Antoine Soubret

Gérard Berginc

Claude Bourrely

Backscattering enhancement of an electromagnetic wave scattered by two-dimensional rough layers

The problem of an electromagnetic wave scattering by a slab with two rough boundaries is solved by a small-perturbation method under the Rayleigh hypothesis. In order to obtain a perturbative development, we use a systematic procedure which involves integral equations called the reduced Rayleigh equations. Then we will show for a dielectric slab deposited on a silver film that the backscattering enhancement can be produced by guided waves which interact with the two rough surfaces.

I. INTRODUCTION

Scattering of electromagnetic waves from multilayer structures is a phenomenon which is of interest in many area of physics such as remote sensing or optical industry, where for example metallic surfaces have a dielectric coating. An extended review of experimental and theoretical works on optical multilayers can be found in [START_REF] Bourrely | Scattering of light from multilayer systems with rough boundaries[END_REF] , and references on recent papers are given in [START_REF] Fuks | Wave diffraction by rough interfaces in an arbitrary plane-layered medium[END_REF] . Many works 2,3,4,5 deal with the small perturbation method to investigate the behaviour of multilayer structure. However, due to calculations complexity, the analytical results are given only to first order in the rms-height of each rough boundary, thus the interaction between the rough surfaces can only be taken into account if the rough surfaces are correlated [START_REF] Elson | Multilayer-coated optics: guided-wave coupling and scattering of interface random roughness[END_REF] in the mean procedure. Our purpose is to show how light can interact with several rough surfaces, to this end, we have choosen the most simple system depicted in Fig. 1, where we have three regions with different permitivities separated by two rough surfaces. The calculations have been made under the hypothesis of the small perturbation method, initially developped by Rice 6,[START_REF] Voronovich | Wave Scattering from Rough Surfaces[END_REF] . Due to the great complexity to derive high orders of the perturbative development, Rice's original method is rather difficult to apply. One way to overcome these difficulties is to use integral equations called the reduced Rayleigh equations, they were first obtained by Celli et al [START_REF] Brown | Vector theory of light scattering from a rough surface : unitary and reciprocal expansions[END_REF] , and later have been generalized in Ref. [START_REF] Soubret | A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces[END_REF] to take into account upward fields in the slab medium. When combining these equations we obtain an integral equation where only the scattering matrix of the whole structure has to be determined, next we have developed a systematic method to calculate the perturbative development. We have shown by means of numerical simulations, how different mechanisms responsible of the enhanced backscattering occur. This phenomenon which has been predicted [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF][START_REF] Celli | Some aspects of light scattering from a randomly rough metal surface[END_REF][START_REF] Mcgurn | Localization effects in the elastic scattering of light from a randomly rough surface[END_REF][START_REF] Maradudin | Enhanced backscattering of light from weakly rough, random metal surfaces[END_REF] and observed [START_REF] West | Observations of backscattering enhancement from polaritons on a rough metal surface[END_REF] in the case of a random metal surface, manifests itself as a well-defined peak in the retroreflection direction through the angular dependence of the intensity related to the diffuse component of the scattered field. For a metallic surface, the phenomenon is produced by the interference of waves which excite surface plasmon polariton along a certain path and then follow the same path but in the reverse direction. For a dielectric bounded by metallic plate with one rough surface [START_REF] Freilikher | Coherent scattering enhancement in systems bounded by rough surfaces[END_REF] , the enhanced backscattering is produced by a similar mechanism where the surface plasmon polariton is replaced by a slab guided wave. In this case the incident wave excites in a first time a guided mode due to surface roughness, and then the roughness transforms the surface wave into a bulk wave. Furthermore, if the slab supports several guided modes, recent investigations [START_REF] Soubret | A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces[END_REF][START_REF] Freilikher | Coherent scattering enhancement in systems bounded by rough surfaces[END_REF] have shown the presence of additionnal peaks, called satellite peaks, in the angular distribution of the incoherent intensity. When the slab has two rough boundaries, the enhancement of backscattering can be produced by two kinds of interaction. The classic one, where the wave is scattered two times by the same boundary, and a new one (see Fig. 1) where the wave is transmitted at a point A of the surface without being diffused, then the wave is scattered by the second rough surface at a point B and by the first one in C. If the wave follows the same path in the reverse direction and excites the same guided mode, then phase difference between the two path is ∆φ = r BC • (k + k i ), where r BC is the distance between B and C. Thus, in the anti-specular direction (k = -k i ), this phase difference is independent of the random position of B and C which produce the backscattering peak.

The outline of the paper is as follows. In Section II we present the system under study and a description of the rough surfaces statistics. In Section III we define the plane wave representation of the electric field in the polarization basis and also the scattering matrix. Section IV is devoted to the calculation of the incoherent cross-section as a function of the perturbative development. In Section V we derive an integral equation from the reduced Rayleigh equations, which provides a systematic way to obtain the perturbative development. The resulting expressions are numerically computed in the case of a dielectric slab deposited on a rough silver surface in Section VI. The calculated incoherent intensity will show a narrow enhanced backscattering. Conclusions drawn from our results are presented and discussed in Section VII. In an Appendix, we collect matrices used in the calculations.

II. THE RANDOM SURFACE

The system which is considered in this paper is depicted in Fig. 2. The three regions are characterized by an isotropic, homogeous dielectric constant ǫ 0 , ǫ 1 and ǫ 2 respectively. The two boundaries are located at z = h 1 (x) and z = -H + h 2 (x), x = (x, y), and these three media are separated by rough surfaces described statistically. In fact, we assume that h 1 (x) and h 2 (x) are stationary, isotropic uncorrelated Gaussian random processes defined by their moments :

< h i (x) > = 0 , (1)
< h i (x) h i (x ′ ) > = W i (x -x ′ ) , (2)
< h 1 (x) h 2 (x ′ ) > = 0 (3)
where i = 1, 2, and the angle brackets denote an average over the ensemble of realizations of the function h 1 (x) and h 2 (x). In this work we will use a Gaussian form for the surface-height correlation function W 1 (x) and W 2 (x):

W i (x) = σ 2 i exp(-x 2 /l 2 i ) , (4)
where σ i is the rms height of the surface h i (x), and l i is the transverse correlation length. In momentum space we have:

< h i (p) > = 0 , (5)
< h i (p) h i (p ′ ) > = (2π) 2 δ(p + p ′ ) W i (p) , (6)
< h 1 (p) h 2 (p ′ ) > = 0 . (7)
with

W i (p) ≡ d 2 x W i (x) exp(-ip • x) (8) = π σ 2 i l 2 i exp(-p 2 l 2 i /4) . (9)

III. THE SCATTERING MATRIX

We suppose that the slab is illuminated from the media 0 by an electromagnetic wave of pulsation ω. In the following we will omit the time dependence exp(-i ω t). The field E 0 in the media 0 can be written as a superposition of an incident and scattered fields :

E 0 (x, z) = E i (p 0 ) exp(ip 0 • x -iα 0 (p 0 ) z) + d 2 p (2π) 2 E s (p) exp(ip • x + iα 0 (p) z) . (10)
where

α 0 (p) ≡ (ǫ 0 K 2 0 -p 2 ) 1 2 , (11)
E i (p 0 ) = E i V (p 0 ) ê0- V (p 0 ) + E i H (p 0 ) êH (p 0 ) (12)
E s (p) = E s V (p) ê0+ V (p) + E s H (p) êH (p) . (13)
The subscript H refers to the horizontal polarization (T E) and V to the vertical polarization (T M ), and are defined by the two vectors:

êH (p) = êz × p , (14) ê0±
V (p) = ± α 0 (p) √ ǫ 0 K 0 p - ||p|| √ ǫ 0 K 0 êz , (15)
where the minus sign refers to incident wave and the plus sign to the scattered wave. It has to be noticed that the vector E s (p) and E i (p 0 ) are expressed in different basis due the fact that ê0± V (p) and ê1± V (p) depend on p. In medium 1, we have a similar expression namely:

E 1 (r) = d 2 p (2π) 2 E 1-(p) exp(ip • x -i α 1 (p)z) + d 2 p (2π) 2 E 1+ (p) exp(ip • x + i α 1 (p)z) , (16)
where

α 1 (p) ≡(ǫ 1 K 2 0 -p 2 ) 1 2 . ( 17
)
The field E 1-is decomposed in the basis (ê - V (p), êH (p)), and E 1+ in the basis (ê + V (p), êH (p)) with

êH (p) = êz × p , (18) ê1±
V (p) = ± α 1 (p) √ ǫ 1 K 0 p - ||p|| √ ǫ 1 K 0 êz . (19)
We now introduce the definition the scattering matrix :

E s (p) ≡ R(p|p 0 ) • E i (p 0 ) , (20)
where R(p|p 0 ) is a two dimensional matrix which can be written in the following form:

R(p|p 0 ) = R V V (p|p 0 ) R V H (p|p 0 ) R HV (p|p 0 ) R HH (p|p 0 ) .

