Reciprocity, unitarity, and time-reversal symmetry of the S matrix of fields containing evanescent components

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Reciprocity, unitarity, and time-reversal symmetry of the S matrix of fields containing evanescent components

R. Carminati, J Sáenz, J.-J Greffet, M. Nieto-Vesperinas

To cite this version:

R. Carminati, J Sáenz, J.-J Greffet, M. Nieto-Vesperinas. Reciprocity, unitarity, and time-reversal

symmetry of the S matrix of fields containing evanescent components. Physical Review A, American

Physical Society, 2000, 62, pp.012712-012718 �10.1103/PhysRevA.62.012712�. �hal-01624615�

Reciprocity, unitarity, and time-reversal symmetry of the S matrix of fields containing evanescent components

R. Carminati,¹J. J. Sa´enz,²J.-J. Greffet,¹ and M. Nieto-Vesperinas³

1Laboratoire d’Energe´tique Mole´culaire et Macroscopique, Combustion, Ecole Centrale Paris, Centre National de la Recherche Scientifique, 92295 Chaˆtenay-Malabry Cedex, France

2Departamento de Fı´sica de la Materia Condensada and Instituto de Ciencia de Materiales ‘‘Nicola´s Cabrera,’’

Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain

3Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cientı´ficas, Cantoblanco, 28049 Madrid, Spain

共Received 7 February 2000; published 16 June 2000兲

We derive general relationships of the S matrix of fields containing evanescent components. Our formalism covers time-independent quantum scattering as well as scattering of classical scalar waves. We show that reciprocity, energy共or probability兲 conservation, and time-reversal symmetry in the presence of evanescent waves lead to relationships that extend the well-known relations previously derived in asymptotic scattering.

On this basis, we discuss the link between reciprocity and time-reversal symmetry. We also address the experimental feasibility of time reversal of a field containing evanescent components.

PACS number共s兲: 03.65.Nk, 03.50.⫺z, 42.25.Fx, 11.55.⫺m I. INTRODUCTION

The scattering matrix (S matrix兲 was introduced by Heisenberg to describe a scattering process without any as- sumption about the details of the interaction关1兴. In this for- malism, the process is thought of as a transformation of an incoming state⌿in into an outgoing state⌿out, which de- scribe the system far away from the interaction potential.

Hence, the S matrix describes the scattering process asymp- totically. The mathematical transcription of this transforma- tion is an operator relationship ⌿out⫽S⌿in, where S is called the S matrix 关2,3兴. It is well known that the S matrix exhibits some properties that are independent of the specific problem under study. In particular, it is unitary and symmet- ric, these two properties reflecting probability 共or energy兲 conservation in elastic scattering and reciprocity, respec- tively 关3兴. The general aim being to get maximum informa- tion about the S matrix with minimum knowledge about the interaction itself, other properties may be derived, based, for example, on dispersion relations and causality conditions关4兴. The existence of such general properties of the S matrix is the reason why it has become a fundamental tool in most areas of theoretical physics, e.g., in quantum scattering关2,3兴, in particle physics关5兴, in field theory, and in statistical phys- ics关6兴. Its definition and its use have also been extended to scattering of classical 共acoustic and electromagnetic兲waves 关7,8兴. For example, the S matrix has become a fundamental tool共as well as a practical one兲to compute scattered fields in physical optics 关9兴. This formalism has also found a wide range of applications with the development of random- matrix theory关10兴, which has recently acquired renewed in- terest through its use in quantum- and classical-wave trans- port in random media关11,12兴.

