Generalized Optical Theorem for Reflection, Transmission, and Extinction of Power for Scalar Fields

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September 2004

Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields

P. Scott Carney University of Illinois

John C. Schotland University of Pennsylvania, [email protected]

Emil Wolf University of Rochester

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Recommended Citation Carney, P. S., Schotland, J. C., & Wolf, E. (2004). Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields. Retrieved from https://repository.upenn.edu/be_papers/54

Copyright American Physical Society. Reprinted from Physical Review E, Volume 70, Issue 3, Article 036611, September 2004, 7 pages. Publisher URL: http://dx.doi.org/10.1103/PhysRevE.70.036611

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/be_papers/54 For more information, please contact [email protected]. Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields

Abstract We present a derivation of the optical theorem that makes it possible to obtain expressions for the extinguished power in a very general class of problems not previously treated. The results are applied to the analysis of the extinction of power by a scatterer in the presence of a lossless half space. Applications to microscopy and tomography are discussed.

Comments Copyright American Physical Society. Reprinted from Physical Review E, Volume 70, Issue 3, Article 036611, September 2004, 7 pages. Publisher URL: http://dx.doi.org/10.1103/PhysRevE.70.036611

This journal article is available at ScholarlyCommons: https://repository.upenn.edu/be_papers/54 PHYSICAL REVIEW E 70, 036611 (2004)

Generalized optical theorem for reﬂection, transmission, and extinction of power for scalar ﬁelds

P. Scott Carney Department of Electrical and Computer Engineering and The Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

John C. Schotland Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

Emil Wolf Department of Physics and Astronomy and The Institute of Optics, University of Rochester, Rochester, New York 14627, USA and The School of Optics/CREOL, University of Central Florida, Orlando, Florida 32816, USA (Received 15 March 2004; published 22 September 2004)

We present a derivation of the optical theorem that makes it possible to obtain expressions for the extinguished power in a very general class of problems not previously treated. The results are applied to the analysis of the extinction of power by a scatterer in the presence of a lossless half space. Applications to microscopy and tomography are discussed.

DOI: 10.1103/PhysRevE.70.036611 PACS number(s): 42.25.Ϫp, 43.20.ϩg, 03.50.Ϫz

I. INTRODUCTION ture of the scattering object and even to reconstruct the spa- tial structure of the scatterer from the data obtained from The conservation of energy for electromagnetic or acous- power extinction experiments [8,9]. tic waves, or conservation of probability in quantum me- In Sec. II, we derive an expression for extinguished power chanics, leads to a remarkable identity known as the optical that is the progenitor of the generalized optical theorem. An theorem. The optical theorem relates the power extinguished expression is obtained that relates the extinguished power to from a plane wave incident on an object to the scattering a volume integral over the domain of the scatterer. Unlike amplitude in the direction of the incident field. Explicitly, the other forms of the optical theorem, this result is obtained optical theorem may be expressed by the formula [1–5] without invoking the asymptotic behavior of the scattered field. Furthermore, this formulation allows for the case that 4␲ ␴ ͑ ͒ ͑ ͒ the scattering object is embedded in an arbitrary background e = Im A k,k , 1 k0 with inhomogeneous optical properties. In Sec. III, scattering from an object in free space is reconsidered and the equiva- ␴ where e is the extinction cross section, k0 is the wave num- lence of our result and the generalized optical theorem [6,7] ber of the field, k is the wave vector of the incident plane is established. In Sec. IV, the problem of scattering from an wave, and A͑k,k͒ is the amplitude of the field scattered in object in a half space is addressed. It is found that the extin- the forward direction, i.e., in the direction of the incident guished power is related to the field that is scattered in the field. The extinction cross section is the total extinguished directions of the components of the incident field in both half power (i.e., the power lost to scattering and absorption of the spaces. The half-space problem is of practical importance in incident field) normalized by the power per unit area incident imaging and tomography when the sample is supported on a on the scatterer. This theorem implies that the total power slide or other flat platform. extinguished from the incident field is removed from the incident field by means of interference between the incident field and the forward scattered field. II. GENERAL RESULTS The optical theorem, expressed by Eq. (1), may be seen to be a special case of a more general theorem that applies to Consider the reduced wave equation for a monochro- ␺͑ ͒ −i␻t ␻ the scattering of an arbitrary incident field and relates the matic, scalar field r e , =ck0, in the presence of a ␩͑ ͒ extinguished power to a weighted integral of the scattering scattering object described by a susceptibility r , embed- amplitude [6,7]. This result accounts for the contribution not ded in a non-absorbing background medium, characterized ͑ ͒ only of the homogeneous components but also of the evanes- by the spatially dependent wave number k r , cent components of the incident field which have to be taken ٌ2␺͑r͒ + k2͑r͒␺͑r͒ =−4␲k2͑r͒␩͑r͒␺͑r͒. ͑2͒ into account when the source of illumination is in the near zone of the scatterer. Equation (2) is ubiquitous in physics. It is encountered, for The generalized optical theorem also applies to the scat- example, as the linear, single particle, time independent, non- tering of partially coherent fields and scattering from random relativistic Schrödinger equation [[10], Sec. 15]. It is also the objects. It has been shown that the generalized optical theo- governing equation for the propagation of sound waves [[11], rem may be used to relate extinguished power to the struc- Sec. 6.2].

