- Email: [email protected]
Manipulation of gravitational waves for communications applications using superconductors
Physica C 433 (2005) 101–107 www.elsevier.com/locate/physc
Manipulation of gravitational waves for communications applications using superconductors R.C. Woods
*
Department of Electrical and Computer Engineering and Microelectronics Research Center, 2128 Coover Hall, Iowa State University, Ames, IA 50011–3060, USA Received 31 August 2005; accepted 4 October 2005
Abstract Previously published calculations claim that gravitational waves propagate inside superconductors with a phase velocity reduction of 300 times and a wavenumber increase of 300 times. Subsequent claims that this result is not credible appear to be either not tenable or not complete. This result has major consequences for the design of instruments to generate and detect gravitational waves, in particular high-frequency gravitational waves (HFGWs) having wavelengths on the same order as the dimensions of typical superconductive components. It is generally assumed that in free space the velocity of an HFGW is the same as that of light and so the free space wavelength of an HFGW at 3 GHz will be 10 cm. Inside a superconductor, the corresponding 3 GHz HFGW wavelength will therefore be 300 lm. The present paper will discuss the technical consequences of this surprising result. In particular, such a large mismatch in HFGW propagation impedance inevitably results in large Fresnel reflections from superconductor-air interfaces. This will cause a number of design problems in equipment proposed for HFGW generation and detection. For example, the superconductor thickness used in HFGW detectors will be critical. It may also be possible with care to exploit the result to obtain HFGW resonators and focusing reflectors. 2005 Elsevier B.V. All rights reserved. PACS: 85.25.Am Keywords: Blooming; Fresnel reflections; Gravitational waves; HFGWs; Optical coating; Optics; Refraction; Superconductors
1. Introduction
*
Tel.: +1 515 294 3310; fax: +1 515 294 8432. E-mail address: [email protected]
Li and Torr [1,2] published calculations of the propagation behavior of gravitational waves inside a superconductor (SC). They claimed that the
0921-4534/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.10.003
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phase velocity of gravitational waves in any SC material would be 106 m s1, i.e., 300 times less than the phase velocity of gravitational waves in other materials, generally assumed equal to the speed of light in a vacuum (3 · 108 m s1). This corresponds to a wavenumber increase of 300 times in SC materials. This result has major consequences for the design of future instruments to generate and detect gravitational waves, which may need to be able to focus, refract, reflect, or manipulate gravitational waves for efficient coupling to detectors, transmitters, generators, resonant chambers, and other transducers. In particular, high-frequency gravitational waves (HFGWs) have wavelengths on the same order as the dimensions of typical SC components envisaged for HFGW use and so any suggestion of phase velocity change will affect the design of those components typically to be used in a future communications link based upon this technology. Kowitt [3] and Harris [4] independently argued that the Li and Torr results [1,2] are not credible. KowittÕs objection [3] was that Li and Torr [1] assumed a value of magnetic permeability of zero (or, equivalently, a magnetic susceptibility of 1) inside a SC. This is an experimental result, widely accepted, for macroscopic samples but the question appears to be that of whether the macroscopic experimental result is true microscopically. It is perfectly true that a value of magnetic permeability different from that of free space may be obtained macroscopically in magnetically active samples in which individual magnetic moments of atoms align with the applied field. Since this enhanced permeability arises from the aggregate effect of localized magnetic moments, it follows that a renormalized permeability is not definable microscopically. However, zero permeability in a SC results from cancellation of the magnetic field H within the SC by distributed current loops giving a distributed magnetization M. In a typical type-II SC, the circulating supercurrents causing this distributed magnetic moment must be outside the magnetic vortices, of course. These circulating supercurrents extend throughout the material, whereas in a conventional magnetic material the notional currents producing the
magnetic moments are those caused by orbiting electrons bound to atoms. While it is the case that in a conventional magnetic material it is not possible to define an effective permeability that has local microscopic significance, this argument does not apply to the extended supercurrents found in SC materials. For these reasons, KowittÕs objection [3] appears not to be tenable. HarrisÕs objection [4] must be taken more seriously; he claims that Li and Torr assume an arbitrary value for the observerÕs distance of 106 cm in one place, and 1013 cm at another point in their calculation, and that both values are unreasonable. This objection is harder to refute but applies only to Torr and Li [2]. Li and Torr [1] predict a changed HFGW phase velocity and establishing its precise value may require an experimental test. However, Fontana [5] has confirmed that the phase velocity of HFGWs is slowed in a SC, and Modanese and Fontana [6] also imply the same point. It appears that at the current time, this issue may be classified as unresolved; that should not prevent deduction of further consequences if it were true, which may provide further tests of the result. The present paper discusses some direct technical consequences of assuming that the Li and Torr result is correct, as those consequences may suggest further possible tests of a prediction that is rather difficult to test experimentally with current technology. In particular, such a large mismatch in HFGW phase velocity results in a huge mismatch in HFGW propagation wave impedance at SC-air interfaces. This in turn inevitably causes large Fresnel reflections from those interfaces. These will produce a number of design problems in SC lenses proposed for HFGW generation and detection [7]. In practice, it seems that it may be easier to exploit the large reflection coefficient of HFGW from SC surfaces, by fabricating focusing HFGW mirrors rather than lenses. These would be designed in a similar manner to existing optical mirrors, but heavy supporting structures would be easy to incorporate since any non-SC material is expected not to affect the HFGWs in any way and so structural frameworks will not vignette the aperture at all.
