M Arminjon

POST-NEWTONIAN APPROXIMATION OF A SCALAR THEORY OF GRAVITATION AND APPLICATION TO LIGHT RAYS

A scalar, preferred-frame theory of gravitation is summarized. Space-time is endowed with both a flat metric and a curved, "physical" metric. Motion is governed by a natural extension of Newton's second law, which implies geodesic motion only for a static field. The theory predicts Schwarzschild's exterior metric in the spherical static situation. It also predicts gravitation waves with the velocity of light. The equations of motion are recast into the "flat space -uniform time" form, and compared with the geodesic equations of motion. The principles of the post-Newtonian approximation of this theory are given, including the way to account for preferred-frame effects. This approximation is then developed more particularly for photons. It is found that the preferred-frame effects do not occur in this case, nor does the difference between Newton's second law and geodesic assumption. Thus, the post-Newtonian predictions of this theory for photons are indistinguishable from the standard post-Newtonian predictions of general relativity.

Introduction

One of the most striking features of Einstein's general relativity is that it predicts three effects of a gravitation field on light rays (or photons), and that these effects are experimentally confirmed. In this paper, it will be shown that a certain scalar theory of gravitation, which has a preferred frame, predicts post-Newtonian gravitational effects on photons -namely, the redshift, the deflection, and the echo delay -which are just the same as the standard post-Newtonian predictions of general relativity (GR). Although this does represent an important step for any theory, it is still not enough for claiming that the investigated theory is viable. However, the simplicity of this scalar theory, and the great difference with GR in its physical concept, make this result at least curious, because the effects on light rays often appear as the most "relativistic" predictions of GR. In fact, this theory predicts all the same exact effects as the standard effects of GR in the static spherically symmetric situation. This is because, in that situation, the theory predicts the Schwarzschild exterior solution and asserts that free particles follow space-time geodesics -and this is true for mass points [1] as well as for photons [2]. But since the theory has a preferred frame, one must account for the effects of the velocity of the massive attracting body through this "ether": this is done in this paper, for photons and in the first post-Newtonian approximation. The non-covariance of the theory is a necessary consequence of the fact that it is a scalar theory which accounts for special relativity (SR). Thus, the source of the gravitation field must be an energy and the energy concept of SR does not lead to an invariant scalar, instead it leads to a covariant tensor T, the energy-momentum of matter and non-gravitational fields.

One may ask, of course, why it should be interesting to investigate a non-covariant theory, knowing the risks it implies (especially for the explanation of Mercury's advance in perihelion). We mention the following reasons:

(i) This scalar theory is a natural modification of the Newtonian theory to account for "relativistic" effects, when one sees these effects as due to the "absolute" Lorentz contraction of bodies that move through the "ether" -as Lorentz and Poincaré saw these effects. Jánossy [3][START_REF] Jánossy | Theory of relativity based on physical reality[END_REF], Prokhovnik [START_REF] Prokhovnik | The Logic of Special Relativity[END_REF][START_REF] Prokhovnik | Light in Einstein's Universe[END_REF][START_REF] Prokhovnik | [END_REF], and others, proved in detail that this "absolute" conception is entirely compatible with the usual formalism and the physical predictions of SR. The Lorentz-Poincaré interpretation of SR was also emphasized by Bell [START_REF] Bell | Speakable and Unspeakable in Quantum Mechanics[END_REF]. The theoretical impossibility in measuring the one-way velocity of light independently of a synchronization convention (see Prokhovnik [START_REF] Prokhovnik | The Logic of Special Relativity[END_REF] and Zhang [START_REF] Zhang | [END_REF]) is important in this interpretation.

(ii) This theory does not predict any singularity or infinite density for the case of gravitational collapse in "free fall" with spherical symmetry: instead, the theory predicts a "bounce", i.e. the initial implosion is stopped and is followed by an explosion [2].

(iii) It does lead to a "quadrupole formula" for the gravitational radiation, in the way briefly outlined in ref. [2].

(iv) Some gravitational effects of motion through an "ether" might contribute to explain the strange dynamical behaviours at the scale of galaxies or clusters of galaxies (e.g. the rotation curves). The currently favoured explanation of such behaviours appeals to large amounts of unseen matter [10].

In order to show that the "relativistic" effects on light rays are not masked by the motion of the attracting body through the ether of this theory, we shall proceed thus: in Section 2, we summarize the theory and give the relevant exact equations -in particular, we give the equations of motion of a particle (mass point or photon) in the form of the "flat space -uniform time" acceleration, as is convenient for applications. In Section 3, we present the principles of the asymptotic expansion for the equations of the theory; we apply these principles to expand, first, the metric and the field equation, then the equations of motion for a photon. We find that the effects of absolute motion on photons do not appear in the first post-Newtonian approximation, which is the approximation currently used [START_REF] Weinberg | Gravitation and Cosmology[END_REF][START_REF] Misner | Gravitation[END_REF][START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF] to evaluate these effects in GR and other "metric theories". By the way, we already underline here an important point: although the investigated theory endows the space-time with a curved metric, it is not truly a metric theory in the usual sense [START_REF] Weinberg | Gravitation and Cosmology[END_REF][START_REF] Misner | Gravitation[END_REF][START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF], because "free" particles do not generally follow space-time geodesics (except for a static gravitation field). This means that the parametrized post-Newtonian formalism does not a priori apply to this theory.

Summary of the Theory and its Main Exact Equations

The starting point of the theory is simple: it is the heuristic interpretation of gravity as the pressure force exerted on matter at the scale of elementary particles by an imagined perfect fluid or "micro-ether". Thus, according to this heuristic view, gravity would be simply "Archimedes' thrust in the ether". It turns out that Newtonian gravity (NG) was interpreted in this way by Euler in 1746 [START_REF] Euler | Opera omnia III1 Commentationes Physicae[END_REF]. However, the point here is not NG, although the present theory does reduce to NG in the limit of weak and slowly varying fields [2]. The construction of the theory was done in refs. [1] and [START_REF] Arminjon | [END_REF] (see ref. [2] for a synthetic account, including new results), except for the general formulation of motion, which has been modified in order to obtain a true conservation equation for the energy [16]. Hereafter, the relevant assumptions and equations are summarized directly in the self-consistent form where they will be used, although this makes them appear less natural.

Basic concepts, space-time metric, field equation

The equations are given in the preferred frame E, which is the rest frame of the reference body M or "macro-ether". 1 The reference body M is assumed to be equipped with an Euclidean space metric g 0 . Observers bound to the frame E would measure the spatial distances defined with g 0 , and would also measure the "absolute time" t, if there were no gravitation. An "absolute" version of Einstein's equivalence principle implies that physical standards of space (resp. time) are contracted (resp. dilated) in a gravitation field [1][2]. The scalar field of the theory is the contraction factor  affecting the time interval dtx measured by a clock fixed at point x bound to the frame E (due to the dilation of the clock period in the gravitation field), or the square f =  2 . The latter defines the 00 component of the physical, curved space-time metric  in any coordinates (x  ), with x 0 = ct (and c the velocity of light), that are bound to the frame E:

  dt dt f x / .       00 1 E (1)
The interpretation of gravity as a pressure force leads to define the acceleration vector g of gravitation [1][2][START_REF] Arminjon | [END_REF]. When reexpressed in terms of the Euclidean metric g 0 and the scalar f, the vector g turns out to be simply [16] g

     c f f f 2 0 0 2 0 , grad g . ( 2
)
Due to the "absolute" version of the equivalence principle, the field  defines also the dilation of an infinitesimal distance dl measured in the direction of vector g, as compared with the distance dl 0 evaluated using the Euclidean metric: dl = dl 0 /, while the infinitesimal distances in directions perpendicular to g are unaffected. This assumption may be reexpressed in saying that the physical space metric g in the frame E has the following relation to the Euclidean metric g 0 (in any coordinates (x  ) bound to the frame E):

2 g g h ij ij ij f          0 1 1 ,     h g g ij i j k l i j k l kl kl f f f f       , , , , , , , , 0 0 (3)
with ((g 0 ) kl ) the inverse matrix of (g 0 kl). Indeed, Eq. ( 3) is space-covariant and implies that, in coordinates orthogonal for the Euclidean metric:

(g 0 ij) = diag (a 0 1, a 0 2, a 0
3), and such that g i =0 for i  1, the physical metric is simply

(gij) = diag (a 0 1/f, a 0 2, a 0
3).

The space-time metric writes, in any coordinates (x  ) bound to the frame E, with x 0 = ct:

ds 2 =  dx  dx  = f (dx 0 ) 2  dl 2 , dl 2 = gij dx i dx j  00 = f, ij =  gij, 0i = 0. (4)
That the 0i components are always zero in the ether frame E, is the natural translation of the assumption that this frame admits a global simultaneity defined with the absolute time t. Hence, a gravitation field that is constant in this frame is necessarily a static field. This implies, in turn, that a stationary rotation of a massive body with respect to the ether frame will produce no Lense-Thirring effect. One may perhaps infer that this will remain true if, in addition, the mass center b of the body has a constant velocity with respect to the ether frame, db/dt V = Const. However, in that case, the gravitation field cannot be static (even in the frame EV that follows the translation) since, due to the Lorentz transformation of the non-Galilean metric , the 0i components cannot reduce to zero in the frame EV [see Eqs. ( 42)-( 43) and ( 55)-( 56) below]. This is independent of the field equation.