Assuming that the background is nonabsorbing [k͑r͒ is By analogy with Eq. (5), the flux density vector of the real], the reduced wave equation (2) then implies that scattered field Fs is defined as ␺*͑rٌ͒2␺͑r͒ − ␺͑rٌ͒2␺*͑r͒ 1 ␺*͑r͖͒. ͑10͒ ١ ␺ ͑r͒ − ␺ ͑r͒ ١ F ͑r͒ = ͕␺*͑r͒ 2 * 2 s s s s s =−4␲k͑r͒ ͕␩͑r͒ − ␩ ͑r͖͉͒␺͑r͉͒ . ͑3͒ 2ik0 By integrating both sides of Eq. (3) over any volume V The power carried by the scattered field is then given by the which is bounded by a closed surface S with unit outward expression normal n, and applying Green’s theorem [[12], Sec. 1.8],itis readily found that ͵ 2 ͑ ͒ ͑ ͒ Ps = d r Fs r · n, 11 S ͖͒ ͑*␺ ١ ͒ ␺͑ ͒ ␺͑ ١ ͒ ͑*␺͕ 2 ͵ d r r r − r r · ndS where S is any closed surface which completely encloses the S scattering volume. It may be seen that ͒ ͑*␲ ͑ ͒2͕␩͑ ͒ ␩*͑ ͖͉͒␺͑ ͉͒2 ͑ ͒ ␺*͑ ٌ͒2␺ ͑ ͒ ␺ ͑ ٌ͒2␺ 3 ͵ =− d r4 k r r − r r . 4 s r s r − s r s r V ␲ 2͑ ͒ ͕␩͑ ͒␺*͑ ͒␺͑ ͖͒ ͑ ͒ =−8 ik r Im r s r r . 12 In the classical wave theory, the expression in the curly brackets {}on the left hand side of Eq. (4) is usually iden- Making use of Eqs. (9) and (12), the power carried by the tified with the energy flux density of the field [[11],p.199]; scattered field, Eq. (11), may also be expressed in terms of in quantum mechanics such an expression is identified with the volume integral as the probability current density 10 , Sec. 17 . The flux den- [[ ] ] 4␲ sity vector 5 , Appendix 11 , with normalization taken so ͵ 3 2͑ ͒␺*͑ ͒␺͑ ͒␩͑ ͒ ͑ ͒ [[ ] ] Ps =− Im d rk r s r r r . 13 that a unit amplitude plane wave has a unit magnitude flux k0 V density vector, is thus defined as The absorbed power and the scattered power are supplied 1 by the incident field. The sum of the absorbed and the scat- ͒ ͑ ͖͒ ͑*␺ ١ ͒ ␺͑ ͒ ␺͑ ١ ͒ ͑*␺͕ ͒ ͑ F r = r r − r r . 5 tered power, referred to as the extinguished power 2ik0 ͑ ͒ The absorbed power is given by the net flux passing through Pe = Pa + Ps, 14 any surface enclosing the scatterer (but not enclosing the represents the power depleted from the incident field because sources of the incident field): of the presence of the scattering body. From Eqs. (7), (13), and (14) it can be seen that the extinguished power is given ͵ 2 ͑ ͒ ͑ ͒ Pa =− d r F r · n, 6 in terms of the incident and the scattered fields by the ex- S pression where P is the power absorbed by the scattering medium. It a 4␲ can be seen from Eq. 4 that P is also given by the expres- ͵ 3 2͑ ͒␺*͑ ͒␺͑ ͒␩͑ ͒ ͑ ͒ ( ) a Pe = Im d rk r i r r r . 15 sion k0 V 4␲ Formula (15) is our main result. It gives the power depleted P = Im ͵ d3rk2͑r͉͒␺͑r͉͒2␩͑r͒, ͑7͒ a k from the incident beam as a consequence of interference be- 0 V tween the incident and the scattered fields in the domain of where Im denotes the imaginary part. When ␩ is real, the the object. scatterer is nonabsorbing and Pa is identically zero. A distinction between the background medium and the scattering object is made on physical grounds, although the III. FREE SPACE effect of the scatterer itself might also be included in the We will now show that, when the scatterer is located in function k͑r͒. The distinction is physically important because free space, the general result expressed in Eq. (15) reduces to the field is taken to be composed of two parts, a scattered results derived previously in Refs. [6,7]. Moreover, the dis- field ␺ and an incident field ␺ with ␺ +␺ =␺. The incident s i i s cussion will provide a template for the calculations of the field may be identified as the field which is present in the subsequent section where the more difficult problem of scat- absence of the scattering object. The incident field evidently tering in a half space will be considered. obeys the Helmoltz equation Let us choose a coordinate system such that the scattering 2␺ ͑ ͒ 2͑ ͒␺ ͑ ͒ ͑ ͒ ജٌ i r + k r i r =0, 8 object is located in the half space z 0 and the sources of the incident field are located in the half space zϽ0. The most except in the region occupied by the source, and it is as- general case, where sources may be located anywhere out- sumed that the source is located outside the domain occupied side of the object, may be treated by an obvious extension by the scatterer. From Eqs. (2) and (8) it is seen that the ␺ the analysis presented here. scattered field s obeys the equation The incident field may be expressed in the form of an ͒ ͑ ͒ 2␺ ͑ ͒ 2͑ ͒␺ ͑ ͒ ␲ ͑ ͒2␩͑ ͒␺ٌ͑ s r + k r s r =−4 k r r r . 9 angular spectrum of plane waves [[13], Sec. 3.2],