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2. Superconducting HFGW components and Fresnel reflection Exploitation of the Li and Torr result [1,2] in HFGW detector design was first suggested by Woods and Baker [8]. Since a SC is a slowing material for HFGW, in principle many geometrical optical components might be designed by analogy with conventional optics. If HFGW are capable of being focused, then the usual results from geometrical and physical optics would be expected to carry over in virtually unchanged form. This means that the minimum diffractionlimited focused spot diameter is 2k/p (corresponding to the best case of focusing from a convergence angle of 90), where k is the HFGW wavelength in the material at the focusing point [9]. Hence, a diffraction-limited spot has area 3002 = 90 000· smaller inside SC than in free space, and so the corresponding energy density is 90 000· (=50 dB) larger inside a SC than in free space. Clearly this result alone merits further attention in the case of noise-limited signal detectors, since it implies considerably easier signal detection under adverse conditions, as well as the notion that HFGW may be focused by suitably shaped SC. For a HFGW having a typical frequency of 4.9 GHz [8], the wavelength k0 = 6.1 cm in air, and so k = 200 lm in SC material. One immediate consequence is that SC components for manipulation of HFGW at this frequency must be made to a tolerance of 20 lm or better for effective optical performance (using the well-known k/10 practical criterion). In view of the current difficulty of fabricating the high-temperature SC ceramic materials to tight tolerances, it may be more cost-effective to use conventional low-temperature SC alloys instead, for which normal machining may be employed. The disadvantage, of course, is that liquid He refrigerant must be used in this case. The relations between the components of waves incident upon a plane boundary between two different optical media are given by the Fresnel relations [9]. These relations are derived by assuming phase-matching at the interface boundary, and so the results are dependent only upon the phase velocities in the materials. The full statement of
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the Fresnel relations depends upon the wave angle of incidence to the normal vector, hi, as well as the refractive index, n. However, generally the results for normal incidence (hi = 0) are sufficient to deduce the important behaviors concerned. At normal incidence, the reflection coefficient is r = (1 n)/(1 + n), and the transmission coefficient is t = 2/(1 + n). (For comparison, the results for grazing incidence, hi = 90, are r? = 1, rk = + 1, and t = 0, where the subscripts ? and k denote the polarization perpendicular to and parallel to the plane defined by the incident, transmitted, and reflected rays.) In almost all cases of interest the transmittance, T = nt2 = 4n/(1 + n)2, is of greater relevance as it gives the ratio of the transmitted intensity to the incident intensity taking correct account of the different wave impedance in the second medium. Using these general results, for HFGW traveling from air to SC, n = 300, and so r = 299/ 301 1 (i.e., corresponding to 180 phase change on reflection at a boundary of air to SC, with almost no amplitude change on reflection). The corresponding transmittance is T = 1.3%. Conversely, for HFGW traveling from SC to air, n is replaced by the inverse value 1/300, and so r = 299/301 + 1 (i.e., corresponding to no phase change on reflection at a boundary of SC to air, with almost no amplitude change on reflection). The corresponding transmittance is T = 1.3%, exactly the same as for HFGWs traveling from air to SC. These results have importance in designing a HFGW lens structure using SC material. Since the transmittance is only 1.3% at each air-SC interface in either propagation direction, it is expected that a simple SC lens analog of a simple optical glass lens would have a transmittance of only 0.017%, i.e., the combined transmittance of two air-SC boundaries, one each side of the lens. In other words, a simple HFGW lens made of SC material will introduce an energy loss of 99.98%, or an insertion loss of 38 dB. This is, of course, an intolerable overhead on the use of a lens, since it squanders most of the 50 dB improvement originally expected using a SC focusing arrangement. A simple lens analog thus appears impractical without further improvement.
