and the l-type polarizations) of the HFGWs can be calculated and completely confirmed [e.g., see Eqs. (52), (56), (64), (68), (69) and (70), respectively]. In other words, the six polarizations of the HFGWs can be clearly displayed and distinguished in the EM response of our 3DSR system.

V. NUMERICAL ESTIMATIONS OF THE PERTURBATIVE PHOTON FLUXES
1. Perturbative photon fluxes (signal EM waves) in the 3DSR system.
Unlike the EM response to the HFGWs in the galactic-extragalactic background EM fields, the 3DSR is a closed cryogenic system with vacuum, which is shielded and isolated by superconductor materials from outside world. The EM response to the HFGWs in the 3DSR system has following important characteristics: (i) Since shielding of the 3DSR system to the EM fields from outside region, the large space-accumulation effect of the PPFs (the signal EM power fluxes) cannot enter and influence the EM fields inside the 3DSR system. However, the superconductor and the shell of the 3DSR system are transparent to the HFGWs. Thus, the PPFs inside the 3DSR system should be calculated by the HFGW amplitudes at the Earth (the far-field amplitudes, which are much less than the amplitudes in the near-field region of the HFGW source).
(ii) Unlike the galactic-extragalactic background magnetic fields (B (0) x ∼ 10 −9 to 10 −11 T ), the background static magnetic fieldsB (0) of the 3DSR system can reach up to ∼ 10T or larger. The cooperation institute (High Magnetic Field Laboratory, Chinese Academy of Sciences) of our research team has been fully equipped with the ability to construct the superconducting magnet [61], and it is also the builder of the superconducting magnet for the Experimental Advanced Superconducting Tokamak (EAST) for controlled nuclear fusion. The magnet can generate a static magnetic field withB (0) = 10T in an effective cross section with diameter of at least 80cm to 100cm, and operation temperature can be reduced to 1K or less. Obviously, such magnetic field is much stronger than the galactic-extragalactic background magnetic fields, although typical spatial dimension of the former is only of the order of magnitude of a meter (typical laboratory dimension).
(iii) The major noise sources of the EM signals from the case of galactic-extragalactic EM fields (see below), would be from the space microwave background, and the key noises of the case of laboratory based high magnet are from the microwave photons inside the 3DSR system, which are almost independent of the space background EM noise and the cosmic dusts.
(iv) Because the PPFs are the first-order perturbations to the background EM fields and not the second-order perturbations, i.e., the PPFs (signal photon fluxes), Eqs. (52) to (55), (56) to (62) and (63) to (66), in the 3DSR system are proportional to the amplitudes themselves (h) of the HFGWs and not their square h 2 , e.g., see Eqs. (42) and (45) etc. (i.e., could be the PPFs in the galactic-extragalactic background magnetic fields). Then the parameter hB (0) in the first-order PPFs [e.g. see Eqs. (52) and (53)] will be much larger than the parameter (hB (0) ) 2 in the second-order PPFs [e.g. see Eqs. (30) and (31)]. This property effectively compensates the weakness of the far-field amplitudes of the HFGWs in the 3DSR system. Of course, since the PPFs in the 3DSR system are always accompanied by the noise photons inside the system, which mainly are from the background photon fluxes caused by the GB. Thus, in order to identify the total signal photon flux at an effective receiving surface ∆s, the time accumulation effect of the PPF must be larger than the effect of the noise photon flux fluctuation at the receiving surface ∆s.
As mentioned above, in order to display the relatively weak PPFs in the background noise photon fluxes (BPFs), we need an long enough accumulation time of the signal [see Appendix B]. For the typical parameters of the HFGWs predicted by the braneworld scenarios [18,19], the observing and distinguishing of the HFGWs would be quite possible due to their large amplitudes, higher frequencies and the discrete spectrum characteristics. The measurement of the relic HFGWs will face to big challenge, but it is not impossible (see Table II). Table II shows the displaying conditions of the HFGWs for some typical cosmological models and high-energy astrophysical processes, where n (1) φ(total) is the total signal photon flux at the receiving surface ∆s (∆s ∼ 3 × 10 −2 m 2 ), and n (0) total is the allowable upper limit of the total noise photon flux at the surface ∆s for various values of the HFGW amplitudes, and the ∆t min [see Eq.(B1) in Appendix B] is the requisite minimal accumulation time of the signals; ν e = ν g = 3 × 10 9 Hz or 3 × 10 8 Hz (the resonance frequency); the background static magnetic field B (0) is 10T; the interaction dimension ∆z is 60cm (typical dimension of the static magnetic field in our 3DSR); the power of the Gaussian beam is ∼ 10W and the operation temperature should be less than 1K.
