Roman Tomaschitz, a,
The diffraction of superluminal radiation fields in crystal lattices is studied. The negative mass-square of the tachyonic wave modes affects the modulation function of diffraction gratings and the scattering amplitude. The Bragg condition for tachyon diffraction as well as the longitudinal and transversal cross sections are derived. Scalar and vectorial Kirchhoff identities for superluminal Proca fields are obtained from Sommerfeld’s dipole functionals, in analogy to electromagnetic theory. These surface-integral representations of the tachyon potential and the tachyonic field strengths are used to calculate the asymptotic diffracted modes and the intensity ratios. The dependence of the primary and secondary intensity peaks on the tachyon mass is analyzed in the reciprocal lattice, and the conversion of transversal into longitudinal radiation by way of Bragg scattering is explained. Specifically, tachyonic spectral fits are performed to the TeV spectra of three active galactic nuclei, H2356 − 309, 1ES 1218 + 304, and 1ES 1101 − 232, obtained with the imaging atmospheric Cherenkov detectors HESS, MAGIC, and VERITAS. The curvature in the spectral maps of these blazars is shown to be intrinsic, generated by ultra-relativistic electron populations in the galactic nuclei rather than by intergalactic absorption, and is reproduced by a tachyonic cascade fit.
Keywords: Tachyon diffraction; Transversal and longitudinal polarization; Green’s function for superluminal wave propagation; Tachyonic Maxwell equations and Fraunhofer far-field approximation; Cross sections and intensity ratios of tachyonic Bragg scattering; Superluminal cascade spectra of TeV blazars
PACS classification codes: 42.25.Fx; 42.25.Ja; 61.05.cc; 95.30.Gv
- 1. Introduction
- 2. Superluminal radiation fields
- 2.1. Proca equation with negative mass-square
- 2.2. Tachyonic dipole fields
- 2.3. Kirchhoff representation of the tachyon potential and the field strengths
- 3. Fraunhofer diffraction of superluminal radiation at a plane aperture
- 3.1. Tachyonic energy flux
- 3.2. Polarized superluminal modes: conversion of transversal into longitudinal tachyons by diffraction
- 3.3. Intensity ratios for the conversion of longitudinal into transversal radiation
- 4. Tachyonic Bragg scattering
- 4.1. Diffraction gratings: negative mass-square and Bragg condition
- 4.2. Tachyon diffraction in crystal lattices: transversal and longitudinal scattering cross sections
- 5. Tachyonic flare spectra of TeV blazars
- 6. Conclusion: tachyonic X-rays and Bragg spectrometers
1. IntroductionWe investigate the diffraction of superluminal wave modes, outlining a theory of tachyon diffraction based on Kirchhoff’s surface-integral representation of Proca fields. We work out two specific examples, diffraction at plane apertures such as diffraction gratings, and tachyonic Bragg scattering in crystal lattices. The formalism is developed in close analogy to electromagnetic diffraction theory, even though there are substantial differences owing to the negative mass-square of tachyons , , ,  and  and the occurrence of longitudinally polarized modes ,  and .
The negative mass-square refers to the radiation rather than the source. This in strong contrast to the traditional approach based on superluminal source particles emitting electromagnetic radiation . The tachyonic radiation discussed here implies superluminal energy transfer, the radiation quanta moving faster than light, in contrast to the rotating superluminal light sources studied in Refs. ,  and  and the vacuum Cherenkov radiation suggested in , ,  and . Tachyons are radiation modes, a kind of photons with negative mass-square, coupled by minimal substitution to the electron current, cf. Section 2. The tachyonic Maxwell equations admit a static potential analogous to the Coulomb potential, but oscillating because of the negative mass-square, and much weaker due to the small tachyonic fine structure constant . Photons can only be radiated by accelerated charges, in contrast to tachyonic quanta, where the emission rate primarily depends on the electronic Lorentz factor rather than on acceleration . Here, we investigate how the intensity peaks and scattering cross sections are affected by the tachyon mass, and study the effect of diffraction on the polarization of superluminal modes. We disentangle the transversal and longitudinal polarization components in the spectral maps of the BL Lacertae objects H2356 − 309, 1ES 1218 + 304, and 1ES 1101 − 232, and show that the TeV spectra of these blazars can be fitted with tachyonic cascades radiated by the thermal electron plasma in the active galactic nuclei. In the spectral maps, the tachyon–electron mass ratio enters in the cutoff energy of the cascades.