In what follows we assume that the "reference ether pressure" pe  (t) [2] does not actually depend on t, which means that we leave aside cosmological problems. [Such problems are considered in ref. 22, written in 1999.] Then the field equation may be written as:

 0 0 0 2 1 8 f f f f G c         , ,   , (5)
with 0  div0 grad0 the usual Laplace operator, defined with the Euclidean metric g 0 , and where f,0 = f/x 0 , G is Newton's gravitation constant, and   (T 00 )E is the mass-energy density in the ether frame [16]. This equation, which is merely space-covariant, implies that gravitation propagates with the velocity of light (as measured with physical standards affected by gravitation). On the other hand, in any static situation, it associates exactly the Newtonian gravity acceleration g = gN with the given constant mass-energy density  = (x). But   T 00 does not only involve the density pm of the pure matter energy (e.g., for "dust", it does not only involve the rest-mass and kinetic energy), it is also increased by the gravitation field itself -indeed, one has  = pm/ = pm/ f [16]. Finally, in the spherical static situation, the space-time metric (4) predicted from the field equation was given in refs. [1][2] (using Eq. ( 5), it is still easier to get):

ds U r c c dt U r c dr r d d d d N N 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 1 2              ( ) ( ) , sin      , (6)
where UN(r) is the Newtonian potential associated with the density  = (r):

U r GM u u du N r ( ) ( ) ,    2 M r u u du r ( ) ( )   4 2 0    (7)
Again, as a static solution, this solution assumes that the mass-energy density is timeindependent (and spherically symmetric) in the ether frame, i.e. the massive body is at rest in the ether or in a stationary rotation with respect to the ether. If (r) = 0 for r > R, then UN(r) = GM/r for r > R (with M = M(R)), this is the Schwarzschild exterior solution. As to the Schwarzschild interior solution of GR, it is only known when, in addition, (r) = Const. for 0  r < R, but Eq. ( 6) does not give the Schwarzschild interior solution in that case.

Equations of motion

Motion is governed by a natural extension of Newton's second law. In the case of a constant gravitation field, it is relatively straightforward to extend the special-relativistic form of Newton's second law (F = dP/dt with P the momentum) to the situation with gravitation, because the time-derivative of a vector (like P) must undoubtedly be defined as the "absolute derivative", the latter being induced from the covariant derivative with respect to the constant space metric The absolute derivative of a vector w depending on the parameter  along a trajectory in a Riemannian space equipped with a fixed metric g is given by:

D D d w d w dx d i i j k i j k w             , ( 8
)
with  i jk the Christoffel symbols of metric g. This derivative was used by Landau & Lifchitz [START_REF] Landau | Théorie des Champs rd French edn[END_REF] to rewrite the spatial part of Einstein's geodesic assumption as Newton's second law, for a constant gravitation field. In the case of a variable gravitation field, we still have, in the frame E (as actually in any other possible reference frame), a well-defined space metric g, but now it depends on time, and we have a local time tx. It turns out that a unique definition may be given for the time-derivative Dw/D of a space vector w depending on the parameter  along a trajectory in the given frame, under the following essential requirements: (i) it must be a space vector depending linearly on w, (ii) it must reduce to the absolute derivative if g does not depend on time, (iii) it must be expressed in terms of the space metric g and its derivatives, and (iv) it must obey Leibniz' derivation rule for a scalar product, i.e.

 

d d D D D D    g g g ( , ) ( , ) ( , ) w z w z w z   .
In coordinates, this unique definition is as follows [16] :

D D d w d w dx d dx d w i i j k i j k ij jk k w               1 2 0 0 g g , , (9)
with  i jk the Christoffel symbols of the space metric g. To define Newton's second law, the trajectory of the test particle is parametrized with the local time  = tx, synchronized along the trajectory as defined by Landau & Lifchitz [START_REF] Landau | Théorie des Champs rd French edn[END_REF]. Actually, definition ( 9) is also valid (and compelling under the above requirements) for any reference frame in any theory with a curved space-time metric [START_REF] Arminjon | [END_REF]. In the present case, the synchronized local time is simply obtained by integration of Eq. ( 1) along the trajectory, because the absolute time t is globally synchronized in the ether frame E (i.e., because the 0i components are assumed zero in E).

There is also no ambiguity for the definition of the momentum. This is

P  m(v) v, m(v)  m(v = 0).v  m(0).(1  v 2 /c 2 ) 1/2 , ( 10
)
the velocity v of the test particle (relative to the frame E) being measured with the local time tx and its modulus v being defined with the space metric g:

v i  dx i /dtx , v  [g(v, v)] 1/2 = (gij v i v j ) 1/2 . ( 11
)
The gravitation force is assumed to be Fg = m(v)g with g given by Eq. ( 2), and there may be also a non-gravitational force F0. Thus, the extension of Newton's second law is obtained as

F0 + m(v)g = DP/Dtx . ( 12
)
For a photon, the velocity v is still defined by [START_REF] Weinberg | Gravitation and Cosmology[END_REF] (and now its modulus is constrained to v = c), but the mass content of the energy E = h is substituted for the inertial mass, the frequency  (number n of oscillations per unit time) being measured with the local time:

 dn/dtx , E  h, (13)
F0 + (E/c 2 ) g = DP/Dtx , P  (E/c 2 ) v. ( 14
)
Defining the "pure energy" of a mass point (i.e., not accounting for the potential energy in the gravitation field) as E = m(v) c 2 , we see that Eq. ( 14) applies to both mass points and photons. Together with the assumed vector g (Eq. ( 2)) and the assumed metric (Eqs. ( 1), ( 3) and ( 4)), Eq. ( 14) implies that free particles (F0 = 0) follow space-time geodesics in a static gravitation field (f,0 = 0): see ref. [1] or ref. [START_REF] Arminjon | [END_REF] for mass points and see ref. [2] for photons.

We now compute the "coordinate acceleration" for a free particle (mass point or photon) in a general gravitation field:

a i  (du i /dt)i=1,2,3, u i  dx i /dt . ( 15
)
Note that u = (u i ) is a space vector, the "absolute velocity", but (a i ) is not a space vector. The "absolute acceleration" a0 may be defined in arbitrary space coordinates as the absolute derivative of u with respect to (the time t and) the Euclidean metric g 0 , and its components are indeed a i in locally Cartesian coordinates x i , i.e. such that g 0 ij = ij and g 0 ij,k = 0 at the point considered. Using definitions (1), ( 2) and (9) in Eq. ( 14) and multiplying by

f =  2 obtains            c f f E d dt E u u u c u i i j k i j k ij jk k 2 0 0 2 1 2 ( ) ,    g g . ( 16
)
To evaluate (d/dt)(E/), we use the energy equation of the theory [16], which is deduced from the extended Newton law (14) for a free particle (mass point or photon):

de dt e t m m      , e E E f e m m      1 . ( 17
)
Inserting Eq. ( 17) into Eq. ( 16) gives after rearranging

du dt t u u i j j i          1 2      ,  j k i j k ij jk k u u t u  1 2 g g     c f f i 2 0 2 ( ) . (18)
This is the form of the equation of motion that we shall use for the pN approximation. However, it will be useful also to have this exact equation in a form that allows a comparison with the following equation of motion, deduced (for arbitrary space-time coordinates) by Weinberg [START_REF] Weinberg | Gravitation and Cosmology[END_REF] from Einstein's assumption that particles follow space-time geodesics:

du dt u i i j i j       ' ' 00 0 2  ' j k i j k u u         ' ' ' 0 0 0 0 2 j j j k j k i u u u u . (t  x 0 ) (19)
The Christoffel symbols  ' of the space-time metric (4), with x 0 = ct, are [START_REF] Arminjon | [END_REF] :

   ' , ' , ' , , 00
0 0 0 0 00 2 1 1 2          j j i i c g f f ( )  0 f i ,  ' 0 0 1 2 k i ij jk  g g , . (20)
Therefore, when changing the time coordinate to x 0 = t (which is convenient for the pN expansion), the existing c or c 2 factors disappear and we may rewrite Eq. ( 18) as

du dt u i i j i j       ' ' 00 0  j k i j k u u       ' ' 00 0 0 0 2 j j i u u (x 0 = t) . ( 21
)
This equation is only valid in coordinates (x  ) bound to E, with x 0 = t, thus 0i 0, whence ' i jk =  i jk . Thus, the difference between ( 19) and ( 21) is that: (i) the  ' i 0j u j term is not multiplied by 2 in Eq. ( 21) as it is in Eq. [START_REF] Arminjon | [END_REF], and (ii) the last term of Eq. ( 19) is missing in Eq. ( 21).

Post-Newtonian Approximation

Principle of the approximation

The principle of the post-Newtonian (pN) expansion is classical (see Fock [START_REF] Fock | The Theory of Space, Time and Gravitation[END_REF], Chandrasekhar [START_REF] Chandrasekhar | [END_REF], Weinberg [START_REF] Weinberg | Gravitation and Cosmology[END_REF], Misner et al. [START_REF] Misner | Gravitation[END_REF], Will [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF]) : it is assumed (i) that the gravitation field is weak so that the Newtonian approximation gives the main terms and moreover the Newtonian potential U is small as compared with the square c 2 , thus Umax /c 2   2 with  a small parameter. (ii) That the velocities of the massive bodies or mass test particles are small as compared with c; this assumption is not independent from (i) since, in a frame moving with the gravitationally bound system, the square of the orbital velocity will be of the order U. Hence, in such a frame, the velocities will be at most of the order c. So any relevant quantity will be expanded in powers of the small parameter . The derivative of any relevant quantity A, with respect to any space coordinate, will be of the same order as A with respect to , of course. It is still assumed that (iii) the derivative of any relevant quantity A with respect to the time coordinate x 0 = ct is of the order   (order of A). This amounts to assuming that |A/x 0 |   O(|A|). It is not valid in the "wave zone", i.e. for distances from the center of the system that are large as compared with the wave length of the gravitation waves emitted by the system,   2lc/v 2 l  with l the typical radius of the system and v the typical orbital velocity (see Landau & Lifchitz [START_REF] Landau | Théorie des Champs rd French edn[END_REF] and Misner et al. [START_REF] Misner | Gravitation[END_REF]; note that the typical dimension of the system is thus small as compared with ). All these assumptions are taken into account automatically if we simply expand all equations with '  1/c as the small parameter, and with x 0 = t as the time coordinate, as proposed by Fock [START_REF] Fock | The Theory of Space, Time and Gravitation[END_REF]. This is what we shall do in the actual calculations. We note that, choosing the units such that the maximum value of the Newtonian potential is Umax = 1, we get indeed  = '.