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The technique of ‘‘optical blooming’’ or ‘‘optical coating’’ was developed many years ago to minimize Fresnel loss in conventional lenses, and merits examination in the present context. The coating material is added in a uniform film over a highly reflective interface between two non-opaque materials. The film thickness is exactly onequarter of a wavelength of the HFGW traveling in the coating material, and the refractive index of the coating is exactly equal to the geometric mean of the two surrounding materials. If one material is air (refractive index unity) and the other is SC, then p the coating material must have refractive index n = 17.3. The value of the refractive index ensures that Fresnel reflections from the air-coating and coating-SC interfaces have equal amplitudes, and the coating thickness ensures that at normal incidence the two reflections are out of phase by 180 and so exactly cancel by destructive interference. There is therefore no mechanism for losing energy at the interface, and so the overall transmittance must be 100%. Although this scheme is very effective for conventional optics and is widely used in all but the very simplest of lens designs, there are a number of major problems with using it for HFGW lenses made from SC. p Firstly, how can a coating material having nc = n = 17.3 be made? At present, the only known HFGW refractive indices are for SC (n = 300, assuming the Li and Torr prediction [1]) and for everything else (n = 1). One method to try is to make a suspension of SC particles held in an inert filler such as epoxy. For this to give an intermediate refractive index, the SC particle size must be much less than k0/(4nc) = 0.9 mm for HFGW frequency 4.9 GHz. Also, to act substantially as a SC, the SC particle size must be much greater than the superconductive coherence length, which is typically a few lm. The best particle size will be the geometric mean of these values, or around 60 lm. To give the precise refractive index value needed, assuming that the overall refractive index is the weighted mean of the contributing components by volume, the SC filling factor would be 5.5% by volume. However, herein lies the second difficulty: optical blooming relies on precise cancellation of reflections, and the coating material must be formulated very precisely to ensure
p that its refractive index is exactly n = 17.3 so that the Fresnel reflections from the two interfaces do indeed cancel precisely. The large standard error currently on the value of n indicates that this may be difficult to achieve in practice. Thirdly, the coating thickness must be precisely k/(4nc). Using precision manufacturing techniques now available, this may be easier to achieve than the allied condition on the required refractive index of the coating. However, the blooming action only works precisely at one wavelength and hence at one frequency, and only for normal incidence. A further important implication of these results is their application to HFGW detectors using a SC coating to enhance the electrical resonant Q of an RF cavity [7,8]. The SC thickness used will be critical. The incident HFGW must penetrate this cavity without significant loss (for efficient HFGW detection) and so to avoid reflection of HFGW, the SC coating must have a thickness of precisely k/2 measured in the SC. This gives a round-trip optical path difference of k between the reflections from the inside and outside surfaces of the SC, but also there is a 180 phase change on reflection at the boundary from air to SC (not present on reflection from the SC to air boundary) so that normal reflections from the inside and outside surfaces will interfere destructively. For a typical frequency of 4.9 GHz, the HFGW wavelength is k = 200 lm in the SC and so the SC coating around the corresponding RF resonant cavity must be 100 lm thick. This should be achievable using traditional low-temperature SC materials where the disadvantage will be the use of liquid helium refrigerant; high-temperature SC materials are easier to refrigerate but harder to fabricate to a close tolerance. It is also acceptable to make the SC thickness an integer multiple of k/2 if this eases fabrication problems, although the precision required of this thickness is unchanged.
3. Simple superconducting HFGW lens design From SnellÕs law of refraction, [9] for HFGW traveling from air to SC, the maximum possible angle of refraction is hr = arcsin (1/300) = 0.2. This means that, to a good approximation, all
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refracted rays inside the SC are normal to the surface. As a direct consequence, the critical angle, hc, for total internal reflection (reflection of HFGW rays incident from the inside of SC onto a SC-air surface) is also arcsin (1/n) = 0.2. Any HFGW incident on a SC-air surface at more than 0.2 to the normal will be reflected with virtually no power loss. These results are unique to HFGW optics in SC materials. For example, the ‘‘lens-makerÕs formula’’ for a thin plano-convex lens is: f = R/ (n 1), where f is the focal length and R is the convex surface radius of curvature [9]. Since total internal reflection occurs for HFGW incident at no more than 0.2 to the normal, the maximum aperture of such a simple thin lens is 2R/n. It follows that the minimum f-number for this lens is nf/(2R) = (1/2)n/(n 1) or very close to f/0.5, a very fast lens. The value f = 1 m gives R = 299 m; this means that, for a lens aperture of 0.1 m (= f/10), the lens differs from flat by only 8 lm. Or, a deviation of 1 lm gives an error in the focal point of 14 cm. Achieving such tolerances is feasible with the traditional low-temperature SC materials, though may be difficult with the newer high-temperature SC materials. As an alternative lens geometry, consider the structure in Fig. 1. In this structure, since all refracted rays are perpendicular to the SC surface, all axial rays are focused to the geometric center of the hemisphere. This gives an effective focal length of f = R, and the ‘‘lens-makerÕs formula’’ no longer applies as this lens is not thin. Such a structure unfortunately has little practical utility as all rays
Incident HFGW flux
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apart from a bundle almost coincident with the optical axis will hit the right-hand side of the hemisphere at greater than critical angle, and so will be reflected back on the opposite side of the optical axis as shown. In fact the maximum aperture of such a simple design is equal to 2R sin hc = 2R/ n = 2f/n, corresponding to an ‘‘f-number’’ of n/ 2 = 150. This would be a very slow lens indeed. However, this design may be considerably improved as shown in Fig. 2. In this design, the inner spherical curved surface is NOT concentric with the outer surface but is shifted right by R/ n = f/n, or 1/300 of the outer lens radius. This is required because the refracted rays are not precisely normal to the outer SC surface, and so if the inner surface were precisely concentric then the rays would be refracted in undesirable directions at the inner surface. The inner surface must, of course, include an anti-reflection coating to reduce reflections. For this design, since in principle all rays up to a radius R = f from the optical axis are converged, the maximum aperture may approach 2f or an ‘‘f-number’’ of f/0.5—a very fast lens. However, this is deceptive since at the extreme edges of the aperture the incident rays are no longer normal to the surface and so the anti-reflection coating of thickness k/4 is not effective. Thinner optical coating at the lens outer edges may improve the performance of this lens. Although aberrations from this lens will be low, other problems such as the precision fabrication necessary (well within 0.2) and the coating refractive index, thickness, and uniformity, mean that in
SC
SC
coating
n
n
√n
Incident HFGW flux
air
air
coating √n
Fig. 1. Alternative geometry for simple HFGW lens based upon a SC hemisphere (or part of one).
