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Wave scattering from a bounded medium with disorder
17 Octobor 1994 PHYSICS LETTERS A ELSEVIER
Physics Letters A 193 (1994) 467-470
Wave scattering from a bounded medium with disorder V. F r e i l i l d a e r , M . P u s t i l n i k , I. Y u r k e v i c h
TheJackandPearlResnickInstituteofAdvancedTechnology,DepartmentofPhysics,Bar-#anUniversity,Ramat-Gan52900,Israel Received 15 March 1994; accepted for publication 4 August 1994 Communicated by A. Lagendijk
Abstract We study theoretically the angular distribution of the intensity of waves reflected from a slab with surface or volume random
scatterers in the case when the thickness of the slab is small in comparison to the mean free path. We predict the existence of enhanced scattering peaks in some special directions caused by "degenerate time-reversal symmetry" in a bounded system with a discrete spectrum.
Much interest has been generated recently in multiple scattering of waves and quantum particles in media with volume and surface disorder [ 1,2 ]. This has been to a large extent due to the discovery of coherence effects in random media, such as backscattering enhancement, weak localization, memory effect etc. It is now well known than an enhanced backscattering peak arises as the result of the constructive interference of multiply scattered waves which occurs in spite of the random character ofeither volume [3] or surface [4] scattering, due to timereversal symmetry. In the case of half-infinite random media such a constructive interference can only occur in the direction opposite to to incident wave since only in this direction for each random path ABCD (Fig. l a) there is a time-reversed DCBA path which has exactly the same phase factor. In order that there be this equality of phases, the velocities of propagation from C to B and from B to C must be equal. In the case of a bounded system the situation is more complicated. If the refractive index n = x/~ inside the slab (0 < z < d) is larger than that outside (in the case of a Schr'6dinger equation this corresponds
A
D
A
A'
D'
\\\ ///
ii ~\ B
C
(a)
D
/
i?k,c
1/11/'/117/i)/// (b)
Fig. 1. Paths with the same phase factor:. (a) half-infinite medium; (b) bounded medium.
to a potential well) then the z-component kz of the wave vector k is quantized inside the slab. The field, corresponding to the discrete part of spectrum, is a sum of eigenmodes, each of them being a standing wave in the z-direction, propagating along the layer with its own (quantized) phase velocity vn-and k are the velocity and wavenumber in the infinite dielectric media, k. = ~ ).
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSD10375-9601 (94)00670-9
vk/kn (v
468
V. Freilikher et al. / Physics Letters A 193 (1994) 467-470
Distinct from infinite media, each ABCD trajectory (Fig. lb) is now N-fold degenerated (N is the number of discrete modes which is proportional to kod, ko being the wavenumber of the field propagating in vacuum), in the sense that on the segment BC there are N "channels" with different velocities. The phase difference for two channels (ABCD). and (A'CBD')m (they differ in the velocities on the BC segment) is equal, A(bnrn = ( k i n + k s c ) " r a t + Iracl (k. -kin) •
One can see that constructive interference (A0.,. = 0 ) can occur not only near the retroreflection angle (n=m, k~= -ki.) which is the case in infinite media, where k.=km=k, but also in other directions k~ # - ki., for which AO,,m= 0 for some n ¢- m. To calculate analytically the form of the peaks, let us consider a plane wave with wave vector k~. incident from the vacuum ( z > d ) on a scattering slab 0 < z < d . It was shown [5] that the intensity ld~f of the diffuse scattered field can be expressed through the Green function G(R, Ro) (R={r, z}, r = (x, y) are the coordinates along the slab) as follows, Idif(Psc,Pin) =4ao(P~)Ot0(Pin) ( [ G(p~,Pin)[2>dif,
(1) where ao(q) =x/~o2 - q 2, ko=k/n,p.~ and pi. are the projections on the (x, y)-plane of vectors k~ and kin respectively,
G(p,p')= Xexp(
'f
(2g) 2
drdr'
-ip.r+ip'.r')G(r,
O; r', O),
and ( > denotes ensemble < lGl2>dif= (IG]2> - (IGl >2.
