Romain Dezert

Philippe Richetti

Alexandre Baron
email: alexandre.baron@u-bordeaux.fr

COMPLETE MULTIPOLAR DESCRIPTION OF REFLECTION AND TRANSMISSION ACROSS A METASURFACE FOR PERFECT ABSORPTION OF LIGHT A PREPRINT

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Introduction

When an electromagnetic wave impinges on a single layer of particles or atoms, local currents are generated within the particles that in turn radiate a field that will interfere with the incoming wave to produce a reflection and a transmission [START_REF] David | Classical electrodynamics[END_REF]. The transmission and reflection coefficients, also known as the scattering parameters, of a two-dimensional material are essential observables in physics and chemistry. They are often used to characterize the effective properties of a material. They may also be engineered to target a desired application. In recent years, many two-dimensional materials composed of sub-wavelength polarizable units or scatterers have been designed and modelled to generate extraordinary and tailored reflection and transmission properties. The recent surge in metasurfaces is a good example of arrays of particles designed to control waves both in amplitude and in phase for wave-front shaping [START_REF] Arbabi | Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission[END_REF][START_REF] Decker | High-efficiency dielectric huygens' surfaces[END_REF][START_REF] Khorasaninejad | Metalenses: Versatile multifunctional photonic components[END_REF], filtering [START_REF] Stanislav B Glybovski | Metasurfaces: From microwaves to visible[END_REF] or perfect absorption [START_REF] Alaee | Theory of metasurface based perfect absorbers[END_REF][START_REF] Thongrattanasiri | Complete optical absorption in periodically patterned graphene[END_REF][START_REF] Liu | Experimental realization of a terahertz all-dielectric metasurface absorber[END_REF].

The field radiated by subwavelength individual systems is often decomposed into multipoles. Very recently, multipolar decompositions have become an essential tool in nanophotonics and metamaterials engineering. Alaee et al. provide an interesting perspective on the matter in [START_REF] Alaee | Exact multipolar decompositions with applications in nanophotonics[END_REF]. Here we shall consider the multipolar decomposition on a vector spherical harmonic basis. This is convenient as it reduces the problem to the sole determination of the multipolar scattering coefficients a n,m and b n,m that fully describe how the field is emitted [START_REF] David | Classical electrodynamics[END_REF][START_REF] Craig | Absorption and scattering of light by small particles[END_REF][START_REF] Grahn | Electromagnetic multipole theory for optical nanomaterials[END_REF]. Numerous problems requiring the expression of the field radiated by an infinite array of particles have been treated in the past. As early as 1932, Strachan tried to develop a theory of the light reflected by a monomolecular layer [START_REF] Strachan | The reflexion of light at a surface covered by a monomolecular film[END_REF], though we now know that his theory was incomplete as it neglected the magnetic polarization density [START_REF] Edward F Kuester | Averaged transition conditions for electromagnetic fields at a metafilm[END_REF]. Later arrays were mostly described in terms of electric and magnetic effective dipole moments [START_REF] Tretyakov | Analytical modeling in applied electromagnetics[END_REF][START_REF] Andrey B Evlyukhin | Optical response features of si-nanoparticle arrays[END_REF][START_REF] Pt Bowen | Effective-medium description of a metasurface composed of a periodic array of nanoantennas coupled to a metallic film[END_REF], but also recently including multipolar contributions up to the octupoles [START_REF] Savinov | Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials[END_REF][START_REF] Pavel D Terekhov | Multipole analysis of dielectric metasurfaces composed of nonspherical nanoparticles and lattice invisibility effect[END_REF]. These works are based on expressing the moments in cartesian coordinates but are usually quite involved as the expansions become complex for increasing multipole orders. For this reason, they are usually truncated at low orders, which potentially limits the domain of application of these methods. We may wonder why cartesian coordinates are used in metasurface analysis, since multipolar decompositions up to any order using vector spherical harmonics are widely used and adapted to the study of individual particles.

In this work and in contrast to what has been done thus far, we directly relate the complex reflection and transmission coefficients of a two-dimensional array composed of single subwavelength scatterers of arbitrary shape to the multipolar coefficients of the particles in the array. These coefficients can be directly computed from the currents generated inside a single particle of the array using Grahn's decomposition [START_REF] Grahn | Electromagnetic multipole theory for optical nanomaterials[END_REF]. These currents can be calculated by various numerical methods. This formalism is simple, powerful, totally general and gives a lot of physical insight on the interaction of light with particles of arbitrary complexity. To demonstrate its use, we apply the formalism to absorption, which enables us to derive two generalized conditions that provide the basic recipe to achieve total absorption of an electromagnetic plane wave under normal incidence by an array of multipolar particles. It should be noted that both these conditions have already been given in the literature for arrays composed of dipoles [START_REF] Radi | Full light absorption in single arrays of spherical nanoparticles[END_REF][START_REF] Alaee | Theory of metasurface based perfect absorbers[END_REF]. However, here it is generalized. It turns out that many multipolar systems may act as perfect absorbers which is a definite improvement made possible by our formalism. We are able to provide a generalized interpretation of critical coupling in terms of spherical multipoles as well as provide two examples of perfect absorbing infinite arrays based on very different systems: single silica microparticles that absorb in the far-infrared due to phonon excitations and clusters of germanium particles that absorb at optical frequencies. In this last case, the absorption is obtained via an engineering of the multipolar behavior tailored with the cluster geometry.

Theoretical formalism

The problem we are considering is illustrated on Fig. 1, which depicts a two-dimensional infinite periodic array of particles of period a. Particles are represented as spheres, although the formalism is applicable to particles of arbitrary shape. The array is excited by a plane electromagnetic wave E i with wavevector k. Currents are generated in the volume of each particle due to both the interaction of the incoming wave with the particle as well as its interaction with the wave scattered by all neighbouring particles. These currents are source fields in Maxwell's equations. Each particle emits a scattered field E s in the positive and negative directions. The superposition of all scattered fields in the negative z direction produces a reflected plane wave E r , while the superposition of all scattered fields in the positive z direction interferes with the incoming field and produces a transmitted plane wave E t . The current density field in the unit cell can then be expended using the spherical basis allowing an exact description of the fields with a full account of multipole orders. This enables particles to be both electrically and magnetically polarizable and to sustain multiple resonances (dipole, quadrupole, octupole, etc...). The surrounding medium is considered to be vacuum in what follows for simplicity, but any host medium to the array is applicable.

With these assumptions, the reflection and transmission coefficients of the array can be derived from the expression of the electric field scattered by a single particle of the lattice. Kruk et al. have derived the relationship between the field emitted by an array and the field scattered by a single particle or unit cell of the lattice, using Green's function [START_REF] Kruk | Invited article: Broadband highly efficient dielectric metadevices for polarization control[END_REF]

E a (z) = 2πi|z| kS E s (z) (1)
where S = a 2 is the unit cell area of the array, E a is the field emitted by the array of particles, E s is that emitted by a unit cell, and k is the free-space wavevector. This expression is valid when considering far-field scattering and for a subwavelength square lattice spacing only. Next we expand the field emitted by a unit cell onto vector spherical harmonics using spherical coordinates. The full derivation is available in the appendix (see subsection Reflection and transmission coefficients), but in the interest of conciseness, we directly provide the expressions of the specular reflection and transmission coefficients in terms of symmetric or even (E n ) and anti-symmetric or odd (O n ) multipoles of order n. where the even and odd multipoles can be expressed as a function of the multipole coefficients as follows

r = - π 2k 2 S ∞ n=1 [E n -O n ] (2)
t = 1 - π 2k 2 S ∞ n=1 [E n + O n ] (3)
E n = +1 m=-1 [m(4n + 1)a 2n,m + (4n -1)b 2n-1,m ] (4)
O n = +1 m=-1 [m(4n -1)a 2n-1,m + (4n + 1)b 2n,m ] (5)
Writing the coefficients in this way is quite convenient as a clear distinction can be made in the role of odd and even modes in the reflection and transmission. This role was already underlined by Kruk et al. in the so-called generalized Kerker regime that enables highly transmitting arrays [START_REF] Kruk | Invited article: Broadband highly efficient dielectric metadevices for polarization control[END_REF]. We also note that similar formulas had already been used and discussed in the case of arrays of cylinders by using a cylindrical harmonic decomposition [START_REF] Liu | Generalized magnetic mirrors[END_REF].