Notice that for the GB of P ∼ 10W in the 3DSR, the maximum of n  Table II). In fact, in many cases discussed in this paper, the peak position of such two kinds of photon fluxes do not appear at the same receiving surface, and especially, the peak value positions (e.g., see Figs. 10a, 10c and 10d) of the signal photon fluxes are just the zero value areas (φ = 0, π/2, π, 3π/2) of the background noises photon flux n (0) φ (see Fig. 9). In this case the displaying condition can be further relaxed.
It is very interesting to compare the displaying and distinguishing conditions for the case in the galacticextragalactic fields (e.g. see part 2 of this section and section III) and in the 3DSR (section IV). In section III, the large space accumulation effect caused by the galactic-extragalactic fields is discussed, and the section IV shows the longer time accumulation effect in the strong background static magnetic field and the resonance response of the GB in the 3DSR. They all can effectively compensate the weakness of the HFGW amplitudes. Clearly, they would be highly complementary to each other. Moreover, the PPFs 3 × 10 9 Hz ∼ 10 2 ∼ 10 6 ∼ 10 10 Quintessential inflationary [28][29][30] or upper limit of ordinary inflationary [24,25] produced in the 3DSR system by the relic HFGWs in the quintessential inflationary, the pre-big-bang and the short-term anisotropic inflationary models, would only be 10 2 s −1 to 10 6 s −1 (see Table II), which are much less than the maximum of the transverse BPF (noise photon flux). However, using very different physical behaviors between the PPFs and the BPFs, especially their very different strength distributions and propagating directions [e.g., see Figs. 9, 10a, 10b, 10d, and Eqs. (B3) and (B4) in Appendix B], it is still possible to reduce the BPF to the allowable upper limits at a suitable receiving surface (see Table II). Considering their very different wave impedances (four orders of magnitude at least [58]), the displaying of the transverse PPFs produced by the relic HFGWs may also be possible though face to big challenge.
2. The perturbative photon fluxes (PPFs) in the EM response of galactic-extragalactic background stable EM fields to the HFGWs.
Because there are very widely distributed galactic-extragalactic stable or quasi-stable stable EM fields (e.g. especially, background magnetic fields [52]), the interaction of the HFGWs with the galactic-extragalactic stable magnetic fields could generate the PPFs when the HFGWs passing through such background magnetic fields.
In the above discussion, we assume that the GWs are planar waves. For the observers and detection systems in the far field region from the GW sources, such assumption, is obviously reasonable. However, if we study a large space accumulation effect in the EM response to the HFGWs, which is in large distance from their sources to the Earth, then such HFGWs should be considered as the spherical GWs for GW sources in the local regions, i.e., these HFGW sources would be in approximately pointlike distribution. This means that the amplitudes of the HFGWs emitted by such sources would be inversely proportional to the propagating distance z (i.e., along the zaxis in our coordinate system). Then the amplitudes A ⊕ , A ⊗ , A x , A y , A b and A l of the HFGWs in the Eqs. (30), (31), (32), (33), (34), (35), (36), (37) and (42) to (45) should be replaced by a ⊕ z 0 /z, a ⊗ z 0 /z, a x z 0 /z, a y z 0 /z, a b z 0 /z and a l z 0 /z, respectively, where z 0 is the reasonable reference distance of the near-field region, which is much less than the distance between the Earth and the HFGW sources, and the a ⊕ , a ⊗ , a x , a y , a b and a l are the amplitudes in the near-field region of the HFGW sources. Obviously, a ⊕ , a ⊗ , a x , a y , a b and a l are much larger than the amplitudes A ⊕ , A ⊗ , A x , A y , A b and A l in the farfield region.