Fig. 2. Improved geometry for simple HFGW lens based upon a SC hemisphere (or part of one). Inner spherical surface is not precisely concentric but is shifted right by approximately r/n.
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many cases a refractive lens will not be the best choice for HFGW focusing applications.
SC
n
4. Design of superconducting reflectors for HFGW manipulation In view of the difficulties in designing effective HFGW refractive SC components as described in Section 3, it appears more practical to utilize SC reflectors. These components may closely mimic equivalent conventional optical designs. For normal incidence the HFGW Fresnel reflectance from an uncoated air-SC interface is 98.7% (i.e., an insertion loss of only 0.06 dB), and this may be adequate for many applications involving one or perhaps two reflections. If not, a reflectionenhancing coating may be used. This will be exactly the same material (SC particles in an inert matrix) proposed in Section 2 for anti-reflection coating, compounded with the same proportions p of constituents and thus again having nc = n, but applied now at a thickness kc/2 (for HFGW at normal incidence). The reflections from the two surfaces add by constructive interference, giving an overall reflectance of 100%. Since the uncoated reflectance is already close to 100% it appears entirely feasible to improve the reflectance using a coating in this manner, provided a suitable coating material can be developed. Fig. 3 shows an example of a SC focusing reflector design. The focal length will be the same as for an equivalent optical mirror, and it does not produce chromatic aberration. A more sophisticated use of SC as HFGW reflectors is illustrated in Fig. 4, showing a Fabry-Perot e´talon [9]. The e´talonÕs finesse is jpr/ (1 r2)j. Using the previous uncoated value of r = 299/301 gives a typical SC cavity HFGW finesse on the order of 240. This is significantly better than typically achieved in an electromagnetic/ optical e´talon (for which typical finesse values are 100). Clearly the finesse can be increased enormously by using even a moderately successful reflection-enhancing coating to bring r closer to unity, so that even greater resonance can be achieved. As a result of the resonant enhancement, the HFGW intensity inside such an e´talon is
HFGW flux
coating √n
Fig. 3. SC used as a HFGW reflector. The SC surface may use a reflection-enhancing coating, of thickness kc/2.
SC n
SC n
Fig. 4. SC used to form a Fabry-Perot e´talon or high-Q resonant cavity using parallel plane walls. The SC surface may use a reflection-enhancing coating, of thickness kc/2.
increased by a factor 1/(1 r2)2, so that a considerable improvement in detector sensitivity and selectivity could thereby be achieved.
5. Conclusions Li and Torr predicted [1,2] a substantial reduction in phase velocity for HFGWs traveling within a SC material. This result has been disputed but
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currently has not been conclusively confirmed or disproved, and so is open to confirmation or refutation. One way to shed light upon this controversy is to deduce some consequences that may be tested rather more easily than may be the basic result concerning phase velocity, and this is the focus of the present paper. Fresnel reflection at the air-SC surfaces will be almost 100% unless an effective anti-reflective coating material can be developed; this material would need a precisely defined HFGW refractive index and also a precisely controlled thickness of one quarter of the HFGW wavelength inside the coating. The tolerances on fabricating such a coating may be prohibitive even if a suitable material (such as a suspension of SC in an inert carrier) could be found and developed. If such a coating were feasible, the unique refractive properties of the SC indicate that all incident HFGW rays will be refracted within 0.2 of normal, and the critical angle for total internal reflection will also be 0.2. Such a small critical angle leads to unique lens geometries for HFGW. High-performance HFGW lenses may be possible using SC, but only if a suitable material for ‘‘optical blooming’’ can be developed. Given all of these difficulties, it appears to be more practical for HFGW optics to use reflection techniques universally, since even without using a reflection-enhancing coating the Fresnel reflection coefficient is very close to unity. For use in multiple-reflection instruments it may be useful to develop a reflection-enhancing coating applied at a uniform thickness of one half the HFGW wave-
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length inside the coating. Since the reflection coefficient is already so close to unity, the requirements on reflection-enhancing coating are not nearly so stringent as those for anti-reflection coating. Both high-performance focussing mirrors and resonant cavities appear to be feasible components relying upon reflection of HFGW from SC surfaces. In particular, a HFGW Fabry-Perot e´talon designed for HFGW use seems likely to outperform an equivalent optical e´talon and may be used in applications requiring HFGW tuning. The implications of this study will be important in the design of future communications links based upon HFGW technology.
References [1] [2] [3] [4] [5] [6]
[7]
[8]
[9]
N. Li, D.G. Torr, Phys. Rev. B 46 (1992) 5489. D.G. Torr, N. Li, Found. Phys. Lett. 6 (1993) 371. M. Kowitt, Phys. Rev. B 49 (1994) 704. E.G. Harris, Found. Phys. Lett. 12 (1999) 201. G. Fontana, personal communication (e-mail message to R.M.L. Baker) July 5, 2004, quoted in Ref. [7]. G. Modanese, G. Fontana, in: Proc. Space Technology and Applications Int. Forum, AIP Conference Proc., Melville, NY, 2004. R.M.L. Baker, E.W. Davis, R.C. Woods, in: Proc. Space Technology and Applications Int. Forum, AIP Conference Proc., Melville, NY, 2005. R.C. Woods, R.M.L. Baker, in: Proc. Space Technology and Applications Int. Forum, AIP Conference Proc., Melville, NY, 2005. B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley-Interscience, New York, NY, 1991, pp. 6, 59, 95, 205, 312.
