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The scattering potential for imaging in the reflection geometry
cm
15 May 1995
. __ .__
OPTICS COMMUNICATIONS
Rf! Optics Communications 117 (1995) 16-19
The scattering potential for imaging in the reflection geometry C.J.R. Sheppard, T.J. Connolly, Min Gu Physical Optics Department,
School of Physics, University of Sydney, Sydney, NSW 2006, Australia
Received 30 May 1994; revised version received 28 September 1994
Abstract Using a generalization of results for scattering by stratified media and by rough surfaces, a new form of the scattering potential, for three-dimensional imaging in the reflection geometry, is proposed.
1. Introduction The concept of the scattering potential, based first Born approximation, for reconstruction of dimensional (3D) refractive index variations holographic data was introduced by Wolf [ 1] form V(r) = -P(n2(t)
-n;(r))
on the threefrom in the
(1)
where n is the (complex) refractive index of the medium and k = 2rrf/h, with h the wavelength in a medium no. This approach has been widely used in a range of disciplines for tomographic reconstruction within the diffraction regime, with acoustic radiation in medical [ 21, non-destructive [ 31, and geophysical [ 41 diagnostics, as well as for optical characterization of density fluctuations in a plasma [ 51. It has also been used to describe 3D image formation in optical instruments, which may be coherent [ 1] or partially coherent [S-10] in nature. The object spectrum is multiplied by an appropriate transfer function, and inverse Fourier transformed to give the image amplitude. This linearized model can then also be used for to solve the inverse problem of reconstructing the object refractive index variation. For imaging. in the reflection, rather than the transmission mode, the transfer function has been investi0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZOO30-4018(95)00107-7
gated for conventional [ 91 as well as confocal imaging [ 1 l-141, but in this case the scattering angles are necessarily large so that the form of the scattering potential needs to be reexamined. For example, the scattering potential in Eq. ( 1) predicts an infinite reflection from a semi-infinite solid, which we know cannot be the case in practice. In fact reflections originate from changes in refractive index, not the absolute value relative to the environment. Recently we have explored two particular forms of object in reflection imaging: stratified media of continuously varying refractive index [ 151, and rough surfaces on a perfect conductor [ 16,171. The aim of the present work is to generalize these results.
2. The stratified medium
For the case of a stratified medium, the angular reflectivity can be derived from Maxwell’s equations [ 151, or alternatively the medium can be regarded as a succession of incremental slabs, the reflectivity of which follow directly from the Fresnel coefficients. Thus for parallel and perpendicular polarizations, respectively, we have
C.J.R. Sheppard et al. /Optics
rll
=
%Cl
-n1c2
n2c1
+n1c2
,
rI=
n2c2
-n1c1
n2c1
+n1cz
9
(2)
so we can write simply Ar,,(c) = $Aln(nlc),
Ar,(c)
= fAln(nc)
,
(4)
We see that the reflectivity is independent of c, i.e. it is a property only of the medium. This conclusion relies on our assumption of axial symmetry. Assuming that the illumination does not deplete in travelling through the medium, neglecting multiple reflections, and also neglecting the depth distortions caused by the refractive index variations (these are the usual assumptions of the first Born approximation), we can integrate over the increments of reflectivity to give for the total reflectivity
r(c) =
+-1
d 2 G (In n) exp( - 2ikzn,c)
I -cc
dz ,
(5)
where the limits can be taken as +m as n can be assumed constant and equal to izOfor z < 0. Now the reflectivity is related to the object spectrum S(s), the 3D Fourier transform of the scattering potential, by [ 111 s S(s) = - r(s) , 2
117 (1995) 16-19
17
This result is similar to those presented several times previously for the 1D inverse scattering problem [ 20241. Although we have taken the stratified medium to have an axis in the z direction, this is completely arbitrary. So Eq. (7) can be generalized to give
(3)
where c = cos 13and 19is the angle subtended at the optic axis. For an axially symmetric system we must integrate the amplitude around the pupil, which has the effect of giving the resultant reflection coefficient for plane polarized illumination as the average of the reflectivities for parallel and perpendicular polarizations [ 181 Ar(c) = $Aln rz .