On the other hand, all of the perturbative EM fields are proportional to the propagating distance z due to the space accumulation effect caused by the same or almost the same propagating velocity between the perturbative EM waves (PPFs) and the HFGWs (gravitons) [also, see Eqs. (30), (31), (32), (33), (34), (35), (36), (37) and (42) to (45)]. Then the parameter z in the numerator and in the denominator in these equations will be canceled each other. In other words, the strength of the perturbative EM fields (i.e., the PPFs) would be a composite effect of the space accumulation of the EM signals and the decay of the HFGWs. According to previous treatment method [62] of such effect and the above equations, in the same way, the PPFs generated by the EM response in the background stable magnetic fields, can be reduced to following forms, respectively,  [18] in the galactic-extragalactic magnetic fields, where N (2) is the perturbative photon flux densities (m −2 ), P is corresponding EM signal power flux densities (W m −2 ) at the Earth, and here we assume that the near-field amplitudes a ⊕ , a ⊗ , a x , a y , a b and a l of the HFGWs in Eqs. (71) to (75), have the same strengths, then the PPFs calculated by such equations have approximately the same order of magnitude. Here, "a" represents typical near-field amplitude of the a ⊕ , a ⊗ , a x , a y , a b , a l .
Background magnetic fieldB (0) ν g ∼ 10 9 Hz ν g ∼ 10 12 Hẑ B where z 0 is the reference distance in the near-field region. If such HFGWs are from point-like objects orbiting a braneworld black hole, then z 0 is at least in the scale of the horizon size of the black hole, and z 0 ≈ 10 4 m due to the horizon size of a black hole with a mass of 10M ⊙ [18,62].
Notice that a ⊕ , a ⊗ , a x , a y , a b and a l in Eqs. (71) to (75) are the amplitudes of the HFGWs in the near-field region of the HFGW source, which are much larger than the amplitudes A ⊕ , A ⊗ , A x , A y , A b and A l in the far-field region (e.g., the Earth) of the HFGW source. As mentioned above, this is because the large space accumulation effect would compensate the decay of the spherical HFGWs and their weakness of the far-field amplitudes.
Fortunately, related observation shows that the distribution region of the stable background magnetic field (see Fig. 8) in our galaxy is nearly few thousand light-year distance (∼ 10 19 m [52,53]), and such magnetic field basically keeps fixed direction and intensity ∼ 10 −9 to 10 −10 Tesla, thus this background magnetic field would provide an effective space-accumulation effect to the PPFs produced by the HFGWs.
According to related estimation for the distance of possible HFGW sources of the braneworld scenarios [e.g., see Ref. [18]] in our galaxy, they may be ∼ 10 18 m to ∼ 10 19 m away from the Earth. In this case, if the amplitudes of the HFGWs (ν ∼ 10 9 to 10 14 Hz) can reach up to the predicted values once they arrive at the Earth, i.e., h ∼ 10 −25 (lower bound) to h ∼ 10 −21 (upper bound), then the near-field amplitude of the HFGWs would be h 0 ∼ 10 −6 to 10 −10 due to the horizon size of the black hole with a mass of 10M ⊙ . Table III shows the estimations of the PPFs (the signal EM power flux densities) generated by the EM response of the background stable magnetic fields to the HFGWs. HereB (0) ∼ 10 −11 T to ∼ 10 −10 T ; the propagating distance ∆z ∼ 10 19 m of the HFGWs from the source to the Earth; the amplitudes of the HFGWs at the Earth are h ∼ 10 −21 to 10 −25 [18]; then the near-field amplitudes of such HFGWs would be a ∼ 10 −6 to 10 −10 , roughly.
Obviously, the PPFs (the EM signals) in Table III are larger than the minimal detectable EM power ∼ 10 −22 W m −2 to ∼ 10 −24 W m −2 under current technological condition [e.g., the Five-hundred-meter Aperture Spherical Telescope (FAST) [63,64], which has been constructed in 2017 in Guizhou province of China. However, above EM signals and the space background EM noise are often accompanied together. Thus distinguishing of the EM signals from the background EM noise at the Earth will be a major challenge. Fortunately, the PPFs (the EM signals) produced by the HFGWs from the braneworld also contain important characteristics of the HFGWs, such as the discrete spectrum property, special waveform, strength distribution and very-high frequency features. Thus, it is always possible to distinguish and display the PPFs (the EM signals) from the background EM noise.