Manipulation of gravitational waves for communications applications using superconductors R.C. Woods
*
Department of Electrical and Computer Engineering and Microelectronics Research Center, 2128 Coover Hall, Iowa State University, Ames, IA 50011–3060, USA Received 31 August 2005; accepted 4 October 2005
Abstract Previously published calculations claim that gravitational waves propagate inside superconductors with a phase velocity reduction of 300 times and a wavenumber increase of 300 times. Subsequent claims that this result is not credible appear to be either not tenable or not complete. This result has major consequences for the design of instruments to generate and detect gravitational waves, in particular high-frequency gravitational waves (HFGWs) having wavelengths on the same order as the dimensions of typical superconductive components. It is generally assumed that in free space the velocity of an HFGW is the same as that of light and so the free space wavelength of an HFGW at 3 GHz will be 10 cm. Inside a superconductor, the corresponding 3 GHz HFGW wavelength will therefore be 300 lm. The present paper will discuss the technical consequences of this surprising result. In particular, such a large mismatch in HFGW propagation impedance inevitably results in large Fresnel reflections from superconductor-air interfaces. This will cause a number of design problems in equipment proposed for HFGW generation and detection. For example, the superconductor thickness used in HFGW detectors will be critical. It may also be possible with care to exploit the result to obtain HFGW resonators and focusing reflectors. 2005 Elsevier B.V. All rights reserved. PACS: 85.25.Am Keywords: Blooming; Fresnel reflections; Gravitational waves; HFGWs; Optical coating; Optics; Refraction; Superconductors
1. Introduction
*
Tel.: +1 515 294 3310; fax: +1 515 294 8432. E-mail address: [email protected]
Li and Torr [1,2] published calculations of the propagation behavior of gravitational waves inside a superconductor (SC). They claimed that the
0921-4534/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.10.003
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phase velocity of gravitational waves in any SC material would be 106 m s1, i.e., 300 times less than the phase velocity of gravitational waves in other materials, generally assumed equal to the speed of light in a vacuum (3 · 108 m s1). This corresponds to a wavenumber increase of 300 times in SC materials. This result has major consequences for the design of future instruments to generate and detect gravitational waves, which may need to be able to focus, refract, reflect, or manipulate gravitational waves for efficient coupling to detectors, transmitters, generators, resonant chambers, and other transducers. In particular, high-frequency gravitational waves (HFGWs) have wavelengths on the same order as the dimensions of typical SC components envisaged for HFGW use and so any suggestion of phase velocity change will affect the design of those components typically to be used in a future communications link based upon this technology. Kowitt [3] and Harris [4] independently argued that the Li and Torr results [1,2] are not credible. KowittÕs objection [3] was that Li and Torr [1] assumed a value of magnetic permeability of zero (or, equivalently, a magnetic susceptibility of 1) inside a SC. This is an experimental result, widely accepted, for macroscopic samples but the question appears to be that of whether the macroscopic experimental result is true microscopically. It is perfectly true that a value of magnetic permeability different from that of free space may be obtained macroscopically in magnetically active samples in which individual magnetic moments of atoms align with the applied field. Since this enhanced permeability arises from the aggregate effect of localized magnetic moments, it follows that a renormalized permeability is not definable microscopically. However, zero permeability in a SC results from cancellation of the magnetic field H within the SC by distributed current loops giving a distributed magnetization M. In a typical type-II SC, the circulating supercurrents causing this distributed magnetic moment must be outside the magnetic vortices, of course. These circulating supercurrents extend throughout the material, whereas in a conventional magnetic material the notional currents producing the
magnetic moments are those caused by orbiting electrons bound to atoms. While it is the case that in a conventional magnetic material it is not possible to define an effective permeability that has local microscopic significance, this argument does not apply to the extended supercurrents found in SC materials. For these reasons, KowittÕs objection [3] appears not to be tenable. HarrisÕs objection [4] must be taken more seriously; he claims that Li and Torr assume an arbitrary value for the observerÕs distance of 106 cm in one place, and 1013 cm at another point in their calculation, and that both values are unreasonable. This objection is harder to refute but applies only to Torr and Li [2]. Li and Torr [1] predict a changed HFGW phase velocity and establishing its precise value may require an experimental test. However, Fontana [5] has confirmed that the phase velocity of HFGWs is slowed in a SC, and Modanese and Fontana [6] also imply the same point. It appears that at the current time, this issue may be classified as unresolved; that should not prevent deduction of further consequences if it were true, which may provide further tests of the result. The present paper discusses some direct technical consequences of assuming that the Li and Torr result is correct, as those consequences may suggest further possible tests of a prediction that is rather difficult to test experimentally with current technology. In particular, such a large mismatch in HFGW phase velocity results in a huge mismatch in HFGW propagation wave impedance at SC-air interfaces. This in turn inevitably causes large Fresnel reflections from those interfaces. These will produce a number of design problems in SC lenses proposed for HFGW generation and detection [7]. In practice, it seems that it may be easier to exploit the large reflection coefficient of HFGW from SC surfaces, by fabricating focusing HFGW mirrors rather than lenses. These would be designed in a similar manner to existing optical mirrors, but heavy supporting structures would be easy to incorporate since any non-SC material is expected not to affect the HFGWs in any way and so structural frameworks will not vignette the aperture at all.