The S matrix was originally defined as an operator acting on asymptotic states. In scattering by a time-independent po- tential 共we shall restrict our discussion to this case兲, this means that the S matrix relates the far-field amplitudes of the incoming and outgoing fields 关2,7,8兴. Nevertheless, in the last ten years, new effects have been observed and new tech-

niques developed that involve scattering and/or direct mea- surement of evanescent waves 共near fields兲. For example, evanescent-wave scattering is involved in the emission pro- cess of an atom or a molecule close to a surface关13兴or in the surface-plasmon-polariton 共SPP兲 mechanism which leads to the enhanced backscattering of light on slightly rough metal surfaces关14兴. It is also the basic principle of near-field scan- ning probe microscopies, using either electron关15兴or photon 关16兴 tunneling. Modeling the image formation in scanning near-field optical micoscopy 共SNOM兲requires a precise de- scription of a mechanism involving scattering of evanescent waves 关17兴. The advent of SNOM has also allowed a direct experimental study of SPP excitation and scattering关18兴and Anderson localization of surface excitations 关19兴and stimu- lated theoretical works on SPP scattering by surface rough- ness or localized objects关20兴. In all these fields, the descrip- tion of the coupling between an incident evanescent wave and a scattered propagating or evanescent wave is of funda- mental interest. An S-matrix formalism, with a definition in- cluding the near-field components, should be very useful in this context. The S matrix also provides a useful formalism to discuss time reversal of wave fields, and especially its link with reciprocity 共symmetry of the S matrix兲and probability or energy conservation共unitarity兲. In particular, the question of time reversal of fields containing evanescent components has recently received increasing attention, with the demon- stration of phase conjugation of optical near fields 关21兴and of time reversal of acoustic waves关22兴. In this last case, the S- matrix formalism was used to discuss the properties of a time-reversal acoustic cavity, without taking into account the role of evanescent waves关23兴. Nevertheless, the question of subwavelength focusing of a field by time reversal was raised. This is an important issue, whose discussion requires the use of a formalism including the evanescent components of the field.

Finding general properties of the S matrix, extended to evanescent waves, is of major importance in understanding and modeling all phenomena and devices involving near- field scattering. To our knowledge, this problem has received

little attention until now, except in electromagnetic wave- guide theory where reciprocity and unitarity relations for the S matrix in the presence of evanescent modes have been briefly discussed关24兴. In this work, we concentrate on reci- procity, unitarity, and time-reversal symmetry in the frame- work of scattering from a localized potential, in the presence of evanescent waves in both the incoming and outgoing fields. The formalism we use covers time-independent quan- tum scattering as well as scattering of classical scalar waves.

In Sec. II, we define the S matrix based on the angular rep- resentation of the field, sometimes called the partitioned S matrix in the literature关9,25兴. In Sec. III, we give a general derivation of the S matrix reciprocity for scalar共quantum and classical兲fields, in the presence of evanescent waves. In Sec.

IV, we show that energy 共or probability兲 conservation in scattering of fields containing evanescent components leads to generalized unitarity relations of the S matrix. These rela- tions extend those previously derived for source-free fields 共i.e., fields without evanescent components兲 关9,25兴, and those obtained in electromagnetic waveguide theory 关24兴. In Sec.

V, we address the problem of time-reversal symmetry of wave fields containing evanescent components. We show that the time-reversal invariance condition leads to a differ- ent relationship for the S matrix. On this basis, the link be- tween reciprocity, unitarity, and time-reversal invariance is discussed. This problem is of fundamental and practical im- portance, due, for example, to its potential application to time reversal of acoustic waves关22兴. Finally, we give a sum- mary and a general conclusion in Sec. VI.

II. DEFINITION OF THE S MATRIX

Let us consider the scattering problem depicted in Fig. 1.