036611-2 GENERALIZED OPTICAL THEOREM FOR REFLECTION,… PHYSICAL REVIEW E 70, 036611 (2004)

␺ ͑ ͒ ͵ 2 ͑ ͒ ik·r ͑ ͒ i r = d kʈ a k e , 16

where kʈ is a vector parallel to the plane z=0 and k=kʈ +zˆkz, with zˆ being the unit vector in the increasing z direction and ͱ 2 2 ͑ ͒ kz = k0 − kʈ . 17

When the modulus ͉kʈ͉ exceeds the free-space wave number, kz becomes purely imaginary. Such values of kz correspond to modes of the incident field that decay exponentially on FIG. 1. Illustrating the notation for scattering in free space. The propagation and are known as evanescent waves. Since the incident field, here taken to be a single plane wave, is represented system under consideration is linear in the fields, the total by a solid line in the direction of the wave vector of the plane wave. field is given by the expression A plane wave component of the scattered field is represented by a dashed line and is labeled with the appropriate amplitude. ␺͑ ͒ ͵ 2 ͑ ͒␾͑ ͒ ͑ ͒ r = d kʈ a k r;k , 18 vectors which are relevant in the near field for evanescent plane wave modes. An explicit form for A may be found by where ␾͑r,k͒ is the total field produced at a point r due to ͓ ͑ ͒ ͔ considering the field in any plane z=z0 (see Fig. 1) such that scattering of a plane wave of unit amplitude a k =1 with the scatterer lies entirely in the region zϽz , wave vector k. The total field ␾ may be separated into inci- 0 dent and scattered fields, k2z ͑ ͒ ͵ 2 −ik2·r␾ ͑ ͒ ͑ ͒ A k1,k2 = d re s r;k1 . 25 ␾͑ ͒ ik·r ␾ ͑ ͒ ͑ ͒ 2␲i r;k = e + s r;k , 19 z=z0 ␾ ͑ ͒ ␾ s r;k being the scattered field produced at a point r on Substitution of the expression for s given in Eq. (20) in the scattering of a plane wave of unit amplitude with wave vec- right hand side of Eq. (25) and use of Eq. (22) yields the tor k, satisfying the integral equation relation