averaging
(2) and
The function G (p,p' ) obeys (before averaging) the Dyson integralequation, which iswritten in the form
The Green function Go(p) has (due to the finiteness of the slab in the z-direction) a set of poles {p,}, corresponding to size quantization. In the case of weak scattering 1>> ( A p , ) - 1 (1 is the mean free path; Ap, is a characteristic distance between poles p,), the averaged Green function G(p) ( G(p)J(p-p')= ( G(p, p') > ) also has a well emphasized pole structure and can be represented in the form
G(p) = Gres(P) + Gsi,s(P), (4) where Gres(P) is a smooth related to the continuous part of the spectrum and Gsins(P) describes the behavior of G (p) in the vicinity of poles [ 6 ], y
(o
)
C,= O~p~Gfft(p) p=,. To allow the result to be physically clear without burdening it with undue details we will restrict our further consideration to the case of one-dimensional disorder with a random potential v(p, q) = v (Px- qx). When calculating the imaginary part of the mass operator lm Z" we assume, for simplicity, that v(p) is a Gaussian &correlated random process, which means that (v(p)v(p'))=W~(p+p'), W being independent ofp. Treating Was a small parameter, it is then possible to calculate lair upon a summarizing ladder and maximally crossed diagrams [ 6 ] using the pole approximation ( 5 ), which results in
laif(P~, Pin) = 40to(P~) cto(Pin) [G2(pse)G2(pi.) [ W( u M(Q) ~ -~. + 1-M(Q)]'
where
=Go(p)(J(p_p')+ f dqv(p,q)G(q,p')).
(3)
Here Go(p) is the Green function of the unperturbed problem; v(p, q) is a scattering potential. Expression (3) is a universal one, and does not depend on the nature of scatterers (volume or surface ), which affects the explicit form of the potential v only.
(5)
where _r(p) is the mass operator of Eq. (3) and the coefficients C, are defined from the unperturbed Green function Go (p) as follows,
X
G(p,p')
I
Gsins(P)= n=, ~ (p2-p2)/Cn-z~,(P)'
Q=Pin +Psc,
M(Q) = w j dp G(p+ ½Q)G*(p- ½Q)
w(~
= n --u-
a.
1 + (Q/u..)2
(6)
v. Freilikheret al. I PhysicsLettersA 193 (1994)467-470
:_7(l ia. + l l , ' , . ) - ' + V .~,. 1 + [ ( Q + p , , - p , . ) l . . , . ] 2 ] '
(7)
2
1.9
c = 1.026
~,, = - W J dp Im G.i.4(P'),
I.B
Wd = 0.032 Oin = 20.35 °
v~ = - W J dp Im G ~ ( p ' ) ,
1.7
~,= - I m Z = is. + y.,
koa=28.75
:g
a.=C./p..
1.6
The angular distribution of the scattered intensity ldif(P,c, Pin) exhibits peaks when the quantity M ( Q ) has its maxima. When IQI ~ v., I Q + p . - p m l ~ z,.m, the function M ( Q ) .~: 1 and Iaif is a smooth function of the scattering angles, determined by v/~v (ladder approximation). In the vicinity of the retroreflection direction (IQI ~ z,., I Q + p . - p , . I :*" ~.m) the main contribution to M comes from the first term in (7) which is equal to M ~ (roW/v) Y~a . = ~ v , / v and describes the well-known backscattered peak. When the direction of scattering occurs in the vicinity of one of the angles, determined by the equation I Q] = [p.-pml ( n ~ m ) , the first term in M ( Q ) can be neglected, whereas the second one contributes essentially and, being equal to rcWa.a,./~.m, prorides (in accordance with the qualitative consideration above) well-pronounced additional peaks of the scattered intensity. For example, in the case of an S-polarized wave scattered from a slab with a slightly rough interface z = d + ~ ( x ) , the potential v(p, q) can be represented in the form [7] v(p, q) = k ~ ( ¢ - 1 ) ~ ( p - # ) . For this case, numerical calculations were carried out to illustrate the predictions made above. The results are shown in Fig. 2 and demonstrate two additional wellpronounced peaks for the intensity of a wave scattered from a two-mode slab. It should be pointed out that the scattering features displayed in the second moments of reflected field are also expected to be observed in intensity-intensity correlations as well [ 8-10 ]. To demonstrate the fact that the features of the reflection predicted above from the bounded disordered media manifest themselves in intensity-intensity correlation functions, we consider the so-called C (~) function related simply to the square of the amplitude-amplitude correlation function,
1.5
vnm=½v(a.+a,.),
469
i : I
!i
....... I
I
-'21.5
-21
!.4 -22
. . . . . J=
_7\
I
I
20.5
20
I
-19.5
-19
0~, (o) Fig. 2. Angular distribution of intensity of an S-polarizedwave scattered from a dielectricslab with a rough surface.The angles of incidenceand scatteringare defined as 0t.(=)= arcsin(pi.(=)/ ~). C (~)(P,Po; P', P'o) = 6 ( p - p o - p ' + p ' o ) G ( p ) G * ( p ' )
XF(p, Po; P', p'o)G(po)G*(p'o) ,
(8)
where the function F(p, Po; P', Pb), being calculated in the same approximation a s ]dif can be expressed through the function M introduced above,
F(p, Po; P',P'o)
= w("
M(p-p')
-~ + l - M ( p - p ' ) q= ½(P+Po +P'+P'o)
M(q) + 1------~q)J' (9)
It is clear from (13) that the intensity-intensity correlation function has well-pronounced peaks when I p - p ' l ~ IP.-p,.[ and IP+Po+P'+P'ol 21p.-p,.I. It is important to stress that the peaks described above originate from the interference of multiply scattered random fields and do not coincide with the regular angular oscillations of the reflection coefficient from the unperturbed layer. To conclude, the size quantization affects drastically the angular distribution of the intensity and the angular intensity-intensity correlation of a wave reflected from a bounded scattering medium. The enhanced reflection occurs not only in the retroreflection direction, but also in some additional scattering angles, determined by the additional time-reversal
470
V. Freilikher et al. / Physics Letters A 193 (1994) 467-4 70
symmetry of the paths of the wave in the systems with a discrete spectrum. This work was supported by the Wolfson Foundation and the Ministry of Science and Technology of Israel.