There is something more to be said in a preferred-frame theory like the present one: the absolute velocity V of the mass-center of the system with respect to the ether should not exceed the order c, as is the case for the typical orbital velocity v in the mass-center frame. For the solar system, we have  2  Umax /c 2  10 5 if we take the absolute maximum, at the center of the Sun [START_REF] Misner | Gravitation[END_REF], whereas the maximum of the "useful" values of U is rather the value at the surface of the Sun, which would give Umax /c 2  2.10 6 . On the other hand, we may take as a working assumption that the ether frame is bound to the mean motion of matter, which is consistent with the present theory [16]. To take this assumption into account, we may further assess V by assuming that the ether frame is at rest with respect to the cosmic microwave background. Then V is approximately 300 km/s [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF], so that V/c  10 3 is indeed smaller than. Strictly speaking, however, the exact value of V should be determined internally to the present theory, from the pN effects of V on celestial mechanics that this theory will predict: as it will be shown here, the effect on photons does not appear in the pN approximation. But, for celestial mechanics, we have also to account for what is known about the motion of the solar system itself with respect to the Galaxy, etc. That is, we could not simply come to the conclusion that V  0, for example, unless it were found to be consistent with that kinematics in the framework of the present theory (although this seems a priori unlikely). The determination, internal to this theory, of the motion (including, perhaps, some straining) of an astronomical reference frame with respect to the ether might thus demand a careful analysis and somewhat complex calculations. However, it is very reasonable to assume that the velocity V of the mass center of the solar system is constant, i.e. does not vary significantly over the relevant time scales (at most a century, say), and is small enough so that the pN approximation is valid.

As to the allowable powers of the small parameter  that must enter the actual expansions, it is known that, from the behaviour of terms under time reversal, some quantities are expected to take only even powers of , e.g. 00 and ij, and other ones only odd powers, e.g. 0i [START_REF] Misner | Gravitation[END_REF]. In the present theory, there is an additional argument: in it, the non-Newtonian effects all come from the "ether compressibility", K  1/c 2 [1][2][START_REF] Arminjon | [END_REF]. So K itself (or Umax K, if the units are not arranged so that Umax =1) could be considered as the small parameter, whence the appearance of only one from two successive terms in any expansion with respect to 1/c = K -the leading (Newtonian) term giving the parity.

Expansion of the metric and the field equation in the preferred frame

The leading expansion is that of the scalar field,  or f =  2 :

     1 2 4 U c S c / / ... , f U c U S c      1 2 2 2 2 4 / ( ) / ..., (22)
whence the space metric by Eq. ( 3) (and with g 0 ij = g 0 ij = ij in Cartesian coordinates):

hij = h (1) ij + O(1/c 2 ), h (1) ij  (U,i U,j)/(g 0 kl U,k U,l), ( 23
) gij = g 0 ij + (2U/c 2 )h (1) ij + O(1/c 4 ), (24a)
g ij = g 0 ij  (2U/c 2 )m ij + O(1/c 4 ), m ij  g 0 ik h (1) kl g 0 lj (24b)
and the space-time metric by Eqs. (4a) and (4c) (remembering that now x 0 = t):

 00 = c 2 f = c 2 [1  2U/c 2 + (U 2 + 2S)/ c 4 + ...], 0i = 0, ij =  gij, (25)
  00 = 1/00 = (1/c 2 )[1 + 2U/c 2 + (3 U 2  2S)/ c 4 + ...],  0i = 0 ,  ij =  g ij . ( 26
)
To expand the field equation, we must also expand the mass-energy density  = (T 00 )E. For the present purpose, it will be enough to recognize that this may be written in the form

  =  N + w 1 /c 2 + ..., (27)
with  N the Newtonian mass density, which only involves rest mass and obeys the usual continuity equation involving the velocity vN found at the Newtonian approximation:

      N N N div t   0 0 v , div0 div g 0 , ( 28
)
and where w 1 is a correction (involving the Newtonian energy). This is a priori obvious, and may be readily verified from the local conservation equation for the energy which applies, in this theory, to a continuous medium [16].

The expansion of the field equation ( 5) follows easily from Eqs. ( 22) and ( 27): we have

 0f = (1/c 2 ) [  2 0U + (1/c 2 )0(U 2 + 2S) + O(1/c 4 )] , (1/f)(f,0/f),0 =  2 U,0,0/c 2 + O(1/c 4 ),
whence (remembering that now x 0 = t):

 0U =  4G  N , (29a)
0S = 4G w 1  0U 2 /2   2 U/ t 2 . ( 29b
)
Thus, what we have noted U in the expansion (22) of the scalar field is indeed the Newtonian potential associated with the Newtonian rest-mass density. The second-order coefficient S is obtained by solving a Poisson equation, too, which means that the propagation of gravity is still instantaneous at the (first) pN approximation, in this theory as also in GR.

Post-Newtonian equations of motion for a photon in the preferred frame

For a photon, the velocity u i is O(c), of course. Using Eqs. ( 22) and ( 24), one gets easily the main part of the coefficients in Eq. ( 18):

1 1 1 2 4       t c U t c          O ,   , , j j c U c          1 1 2 4 O , ( 30
) g g ij jk t c          O 1 2 ,   c f f U c i ij j 2 0 0 2 2 1           g , O , (31)
although one of them has a somewhat complex expression in general space coordinates x i :

        jk i jk i jk i il l jk c U c c           0 2 2 0 4 2 2 1 m O , ( 32
   jk i il lj k lk j jk l U U U    1 2 0 1 1 1 g h h h ( ) ( ) ( ) , , , ( ) ( ) ( ) . (33)
In Cartesian coordinates, this becomes:

   jk i ij k ik j jk i c U U U    1 2 1 1 1 ( ) ( ) ( ) , , , h h h ( ) ( ) ( )
+ O 1 4 c      . (34)
Inserting Eqs. ( 30)-(32) into Eq. ( 18), we get (remembering that

u i = O(c)):     du dt u u a U U u u c i jk i j k i ij j j j i      0 0 0 2 2 g , ,           2 2 0 c U m u u jk i il l jk j k          O 1 c . (35)
(Note that the left-hand side is indeed the absolute acceleration.) In Cartesian coordinates:

  du dt a U U u u c i i i j j i    0 2 2 , ,             ( ) ( ) ( ) , , , U U U u u c c ij k ik j jk i j k h h h ( ) ( ) ( ) O 1 1 1 2 1 . (36a)
This equation of motion has to be augmented with the constraint ds 2 = 0, or v 2  gij v i v j = c 2 , of course. To the required order, this may be written in arbitrary space coordinates as

gij u i u j = c 2 + O(1). (36b)
It is interesting to notice that this same pN equation of motion for a photon would be obtained the present theory if, in the place of Newton's second law [START_REF] Euler | Opera omnia III1 Commentationes Physicae[END_REF], we would postulate that photons follow the (null) space-time geodesics. Indeed, only two terms make the difference between the "Newtonian" acceleration [START_REF] Chandrasekhar | [END_REF] and the "Einsteinian" acceleration [START_REF] Arminjon | [END_REF], and both terms are of order O(1/c): this is true for the  ' i 0j u j term, because (see Eq. ( 31))

 ' i 0j = (g ik  gjk / t)/2 = O(1/c 2 );
and this is true for the  ' 0 jk u j u k u i term, because (see Eqs. (24a) and ( 26))   ' 0 jk = ( 00  gjk / t)/2 = O(1/c 4 ).

Transition to a moving frame: principle and exact equations

In the investigated theory, there is a global flat space-time metric  0 , with line element

(ds 0 ) 2   0  dy  dy  (37)
in arbitrary space-time coordinates (y  ). By definition, this metric is such that

(ds 0 ) 2 =  dx  dx  = c 2 dt 2  (dl 0 ) 2 , (38a) i.e.   0  =  with ()  diag (1, 1, 1, 1), (38b)
when expressed in space-time coordinates (x  ) that are bound to the preferred frame E, with x 0 = ct and with (x i ) being global Cartesian space coordinates on the reference body M (i.e. g 0 ij = ij). Hence, there are also global Lorentz transformations of the space-time RM, which are those transformations of the space-time coordinates (x  ) that leave the flat metric ( 0 ) in the standard form,  0  = . These transformations give transition between space-time coordinates (x  ) that are bound to the preferred frame E , with x 0 = ct and with the x i 's being Cartesian space coordinates on the reference body M, and space-time coordinates (x'  ) that are bound to a uniformly moving frame E V, with x' 0 = ct' and with the x' i 's being Cartesian space coordinates on the reference body MV moving with uniform velocity V with respect to E (the x' i 's are such that g' 0 ij = ij with g' 0 the Euclidean space metric on MV, associated with  0 ). Of course, one may also pass from a uniformly moving frame E V to another one, E V', by a such transformation.

Thus, in order to evaluate the preferred-frame effects that this theory predicts, we pass from E to that frame E V which is assumed to follow the motion of the mass-center of the gravitating system (see § 3.1), by using a special Lorentz transformation of the flat metric  0 . That is, V being the absolute velocity of the mass center, we take Cartesian coordinates on M such that V 1 = V, V 2 = V 3 = 0, and we define a new coordinate system thus (using low indices for Cartesian coordinates from now on):

x'1 = V (x1  Vt), t' = V (t  Vx1 /c 2 ), x'2 = x2 , x'3 = x3 . ( 39
)
The Euclidean space metric g' 0 attached with the moving reference body MV is such that g'

0 ij = ij, hence (dl' 0 ) 2 = dx'i dx'i. (40)
We transform the physical, curved space-time metric  [Eqs. (3) and (4c)] to the x'coordinates (with x'0 = ct' ):

'00  V 2 [00 + 2 (V/c)01 + (V 2 /c 2 )11] = V 2 {f  (V 2 /c 2 )[1 + 1 1 f       h11]}, (41)
 '01  V 2 [(V/c)00 + (1 + V 2 /c 2 ) 01 + (V/c)11] = V 2 (V/c) [f 1  1 1 f       h11], (42)  '0i  V [ 0i + (V/c)1i] =  V (V/c) 1 1 f       h1i (i = 2,3), ( 43
)  '11  V 2 [(V 2 /c 2 )00 + 2 (V/c)01 + 11] = V 2 [(V 2 /c 2 )f  1  1 1 f       h11], (44)
 '1i  V [(V/c)0i + 1i] =  V 1 1 f       h1i (i = 2,3), ( 45
)  'ij ij =  1  1 1 f       hij (i, j = 2, 3). ( 46
)
At large distance from the massive bodies, we have f  1 and h11 = O(1) [see Eqs. ( 22) and ( 23)], hence '00  1. Thus, the time measured by a clock bound to E V and outside the gravitation field (far enough) is indeed t'. The time measured by a clock bound to E V and in the gravitation field is affected in the ratio dt'x' /dt' =  ' 00 .