The regions 0⬍z⬍z₁ and z₂⬍z⬍L, denoted by R^⫺ and R^⫹, respectively, are assumed to be of constant potential, so that the wave field in these regions obeys the time- independent wave equation

ⵜ²⌿共r兲⫹k²⌿共r兲⫽0, 共1兲 where r⫽(x, y ,z). In Eq. 共1兲, ⌿(r) is either the time- independent wave function of a state of energy E 关k²

⫽2m/ប²(E⫺V), where V is the potential and m the mass of the particle兴, or a monochromatic classical wave of fre- quency ␻ ^(k⫽␻/c, where c is the phase velocity in the medium兲. The wave number k² is real, but can be negative, e.g., in a tunneling barrier. The scattering potential is as-

sumed confined within the strip z₁⬍z⬍z₂, and independent of the field 共linear scattering兲. The regions z⬍0 and z⬎L contain sources, the presence of which, at finite distance from the scattering region, is necessary to ensure the exis- tence of incident evanescent waves in both R^⫺ and R^⫹. The evanescent waves explicitly appear when the angular- spectrum representation 共or plane-wave expansion兲 of the field is used. In this representation, the fields⌿^⫺inR^⫺and

⌿^⫹ inR^⫹ are written关9兴

⌿^⫾共r兲⫽

冕

^{a⫾共K兲exp共iK•R⫹i␥z兲d2K}

⫹

冕

^{b⫾共K兲exp共iK•R⫺i␥z兲d2K, 共2兲}

where␥^(K)⫽冑^k2⫺K² for K²⭐k² 共homogeneous or propa- gating components兲and␥^(K)⫽i冑^K2⫺k²for K²⬎k²共inho- mogeneous or evanescent components兲. We use the notations R⫽(x, y ) and K⫽兩K兩. Except when the integration domain is specified, all integrals in this paper are extended to ⫺⬁

⬍(K_x,K_y)⬍⫹⬁. Note that Eq.共2兲is a representation of the field valid in regionsR^⫺ andR^⫹ where兩z兩 remains finite, so that there is no divergence of evanescent waves when兩z兩 increases.

In the angular-spectrum representation共2兲, the partitioned S matrix relates the outgoing vector ⌿^out(K)

⫽关b^⫺(K) a^⫹(K)兴 to the incoming vector ⌿ⁱⁿ(K)

⫽关a^⫺(K) b^⫹(K)兴 by the relation关9,25兴

⌿^out共K兲⫽

冕

^S共K,K

^⬘

^兲⌿in共K

^⬘

^兲d2K

^⬘

^{, 共3兲}

where S is a 2⫻2 matrix, sometimes called the partitioned S matrix 关9,25兴, which can be written in the form

S共K,K

⬘

兲⫽

冋

^{rt共共K,KK,K}

^{⬘ ⬘}

^{兲兲 ␳␶共共K,KK,K}

^{⬘ ⬘}

^兲兲

册

^{. 共4兲}

The four elements r,t,␳^,␶ have the meaning of generalized reflection and transmission coefficients 关9,25兴. Their defini- tion can easily be extended to vector fields, as in electromag- netic scattering. In this case, the four coefficients become tensor operators 关26兴.

III. RECIPROCITY RELATION

For incoming and outgoing fields without evanescent components 共source-free fields兲, reciprocity and unitarity re- lations for the partitioned S matrix are well established 关9,25兴. They were derived as a consequence of the symmetry and unitarity of the asymptotic共far-field兲S matrix. Extend- ing these relations to general wave fields with evanescent components requires a different procedure. Note that reci- procity of evanescent waves was derived previously in spe- cific cases, such as electromagnetic waveguide theory 关24兴, electromagnetic 共vector field兲 scattering 关26兴, and elastic- wave scattering at a solid-solid interface 关27兴. In these ex- amples, a suitable formulation of the reciprocity theorem was used for each particular case. In this paper, we give a proof of reciprocity of the S matrix for scattering of both homoge- FIG. 1. Scattering geometry and notation.

R. CARMINATI et al. PHYSICAL REVIEW A 62 012712

012712-2

neous and evanescent scalar waves from a localized poten- tial, starting from a general formulation of reciprocity valid for any kind of scalar wave.

Let ⌿1 and ⌿2 be two fields that are solutions of the scattering problem depicted in Fig. 1. With reference to Fig.