In Section 2, we discuss the tachyonic Maxwell equations in Fourier space, including the material equations relating tachyonic inductions and field strengths. We calculate the superluminal radiation fields generated by dipole currents, and introduce dipole functionals to derive the Kirchhoff identities for the scalar and vector potentials as well as the tachyonic field strengths. In Section 3, we study tachyon diffraction at a plane aperture, calculate the asymptotic diffracted wave modes in the far-field regime, and assemble the intensity ratios determining the conversion efficiency from transversal to longitudinal radiation and vice versa. In Section 4, we consider the specific case of a grating aperture, and calculate the modulation function, from which the intensity peaks of the diffracted superluminal wave fields can be read off. We discuss Bragg diffraction of tachyons in crystal lattices, in particular the effect of the negative mass-square on the transversal and longitudinal scattering cross sections. In Section 5, we perform tachyonic cascade fits to blazar spectra, separate the transversal and longitudinal flux components, and derive estimates of the electronic source populations in the active galactic nuclei. In Section 6, we present our conclusions with regard to tachyonic X-ray spectra obtained with Bragg gratings.
2. Superluminal radiation fields
2.1. Proca equation with negative mass-square
The tachyonic radiation field in vacuum is a real vector field
with negative mass-square, satisfying the Proca equation ,
subject to the Lorentz condition Aμ,μ=0
is the mass of the superluminal Proca field Aμ,
and q the tachyonic charge carried by the
subluminal electron current jμ = (ρ, j).
In the Proca equation, the mass term is added with a positive sign, and
the sign convention for the metric defining the d’Alembertian ∂ν∂ν
is diag(−1, 1, 1, 1), so that
is the negative mass-square of the radiation field. The 3D version of
Proca’s equation is a set of Maxwell equations,
In a permeable medium, the potential and field strengths in
the inhomogeneous vacuum equations are replaced by inductions, (A0, A) → (C0, C),
(E, B) → (D, H),
defined by material equations ,  and . We will
mostly consider monochromatic waves, ,
and analogously for the scalar potential A0,
the current, charge density, field strengths, and inductions. Fourier
amplitudes are denoted by a hat. The tachyonic Maxwell equations (2.1) read in
Fourier space as
as well as and are related to the primary fields by frequency-dependent dielectric and magnetic permeabilities. In an anisotropic medium, we have to use tensorial permeabilities, e.g. . The Fourier amplitudes of the field strengths and potentials are connected by and . Current conservation, , implies the Lorentz condition . In a dissipative medium, the permittivities (ε0, ε) and permeabilities (μ0, μ) are complex, resulting in exponential attenuation of the wave fields .
2.2. Tachyonic dipole fields
We substitute the potential representation of the field
strengths into the inhomogeneous field equations in (2.2), and make
use of the Lorentz condition to find
, but then we restrict to a frequency range with positive squared wave numbers. The Green function inverting the wave equations (2.4) is
(2.7), symmetric with respect to the first and second argument.
The field strengths are found via the potential representation
stated after (2.3). We define
G=O(1/r), with r=|x − x′| and n = (x − x′)/r.
We consider a dipole current and the corresponding charge
field generated by the dipole is , or
(2.6) for G(x, x0; ω), we write the dipole field in (2.14) as
for the field (2.14) generated by a dipole p, and analogously in (2.15). If q is another arbitrary constant dipole vector, then , and the same holds true for in (2.13). We also note the anti-symmetry . If the dipole vectors q and p depend on x rather than x0, these symmetries remain valid, provided that the gradients and rotors are replaced by the substitutions indicated after (2.16).
2.3. Kirchhoff representation of the tachyon potential and the field strengths
We denote the singular dipole current (2.12) and the
dipole fields (2.13), (2.14), (2.15) and (2.16) by a
subscript δ, e.g.