(47)

It is easy to verify in the same way that dl' 0 is also the distance which is locally measured by observers that are bound to the moving frame EV and remote from the massive bodies. The relative velocity u'i  dx'i /dt' and the relative acceleration a'i  du'i /dt', if needed, may be deduced from the absolute velocity ui  dxi /dt and the absolute acceleration a0i = ai  dui /dt by using the usual transformations of SR, which are obtained simply by differentiating (39). These formulae are valid independently of the velocity measured with "unaffected" standards, u  (ui ui) 1/2 , hence they are valid for photons (V must be smaller than c, but the unique condition on u is that Vu1  c 2 ). To transform the acceleration completely, one would also have to reexpress the space and time derivatives as relative to the coordinates x'in the moving frame. If, in the preferred frame, the derivatives of some function (x) play a rôle, we first define a new function, say '(x'):

' (x'0, ..., x'3)  (x0(x'0, ..., x'3), ..., x3(x'0, ..., x'3))

and we obtain from Eq. (39):

          t t V x V         ' ' ' ' 1 , (49a)
          x x V c t V 1 1 2         ' ' ' ' , (49b)
      x x i i i   ' ' ( , ) 2 3 .
(49c)

"Post-Newtonian Lorentz-invariance" of the pN approximation for photons

We begin with the gravitational red-shift. As is known, this effect is calculated, in GR, for a stationary metric (,0 = 0), using the fact that ds 2 = 0 for a photon, and the assumption that the frequency measured at the place of emission (proper frequency) is independent of the gravitation field. It is first shown that the frequency 2, measured at point 1, of light emitted at point 2, differs from the proper frequency 0 = 1 (emitted and measured at point 1) in the ratio

    2 1 00 2 00 1 1 2        ( ) ( ) / x x (50)
(see Weinberg [START_REF] Weinberg | Gravitation and Cosmology[END_REF]). This first reasoning holds exactly the same in the present theory since the "equivalence principle" is true in the theory, in the sense that the laws of non-gravitational physics (that rule the behaviour of light emitters and detectors) are expressed in terms of the curved physical space-time metric  [START_REF] Arminjon | [END_REF]. However, the metric cannot a priori be stated to be stationary (to an approximation equivalent to that in GR) in the preferred frame, but instead in some uniformly moving frame E V. The second step, in evaluating the red-shift in GR, is to show that the 00 term has the following expansion:

 00 = 1  2U/c 2 + O(1/c 4 ), (51)
with U the Newtonian potential. In the present theory, we enter the expansion of f [Eq. ( 22)] into the expression of ' 00 in the moving frame EV [Eq. ( 41)] and obtain also

 ' / 00 2 2 2 4 2 2 2 2 4 1 1 1 2 1 1 1 1 2 1                                        V c U c c V c c U c c O O O . ( 52
)
To be complete, we mention that Eq. ( 49) gives for V/c <   (Umax /c 2 ) 1/2 (see § 3.1):

      x x c 1 1 2 1         ' ' O ,       x x i i i   ' ' ( , ) 2 3
(53) so that the Poisson equation (29a), obtained in the preferred frame, is also Lorentz-invariant, if one neglects O(1/c 2 ) terms (since the same is true for the Newtonian mass density, of course). Hence, at this approximation, U is indeed the Newtonian potential as defined in EV. We conclude that, to just the same level of approximation as in the pN approximation of GR, more precisely: neglecting O(1/c 4 ) terms in the basic equations (50) and (52), the present theory predicts exactly the same gravitational red-shift as does general relativity, and this is true accounting for the preferred-frame effects (which do not appear at this approximation).

The other two "classical" effects of gravitation on photons are the deflection of electromagnetic radiation by a gravitation field, and the gravitational increase in the time of flight (the delay of the radar echo on a forth-and-back travel, as compared with the echo "in the absence of the gravitation field"; in practice the comparison is with a flight in a weaker gravitation field). Both effects are calculated, in GR, using the pN equation of motion for a photon, which depends only on the Newtonian potential, and using the further approximation echo delay as does standard general relativity, and this is true accounting for the preferredframe effects (which do not appear at this approximation). Here, "standard" GR means that the present theory recovers the predictions made with the Schwarzschild metric, as given, e.g., by Weinberg [11,chapter 8].

Conclusion

It has been shown that the present theory is simply indistinguishable from standard GR for the post-Newtonian approximation of gravitational effects on photons, which is the approximation used to compare GR with the observation of these effects. This is not enough to qualify this theory as viable, although the effects of gravitation on photons are among the most striking and the best established predictions of GR. It is interesting to note that this set of effects, which may be regarded as a brilliant confirmation of Einstein's theory, can also be accounted for by a scalar theory with a preferred frame, which derives motion from an extension of Newton's second law instead of assuming that free particles follow space-time geodesics.

Endnotes 1. A frame can be physically thought as a spatial network of observers equipped with measuring rods and clocks, allowing to define space-time coordinates (z  ) ( = 0,...,3) such that each observer has constant space coordinates z i (i = 1, 2, 3). Such coordinates are called "adapted to the frame", see Cattaneo [17]. The 3-D reference body associated with a frame may be defined as the set of the world-lines of the observers, and may be described by the space coordinates (z i ).

2. Equation (3) does not apply when 0 f = 0. The point is that the assumed contraction occurs in the direction of the gravity acceleration g, and thus is a priori undefined at those places where g = 0 [unless the place is one where the scalar potential f has the same value (f = 1) it has "at infinity", in which case there is no contraction at all]. In the generic, physically realistic case, the vector g vanishes merely at isolated points, e.g. at one point between two massive bodies, and also at one point near the centre of each massive body. At such isolated points where g = 0, the direction of contraction cannot be defined uniquely. But Eq. (3a) applies and shows that the metric g is discontinuous at this isolated point, but remains bounded in its neighborhood. Hence, geometrical elements such as the length of a line are defined as with a continuous metric. Moreover, the examination of the case with spherical symmetry shows that no difficulty occurs in the dynamical equations either. A more difficult case occurs in the highly idealized situation of "collapse in free fall" : g vanishes in a bounded volume domain [2]. This case was easily solved in ref. [2] for the relevant subcase of spherical symmetry, and the way of solution can be extended to the case without any symmetry. 3 We may use Eq. ( 23) if we neglect O(1/c 2 ) terms in h, for the following reason. In the Cartesian coordinates (xi) in the preferred frame, we have

h (1) ij = U,i U,j /(U,k U,k ).
Then Eq. (53) gives us the same expression, plus O(1/c 2 ) terms, in the Lorentz-transformed Cartesian coordinates (x'i) in the moving frame, and this may be rewritten as h (1) ij = (U,i U,j )/(g' 0 kl U,k U,l) + O(1/c 2 ) in arbitrary space coordinates of the moving frame.

  and second kind) Christoffel symbols of the Euclidean metric g 0 in the space coordinates x i , and where

of the Newtonian potential of the main massive body by the spherical (and constant) one, i.e. U = GM/r (see Weinberg [START_REF] Weinberg | Gravitation and Cosmology[END_REF], Misner et al. [START_REF] Misner | Gravitation[END_REF], and Will [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF]). Actually, Weinberg [11, chapter 8] calculates both effects with the Schwarzschild metric (retaining the pN terms in the geodesic equations of motion), whereas Misner et al. [START_REF] Misner | Gravitation[END_REF] and Will [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF] present the calculation in the more general context of the "parametrized pN formalism", with the additional setting U = GM/r.

First, we recall that, to pN approximation, photons obey the geodesic equation in the present theory (see after Eq. 36), and this is generally covariant. Therefore, preferred-frame effects could exist in the pN approximation only in the case that they would affect the space-time metric in the moving frame EV. As Eqs. ( 41)-( 46) show, this is certainly the case for the exact metric (i.e., the velocity V enters the exact metric). However, expanding these equations with the help of Eqs. ( 22) and (24a), one finds immediately that (with x'0 = ct' )

Putting U = GM/r in these equations (as is done in GR), we verify that this is the expansion of the Schwarzschild metric, when one neglects O(1/c 3 ) terms. Indeed, Eq. ( 54), written in Cartesian coordinates for the metric g' 0 , i.e. such that g' 0 ij = ij, is immediately rewritten in arbitrary space coordinates (y' i ) as

In spherical coordinates, y' 1 = r, y' 2 = , y' 3 = , we have

With U = U(r), Eq. ( 23) gives thus

So, with U = GM/r, the space-time metric writes in spherical coordinates:

which is the Schwarzschild metric (6), at the corresponding approximation. The order of the neglected terms is actually O(1/c 4 ), except for the neglected'0i components, which are indeed O(1/c 3 ). Such '0i components of order O(1/c 3 ) also exist in the pN approximation of GR (they are obtained from the well-known vector potential), and they do not play any rôle in the pN approximation for photons [START_REF] Weinberg | Gravitation and Cosmology[END_REF][START_REF] Misner | Gravitation[END_REF][START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF]. We conclude that, to just the same level of approximation as in the pN approximation of GR, in particular neglecting the 0i components of the metric, which are O(1/c 3 ), the present theory predicts exactly the same light deflection and the same


Massinissa Ziane
email: massinissa.ziane@univ-rennes.fr

Senior Member, IE Artem Boriskin

Cecile Leconte

Senior Member, IEEE Laurent Le Coq

Senior Member, IEEE Maxim Zhadobov
email: maxim.zhadobov@univ-rennes.fr

Novel Technique for In-Body Absorbed Power Density Assessment Based on Free-Space E-Field Measurement

Keywords: Absorbed power density (APD), dosimetry, millimeter waves (mmWaves), near-field measurements

We introduce a novel method for measurement of the absorbed power density (APD) above 6 GHz, accounting for antenna/body interaction. A reflectivity-based thin planar solid phantom is used to mimic the scattering characteristics of human skin. This allows to perturb the antenna under test (AUT) in a similar way as it would be perturbed by the presence of the human body. The lower electromagnetic losses in the phantom medium result in an enhanced transmission through the phantom facilitating free-space E-field vector measurement on the opposite to the AUT side of the phantom. The measured E-field is backpropagated in space to retrieve APD using a plane-wave spectrum (PWS) technique. The proposed method is validated at 60 GHz using reference antennas, namely, a cavityfed dipole array and pyramidal horn with a slot array. A very good agreement is demonstrated between the target APD at the air/skin interface (peak, averaged, and spatial distribution) and the one reconstructed from the simulated and measured E-field. The maximum relative deviations between the computed and measured peak ( pAPD), 1 cm 2 peak spatial average ( psAPD 1 cm 2 ), and 4 cm 2 peak spatial average ( psAPD 4 cm 2 ) are 13.2% (0.55 dB), 14.9% (0.6 dB), and 15.7% (0.63 dB), respectively. The results demonstrate that the proposed technique is promising for experimental dosimetry and compliance testing of wireless devices operating above 6 GHz.