2, let us consider the volume V delimited by the closed sur- face ⌺ composed of two planes z⫽z^⫺ and z⫽z^⫹ and a sphere of radius R centered at the center of the potential region. The application of Green’s second identity leads to

冕

V共⌿2⌬⌿1⫺⌿1⌬⌿2兲dV⫽

冕

⌺

冉

^⌿2⳵⳵^⌿n1⫺⌿1

⳵⌿2

⳵ⁿ

冊

^dS,

共5兲 where⳵^/⳵ⁿ⫽n•ⵜ and n is the outward normal on the sur- face⌺. Both⌿1 and⌿2 satisfy Eq.共1兲inR^⫺andR^⫹, so that the integrand in the left-hand side in Eq. 共5兲 vanishes.

Moreover, in the far-field asymptotic limit兩k兩r→⬁, one has

⳵⌿j/⳵^r⫽ikr^⫺1⌿j, so that ⌿2⳵⌿1/⳵ⁿ⫺⌿1⳵⌿2/⳵^{n van-} ishes identically on the sphere surface when its radius R tends to infinity. Finally, Eq.共5兲leads to the following equal- ity:

冕

^z⫽z^⫺

冉

^{⌿2⳵⳵⌿z1⫺⌿1⳵⳵⌿z2}

冊

^d2R

⫽

冕

z⫽z^⫹

冉

^{⌿2⳵⳵⌿z1⫺⌿1⳵⳵⌿z2}

冊

^{d2R. 共6兲}

Equation共6兲is a scalar version of Lorentz’s reciprocity theo- rem, originally derived for the electromagnetic field 关28兴. In order to obtain a reciprocity theorem for the S matrix, we introduce the angular-spectrum representation Eq.共2兲of the fields ⌿1 and⌿2 into Eq.共6兲. After some algebra, one ob- tains the following expression:

冕 冕

^{关a1⫺共K兲 b1⫹共K兲兴兵␥共K兲S共K,K}

^⬘

^兲

⫺␥共K

⬘

兲S^T共⫺K

⬘

^,⫺K兲其

冋

^{ab2⫺2⫹共共KK}

^{⬘ ⬘}

^兲兲

册

^{d2K d2K}

^⬘

^{⫽0, 共7兲}

where the superscript T denotes the transposed matrix. Be- cause Eq. 共7兲 must be satisfied for any incoming vectors 关a₁^⫺(K) b₁^⫹(K)兴 in situation 1 and 关a₂^⫺(K) b₂^⫹(K)兴 in situation 2, one finally obtains

␥共K兲S共K,K

⬘

兲⫽␥共K

⬘

兲S^T共⫺K

⬘

^,⫺K兲. 共8兲 Equation 共8兲 is valid for 0⬍兩K兩⬍⫹⬁ and 0⬍兩K

⬘

兩⬍⫹⬁, i.e., for propagating and evanescent waves. Note that the presence of the factors ␥ ^{in Eq.} 共8兲is a consequence of the definition of the angular spectrum of the field 关Eq. 共2兲兴 by integration over the parallel wave vector K. When using an integration over the solid angle⍀, with k d⍀⫽d²K/␥^{, these} factors disappear from the reciprocity relations. Neverthe- less, the presence of evanescent waves would involve com- plex angles in the ⍀ representation, so that the K represen- tation looks more appropriate.

IV. EXTENDED UNITARITY RELATIONS

The second basic property of the S matrix is unitarity. It is satisfied by the S matrix of a lossless system共elastic scatter- ing兲. It is often assumed that the unitarity condition involves only propagating waves. This belief is, in fact, based on the use of asymptotic fields in the derivation. Indeed, unitarity has been studied extensively in far-field共asymptotic兲scatter- ing 关2兴 or scattering of source-free fields in the angular- spectrum representation 关9,25兴. Conversely, the extension of unitarity relations to wave fields containing evanescent com- ponents has received little attention until now, except in the context of electromagnetic waveguide theory关24兴. Neverthe- less, as discussed in the Introduction of the present paper, extended unitarity relations could be helpful in various re- cent applications, such as time-reversed acoustics, near-field optics, or propagation through random media from the view- point of random-matrix theory. In this section, we show how such relations may be derived in the framework of free-space scattering from a localized potential.