References [ I ] M. Nieto-Vesperinas and J.C. Dainty, eds., Scattering in volumes and surface (North-Holland, Amsterdam, 1990). [2]C.M. Soukoulis, ed., Localization and propagation of classical waves in random and periodic structures (Plenum, New York, 1993).
[3] M.P. van Albada and L. Lagcndijk, Phys. Rev. Lett. 55 (1985) 2692. l 4 ] M.-J. Kim, J.C. Dainty, A.T. Friborg and A.J. Sant, J. Opt. Soc. Am. A 7 (1989) 569. [5] A.R. McGurn, A.A. Maradudin and V. C.elli, Phys. Rev. B 31 (1985) 4866. [6]V. Freilikher and I. Yurkevich, Phys. Lett. A 183 (1993) 247. [7]V.D. Freilikher and I.M. Fuks, Izv. Vissh. Uch. Zav. Radiofiz. 13 (1970) 98. [ 8 ] M. Nieto-Vesperinas and J.A. Sanchez-Gil, Phys. Rev. B 46 (1992) 3112. [9] A. Arsenieva and S. Feng, Phys. Rev. B 47 (1993) 13047. [ 10] V. Freilikher and I. Yurkevich, Phys. Lett. A 183 (1993) 253.
Physics Letters A 193 (1994) 467-470
Wave scattering from a bounded medium with disorder V. F r e i l i l d a e r , M . P u s t i l n i k , I. Y u r k e v i c h
TheJackandPearlResnickInstituteofAdvancedTechnology,DepartmentofPhysics,Bar-#anUniversity,Ramat-Gan52900,Israel Received 15 March 1994; accepted for publication 4 August 1994 Communicated by A. Lagendijk
Abstract We study theoretically the angular distribution of the intensity of waves reflected from a slab with surface or volume random
scatterers in the case when the thickness of the slab is small in comparison to the mean free path. We predict the existence of enhanced scattering peaks in some special directions caused by "degenerate time-reversal symmetry" in a bounded system with a discrete spectrum.
Much interest has been generated recently in multiple scattering of waves and quantum particles in media with volume and surface disorder [ 1,2 ]. This has been to a large extent due to the discovery of coherence effects in random media, such as backscattering enhancement, weak localization, memory effect etc. It is now well known than an enhanced backscattering peak arises as the result of the constructive interference of multiply scattered waves which occurs in spite of the random character ofeither volume [3] or surface [4] scattering, due to timereversal symmetry. In the case of half-infinite random media such a constructive interference can only occur in the direction opposite to to incident wave since only in this direction for each random path ABCD (Fig. l a) there is a time-reversed DCBA path which has exactly the same phase factor. In order that there be this equality of phases, the velocities of propagation from C to B and from B to C must be equal. In the case of a bounded system the situation is more complicated. If the refractive index n = x/~ inside the slab (0 < z < d) is larger than that outside (in the case of a Schr'6dinger equation this corresponds
A
D
A
A'
D'
\\\ ///
ii ~\ B
C
(a)
D
/
i?k,c
1/11/'/117/i)/// (b)
Fig. 1. Paths with the same phase factor:. (a) half-infinite medium; (b) bounded medium.
to a potential well) then the z-component kz of the wave vector k is quantized inside the slab. The field, corresponding to the discrete part of spectrum, is a sum of eigenmodes, each of them being a standing wave in the z-direction, propagating along the layer with its own (quantized) phase velocity vn-and k are the velocity and wavenumber in the infinite dielectric media, k. = ~ ).