Distance between phantom and probe.

I. INTRODUCTION

T HE continuous evolution of mobile terminals, such as smartphones, tablets, and body-worn devices, has drastically increased the wireless data traffic, which keeps growing due to video streaming applications and cloud computing. The growing need for high-performance mobile communications leads to a progressive shift of operating frequencies toward the millimeter-wave (mmWave) band. In particular, the fifthgeneration (5G) heterogeneous cellular mobile networks will ACCEPTED MANUSCRIPT / CLEAN COPY exploit the FR2 bands around 26 GHz, 39 GHz, and in some countries 60 GHz [START_REF]5G Spectrum Public Policy Position[END_REF], [START_REF]5G Spectrum GSMA Public Policy Position[END_REF], [START_REF] Dehos | Millimeter-wave access and backhauling: The solution to the exponential data traffic increase in 5G mobile communications cystems[END_REF], [START_REF]FP7 CORDIS European Commission. Beyond 2020 Heterogeneous Wireless Network With Nillimeter Wave Small Cell Access and Backhauling[END_REF], [START_REF] Baykas | IEEE 802.15.3c: The first IEEE wireless standard for data rates over 1 Gb/s[END_REF].

The upcoming use of FR2 technologies will involve user exposure to mmWave radiation. This includes the near-field exposure by mobile devices operating in the vicinity of human body. mmWave radiation is not present in our natural background due to the strong atmospheric absorption, and the accurate control of user exposure is of upmost importance from the environmental safety viewpoint. To prevent overexposure risks, safety exposure limits are set by the International Commission on Non-Ionizing Radiation Protection (ICNIRP) and the Institute of Electrical and Electronics Engineering (IEEE C95.1) [START_REF]Guideliens for limiting exposure to electromagnetic fields (100 kHz to 300 GHz)[END_REF], [START_REF]safety Levels With Respect to Human Exposure to Electric, Magnetic, and Electromagnetic Fields, 0 Hz to 300 GHz[END_REF].

Below 6 GHz, the specific absorption rate (SAR) is used as the main dosimetric quantity. Above 6 GHz and up to 300 GHz, in-tissue power density at the skin surface, referred as the absorbed power density (APD) by ICNIRP and as the epithelial power density by IEEE C95.1, is used as the main dosimetric quantity. The limits (basic restrictions) are set to 10 mW/cm 2 for occupational environments (referred as restricted environments by IEEE) and 2 mW/cm 2 for the general public (referred as unrestricted environments by IEEE). The APD is to be averaged over 4 cm 2 and 6 min. To account for focal beam exposure, from 30 to 300 GHz APD is also to be averaged over 1 cm 2 and must not exceed two times the limits for 4 cm 2 . Note that the algorithms for APD averaging in compliance testing are currently under discussion [START_REF] Li | Calculated epithelial/absorbed power density for exposure from antennas at 10-90 GHz: Intercomparison study using a planar skin model[END_REF], [START_REF] Kapetanovic | Areaaveraged transmitted and absorbed power density on a realistic ear model[END_REF].

Compliance testing of wireless devices with respect to the exposure limits is currently performed based on free-space measurement of the incident power density (IPD) close to a device under test according to the IEC/IEEE 63195 standard [START_REF]Assessment of Power Density of Human Exposure to Radio Frequency Fields from Wireless Devices in Close Proximity to the Head and Body (Frequency Range of 6 GHz to 300 GHz)[END_REF]. In [START_REF] Pfeifer | Total field reconstruction in the near field using pseudovector E-field measurements[END_REF], IPD is assessed using a miniaturized diode-loaded probe designed to measure the magnitude of the E-field deriving the polarization ellipse from different probe rotation angles. The phase, which cannot be directly measured by such probes, is retrieved using the plane-to-plane phase reconstruction method [START_REF] Hislop | Phase retrieval of scattered fields[END_REF]. It consists in: 1) amplitude measurement in two parallel planes at different distances from the source (the phase distribution is arbitrary set in both the planes) and 2) iterative forward and backward near-field to near-field transformation until the phase distribution converges.

For accurate exposure assessment under realistic conditions, it is important to account for antenna/body interaction, including standing wave and antenna detuning phenomena that may result in enhancement of user exposure levels [START_REF] Ziane | Antenna/body coupling in the near-field at 60 GHz: Impact on the absorbed power density[END_REF], [START_REF] Derat | Influence of source-phantom multiple interactions on the field transmitted in a flat phantom[END_REF]. To account for this, the microwave dosimetry methods, based on in-phantom E-field measurements of SAR, are commonly used below 10 GHz. However, at higher microwave frequencies they are facing physical limitations due to prohibitively high losses in biological tissues [START_REF] Cozza | System for Measuring an Electromagnetic Field[END_REF], [START_REF] Derat | Near-field and vector signal analysis techniques applied to specific absorption rate measurement[END_REF], [START_REF] Speag | DASY 8 SAR Measurement[END_REF], [START_REF] Mvg | ComoSAR SAR Measurement System[END_REF]. This substantially complicates measurement of the transmitted Efield using a scanning probe (or an embedded array of probes) inside a tissue-simulating liquid. Some attempts have been undertaken to scale the in-phantom measurement methods to frequencies above 10 GHz [START_REF] Jafari | Absorbed/epithelial power density assessment using plane-wave spectrum method from inside the skin tissue above 6 GHz[END_REF], [START_REF] Karimi | Method and implementations to measure the absorbed power density[END_REF], [START_REF] Speag | DASY 8 APD Module (10-45 GHz)[END_REF], [START_REF] Chitnis | Tracable absorbed power density assessment system in the 28 GHz band[END_REF]. The vector E-field is probed inside the liquid phantom and then backpropagated in space to retrieve APD at the phantom surface. As an alternative, a temperature-based method for APD assessment was proposed in [START_REF] Ziane | High-resolution technique for near-field power density measurement accounting for antenna/body coupling at millimeter waves[END_REF], [START_REF] Ziane | Near-field power density mapping of close-to-body low-power mmWave devices[END_REF], and [START_REF] Ziane | Multi-physics technique for absorbed power density measurement above 6 GHz[END_REF] to allow APD retrieval from the mmWave-induced temperature rise on the phantom surface measured using an infrared (IR) camera with the spatial resolution of the order of 100 µm.

This article introduces a novel technique for APD assessment based on free-space E-field measurements, accounting for antenna/body interaction. The method uses a reflectivitybased thin planar solid phantom designed to mimic the scattering characteristics of human skin and maximize the through-phantom transmission to guarantee a measurable E-field behind the phantom. A tapered open-ended waveguide is used as a probe to measure the amplitude and phase of the transmitted E-field. Free-space measurements ensure enhanced robustness with respect to probe positioning compared with the measurements in lossy liquid phantoms with a strong spatial APD gradient. The measured E-field is then backpropagated in space to retrieve APD at the phantom surface using a plane-wave spectrum (PWS) technique. The method is experimentally validated at 60 GHz using a near-field scanner for two reference feeds: a cavity-fed dipole array and a pyramidal horn with slot array [START_REF]Assessment of Power Density of Human Exposure to Radio Frequency Fields from Wireless Devices in Close Proximity to the Head and Body (Frequency Range of 6 GHz to 300 GHz)[END_REF].

II. METHOD OVERVIEW

The proposed APD assessment method is schematically represented in Fig. 1. The antenna under test (AUT) is placed at a distance d from a thin phantom reproducing the scattering properties of human skin. This reflectivity-based phantom, described in Section II-A, perturbs the AUT in the same way as it would be perturbed by the presence of the human body [START_REF] Ziane | High-resolution technique for near-field power density measurement accounting for antenna/body coupling at millimeter waves[END_REF], [START_REF] Ziane | Near-field power density mapping of close-to-body low-power mmWave devices[END_REF]. Note that the complex permittivity of the phantom is different from the one of skin. It is chosen to reduce the losses in the phantom medium and thus enhance transmission through the phantom and ensure measurable E-field tangential component (E || p ) at z > s. E || p is sensed using a free-space probe at z = s + d p where s is the phantom thickness and d p is the distance between the phantom and the probe. Using the PWS theory [START_REF] Scott | The Spectral Domain Method in Electromagnetics[END_REF], E || p is backpropagated in the negative zdirection to retrieve the E-field (E 0 ) and then APD at z = 0 (as detailed in Section II-B). The accuracy of the reconstruction technique is numerically assessed as detailed in Section II-C.

A. Reflectivity-Based Skin-Equivalent Phantom

At mmWaves, skin is characterized by relatively high dielectric losses (ϵ r = 7.98 -j11.90 at 60 GHz [START_REF] Gabriel | The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz[END_REF]). In phantoms reproducing the complex permittivity of skin, and therefore simulating both the absorption and scattering properties of skin, the electromagnetic power is absorbed at the phantom surface (penetration depth is about 0.5 mm at 60 GHz). The shallow penetration at mmWaves complicates the APD reconstruction from E-field measurement inside a lossy liquid phantom.

To overcome this problem, an alternative approach consists in using a thin phantom reproducing the power reflection coefficient of skin (i.e., ≈37.6% for the normal incidence, Gabriel's dry skin model [START_REF] Gabriel | The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz[END_REF] at 60 GHz). Such thin electromagnetic skin phantom, initially introduced in [START_REF] Guraliuc | Solid phantom for body-centric propagation measurements at 60 GHz[END_REF], reproduces the antenna/body interaction thus ensuring realistic measurements representative of close-to-body use cases. According to the power conservation principle, the APD at z = 0 is equal to the one at the skin surface since the reflected from the phantom power is the same as the power reflected from skin (within the measurement accuracy). Note that the single-layer phantom is enough for accurate APD evaluation, and introducing additional tissue layers would unnecessarily overcomplicate the phantom design and substantially complicate its fabrication, introducing additional sources of uncertainty. To ensure a measurable amplitude and phase of E-field in the scanning plane located at the opposite of the AUT side (Fig. 1), the complex permittivity and thickness of the phantom are jointly adjusted to minimize losses inside and maximize the throughphantom transmission.