IV. DIFFUSE CROSS-SECTION

In a previous work [START_REF] Soubret | A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces[END_REF] we have defined a new product ⊙ between two dimensional matrices in the form :

f ⊙ g = f V V f V H f HV f HH ⊙ g V V g V H g HV g HH (21) ≡    f V V g * V V f V H g * V H Re(f V V g * V H ) -Im(f V V g * V H ) f HV g * HV f HH g * HH Re(f HV g * HH ) -Im(f HV g * V H ) 2Re(f V V g * HV ) 2Re(f V H g * HH ) Re(f V V g * V V + f HV g * V H ) -Im(f V V g * HH -f V H g * HV ) 2Im(f V V g * HV ) 2Im(f V H g * HH ) Im(f V V g * V V + f HV g * V H ) Re(f V V g * HH -f V H g * HV )    .
which allows to write the incoherent Muller matrix 16 in a condensed expression:

M incoh (p|p 0 ) = K 2 0 cos 2 θ (2π) 2 < R(p|p 0 ) ⊙ R(p|p 0 ) > -< R(p|p 0 ) > ⊙ < R(p|p 0 ) > . (22)
Furthermore, we define the generalization of the classical bistatic coefficient 17 also called mean differential coefficient 18 by :

γ incoh (p|p 0 ) ≡ 1 A cos θ 0 M incoh (p|p 0 ) ( 23
)
where A is the area of the illuminated surface and θ 0 the incident angle (see Fig. 3). In this paper we are interested by the perturbative development of the scattered fields as a function of the surface elevations h 1 and h 2 . In a perturbative expansion of the scattering matrix, the terms which contain an expression like h n 1 h m 2 will be denoted R (nm) , so that the perturbative development of R becomes: R = R (00) + R [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] + R (01) + R [START_REF] Celli | Some aspects of light scattering from a randomly rough metal surface[END_REF] + R [START_REF] Ogilvy | Theory of Wave Scattering From Random Rough Surfaces[END_REF] + R

(21) + R (12) + R (22) + R (30) + R (03) + . . . (24)
With this decomposition and using Eqs. (1,3) with the fact that h 1 and h 2 are Gaussian random processes i.e.

< h 2p+1 1 (x) > = 0 , p a positive integer (25) < h 2p+1 2 (x) > = 0 ( 26
)
the incoherent bistatic matrix will be given by the contribution of three terms:

γ incoh (p|p 0 ) = γ incoh u (p|p 0 ) + γ incoh d (p|p 0 ) + γ incoh ud (p|p 0 ) , (27)
where [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] ⊙ R [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] > + < R [START_REF] Ogilvy | Theory of Wave Scattering From Random Rough Surfaces[END_REF] ⊙ R [START_REF] Ogilvy | Theory of Wave Scattering From Random Rough Surfaces[END_REF] > + < R (30) ⊙ R [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] > , (28) corresponds to the incoherent bistatic matrix for the slab where only the upper surface has a roughness (h 2 (x) = 0), and its expansion is made up to order four in the rms-height elevation σ 1 . Similarly,

γ incoh u (p|p 0 ) = K 2 0 cos 2 θ A (2π) 2 cos θ 0 < R (
γ incoh d (p|p 0 ) = K 2 0 cos 2 θ A (2π) 2 cos θ 0 < R (01) ⊙ R (01) > + < R (02) ⊙ R (02) > + < R (03) ⊙ R (01) > , (29)
is associated to a system where only the bottom surface is rough (h 1 (x) = 0), where also the perturbative development is made up to order four in σ 2 . The last term γ incoh ud contains terms which describe the scattering process between the two rough surfaces where only the leading terms are retained: [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] ⊙ R [START_REF] Mcgurn | Localization effects in the elastic scattering of light from a randomly rough surface[END_REF] > + < R [START_REF] Mcgurn | Localization effects in the elastic scattering of light from a randomly rough surface[END_REF] ⊙ R [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] >

γ incoh ud (p|p 0 ) = K 2 0 cos 2 θ A (2π) 2 cos θ 0 < R (
+ < R (01) ⊙ R (21) > + < R (21) ⊙ R (01) > + < R (11) ⊙ R (11) > + . . . , (30)
More precisely all terms up of order σ 2 1 σ 2 2 are included. If the value of σ 1 and σ 2 are of the same order of magnitude, these terms will be comparable to the order four in the expressions (28,29). Thus we can suppose that the following terms in the expansion (30), which are of order σ 4 1 σ 2 2 , σ 2 1 σ 4 2 , σ 4 1 σ 4 2 will be negligible compared to those retained in Eq. (30). However, due to the complexity of the perturbative development, these terms of the sixth order have not been calculated. In the following section, we will show how the perturbative development can be put in the following form: R [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] (p|p 0 ) = α 0 (p 0 ) X [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF]

(p|p 0 ) h 1 (p -p 0 ) , (31)
R (01) (p|p 0 ) = α 0 (p 0 ) X (01) (p|p 0 ) h 2 (p -p 0 ) , (32)
R (11) (p|p 0 ) = α 0 (p 0 ) d 2 p 1 (2π) 2 X (11)12 (p|p 1 |p 0 ) h 1 (p -p 1 ) h 2 (p -p 0 ) +X (11)21 (p|p 1 |p 0 ) h 2 (p -p 1 ) h 1 (p -p 0 ) , (33)
R (20) (p|p 0 ) = α 0 (p 0 ) d 2 p 1 (2π) 2 X (20) (p|p 1 |p 0 ) h 1 (p -p 1 ) h 1 (p 1 -p 0 ) , (34)
R (02) (p|p 0 ) = α 0 (p 0 ) d 2 p 1 (2π) 2 X (02) (p|p 1 |p 0 ) h 2 (p -p 1 ) h 2 (p 1 -p 0 ) , (35)
R (21) (p|p 0 ) = α 0 (p 0 ) d 2 p 1 (2π) 2 d 2 p 2 (2π) 2 X (21)112 (p|p 1 |p 2 |p 0 ) h 1 (p -p 1 ) h 1 (p 1 -p 2 ) h 2 (p 2 -p 0 ) + X (21)121 (p|p 1 |p 2 |p 0 ) h 1 (p -p 1 ) h 2 (p 1 -p 2 ) h 1 (p 2 -p 0 ) +X (21)211 (p|p 1 |p 2 |p 0 ) h 2 (p -p 1 ) h 1 (p 1 -p 2 ) h 1 (p 2 -p 0 ) , (36)
R (12) (p|p 0 ) = α 0 (p 0 ) d 2 p 1 (2π) 2 d 2 p 2 (2π) 2 X (12)221 (p|p 1 |p 2 |p 0 ) h 2 (p -p 1 ) h 2 (p 1 -p 2 ) h 1 (p 2 -p 0 ) + X (12)212 (p|p 1 |p 2 |p 0 ) h 2 (p -p 1 ) h 1 (p 1 -p 2 ) h 2 (p 2 -p 0 ) +X (12)122 (p|p 1 |p 2 |p 0 ) h 1 (p -p 1 ) h 2 (p 1 -p 2 ) h 2 (p 2 -p 0 ) , (37)
R (30) (p|p 0 ) = α 0 (p 0 ) d 2 p 1 (2π) 2 d 2 p 2 (2π) 2 X (30) (p|p 1 |p 2 |p 0 ) h 1 (p -p 1 ) h 1 (p 1 -p 2 ) h 1 (p 2 -p 0 ) , (38)
R (03) (p|p 0 ) = α 0 (p 0 ) d 2 p 1 (2π) 2 d 2 p 2 (2π) 2 X (03) (p|p 1 |p 2 |p 0 ) h 2 (p -p 1 ) h 2 (p 1 -p 2 ) h 2 (p 2 -p 0 ) . (39)
In these expressions we have added superscripts in some terms to indicate the order of apparition of the functions h 1 and h 2 . For example, in X (21)121 the superscript 121 indicates that it is the coefficient associated with the product h 1 (p-p 1 ) h 2 (p 1 -p 2 ) h 1 (p 2 -p 0 ). When we combine Eq. ( 6), δ(0) = A/(2π) [START_REF] Fuks | Wave diffraction by rough interfaces in an arbitrary plane-layered medium[END_REF] and the previous development, we obtain the following expression for the quantities (28-30): -10) (p|p 0 ) + I (p|p 0 ) + I (30-10) (p|p 0 ) , (40) -10) (p|p 0 ) + I

