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Electromagnetic scattering by large rotating particles in the eikonal formalism
Volume 70, number 3
OPTICS COMMUNICATIONS
ELECTROMAGNETIC SCATTERING IN THE EIKONAL FORMALISM
1 March 1989
BY LARGE ROTATING PARTICLES
C. BOURRELY, P. CHIAPPETTA and T. LEMAIRE Centre de Physique Thkorique 2, CNRS - Luminy, Case 907, F-13288 Marseille Cedex 09, France
Received 12 September 1988
We extend the eikonal formalism to the case of large scattering objects whose axis of symmetry is not parallel to the direction of the incident wave vector.
1. Introduction
A theoretical description of electromagnetic scattering by particles of arbitrary shape, although in principle exactly solvable from the Helmoltz equation, cannot be computed in practice for rough surfaces since an exact solution requires the knowledge of boundary conditions at the scattering surface [ 11. For rough particles whose dimensions are greater than the incident wavelength, a modelisation of the diffracted amplitude, valid in the whole angular domain for scattering potentials with compact support, has been obtained [2] in the framework of the eikonal formalism [ 31. Recently [ 41, an expression for the diffracted amplitude taking exactly into account the shape of the particle has been derived for scattering potentials having an axis of symmetry parallel to the direction of the incident wave vector k. The purpose of this paper is to remove this last assumption. The main motivation for such a study is the measurement [ 5 ] of the intensity backscattered by targets whose surface section is made of quarter circles of radius 2 cm having n= 1.89 + iO.01 as a complex index of refraction (see fig. 1). Experimental radar backscattering cross sections versus the orientation angle f3,(angle between the axis of symmetry of the object and the direction of the ‘) Allocataire MRES. 2, Laboratoire Propre LP.706 1, Centre National de la Recherche Scientifique.
Fig. 1. Global three dimensional shape of the scattering object.
incident wave vector) were recorded for horizontal and vertical polarizations at two frequencies IJ,= 35 GHz and v2= 94 GHz. Although polarization effects are not negligible, since our scalar solution of the wave equation does not take into account these effects, as a first approximation we will make a qualitative comparison.
2. Generalization of the eikonal picture Let us first describe the eikonal picture used in the calculations. We consider the scattering of an electromagnetic wave of wavelength I and wavevector k
0 030-4018/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
173
Volume 70, number
3
OPTICS
by an object having an axis of symmetry which does not coi’ncide with the direction of k (along the zaxis). The angle between the two vectors is denoted as 0, (see fig. 1). We want to solve the Helmholtz equation [d+n2(r)k2]
ty(r)=O,
I March
COMMUNICATIONS
(1)
where n(r) is the complex index of refraction. Eq. ( 1) has the general solution
(where @(r)=exp(-iksr) y(r)) varies slowly on distances less than A. G3: for a point r within Sz, the product V(r) @(r) varies appreciably only on a distance much larger than ;1. We get in spherical coordinates with the z axis parallel to k: @(r)=l-
ETdr”
id@’
v(r) =exp(ik*r)
@(r-r”)
(2) where B is the volume inside which 1 - n2 (r) # 0. For T-+M the scattering amplitude becomes
[l-n’(r-r”)],
0
0
-~~exp~~!~~,r”) [1-n2(y’)]~(r’)dr’, R
1989
Ir.,lik,
(4)
where r” = r - r’ . Assuming that V(r) @(r) varies slowly as a function of p”, we get the expression of@(r) within Sz: @(x,Y,z)=exp(-ik/2)
j
dz’ [l-n’(x,y,z’)].
-co
f(k,, k)= - g
x[l-n2(r’)l
(5)
J‘ d3r’ exp( -ik;r’) R
v(f),
(3)
where k,= kr/r is the scattered wavevector. Such a solution requires the knowledge of w( r) inside 52. Using the following Glauber assumptions: Gl:ifaisthemeanrangeof V(r)=l-n2(r),we impose ka 2 1. G2: for a point r within L?, the product V(r) $(r)
Fig. 2. Intensity backscattered by an ellipsoid of minor axis/J= cm as a function of the orientation angle 0,. 174
Putting r=b+ze, where b is the impact parameter and 8 the angle between k, and k, we get f(b,k)=-sjdplbdbdz
[I-n2(b,z)] R
xexp(
-ik/2)
1 dz’ [ 1 -n2(b,
z’)]
.