Test on a monolayer of high index particles

To illustrate this formalism, we apply it to an infinite square array composed of spherical particles made of a fictitious material with loss that has an index of refraction N = 4 + 0.05i. The surface fill fraction is taken equal to 0.4. The reflectance (R) and transmittance (T ) spectra of the array are represented on Fig. 2(a) as a function of the normalized wavelength (λ/a) and are calculated using eqs. 2 and 3 after decomposing the current fields obtained numerically using the eletromagnetics module of the finite-element based software COMSOL Multiphysics. These quantities are compared to the reflectance and transmittance calculated from the input and output ports in COMSOL through a reflected and

n = b n,1 = b n,-1
). The absorption (A) spectrum is also calculated integrating the ohmic losses in the volume of the particles and the total energy balance is verified by ensuring R + T + A = 1. The example of an array composed of particles that exhibit no symmetry is given in the appendix (see subsection Particle of arbitrary symmetry).

Application to Perfect Absorption

Next, we apply the formalism to the problem of complete absorption of a plane wave under normal incidence on a sheet of particles. Many perfect absorbers have been proposed in recent years that are systematically interpreted in terms of critical dipole coupling. The conditions for perfect absorption with an array of dipoles are given by numerous authors [START_REF] Ra | Thin perfect absorbers for electromagnetic waves: theory, design, and realizations[END_REF][START_REF] Alaee | Theory of metasurface based perfect absorbers[END_REF]. These conditions can be generalized to a multipolar system using eqs. 2 and 3. Complete absorption occurs when both the reflection and transmission coefficients are simultaneously cancelled, that is when

+∞ n=1 O n = +∞ n=1 E n = k 2 S π (6)
This last equation contains two distinct conditions:

1. The generalized Kerker condition. As already noticed by [START_REF] Kruk | Invited article: Broadband highly efficient dielectric metadevices for polarization control[END_REF] in the context of hightly transmitting metasurfaces, this occurs when odd modes are equal to even modes, which we translate here in terms of multipole coefficients as O n = E n . When this condition is realized, we have r = 0. It should be noted that this general condition only requires the sums of odd and even modes to be equal. It does not require that for every n, each symmetric mode be equal to the anti-symmetric odd mode, which would be a much stronger requirement. 2. The zero transmission condition. This occurs when [E n + O n ] = 2k 2 S/π. Since the relation needs to be satisfied in the complex plane it actually means that the overall sum of odd and even modes needs to be a purely real number and reach the exact value that enables a destructive interference with the incoming plane wave. Equation 6 translates into a critical coupling condition equivalent to the classical dipolar critical coupling condition [START_REF] Alaee | Theory of metasurface based perfect absorbers[END_REF].

The power scattered by odd modes is equal to that scattered by even modes and the total scattered power is equal to the absorbed power (see appendix in subsection Critical coupling equivalent formulation with multipole coefficients and optical theorem). An equivalent formulation is that

σ O s = σ E s = σ a 2 (7)
where σ O s (σ E s ) and σ a are respectively the scattering cross-sections of odd (even) modes and the absorption crosssection of a particle within the array. We find two of the main features of perfect absorption, that the total scattering is equal to the absorption and that even and odd modes contribute equally and to half of the total scattering each.

Under the assumption that the multipoles of a single particle are only weakly perturbed by the array, this multipolar description becomes very powerful for design as an initial parameter exploration may be done on an individual particle before resorting to a periodic calculation. To illustrate this concept, we shall now propose two designs of perfect absorbers composed of a periodic arrangement of particles that operate in a multipolar regime in contrast to most absorbers investigated thus far [START_REF] Odebo Lank | Large-scale silicon nanophotonic metasurfaces with polarization independent near-perfect absorption[END_REF][START_REF] Viktar S Asadchy | Broadband reflectionless metasheets: frequency-selective transmission and perfect absorption[END_REF][START_REF] Liu | Experimental realization of a terahertz all-dielectric metasurface absorber[END_REF][START_REF] Tian | Near-infrared super-absorbing all-dielectric metasurface based on single-layer germanium nanostructures[END_REF][START_REF] Valagiannopoulos | Symmetric absorbers realized as gratings of pec cylinders covered by ordinary dielectrics[END_REF][START_REF] Ming | Degenerate critical coupling in all-dielectric metasurface absorbers[END_REF]. First we consider a homogenous microsphere made of SiO 2 with a radius r = 4.12µm. The sphere is initially considered individually through Mie theory and r is tuned in the mid-infrared region to exploit the phonon resonance that occurs near 10 µm. The size is chosen to satisfy the generalized Kerker condition and make sure that the sum of odd modes and the sum of even modes are real and larger than k 2 S * /π, where S * = (2r) 2 . This last requirement may be considered a threshold to reach the zero transmission condition when the particle will be placed in the array as it would correspond to a hypothetical dense square array. We see from Fig. 3(a) that the conditions are reached since both O n and E n (priming denotes that the coefficients are those of the isolated particle) become purely real and go beyond the k 2 S * /π value near 10.1 µm in the red shaded area. Next, an array of these particles is simulated and the pitch is fixed to find a value of S that will satisfy eq. 6 as can be seen on Fig. 3(b). This condition is fulfilled for an array with a fill fraction πr 2 /S = 0.7 . We note that the array influences both the amplitude of the scattering as well as the wavelength at which the conditions are satisfied in comparison to the isolated particle. Figure 3(c) shows the spectra of R, T , and A of the array. As for Fig. 2(a), the parameters calculated directly from COMSOL through planewave projections are compared to those calculated using eqs. 2 and 3. Perfect absorption is reached at a wavelength close to 10.2 µm. We see from the multipole decomposition in the array (Fig. 3(d)) that the perfect absorption condition given by eq. 6 is almost perfectly preserved showing that our assumption is reasonable and our design strategy is valid. As revealed by Fig. 3(d), we find that the system operates in a heavily multipolar regime (see gray region), where at least 8 multipoles are required to account for the perfect absorption behavior. To our knowledge, no multipolar perfect absorber has been proposed up to now. To further reveal the generality of the concept, we also consider a cluster built with spherical particles following an assembly method proposed in [START_REF] Dezert | Isotropic huygens dipoles and multipoles with colloidal particles[END_REF]. Such clusters offer an interesting alternative to spheres made of a homogenous material for which the optical properties are limited by the dispersion of the material. Tuning the volume fraction of inclusions within the clusters enables a tuning of the effective parameters of the cluster. Figure 3(e-h) shows the same quantities as those calculated for the silica system and are computed for a particle cluster composed of thirteen Ge particles that are 75 nm in radius and are pseudo-homogenously distributed in a sphere through repulsive interaction, with a volume fraction of 0.35. The square array made of these clusters has a surface fill fraction of 0.6. Here the resonator is designed to operate at visible frequencies. We see that near perfect absorption is reached at a wavelength close to 790 nm in a heavily multipolar regime once again.

Conclusion

Through a multipolar description of the reflection and transmission coefficients of a monolayer of arbitrary particles, we show that it is possible to infer properties of the metasurface by directly considering heavily multipolar individual particles. We have applied the formalism introduced to perfect absorption theory and given the generalized condition for total absorption of a plane electromagnetic wave under normal incidence on the surface. This phenomenon occurs when the superposition of odd multipoles is equal to that of even multipoles and are equal to the purely real number k 2 S/π, where k is the free-space wavevector of the impinging wave and S is the surface area of a unit period in the array. We use this result to propose two designs of perfect absorbers, one that operates in the mid-infrared region and is made of a infinite periodic array of resonant silica microspheres and another that operates at visible frequencies and is made of an infinite periodic array of germanium clusters. These results are of great interest to the metasurfaces community that has based a lot of its designs on dipolar and sometimes weakly mutlipolar systems. Furthermore it may interest the self-assembly and nanochemistry community who separate the fabrication of individual particles from the two-dimensional or three-dimensional assembly of materials [START_REF] Matthew J Rozin | Colloidal metasurfaces displaying near-ideal and tunable light absorbance in the infrared[END_REF][START_REF] Gómez-Graña | Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies[END_REF][START_REF] Baron | Self-assembled optical metamaterials[END_REF][START_REF] Ponsinet | Selfassembled nanostructured metamaterials[END_REF]. two directions, due to the polarization of the incident wave, E s has only one non-zero component along e θ . Our starting point is therefore the expression of the θ-component of the scattered field given in [START_REF] Grahn | Electromagnetic multipole theory for optical nanomaterials[END_REF] expanded in terms of spherical vector wave functions that takes the following form in the far field domain:

E s (θ, φ) =iE 0 e ikr kr ∞ n=1 +n m=-n π(2n + 1) n(n + 1) Q nm × [τ nm (cos θ)a nm + π nm (cos θ)b nm ] e imφ (8)
where Q nm is the normalization constant related to the associated Legendre functions:

Q nm = 2n + 1 4π (n -m)! (n + m)! (9)
τ nm and π nm are the angular scattering functions. They can be expressed using the associated Legendre functions P m n (cos θ) [START_REF] Doicu | Light scattering by systems of particles: null-field method with discrete sources: theory and programs[END_REF]:

π nm (cos θ) = m sin θ P m n (cos θ) (10)
τ nm (cos θ) = d dθ P m n (cos θ) (11)
The angular scattering function evaluated for θ = 0 and θ = π can for example be found in [START_REF] Xu | Electromagnetic scattering by an aggregate of spheres[END_REF] and read:

π n,m (±1) =            (±1) n-1 n(n + 1) 2 m = 1 (±1) n-1 2 m = -1 0 otherwise (12)
τ n,m (±1) =            (±1) n n(n + 1) 2 m = 1 - (±1) n 2 m = -1 0 otherwise (13)
The angular scattering functions cancel for all m / ∈ {1, -1} for both the forward and backward directions. Using these values, the scattered electric field reads in the forward direction:

E s (0, 0) = i E 0 4 e ikr kr ∞ n=1 (2n + 1) [a n,1 -a n,-1 + b n,1 + b n,-1 ] (14)
and in the backward direction:

E s (π, π) = i E 0 4 e ikr kr ∞ n=1 (2n + 1)(-1) n+1 [a n,1 -a n,-1 -b n,1 -b n,-1 ] (15)
Combining 2 and 3 with 8, the field leaving the array can be expressed:

E a (z + ) = - π 2k 2 S ∞ n=1 (2n + 1) (a n,1 -a n,-1 +b n,1 + b n,-1 ) E 0 (16) E a (z -) = - π 2k 2 S ∞ n=1 (-1) n+1 (2n + 1) (a n,1 -a n,-1 -b n,1 -b n,-1 ) E 0 ( 17
)
Inserting these 2 expressions in E t = E i +E a (z + ) and E r = E a (z -) allows us to finally write compactly the reflection and transmission coefficients expressed as a function of the spherical multipole coefficients of a single resonator (or unit cell) of the array:

t = 1 - π 2k 2 S ∞ n=1 m=-1,+1 (2n + 1) (ma n,m + b n,m ) (18)
r = - π 2k 2 S ∞ n=1 m=-1,+1 (-1) n+1 (2n + 1) (ma n,m -b n,m ) (19)
𝝀/𝒂

Particle of arbitrary symmetry

The introduced multipole-based formalism can be used to study periodic arrays of particles with arbitrary shape. To illustrate its large domain of validity, we apply it to arrays of particles consisting of truncated spherical particles with refractive index N = 4 + 0.05i and radius r = 150 nm. Truncations by 2 random planes and a curved surface were used to completely break the symmetry of the final structure as illustrated in the inset of Fig. 4(a). The surface fill fraction is taken equal to 0.4 as in Fig. 2 in the main text. The specular reflectance (R) and transmittance (T ) as well as the absorption spectra (A) of the array are represented on Fig. 4(a) as a function of the normalized wavelength (λ/a). These quantities are well reproduced by the reflectance and transmittance calculated using the extracted multipoles with eqs. 2 and 3 in the main text. The spectra of the modulus of each multipole coefficients of order m = 1 (m = -1) up to the fifth degree are represented on Figure 4(b) (Figure 4(c)).

Critical coupling equivalent formulation with multipole coefficients and optical theorem

In this section we provide the derivation of eq.7 i.e. the energy balance for the metasurface translating the critical coupling of both the symmetric and anti-symmetric excited modes.

By definition, the fraction of incoming power scattered in the backward (forward) direction by the array is equal to |r| 2 (|t -1| 2 ). Using these definitions and the expressions of the reflection and transmission coefficients (i.e. eqs. 18 and 19) the scattering cross sections of even and odd multipoles of a unit cell within the array may be defined by multiplying those quantities by the surface area of a unit cell S, which yields

σ E s = π 2 2k 4 S ∞ n=1 E n 2 (20) σ O s = π 2 2k 4 S ∞ n=1 O n 2 (21)
Assuming perfect absorption, with the condition

+∞ n=1 O n = +∞ n=1 E n = k 2 S
π , the two contributions simply become

σ O s = σ E s = σ a 2 (22)
where the absorption cross-section is actually equal to S since all of the energy impinging a unit cell is absorbed.

We can also show that the optical theorem holds by considering the total absorption A = 1 -R -T , which can be expressed with the multipole coefficients and provides the proportion of absorption due respectively to the symmetric and anti-symmetric modes. Using eqs. 19 and 18 the energy balance then reads:

A = -2 π 2k 2 S 2   ∞ n=1 E n 2 + ∞ n=1 O n 2   + π k 2 S ∞ n=1 Re (E n + O n ) (23)
or equivalently

σ a = -σ s + σ ext (24)
where σ ext is the extinction cross section of the particles in the array, which in the case of an x-polarized incident plane wave becomes [START_REF] Grahn | Electromagnetic multipole theory for optical nanomaterials[END_REF]:

σ ext = π k 2 ∞ n=1 m=-1,+1 (2n + 1)Re (M a n,m + b n,m ) (25)
As for the scattering cross section, the extinction cross section expression can be broken down into even and odd contributions as:

σ E ext = π k 2 ∞ n=1
Re(E n ) (26)

σ O ext = π k 2 ∞ n=1 Re(O n ) (27)
Figure 5: Cross section decomposition in terms of even and odd contributions apply to a cluster-based absorbing metasurface. Each cluster consist of thirteen Ge particles that are 75 nm in radius distributed in a sphere of radius R = 250nm. The square array has a surface fill fraction of 0.6.

In the same way the absorption cross section can be decomposed into even and odd terms obtained by the differences

σ O a = σ O ext -σ O s ( 28
)
σ E a = σ E ext -σ E s ( 29
)
We see that the optical theorem holds for odd and even superpositions independently.

Accessing this decomposition would usually require a configuration with double illumination from both sides of the array as described in [START_REF] Ming | Degenerate critical coupling in all-dielectric metasurface absorbers[END_REF] with co-or cross-polarization allowing for a selective excitation of either the symmetric or anti-symmetric modes of the structure. Instead, using the multipole decomposition allows for a one step procedure to retrieve both the emission and absorption due to the even or odd excited modes.

To illustrate this approach, we present in Fig. 5 the decomposition of the scattering and absorption cross-sections in terms of even and odd multipoles applied to the Ge cluster-based absorber presented in the main text. The absorption σ * a and emission σ * s cross section calculated with respectively the absorbed power obtained via the integration of the ohmic losses and the reflection and transmission coefficients are perfectly reproduced by those reconstructed from the multipole coefficients. In the range of high absorption, the cross sections satisfy:

   σ O s ≈ σ O a ≈ σ 0 2 σ E s ≈ σ E a ≈ σ 0 2 (30)
where σ 0 = S. Strict equalities would be obtained in the case of an absorption reaching 100% while in the present case the maximum absorption exhibited by the metasurface is 98%.

The presented cross sections are quantities of interest as they carry the same information than the scattering, extinction and absorption cross section defined for isolated particles in homogeneous space [START_REF] David | Classical electrodynamics[END_REF][START_REF] Grahn | Electromagnetic multipole theory for optical nanomaterials[END_REF]. Therefore they allow for a direct comparison between the optical properties of particles isolated or placed in a periodic array.

Figure 1 :

 1 Figure1: Artistic rendition of the multipolar decomposition of the field scattered by an array of particles. The particles are reprensented as spheres but the shape can be arbitrary. An electromagnetic plane wave E i with wavevector k is incident on the metasurface. Current densities J are produced within the particles which radiate a scattered field E s in both the top and bottom directions. The interference of the incident field with the scattered field produces a reflected (E r ) and transmitted field (E t ). The current density within a single particle in the array is decomposed on a vector spherical harmonic basis which enables a determination of the multipole coefficients a n,m and b n,m of the unit cell.

Figure 2 :

 2 Figure 2: Test of the multipole decomposition. (a) Reflectance (R), transmittance (T) and absorption (A) spectra of an infinite square periodic array of period a composed of a spherical particle of index N = 4 + 0.05i shown in the inset.The surface fill fraction is 0.4. The continous lines are the spectra calculated from the input and output ports, while the dotted curves are computed from the current decompositions inside the particle and introduced in eqs. 2 and 3. The absorption is caculated directly from the currents. The green curve equal to 1 across the spectrum is R + T + A and ensures energy conservation. (b) Spectra of the magnitude of the multipolar coefficients. Continuous (dashed) lines represent the electric (magnetic) multipoles.