γ incoh u (p|p 0 ) = K 4 0 cos 2 θ cos θ 0 (2π) 2 I ( 10
γ incoh d (p|p 0 ) = K 4 0 cos 2 θ cos θ 0 (2π) 2 I (01-01) (p|p 0 ) + I (02-02) (p|p 0 ) + I (03-01) (p|p 0 ) , (41)
γ incoh ud (p|p 0 ) = K 4 0 cos 2 θ cos θ 0 (2π) 2 I ( 12
(p|p 0 ) + I (21-01) (p|p 0 ) , (42)
where [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] (p|p 0 ) ⊙ X [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] [START_REF] Mcgurn | Localization effects in the elastic scattering of light from a randomly rough surface[END_REF] (p|p 0 ) ⊙ X [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] (p|p 0 ) + X [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] (p|p 0 ) ⊙ X [START_REF] Mcgurn | Localization effects in the elastic scattering of light from a randomly rough surface[END_REF] (p|p 0 ) ,

I (10-10) (p|p 0 ) = W 1 (p -p 0 ) X (
(p|p 0 ) (43)
I (20-20) (p|p 0 ) = d 2 p 1 (2π) 2 W 1 (p -p 1 )W 1 (p 1 -p 0 ) X (20) (p|p 1 |p 0 ) ⊙ X (20) (p|p 1 |p 0 ) + X (20) (p|p + p 0 -p 1 |p 0 ) (44)
I (30-10) (p|p 0 ) = W 1 (p -p 0 ) X (10) (p|p 0 ) ⊙ X (30) (p|p 0 ) + X (30) (p|p 0 ) ⊙ X (10) (p|p 0 ) , (45)
I (01-01) (p|p 0 ) = W 2 (p -p 0 ) X (01) (p|p 0 ) ⊙ X (01) (p|p 0 ) (46)
I (02-02) (p|p 0 ) = d 2 p 1 (2π) 2 W 2 (p -p 1 )W 2 (p 1 -p 0 ) X (02) (p|p 1 |p 0 ) ⊙ X (02) (p|p 1 |p 0 ) + X (02) (p|p + p 0 -p 1 |p 0 ) (47)
I (03-01) (p|p 0 ) = W 2 (p -p 0 ) X (01) (p|p 0 ) ⊙ X (03) (p|p 0 ) + X (03) (p|p 0 ) ⊙ X (01) (p|p 0 ) , (48)
I (12-10) (p|p 0 ) = W 1 (p -p 0 ) X (
I (21-01) (p|p 0 ) = W 2 (p -p 0 ) X (21) (p|p 0 ) ⊙ X (01) (p|p 0 ) + X (01) (p|p 0 ) ⊙ X (21) (p|p 0 ) , (49)
I (11-11) (p|p 0 ) = d 2 p 1 (2π) 2 W 1 (p -p 1 ) W 2 (p 1 -p 0 ) X (11)12 (p|p 1 |p 0 ) (50)
⊙ X [START_REF] Celli | Some aspects of light scattering from a randomly rough metal surface[END_REF]12 (p|p 1 |p 0 ) + X [START_REF] Celli | Some aspects of light scattering from a randomly rough metal surface[END_REF]21 (p|p + p 0p [START_REF] Celli | Some aspects of light scattering from a randomly rough metal surface[END_REF]21 (p|p 1 |p 0 ) ⊙ X [START_REF] Celli | Some aspects of light scattering from a randomly rough metal surface[END_REF]21 (p|p 1 |p 0 ) + X [START_REF] Celli | Some aspects of light scattering from a randomly rough metal surface[END_REF]12 (p|p

1 |p 0 ) + W 2 (p -p 1 ) W 1 (p 1 -p 0 ) X (
+ p 0 -p 1 |p 0 ) (51)
with [START_REF] Mcgurn | Localization effects in the elastic scattering of light from a randomly rough surface[END_REF]221 (p|p 1 |p|p 0 ) + X [START_REF] Mcgurn | Localization effects in the elastic scattering of light from a randomly rough surface[END_REF]212 (p|p

X (30) (p|p 0 ) = d 2 p 1 (2π) 2 W 1 (p 1 -p 0 ) X (30) (p|p 0 |p 1 |p 0 ) +W 1 (p -p 1 ) (X (30) (p|p 1 |p 0 -p + p 1 |p 0 ) + X (30) (p|p 1 |p|p 0 )) , (52)
X (03) (p|p 0 ) = d 2 p 1 (2π) 2 W 2 (p 1 -p 0 ) X (03) (p|p 0 |p 1 |p 0 ) +W 2 (p -p 1 ) (X (03) (p|p 1 |p 0 -p + p 1 |p 0 ) + X (03) (p|p 1 |p|p 0 )) , (53)
X (12) (p|p 0 ) = d 2 p 1 (2π) 2 W 2 (p -p 1 ) X (
1 |p 0 -p + p 1 |p 0 ) +W 2 (p 1 -p 0 ) X (12)122 (p|p 0 |p 1 |p 0 ) , (54)
X (21) (p|p 0 ) = d 2 p 1 (2π) 2 W 1 (p -p 1 ) X (21)112 (p|p 1 |p|p 0 ) + X (21)121 (p|p 1 |p 0 -p + p 1 |p 0 ) +W 1 (p 1 -p 0 ) X (21)211 (p|p 0 |p 1 |p 0 ) , (55)

V. PERTURBATIVE DEVELOPMENT AND REDUCED RAYLEIGH EQUATIONS

In order to obtain the development (24) for the scattering matrix, a practical method is provided by the use of the reduced Rayleigh equations first obtained by Brown et al 8 for a single rough surface, and then extended in 9 for a more general system. These equations are exact, under the Rayleigh hypothesis, and their main advantages are that one of the electric fields E i , E s , E 1-, E 1+ (see Fig. 2) of the problem has been eliminated. These equations are derived by taking linear combinations of the electromagnetic boundaries conditions at the first surface h 1 , where the Fourier transform of the fields has been introduced. In particular, we obtain the two following equations (see Eqs.(99-100) in 9 ):

d 2 p (2π) 2 M 1+,0+ h (u|p) • R(p|p 0 ) • E i (p 0 ) + M 1+,0- h (u|p 0 ) • E i (p 0 ) = 2 (ǫ 0 ǫ 1 ) 1 2 α 1 (u) (ǫ 1 -ǫ 0 ) E 1+ (u) , (56)
d 2 p (2π) 2 M 1-,0+ h (u|p) • R(p|p 0 ) • E i (p 0 ) + M 1-,0- h (u|p 0 ) • E i (p 0 ) = - 2 (ǫ 0 ǫ 1 ) 1 2 α 1 (u) (ǫ 1 -ǫ 0 ) E 1-(u) , (57)
where