(6)
--cc
8 cm, major axis LY= 17.6 cm, refractive
index n= 1.89+i10S2
forl=0.32
1 March 1989
OPTICS COMMUNICATIONS
Volume 70, number 3
0
40
20
60
60
60
60
rD@Wa@s 6i (degrees)
0
40
20
rotatiOn qls
ai (degrees)
Fig. 3. (a) Intensity backscattered by the object shown in fig. 1 of refractive index n = 1.89 + i 10-l for I= 0.86 cm as a function of the orientation angle 0,. (b) Same characteristics as in (a) for 1=0.32 cm.
Of course the volume of integration Q depends on 0,. Let us first perform the integration over the z variable, b and p being fixed to b. and vo. If S(z) is the expression of the boundary at the surface of the scatterer, we introduce the intersecting points zj( boypo) between the line b = b. at Q= p. and S(z) . We get f(&, k) = - g j$, 7 d yl 0
“?”
b db “‘r
bidco)
‘) dz
.a-l(b,
(P)
Xexp(-ikbsin8cos8+2ikzsin28/2)[l-n2(8,z)] Xexp( -ik/2)
j dz’ [ 1 -n2(6,z’)], z2,-L(b,p)
(7)
where n2(b,q, z)= 1 for ZER\U,N~[Z~~_~,Z~~]. We define the scattered intensity as 1=k21f(S,
k) 12.
(8)
3. Numerical results We will now give numerical results for the intensity backscattered by an ellipsdid with an index of refraction n = 1.89 2 iO.01 illuminated by an electromagnetic wave of wavelength A= 0.32 cm whose major axis makes an angle ei with the direction of the incident wave. The minor axis /3 is equal to 8 cm and the major axis a~2.2 8. 175
As shown in fig. 2, the scattered intensity increases when the surface of the scatterer seen by the incident wave grows. The intensity backscattered by the object depicted in fig. 1 is shown in figs. 3a and 3b for two incident wavelengths h~O.32 cm and A~0.86 cm. The comparison to experimental preliminary data can only be qualitative since we do not have up to now a method to include the polarization. Nevertheless the number of oscillations and the location of minima is consistent with data for vertical polarization which should slightly differ from our scalar solution according to ref. [ 6 1. We see on figs. 3a and 3b that there are all the more oscillations as the wavelength is small, since the shorter the wavelength is, the finer is the detailed structure of the object observed. Each discontinuity of the derivative of the surface between two quarter circles manifests itself by a maximum of backscattered intensity. A comparison with a previous calculation [ 71 shows that the diffracted intensity detected around the object exhibits oscillations of larger magnitude than in the present case.
176
4. Conclusion To conclude, we have extended the eikonal formalism to the case of scattering objects whose axis of symmetry is not parallel to the direction of the incident wavevector. We hope that the predictions we can make in the whole angular domain will be soon experimentally tested in order to check the validity of this extended version of the eikonal formalism.
References [ 1 ] 3. Stratton, Electromagnetic theory (McGraw Hill, New York, London 1941). [2] J.M. Pert-in and P. Chiappetta, Optica Acta 32 (1985) 907. [ 31 R.J. Glauber, Lectures in Theoretical Physics, Vol. 1, eds. W.E. Brittin and L.C. Dunham (Interscience, New York, 1958). [ 41 C. Bourrely, P. Chiappetta and B. Torresani, J. Opt. Sot. Am. A3 (1986) 250. [ 5] R. Deleuil, Optica Acta 16 ( 1969) 23; C. Bourrely, P. Chiappetta, R. Deleuil and B. Torresani, Preprint CPT-88/p.2086. [6] J.M. Perrin and P.L. Lamy, Optica Acta 33 (1986) 1001. [ 71 C. Bourrely, P. Chiappetta and B. Torresani, Optics Comm. 58 (1986) 365.