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2. Superconducting HFGW components and Fresnel reflection Exploitation of the Li and Torr result [1,2] in HFGW detector design was first suggested by Woods and Baker [8]. Since a SC is a slowing material for HFGW, in principle many geometrical optical components might be designed by analogy with conventional optics. If HFGW are capable of being focused, then the usual results from geometrical and physical optics would be expected to carry over in virtually unchanged form. This means that the minimum diffractionlimited focused spot diameter is 2k/p (corresponding to the best case of focusing from a convergence angle of 90), where k is the HFGW wavelength in the material at the focusing point [9]. Hence, a diffraction-limited spot has area 3002 = 90 000· smaller inside SC than in free space, and so the corresponding energy density is 90 000· (=50 dB) larger inside a SC than in free space. Clearly this result alone merits further attention in the case of noise-limited signal detectors, since it implies considerably easier signal detection under adverse conditions, as well as the notion that HFGW may be focused by suitably shaped SC. For a HFGW having a typical frequency of 4.9 GHz [8], the wavelength k0 = 6.1 cm in air, and so k = 200 lm in SC material. One immediate consequence is that SC components for manipulation of HFGW at this frequency must be made to a tolerance of 20 lm or better for effective optical performance (using the well-known k/10 practical criterion). In view of the current difficulty of fabricating the high-temperature SC ceramic materials to tight tolerances, it may be more cost-effective to use conventional low-temperature SC alloys instead, for which normal machining may be employed. The disadvantage, of course, is that liquid He refrigerant must be used in this case. The relations between the components of waves incident upon a plane boundary between two different optical media are given by the Fresnel relations [9]. These relations are derived by assuming phase-matching at the interface boundary, and so the results are dependent only upon the phase velocities in the materials. The full statement of
103
the Fresnel relations depends upon the wave angle of incidence to the normal vector, hi, as well as the refractive index, n. However, generally the results for normal incidence (hi = 0) are sufficient to deduce the important behaviors concerned. At normal incidence, the reflection coefficient is r = (1 n)/(1 + n), and the transmission coefficient is t = 2/(1 + n). (For comparison, the results for grazing incidence, hi = 90, are r? = 1, rk = + 1, and t = 0, where the subscripts ? and k denote the polarization perpendicular to and parallel to the plane defined by the incident, transmitted, and reflected rays.) In almost all cases of interest the transmittance, T = nt2 = 4n/(1 + n)2, is of greater relevance as it gives the ratio of the transmitted intensity to the incident intensity taking correct account of the different wave impedance in the second medium. Using these general results, for HFGW traveling from air to SC, n = 300, and so r = 299/ 301 1 (i.e., corresponding to 180 phase change on reflection at a boundary of air to SC, with almost no amplitude change on reflection). The corresponding transmittance is T = 1.3%. Conversely, for HFGW traveling from SC to air, n is replaced by the inverse value 1/300, and so r = 299/301 + 1 (i.e., corresponding to no phase change on reflection at a boundary of SC to air, with almost no amplitude change on reflection). The corresponding transmittance is T = 1.3%, exactly the same as for HFGWs traveling from air to SC. These results have importance in designing a HFGW lens structure using SC material. Since the transmittance is only 1.3% at each air-SC interface in either propagation direction, it is expected that a simple SC lens analog of a simple optical glass lens would have a transmittance of only 0.017%, i.e., the combined transmittance of two air-SC boundaries, one each side of the lens. In other words, a simple HFGW lens made of SC material will introduce an energy loss of 99.98%, or an insertion loss of 38 dB. This is, of course, an intolerable overhead on the use of a lens, since it squanders most of the 50 dB improvement originally expected using a SC focusing arrangement. A simple lens analog thus appears impractical without further improvement.