It should be pointed out again that if the propagating direction of the GWs (including the HFGWs) and the pointing direction of the background EM fields are parallel to each other (i.e., only the longitudinal EM fieldsÊ (0) z ,B (0) z exist here), then GWs in the GR framework do not generate any perturbative effect to the EM fields [49,50]. Unlike the physical behavior of the GWs in the GR framework, the GWs having additional polarization states will generate perturbative EM fields to the background EM fields in any case, even if the pointing direction of the background EM fields and the propagating direction of the HFGWs are parallel to each other [see Eqs. (42), (43), (74), (75)], which are just the perturbation effects produced by the pure additional polarization states (the x-type and the y-type polarizations, i.e., the vector mode gravitons) of the HFGWs.
In this case, displaying condition of the HFGWs can be greatly relaxed. In other words, this is an important symbol to display and distinguish the HFGWs in the GR framework and the HFGWs beyond the GR, whatever in the astrophysical scale (e.g. see section III) or in the laboratory dimension (e.g. in 3DSR, section IV).
HFGWs almost have the same order of magnitude in the frequency band of ∼ 10 9 Hz to 10 12 Hz, and because of ω g = KT (K is the Boltzmann constant), the power of the PPFs of 10 12 Hz will be roughly 10 3 times larger than that of the PPFs of 10 9 Hz, and requirement of operating temperature can be relaxed to 140K from 0.14K. In this case, for the background photon flux n Besides, we need to mention two points: (1) Since the 3DSR is a detection system fixed on the Earth, so the 3DSR will have a rotation period of 24 hours to any GW sources in the local space region due to the rotation of the Earth. Thus the background "transverse" and "longitudinal" EM fields in the 3DSR will be periodically changed with the rotation of the Earth. In fact, this is an issues on the relationship between the characteristic parameters (i.e., each polarization component) of the HFGWs and the coordinate rotation. In our 3DSR system, this is an issue on the relationship between the EM response to the HFGWs and the coordinate rotation. Nevertheless, the 3DSR has a special and definite direction, namely, the positive direction of the symmetrical axis of the GB (i.e. the z direction in section IV). We have showed [55] that only when the propagating direction of the HFGWs and the positive direction of the symmetrical axis of the GB are the same, the PPFs reach their maximum. If they are perpendicular or opposite to each other, then the PPFs will be one or two orders of magnitude smaller than their peak values. Thus we can determine the propagating direction of the HFGWs by the peak moments of PPFs. The preliminary calculations show that within about two hours around the peak moments, the PPFs can basically maintain the intensity at the same orders of magnitude as the peak values. This means that we can maintain an effective signal accumulation time of ∼ 7 × 10 3 s for the PPFs, i.e., it basically approaches 10 4 s in Table II. This is satisfactory. Moreover, the 3DSR is an EM detection system in small scale (order of magnitude of a meter). Thus its spatial orientation can also be effectively adjusted to the best or near optimal direction for the possible HFGW sources.
(2) The detection frequency band of the 3DSR is ∼ 10 8 Hz to 10 12 Hz or higher, and the observational frequencies (for EM signals) of the FAST is ∼ 7 × 10 7 Hz to 3 × 10 9 Hz [64,65]. This means that the detection frequency bands of the 3DSR and the FAST are overlapping partly in the GHz band. Therefore, the cooperation and coincidence experiments of them will have very strong complementarity to distinguish and display the possible all six polarizations of the HFGWs in microwave frequency band.
The two points mentioned above will be discussed and studied in detail elsewhere.
Appendix A: A new group of special solutions of the Gaussian-type photon flux for the Helmholtz equation In this paper, the 3DSR is actually a coupling system between the background static EM fields and the Gaussian-type photon flux (Gaussian beam). Under the condition of the resonance response to the HFGWs, the PPFs (the signal photon fluxes) and the BPFs (including other noise photons) have very different physical behaviors in the local regions. Thus it makes the 3DSR system having very low standard quantum limit. In Refs. [51,55] we have given several coupling forms between the GB and the background static magnetic field. Here we select a new group of solutions of the Helmholtz equation, and give their complete expressions. Moreover, in order to display and distinguish effectively all six polarization states of the HFGWs, here such GB does not only couple with the transverse static EM fields, but also the longitudinal static EM fields. In this case, it is possible to display and distinguish all of the different polarization states of the HFGWs.