OPTICS COMMUNICATIONS
ELECTROMAGNETIC SCATTERING IN THE EIKONAL FORMALISM
1 March 1989
BY LARGE ROTATING PARTICLES
C. BOURRELY, P. CHIAPPETTA and T. LEMAIRE Centre de Physique Thkorique 2, CNRS - Luminy, Case 907, F-13288 Marseille Cedex 09, France
Received 12 September 1988
We extend the eikonal formalism to the case of large scattering objects whose axis of symmetry is not parallel to the direction of the incident wave vector.
1. Introduction
A theoretical description of electromagnetic scattering by particles of arbitrary shape, although in principle exactly solvable from the Helmoltz equation, cannot be computed in practice for rough surfaces since an exact solution requires the knowledge of boundary conditions at the scattering surface [ 11. For rough particles whose dimensions are greater than the incident wavelength, a modelisation of the diffracted amplitude, valid in the whole angular domain for scattering potentials with compact support, has been obtained [2] in the framework of the eikonal formalism [ 31. Recently [ 41, an expression for the diffracted amplitude taking exactly into account the shape of the particle has been derived for scattering potentials having an axis of symmetry parallel to the direction of the incident wave vector k. The purpose of this paper is to remove this last assumption. The main motivation for such a study is the measurement [ 5 ] of the intensity backscattered by targets whose surface section is made of quarter circles of radius 2 cm having n= 1.89 + iO.01 as a complex index of refraction (see fig. 1). Experimental radar backscattering cross sections versus the orientation angle f3,(angle between the axis of symmetry of the object and the direction of the ‘) Allocataire MRES. 2, Laboratoire Propre LP.706 1, Centre National de la Recherche Scientifique.
Fig. 1. Global three dimensional shape of the scattering object.
incident wave vector) were recorded for horizontal and vertical polarizations at two frequencies IJ,= 35 GHz and v2= 94 GHz. Although polarization effects are not negligible, since our scalar solution of the wave equation does not take into account these effects, as a first approximation we will make a qualitative comparison.
2. Generalization of the eikonal picture Let us first describe the eikonal picture used in the calculations. We consider the scattering of an electromagnetic wave of wavelength I and wavevector k
0 030-4018/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
173
Volume 70, number
3
OPTICS
by an object having an axis of symmetry which does not coi’ncide with the direction of k (along the zaxis). The angle between the two vectors is denoted as 0, (see fig. 1). We want to solve the Helmholtz equation [d+n2(r)k2]
ty(r)=O,
I March
COMMUNICATIONS
(1)
where n(r) is the complex index of refraction. Eq. ( 1) has the general solution
(where @(r)=exp(-iksr) y(r)) varies slowly on distances less than A. G3: for a point r within Sz, the product V(r) @(r) varies appreciably only on a distance much larger than ;1. We get in spherical coordinates with the z axis parallel to k: @(r)=l-
ETdr”
id@’
v(r) =exp(ik*r)
@(r-r”)
(2) where B is the volume inside which 1 - n2 (r) # 0. For T-+M the scattering amplitude becomes
[l-n’(r-r”)],
0
0
-~~exp~~!~~,r”) [1-n2(y’)]~(r’)dr’, R
1989
Ir.,lik,
(4)
where r” = r - r’ . Assuming that V(r) @(r) varies slowly as a function of p”, we get the expression of@(r) within Sz: @(x,Y,z)=exp(-ik/2)
j
dz’ [l-n’(x,y,z’)].
-co
f(k,, k)= - g
x[l-n2(r’)l
(5)
J‘ d3r’ exp( -ik;r’) R
v(f),
(3)
where k,= kr/r is the scattered wavevector. Such a solution requires the knowledge of w( r) inside 52. Using the following Glauber assumptions: Gl:ifaisthemeanrangeof V(r)=l-n2(r),we impose ka 2 1. G2: for a point r within L?, the product V(r) $(r)
Fig. 2. Intensity backscattered by an ellipsoid of minor axis/J= cm as a function of the orientation angle 0,. 174
Putting r=b+ze, where b is the impact parameter and 8 the angle between k, and k, we get f(b,k)=-sjdplbdbdz
[I-n2(b,z)] R
xexp(
-ik/2)
1 dz’ [ 1 -n2(b,
z’)]
.
