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Generalized Optical Theorem for Reflection, Transmission, and Extinction of Power for Scalar Fields University of Pennsylvania ScholarlyCommons Departmental Papers (BE) Department of Bioengineering 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 Follow this and additional works at: https://repository.upenn.edu/be_papers 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 extin- guished 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]. 1539-3755/2004/70(3)/036611(7)/$22.5070 036611-1 ©2004 The American Physical Society CARNEY, SCHOTLAND, AND WOLF PHYSICAL REVIEW E 70, 036611 (2004) 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 .
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