M 1b,0a h (u|p) ≡ I(bα 1 (u) -aα 0 (p)|u -p) bα 1 (u) -aα 0 (p) M 1b,0a (u|p) (58) with M 1b,0a (u|p) = ||u||||p|| + ab α 1 (u) α 0 (p) û • p -b ǫ 1 2 0 K 0 α 1 (u) (û × p) z a ǫ 1 2 1 K 0 α 0 (p) (û × p) z (ǫ 0 ǫ 1 ) 1 2 K 2 0 û • p , (59)
and

I(α|p) ≡ d 2 x exp(-ip • x -iα h 1 (x)) , (60)
the symbols a = ± and b = ± in Eqs. (59-60) represent a given choice linked to the field propagation.

In order to obtain a single equation for R(p|p 0 ), we have to find a relation between E 1-and E 1+ . To this end we already know an expression of the scattering matrix for a single rough surface separating two homogenous media of permittivity ǫ 1 and ǫ 2 , which is translated along the z-axis to the height z = -H, and illuminated by a plane wave E 1-(p). This scattering matrix denoted R H s ǫ1,ǫ2 is given by:

R H s ǫ1,ǫ2 (p|p 0 ) = exp(i(α 1 (p) + α 1 (p 0 )) H) R s ǫ1,ǫ2 (p|p 0 ) , (61)
where R s ǫ1,ǫ2 (p|p 0 ) can be found in Refs. [START_REF] Soubret | A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces[END_REF][START_REF] Mcgurn | Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces[END_REF][START_REF] Johnson | Third-order small-perturbation method for scattering from dielectric rough surfaces[END_REF] . The phase term in Eq. ( 61) comes from the translation z = -H of the rough surface h 2 (x) (see Refs. [START_REF] Voronovich | Wave Scattering from Rough Surfaces[END_REF][START_REF] Soubret | A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces[END_REF] ). Thus we have the following relation:

E 1+ (u) = d 2 u 1 (2π) 2 R H s ǫ1,ǫ2 (u|u 1 ) • E 1-(u 1 ) . ( 62
)
Now combining Eq. ( 62) with Eqs. (56-57), we obtain an integral equation where R(p|p 0 ) is the only unknown:

d 2 p (2π) 2 M 1+,0+ h (u|p) + d 2 u 1 (2π) 2 α 1 (u) α 1 (u 1 ) R H s ǫ1,ǫ2 (u|u 1 ) • M 1-,0+ h (u 1 |p) • R(p|p 0 ) = -M 1+,0- h (u|p 0 ) + d 2 u 1 (2π) 2 α 1 (u) α 1 (u 1 ) R H s ǫ1,ǫ2 (u|u 1 ) • M 1-,0- h (u|p 0 ) . (63)
Expanding I(α|p) in Eq. ( 60) in power of h 1 :

I(α|p) = (2π) 2 δ(p) -iα h (1) 1 (p) - α 2 2 h (2) 1 (p) - iα 3 3! h (3) 1 (p) + • • • , (64)
h (n) 1 (p) ≡ d 2 x exp(-ip • x) h n 1 (x) , (65)
and using the perturbative development

9 of R H s ǫ1,ǫ2 in power of h 2 : R H s ǫ1,ǫ2 (p|p 0 ) = (2π) 2 δ(p -p 0 ) X H(0) s ǫ1,ǫ2 (p 0 ) + α 0 (p 0 ) X H(1) s ǫ1,ǫ2 (p|p 0 ) h 2 (p -p 0 ) +α 0 (p 0 ) d 2 p 1 (2π) 2 X H(2) s ǫ1,ǫ2 (p|p 1 |p 0 ) h 2 (p -p 1 )h 2 (p 1 -p 0 ) +α 0 (p 0 ) d 2 p 1 (2π) 2 d 2 p 2 (2π) 2 X H(3) s ǫ1,ǫ2 (p|p 1 |p 2 |p 0 ) h 2 (p -p 1 )h 2 (p 1 -p 2 )h 2 (p 2 -p 0 ) , (66)
we finally obtain the expansions (24,31-39). The expression for the scattering matrix when only one rough surface is involved was given in Ref. [START_REF] Soubret | A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces[END_REF] , here a small change have been made in the notations:

R (n0) (p|p 0 ) = R (n) u (p|p 0 ) , (67)
R (0n) (p|p 0 ) = R (n) d (p|p 0 ) (68)
n being an integer ranging from 0 to 3. For the others coefficients we have :

X (11)21 (p|p 1 |p 0 ) =T 10 (p) • U (0) (p) • X H(1) s ǫ1,ǫ2 (p|p 1 ) • -ǫ • D - 10 (p 1 ) • X (10) (p 1 |p 0 ) + iS + (p 1 |p 0 ) , (69)
X (11)12 (p|p 1 |p 0 ) =i P + (p|p 1 ) • X (01) (p 1 |p 0 ) , (70)
X (21)112 (p|p 1 |p 2 |p 0 ) =i P + (p|p 1 ) • X (11)12 (p 1 |p 2 |p 0 ) + 1 2 α 1 (p) P -(p|p 2 ) -α 0 (p 2 ) P + (p|p 2 ) • X (01) (p 2 |p 0 ) (71) X (21)121 (p|p 1 |p 2 |p 0 ) =i P + (p|p 1 ) • X (11)21 (p 1 |p 2 |p 0 ) , (72)
X (21)211 (p|p 1 |p 2 |p 0 ) =T 10 (p) • U (0) (p) • X H(1) s ǫ1,ǫ2 (p|p 1 ) • -ǫ • D - 10 (p 1 ) • X (20) (p 1 |p 2 |p 0 ) + i (ǫ 1 -ǫ 0 ) 2 (ǫ 0 ǫ 1 ) 1/2 M 1-,0+ (p 1 |p 2 ) • X (10) (p 2 |p 0 ) - 1 2 α 1 (p 1 ) S + (p 1 |p 0 ) + α 0 (p 0 ) S -(p 1 |p 0 ) (73)
X (12)122 (p|p 1 |p 2 |p 0 ) =i P + (p|p 1 ) • X (02) (p 1 |p 2 |p 0 ) , (74)
X (12)212 (p|p 1 |p 2 |p 0 ) =T 10 (p) • U (0) (p) • X H(1) s ǫ1,ǫ2 (p|p 1 ) • -ǫ • D - 10 (p 1 ) • X (11)12 (p 1 |p 2 |p 0 ) + i (ǫ 1 -ǫ 0 ) 2 (ǫ 0 ǫ 1 ) 1/2 M 1-,0+ (p 1 |p 2 ) • X (01) (p 2 |p 0 ) , (75)
X (12)221 (p|p 1 |p 2 |p 0 ) =T 10 (p) • U (0) (p) • -X H(1) s ǫ1,ǫ2 (p|p 1 ) • ǫ • D - 10 (p 1 ) • X (11)21 (p 1 |p 2 |p 0 ) + X H (2)
s ǫ1,ǫ2 (p|p

1 |p 2 ) • -ǫ • D - 10 (p 2 ) • X (10) (p 2 |p 0 ) + i S + (p 2 |p 0 ) , (76) (77)
where the matrices T [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] , U (0) , P ± , D ± 10 , X [START_REF] Mcgurn | Localization effects in the scattering of light from a randomly rough grating[END_REF] , X [START_REF] Ogilvy | Theory of Wave Scattering From Random Rough Surfaces[END_REF] , X (01) , X (02) , X

H (1) s ǫ1,ǫ2 , X H (2)
s ǫ1,ǫ2 are respectively defined by Eqs.(71),(87),(114,115),(57),(104,105),(96,97),(60,61) in Ref. [START_REF] Soubret | A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces[END_REF] , and ǫ, S ± are given in Appendix A.