Figure 3 :

 3 Figure 3: Design of perfect absorbers using the multipolar behavior of individual particles. (a) Wavelength dependence of the sum of even and odd multipole coefficients of an individual SiO 2 microsphere in vacuum that is 4.12 µm in radius, calculated using Mie theory. (b) Wavelength dependence of the sum of even and odd multipole coefficients of the SiO 2 microsphere in an infinite square array with a surface fill fraction of 0.7. (c) Spectra of R, T and A calculated for the square array of SiO 2 microspheres, revealing the perfect absorption behavior calculated using the same methods as those used for Fig. 2. (d) Spectra of the modulus of the multipole coefficients retrieved from the current decomposition for the SiO 2 microsphere array.(e-h) are respectively the same quantities as those reprensented on panels (a-d) but for a particle cluster composed of thirteen Ge particles that are 75 nm in redius and are pseudo-homgenously distributed in a sphere through repulsive interaction, with a volume fraction of 0.35. The square array made of these clusters has a surface fill fraction of 0.6. The individual cluster was calculated and explored using the T-matrix method.

a 1 ,1 a 2 ,1 a 3 ,1 a 4 ,1 a 5,1 b 1 ,1 b 2 ,1 b 3 ,1 b 4 ,1 b 5 a 1 ,-1 a 2 ,-1 a 3 ,-1 a 4 ,-1 a 5,-1 b 1 ,-1 b 2 ,-1 b 3 ,-1 b 4 ,-1 b 5 Figure 4 :

 1234123451234123454 Figure 4: Validation of the multipole decomposition with unsymmetrical particles. (a) Reflectance (R), transmittance (T ) and absorption (A) spectra of an infinite square periodic array of period a composed of a 3 times truncated spherical particle. The continuous lines are the spectra of the specular reflection and transmission calculated from the input and output ports, while the dashed curves are computed from the extracted multipoles as introduced in eqs. 19 and 18. The green curve equal to 1 across the spectrum is R + T + A ensuring energy conservation. (b) Spectra of the magnitude of the multipole coefficients of order m = 1. Continuous (dashed) lines represents the electric (magnetic) multipoles.(c) Spectra of the magnitude of the multipole coefficients of order m = -1.

Acknowledgements

The authors aknowledge financial support from the LabEx AMADEus (ANR-10-LABX-42) in the framework of the IdEx Bordeaux (ANR-10-IDEX-03-02).

Appendix Reflection and transmission coefficients

In this section we provide the derivation of eqs. 2 and 3, i.e. the decomposition of r and t as a function of vector spherical harmonic coefficients.

We consider E i , an e x polarized plane wave of magnitude E 0 propagating along the e z direction, normally incident on the metasurface consisting in a square array of particles. Using a multipolar decomposition and spherical coordinates we want to express E s (z) the field scattered by a single particle of the array along the e z axis for both the forward (θ = 0, φ = 0) and backward (θ = π, φ = π) directions , where φ is the azimuthal angle and θ the polar one. For these


Bernard Guy
email: guy@emse.fr

Keywords: Lorentz transformation, clock, space, time, movement,  factor, factor  = v/c, angle , light, atomic clock, optical clock, reference frame at rest, moving reference frame, second postulate, relativity theory

Our purpose is to examine the articulation between the Lorentz transformation (theory of relativity) and the very construction of the concept of time. The latter does not exist ready, waiting to be enrolled in the equations of mathematical physics. However, when deriving the transformation, the existence of time is not contested. Under these conditions, the notion of a clock is not the subject of any special discussion. We want to criticize this classic approach because, if we want to deepen and contest this very existence of time, we are led to "open" the clocks and look at what happens there... We realize that we always transform a movement into time (a clock is a point of view on a movement). In these conditions, it is a question of taking up in a concrete way, i. e. by looking at this or that particular clock, what is happening about time. In the case of an atomic clock like today, it is a movement of light. In this context, we are led to compare the relation of the same photon to two reference frames (at rest / mobile; this is how we define the two clocks) while assigning it the same speed (second postulate of relativity). The new approach leads to the study the writing of the transformation according to the orientation of the movement defining time. The meaning of the  relativity factor is extended, making it depend not only, as in the standard case, on the ratio  = v/c of the velocity modules (the relative movement of the reference frames and light), but also on the angle  between these two movements. A general relation  = (, ) is proposed which makes it possible to find in the same framework various transformations already known, including Lorentz's (the intelligence of the word transformation is renewed in the sense of accommodating a particular movement of light, and not all possible movements).

The Lorentz transformation, time and space

Generalization of the  factor as a function of the direction of the hidden movement in the clocks

Introduction

In relativity theory, the Lorentz transformation makes it possible to move from a collection of space and time variables (x, y, z, t), evaluated in a reference frame R said at rest, to another collection (x', y', z', t') evaluated in a moving reference frame R' at a speed v with respect to the first1 . Our purpose here is to examine the articulation between this transformation and the construction of the concept of time. The latter does not exist ready, waiting to be enrolled in the equations of mathematical physics. However, when deriving the transformation, the existence of time is not contested: it is then a question of establishing its measurement and comparing the estimated values in the two reference frames in relative motion, under various constraints, in particular that of an equal "speed" c of light in each reference frame (the consequence is the relativity of time). In these circumstances, the notion of a clock is not the subject of any special discussion: it is assumed that it is available as much as necessary.

Located at the various locations of the reference frames, the clocks give the local time; they are even assumed to be punctual for the proper functioning of the reasoning (after the socalled synchronization process, we can limit ourselves to one clock per reference frame).

We want to criticize this classic approach because, if we want to dig a little deeper and challenge this very existence of time, we are led to "open" the clocks and look at what is happening there... We realize that we always transform a movement into time (a clock is a point of view on a movement). In the case of an atomic clock (also known as an optical clock) used as a reference as today, it is a movement of light that is involved. The question then arises as to how this movement of light is or is not in harmony with the movement of "external" light that we have just discussed.

The answer assumes that the same choices are made for the two or three (including the two clocks associated with the two reference frames) lights, which are now only one from a conceptual point of view. Indeed, we are logically led to make the same assumption "c = cte" for the two internal photons of the two clocks as for the external photon. As a result, all three photons play the same role and only one photon is required. The same photon serves both as a clock, or as plural clocks (each point of view on it is a clock; we have one for each reference frame) and as support for the postulate "c = cte". There is no other photon to consider, nor is there any time that would flow locally here or there, to measure here or there.

Because of the relative displacement between the two reference frames, the direction of the movements will then intervene: the movement in R does not have the same orientation as in R'. Previously, there was no interest in the direction of the movement for two additional reasons: -we did not open the clocks: a scalar time came out of them as if by magic; -we were interested in all possible directions for the "external" photon (whose "constant" speed was taken). The orientation of the relative displacement between the two reference frames was relevant for the offset of the origins, but not in the  distortion factor.

It is ultimately the understanding in terms of movement of time itself that presents itself to us in Lorentz transformation. There is no longer time on one side (hidden in the clock) and light on the other, it is the same thing; time being marked by the movement of light. This statement ipso facto particularizes the direction of light in the "real" "clock", and asks us not to accommodate all possible movements (a summary of our approach is given in Figure 1).

In the context of today's optical clocks, expressing the identity of the properties of light propagation in the two reference frames amounts to considering two points of view (that of R and that of R') on a photon to which the same velocity is assigned, in a reasoning where the implicit geometric content of the equations is as important as the explicit algebraic content.

The new approach therefore leads to the study of the writing of the transformation as a function of the orientation of the photon motion defining time; various transformations are then possible according to the directions of the axes of the reference frames with respect to the displacements in play (reference frames, light). In doing so, the understanding of the word transformation is renewed in the sense, as we have noticed, of accommodating a singular movement of light and not all imaginable movements.

Here is what our plan will be. We will start with a few brief reminders on the standard Lorentz transformations (closed clocks), section 2. We will then focus our discussion on the  transformation factor (it is it that retains its originality), by presenting a new formulation: it will take into account a temporal movement of any direction, both in relation to the coordinate axes and the direction of the relative movement of the reference frames (section 3). We will then look at some remarkable special cases (section 4), before making proposals for complete general transformations explaining the offsets in the origins of the reference frames, in addition to the scale factor  (section 5). We will then be able to discuss some problems related to the composition of non-collinear Lorentz transformations (section 6), before concluding and briefly discussing what this approach brings to the approach to natural phenomena (section 7). Some borrowings from older texts will be highlighted, with the present set providing new formulations and conceptual insights. The reference [START_REF] Guy | Linking quantum mechanics and general relativity? Reflections and proposals[END_REF] gives access to many of our works (see also [START_REF] Guy | Time and space are of the same stuff[END_REF]Guy, , 2017[START_REF] Guy | Space = Time. Dialogue on the system of the world[END_REF].

Some reminders on the Lorentz transformation

There is no question of proposing a panorama, even partial, of everything that has been written about Lorentz transformation. We will simply comment on it quickly through the delicate points already discussed by us: these concern on the one hand the composition of non-collinear transformations, with the related question of additional rotations [START_REF] Guy | Les relations de Lorentz et le temps: proposition d'utilisation d'un paramètre tri-dimensionnel défini par un déplacement[END_REF], and on the other hand the twins paradox, the relativity of time and the expected time gaps [START_REF] Guy | On the comparative age and aging of two twins (relativity theory[END_REF], the previous references giving access to an already abundant literature.