␾ ͑ ͒ 2 ͵ 3 Ј ͑ Ј͒␩͑ Ј͒␾͑ Ј ͒ ͑ ͒ ͑ ͒ 2͵ 3 −ik2·r␩͑ ͒␾͑ ͒ ͑ ͒ s r;k = k0 d r G r,r r r ;k . 20 A k1,k2 = k0 d re r r;k1 . 26 V In Eq. (20) G is the retarded (outgoing) Green’s function, We now return to the extinguished power. Substituting which satisfies the equation Eqs. (16) and (18) in Eq. (15) we find that ͒ ͑ Ј͒ 2 ͑ Ј͒ ␲␦ ͑3͒͑ Ј͒ ͑ 2ٌ G r,r + k0G r,r =−4 r − r , 21 ␲ ͵ 2 2 ͑ ͒ *͑ ͒ Pe =4 k Im d k1ʈd k2ʈ a k1 a k2 ␦ ͑3͒ denoting the three-dimensional Dirac delta function. The Green’s function may be represented as a superposition of * plane wave modes, ϫ͵ 3 −ik2·r␩͑ ͒␾͑ ͒ ͑ ͒ d re r k1,r . 27 2 V i d kʈ ͑ Ј͒ ͵ ͓ ͑ Ј͒ ͉ Ј͉͔ G r,r = exp ikʈ · r − r +ik z − z . 2␲ k z Thus, making use of Eq. (26), the results originally presented z in Refs. [6,7] are recovered: ͑22͒ 4␲ Ͻ ͵ 2 2 ͑ ͒ *͑ ͒ ͑ *͒ ͑ ͒ The scatterer lies entirely in the region z z0 as shown in Pe = Im d k1ʈd k2ʈ a k1 a k2 A k1,k2 . 28 ␾ Ͼ k0 Fig. 1. The field s, in the region z z0, may be represented in the form of an angular spectrum of plane waves The case of illumination by a single plane wave of amplitude ͑ ͒ ␦͑2͒͑ ͒ 2 a may be recovered by taking a k =a kʈ −k0ʈ . One finds i d k2ʈ ␾ ͑r;k ͒ = ͵ A͑k ,k ͒eik2·r, ͑23͒ s 1 ␲ 1 2 that 2 k2z 4␲͉a͉2 where A is the usual scattering amplitude between states of ͑ *͒ ͑ ͒ Pe = Im A k0,k0 . 29 real momenta, as it may be seen that in the far zone of the k0 Ͼ ␾ scatterer, for z 0, the scattered field s takes the asymptotic This expression was originally derived for real k by Feen- form berg [1] in the context of quantum mechanics, with the re- ik r e 0 strictions that Pa =0 and with implied cylindrical symmetry. ␾ ͑ ͒ϳ ͑ ͒ ͑ ͒ s r;k1 A k1,k0rˆ , rˆ = r/r. 24 An heuristic argument was given by van de Hulst 2 for the r [ ] theorem in the context of electromagnetic scattering. A rig- It should be noted that, while A gives the asymptotic behav- orous derivation for the electromagnetic case was given by ior of the field, it is not defined only by the asymptotic be- Jones [3]. When the incident field is a plane wave of ampli- havior of the field. It is also well defined for complex wave tude a, it is common practice to normalize the power by the

036611-3 CARNEY, SCHOTLAND, AND WOLF PHYSICAL REVIEW E 70, 036611 (2004)

intensity of the incident ﬁeld. This normalized quantity, de- ␴ ͉ ͉2 ﬁned by the expression e = Pe / a , and Eq. (1), is referred to as the extinction cross section [[4] Sec. 1.3]. For real k, Eq. (1) is the optical (cross section) theorem.