VI. NUMERICAL RESULTS

As an application of the previous formalism we consider a system made of a air-dielectric film whose dielectric constant is ǫ 1 = 2.6896 + i 0.0075(ǫ 0 = 1), deposited on silver surface with ǫ 2 = -18.3 + i 0.55. The vacuum-dielectric interface is a two-dimensionnal rough surface, whose parameters are σ 1 = 15 nm and l 1 = 100 nm. The dielectric-silver boundary is also rough and defined by σ 2 = 5 nm and l 2 = 100 nm. The incident wave has an arbitrary polarization and his wavelength is λ = 632.8 nm. With this set of parameters, the conditions of validity of the small-perturbation theory are satisfied 20 namely:

2π ǫ 1 ǫ 0 1/2 σ 1 λ ≪ 1 , 2π ǫ 2 ǫ 1 1/2 σ 2 λ ≪ 1 (78) σ 1 l 1 ≪ 1 , σ 2 l 2 ≪ 1 . (79)
The thickness of the film is H = 500 nm and support two-guided wave polaritons for the (T E) polarizations at p 1 T E = 1.5534 K 0 , and p 2 T E = 1.2727 K 0 , and three guided-modes for the (T M ) polarizations at p 1 T M = 1.7752 K 0 , p 2 T M = 1.4577 K 0 and p 3 T M = 1.034 K 0 . We have computed the incoherent bistatic coefficient γ incoh (p|p 0 ), where the integrals involved in Eqs. (44,47,51, 52-55) are evaluated using Legendre quadrature. The results are shown in Figs. (4-7), where the incoherent bistatic coefficient is drawn as of function of the scattering angle θ for two different angles of incidence and the incident wave is linearly polarized. In Fig. 4, the wave is normally incident and the scattered field is observed in the incident plane (φ = 0 o ). The single scattering contribution on each surface, associated with the terms I (10-10) +I (01-01) , is plotted as a dotted line, the double-scattering contribution I + I (02-02) + I as a dashed line, the other terms I (30-10) + I (03-01) + I + I (21-01) as a dash-dotted line, and the total contribution γ incoh by the solid curve. We observed an enhancement of the backscattering which corresponds to the physical process in which the incident light excites a guided-mode through the roughness of the slab and then is scattered into a volume wave which is also due to the roughness effect. During the same process,the light can follow this path in the opposite direction where one possible configuration is shown in Fig. 1. These two paths can interfere constructively near the backscattering direction to produce a peak. As these paths are identical for the two waves under consideration, they should have the same degree of interaction with the rough surface, thus a term like I (30-10) cannot produce the peak because the first wave interact three times with the upper rough surface while the second wave only one time.

The effect comes only from the terms I + I (02-02) + I , which contain the paths indicated in Fig. 1. However, it has to be noticed that these terms contain also paths that do not produce enhanced backscattering, for example in I we have the scattering process where the incident wave is only scattered one time by the upper rough surface but the scattering process is of order two in h 1 . This is the reason why the terms I + I (02-02) + I are not zero away from the anti-specular direction. In order to separate the different contributions to the backscattering peak, we have drawn the contributions of I , I (02-02) , I separately in Fig. 5 as a dashed-line, dotted-line and solid curve respectively, we see that each term produces an enhancement near the anti-specular direction. The terms I , I (02-02) are the classic one(see Refs. [START_REF] Soubret | A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces[END_REF][START_REF] Freilikher | Coherent scattering enhancement in systems bounded by rough surfaces[END_REF] ) where the fields do not interact with both rough surfaces but produce the peak due to the scattering on the same rough surface: the upper one for I (20-20) and the bottom one for I (02-02) . The contribution I is the new result of this work, because we see that the mechanism of Fig. 1 exists and has the same magnitude as the other terms for the choosen parameters. In order to clearly show the displacement of the backscattering enhancement as the angle of incidence is varied, we show in Figs. 6 and 7 the numerical results of the perturbation method when θ 0 = 20 o . As expected, we clearly see the peak which is now located at θ 0 = -20 o . Recent papers [START_REF] Soubret | A new application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces[END_REF][START_REF] Freilikher | Coherent scattering enhancement in systems bounded by rough surfaces[END_REF] have also explored the satellite peaks phenomenon which occurs when the wave follows two reverse paths but with different guided-mode excitations. In Fig. 5, this phenomenon appears for the (T M ) to (T M ) polarization due to the term I (02-02) . Although a similar phenomenon occurs for the other terms it is too weak to contribute significantly..

VII. CONCLUSION

In conclusion, the results in this paper clearly show that the backscattering enhancement produced by a rough slab due to the guided-wave polaritons has several origin. At small order in the perturbative development, the peak is produced by a double scattering mechanism. The new results we would like to emphasize, is that not only processes where the two scattering events take place on the same rough surface are capable to produce the backscattering peak, but it is also due to the fact that the wave can be scattered first by one of the rough surface and second by the other surface. To carry the proof, we have performed a perturbative development up to order four in the rms-height of the surfaces which has been possible using an integral equation called reduced Rayleigh equation, in which only the scattering matrix of the whole structure is unknown.

k 0 k 0 k k A B C ǫ 0 ǫ 1 ǫ 2
FIG. 1: New mechanisms responsible of enhanced backscattering. The incident wave is transmitted in A as perfect interface between the media 0 and 1, then the wave is scattered by the second rough surface in B and by the first one in C. But the wave can now follow this path the other way round. The phase difference between these two waves is zero in the anti-specular direction which produces the peak.

x

z = h 1 (x) ǫ 0 ǫ 1 z ê0- V (p 0 ) êH (p 0 ) ê0+ V (p) êH (p) H ǫ 2 z = -H + h 2 (x) E 1+ (p) E 1-(p) E i (p 0 ) E s (p)
FIG. 2: A rough surface with an incident wave coming from the medium 0 and scattered by a slab with two rough surfaces. FIG. 4: The bistatic coefficients for an horizontal (T E) and a vertical (T M ) polarized normally incident light of wavelength λ = 632.8 nm, on a slab with an upper two-dimensional randomly rough surface, characterized by the parameters σ 1 = 15 nm, l 1 = 100 nm, and a bottom rough surface characterized by σ 2 = 5 nm, l 2 = 100 nm. The dielectric constants are ǫ 1 = 2.6896 + i 0.0075 and ǫ 2 = -18.3 + i 0.55. The thickness of the film is H = 500 nm. The scattered field is observed in the incident plane. For each figure are plotted : the total incoherent scattering γ incoh (solid curve), the first order given by I + I (01-01) (dotted curve), the second order I + I (02-02) + I (dashed curve), and the third order I (30-10) + I (03-01) + I + I (21-01) (dash-dotted curve).

-50 0 50

FIG. 3 :

 3 FIG.3: Definition of the scattering vector.

FIG. 5 :FIG. 6 :FIG. 7 :

 567 FIG.5: Only the second order contributions of Fig.4are depicted. I is the solid curve, I(20-20)

ACKNOWLEDGMENTS

(AS) thanks ANRT for financial support during the preparation of his thesis (contract CIFRE-238-98).