Let us first look at the classic situation where the question of clocks is not yet raised. The Lorentz transformation is first presented for a relative displacement of the reference frames parallel to a coordinate axis. We are brought back to study the transformation of a pair (x, t) in the reference frame at rest into the pair (x', t') in the reference frame moving at the speed v with respect to the first. The other coordinates are not modified (y = y'; z = z'). The constant speed of light constraint, recalled above, is written along the x-axis (if x =  ct, then x' =  ct'). The vector v is parallel to the x-axis; we obtain the well-known formulas

x' = (x -vt) t' = (t -vx/c 2 ) with  = (1 -v 2 /c 2 ) -1/2 (1)
The  factor

The previous transformation shows the product of two different terms: -one expresses the offset of the origins between the two reference frames (in the spatial term (x -vt) and its temporal correspondent (t -vx/c2 )), -the other a distortion of scale in the  factor; this depends only on the velocity ratio = v/c. This two-fold separation results from the different constraints imposed on the structure of the equations of relativity (homogeneity, linearity, etc.) and detailed in the original articles and treatises (e. g. [START_REF] Lorentz | Electromagnetic phenomena in a system moving with any velocity smaller than that of light[END_REF][START_REF] Poincaré | Sur la dynamique de l'electron[END_REF][START_REF] Einstein | On the electrodynamics of moving bodies[END_REF][START_REF] Lévy-Leblond | One more derivation of the Lorentz transformation[END_REF]Rougé, 2002). As we have already argued, it is the  factor, discussed in the next section, that gives the Lorentz transformation its originality.

In the first derivation proposed above, we observe a perfect symmetry between the two variables x and t (x' and t'), already noted by Poincaré and clearly visible by taking c = 1. The corresponding transformations are called special transformations or "boosts". They define a group; i.e. the composition of two transformations with respective velocities v 1 and v 2 is a transformation of the same type, of velocity v 3 , a function of v 1 and v 2 (we write

v 3 = v 1  v 2 ).
The next natural step is to extend the previous result to relative movements between any reference frames, i. e. such that the velocity v is not parallel to the coordinate axes. The approach consists in making a first rotation of the rest reference frame to position its x-axis along the speed v. A special Lorentz transformation is then performed as just now. We then return to the initial directions of the reference frames by reversing the previous rotation. It so happens that this process leads to a series of difficulties: in our analysis, they are due to the fact that we no longer have the symmetry between spatial and temporal variables that we had for the couple (x, t) of the special transformation; the group structure is lost. The solution proposed in the literature consists in releasing the constraint of light propagation at a constant speed along a single direction (the x-axis containing the speed v in the boosts) and writing it in all directions; this is done by preserving the quadratic form ds 2 = dx 2 + dy 2 + dz 2 -c 2 dt 2 . In addition to translations, the geometric operations allowed by the constraints then include a priori rotations of the axes. With regard to the composition of several non-collinear Lorentz transformations, we find a mathematical coherence (group structure), adding new rotations (in addition to the previous ones) depending on the speeds of the different displacements to be composed (they are called Thomas rotations, cf. [START_REF] Guy | Les relations de Lorentz et le temps: proposition d'utilisation d'un paramètre tri-dimensionnel défini par un déplacement[END_REF]. This approach, which has been extensively analysed in the literature, raises many questions, particularly about the physical nature of what is written 2 .

New approach. Proposal of a  factor taking into account the direction of light propagation

We do not repeat our contribution to this discussion, to which the reader is invited to refer.

We are not encouraged to retain the various rotations we have just mentioned, insofar as we feel that a resumption of the Lorentz transformation is necessary. This must be done on the occasion of an "opening" of the clocks and the consideration that time is associated with a movement; as we have said, it is now that of a photon in an atomic clock. The position of this photon is described by three space coordinates, t x , t y and t z , and the scalar time corresponds, at least momentarily and locally, to the norm t = (t x 2 + t y 2 +t z 2 ) 1/2 . A slightly different way of understanding the three parameters useful for defining time is to see them as defining the direction of photon motion (see below). We thus limit ourselves to a particular clock, i.e. the movement of light that travels through it (the clock is here the point of view on the standard photon). We can also specify in which reference frame we locate the direction of the standard movement (the orientation in the other reference frame derives from it, taking into account the relative movement). We will consider it here in the moving reference frame R', and will also look at the movement of the photon from the reference frame at rest R, while decreeing it to keep the speed c in the two reference frames (reversing the points of view does not only mean changing v tov, but changing the reference frame where we start to designate a direction of light -we have done so here in R' ; by reversing the points of view, it is in R that we must designate the direction of the speed of light-; see also [START_REF] Guy | On the comparative age and aging of two twins (relativity theory[END_REF]. In total, it is no longer a question of postulating the conservation of the "speed" of light in all directions 3 .

Notations

Let us set the notations that we will use later. We will use, in addition to the vector v of the standard Lorentz transformation, a vector c' for the movement of light in R', corresponding to a vector c in R. The orientation of the velocity v is defined by the direction cosines 4 , or ratios,

 i such that v x =  x .v, v y =  y .v, v z =  z .v.
The coordinates v x , v y and v z make it possible to define the module v (we will only rarely distinguish vector and module by different notations, the context allowing to understand). The two vectors c and c' have the same norm (it is the expression of Einstein's second postulate), but different directions a priori. The corresponding 3 We can continue to use the interval ds 2 by adding constraints; we no longer envisage a multiplicity of movements of light, with a time on its own side, but a single movement to define a time that respects the "logic" of a clock. 4 The direction cosines  i define the orientation of a unit vector carried by v and check

 x 2 +  y 2 +  z 2 = 1.
times t and t' are the scalars that scale the progression of the photon along the directions c and c'. The associated t and t' vectors are carried by the c and c' vectors respectively in the two reference frames. The link between them is written

t ⃗ = 1 c c ⃗t and t′ ⃗ ⃗⃗ = 1 c c ⃗′t′
where it is important for a good understanding to distinguish between vectors c ⃗ and t ⃗ and scalars c and t (vector modules), as well as for c′ ⃗⃗⃗ and t′ ⃗ ⃗⃗ with respect to c' = c and t'. The orientation of the vectors c or t, on the one hand, c' or t' on the other hand is defined by the direction cosines  i and ' i respectively. These ratios check: these relations by the constant c, (ct) x = ct x =  x .ct (t is a vector in the first term, and its module in the third one) and also for the other components and in both reference frames.

t x =  x .
The last relations can still be written c x =  x .c, c y =  y .c, c z =  z .c for vector c, and similarly with the ratios ' x , ' y and ' z for vector c'. In order to formulate Lorentz transformations for the vector t (t x , t y , t z ), in association with the position vector (x, y, z), we will then reason about the vectors t and t' rather than the vectors c and c'.

Case of two dimensions of space

To simplify the calculations, we will reason in the case of two space dimensions x and y as in

Expression of the second postulate for an optical clock

It is therefore for the movement of a photon P marking time, another name of a clock, that we will formulate the second postulate of relativity. Looking at the photon from O, time in R is identified by OP = ct; looking at it from O', time in R' is identified by O'P = ct' (relational and not substantial approach). In both cases, the factor c is the same; that is the expression of the second postulate. The offset of the origins is marked by OO' = vt. Refer to Figure 2.

By comparing the two times t and t' in the ratio  (as in classical relativity), we can simply say that the ratio is lower or greater than one. Avoiding a misunderstanding, because of the reflexes acquired by the standard Lorentz transformations5 , we will not ask ourselves what is seen deformed or not deformed in one reference frame or another, thinking in terms of dilation and contraction of time and length: there is no substantial time or space, waiting to be dilated or contracted; no influence of a movement on the flow of time (which does not exist).

We only see movements compared to each other (there are no longer clocks on one side and rulers on the other, positions on one side and times on the other; cf. [START_REF] Guy | On the "speed" of light and its measurement: disappearance of space and time standards; the standard of motion; <hal-00814874[END_REF]. Seeking a stable basis for comparison, we are drawn into an infinite regression, which is "temporarily" stopped by simultaneously decreeing the construction of a reference frame and the adoption of a standard signal within it (the photon).