ACCEPTED MANUSCRIPT / CLEAN COPY

The phantom is fabricated using carbon black powder and PDMS [Fig. 2(a)] as detailed in [START_REF] Guraliuc | Solid phantom for body-centric propagation measurements at 60 GHz[END_REF]. The dimensions of the phantom are chosen to be large enough to avoid surface waves (10 × 10 cm corresponding to 20 × 20λ 0 , where λ 0 is the free-space wavelength). In brief, the PDMS and carbon powder are mixed and stirred thoroughly, then degassed under vacuum, and finally dried in an oven at 110 • C. For the carbon concentration of 40%, the optimal phantom thickness was found to be 1.2 mm. The complex permittivity of the carbon-PDMS mixture was measured using the free-space transmission method [START_REF] Guraliuc | Solid phantom for body-centric propagation measurements at 60 GHz[END_REF]. It uses a vector network analyzer (VNA), a transmission/reflection quasi-optical setup with two focusing lenses, and two reference WR-15 horn antennas. The measured complex permittivity of the phantom bulk material at 60 GHz is ϵ r = 12.5 -j3.6.

The power reflection coefficient transmitted through the phantom power was calculated at 60 GHz using Fresnel equations [Fig. 2 coefficient and transmitted through the phantom power were calculated was 36.6% for the normal incidence, which is ACCEPTED MANUSCRIPT / CLEAN COPY in excellent agreement with Gabriel's dry skin model (<1% deviation). The maximum difference of 2% is found for the 0 • -45 • range, which implies that APD at the air/phantom interface is the same as at the air/skin interface (with discrepancy less than 4%) according to the power conservation principle. The power transmission through the phantom is 9.94% at the normal incidence. It is 29× higher compared with the skin-equivalent model with the same thickness. Accounting for APD exposure limits (2 and 4 mW/cm 2 ) [START_REF]Guideliens for limiting exposure to electromagnetic fields (100 kHz to 300 GHz)[END_REF], [START_REF]safety Levels With Respect to Human Exposure to Electric, Magnetic, and Electromagnetic Fields, 0 Hz to 300 GHz[END_REF], required 10 dB of dynamic range [START_REF]Assessment of Power Density of Human Exposure to Radio Frequency Fields from Wireless Devices in Close Proximity to the Head and Body (Frequency Range of 6 GHz to 300 GHz)[END_REF], and transmission through the phantom (≈10%), the sensitivity of the measurement system should be at least -17 dBm.

B. Reconstruction of APD Using PWS

The E-field can be propagated in space using the spectral-domain formulation defined by the PWS theory [START_REF] Scott | The Spectral Domain Method in Electromagnetics[END_REF], [START_REF] Gregson | Principles of planar near-field antenna measurements[END_REF]. This approach is widely used to retrieve SAR below 6 GHz [START_REF] Derat | Near-field and vector signal analysis techniques applied to specific absorption rate measurement[END_REF], [START_REF]Measurement Procedure for the Assessment of Specific Absorption Rate of Human Exposure to Radio Frequency Fields From Hand-Held and Bodyworn Wireless Communication Devices-Human Models, Instrumentation and Procedures (Frequency Range of 4 MHz to 10 GHz)[END_REF] as well as IPD [START_REF] Omi | Performance analysis of incident power density evaluation by inverse source method for compliance assessment at quasi-millimeter and millimeter wave bands[END_REF], [START_REF] Sasaki | Error analysis of a near-field reconstruction technique based on plane wave spectrum expansion for power density assessment above 6 GHz[END_REF], [START_REF] Xu | Investigation of planar near-field measurement of millimeter-wave antenna for 5G application[END_REF] and APD [START_REF] Jafari | Absorbed/epithelial power density assessment using plane-wave spectrum method from inside the skin tissue above 6 GHz[END_REF] above 6 GHz. First, the Fourier transformation is applied to the measured electric field. This allows one to represent the spatial distribution of each field component over a transverse plane as a superposition of plane waves, referred as the PWS representation. In other words, the electromagnetic field on a transversal plane can be represented as an infinite sum of plane waves having different directions of propagation defined by couplet K = k x x + k y ŷ. The PWS of the Efield phasor component E( P, z 0 ) over the plane z = z 0 and P = x x + y ŷ is expressed as

Ê(K , z 0 ) = R E( P, z 0 )e j K P d P. (1)
The PWS over any plane parallel to plane z = z 0 located at distance l in a homogeneous medium is computed by multiplying the PWS at z = z 0 by the propagator e -jk z l

Ê(K , z 0 + l) = Ê(K , z 0 )e -jk z l (2)
where k z is the longitudinal propagation constant given as

k z = (k 2 -|K | 2 ) 1/2
, and k is the propagation constant.

The PWS is a convenient way to represent the propagation of the field in complex media (e.g., from the scanning plane z = s + d p to the surface of the phantom z = 0). Ê|| 0 (K ), the PWS of E || 0 (K ), is given as the superposition of the total E-field spectra propagating in the positive and negative directions along the z-axis and expressed as follows:

Ê|| 0 (K ) = Ê||+ 0 (K ) + Ê||- 0 (K ). (3)
Ê||+ 0 (K ) = T ad (I -e -j2k d z s R 2 da ) -1 × T ad e -k d z s T da I -e -j2k d z s R 2 da -1 -1 × Ê|| p (K )e jk 0 z d p (4) Ê||- 0 (K ) = T ad (I -e -j2k d z s R 2 da ) -1 × T ad e -k d z s T da I -e -j2k d z s R 2 da -1 -1 × e -j2k d z s R da Ê|| p (K )e jk 0 z d p (5)
where T ad and T da are the spectral transmission coefficients at the air/dielectric and dielectric/air interfaces [START_REF] Scott | The Spectral Domain Method in Electromagnetics[END_REF], respectively, R da is the spectral reflection coefficient at the dielectric/air interface [START_REF] Scott | The Spectral Domain Method in Electromagnetics[END_REF], I is the identity matrix, and k d z and k 0 z are the longitudinal propagation constant of the dielectric and air, respectively. The spectral transmission and reflection operators, respectively, T and R, for a dielectric interface between media 1 and 2 are defined as [START_REF] Hislop | Phase retrieval of scattered fields[END_REF], [START_REF] Scott | The Spectral Domain Method in Electromagnetics[END_REF]

T = 2X -1 (Y 1 + Y 2 ) -1 Y 1 X (6) R = X -1 (Y 1 + Y 2 ) -1 (Y 1 -Y 2 )X (7)
Y i = - 1 ωµ 0 k z,i k 2 i -k 2 x -k i k y -k i k y k 2 i -k 2 x ( 8
)
X = 0 1 -1 0 (9)
where i ∈ [START_REF]5G Spectrum Public Policy Position[END_REF][START_REF]5G Spectrum GSMA Public Policy Position[END_REF]. I is given as follows:

I = 1 0 0 1 . ( 10
)
The normal field spectrum component is obtained from the tangential field spectra Ê|| using the Gauss law

Ê+ z (K ) = K • Ê||+ k z , Ê- z (K ) = - K • Ê||- k z . (11)
The H-field spectrum is calculated as [30]

Ĥ = - K × Ê+ ωµ -- K × Ê- ωµ . (12)
The spatial field components (E and H) are retrieved using the inverse Fourier transform of the field spectra. The APD is calculated as

APD = 1 2 Re(E × H * ) • n ( 13
)
where n is the unit vector normal to the APD evaluation surface.

Finally, the spatial peak APD ( pAPD) is given as

pAPD = max(APD). ( 14
)
The peak spatial average APD ( psAPD) equals to the maximum spatial average APD (sAPD) where sAPD is defined as the mean APD over a planar square shape area

A [6] psAPD = max 1 2A A Re(E × H * ) • ds ( 15
)
where A is the averaging square area equal to 1 or 4 cm 2 ( psAPD 1 cm 2 or psAPD 4 cm 2 , respectively), and ds is the integral variable vector normal to the averaging area A.

C. APD Reconstruction Accuracy

To assess the accuracy of the reconstruction technique, two types of simulations were performed using the time-domain solver of CST Microwave Studio 2022 using the finite integration technique (FIT). The total number of mesh cells reaches about 13 and 21 million for the reference horn and dipole array antennas, respectively (in the presence of the phantom). The smallest mesh cell dimension is 316 µm (≈λ 0 /15) in ACCEPTED MANUSCRIPT / CLEAN COPY free-space and 92 µm inside the phantom (≈λ d /15). Open boundaries [perfectly matched layers (PMLs)] were used. In simulations, the antenna input power is set to 10 mW.

In one simulation, the E-field monitor is set to compute the E-field vector in free-space (without OWG probe) at the opposite side of the phantom with respect to the AUT (z = s + d p ) (E ( p,s) ). E ( p,s) is used to reconstruct APD at the surface of the phantom (APD (r,s) ). In another simulation, a half-space skinequivalent phantom is used to compute the APD at its surface using the power monitor (APD (s) ) (ϵ r = 7.98 -j11.90). The relative error of the reconstruction technique is assessed as follows:

s i = APD (r,s) APD (s) -1 ( 16
)
where i denotes the relative error for pAPD, psAPD 1 cm 2 , or psAPD 4 cm 2 ( peak , 1 cm 2 , and 4 cm 2 , respectively).

III. MEASUREMENT SYSTEM

A. Near-Field Scanner

The measurement system is schematically represented in Fig. 3(a) and(b). VNA (Agilent N5242A) operating at 10 MHz-26.5 GHz is used to generate a radio frequency (RF) signal. The frequency is upconverted to the 50-75-GHz band using a VNA extender (VNAX) (Vadiodes WR15). The local oscillator (LO) of the VNAX is controlled using a microwave analog signal generator (Keysight N5173B). The output signal from the VNAX is amplified and used to feed the AUT through a set of WR15 waveguides. The phantom is set at a distance d from the AUT. The amplitude and phase of the transmitted through the phantom E-field are measured using a linearly polarized tapered OWG [Fig. 3(c)]. The use of amplifiers at the emission and reception stages results in measurement sensitivity better than -60 dBm. The amplitude and phase of the E-field are derived from the amplitude and phase difference between the two ports of the VNA.