APPENDIX A: SCATTERING MATRIX COEFFICIENTS

Here we collect the expression of the matrices not given in 9 : ǫ ≡ 1 2

(ǫ 0 ǫ 1 ) -1/2 0 0 1 (A1)

After some calculations we obtain :

LIST OF FIGURES The bistatic coefficients for an horizontal (T E) and a vertical (T M ) polarized normally incident light of wavelength λ = 632.8 nm, on a slab with an upper two-dimensional randomly rough surface, characterized by the parameters σ 1 = 15 nm, l 1 = 100 nm, and a bottom rough surface characterized by σ 2 = 5 nm, l 2 = 100 nm. The dielectric constants are ǫ 1 = 2.6896 + i 0.0075 and ǫ 2 = -18.3 + i 0.55. The thickness of the film is H = 500 nm. The scattered field is observed in the incident plane. For each figure are plotted : the total incoherent scattering γ incoh (solid curve), the first order given by I (10-10) +I (01-01) (dotted curve), the second order I + I (02-02) + I (dashed curve), and the third order


A M Buryakov

A V Gorbatova

P Y Avdeev

E D Lebedeva

K A Brekhov

A V Ovchinnikov

N S Gusev

E A Karashtin

M V Sapozhnikov

E D Mishina

N Tiercelin

V L Preobrazhensky

Efficient Co/Pt THz spintronic emitter with tunable polarization

We report on the design of a spintronic emitter based on the Pt(3nm)/Co(3nm) structure, which enables the control over terahertz radiation polarization. Utilizing the field-induced magnetization rotation that takes place at low magnetic fields of up to 250 Oe at room temperature, we have achieved the full range of terahertz polarization rotation from 0 to 360. This rotation became possible due to the uniaxial magnetic anisotropy induced in the plane of the cobalt film during its fabrication. We evaluated the efficiency of the Co/Pt structure in generating terahertz radiation and found that the terahertz pulse energy flux reaches 160 nJ/cm 2 at an excitation flux of 4 mJ/cm 2 .

In recent years, advances in terahertz (THz) technologies and magnetism have unveiled the high potential of spintronic THz emitters (STE) based on bilayer structures comprising of magnetic and non-magnetic metal films. [START_REF] Seifert | Efficient metallic spintronic emitters of ultrabroadband terahertz radiation[END_REF] Excitation of such structures by femtosecond pulses leads to the generation of sub-picosecond spin currents within them, consequently resulting in the emission of broadband terahertz radiation. [START_REF] Chen | Generation and manipulation of chiral broadband terahertz waves from cascade spintronic terahertz emitters[END_REF] One of the key advantages of STE is their ability to control a diverse set of THz radiation parameters (polarization, amplitude, and bandwidth), by applying external magnetic and/or electric fields. [START_REF] Chen | Generation and manipulation of chiral broadband terahertz waves from cascade spintronic terahertz emitters[END_REF][START_REF] Li | Spintronic terahertz emission with manipulated polarization (STEMP)[END_REF][START_REF] Kong | Broadband spintronic terahertz emitter with magnetic-field manipulated polarizations[END_REF] Such magnetic-field driven solid-state THz sources, with con-trolled polarization states, show significant promise for the development of functional on-chip devices for THz opto-spintronics. [START_REF] Kong | Broadband spintronic terahertz emitter with magnetic-field manipulated polarizations[END_REF] Optimization of such devices is impossible without a thorough under-standing the physical mechanisms and origin of spin dynamics induced by femtosecond radiation. [START_REF] Unikandanunni | Inertial spin dynamics in epitaxial cobalt films[END_REF][START_REF] Gang | Element-selective investigation of femtosecond spin dynamics in NiPd mag-netic alloys using extreme ultraviolet radiation[END_REF][START_REF] Unikandanunni | Anisotropic ultrafast spin dynamics in epitaxial cobalt[END_REF][START_REF] Rudolf | Ultrafast magnetization enhancement in metallic multilayers driven by superdiffusive spin current[END_REF][START_REF] Medapalli | Femtosecond photocurrents at the FeRh/Pt interface[END_REF] Among the most promising applications are ultra-fast data transfer, [START_REF] Chen | Generation and manipulation of chiral broadband terahertz waves from cascade spintronic terahertz emitters[END_REF][START_REF] Kong | Broadband spintronic terahertz emitter with magnetic-field manipulated polarizations[END_REF][START_REF] Withayachumnankul | Integrated silicon pho-tonic crystals toward terahertz communications[END_REF] next generation wireless communication (6G), [START_REF] Deng | On-chip polarizationand frequency-division demultiplexing for multidimensional terahertz communication[END_REF][START_REF] Zhang | Active control of terahertz waves using vanadium-dioxide-embedded metamaterials[END_REF][START_REF] Yang | Terahertz topological photonics for on-chip communication[END_REF] and ultrafast electromagnetic recording. [START_REF] Kimel | Writing magnetic memory with ultrashort light pulses[END_REF] In addition, THz emitters are simple, easy-tomanufacture structures with high for this type of emitters optical-THz conversion efficiency. These attributes allow STE to compete with conventional sources of THz radiation, such as photoconductive antennas and inorganic non-linear optical crystals. [START_REF] Cheng | Studying spin-charge conversion using terahertz pulses[END_REF] According to the 2023 roadmap, [START_REF] Leitenstorfer | The 2023 terahertz science and technology roadmap[END_REF] THz spintronics offers not only a unique approach for performing fast spintronic operations but also new functionalities for THz photonics in the field of generation, modulation, and detection of THz radiation. Polarization rotation is one of the functions required in the THz optical schemes. Several works [START_REF] Khusyainov | Polarization control of THz emission using spin-reorientation transition in spintronic heterostructure[END_REF][START_REF] Khusyainov | Composite multiferroic terahertz emitter: Polarization control via an electric field[END_REF][START_REF] Hewett | Spintronic terahertz emitters exploiting uniaxial magnetic anisotropy for field-free emission and polarization control[END_REF][START_REF] Agarwal | Electric-field control of non-linear THz spintronic emitters[END_REF] have already shown the possibility of THz radiation polarization rotation without mechanical action on the emitter. The proposed emitters possess uniaxial magnetic anisotropy, allowing for magnetic field-induced magnetization rotation and consequently resulting in THz polarization rotation. By applying a magnetic field along the "hard axis," a full 360 rotation of THz radiation polarization has been achieved, as initially demonstrated in Ref. 21. However, the issue of low power output remains a challenge for controllable THz emitters. Currently, the highest energy per pulse achieved at the STE is 90 nJ. [START_REF] Rouzegar | Broadband spintronic terahertz source with peak electric fields exceeding 1.5 MV/cm[END_REF] The bilayer films consisting of Co (3nm) and Pt (3nm) were deposited by magnetron sputtering on a glass plate (200lm thickness) in an argon atmosphere at a pressure of 4.10 3 Torr and room temperature. Prior to deposition, the vacuum chamber was evacuated to 10 6 Torr. The growth rate is 0.25 nm/s for Pt and 0.125 nm/s for Co. The Si top layer was 5 nm to protect the structure from oxidation. The accuracy in thickness of layer deposition is 10%. The deposition process was carried out under an applied magnetic field of 400 Oe, aligned in the plane of the samples. Magnetic field assisted fabrication enabled the creation of an easy magnetization axis in the same as magnetic field direction, and facilitated the occurrence of field-induced magnetization rotation. The selection of the thicknesses for the platinum and cobalt layers was guided by the findings of Ref. 22, which demonstrated the maximum optical-terahertz conversion efficiency precisely for these thicknesses. To measure the parameters of THz generation, including the rotation of the polarization plane, we utilized the conventional method of THz time-domain spectroscopy (THz-TDS). 17,23 A femtosecond laser system, consisting of a Ti:Sapphire laser and a regenerative amplifier (Avesta Project Ltd.), delivered radiation at wavelength of 800 nm, a pulse repetition rate of 3 kHz, and a pulse duration of 100 fs. The pump beam aperture on the sample surface was about 1 mm. The Co/Pt structure was placed on a nonmagnetic holder between the poles of an electromagnet to control the magnetization. The magnetic field was applied along the X-axis of the laboratory frame in the film plane [Fig. 1(a)]. THz radiation generated by the Co/Pt structure was detected by electro-optical sampling in a setup with a nonlinear optical ZnTe crystal. The polarization of both pump and probe beams was parallel to the [110] axis jj Z of the ZnTe crystal. In this geometry, only EX-component of the THz field is detected (see Ref. 23 for details). To determine the polarization of the THz radiation, a wire-grid polarizer (WGP) was installed in front of the ZnTe analyzer. In order to assess the emitter efficiency, a high-energy chromium-forsterite laser with amplified chirped pulses was employed. This laser system exhibited a pulse duration of 100 fs, a central wave-length of 1240 nm, and a maximum output energy of 20 mJ. It operated at a repetition rate of 10Hz. The excitation pump beam was collimated to a width of 2.5 cm (FWHM), and the size of the excitation region on the Co/Pt surface was restricted by a 1.2 cm diameter aperture due to the limited sample size. After the pump pulse passed through the diaphragm, the average fluence ranged from 0.78 to 10 mJ/cm 2 . Pulse energy was measured using a Golay cell, and the absolute calibration of the spintronic THz emitter was carried out with the setup described in Ref. 24. The Co/Pt emitters operate on the inverse spin Hall effect (ISHE). The direction of polarization for THz waves emitted due to the ISHE is determined by the directions of the injected spin current and the spin polarization of electrons. [START_REF] Seifert | Spintronics with terahertz radiation: Probing and driving spins at highest frequencies[END_REF] The latter depends on the magnetization direction, which is influenced by the magnitude and orientation of the magnetizing field. As a result, the phase of the generated THz waves depends on both the sample orientation relatively to the optical beam and the external magnetic field direction. This dependence is illustrated schematically in Fig. 1(a). Experimental data for THz signals with a pump energy flux of 0.97 mJ/cm 2 and two magnetizing field strengths of þ2 and 2 kOe, directed parallel to the easy magnetization axis (E.A.), are shown in Fig. 1(b). These figures depict how the phase of the terahertz wave changes depending on the sample orientation and the applied magnetic field direction. Namely, both a change in the direction of the magnetic field and a change in the orientation of the sample changed the polarity of the terahertz pulse. Notably, there is a difference in the delay time of terahertz pulses for the two distinct directions of optical beam incidence, "front side" and "back side," as it is seen in Fig. 1(b). This difference is attributed to the optical (n = 1.45 [START_REF] Arosa | Refractive index spectroscopy and material dis-persion in fused silica glass[END_REF] ) and THz pulses (n »2.1 27 ) delay in the SiO2 substrate, which thickness is 0.18mm. The decrease in the signal amplitude recorded when irradiating the sample from the Co/Pt film side (front side) can be explained by the high absorption of THz radiation in the glass substrate. Phase flip while changing the geometry from front side to back side confirms the ISHE for THz emission in the studied STE. The dependence of the peak THz amplitude on the applied magnetic field exhibits behavior similar to the standard magnetic hysteresis. Figure 3(a) displays the magnetic hysteresis loops obtained by the THz-TDS method and measured at maximum electric field amplitude, correspondent to the delay time t1⁄40 ps [see Fig. 2(a)]. Hysteresis loops are presented in two different geometries when the applied magnetic field vector H is parallel to the easy magnetization axis (E.A.) or perpendicular to it (H.A.). The shape of the hysteresis loops points to the presence of uniaxial magnetic anisotropy in the sample with an anisotropy field HA 120 Oe. The coercive field of the sample does not exceed 80 Oe. The dependence of the THz polarization angle on the magnetic field is obtained from the approximation of angular diagrams (Fig. S2) in the geometry of hard axis and "easy axis" (Fig. S3); The circle and square points are experimental results, while the blue and violet lines represent a model that uses Eq. ( 1).