This discussion refers to the debate between Bergson and Einstein on the interpretation of the "flow" of time, and its changes compared between two reference frames (cf. the twins paradox). Without going into detail on the positions of the two scholars (see some elements in [START_REF] Guy | On the comparative age and aging of two twins (relativity theory[END_REF], we dare to say that these two authors were both wrong in their belief in a substantial time (which the first considered unique, the second multiple). Bergson was somehow right when he spoke of a parallax or perspective effect to explain the differences expressed by the  factor; there is indeed a change according to the point of view (OP and O'P are different) but the parallax is original because the effect is not necessarily symmetrical (see above and Guy, op. cit.). And he was wrong in denying any reality at one time in the moving reference frame (he was talking about its fictional, imaginary, fantastical character!): there is something that can be attributed to the R' reference frame. On this point, Einstein was right:

the distance of the photon from its origin, i.e. the time in this reference frame, is not illusory (by repeating it in our own words).

Derivation of the  factor

To connect t and t', in the  factor, the Pythagoras theorem gives us: OP 2 = OH 2 + HP 2 = (OK

+ KH) 2 + (PH' + H'H) 2
, what is written :

c 2 t 2 = (vtcos + ct'cos') 2 + (vtsin + ct'sin') 2 c 2 t 2 = v 2 t 2 cos 2  + c 2 t' 2 cos 2 ' + 2vctt'coscos + v 2 t 2 sin 2  + c 2 t' 2 sin 2 ' + 2vctt'sinsin c 2 t 2 = v 2 t 2 (cos 2  + sin 2 ) + c 2 t' 2 (cos 2 ' + sin 2 ') + 2vctt'(coscos'+ sinsin') c 2 t 2 = v 2 t 2 + c 2 t' 2 + 2vctt'cos('-) (c 2 -v 2 )t 2 -2vctt'cos(' -) -c 2 t' 2 = 0
Where traditional trigonometric relations have been used. We obtain an equation of the second degree of t as a function of t' and the parameters v, c, and angle (' -). The reduced discriminant ' is equal to :

' = v 2 c 2 t' 2 cos 2 (' -) + (c 2 -v 2 )c 2 t' 2 = c 2 t' 2 (v 2 cos 2 (' -) + c 2 -v 2 ) = c 2 t' 2 (c 2 -v 2 sin 2 (' -))
This discriminant is always positive and the problem has two solutions. The square root of the discriminant is ct'(c 2 -v 2 sin 2 (' -)) 1/2 , and the solutions are

𝑡 = 𝑣𝑐𝑡 ′ cos( ′ -)  𝑐𝑡′√𝑐 2 -𝑣 2 𝑠𝑖𝑛 2 ( ′ -) 𝑐 2 -𝑣 2
That we can write t = t' with

 = 𝑣 𝑐 cos(' -)  √ 1 - 𝑣 2 𝑐 2 𝑠𝑖𝑛 2 (' -) 1 - 𝑣 2 𝑐 2
Where  is a function of v, c, and ' -. Using the relative notations  = v/c and  = ' - we write :

 =  cos   √ 1- 2 sin 2  1- 2 (2)
This fundamental relation (2)  = (,) is a function of the ratio of the modules of v (relative movement of the reference frames) and c (movement of light defining time), and of the angle between these movements;  does not depend on  which will play in terms of the offset between the origins. For the time being, we consider that this is indeed the  factor of a

Lorentz transformation (in a modified sense as explained above), although we have not completely derived it. There are two arguments for this: -the similarity of approach with that derived from the standard transformation (in terms of time "dilation"), -that of the connection with the standard expression (see below). These points may be developed in the future. (the cosine then changes sign). In total, we will consider that the choice of the sign in the expression of  refers to questions of orientation of the axes bearing v, t and t', as well as the angle . We can choose a sign for one  from which to find others for other choices of direction.

Discussion

Three-dimensional calculation

Our previous reasoning focused on two dimensions of space x and y. The calculation for three coplanar by construction (the path of the same photon P is followed), the useful calculation can be done in two dimensions in a plane, even if this plane is immersed in a threedimensional space. In this case, the value of the angle  as a function of the direction cosines of vectors c' and v is given in Annex 2, relation (A2).

Study of some special cases (

 = 0,  = /2)
We are not going to do a general study of the function () to given  (relation (2)) for the moment. A first way to assess the relevance of our fundamental expression is to report specific values of the angle .

Parallel movement

In the case of a movement of light parallel to the vector v, we have  = 0, i. e.  = ', (we will also have  = '); we then have cos = 1 and sin = 0. These results are also valid for both angles  and ' simultaneously zero. There remains

 =   1 1- 2 (3)
By taking the + sign, we have

 = (1 -β) -1 = (1 -v/c) -1 (4)
This is the value already found for parallel transport in [START_REF] Guy | On the comparative age and aging of two twins (relativity theory[END_REF], corresponding to [START_REF] Elbaz | L'onde stationnaire et la transformation de Lorentz[END_REF]; the formulations given by these two authors contribute to transformations in the renewed sense announced in our introduction.

If we take the sign -, we have  = -(1 + β) -1 = -(1 + v/c) -1 by changing v to -v and t or t' to -t or -t' in the previous expression (value also proposed by the previous authors).

Perpendicular movement

In the case of a movement of light perpendicular to the direction v of the relative movement of the reference frames, we have  = ' - = /2. Then cos = 0 and sin = 1, and:

 =  √ 1- 2 1- 2 (5)
Or

 =  (1 - 2 ) -1/2 =  (1 -v 2 /c 2 ) -1/2 (6)
The sign  referring to the choice of orientation of the t or t' axes. This value corresponds to that given in [START_REF] Guy | On the comparative age and aging of two twins (relativity theory[END_REF] for perpendicular transport, with or without round trip6 . It is also the classic value of Lorentz transformation (1904)7 . There is then a coincidence, for the formulation of , between the choice of a particular path for light (our point of view) and that of accommodating all directions (classical point of view). Various authors (e.g. [START_REF] Hoffman | Histoire d'une grande idée, la relativité[END_REF]Rougé, 2002) have chosen this particular perpendicular path when they want to find a pedagogical way to make the standard Lorentz transformation understood. They then identify without saying the "passage of time" with the motion of the photon. One would be tempted to ask these authors why they did not choose parallel transport (which does not give back the standard expression)? In a similar intellectual approach, effective measurements on an optical clock displaced perpendicular to the movement of light give a very interesting return to the standard  factor (Chou et al., 2010) 8 .

It is precious for us to find the  factor of Lorentz classical formula, as well as that of Elbaz or our own work, as particular cases of a more general formula where the angle between the directions of the movements of interest of the problem now appears. Table 1 gives various cases of  factors that we believe can all be found by using the fundamental formula (2) 9 .

Positioning of the movement associated with time with respect to the relative movement of the reference frames Angle value 

Lorentz Transformation

 factor

With = v/c

Bibliographical references

Parallel movement in the same direction

 = 0 (1 -) -1 Elbaz (1984) Parallel movement in opposite direction  = 0 (1 + ) -1 Elbaz (1984) Parallel round trip  = 0 (1 - 2 ) -1 McCarthy (1999) Parallel round trip Double (x =  ct, x' =  ct')  = 0 All directions (3D) (1 - 2 ) -1/2 Lorentz (1904)
Perpendicular displacement (with or without round trip) 2002), [START_REF] Guy | On the comparative age and aging of two twins (relativity theory[END_REF] Table 1 Some special cases of the  factor depending on the angle  and the type of light movement chosen (single, multiple, with or without round trip). The sign of each factor  and  can be changed according to the orientation choices discussed in the text. 8 To our knowledge, this type of experiment has not been done on a parallel trip? 9 McCarthy's article quoted discusses the general case of physical phenomenon of propagation, measured by clocks based on the same phenomenon of propagation; said brutally, the author's aim is to disqualify the theory of relativity. We will not go that far, the underlying problem of recursivity, posed by all measurements, being inevitable because of our situation inside the world (there is no clock independent of phenomena...): relativity is a way to face it.

 = /2 (1 - 2 ) -1/2 Hoffman (1980), Rougé (
The  factors can also take the inverse values from those reported in the second column of Table 1 if, during the exchange between v and -v, the angle ' is still imposed in the same reference frame R', without then offering this possibility to the angle  of the reference frame R. A discussion on the symmetry or asymmetry of the two points of view on the two reference frames can be found in [START_REF] Guy | On the comparative age and aging of two twins (relativity theory[END_REF].

Group structure

We will discuss later on the question of the group structure of the complete transformation which integrates the  factor and the offset in origins. The composition of several offsets is especially important when the different directions of relative movement are not parallel. If we stick to the  factor that is a scalar, we do not have at its strict level any composition problem;

we check that we have the group structure, i. e. that it is possible to find a v 3 such that

(v 3 ) = (v 1 v 2 )
, where  is the law of relativistic addition of the speeds of modules v i . One will

check on various examples that (v 3 ) = (v 1 v 2 ) = (v 1 ).(v 2 )(1 + v 1 v 2 )
, in particular for parallel and perpendicular transport as defined above 10 .