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSD10375-9601 (94)00670-9
vk/kn (v
468
V. Freilikher et al. / Physics Letters A 193 (1994) 467-470
Distinct from infinite media, each ABCD trajectory (Fig. lb) is now N-fold degenerated (N is the number of discrete modes which is proportional to kod, ko being the wavenumber of the field propagating in vacuum), in the sense that on the segment BC there are N "channels" with different velocities. The phase difference for two channels (ABCD). and (A'CBD')m (they differ in the velocities on the BC segment) is equal, A(bnrn = ( k i n + k s c ) " r a t + Iracl (k. -kin) •
One can see that constructive interference (A0.,. = 0 ) can occur not only near the retroreflection angle (n=m, k~= -ki.) which is the case in infinite media, where k.=km=k, but also in other directions k~ # - ki., for which AO,,m= 0 for some n ¢- m. To calculate analytically the form of the peaks, let us consider a plane wave with wave vector k~. incident from the vacuum ( z > d ) on a scattering slab 0 < z < d . It was shown [5] that the intensity ld~f of the diffuse scattered field can be expressed through the Green function G(R, Ro) (R={r, z}, r = (x, y) are the coordinates along the slab) as follows, Idif(Psc,Pin) =4ao(P~)Ot0(Pin) ( [ G(p~,Pin)[2>dif,
(1) where ao(q) =x/~o2 - q 2, ko=k/n,p.~ and pi. are the projections on the (x, y)-plane of vectors k~ and kin respectively,
G(p,p')= Xexp(
'f
(2g) 2
drdr'
-ip.r+ip'.r')G(r,
O; r', O),
and ( > denotes ensemble < lGl2>dif= (IG]2> - (IGl >2.
averaging
(2) and
The function G (p,p' ) obeys (before averaging) the Dyson integralequation, which iswritten in the form
The Green function Go(p) has (due to the finiteness of the slab in the z-direction) a set of poles {p,}, corresponding to size quantization. In the case of weak scattering 1>> ( A p , ) - 1 (1 is the mean free path; Ap, is a characteristic distance between poles p,), the averaged Green function G(p) ( G(p)J(p-p')= ( G(p, p') > ) also has a well emphasized pole structure and can be represented in the form
G(p) = Gres(P) + Gsi,s(P), (4) where Gres(P) is a smooth related to the continuous part of the spectrum and Gsins(P) describes the behavior of G (p) in the vicinity of poles [ 6 ], y
(o
)
C,= O~p~Gfft(p) p=,. To allow the result to be physically clear without burdening it with undue details we will restrict our further consideration to the case of one-dimensional disorder with a random potential v(p, q) = v (Px- qx). When calculating the imaginary part of the mass operator lm Z" we assume, for simplicity, that v(p) is a Gaussian &correlated random process, which means that (v(p)v(p'))=W~(p+p'), W being independent ofp. Treating Was a small parameter, it is then possible to calculate lair upon a summarizing ladder and maximally crossed diagrams [ 6 ] using the pole approximation ( 5 ), which results in
laif(P~, Pin) = 40to(P~) cto(Pin) [G2(pse)G2(pi.) [ W( u M(Q) ~ -~. + 1-M(Q)]'
where
=Go(p)(J(p_p')+ f dqv(p,q)G(q,p')).
(3)
Here Go(p) is the Green function of the unperturbed problem; v(p, q) is a scattering potential. Expression (3) is a universal one, and does not depend on the nature of scatterers (volume or surface ), which affects the explicit form of the potential v only.
(5)
where _r(p) is the mass operator of Eq. (3) and the coefficients C, are defined from the unperturbed Green function Go (p) as follows,
X
G(p,p')
I
Gsins(P)= n=, ~ (p2-p2)/Cn-z~,(P)'
Q=Pin +Psc,
M(Q) = w j dp G(p+ ½Q)G*(p- ½Q)
w(~
= n --u-
a.
1 + (Q/u..)2
(6)
v. Freilikheret al. I PhysicsLettersA 193 (1994)467-470
:_7(l ia. + l l , ' , . ) - ' + V .~,. 1 + [ ( Q + p , , - p , . ) l . . , . ] 2 ] '
(7)
2
1.9
c = 1.026
~,, = - W J dp Im G.i.4(P'),
I.B
Wd = 0.032 Oin = 20.35 °
v~ = - W J dp Im G ~ ( p ' ) ,
1.7
~,= - I m Z = is. + y.,
koa=28.75
:g
a.=C./p..