The presence of magnetic anisotropy with a pronounced hard axis in the Co/Pt sample indicates the possibility of precisely controlled manipulation of the THz polarization by low magnetic fields (HA < H < HA up to 120 Oe). To prove that the rotation of THz polarization is the result of a field-induced magnetization rotation induced by an external magnetic field, we studied the angular dependences of the THz signal. In the used optical scheme, polarization rotation angle cannot be seen directly from experimental dependences. The experimental dependences are obtained by fitting them (in Figs. S2 andS3) with the function . [START_REF] Khusyainov | Polarization control of THz emission using spin-reorientation transition in spintronic heterostructure[END_REF][START_REF] Zainullin | Polarization analysis of THz radiation using a wire grid polarizer and ZnTe crystal[END_REF] In this function, 𝜃 is the phase difference between ∆𝑆(𝑡) and the reference signal, t represents the delay time, [START_REF] Jiang | Electro-optic sampling near zero optical transmission point[END_REF] and is the function describing the dependence of the THz electric field, [START_REF] Brunken | Electro-optic sampling at the TESLA test accelerator: Experimental setup and first results[END_REF] 𝜙 ! is the angle of the WGP polarizer, and 𝜙 " is the angle between THz field and the [1,1,0] axis of the ZnTe crystal.

We measured the temporal dependence of the THz signal [Fig. 2(a)] on the rotation angle 𝜙 ! of the WGP polarizer installed in the path of the THz beam. The measurements were made in a magnetic field directed along E.A. or H.A. All measured angular dependences of the THz signal are shown in the supplementary material (Fig. S2 for the hard axis and Fig. S3 for the easy axis). The extracted from each polarization dependence fitting parameter, angle 𝜙 " as function of magnetic field is shown in Fig. 3(b). The measurements were carried out with a slight deviation of the magnetizing field from the hard axis (~1°), which makes it possible for the magnetization vector to fall into one of two opposite directions (see also S3). Otherwise, a domain structure was formed and the effect disappeared. [START_REF] Khusyainov | Polarization control of THz emission using spin-reorientation transition in spintronic heterostructure[END_REF][START_REF] Kolejak | 360 polarization control of terahertz spintronic emitters using uniaxial FeCo/TbCo2/FeCo trilayers[END_REF] Note that, as showed earlier, [START_REF] Khusyainov | Polarization control of THz emission using spin-reorientation transition in spintronic heterostructure[END_REF][START_REF] Hewett | Spintronic terahertz emitters exploiting uniaxial magnetic anisotropy for field-free emission and polarization control[END_REF] the hysteresis loops for the measured Ex-component of the THz field completely coincide with the hysteresis loops for the magnetization component My. Along with this, the magnetization modulus remains constant, as does the THz field amplitude.

In the framework of ISHE mechanism, the observed rotation of THz polarization is controlled by orientation of the electron spin polarization antiparallel to the magnetic moment of the Co film. The thermodynamic potential Fm of magnetic subsystem of uniaxial magnetic film consist of Zeeman energy of the magnetic moment 𝑀 ++⃗ inter-action with the magnetizing field 𝐻 + +⃗ and anisotropy energy characterized by the anisotropy field HA ,