(6)
--cc
8 cm, major axis LY= 17.6 cm, refractive
index n= 1.89+i10S2
forl=0.32
1 March 1989
OPTICS COMMUNICATIONS
Volume 70, number 3
0
40
20
60
60
60
60
rD@Wa@s 6i (degrees)
0
40
20
rotatiOn qls
ai (degrees)
Fig. 3. (a) Intensity backscattered by the object shown in fig. 1 of refractive index n = 1.89 + i 10-l for I= 0.86 cm as a function of the orientation angle 0,. (b) Same characteristics as in (a) for 1=0.32 cm.
Of course the volume of integration Q depends on 0,. Let us first perform the integration over the z variable, b and p being fixed to b. and vo. If S(z) is the expression of the boundary at the surface of the scatterer, we introduce the intersecting points zj( boypo) between the line b = b. at Q= p. and S(z) . We get f(&, k) = - g j$, 7 d yl 0
“?”
b db “‘r
bidco)
‘) dz
.a-l(b,
(P)
Xexp(-ikbsin8cos8+2ikzsin28/2)[l-n2(8,z)] Xexp( -ik/2)
j dz’ [ 1 -n2(6,z’)], z2,-L(b,p)
(7)
where n2(b,q, z)= 1 for ZER\U,N~[Z~~_~,Z~~]. We define the scattered intensity as 1=k21f(S,
k) 12.
(8)
3. Numerical results We will now give numerical results for the intensity backscattered by an ellipsdid with an index of refraction n = 1.89 2 iO.01 illuminated by an electromagnetic wave of wavelength A= 0.32 cm whose major axis makes an angle ei with the direction of the incident wave. The minor axis /3 is equal to 8 cm and the major axis a~2.2 8. 175
As shown in fig. 2, the scattered intensity increases when the surface of the scatterer seen by the incident wave grows. The intensity backscattered by the object depicted in fig. 1 is shown in figs. 3a and 3b for two incident wavelengths h~O.32 cm and A~0.86 cm. The comparison to experimental preliminary data can only be qualitative since we do not have up to now a method to include the polarization. Nevertheless the number of oscillations and the location of minima is consistent with data for vertical polarization which should slightly differ from our scalar solution according to ref. [ 6 1. We see on figs. 3a and 3b that there are all the more oscillations as the wavelength is small, since the shorter the wavelength is, the finer is the detailed structure of the object observed. Each discontinuity of the derivative of the surface between two quarter circles manifests itself by a maximum of backscattered intensity. A comparison with a previous calculation [ 71 shows that the diffracted intensity detected around the object exhibits oscillations of larger magnitude than in the present case.
176
4. Conclusion To conclude, we have extended the eikonal formalism to the case of scattering objects whose axis of symmetry is not parallel to the direction of the incident wavevector. We hope that the predictions we can make in the whole angular domain will be soon experimentally tested in order to check the validity of this extended version of the eikonal formalism.
References [ 1 ] 3. Stratton, Electromagnetic theory (McGraw Hill, New York, London 1941). [2] J.M. Pert-in and P. Chiappetta, Optica Acta 32 (1985) 907. [ 31 R.J. Glauber, Lectures in Theoretical Physics, Vol. 1, eds. W.E. Brittin and L.C. Dunham (Interscience, New York, 1958). [ 41 C. Bourrely, P. Chiappetta and B. Torresani, J. Opt. Sot. Am. A3 (1986) 250. [ 5] R. Deleuil, Optica Acta 16 ( 1969) 23; C. Bourrely, P. Chiappetta, R. Deleuil and B. Torresani, Preprint CPT-88/p.2086. [6] J.M. Perrin and P.L. Lamy, Optica Acta 33 (1986) 1001. [ 71 C. Bourrely, P. Chiappetta and B. Torresani, Optics Comm. 58 (1986) 365.