and consider a closed surface S around the dipole
at x0. An independent second
set of fields, ,
solves the free tachyonic Maxwell equations (2.2) inside the
cavity defined by S. (These fields are supposed to
be generated by a current distribution outside the enclosure S,
but we will only be concerned with the fields in the interior.) We
define the flux functional  and 
, etc., depend on a point x′ on the boundary S. The Green’s function G(x′, x0; ω) is symmetric with respect to an interchange of x′ and x0. It is also assumed that the substitutions → −0 and rot → −rot0 are performed in the dipole fields. Finally, we write x for x0 (an arbitrary point inside the cavity), and drop the scalar multiplication with the arbitrary dipole vector p, to obtain the surface-integral representation
(2.2) and (2.3) (with ε0 = ε and μ0 = μ) inside the cavity, that is, with zero current and charge density. G(x, x′;ω) is the Green function (2.7), where x ranges inside the cavity, and the integration is over the surface variable x′. The rotor and gradient refer to x. The potentials and in the surface integrals can be replaced by field strengths (via substitution of the field equations), and the operator by (μεω2+div), cf. (2.16). ,
for (rot rot−μω2). The analogous surface-integral representation of the magnetic field is obtained via ,
div). These identities follow from the electric field strength (2.19), by applying the field equations or potential representation, without actually solving differential equations. We also note that the Green function in these identities is the whole-space Green function (2.7), and is not required to satisfy any particular boundary conditions on the closed surface S, for technical simplicity. Therefore, these identities are only valid for solutions of the field equations, so that the boundary values cannot be arbitrarily prescribed. In the next section, however, we will prescribe boundary conditions, and convince ourselves that the fields calculated by means of the above Kirchhoff identities are asymptotic solutions of the free tachyonic Maxwell equations (2.2). This is in fact the practical use of these identities: Even though they will not give exact solutions for arbitrary boundary values, the resulting fields may well be approximate solutions in the far-field regime, which has to be checked on a case-by-case basis, by substitution into the field equations.
3. Fraunhofer diffraction of superluminal radiation at a plane aperture
3.1. Tachyonic energy flux
As in Section 2.2, we put ε0 = ε
and μ0 = μ.
When studying diffraction in the far-field limit  and , it
suffices to use the dipole approximation |x − x′| r(1 − nx′/r),
where r = |x|
and n = x/r,
so that (for large r and |x′| = O(1))
To find the energy flux carried by polarized superluminal
modes incident upon the aperture, we use a plane-wave ansatz in the
tachyonic Maxwell equations (2.2) (with
vanishing charge and current, as well as ε0 = ε
and μ0 = μ),
and analogously for the scalar potential and the field strengths. Here,
k = k(ω)k0,
where k is the wave number in the Green function (2.7), and k0
a constant unit vector. The transversality condition is ,
and the set of transversal modes reads
, and perform a time average, to obtain the transversal and longitudinal components of the energy flux,
3.2. Polarized superluminal modes: conversion of transversal into longitudinal tachyons by diffraction
We consider an incident transversal plane wave, propagating in
the lower half-space toward the aperture A in the (x, y)
plane. The Fourier amplitude of the vector potential is ,
cf. (3.2). By making
use of the Kirchhoff identity (2.19) and the
limit procedure outlined after (3.1), we find
the leading asymptotic order (r → ∞,
Fraunhofer regime ) of the
diffracted wave in the upper half-space as
. The scalar potential reads, cf. (2.20),
and , respectively. Here, εj and n = x/r constitute the orthonormal triad defining the outgoing linear polarizations, n being the unit wave vector of the outgoing spherical wave. The incident transversal wave is linearly polarized, ; the transversal linear polarization vectors ε0,1 and ε0,2 of the incoming wave are real and define with its unit wave vector k0 an orthonormal triad, so that k0 = ε0,1 × ε0,2 cyclically.
As for the outgoing transversal polarization vectors εj
of the diffracted wave in the upper half-space, we choose two real
orthogonal to n, so that ε1
lies in the plane generated by n and the
incident unit wave vector k0,
(3.8) can readily be performed by substitution of these scalar products. The transversal εj projections of (3.8) read
, is indicated as argument. The longitudinal projection of the outgoing wave (3.8) is
(3.15) at normal incidence k0 = e3 reads
In the far-field limit, the outgoing spherical waves are
structured as 
is an arbitrary complex linear combination of the two linear polarization components , cf. (3.14). The plane-wave counterpart to (3.20) is stated in (3.2) and (3.3). The time-averaged superluminal flux vectors ST,L can be assembled with these asymptotic spherical waves as done in (3.4) for plane waves. The transversal and longitudinal components of the diffracted energy flux are
(3.14) and (3.15). The subscripts i and j refer to the respective incoming and outgoing transversal polarization states. These averages are obtained from the asymptotic time-averaged Poynting vectors, cf. after (3.3),
(3.21), we perform the same average/summation over the transversal polarizations as done in (3.16) and (3.18). The ratio of the outgoing flux transversally or longitudinally diffracted into the solid angle element dΩ and the transversal flux incident upon the aperture area is found as
(3.16) and (3.18), we find the intensity ratios for the diffraction of transversal radiation,
3.3. Intensity ratios for the conversion of longitudinal into transversal radiation
We consider an incident longitudinal plane wave, ,
cf. (3.3), and
proceed analogously to the transversal case in Section 3.2. Employing
the surface integrals (2.19), (2.20), (2.21) and (2.22), we find
the diffracted electric field strength in the Fraunhofer regime,
in (3.28) read, cf. (3.14),
(3.11), (3.12) and (3.13). At normal incidence k0 = e3, we find and, cf. (3.16),
(3.31) is parametrized as, cf. (3.18),
(3.32) and (3.33) substituted.