Communications
(6)
V(r) =
&
V2( In n(r) )
.
0
This is our proposed form for the 3D scattering potential for the reflection geometry, giving an associated object spectrum S(p) = %p2
F(ln n(r))
,
(9)
wherep is the normalized spatial frequency vector, and F denotes the Fourier transform operator. In an observation using monochromatic radiation in the far-field we can only observe spatial frequencies for inside the sphere Ip 12< 4 resulting from the Ewald sphere construction.
3. The rough surface In the second case we have investigated recently, that of a rough surface on a perfect conductor, assuming the Kirchhoff approximation and a scalar theory, the scattering can be described by the object spectrum very simply in terms of 3D spatial frequencies in the form [ 17,25,26] S@) = $
S(x, Y)) ,
F(a(z-
(10)
where &x, y) is the surface profile. This expression again has resulted from assuming a particular axis for the surface, but can be written as m
where s is the axial spatial frequency, normalized by k. Eq. (6) describes the well-known property first described by Darwin [ 191 that a plane of atoms (which can be represented by a Mimction) does not reflect radiation independently of angle (i.e. as the Fourier transform of the scattering potential), but as l/cos 8. We thus obtain for the scattering potential V(r)
= - i
-d2 (In n(r)) 4kno dz2
.
s(p) =
2p2F
qz-
d-(x,Y>> dz
2
(11)
-cc
in which the integral can be recognized as the bulk of the scattering material (Fig. 1) . This represents a dipole layer distributed on the surface (Ref. [ 271 p. 5). If the material is not a perfect conductor, but nevertheless the reflectivity is assumed independent of angle of incidence, Eq. ( 11) is still valid but with an appropriate reflection coefficient. Consider a rough surface
C.J.R. Sheppard et al. /Optics
18
Communications
117 (1995) 16-19
medium the model is based on a vector theory including polarization effects but assuming axial symmetry. As the inclination of the normal to the medium is increased this assumption becomes less valid. There is no fundamental distinction between transmission and reflection imaging, but for transmission the assumption of axial symmetry breaks down completely.
Acknowledgements The authors acknowledge support from the Australian Research Council, and the Science Foundation for Physics within the University of Sydney.
References Fig. 1. For a rough surface, (a) shows the surface profile, whereas (b) shows the bulk object function.
on a medium of refractive index n,. Then the logarithm of the refractive index variation can be written ln(n(r))
=ln no 2.
+ln(n,lno) I --m
&z-5(x,
(12)
Y)) dz 9
or assuming Eq. (9) is valid
S(z--S(&
Y>> dz
(13)
--m
as the constant term in Eq. (12) disappears upon differentiation. It is seen that this is identical to Eq. ( 11) apart from a constant factor iln(n,/nO), which is an appropriate form of the amplitude reflection coefficient from a dielectric interface, valid if n is not too large
i2f.31.