1.6
The angular distribution of the scattered intensity ldif(P,c, Pin) exhibits peaks when the quantity M ( Q ) has its maxima. When IQI ~ v., I Q + p . - p m l ~ z,.m, the function M ( Q ) .~: 1 and Iaif is a smooth function of the scattering angles, determined by v/~v (ladder approximation). In the vicinity of the retroreflection direction (IQI ~ z,., I Q + p . - p , . I :*" ~.m) the main contribution to M comes from the first term in (7) which is equal to M ~ (roW/v) Y~a . = ~ v , / v and describes the well-known backscattered peak. When the direction of scattering occurs in the vicinity of one of the angles, determined by the equation I Q] = [p.-pml ( n ~ m ) , the first term in M ( Q ) can be neglected, whereas the second one contributes essentially and, being equal to rcWa.a,./~.m, prorides (in accordance with the qualitative consideration above) well-pronounced additional peaks of the scattered intensity. For example, in the case of an S-polarized wave scattered from a slab with a slightly rough interface z = d + ~ ( x ) , the potential v(p, q) can be represented in the form [7] v(p, q) = k ~ ( ¢ - 1 ) ~ ( p - # ) . For this case, numerical calculations were carried out to illustrate the predictions made above. The results are shown in Fig. 2 and demonstrate two additional wellpronounced peaks for the intensity of a wave scattered from a two-mode slab. It should be pointed out that the scattering features displayed in the second moments of reflected field are also expected to be observed in intensity-intensity correlations as well [ 8-10 ]. To demonstrate the fact that the features of the reflection predicted above from the bounded disordered media manifest themselves in intensity-intensity correlation functions, we consider the so-called C (~) function related simply to the square of the amplitude-amplitude correlation function,
1.5
vnm=½v(a.+a,.),
469
i : I
!i
....... I
I
-'21.5
-21
!.4 -22
. . . . . J=
_7\
I
I
20.5
20
I
-19.5
-19
0~, (o) Fig. 2. Angular distribution of intensity of an S-polarizedwave scattered from a dielectricslab with a rough surface.The angles of incidenceand scatteringare defined as 0t.(=)= arcsin(pi.(=)/ ~). C (~)(P,Po; P', P'o) = 6 ( p - p o - p ' + p ' o ) G ( p ) G * ( p ' )
XF(p, Po; P', p'o)G(po)G*(p'o) ,
(8)
where the function F(p, Po; P', Pb), being calculated in the same approximation a s ]dif can be expressed through the function M introduced above,
F(p, Po; P',P'o)
= w("
M(p-p')
-~ + l - M ( p - p ' ) q= ½(P+Po +P'+P'o)
M(q) + 1------~q)J' (9)
It is clear from (13) that the intensity-intensity correlation function has well-pronounced peaks when I p - p ' l ~ IP.-p,.[ and IP+Po+P'+P'ol 21p.-p,.I. It is important to stress that the peaks described above originate from the interference of multiply scattered random fields and do not coincide with the regular angular oscillations of the reflection coefficient from the unperturbed layer. To conclude, the size quantization affects drastically the angular distribution of the intensity and the angular intensity-intensity correlation of a wave reflected from a bounded scattering medium. The enhanced reflection occurs not only in the retroreflection direction, but also in some additional scattering angles, determined by the additional time-reversal
470
V. Freilikher et al. / Physics Letters A 193 (1994) 467-4 70
symmetry of the paths of the wave in the systems with a discrete spectrum. This work was supported by the Wolfson Foundation and the Ministry of Science and Technology of Israel.
References [ I ] M. Nieto-Vesperinas and J.C. Dainty, eds., Scattering in volumes and surface (North-Holland, Amsterdam, 1990). [2]C.M. Soukoulis, ed., Localization and propagation of classical waves in random and periodic structures (Plenum, New York, 1993).
[3] M.P. van Albada and L. Lagcndijk, Phys. Rev. Lett. 55 (1985) 2692. l 4 ] M.-J. Kim, J.C. Dainty, A.T. Friborg and A.J. Sant, J. Opt. Soc. Am. A 7 (1989) 569. [5] A.R. McGurn, A.A. Maradudin and V. C.elli, Phys. Rev. B 31 (1985) 4866. [6]V. Freilikher and I. Yurkevich, Phys. Lett. A 183 (1993) 247. [7]V.D. Freilikher and I.M. Fuks, Izv. Vissh. Uch. Zav. Radiofiz. 13 (1970) 98. [ 8 ] M. Nieto-Vesperinas and J.A. Sanchez-Gil, Phys. Rev. B 46 (1992) 3112. [9] A. Arsenieva and S. Feng, Phys. Rev. B 47 (1993) 13047. [ 10] V. Freilikher and I. Yurkevich, Phys. Lett. A 183 (1993) 253.