IV. HALF SPACE We now consider the problem of scattering from an object in the presence of a planar interface separating two homogeneous half spaces. It is assumed that the scatterer is of finite ജ ജ extent and is located entirely in the region z1 z z2 as shown in Fig. 2. The half space zϾ0 is assumed to be vacuum. The other half space zϽ0 is taken to consist of a material whose index of refraction is nϾ1. The fields satisfy Eqs. (2) and (9), with k for z ജ 0, k͑r͒ = ͭ 0 ͮ ͑30͒ Ͻ nk0 for z 0. The Green’s function G͑r,rЈ͒ for this situation satisfies the equation Ј͒ 2͑ ͒ 2 ͑ Ј͒ ␲␦͑3͒͑ Ј͒͑͒ ͑ 2ٌ G r,r + n z k0G r,r =−4 r − r 31 and obeys the boundary conditions ͉ ͑ Ј͉͒ ͉ ͑ Ј͉͒ ͑ ͒ G r,r z=0+ = G r,r z=0−, 32

͒ ͑ ͉͒ ͑ ١ ͉ ͉͒ ͑ ١ ͉ zˆ · G r,rЈ z=0+ = zˆ · G r,rЈ z=0−. 33 G͑r,rЈ͒ admits the plane wave decomposition 2 i d kʈ ͑ Ј͒ ͵ ͓ ͑ Ј͔͒ ⌰͑ ͒⌰͑ Ј͒ G r,r = exp ikʈ · r − r ͩ z z ␲ 2 kz ϫ͕ ͑ ͉ ͉͒ ͑ ͒ ͓ ͑ ͔͖͒ exp ikz z − zЈ + R k,kЈ exp ikz zЈ + z ⌰͑ ͒⌰͑ Ј͒ ͑ Ј͒ ͓ Ј Ј ͔ + − z z T k,k exp ikzz −ikz z k ⌰͑ ͒⌰͑ Ј͒ z Ј͑ Ј͒ ͓ Ј Ј͔ + z − z T k,k exp ikzz −ikz z FIG. 2. Illustrating the notation for the half-space problem. In kЈ ␾+͑ Ј͒ z (a), an incident mode i r,k1 ,k1 associated with sources in the k lower half space is represented by a solid line indicating the wave ⌰͑ ͒⌰͑ Ј͒ z ͕ ͑ Ј͉ Ј͉͒ vector of the three planewave components of the mode. A mode of + − z − z Ј exp ikz z − z kz the scattered field is represented by a dashed line. The two plane wave components of the scattered field in the upper half space Ј͑ Ј͒ ͓ Ј͑ Ј ͔͖͒ͪ ͑ ͒ + R k,k exp −ikz z + z , 34 combine to produce an outgoing plane wave with wave vector k2 +͑ Ј Ј͒ and amplitude proportional to A+ k1 ,k1 ,k2 ,k2 . The plane wave where ⌰ is the Heaviside step function, i.e., ⌰͑x͒=1 for component of the scattered mode in the lower half space is proportional to A+͑k ,kЈ,k ,kЈ͒.In b , the notation is similarly illus- xϾ0 and ⌰͑x͒=0 for xഛ0. The factors R and T are the − 1 1 2 2 ( ) trated, here with a different mode of the incident field, ␾−͑r,k ,kЈ͒, reflection and transmission coefficients, respectively, i 1 1 generated by sources in the upper half space. k − kЈ 2k ͑ Ј͒ z z ͑ Ј͒ z ͑ ͒ flection and transmission coefficients is somewhat redundant R k,k = Ј, T k,k = Ј , 35 kz + kz kz + kz and not as compact as it might be. However, this expanded notation will prove useful shortly. Ј kz The source of the field is assumed to be entirely outside RЈ͑k,kЈ͒ =−R͑k,kЈ͒, and TЈ͑k,kЈ͒ = T͑k,kЈ͒, ͑36͒ ജ k the region of the scatterer, that is, outside the region z1 z z ജ z2. In the region of the scatterer the incident field obeys with the wave vectors, in the upper and lower half spaces, Eq. (8) and may be represented as a superposition of modes respectively, given by Eq. (17), and of this equation analogous to the angular spectrum represen- ͱ 2 2 2 tation for the free-space problem. We will exclude from our kЈ = kʈ + zˆkЈ, and kЈ = n k − kʈ . ͑37͒ z z 0 consideration the case that sources are located in the region Ͼ Ͼ ␳␳ The validity of Eq. (34) may be verified by direct substitu- z1 z z2. Writing r= ˆ +zzˆ, the reflection of the point r tion. It might be noted that the chosen notation for the re- through the z=0 plane is given by ˜r=␳␳ˆ −zzˆ. The modes

036611-4 GENERALIZED OPTICAL THEOREM FOR REFLECTION,… PHYSICAL REVIEW E 70, 036611 (2004)