4. Discussion Thus Eq. ( 11) gives the object spectrum and Eq. (8) the scattering potential, for two different object models. For the case of a rough surface the model is based on a scalar assumption. For the case of a stratified
[I] E. Wolf, Optics Comm. 1 (1969) 153. [ 21 R.F. Mueller, M. Kaveh and G. Wade, Proc. IEEE 67 ( 1979) 567. [3] G.S. Kino, Proc. IEEE 67 (1979) 510. [4] A.J. Devaney, IEEE Trans. Geoscience Remote Sensing 22 (1984) 3. [5] .I. Howard, J. Opt. Sot. Am. A 8 (1991) 1955. [6] N. Streibl, J. Opt. Sot. Am. A 2 (1985) 121. [7] C.J.R. Sheppard, Optik 72 (1986) 131. [8] C.J.R. Sheppard, Optik 74 (1986) 128. [9] C.J.R. Sheppard and X.Q. Mao, J. Opt. Sot. Am. A 6 (1989) 1260. [lo] C.J.R. Sheppard, European J. Cell Biol. Suppl. 25 48 (1989) 29. [ 1 l] C.J.R. Sheppard and C.J. Cogswell, J. Microscopy 159 (1990) 179. [ 121 C.J.R. Sheppard and Min Gu, J. Microsc. 165 (1991) 377. [13] C.J.R. Sheppard, Min Gu and X.Q. Mao, Optics Comm. 81 (1991) 281. [ 141 C.J.R. Sheppard, Mm Gu, Y. Kawata and S. Kawata, J. Opt. Sot. Am. A 11 ( 1994) 593. [ 151 C.J.R. Sheppard, T.J. Connolly, J. LeeandC.J. Cogswell,Appl. Optics 33 (1994) 631. [ 161 C.J.R. Sheppard, T.J. Connolly and Min Gu, Phys. Rev. Lett. 70 (1993) 1409. [ 171 C.J.R. Sheppard, T.J. Connolly and Min Gu, J. Mod. Optics40 (1993) 2407. [ 181 C.J.R. Sheppard and Min Gu, Optics Comm. 87 ( 1992) 180. [19] C.G. Darwin,Phil. Mag.27 (1914) 315,645. [20] LM. Gel’fand and B.M. Levitan, Am. Math. Sot. Trans. Ser. 2 (1955) 253. [21] C.H. Greenewalt, W. Brandt and D.D. Friel, J. Opt. Sot. Am. 50 (1960) 1005. [22] L.M. Brekhovsl&h, Waves in Layered Media (Academic, New York, 1960).
C.J.R. Sheppard et al. /Optics Communications 117 (1995) 16-19 [23] E. Vogelzang, H.A. Ferwerda and D. Yevick, Optica Acta 31 (1984) 541. [24] L. Sossi and P. Kard, On the theory of the reflection and transmission of light by a thin inhomogeneous dielectric film, National Research Council of Canada Tech. Trans. 1905 (1968).
19
1251 P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Ma, 1987). [26] R.J. Wombell and J.A. DeSanto, J. Opt. Sot. Am. A 8 (1991) 1892. [27] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory (Springer, Berlin, 1992). [28] C.J.R. Sheppard and T.J. Connolly, Imaging of random surfaces, J. Mod. Optics, to be published.
15 May 1995
. __ .__
OPTICS COMMUNICATIONS
Rf! Optics Communications 117 (1995) 16-19
The scattering potential for imaging in the reflection geometry C.J.R. Sheppard, T.J. Connolly, Min Gu Physical Optics Department,
School of Physics, University of Sydney, Sydney, NSW 2006, Australia
Received 30 May 1994; revised version received 28 September 1994
Abstract Using a generalization of results for scattering by stratified media and by rough surfaces, a new form of the scattering potential, for three-dimensional imaging in the reflection geometry, is proposed.
1. Introduction The concept of the scattering potential, based first Born approximation, for reconstruction of dimensional (3D) refractive index variations holographic data was introduced by Wolf [ 1] form V(r) = -P(n2(t)
-n;(r))
on the threefrom in the
(1)
where n is the (complex) refractive index of the medium and k = 2rrf/h, with h the wavelength in a medium no. This approach has been widely used in a range of disciplines for tomographic reconstruction within the diffraction regime, with acoustic radiation in medical [ 21, non-destructive [ 31, and geophysical [ 41 diagnostics, as well as for optical characterization of density fluctuations in a plasma [ 51. It has also been used to describe 3D image formation in optical instruments, which may be coherent [ 1] or partially coherent [S-10] in nature. The object spectrum is multiplied by an appropriate transfer function, and inverse Fourier transformed to give the image amplitude. This linearized model can then also be used for to solve the inverse problem of reconstructing the object refractive index variation. For imaging. in the reflection, rather than the transmission mode, the transfer function has been investi0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZOO30-4018(95)00107-7
gated for conventional [ 91 as well as confocal imaging [ 1 l-141, but in this case the scattering angles are necessarily large so that the form of the scattering potential needs to be reexamined. For example, the scattering potential in Eq. ( 1) predicts an infinite reflection from a semi-infinite solid, which we know cannot be the case in practice. In fact reflections originate from changes in refractive index, not the absolute value relative to the environment. Recently we have explored two particular forms of object in reflection imaging: stratified media of continuously varying refractive index [ 151, and rough surfaces on a perfect conductor [ 16,171. The aim of the present work is to generalize these results.