In eitherR^⫺ or R^⫹, the current density associated with the field is J(r)⫽A Im兵⌿*(r)ⵜ⌿(r)其, where A is a con- stant, Im denotes the imaginary part, and the asterisk is the complex conjugate. With reference to Fig. 2, energy 共or probability兲 conservation states that the total flux ␾

⫽兰J•n dS flowing outside the volume V vanishes. When the radius R of the sphere tends to infinity, the contribution of the flux through the portions of the sphere surface between the two planes z⫽z^⫺ and z⫽z^⫹ vanishes. Finally, energy 共or probability兲conservation reads

␾z⫺⫽␾z⫹, 共9兲 where

␾z^⫾⫽

冕

z⫽z^⫾

J共R,z^⫾兲•n d²R. 共10兲 FIG. 2. Closed volume used for the application of Green’s iden- tity and the energy共or probability兲balance.

Using the angular-spectrum representation Eq. 共2兲, the cur- rent ␾z⫾ accross a plane z⫽z^⫾ in R^⫾ can be cast in the following form:

␾z⫾⫽A

冕

K²⭐k²␥关兩a^⫾共K兲兩²⫺兩b^⫾共K兲兩²兴d²K

⫹A

冕

^K2⬎k²␥关a^⫾共K兲b^⫾*_共K兲⫺a^⫾*_共K兲b^⫾共K兲兴d²K.

共11兲 This expression for the current deserves some comment. It explicitly shows two separated contributions, one stemming from propagating waves (K²⭐k²), and the other stemming from evanescent waves only (K²⬎k²). Note that the latter is a crossed term between counterdecaying evanescent waves.

For a given K such that K²⬎k², if either a(K) or b(K) vanishes, then the associated current also vanishes. For the aim pursued in this section, it is precisely the existence of this contribution to the current that leads to the extended unitarity relations of the S matrix. Introducing Eq.共11兲into Eq. 共9兲, and using the definition of the S matrix 共3兲, one obtains the three following relations, involving scattering be- tween two propagating waves, between two evanescent waves, and between one evanescent and one propagating wave:

冕

K⬘^⭐k

␥共K

⬙

兲

␥共K

⬘

兲S共K,K

⬘

兲S^†共K

⬙

^,K

⬘

兲d²K

⬘

⫽␦共K⫺K

⬙

兲U for K⭐k,K

⬙

^{⭐k, 共12兲}

⫽S共K,K

⬙

兲 for K⭐k,K

⬙

⬎k, 共13兲

⫽S共K,K

⬙

兲⫺S*_共⫺K,⫺K

⬙

兲

for K⬎k,K

⬙

⬎k, 共14兲

where U is the 2⫻2 unit matrix. The superscript † denotes the conjugated and transposed matrix.

Equation共12兲is the well-known unitarity condition of the S matrix restricted to the homogeneous components of the fields, which was obtained previously 关2,3,25兴. Using the partitioned form of the S matrix Eq.共4兲, this condition can be developed in terms of the generalized reflection and trans- mission coefficients. The resulting four expressions are the generalized Stokes relations of surface optics, which were derived in 关25兴. Equations 共13兲 and 共14兲 express, in the S matrix formalism, probability 共or energy兲conservation in a scattering process in which the incoming and/or the outgoing fields contain evanescent components. Hence, they can be considered as extended unitarity conditions for the S matrix of fields containing evanescent components. Because the S- matrix formalism is used in many fields of theoretical phys- ics, and because there has been increasing interest in phe- nomena involving direct use and measurement of evanescent fields, we believe that this result may have important conse- quences and applications. For example, the development of Eqs. 共13兲 and 共14兲 in terms of generalized reflection and transmission coefficients leads to relationships that extend

the generalized Stokes relations to evanescent fields. Such relations may be useful, e.g., in the context of surfaces, thin films, and multilayers optics. The generalized Stokes rela- tionships and their extension to fields containing evanescent components are given in the Appendix.