±͑ Ј Ј͒ incident from the lower half space may be expressed as and A− k1 ,k1 ,k2 ,k2 is the amplitude for the scattering of the incident mode ␾±͑r;k ,kЈ͒ into the outgoing (in the ␾+͑ Ј͒ ͕⌰͑ ͓͒ ikЈ·r Ј͑ Ј͒ ikЈ·˜r͔ i 1 1 i r;k,k = − z e + R k,k e ͑ Ј ͒ lower half space) plane wave exp ik2 ·˜r . Then in the upper + ⌰͑z͒TЈ͑k,kЈ͒eik·r͖͑38͒ half space the scattered mode may be expressed as 2 and the modes incident from the upper half space are i d k2ʈ ␾±͑r;k ,kЈ͒ = ͵ A±͑k ,kЈ,k ,kЈ͒eik2·r ͑45͒ s 1 1 ␲ + 1 1 2 2 ␾−͑ Ј͒ ͕⌰͑ ͓͒ ik·˜r ͑ Ј͒ ik·r͔ 2 k2z i r;k,k = z e + R k,k e and in the lower half-space, + ⌰͑− z͒T͑k,kЈ͒eikЈ·˜r͖. ͑39͒ 2 ± i d k2ʈ ± Ј ˜ It may be verified that these modes are orthogonal and that ␾ ͑r;k ,kЈ͒ = ͵ A ͑k ,kЈ,k ,kЈ͒eik2·r ͑46͒ s 1 1 ␲ Ј − 1 1 2 2 they satisfy the reduced wave equation Eq. (2) and the 2 k2z boundary conditions (32) and (33). Moreover, the Green’s The normalization has been chosen so that in the upper half function for the case that the points rЈ and r lie, respectively, space Ͼ Ͼ inside and outside the domain z1 z z2, may be represented ik r as a superposition of these incident modes. This set of modes e 0 ␾±͑r;k ,kЈ͒ϳ A±͑k ,kЈ,k ,kЈ͒, k r → ϱ, ͑47͒ is thus complete on the space of allowed incident fields. The s 1 1 r + 1 1 2 2 0 incident field may then be expressed as where k2 is parallel to r. In the lower half space, ␺ ͑ ͒ ͵ 2 ͓ +͑ ͒␾+͑ Ј͒ −͑ ͒␾−͑ Ј͔͒ eink0r r = d kʈ a k r;k,k + a k r;k,k , i i i ␾±͑r;k ,kЈ͒ϳ A±͑k ,kЈ,k , kЈ͒, k r → ϱ, ͑48͒ s 1 1 r − 1 1 2 2 0 ͑40͒ Јʈ where k2 ˜r. or, more compactly, The scattering amplitudes may be determined by considering, for the upper half space, the scattered field in some ␺ ͑ ͒ ͵ 2 n͑ ͒␾n͑ Ј͒ ͑ ͒ plane z=z where z is chosen so that the susceptibility of the r = ͚ d kʈ a k r;k,k . 41 1 1 i i Ͼ n=± scatterer is zero for z z1. Then

␾±͑ Ј͒ −ik2zz1 For convenience, r;k,k will denote the total field gen- −ik2ze A±͑k ,kЈ,k ,kЈ͒ = ͵ d2re−ik2ʈ·r␾±͑r;k ,kЈ͒. erated at a point r by scattering of an incident mode + 1 1 2 2 2␲ s 1 1 ␾±͑ Ј͒ z=z1 i r;k,k . As before, the total field modes are given in ␾± ␾± ␾± ͑49͒ terms of their incident and scattered parts as = i + s . The total field may be represented in terms of the total field The scattering amplitude in the lower half space may be modes by the expression determined by considering the scattered field in some plane ഛ z=z2 0, ␺͑ ͒ 2 n͑ ͒␾n͑ Ј͒͑͒ r = ͚ ͵ d kʈ a k r;k,k 42 Ј Ј ik2zz2 n=± −ik2ze A±͑k ,kЈ,k ,kЈ͒ = ͵ d2re−ik2ʈ·r␾±͑r;k ,kЈ͒. − 1 1 2 2 2␲ s 1 1 and the scattered field by z=z2 ͑50͒ ␺ ͵ 2 n͑ ͒␾n͑ Ј͒ ͑ ͒ s = ͚ d kʈ a k s r;k,k . 43 n=± The scattered field satisfies the integral equation