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The technique of ‘‘optical blooming’’ or ‘‘optical coating’’ was developed many years ago to minimize Fresnel loss in conventional lenses, and merits examination in the present context. The coating material is added in a uniform film over a highly reflective interface between two non-opaque materials. The film thickness is exactly onequarter of a wavelength of the HFGW traveling in the coating material, and the refractive index of the coating is exactly equal to the geometric mean of the two surrounding materials. If one material is air (refractive index unity) and the other is SC, then p the coating material must have refractive index n = 17.3. The value of the refractive index ensures that Fresnel reflections from the air-coating and coating-SC interfaces have equal amplitudes, and the coating thickness ensures that at normal incidence the two reflections are out of phase by 180 and so exactly cancel by destructive interference. There is therefore no mechanism for losing energy at the interface, and so the overall transmittance must be 100%. Although this scheme is very effective for conventional optics and is widely used in all but the very simplest of lens designs, there are a number of major problems with using it for HFGW lenses made from SC. p Firstly, how can a coating material having nc = n = 17.3 be made? At present, the only known HFGW refractive indices are for SC (n = 300, assuming the Li and Torr prediction [1]) and for everything else (n = 1). One method to try is to make a suspension of SC particles held in an inert filler such as epoxy. For this to give an intermediate refractive index, the SC particle size must be much less than k0/(4nc) = 0.9 mm for HFGW frequency 4.9 GHz. Also, to act substantially as a SC, the SC particle size must be much greater than the superconductive coherence length, which is typically a few lm. The best particle size will be the geometric mean of these values, or around 60 lm. To give the precise refractive index value needed, assuming that the overall refractive index is the weighted mean of the contributing components by volume, the SC filling factor would be 5.5% by volume. However, herein lies the second difficulty: optical blooming relies on precise cancellation of reflections, and the coating material must be formulated very precisely to ensure
p that its refractive index is exactly n = 17.3 so that the Fresnel reflections from the two interfaces do indeed cancel precisely. The large standard error currently on the value of n indicates that this may be difficult to achieve in practice. Thirdly, the coating thickness must be precisely k/(4nc). Using precision manufacturing techniques now available, this may be easier to achieve than the allied condition on the required refractive index of the coating. However, the blooming action only works precisely at one wavelength and hence at one frequency, and only for normal incidence. A further important implication of these results is their application to HFGW detectors using a SC coating to enhance the electrical resonant Q of an RF cavity [7,8]. The SC thickness used will be critical. The incident HFGW must penetrate this cavity without significant loss (for efficient HFGW detection) and so to avoid reflection of HFGW, the SC coating must have a thickness of precisely k/2 measured in the SC. This gives a round-trip optical path difference of k between the reflections from the inside and outside surfaces of the SC, but also there is a 180 phase change on reflection at the boundary from air to SC (not present on reflection from the SC to air boundary) so that normal reflections from the inside and outside surfaces will interfere destructively. For a typical frequency of 4.9 GHz, the HFGW wavelength is k = 200 lm in the SC and so the SC coating around the corresponding RF resonant cavity must be 100 lm thick. This should be achievable using traditional low-temperature SC materials where the disadvantage will be the use of liquid helium refrigerant; high-temperature SC materials are easier to refrigerate but harder to fabricate to a close tolerance. It is also acceptable to make the SC thickness an integer multiple of k/2 if this eases fabrication problems, although the precision required of this thickness is unchanged.
3. Simple superconducting HFGW lens design From SnellÕs law of refraction, [9] for HFGW traveling from air to SC, the maximum possible angle of refraction is hr = arcsin (1/300) = 0.2. This means that, to a good approximation, all
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refracted rays inside the SC are normal to the surface. As a direct consequence, the critical angle, hc, for total internal reflection (reflection of HFGW rays incident from the inside of SC onto a SC-air surface) is also arcsin (1/n) = 0.2. Any HFGW incident on a SC-air surface at more than 0.2 to the normal will be reflected with virtually no power loss. These results are unique to HFGW optics in SC materials. For example, the ‘‘lens-makerÕs formula’’ for a thin plano-convex lens is: f = R/ (n 1), where f is the focal length and R is the convex surface radius of curvature [9]. Since total internal reflection occurs for HFGW incident at no more than 0.2 to the normal, the maximum aperture of such a simple thin lens is 2R/n. It follows that the minimum f-number for this lens is nf/(2R) = (1/2)n/(n 1) or very close to f/0.5, a very fast lens. The value f = 1 m gives R = 299 m; this means that, for a lens aperture of 0.1 m (= f/10), the lens differs from flat by only 8 lm. Or, a deviation of 1 lm gives an error in the focal point of 14 cm. Achieving such tolerances is feasible with the traditional low-temperature SC materials, though may be difficult with the newer high-temperature SC materials. As an alternative lens geometry, consider the structure in Fig. 1. In this structure, since all refracted rays are perpendicular to the SC surface, all axial rays are focused to the geometric center of the hemisphere. This gives an effective focal length of f = R, and the ‘‘lens-makerÕs formula’’ no longer applies as this lens is not thin. Such a structure unfortunately has little practical utility as all rays
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apart from a bundle almost coincident with the optical axis will hit the right-hand side of the hemisphere at greater than critical angle, and so will be reflected back on the opposite side of the optical axis as shown. In fact the maximum aperture of such a simple design is equal to 2R sin hc = 2R/ n = 2f/n, corresponding to an ‘‘f-number’’ of n/ 2 = 150. This would be a very slow lens indeed. However, this design may be considerably improved as shown in Fig. 2. In this design, the inner spherical curved surface is NOT concentric with the outer surface but is shifted right by R/ n = f/n, or 1/300 of the outer lens radius. This is required because the refracted rays are not precisely normal to the outer SC surface, and so if the inner surface were precisely concentric then the rays would be refracted in undesirable directions at the inner surface. The inner surface must, of course, include an anti-reflection coating to reduce reflections. For this design, since in principle all rays up to a radius R = f from the optical axis are converged, the maximum aperture may approach 2f or an ‘‘f-number’’ of f/0.5—a very fast lens. However, this is deceptive since at the extreme edges of the aperture the incident rays are no longer normal to the surface and so the anti-reflection coating of thickness k/4 is not effective. Thinner optical coating at the lens outer edges may improve the performance of this lens. Although aberrations from this lens will be low, other problems such as the precision fabrication necessary (well within 0.2) and the coating refractive index, thickness, and uniformity, mean that in
SC
SC
coating
n
n
√n
Incident HFGW flux
air
air
coating √n
Fig. 1. Alternative geometry for simple HFGW lens based upon a SC hemisphere (or part of one).
Fig. 2. Improved geometry for simple HFGW lens based upon a SC hemisphere (or part of one). Inner spherical surface is not precisely concentric but is shifted right by approximately r/n.