4. Tachyonic Bragg scattering
4.1. Diffraction gratings: negative mass-square and Bragg condition
We start with a grating defined as an array of 2N + 1
slits parallel to the e2
axis (y coordinate). The slits are rectangles of
(large) height 2b and width 2a.
The x coordinate along e1
ranges in equidistantly spaced intervals [nd − a, nd + a],
where n = −N, …, 0, …, N,
and d > 2a.
The y coordinate of this grating aperture ranges in
[−b, b], and the
modulation function (3.6) factorizes
. The area of the aperture is 4(2N + 1)ab. In the intensity ratios (3.25) and (3.37), the b factors in area(A) and cancel, so that the height of the slits only enters in δ(2)(kq2, b). Thus the b → ∞ limit is well defined and gives δ(kq2). The principal maxima of M2 are determined by the interference factor, the square of the first ratio in (4.4). They are located at η = nπ for integer n. In between the principal intensity maxima, there are secondary ones, separated by zeros located at η = nπ/(2N + 1), where n is integer but not a multiple of 2N + 1. The square of the second ratio in (4.4) attenuates the maxima at large η. However, since 2a/d < 1, it does not significantly affect the location of the maxima. The principal maxima occur at η = nπ, which we may write, with k = 2π/λ, as a Bragg condition k0e1 − ne1 = nλ/d. We consider normal incidence k0 = e3, and choose the transversal polarization vectors along the coordinate axes, ε0,i = ei. In the (e1, k0) plane orthogonal to the slits, we then have φ = 0 and ne1 = sinθ, so that the principal maxima are recovered at scattering angles defined by , with integer n.
4.2. Tachyon diffraction in crystal lattices: transversal and longitudinal scattering cross sections
We consider a monochromatic superluminal radiation mode, ,
hitting a crystal lattice, and apply this field to the electron density
of the crystal. This generates a tachyonic current j(x, t) = qne(x)v(t),
is the periodic electron density in the crystal lattice. The velocity
of the electrons carrying tachyonic charge q is
determined by .
(In the Heaviside–Lorentz system, q2/(4πc)≈1.0×10-13,
estimated from Lamb shifts in hydrogenic ions .) In dipole
approximation, we may neglect the spatial dependence of the amplitude ,
We thus find the Fourier amplitude of the current
. The tachyonic radiation fields generated by current (4.5) in the crystal lattice can be split into transversally and longitudinally polarized components , like the diffracted waves (3.19). We are interested in the asymptotic radiation fields outside the crystal, determined by the current transform 
is the tachyonic wave number (2.5). The projections of onto a right-handed triad of polarization vectors ε1,2 and n of the radiation field are
(3.10). The outgoing transversal and longitudinal field components are stated in (3.19) and (3.20), where we put ε = μ = 1, replace the superscript T by T(i), and substitute as defined in (4.7) and (4.8).
We further specify the wave field incident upon the crystal as
a plane wave ,
cf. before (3.2),
generating the current
in (4.5). The
of the current transform (4.7) are thus
(3.19) and (3.20), we obtain the asymptotic outgoing field strengths,
ST(i),L in (3.22), and substitute the scattered fields (3.19) and (3.20),
ST is obtained by adding the transversal polarization components ST(i), which amounts to replacing the linearly polarized current transforms in (4.12) by in (4.13), according to (4.8).