The extinguished power [Eq. (15)] may thus be expressed as ␾±͑ Ј͒ 2 ͵ 3 Ј ͑ Ј͒␩͑ Ј͒␾±͑ Ј Ј͒ ͑ ͒ s r;k,k = k0 d r G r,r r r ;k,k . 51 ␲ ͵ 3 ͵ 2 2 Pe =4 k Im d r d k1ʈd k2ʈ Making use of Eqs. (49)–(51), one ﬁnds that

m* n m* n ϫ ͚ a ͑k ͒a ͑k ͒␾ ͑r;k ,kЈ͒␾ ͑r;k ,kЈ͒␩͑r͒. ±͑ Ј Ј͒ 2 ͵ 3 ͓ −ik2·r ͑ Ј͒ −ik2·˜r͔ 2 1 i 2 2 1 1 A+ k1,k1,k2,k2 = k0 d r e + R k2,k2 e m,n=± ͑44͒ ϫ␩͑ ͒␾±͑ Ј͒ ͑ ͒ r r;k1,k1 , 52 ␾± The scattered ﬁeld modes s may be expressed in the and form of angular spectra as in Eq. 23 . The situation is some- ( ) Ј k2z what more complicated now, and some explanation of the ±͑ Ј Ј͒ 2 ͵ 3 ͑ Ј͒ −ik2·˜r A− k1,k1,k2,k2 = k0 d rTk2,k2 e notation is appropiate. The amplitude A is now a function of k2z the wave vectors corresponding to the transverse wave vector ϫ␩͑r͒␾±͑r;k ,kЈ͒. ͑53͒ of the incident mode and the plane wave into which that 1 1 ±͑ Ј Ј͒ mode is scattered. That is, A+ k1 ,k1 ,k2 ,k2 is the amplitude The integrals in Eq. (44) may be identiﬁed with the scat- ␾±͑ Ј͒ for the scattering of the incident mode i r;k1 ,k1 into the tering amplitude in certain directions. It is useful to note that ͑ ͒ Ͼ outgoing (in the upper half space) plane wave exp ik2 ·r , in the region of the scatterer, i.e., zЈ 0,

036611-5 CARNEY, SCHOTLAND, AND WOLF PHYSICAL REVIEW E 70, 036611 (2004)