The writing of the Lorentz transformation in the most general case

We do not repeat here the demonstration of the complete derivation of the Lorentz transformation in the general case, nor do we examine its proper functioning in the context of our new proposal, postponing this necessary examination to a later work. For the moment, we are content to propose expressions that are consistent with the "geometric" choices made and with the whole process. In doing so, we follow a direction of reasoning that is the opposite of the usual direction where the factor  appears as a consequence of the complete derivation.

We have proposed a factor  first. Equipped with this factor, expressing the distortions between durations (or lengths), the complete Lorentz transformations also involve origin offsets for the space and time coordinates in the relative movement of the reference frames at velocity v. This approach in the sense:  factor → Lorentz transformation was described in 10 When a velocity v 2 is expressed as dx'/dt' in the moving reference frame at velocity v 1 (where x' and t' are given by a usual Lorentz transformation) the expression of the velocity composition is independent of the  factor. [START_REF] Guy | Lightning and thunder. Walks between space and time[END_REF]. It leads to a transformation involving on the one hand the space coordinates x, y and z, and on the other hand the coordinates of the photon useful for the definition of time, namely t x , t y and t z . In the end, we have to manipulate a vector with 6 components (x, y, z, t x , t y , t z ). For space coordinates, transformations are written:

r′ ⃗⃗⃗ = ( r ⃗ -v ⃗⃗t) (7)
What is developing in:

x' = (x -v x t) y' = (y -v y t) z' = (z -v z t) (8)
In ( 7), it is the module or scalar t that measures the displacement of R' relative to R in the vector vt, where v is the vector.

For time coordinates, it is necessary to take into account the different directions of t and t' (intervention of  i and ' i ), but also the orientation of v with respect to t' and t' (intervention of  i and  i ). We then have the following equations, where v is a scalar:

𝑡′ 𝑥 = ′ 𝑥  𝑥  (𝑡 𝑥 - 𝑣𝑥 𝑐 2  𝑥  𝑥 ) 𝑡′ 𝑦 = ′ 𝑦  𝑦  (𝑡 𝑦 - 𝑣𝑦 𝑐 2  𝑦  𝑦 ) 𝑡′ 𝑧 = ′ 𝑧  𝑧  (𝑡 𝑧 - 𝑣𝑧 𝑐 2  𝑧  𝑧 ) (9)
For which we do not have a synthetic vector expression: we could write

t ′ ⃗⃗⃗ = ( Proj(t, t ′ )t ⃗ - v c 2 Proj(v, t′)r ⃗) (10)
subject to defining two projection operators of t on t' and v on t', Proj(t, t'), a function of  i and ' i , and Proj(v, t'), a function of ' i and  i (as discussed in the next section).

In the simplified two-dimensional case of space (x, y), already discussed, the previous transformations are written by replacing  x ,  y , by cos and sin, and  x and  y by cos and sin ; so for t' x and t' y as a function of t x , x, and t y and y, we have :

𝑡′ 𝑥 = 𝑐𝑜𝑠′ 𝑐𝑜𝑠  (𝑡 𝑥 - 𝑣𝑥 𝑐 2 𝑐𝑜𝑠 𝑐𝑜𝑠 ) t′ y = sin′ sin  (t y -vy c 2 sin sin ) (11)
In the case where  =  and therefore  = ', we find the formulas already given in [START_REF] Guy | Les relations de Lorentz et le temps: proposition d'utilisation d'un paramètre tri-dimensionnel défini par un déplacement[END_REF][START_REF] Guy | On the comparative age and aging of two twins (relativity theory[END_REF], which we noted :

) ( ' 2 r c v t t   (12) developed in : ) ( ' 2 x c v t t x x   ) ( ' 2 y c v t t y y   ) ( ' 2 z c v t t z z   (13)
With v(v x , v y , v z ) of module v. The scalar time is built by t² = t x ² + t y ² + t z ² and can be enrolled in a four-component vector (x, y, z, t), from the six-component vector (x, y, z, t x , t y , t z ).

Commentary

In all the previous equations, the expression of the  factor is given by the general formula (2).

Established in our two-dimensional reasoning, this relation is valid for three dimensions as we have said. The angle  is then defined in the plane (v, c') and its value as a function of the directions of the vectors in the three-dimensional space is given in Annex 2 as recalled above.

The formulations given in [START_REF] Guy | Les relations de Lorentz et le temps: proposition d'utilisation d'un paramètre tri-dimensionnel défini par un déplacement[END_REF][START_REF] Guy | On the comparative age and aging of two twins (relativity theory[END_REF] deserve discussion. For the space coordinates, we wrote vector terms in vt, where the vector is t and the scalar v, unlike what we said just now in the relation ( 7). This is written vt x , vt y , vt z in the developed expressions. The latter are accurate because, in our past work, we have asked for parallelism between vectors v and c and c', or between v, t and t'; this leads to v x t = vt x , v y t = vt y , v z t = vt z . But the old relations are not valid in general: it is necessary to write v x t and not vt x , because t is the parameter that counts (and not such or such t x ) to move along v, that we project in v x t etc. This explanation reconciles the two points of view (compare to other authors who have used a threedimensional parameter to define time: [START_REF] Franco | Vectorial Lorentz transformations[END_REF][START_REF] Tsabary | An alternative mathematical model for special relativity[END_REF].

Composition of non-collinear Lorentz transformations

What can we say now about the composition of several reference frame displacements? It is now necessary to conceive this composition in a general way, within the framework of a time understood itself as a displacement, and by using its projections according to the various useful directions. When composing any velocity displacements, we will have to correlate oriented changes of standards, associated with changes in velocity directions. We can divide the problem into two sub-problems.

The first is set by the directions in which the clock photon is seen from the different reference frames; it is assumed that the choice of the reference frame where this direction is imposed has been made beforehand, the other directions being deduced from this, taking into account the relative speeds of the reference frames. This problem was addressed in the case of two reference frames, and projections useful for connecting standards were highlighted in the relation (10). The angles ,  and ' appearing in the developed relations (9) show the various projections to be made on the coordinate axes to refer to comparable terms, taking into account the different orientations of the vectors concerning times and displacements.

The second sub-problem concerns the actual composition of the movements of the reference frames at speeds v, v', v''' etc. The two problems are intertwined, but in order not to complicate things and limit ourselves to the composition of movements, we decide to neglect the first problem, or, more exactly, to assume that it does not arise; this is another way of saying that we have taken the different  equal to zero, or that we have already made the projections announced in a relation such as (10). Otherwise, these projections will have to be added to those we will see later. Let's call (C) this choice (the  = 0). If we limit ourselves to the composition of the scalars , we have seen above that the group structure was acquired (the difficulty arises for the offsets of the origins mixing "vector" spaces and times from different directions).

Let us therefore look at how we can compose two Lorentz transformations (as defined in the previous section) along any two directions v and v', knowing that the vectors t and t' of the first and second reference frame are parallel to v, and that the vector t'' of the third reference frame is parallel to v'', according to the choice (C) 11 . The relations we have given can be written in a six-dimensional space for vectors m = (r, t) = (x, y, z, t x , t y , t z ) T with t = (t x 2 +t y 2 +t y 2 +t z 2 ) 2 and m' of analogous expression as a function of r' and t' (the exponent "T" means that a transposition is performed). Transformation is written:

m' = L(v).m
by continuing to use the notation L(v). In reality, we are dealing with a single space of dimension three counted twice, r and t (r' and t') both belonging to it. In the first reference frame, time is defined by the propagation of a signal along a certain direction t, which also makes it possible to define the ruler used to measure distances; t and v have the same direction (choice (C)). If the direction v is changed and displacements along a new direction v' are considered, length and time standards (marked by the movement of the mobile marking time) along this new direction must be considered; thus all useful vectors (having both temporal and spatial significance) must be projected onto the new direction before writing the Lorentz transformation linking the two reference systems. Let us designate by m v (r v , t v ) the vectors considered according to the direction v and by m v' (r v' , t v' ) their projections according to v' (all the components of m are subjected to the projection operation). Let us call Proj(v,v') the matrix giving the projection of a vector considered along the direction v' on the direction v'. It is a diagonal matrix whose coefficients are ratios of the direction cosines of the directions v and v'. Without changing the notation, we consider that Proj(v,v') can be written in a two-times three-dimensional space, repeating the coefficients a second time to 11 The following text, up to but excluding Section 7, is from Guy (2010), with some modifications.

have a 6-dimensional matrix. We check that these projection matrices commute with the matrices L. Thus, when passing from v to v' we write:

m v' = Proj(v,v')m v
The Lorenz transformation concerns two vectors m and m', where the components are measured using the same direction of the standards; for the first transformation we therefore have:

m' v = L(v).
'' v = Proj(v',v).m'' v' it finally comes m'' v = L(v').L(v).m v
where the property that the matrices L and Proj commute is used, as well as the identity Proj(v',v). Proj(v,v') = 1. We see in total that, by composition, we have:

L(v'') = L(v).L(v') (13)
in vector writing. So we can project in any direction we want. We can compose the vector relations giving r' and t' as a function of r and t then r'' and t'' as a function of r' and t'. Each time v and v' multiply a vector, the modules are written. The speed composition law is then:

v" = v' v = 2 ' . 1 ' c v v v v   (14)
where v and v' are vectors of modules v and v'. Each component behaves like a scalar and projects on the coordinate axes (taking into account the remarks given above that t.v x = t x .v etc. -choice (C)). We obtain, for example for the component along the x-axis:

v" x = 2 ' . 1 ' c v v v v x x   (15)
and so on for the other components.