2. The stratified medium
For the case of a stratified medium, the angular reflectivity can be derived from Maxwell’s equations [ 151, or alternatively the medium can be regarded as a succession of incremental slabs, the reflectivity of which follow directly from the Fresnel coefficients. Thus for parallel and perpendicular polarizations, respectively, we have
C.J.R. Sheppard et al. /Optics
rll
=
%Cl
-n1c2
n2c1
+n1c2
,
rI=
n2c2
-n1c1
n2c1
+n1cz
9
(2)
so we can write simply Ar,,(c) = $Aln(nlc),
Ar,(c)
= fAln(nc)
,
(4)
We see that the reflectivity is independent of c, i.e. it is a property only of the medium. This conclusion relies on our assumption of axial symmetry. Assuming that the illumination does not deplete in travelling through the medium, neglecting multiple reflections, and also neglecting the depth distortions caused by the refractive index variations (these are the usual assumptions of the first Born approximation), we can integrate over the increments of reflectivity to give for the total reflectivity
r(c) =
+-1
d 2 G (In n) exp( - 2ikzn,c)
I -cc
dz ,
(5)
where the limits can be taken as +m as n can be assumed constant and equal to izOfor z < 0. Now the reflectivity is related to the object spectrum S(s), the 3D Fourier transform of the scattering potential, by [ 111 s S(s) = - r(s) , 2
117 (1995) 16-19
17
This result is similar to those presented several times previously for the 1D inverse scattering problem [ 20241. Although we have taken the stratified medium to have an axis in the z direction, this is completely arbitrary. So Eq. (7) can be generalized to give
(3)
where c = cos 13and 19is the angle subtended at the optic axis. For an axially symmetric system we must integrate the amplitude around the pupil, which has the effect of giving the resultant reflection coefficient for plane polarized illumination as the average of the reflectivities for parallel and perpendicular polarizations [ 181 Ar(c) = $Aln rz .
Communications
(6)
V(r) =
&
V2( In n(r) )
.
0
This is our proposed form for the 3D scattering potential for the reflection geometry, giving an associated object spectrum S(p) = %p2
F(ln n(r))
,
(9)
wherep is the normalized spatial frequency vector, and F denotes the Fourier transform operator. In an observation using monochromatic radiation in the far-field we can only observe spatial frequencies for inside the sphere Ip 12< 4 resulting from the Ewald sphere construction.
3. The rough surface In the second case we have investigated recently, that of a rough surface on a perfect conductor, assuming the Kirchhoff approximation and a scalar theory, the scattering can be described by the object spectrum very simply in terms of 3D spatial frequencies in the form [ 17,25,26] S@) = $
S(x, Y)) ,
F(a(z-
(10)
where &x, y) is the surface profile. This expression again has resulted from assuming a particular axis for the surface, but can be written as m
where s is the axial spatial frequency, normalized by k. Eq. (6) describes the well-known property first described by Darwin [ 191 that a plane of atoms (which can be represented by a Mimction) does not reflect radiation independently of angle (i.e. as the Fourier transform of the scattering potential), but as l/cos 8. We thus obtain for the scattering potential V(r)
= - i
-d2 (In n(r)) 4kno dz2
.
s(p) =
2p2F
qz-
d-(x,Y>> dz
2
(11)
-cc
in which the integral can be recognized as the bulk of the scattering material (Fig. 1) . This represents a dipole layer distributed on the surface (Ref. [ 271 p. 5). If the material is not a perfect conductor, but nevertheless the reflectivity is assumed independent of angle of incidence, Eq. ( 11) is still valid but with an appropriate reflection coefficient. Consider a rough surface
C.J.R. Sheppard et al. /Optics
18
Communications
117 (1995) 16-19
medium the model is based on a vector theory including polarization effects but assuming axial symmetry. As the inclination of the normal to the medium is increased this assumption becomes less valid. There is no fundamental distinction between transmission and reflection imaging, but for transmission the assumption of axial symmetry breaks down completely.