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many cases a refractive lens will not be the best choice for HFGW focusing applications.
SC
n
4. Design of superconducting reflectors for HFGW manipulation In view of the difficulties in designing effective HFGW refractive SC components as described in Section 3, it appears more practical to utilize SC reflectors. These components may closely mimic equivalent conventional optical designs. For normal incidence the HFGW Fresnel reflectance from an uncoated air-SC interface is 98.7% (i.e., an insertion loss of only 0.06 dB), and this may be adequate for many applications involving one or perhaps two reflections. If not, a reflectionenhancing coating may be used. This will be exactly the same material (SC particles in an inert matrix) proposed in Section 2 for anti-reflection coating, compounded with the same proportions p of constituents and thus again having nc = n, but applied now at a thickness kc/2 (for HFGW at normal incidence). The reflections from the two surfaces add by constructive interference, giving an overall reflectance of 100%. Since the uncoated reflectance is already close to 100% it appears entirely feasible to improve the reflectance using a coating in this manner, provided a suitable coating material can be developed. Fig. 3 shows an example of a SC focusing reflector design. The focal length will be the same as for an equivalent optical mirror, and it does not produce chromatic aberration. A more sophisticated use of SC as HFGW reflectors is illustrated in Fig. 4, showing a Fabry-Perot e´talon [9]. The e´talonÕs finesse is jpr/ (1 r2)j. Using the previous uncoated value of r = 299/301 gives a typical SC cavity HFGW finesse on the order of 240. This is significantly better than typically achieved in an electromagnetic/ optical e´talon (for which typical finesse values are 100). Clearly the finesse can be increased enormously by using even a moderately successful reflection-enhancing coating to bring r closer to unity, so that even greater resonance can be achieved. As a result of the resonant enhancement, the HFGW intensity inside such an e´talon is
HFGW flux
coating √n
Fig. 3. SC used as a HFGW reflector. The SC surface may use a reflection-enhancing coating, of thickness kc/2.
SC n
SC n
Fig. 4. SC used to form a Fabry-Perot e´talon or high-Q resonant cavity using parallel plane walls. The SC surface may use a reflection-enhancing coating, of thickness kc/2.
increased by a factor 1/(1 r2)2, so that a considerable improvement in detector sensitivity and selectivity could thereby be achieved.
5. Conclusions Li and Torr predicted [1,2] a substantial reduction in phase velocity for HFGWs traveling within a SC material. This result has been disputed but
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currently has not been conclusively confirmed or disproved, and so is open to confirmation or refutation. One way to shed light upon this controversy is to deduce some consequences that may be tested rather more easily than may be the basic result concerning phase velocity, and this is the focus of the present paper. Fresnel reflection at the air-SC surfaces will be almost 100% unless an effective anti-reflective coating material can be developed; this material would need a precisely defined HFGW refractive index and also a precisely controlled thickness of one quarter of the HFGW wavelength inside the coating. The tolerances on fabricating such a coating may be prohibitive even if a suitable material (such as a suspension of SC in an inert carrier) could be found and developed. If such a coating were feasible, the unique refractive properties of the SC indicate that all incident HFGW rays will be refracted within 0.2 of normal, and the critical angle for total internal reflection will also be 0.2. Such a small critical angle leads to unique lens geometries for HFGW. High-performance HFGW lenses may be possible using SC, but only if a suitable material for ‘‘optical blooming’’ can be developed. Given all of these difficulties, it appears to be more practical for HFGW optics to use reflection techniques universally, since even without using a reflection-enhancing coating the Fresnel reflection coefficient is very close to unity. For use in multiple-reflection instruments it may be useful to develop a reflection-enhancing coating applied at a uniform thickness of one half the HFGW wave-
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length inside the coating. Since the reflection coefficient is already so close to unity, the requirements on reflection-enhancing coating are not nearly so stringent as those for anti-reflection coating. Both high-performance focussing mirrors and resonant cavities appear to be feasible components relying upon reflection of HFGW from SC surfaces. In particular, a HFGW Fabry-Perot e´talon designed for HFGW use seems likely to outperform an equivalent optical e´talon and may be used in applications requiring HFGW tuning. The implications of this study will be important in the design of future communications links based upon HFGW technology.
References [1] [2] [3] [4] [5] [6]
[7]
[8]
[9]
N. Li, D.G. Torr, Phys. Rev. B 46 (1992) 5489. D.G. Torr, N. Li, Found. Phys. Lett. 6 (1993) 371. M. Kowitt, Phys. Rev. B 49 (1994) 704. E.G. Harris, Found. Phys. Lett. 12 (1999) 201. G. Fontana, personal communication (e-mail message to R.M.L. Baker) July 5, 2004, quoted in Ref. [7]. G. Modanese, G. Fontana, in: Proc. Space Technology and Applications Int. Forum, AIP Conference Proc., Melville, NY, 2004. R.M.L. Baker, E.W. Davis, R.C. Woods, in: Proc. Space Technology and Applications Int. Forum, AIP Conference Proc., Melville, NY, 2005. R.C. Woods, R.M.L. Baker, in: Proc. Space Technology and Applications Int. Forum, AIP Conference Proc., Melville, NY, 2005. B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley-Interscience, New York, NY, 1991, pp. 6, 59, 95, 205, 312.