We also need the flux density of the incident plane wave, ,
decomposed into transversal and longitudinal components, cf. (3.2) and (3.3). The flux
carried by the transversal component, ,
reads, cf. (3.4),
by sin2θ, where θ is the angle between and k′. If the incident transversal radiation is unpolarized, we have to replace sin2θ by the average (1 + cos2)/2, where is the scattering angle between the in- and outgoing wave vectors k and k′. In the longitudinal cross section,
ratio by cos2 . (4.18) are taken over the crystal volume, and give a sizeable contribution only if k − k′ very nearly coincides with a reciprocal lattice vector. We thus arrive at the Laue condition k − k′ = G, so that the respective integral just gives the crystal volume. We square k′ = k − G to arrive at 2kG = |G|2, which is the diffraction condition for the incident wave vector. Alternatively, |G| = |k′ − k| = 2k sin (/2), with scattering angle as above. The parallel lattice planes of the direct lattice orthogonal to a fixed G are equidistantly spaced, at distance d = 2π/|G0|, where G0 is the shortest reciprocal lattice vector parallel to G, the latter being an integer multiple of G0, |G| = n|G0|, cf., e.g., Ref. . Replacing the wave number by frequency via the dispersion relation , we find the Bragg condition for tachyon diffraction,
, which is the maximal wavelength attainable by tachyonic vacuum modes.
5. Tachyonic flare spectra of TeV blazars
Fig. 1, Fig. 2 and Fig. 3 depict
tachyonic cascade fits to the TeV spectra of the γ-ray blazars
H2356 − 309, 1ES 1218 + 304, and
1ES 1101 − 232, obtained with imaging air Cherenkov
telescopes . The
cascades are plots of the E2-rescaled
and ΔL = 0. γ is the electronic Lorentz factor, and αq the tachyonic fine structure constant. We use the Heaviside–Lorentz system, so that αq=q2/(4πc)≈1.0×10-13 and mt ≈ 2.15 keV/c2, as inferred from Lamb-shift estimates . The tachyon–electron mass ratio is mt/m ≈ 1/238, and a spectral cutoff occurs at
0ωωmax(γ) can be radiated by a uniformly moving charge, the tachyonic spectral densities pT,L(ω) being cut off at the break frequency ωmax. A positive ωmax(γ) requires Lorentz factors exceeding the threshold μt.
Fig. 1. Spectral map of the BL Lac object H2356 − 309. HESS data points from Ref. . The solid line T + L depicts the E2-scaled differential tachyon flux dNT+L/dE, obtained by adding the flux densities ρ1,2 of two electron populations, cf. (5.1). The transversal (T) and longitudinal (L) flux densities dNT,L/dE add up to the total unpolarized flux T + L. The exponential decay of the cascades ρ1,2 sets in at about Ecut ≈ (mt/m)kT, cf. after (5.2), implying cutoffs at 0.84 TeV for the ρ1 cascade and 92 GeV for ρ2. The χ2 fit is done with the unpolarized tachyon flux T + L, and subsequently split into transversal and longitudinal components. Temperature and number count of the electron populations are recorded in Table 1.
Fig. 2. Spectral map of the blazar 1ES 1218 + 304. MAGIC data points from Ref. , VERITAS points from Ref. . The upper flux limit in the 0.1–0.2 TeV interval is based on STACEE observations in 2006 and 2007 . The spectral fit T + L = ρ1 + ρ2 is performed with the electron distributions quoted in Table 1; the polarized flux components are labeled T and L. The ρ1 cascade is cut at Ecut ≈ 0.28 TeV, and ρ2 at 46 GeV. Comparing to the BL Lac in Fig. 1, located at a lower redshift, there is no indication of absorption in the spectral slope. The electron densities generating the cascades ρ1,2 are thermal in either case. The spectral curvature is intrinsic, caused by the Boltzmann factor of the electron populations in the galactic nucleus.
Fig. 3. Spectral map of the BL Lac object 1ES 1101 − 232. HESS flux points from Ref. . The plots are labeled as in Fig. 1 and Fig. 2. The ρ1 cascade is cut at 1.6 TeV and ρ2 at 0.21 TeV. The parameters of the electron populations are listed in Table 1. The spectral slope is steeper than that of 1ES 1218 + 304 in Fig. 2, even though these blazars have almost identical redshifts, which suggests that the shape of the plotted density E2dNT+L/dE is intrinsic rather than affected by intergalactic absorption; tachyonic γ-rays do not interact with background photons.
The average pT,L(ω) defining the differential flux (5.1) is taken over thermal ultra-relativistic electron distributions . The least-squares fit is performed with the total unpolarized flux density dNT+L = dNT + dNL. The cascades are labeled ρ1,2 in the figures, and the parameters of the electron populations generating them are listed in Table 1. The details of the spectral fitting have been explained in Ref. . The electron count is calculated as , where defines the tachyonic flux amplitude extracted from the fit. The cutoff parameter of the thermal cascades is related to the electron temperature by kT[TeV] ≈ 5.11 × 10−7/β, and the internal energy estimates of the source populations in Table 1 are obtained from U[erg] 2.46 × 10−6ne/β. The distance estimates of the active galactic nuclei are based on d cz/H0, with c/H0 ≈ 4.4 × 103 Mpc. Hence, d[Mpc] ≈ 4.4 × 103z, and , cf. Table 1.