V. TIME-REVERSAL INVARIANCE

In this section we discuss time-reversal invariance for classical scalar waves 共we exclude from the discussion the question of time reversal in quantum mechanics which is difficult to separate from the measurement problem兲.

A. Time reversal in angular-spectrum representation Let⌿(r,t) be a classical scalar field in the time domain, and ⌿(r,␻) its frequency spectrum. Because ⌿(r,t) is a real function, its frequency spectrum satisfies ⌿(r,␻⁾

⫽⌿*(r,⫺␻). From this condition, it is straightforward to show that the time-reversed field ⌿(r,⫺t) has a frequency spectrum ⌿*(r,␻). Hence time reversal of ⌿(r,t) is equivalent to complex conjugation of⌿(r,␻) throughout all space. Note that time reversal is not equivalent to phase con- jugation in only one plane 关29兴. We shall come back to this point later.

Let us see what time reversal means in terms of the an- gular spectrum of the monochromatic field ⌿(r) 共the vari- able␻ is omitted in the following兲. From complex conjuga- tion of Eq. 共2兲 and the change of variable K→ÀK, one obtains

⌿*_共r兲⫽

冕

^{a*共⫺K兲exp共iK•R⫺i␥*z兲d2K}

⫹

冕

^{b*共⫺K兲exp共iK•R⫹i␥*z兲d2K. 共15兲}

The symbols⫾ have been omitted in Eq.共15兲because we do not need to specify at this stage whether the field propagates inR^⫹or inR^⫺. We see that, in terms of angular spectrum, time reversal is equivalent to the transformation a(K)exp(i␥^z)→a*(⫺K)exp(⫺i␥*z) and b(K)exp(⫺i␥^z)

→b*(⫺K)exp(i␥*z) for all values of z in eitherR^⫹ orR^⫺. Note that Eq.共15兲is valid in regions of space for which 兩z兩 remains finite, so that there is no divergence of time-reversed evanescent waves when 兩z兩 increases.

B. Time-reversal invariance: consequence for the S matrix In order to study the implication of time-reversal invari- ance for the S matrix, let us consider a monochromatic field

⌿1(r) which is a solution of the scattering problem depicted in Fig. 1, the scattering potential being described by the S matrix S. Let⌿2(r) be the time reversal of the field⌿1(r).

In terms of the angular spectrum, this means that a₂^⫹共K兲exp共i␥^z兲⫹b₂^⫹共K兲exp共⫺i␥^z兲

⫽a₁^⫹*_共⫺K兲exp共⫺i␥*z兲⫹b₁^⫹*_共⫺K兲exp共i␥*z兲 共16兲

R. CARMINATI et al. PHYSICAL REVIEW A 62 012712

012712-4

for all values of z inR^⫹and

a₂^⫺共K兲exp共i␥^z兲⫹b₂^⫺共K兲exp共⫺i␥^z兲

⫽a₁^⫺*_共⫺K兲exp共⫺i␥*z兲⫹b₁^⫺*_共⫺K兲exp共i␥*z兲 共17兲 for all values of z inR^⫺. The scattering problem is invariant under time reversal if, and only if,⌿2(r) is also a solution of the scattering problem, described by the same S matrix S.