Acknowledgements The authors acknowledge support from the Australian Research Council, and the Science Foundation for Physics within the University of Sydney.
References Fig. 1. For a rough surface, (a) shows the surface profile, whereas (b) shows the bulk object function.
on a medium of refractive index n,. Then the logarithm of the refractive index variation can be written ln(n(r))
=ln no 2.
+ln(n,lno) I --m
&z-5(x,
(12)
Y)) dz 9
or assuming Eq. (9) is valid
S(z--S(&
Y>> dz
(13)
--m
as the constant term in Eq. (12) disappears upon differentiation. It is seen that this is identical to Eq. ( 11) apart from a constant factor iln(n,/nO), which is an appropriate form of the amplitude reflection coefficient from a dielectric interface, valid if n is not too large
i2f.31.
4. Discussion Thus Eq. ( 11) gives the object spectrum and Eq. (8) the scattering potential, for two different object models. For the case of a rough surface the model is based on a scalar assumption. For the case of a stratified
[I] E. Wolf, Optics Comm. 1 (1969) 153. [ 21 R.F. Mueller, M. Kaveh and G. Wade, Proc. IEEE 67 ( 1979) 567. [3] G.S. Kino, Proc. IEEE 67 (1979) 510. [4] A.J. Devaney, IEEE Trans. Geoscience Remote Sensing 22 (1984) 3. [5] .I. Howard, J. Opt. Sot. Am. A 8 (1991) 1955. [6] N. Streibl, J. Opt. Sot. Am. A 2 (1985) 121. [7] C.J.R. Sheppard, Optik 72 (1986) 131. [8] C.J.R. Sheppard, Optik 74 (1986) 128. [9] C.J.R. Sheppard and X.Q. Mao, J. Opt. Sot. Am. A 6 (1989) 1260. [lo] C.J.R. Sheppard, European J. Cell Biol. Suppl. 25 48 (1989) 29. [ 1 l] C.J.R. Sheppard and C.J. Cogswell, J. Microscopy 159 (1990) 179. [ 121 C.J.R. Sheppard and Min Gu, J. Microsc. 165 (1991) 377. [13] C.J.R. Sheppard, Min Gu and X.Q. Mao, Optics Comm. 81 (1991) 281. [ 141 C.J.R. Sheppard, Mm Gu, Y. Kawata and S. Kawata, J. Opt. Sot. Am. A 11 ( 1994) 593. [ 151 C.J.R. Sheppard, T.J. Connolly, J. LeeandC.J. Cogswell,Appl. Optics 33 (1994) 631. [ 161 C.J.R. Sheppard, T.J. Connolly and Min Gu, Phys. Rev. Lett. 70 (1993) 1409. [ 171 C.J.R. Sheppard, T.J. Connolly and Min Gu, J. Mod. Optics40 (1993) 2407. [ 181 C.J.R. Sheppard and Min Gu, Optics Comm. 87 ( 1992) 180. [19] C.G. Darwin,Phil. Mag.27 (1914) 315,645. [20] LM. Gel’fand and B.M. Levitan, Am. Math. Sot. Trans. Ser. 2 (1955) 253. [21] C.H. Greenewalt, W. Brandt and D.D. Friel, J. Opt. Sot. Am. 50 (1960) 1005. [22] L.M. Brekhovsl&h, Waves in Layered Media (Academic, New York, 1960).
C.J.R. Sheppard et al. /Optics Communications 117 (1995) 16-19 [23] E. Vogelzang, H.A. Ferwerda and D. Yevick, Optica Acta 31 (1984) 541. [24] L. Sossi and P. Kard, On the theory of the reflection and transmission of light by a thin inhomogeneous dielectric film, National Research Council of Canada Tech. Trans. 1905 (1968).
19
1251 P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Ma, 1987). [26] R.J. Wombell and J.A. DeSanto, J. Opt. Sot. Am. A 8 (1991) 1892. [27] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory (Springer, Berlin, 1992). [28] C.J.R. Sheppard and T.J. Connolly, Imaging of random surfaces, J. Mod. Optics, to be published.