General form of the circular mode GB of fundamental frequency is [56] where ψ 0 is the amplitude of electrical field of the GB, f = πW 2 0 /λ e , W = W 0 1 + (z/f ) 2 , R = z + f 2 /z. The W 0 is the minimum spot radius, R is the curvature radius of the wave front of the GB at z, ω e is the angular frequency, λ e is the EM wavelength, the z-axis is the symmetrical axis of the GB, and δ is a phase factor.
According to Eq. (A1) and using the the nondivergence condition ∇ ·Ẽ (0) = 0 in the free space, and B (0) = −i/ω e · ▽ ×Ẽ (0) , a group of special solutions we selected are as follows, and the components of the GB in the Cartesian coordinates can be given bỹ i ω e ∂ψ ez ∂y = sin 2φ ω e F 2 (x, k e , w), −i(2F 1 (x, k e , W ) + cos 2 φF 3 (x, k e , W ))], ]ψ. (A3) where, ]}ψdz, (A5) According to Eqs. (A1) to (A7), we obtained the strength distribution (see Fig. 9) of the transverse background photon flux n (0) φ in the cylindrical polar coordinates. As demonstrated in this paper, the PPFs (signal photon fluxes) and the BPF (the dominated noise photon flux) have very different physical behaviors (such as the strength distribution, propagating direction, decay rate, wave impedance etc.) in the local regions. This is one important physical mechanism of the displayability and separability between the PPFs and the BPF, including the different polarizations of the HFGWs. = 0, the noise photon flux fluctuation in the displaying condition, Eq. (B1), will not be caused by the BPF of this direction, but would be caused by other noise photon fluxes. However, the latter are much less than the former [60] in the 3DSR. In this case, the displaying condition in Table II can be greatly relaxed. In other words, the PPFs produced by the HFGWs in the braneworld [18] and in the short-term anisotropic inflation [31], can almost be instantaneously displayed, and the requisite minimal accumulation time of the signal displaying the HFGWs predicted by the interaction [32] of astrophysical plasma with intense EM waves, by the pre-big-bang [26,27] and by the quintessential inflationary [28][29][30] models, would be further relaxed effectively. Thus, utilizing a highly orientational receiving surface to display the PPFs, will be very useful.
By the way, the wave impedance and wave impedance matching would also play important roles for the displaying condition. For most of the signal photon fluxes discussed in this paper, ratios of the electric components to the magnetic components are much less than that of the background noise photon fluxes. The typical values of the wave impedance of the former are ∼ 10 −4 Ω or less, and the typical values of the latter are ∼ 100Ω to 377Ω [58,59]. This means that the 3DSR system would be equivalent to a "good superconductor" to the PPFs. Thus, the PPFs could be distinguished from the BPF and other noise photons by the wave impedance matching, and this is another important symbol to distinguish them.
It is very interesting to compare the PPFs n (1) x−⊗ , n (1) x−x propagating along the x-direction and the PPF n   x−⊗ , n (1) x−x , Eqs. (B4), Eqs. (B5), is much better than that of n (1) y−⊕,b,l , but distinguishing and displaying the n i.e., the decay rate of n (1) y−⊕,b,l is obviously slower than that of n (0) y , although the peak value of n (0) y is much larger than that of n y−⊕,b,l would be measurable. In this case, due to the background photon flux decays faster than the signal photon flux, in the area around y = 0.25m, the n (0) y already decays into the comparable level to the n (1) y−⊕,b,l , and in the further area, the background noise will decay into lower level than the measurable signal photon flux. This is satisfactory. In this case, the zero value characteristics of the background noise photo flux BPFs n (0) x , n (0) y at the longitudinal symmetric surface, and the peak property of the signal photon fluxes ( n (1) x−⊗ , n (2) x−⊗ ) at the longitudinal symmetric surface, inspire us to utilize the fractal membranes (they are very effective microwave lenses with strong focusing function and "one-way valve" property [55,[66][67][68][69] to the photon fluxes in the GHz to THz band), and they will provide better selecting way for the signal photon fluxes, especially for the signal photon fluxes generated by the stochastic relic HFGWs. These issues will also be discussed and studies in detail elsewhere.