A position controller (Newport XPS) holding the measurement probe is used to scan the field over a planar surface. The dimensions of the scanning surface are 40 × 40 mm (8 × 8 λ 0 ) covering -30 dB area with respect to the peak E-field in the scanning plane. This results in a typical measurement duration of 2 h 40 min. It can be significantly reduced using IRbased measurement (≈few minutes) [START_REF] Ziane | High-resolution technique for near-field power density measurement accounting for antenna/body coupling at millimeter waves[END_REF], [START_REF] Ziane | Near-field power density mapping of close-to-body low-power mmWave devices[END_REF]. The measured signal is amplified, downconverted by a VNAX transceiver, and transmitted to the second port of the VNA. Switching from the measurement of the main-polarization (y-axis) component to the cross-polarization (x-axis) component of the E-field vector (E || ( p,m) ) is performed using a 90 • mechanical rotation of the AUT.

Measured by VNA, the relative amplitude and phase are converted into absolute APD using a probe calibration method recommended in [START_REF]Assessment of Power Density of Human Exposure to Radio Frequency Fields from Wireless Devices in Close Proximity to the Head and Body (Frequency Range of 6 GHz to 300 GHz)[END_REF]. In brief, the method consists in calibrating the measurement probe in a known field radiated by a reference antenna ( Ê(ref,s) ). As reference antenna, we used a V -band linearly polarized conical horn with 21.4-dB gain (QuinStar, CA, USA). The field is computed outside the reactive field region at d = 10 • λ 0 to avoid the influence of the probe on the reference antenna. Then the field is measured at the same location (d = 10 • λ 0 ) using the near-field scanner ( Ê(ref,m) ). A probe correction matrix is calculated as

M c = Ê(ref,m) Ê(ref,s) . ( 17
)
The absolute values of E-field ( Ê(p,m) ) are retrieved from the relative E-field measured by VNA ( Ê(p,r,m) ) as follows:

Ê(p,m) = M c • Ê(p,r,m) . (18)

B. E-Field Probe Effect

The impact of the reflection and standing wave between the probe and phantom on APD was investigated numerically considering two different scenarios. In the first scenario, APD was computed using full-wave simulation accounting for the phantom/probe interaction. In the second scenario, the AUT and phantom were replaced by their equivalent source and used to compute APD. The equivalent source reproduces the radiation of AUT with discrepancy lower than 1% in terms of E-field intensity and power density. The scattered from ACCEPTED MANUSCRIPT / CLEAN COPY the probe field is reflected back by the phantom in the first scenario, whereas it is neglected in the second scenario. The ratio between APD computed using full-wave and equivalent source is given in Fig. 4. This result suggests that the probe should be located far enough from the phantom (for d p ≥ 2.5 mm, the changes in APD are <±6%). Nevertheless, d p should be kept sufficiently short to minimize the numerical error related to E-field free-space back propagation from z = s + d p to z = s plane, which increases with d p . The best tradeoff is found for 2.5 ≤ d p < 5 mm where the maximum error on APD reconstruction due to the standing wave and back propagation is within ±6%. Measurements of the dipole array and horn antennas are performed for d p equal to 2.5 mm.

IV. VALIDATION FOR REFERENCE ANTENNAS

The proposed technique was validated for two reference antennas (pyramidal horn and dipole array antennas) defined by IEC/IEEE 63195 standard for validation of the power density measurement systems (Fig. 5).

A. APD Distribution

The spatial distributions of APD (s) , APD (r,s) , and APD (r,m) are presented in Figs. 6 and7 for d = 2, 5, and 10 mm. They are compared using the correlation coefficient ϒ defined as where X and Y are the distributions to compare, and X and Ȳ are, respectively, the mean values of X and Y. The parameter ϒ is an indicator of how close the two relative distributions fit. The results demonstrate 99.9% fit between the simulated APD (r,s) and APD (s) patterns and around 95% (±1%) between the measured APD (r,m) and the simulated APD (s) for both the antennas.

ϒ = m n (X mn -X )(Y mn -Ȳ ) m n (X mn -X ) 2 m n (Y mn -Ȳ ) 2 (19)
As seen in Figs. 6 and7, the main beam shape and position are well-reproduced for both the reference antennas and all the distances. The decrease in APD reconstruction accuracy for side lobes is mainly attributed to the directional effect of the probe. Advanced probe compensation schemes to improve the side lobes' measurements are out of the scope of this study and constitute one of its perspectives. For the horn antenna, discrepancies of about 3 dB between APD (r,m) and APD (s) distributions are noted in the -1 < x < 1 mm interval (Fig. 6(c)-1-D distribution). They are attributed to the averaging effect of the probe, which integrates the field over the surface of its aperture (1.88 × 3.86 mm).

B. Spatial-Average APD APD (s) , APD (r,s) , and APD (r,m) profiles as a function of d are shown in Fig. 8. The relative error of the APD (r,m) reconstruction from the measured data is calculated as

m i = APD (r,m) APD (s) -1 . (20)
s i and m i are given in Table I. The results show an excellent agreement between APD (s) and APD (r,s) , peak and averaged over 1 and 4 cm 2 . The ACCEPTED MANUSCRIPT / CLEAN COPY maximum s peak of 4.3% (0.18 dB) is found for the horn antenna at d = 2 mm. The maximum s 1 cm 2 and s 4 cm 2 equal to 3.2% (0.14 dB) and 2.9% (0.13 dB), respectively, were found for the dipole array at d = 6.6 and 3.1 mm, respectively. These errors are lower at d = 2, 5, and 10 mm.

A very good agreement is also demonstrated between simulations and measurements (APD (s) and APD (r,m) ). The maximum m peak , m 1 cm 2 , and m 4 cm 2 are equal to 9.2% (0.38 dB), 14.9% (0.6 dB), and 12.5% (0.51 dB), respectively, for the dipole array antenna. For the horn antenna, the maximum m peak , m 1 cm 2 , and m 4 cm 2 are equal to 13.2% (0.54 dB), 10.5% (0.43 dB), and 15.7% (0.63 dB), respectively. In some cases (e.g., dipole array at d = 5 mm), the error on the averaged APD (r,m) exceeds the error on its peak ( m peak < m 1 cm 2 and m peak < m 4 cm 2 ) [Fig. 7(b)]. This is attributed to the directional effect of the OWG probe, which introduces errors in the E-field distribution.

Note that APD strongly changes with d. For example, the horn antenna position shift from d = 2.8 to 3.4 mm results in 116% (3.34 dB), 76% (2.46 dB), and 32% (1.21 dB) decrease in pAPD, psAPD 1 cm 2 , and psAPD 4 cm 2 , respectively. Therefore, determination of the maximum APD would require measurements with a fine spatial sampling resolution (of the order of hundreds of µm).

C. Uncertainty Analysis

The main sources of uncertainty for the proposed method are listed hereafter. II. We used the uncertainty analysis methodology from [START_REF]Assessment of Power Density of Human Exposure to Radio Frequency Fields from Wireless Devices in Close Proximity to the Head and Body (Frequency Range of 6 GHz to 300 GHz)[END_REF]. Uncertainties reported in Table II correspond to the maximum error in APD ( pAPD, psAPD 1 cm 2 , and psAPD 4 cm 2 ) produced by ACCEPTED MANUSCRIPT / CLEAN COPY II are assumed to have a rectangular probability distribution, and hence, the standard uncertainty is obtained as the ratio of the uncertainty over the divisor √ 3 [START_REF]Assessment of Power Density of Human Exposure to Radio Frequency Fields from Wireless Devices in Close Proximity to the Head and Body (Frequency Range of 6 GHz to 300 GHz)[END_REF], [START_REF]Uncertainty of Measurement-Part 3: Guide to the Expression of Uncertainty in Measurement[END_REF]. The weighting factor is set to c i = 1 and the degrees of freedom to v i = ∞. The combined and expanded uncertainties (with the coverage factor k = 2) on APD ( pAPD, psAPD 1 cm 2 , and psAPD 4 cm 2 ) are assessed according to the guidelines in ISO/IEC Guide 98-3 [START_REF] Sasaki | Error analysis of a near-field reconstruction technique based on plane wave spectrum expansion for power density assessment above 6 GHz[END_REF] and found to be below 16% (0.65 dB) and 35% (1.3 dB), respectively.

Uncertainties related to phantom/probe coupling and probe calibration represent the main sources of error. Reducing uncertainties related to these two sources of error would allow to reduce the overall uncertainty. This constitutes one of the perspectives of this study.

D. Free-Space Versus In-Liquid Measurements

The conventional dosimetry methods developed for SAR measurements use liquid tissue-equivalent phantoms [START_REF]Measurement Procedure for the Assessment of Specific Absorption Rate of Human Exposure to Radio Frequency Fields From Hand-Held and Bodyworn Wireless Communication Devices-Human Models, Instrumentation and Procedures (Frequency Range of 4 MHz to 10 GHz)[END_REF]. Due to increasing with frequency losses in tissue-simulating liquids, free-space measurements ensure enhanced robustness with respect to probe positioning compared with the in-liquid measurements. Indeed, as shown in Table II, in free-space the uncertainty related to the probe positioning along the z-axis results in pAPD, psAPD 1 cm 2 , and psAPD 4 cm 2 uncertainties of 2.9% (0.13 dB), 2.1% (0.09 dB), and 2.3% (0.1 dB), respectively. Assuming that the measurements are performed not in free-space but in a liquid with the permittivity of skin (ϵ r = 7.98 -10.98 j), the same positioning tolerance of the probe (i.e., ±100 µm) would result in pAPD, psAPD 1 cm 2 , and psAPD 4 cm 2 uncertainties of 34.7% (1.85 dB), 34.4% (1.83 dB), 34.4% (1.83 dB), respectively. The overall expended uncertainty for psAPD 4 cm 2 would increase from 35% (1.3 dB) to 77% (2.48 dB). This uncertainty rise is due to the sharp E-field gradient in the lossy liquid phantoms at mmWaves. This suggests that performing measurements in free-space, using a thin reflectivity-based phantom, reduces the measurement uncertainty compared with in-liquid measurements (assuming all other parameters of the measurement system remain the same).