Electronic source distributions ρi generating the tachyonic cascade spectra of the active galactic nuclei in Fig. 1, Fig. 2 and Fig. 3. Each ρi stands for a thermal ultra-relativistic Maxwell–Boltzmann density with cutoff parameter β in the Boltzmann factor, cf. after (5.3). determines the amplitude of the tachyon flux generated by the electron density ρi, from which the electron count ne∝d2 is inferred at the indicated distance. d is the distance to the blazar, estimated from the redshift z. kT is the temperature and U the internal energy of the electron populations ρi, cf. Ref. . The distance estimates do not affect the spectral maps in Fig. 1, Fig. 2 and Fig. 3, but the electronic source count ne. Each cascade depends on two fitting parameters β and , extracted from the χ2 fit T + L in the figures.
Fig. 1 shows the tachyonic spectral map of the blazar H2356 − 309 at redshift z ≈ 0.165 . TeV γ-ray spectra of blazars are usually assumed to be generated by inverse Compton scattering, which results in a flux of TeV photons thought to be partially absorbed by interaction with infrared background photons, so that the intrinsic spectrum has to be reconstructed on the basis of intergalactic absorption models. By contrast, the extragalactic tachyon flux is not attenuated by interaction with the background light, there is no absorption of tachyonic γ-rays. Apparently, the curvature present in the TeV spectra of blazars is not correlated with distance, at least there is no evidence to that effect if we compare the spectral slopes in Fig. 1, Fig. 2 and Fig. 3 to the spectral maps of other flaring active galactic nuclei such as the BL Lacertae objects (BL Lacs) H1426 + 428 (z ≈ 0.129, 570 Mpc) and 1ES 1959 + 650 (z ≈ 0.047, 210 Mpc) in Ref. , the blazars 1ES 0229 + 200 (z ≈ 0.140, 620 Mpc) and (z ≈ 0.188, 830 Mpc) in Ref. , and the quasar 3C 279 (z ≈ 0.538, 2.4 Gpc) in Ref. . Fig. 2 shows the tachyonic spectral fit of the BL Lac 1ES 1218 + 304 at z ≈ 0.182 ,  and , and Fig. 3 the spectral map of the blazar 1ES 1101 − 232 at z ≈ 0.186 . There is no correlation between redshift and spectral curvature visible. The curvature in the spectral maps of BL Lacs is intrinsic, generated by the superluminal spectral densities of the thermal electron plasma in the active galactic nuclei .
6. Conclusion: tachyonic X-rays and Bragg spectrometers
We have outlined a diffraction theory of superluminal wave fields based on Kirchhoff identities, cf. Section 2, and discussed the specific case of tachyonic Bragg diffraction, first with regard to a grating aperture and then in crystal lattices, cf. Section 4. We analyzed the effect of diffraction on the polarization of tachyons, cf. Section 3, and separated the transversal and longitudinal flux components in the spectral maps of γ-ray blazars, cf. Section 5. A more detailed summary is given in the Introduction. Here, we briefly sketch how the negative mass-square shows in tachyonic X-ray spectra obtained with Bragg spectrometers.
Tachyonic spectral fits are based on the En-scaled
flux density, cf. (5.1),
To this end, we start with an experimental plot of the
(assumed photonic) spectral density ,
parametrized by wavelength as inferred from the Bragg condition 2dsin(/2)=nλ.
The power radiated over a finite range of wavelengths is .
We reparametrize with energy via the photonic dispersion relation λ = 2π/ωph,
p(ω) in (6.1). More generally, if the photonic spectral density is plotted, the tachyonic density fn(ω) := ωn−1p(ω) is recovered as
. This apparently requires ωphmt, which is not a severe restriction, as the condition λ < 2d for Bragg scattering in a crystal amounts to roughly the same, cf. the end of Section 4.2.
We consider a set of photonic data points
labeled by index i. These flux points are inferred
from Bragg diffraction, that is, from the wavelength of the incident
quanta, and subsequently parametrized by frequency via the photonic
dispersion relation. If a tachyonic spectral fit based on density (6.1) is
performed, we have to use instead the tachyonic dispersion relation for
the energy parametrization. In effect, the photonic data points
are mapped into tachyonic points (ωi, fn(ωi))
The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.