These new relations, thanks to a better symmetry between "vector" time and position, allow us to find an internal composition law and an abelian group structure. There is no need to add rotation after the fact. The proposed coordinate transformation and velocity composition formulas do not pose the same problems as non-collinear velocity compositions and Thomas rotation. We will verify that, in total, the proposed transformations and their compositions do not present any of the difficulties discussed in [START_REF] Guy | Les relations de Lorentz et le temps: proposition d'utilisation d'un paramètre tri-dimensionnel défini par un déplacement[END_REF].

What about natural phenomena? Conclusions

What is the relation between the previous formalism and physical reality? Every physical system carries its own clock, a way of talking about the tracking allowed by the movements of particles and photons that occur within it. The proper "rhythm" of this clock is accessed through the measurements made on the "at rest" system that the observer has in front of him.

If the system is now moving relative to the observer, it will be understood after the previous calculations that the times associated with the starting system are modified, "dilated" or not. A wide variety of movements can be imagined, both for the interplay of phenomena within the system and for the relative movements of the system in relation to the observer. Undoubtedly, the complicated movements within the system of interest to us are difficult to reduce to a single parameter ', which rather represents a kind of average of different movements (likewise does the standard Lorentz transformation reflect an average behavior?).

The quantitative values of temporal distortions, which can be calculated by a factor such as , are then varied: there is no generality and work is to be done in each case. We have given only one new way of posing problems from which to examine at new expense the interpretations already given in the literature (and sometimes subject to debate): from the examination of relativistic muons (whose lifetime is seen to be longer in their displacement than in their resting state) to that of clocks embedded in aircraft and for which the time distortion seems to vary according to the direction of rotation around the globe. We will also have to look at what is derived from the mathematical form of the  factor on the structure of other physical relations (relating to energy for example): do we need a finer representation of the  factor, function of what we know about the system? It is because we have spatialized time, that is, closely associated with a movement, that we have been able to make the proposals contained in this text. A number of points, detailed in the foregoing lines, deserve further elaboration.

Figure 1 Illustration of the steps of our proposal

In 1a, the situation described by the classical approach is represented. Two reference frames R and R' of origin O and O' are relatively moving to the speed v (OO' = vt). Each of the two reference frames is equipped with a clock, H and H'. We consider the displacement of a photon P and assume that its displacement speed is the same viewed from R and from R'. This photon can move in all directions: this possibility has been represented by the wave surface centered on O', i.e. a two-dimensional circle (on which two photons P have been pictured, we could put as many as we want).

In 1b, the clocks H and H' were opened and we found that each uses the movement of a photon, P H and P H' . We now have three photons, P H , P H' and P, counting the latter as an infinity in the different directions.

In 1c, the approach was unified, being led to postulate the same behavior for the different photons that are now one at the conceptual level. To have clocks that are needed to "measure time" in both reference frames, we are thus led to particularize a photon that will serve both for time in R and in R' (while "embodying" the second postulate of relativity). The two clocks H and H' as defined in Figure 1a are destroyed (or forgotten, they were fictions).

In total, this proposal makes the relativistic approach coherent: one could wonder (within the understanding 1a) by what mystery the clocks offered a time that shifted from one reference frame to another, to satisfy the properties of a photon whose journey was external to them, a time that was variable with the relative movement of the reference frames12 .

Figure 2

Two reference frames R and R' are represented in two spatial dimensions, in relative motion at velocity v at an angle  to the Ox O'x' direction. In R' the movement of the clock photon P is done in the direction of angle '; from the reference frame R, this movement is seen with the angle , taking into account the relative movement between the reference frames. During the "time interval" t (measured by a piece of movement), the reference frame R' moved according to OO', the photon P according to O' P in R' and according to OP in R. It is decided / assumed that the photon is seen moving at the same "speed" in both reference frames. We therefore have OP = ct, and O'P = ct', with the same factor c ; the latter being much larger than v, the lengths OP, O'P are much larger than OO' in length vt, which has not been respected for the readability of the figure.

Figure 3 Measurements of the angles and sides of the OO'P triangle

These measurements make it possible to determine the angle ' from the angles  and , and the ratio = v/c. The angles of the OO'P triangle are determined from the definitions of , ' and  given in Figure 2.

Figure 2 .

 2 Figure2. In this case, the notations , ' and  denote angles such as:  x = cos,  y = sin;  x = cos,  y = sin in the rest reference frame; and ' x = cos', ' y = sin' in the moving reference frame. The vector v is seen in R under the angle  and under the same angle  = ' in R' (we will use  for both). Neither the z-direction nor the variables and parameters associated with this direction are involved. It is useful to define the angle  = ' -' between the vectors c' and v (we will not use the equivalent notation ' = '-'), i.e. between the relative displacement of the reference frames and the movement of the photon marking time in the moving reference frame (we have said that we locate this movement first). We have here two factors expressing the relations between two movements: the ratio  of the two modules, the angle  between them. The three angles , ',  (') and the parameter = v/c are linked together by the relation (A1) :  = ' -Arcsin{ sin} (see Annex 1).

  of the sign  in the expression of  What meaning can we give to the choice of sign in the relation (2)? The second term of the numerator and the denominator having a constant positive sign, we must at first sight make a possible change of sign on the first term of the numerator, that is cos. Let us write the factor  as  = a  b. If we take the + sign, we have  = a + b. If we take the sign -, we have  = a -b, that we write = -(-a + b). In this form, we see that, by keeping the same sign for the second term b of the numerator, we can change the sign, not only of the first factor a, but also of the whole expression of . We interpret this as follows: a) the change in sign of the entire expression of  = t/t' means changing orientation of one of the two t or t'; b) the change in sign of the first term a = cos is obtained either by changing v to -v, or by changing to  to  + π

  dimensions x, y and z does not present any difficulty in principle. It brings nothing new from the point of view of the fundamental relation (2) that we have demonstrated. Indeed, what is at stake is on the one hand the ratio of the modules of v and c, and on the other hand the angle  between the vectors c' and v. Since the three vectors v, c and c', or OO', OP and O' P' are

  m v To deal with a new Lorentz transformation according to v', we must first express the previous relation along this new direction: m' v' = Proj(v,v').L(v).m v before expressing the new transformation by: m'' v' = L(v')m' v' By projecting everything onto the initial direction v by m

  t, t y =  y .t, t z = 

z .t (rest reference frame); t' x =' x .t', t' y =' y .t', t' z =' z. t' (moving reference frame); or, by multiplying

We will use the word speed (with or without quotation marks) here, although for light it is not appropriate (see the discussions in the literature and in our work).

Our second announced point of view on Lorentz transformation, on the side of the twins paradox, will be discussed below.

Whatever the interpretation given, the convergence of the two approaches (the classical approach / ours) towards identical equations for particular cases constitutes a certain assurance that we are not getting lost; the functioning of the equations has a meaning in itself.

We talk about round trip when the motion of the photon is not indefinitely rectilinear but is reflected on a wall, for example.

One may wonder why the standard transformation corresponds to this geometric configuration?

The inconsistencies of the relativistic approach have been analyzed by critical physicists, but, in our opinion, without any new proposal.

Acknowledgements

We thank Guy Tsabary (Israel; co-author of the article by Tsabary and Censor quoted) for his encouragement to write this text.

Annex 1 Relation between the angles  , ' and 

The three angles  , ' and , and the ratio  = v/c are related to each other. The trigonometric relations in any triangle are applied here to the POO' triangle (see Fig. 2). The angles and lengths of the sides are shown in Figure 3 and the relations are written:

We specify ' and , and we want to know . Equality is derived from the second and third terms:

Which we can also note (taking into account the definitions of  and ):

Appendix 2. Calculation of the angle  as a function of the direction cosines of vectors v and c' (in the case of three spatial dimensions)

Let  x ,  y, and  z be the direction cosines of vector v, and ' x , ' y and ' z those of vector c'.

The cosine of the sought angle  is equal to the scalar product of the vectors v and c' divided by the product of their modules vc. We derive  from it by the relation :