(1) where 𝑛 +⃗ is the unit vector parallel to the E.A. axis. The magnetization rotation is described by the equation of state that delivers minimum for Fm , (2) Where 𝜓 is the angle between magnetic moment and the hard axis, and 𝛾 is an angle of the magnetic field deflection from the hard axis [Fig. 1(a)]. We assumed the magnetization value constant (/𝑀 ++⃗ /), in agreement with the Landau theory. Since the direction of magnetization determines the direction of THz polarization, then Eq. ( 2) also describes the dependence of the rotation angle 𝜙 " of the THz polarization on the magnetic field strength (supplementary material II). The result of finding the dependence 𝜙 " (𝐻) is shown in Fig. 3(b). Solutions of Eq. ( 1) were found for parameters HA = 120 Oe and 𝜙 " = 0.03. The latter was obtained by fit-ting to experimental data. Near the field-induced magnetization rotation points H = ±HA, characteristic spikes appear on the calculated curve. Figure 4(a) shows the generated THz pulse energy density in the range of optical fluence of (1-10) mJ/cm 2 . The minimum detected THz pulse energy density was about 70 nJ/cm 2 at the optical pump fluence of 1.5 mJ/cm 2 . For lower fluences, the THz energy remained on the noise level. In the range of 1-4 mJ/cm 2 , the THz pulse energy increased linearly and reached the highest value of 160 nJ/cm 2 . For the pump fluence above the threshold value of 4.2 mJ/cm 2 , the THz pulse energy decreased irreversibly (this area is marked as "Damage" in Fig. 4). Enhancing thermal conductivity with a silicon substrate 22 might enable increasing this threshold. Figure 4(b) demonstrates the efficiency of the optical-to-THz conversion in the range of optical fluence of (1-4.5) mJ/cm 2 . The maximal THz emission efficiency from a spintronic emitter was reported in Ref. 22. The peak THz pulse energy was 29 nJ/cm 2 with an excitation beam width of 2cm (full width at half maximum, FWHM, of intensity) and a maximum fluence of 1.1 mJ/cm 2 . At the maximum pump energy of 48 mJ (1.33 mJ/cm 2 ) for the nonlinear ZnTe crystal, the THz pulse energy amounts to 1.5 µJ (or, when normalized to the excitation area, 41.67 nJ/cm 2 ). [START_REF] Blanchard | Generation of 1.5 lJ single-cycle terahertz pulses by optical rectification from a large aperture ZnTe crystal[END_REF] In comparison with that work and normalizing by the effective generation area (the pump beam width), we achieved 2.4-fold increase in the efficiency at the optical pump fluence of 1.1 mJ/cm 2 and 5.5-fold increase at the fluence of 4 mJ/cm 2 . The spintronic emitter demonstrated an optical-to-terahertz conversion of 7x10 3 %. In previous studies, values of 2.6x10 3 % for the spintronic emitter and 3.1x10 3 % for the nonlinear ZnTe crystal were achieved.

FIG. 1 .

 1 FIG. 1. (a) Schematic representation of the spin current direction and polarity of the generated terahertz waves due to the inverse spin Hall effect; 𝜓angle between magnetic moment and the hard axis, and 𝛾 -angle of the magnetic field deflection from the hard axis (b) Time-domain profile of the THz signal DS (see supplementary material S.III) for two directions of the optical beam incidence and an applied planar magnetic field of ±2 kOe.

Figure 2 (

 2 Figure 2(a) displays a comparison of temporal profiles of the THz signal for opposite polarities of the magnetic field when samples are irradiated from the substrate side. The phase inversion of the THz signal upon changing the magnetic field direction indicates a 180 ° rotation of the THz polarization. The inset in Fig. 2(a) presents the spectra of the THz signal. The maximum spectral amplitude is observed at a frequency of 1 THz. The width of the frequency spectrum is limited by 3 THz due to the sensitivity band of the ZnTe detector.Figure 2(b) illustrates the dependence of the THz amplitude on the laser energy flux. The increase in the THz signal with rising pump energy is attributed to the enhanced spin injection from the ferromagnetic layer into the platinum layer.

Figure 2 (

 2 Figure 2(a) displays a comparison of temporal profiles of the THz signal for opposite polarities of the magnetic field when samples are irradiated from the substrate side. The phase inversion of the THz signal upon changing the magnetic field direction indicates a 180 ° rotation of the THz polarization. The inset in Fig. 2(a) presents the spectra of the THz signal. The maximum spectral amplitude is observed at a frequency of 1 THz. The width of the frequency spectrum is limited by 3 THz due to the sensitivity band of the ZnTe detector.Figure 2(b) illustrates the dependence of the THz amplitude on the laser energy flux. The increase in the THz signal with rising pump energy is attributed to the enhanced spin injection from the ferromagnetic layer into the platinum layer.

FIG. 2 .

 2 FIG. 2. (a) THz polarization rotation resulting from a change in the polarity of the external magnetic field. Inset: Correspondent FTT spectra. (b) Dependence of the THz signal amplitude on the optical radiation energy flux. The star denotes the point corresponding the graph in (a).

FIG. 3 .

 3 FIG. 3. (a) Dependences of THz field amplitude and (b) THz polarization angle on the external magnetic field.The dependence of the THz polarization angle on the magnetic field is obtained from the approximation of angular diagrams (Fig.S2) in the geometry of hard axis and "easy axis" (Fig.S3); The circle and square points are experimental results, while the blue and violet lines represent a model that uses Eq. (1).

  31

  1.2cm. The blue star: the maximum THz energy currently achieved in similar STE structures.[START_REF] Rouzegar | Broadband spintronic terahertz source with peak electric fields exceeding 1.5 MV/cm[END_REF] The cyan asterisk denotes the highest THz energy presently attained in ZnTe crystal as per Refs. 22, 30, and 31. (b) The optical-to-terahertz conversion efficiency. In summary, this research highlights the potential for controlled manipulation of THz polarization in a Co/Pt spintronic emitter by applying an external magnetic field. By rotating magnetization within field-induced magnetization rotation by changing magnetizing field value only, the rotation angle of the THz polarization can be controlled by 360°. Studies of the dependences of the peak THz amplitude and the polarization angle on the applied magnetic field yielded results consistent with the theoretical model based on the equation of state for uniaxial magnetic films. Additionally, a linear correlation was observed between terahertz pulse energy and excitation pump intensity for the Co/Pt spintronic emitter. The THz generator achieved its highest pulse energy flux of 160 nJ/cm 2 at the optical pump fluence of about 4 mJ/cm 2 . This was achieved by combination of the most efficient STE structure (Co/Pt) with the creation of uniaxial magnetic anisotropy in it, accompanied by field-induced magnetization rotation. In-plane field-induced magnetization rotation is the tool for rotation of THz polarization. This study offers valuable insight into manipulating THz polarization in spintronic emitters, opening up potential applications in fields, such as communication, imaging, and sensing. By employing advanced materials and enhancing thermal conductivity with a silicon substrate, further improvements in emitter performance and thermal stability may be realized. See the supplementary material for an in-depth examination of the angular dependences derived from terahertz time domain spectroscopy, the findings of which were utilized in the construction of Fig. 3(b). This material presents results depicting the relationship between the intensity of terahertz generation and the polarization angle of the optical pump beam. Additionally, it elucidates the process behind calculating the detected X-component of the terahertz field. A comprehensive discussion regarding the 360° rotation of the magnetization, achieved via two jumps in fields marginally smaller than the critical field H A , is also included. The work was supported by the RSF (Grant No. 23-19-00849). Experimental work of A. Gorbatova was partly supported by RTU MIREA (Grant No. NICH-56). The samples of spintronic terahertz emitters were designed and fabricated using the facilities of the Center of Excellence "Center of Photonics" and funded by The Ministry of Science and Higher Education (Contract No. 075-15-2022-316).

AUTHOR DECLARATIONS Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Arseniy Buryakov: Conceptualization (equal); Writing -original draft (equal). Elena D. Mishina: Writing -review & editing (equal). Nicolas Tiercelin: Conceptualization (equal). Vladimir Preobrazhensky: Conceptualization (equal); Writing -review & edit-ing (equal). Anastasiya V. Gorbatova: Data curation (equal). Pavel Y. Avdeev: Methodology (equal). Ekaterina D. Lebedeva: Software (equal). Kirill A. Brekhov: Formal analysis (equal). Andrey V. Ovchinnikov: Data curation (equal). Nikita S. Gusev: Methodology (equal). Evgeny Karashtin: Formal analysis (equal). Maksim V. Sapozhnikov: Project administration (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.