V. CONCLUSION

In this article, we introduced a novel method for APD assessment, based on free-space E-field measurements, accounting for the effect of the body presence. The system uses a reflectivity-based solid phantom that mimics the skin scattering characteristics at one side (AUT side) while providing an enhanced transmission through the phantom to maximize the E-field magnitude at the opposite side. A tapered open-ended waveguide is used to measure the E-field transmitted through the phantom. The method was validated using the reference cavity-fed dipole array and slotted pyramidal horn antennas at 60 GHz.

An excellent agreement was found between the simulated APD (peak, averaged, and distribution) and the reconstructed one from the simulated E-field. The maximum relative error on the reconstructed pAPD, psAPD 1 cm 2 , and psAPD 4 cm 2 from the simulated tangential components of the E-field was found to be 4.3% (0.18 dB) (d = 2 mm for horn), 3.2% (0.14 dB) (d = 6.6 mm for dipole array), and 2.9% (0.13 dB) (d = 3.1 mm for dipole array), respectively. The correlation between the simulated and numerically reconstructed APDs was found to be higher than 99.9%.

A very good agreement was found between the simulated APD and the one reconstructed from measurements as well. The maximum errors on pAPD, psAPD 1 cm 2 , and psAPD 4 cm 2 for all the phantom-to-probe separation distances d of 2, 5, and 10 mm do not exceed 15.7% (0.63 dB).

The uncertainty budget analysis for the proposed APD measurement method was performed according to IEC/IEEE ACCEPTED MANUSCRIPT / CLEAN COPY 63195. The expanded uncertainty in pAPD, psAPD 1 cm 2 , and psAPD 4 cm 2 is 1.14, 1.27, and 1.3 dB, respectively. This proof-of-concept study demonstrates a strong potential of this new technique for APD assessment and compliance testing of mmWave wireless devices. Scaling the proposed APD approach to other FR2 bands constitutes a perspective of this study.

APPENDIX APD RECONSTRUCTION FROM FREE-SPACE E-FIELD BACKPROPAGATION

A part of the incident field spectrum ( Ê|| inc ) is transmitted to the phantom. The transmitted field propagates from plane z = 0 to z = s where a part of it is transmitted to the air (z > s) and the other part is reflected. The latter propagates toward z = 0 in the reverse direction of the z-axis where it is reflected back toward z = s. This forth and backpropagation keeps going until the amplitude of the field inside the phantom vanishes after n iterations. ( Ê|| p ) is given as the superposition of the total transmitted field at z > s as

Equation ( 19) is a geometric series, also called the Neumann series [START_REF] Hislop | Introduction to Spectral Theory: With Applications to Schrödinger Operators[END_REF], with the common ratio R 2 da e -j2k d z s . If the eigenvalue of R 2 da e -j2k d z s is within ±1, the geometric series converges to the following as n tends to infinity:

The field inside the phantom can be expressed as the superposition of the total E-field spectra propagating in the positive and negative directions along the z-axis. Ê|| a , given as the superposition of Ê||+ a and Ê||a , is expressed using Ê|| inc as follows:

From ( 20), Ê|| inc is expressed using Ê|| p as follows:

Fig. 1 .

 1 Fig. 1. Schematic representation of the proposed APD assessment method from free-space E-field vector measurement.

  Fig. 2. Reflectivity-based phantom. (a) Fabricated prototype. (b) Power reflection coefficient versus incident angle for parallel (TM) and perpendicular (TE) polarizations. (c) Through phantom versus through skin power transmission for normal incidence at 60 GHz (normalized to the incident power).

Fig. 3 .

 3 Fig. 3. Measurement setup. (a) Three-dimensional outline. (b) Full-chain measurement system. (c) OWG field probe (dimensions are in mm).

Fig. 4 .

 4 Fig. 4. Effect of the phantom/probe interaction on APD.

Fig. 5 .

 5 Fig. 5. Reference antennas used as AUT. (a) Pyramidal horn with slot array and (b) cavity-fed dipole array [10].

Fig. 6 .

 6 Fig. 6. APD distribution for the horn antenna at (a) d = 2 mm, (b) d = 5 mm, and (c) d = 10 mm.

Fig. 7 .

 7 Fig. 7. APD distribution for the dipole array antenna at (a) d = 2 mm, (b) d = 5 mm, and (c) d = 10 mm.

Fig. 8 .

 8 Fig. 8. APD for horn (top) and dipole array (bottom) antennas. (a) pAPD. (b) psAPD 1 cm 2 . (c) psAPD 4 cm 2 .

  1) Calibration a) Amplifier Linearity: ±2% (0.09 dB). b) Complex Permittivity (Due to Errors in Permittivity Characterization): ±5% (±0.21 dB) for the real part and ±10% (±0.41 dB) for the imaginary part. c) Probe Positioning: ±3% (±0.13 dB). d) Probe Calibration: ±6% (0.25 dB), ±13.5% (0.55 dB), and ±5.4% (0.23 dB) on peak E p , spatial average of ∥E p ∥ over 1 and 4 cm 2 , respectively. 2) Scanning probe a) Position Along z-Axis: ±100 µm. b) Position Along x-Axis and y-Axis: ±50 µm. c) Amplitude and Phase: ±5% (±0.21 dB). d) Phantom/Probe Interaction: ±5.4% (±0.23 dB). e) AUT/Probe Interaction: ±1% (±0.04 dB). 3) Phantom a) Complex Permittivity (Due to Fabrication Errors): ±2.9% (±0.12 dB) for the real part and ±6% (±0.25 dB) for the imaginary part. b) Thickness: ±50 µm. c) APD in Skin: Differences between APD ( pAPD, psAPD 1 cm 2 , and psAPD 4 cm 2 ) at the surface of the skin [27] and the fabricated phantom are within ±4.95% (±0.21 dB). 4) Postprocessing a) APD Reconstruction: ±4% (±0.17 dB). b) Scanning Area Truncation: Truncation error is negligibly small for the scanning area (8 × 8λ 0 ). c) Sampling: ±1% (±0.04 dB). 5) Device under test (DUT) a) Positioning: ±3% (±0.13 dB). b) Power Drift: ±5% (±0.21 dB). The uncertainty budget is provided in Table

  Ê|| p (K ) = Ê|| inc (K )T ad e -jk d z s T da × I + R 2 da e -

-j2k d z s - 1 e

 1 Ê||p (K ) = T ad e -jk d z s T da I -R 2 da e -jk d 0 d p Ê|| inc (K ).

  Ê|| a (K ) = T ad (I + R da e -j2k d z s ) I -R 2 da e -) = T ad R da e -j2k d z s I -R 2 da e -

1 ×-j2k d z s - 1 ×-j2k d z s - 1 ×- 1 - 1 ×

 11111 Ê||inc (K )= T ad e -jk d z s T da I -R 2 da e -j2k d z s -1 -1 e jk d 0 d p Ê|| p (K ).(26)Replacing Ê|| inc in (21)Ê|| a (K ) = T ad (I + R da e -j2k d z s )(I -R 2 da e -j2k d z s ) -T ad e -jk d z s T da I -R 2 da e -) = T ad I -R 2 da e T ad e -jk d z s T da I -R 2 da e -) = T ad R da e -j2k d z s I -R 2da e T ad e -jk d z s T da I -R 2 da e -j2k d z e jk d 0 d p Ê|| p (K ).

  Simulated APD at the surface of the skin. APD (r,s) Reconstructed APD from E ( p,s) . APD (r,m) Reconstructed APD from E ( p,m) .

	E ( p,s)	Simulated E p (without OWG probe).
	E ( p,m)	Measured absolute E p .
	E ( p,r,m)	Measured relative E p .
	E (ref,s)	Simulated reference E (without OWG
		probe).
	E (ref,m)	Measured reference E.
	M c	Probe calibration matrix.
	H	H-field vector.
	APD (s)
	pAPD	Peak APD.
	sAPD	Spatial average APD.
	psAPD	Peak spatial average APD.
	ϒ	Correlation factor.
	ϵ r	Complex relative permittivity of the phan-
		tom.
	k 0	Free-space propagation constant.
	k d	Propagation constant in the phantom.
	k z	Longitudinal propagation constant.
	k 0 z k d z	Longitudinal propagation constant in the air. Longitudinal propagation constant in the
		phantom.
	λ 0	Free-space wavelength.
	λ d	Wavelength in the phantom.
	T	Spectral transmission coefficient.
	R	Spectral reflection coefficient.
	s	Phantom thickness.
	d	Distance between AUT and phantom.
	d p

j i

Relative error on reconstructed APD.

j s

Relative error on APD (r,s) .

j m

Relative error on APD (r,m) .

TABLE I

 I

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ACKNOWLEDGMENT

The authors would like to thank F. Boutet for the kind assistance in using the quasi-optical setup for complex permittivity measurements. They also acknowledge the Mobile & Wireless Forum (MWF) for providing the reference antennas.

This work was supported in part by the French National Research Program for Environmental and Occupational Health of Agence Nationale de Sécurité Sanitaire de l'alimentation, de l'Environnement et du travail (ANSES) through NEAR 5G Project under Grant 2018/2 RF/07; in part by the Labex CominLabs through EM-ART Project under Grant ANR-10-LABX-07-01; in part by the "Région Bretagne" (ARED Program); in part by the French Ministry of Higher Education and Research (MESR); in part by the Brittany Region, Ministry of Higher Education and Research, Rennes Métropole and Conseil Départemental through CPER Project SOPHIE/STIC and Ondes; and in part by the French National Center for Scientific Research (CNRS).


Imane Skifa

Nicolas Chauchat

Pierre-Henri Cocquet

Yves Le Guer

Optimizing Chaotic Mixing in Open Ponds with Lateral Mixers for Enhanced Spirulina Cultivation

come

• Proposing a dual-system strategy with a single inlet and outlet, our goal is to combine two distinct solutions.

• The methodology entails alternating between these solutions over a designated timeframe.

C1

Outlet: outflow Inlet: velocity-inlet C2 C1