␾+*͑ Ј͒ Ј*͑ Ј͓͒ −ik*·r ͑ * Ј*͒ −ik*·˜r͔ i r;k,k = T k,k e + R k ,k e *͑ Ј͒ n͑ Ј* Ј*͔͒ͮ ͑ ͒ + R k2,k2 A+ k1,k1,k2 ,k2 . 58 * + RЈ*͑k,kЈ͒TЈ͑k*,kЈ*͒e−ik ·˜r ͑54͒ and ␾−*͑ Ј͒ *͑ Ј͓͒ −ik*·r ͑ * Ј*͒ −ik*·˜r͔ V. DISCUSSION i r;k,k = R k,k e + R k ,k e * + TЈ*͑k,kЈ͒T͑k*,kЈ*͒e−ik ·˜r, ͑55͒ The results presented here provide insight into the interference mechanisms that ensure energy conservation in the where we have made use of the identity that T*͑k,kЈ͒ scattering of scalar waves. Equation (15) provides a frame- =T͑k*,kЈ*͒, and similar expressions for R and RЈ. By sub- work in which to obtain a relationship between the scattering stituting these expressions into Eqs. (52) and (53) and com- amplitude and the extinguished power in problems with an paring to Eq. (15), the power extinguished from an incident arbitrary background medium. We have obtained such a re- ␾+͑ Ј ͒ field i k,k ,r is seen to be given by the expression lationship for the case that the background medium consists of a lossless half space of index different from the vacuum. 4␲ ͓ Ј*͑ Ј͒ +͑ Ј * Ј*͒ This problem is relevant to the scattering of a single evanes- Pe = Im T k,k A+ k,k ,k ,k k0 cent plane wave. Such a field can be generated only in the + RЈ*͑k,kЈ͒A+͑k,kЈ,k*,kЈ*͔͒. ͑56͒ half-space geometry. In free space, evanescent modes may be − present in superposition with other modes of the field if the This result has a clear physical interpretation: In the absence source is placed near the scatterer, but a single evanescent of the scatterer, the incident field imparts a certain amount of mode is never present in isolation [7]. The results have a power to the far zone via the outgoing plane waves reflected clear physical meaning, namely, that the extinguished power from and transmitted through the boundary of the half is simply related to the scattering amplitude of the scattered spaces. The scatterer depletes, or extinguishes, some of the field in the direction of the outgoing plane wave components power from the incident field. In order to properly account of the incident field. for the total power, the field produced on scattering must To lowest order in the susceptibility, when the scatterer is interfere coherently with the incident field in order to extin- in vacuum, the extinguished power is the projection of the ␾+͑ Ј͒ guish that field. Thus the incident mode i r;k1 ,k1 delivers incident intensity on the imaginary part of the susceptibility, Ј ˜ power to the far zone through the plane waves eik1·r and ik1·r ␲ ͵ 3 ͉␺ ͑ ͉͒2 ␩͑ ͒ ͑␩2͒ ͑ ͒ e , and the extinguished power is directly related to the Pe =4 k0 d r i r Im r + O . 59 amplitude of the scattered plane waves in those same direc- V tions as may be seen in expression 56 . The situation is ( ) This formula suggests a manner in which object structure illustrated in Fig. 2. In the event that the incident mode con- may be investigated. If the intensity of the incident field sists of a wave totally internally reflected in the zϽ0 half forms the kernel of a transformation that can be inverted, space, the extinguished power is related to the amplitude of then the object structure, as described by Im ␩͑r͒, may be the modes of the field coupled back into the zϽ0 half space, found from power extinction measurements. In Refs. [8,9] propagating in the direction of the beam reflected from the the incident field was taken to consist of two plane waves interface. and consequently the extinguished power was related to a The power extinguished from the incident field Fourier transform [8] or to a Fourier-Laplace transform [9] ␾−͑r;k,kЈ͒ is given by the expression i of the object. The present result shows that the extinguished 4␲ power is a meaningful measure of object structure for certain P = Im͓T*͑k,kЈ͒A−͑k,kЈ,k*,kЈ*͒ e k − forms of the incident field. Two tomographic modalities cur- 0 rently in practice may also be understood as variants of this *͑ Ј͒ −͑ Ј * Ј*͔͒ ͑ ͒ + R k,k A+ k,k ,k ,k . 57 approach. Computed tomography with x rays [14] is accom- This expression may also be interpreted as relating the extin- plished with measurements of the field attenuation along rays guished power to the amplitude of the scattered waves which passing through the sample under investigation. That situa- are coincident with the outgoing parts of the incident field. In tion is described by Eq. (59) when the incident field is assumed to be localized to the ray path. The technique of trans- general, with an incident field given by Eq. (41), mission mode confocal imaging [15] may also be understood 4␲ ͭ͵ 2 2 n͑ ͒ +*͑ ͒ in the context of Eq. (59). There the field intensity is much ͚ ʈ ʈ Pe = Im d k1 d k2 a k1 a k2 higher at the focus than at any other point in the medium, n=± k0 and, as is well known, the data represent a convolution of the ϫ͓ Ј*͑ Ј͒ n͑ Ј * Ј*͒ T k2,k2 A+ k1,k1,k2,k2 object structure with the intensity in the focus. Because the transmitted field is collected in a confocal arrangement, the + RЈ*͑k ,kЈ͒An͑k ,kЈ,k*,kЈ*͔͒ 2 2 − 1 1 2 2 signal is simply related to the extinguished power. Results analogous to those presented here may be obtain ͵ 2 2 n͑ ͒ −*͑ ͒ + d k1ʈd k2ʈ a k1 a k2 for the vector (electromagnetic) field and will be presented in another paper. Layer structures and waveguide geometries ϫ ͓ ͑ Ј͒ n͑ Ј * Ј*͒ T * k2,k2 A− k1,k1,k2,k2 will also be studied in future work.

036611-6 GENERALIZED OPTICAL THEOREM FOR REFLECTION,… PHYSICAL REVIEW E 70, 036611 (2004)

ACKNOWLEDGMENTS Office of Basic Energy Sciences at the U.S. Department of Energy under Grant No. DE-FG02-2ER45992. P.S.C. would E.W. would like to acknowledge supprort by the U.S. Air like to acknowledge support by the National Science Foun- Force Office of Scientific Research under Grant No. F49620- dation under CAREER Grant No. 0239265 and U.S. Air 031-0138 and by the Engineering Research Program of the Force MURI Grant No. F49620-03-1-0379.

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