This means that the outgoing vectors ⌿1 out(K)

⫽关b₁^⫺(K) a₁^⫹(K)兴 and ⌿2

out(K)⫽关b₂^⫺(K) a₂^⫹(K)兴 are connected to the incoming vectors ⌿1

in(K)

⫽关a₁^⫺(K) b₁^⫹(K)兴 and ⌿2

in(K)⫽关a₂^⫺(K) b₂^⫹(K)兴, re- spectively, by relation 共3兲. Introducing these conditions into Eqs. 共16兲and共17兲leads to the following relations:

冕

K⬘^⭐kS共K,K

⬘

兲S*_共⫺K

⬘

^,K

⬙

兲d²K

⬘

⫽␦共K⫹K

⬙

兲U for K⭐k,K

⬙

^{⭐k, 共18兲}

⫽⫺S共K,⫺K

⬙

兲 for K⭐k,K

⬙

⬎k, 共19兲

⫽S*_共⫺K,K

⬙

^兲⫺S共K,⫺K

⬙

兲

for K⬎k,K

⬙

⬎k, 共20兲

where the asterisk denotes the conjugated matrix. These re- lations express the condition of time-reversal invariance in terms of the S matrix of fields containing evanescent compo- nents.

The set of Eqs.共18兲–共20兲is very similar to the set of Eqs.

共12兲–共14兲, which describes energy conservation. In fact, it is easy to see that these two sets of equations are equivalent, provided that the reciprocity relation共8兲is satisfied. Indeed, Eqs.共18兲–共20兲are transformed into Eqs.共12兲–共14兲by using Eq. 共8兲 and changing K

⬙

→⫺K

⬙

. The result we have ob- tained can be summarized as follows: the condition of time- reversal invariance is equivalent to both energy conservation 共extended unitarity condition兲and reciprocity 共symmetry of the S matrix兲. Although this result was already known for source-free fields 关23兴, we have demonstrated that it holds for fields containing evanescent components.

C. Time-reversal invariance and reciprocity

The results in this paper also provide a basis to discuss the link between time-reversal symmetry and reciprocity, which is sometimes confusing in the literature 关3兴 共see also a dis- cussion of this point in Ref.关26兴兲. For a scattering system in which energy is conserved关Eqs.共12兲–共14兲are satisfied兴, the conditions of time-reversal invariance 关Eqs. 共18兲–共20兲兴 and reciprocity关Eq.共8兲兴are equivalent. This is probably the rea- son why time-reversal symmetry and reciprocity are often mistaken. In particular, reciprocity is often presented as a consequence of time-reversal invariance 关3兴. This is in gen- eral incorrect. For example, we have seen that imposing time-reversal invariance leads to Eqs. 共18兲–共20兲and not to

the reciprocity condition Eq.共8兲. Moreover, a scattering sys- tem may be reciprocal, without being conservative关Eq.共8兲is satisfied, but not Eqs. 共12兲–共14兲兴. In this case, the system is not invariant under time reversal 关Eqs. 共12兲–共14兲cannot be satisfied兴. These conclusions hold for fields with or without evanescent components.

D. Experimental feasibility of time reversal

An important problem is the experimental achievement of time reversal in a situation involving wave scattering. In op- tics, the development of phase conjugating mirrors provides a practical tool to produce fields that are conjugates of each other in a given plane. Nevertheless, it has been shown that this type of phase conjugation is not equivalent to time re- versal when the fields involved contain evanescent compo- nents关29兴. The subject of time reversal of fields containing evanescent components is of particular interest in the context of time-reversed acoustics关22兴. In this technique, the acous- tic field in a direct situation is recorded on a given surface⌺, after scattering by an arbitrary object. In the reversed situa- tion, the time-reversed field is emitted from the surface⌺, in the presence of the same scattering object. In the frequency domain, the fields in the two situations are complex conju- gates of each other on⌺. Thus, this experiment is equivalent to achieving acoustic phase conjugation on the surface⌺. In both optics关21兴and acoustics关22兴, the possibility of achiev- ing time reversal of both the homogeneous and evanescent components of the field by phase conjugation may be ques- tioned.

The first part of the answer is given by showing that phase conjugation on the surface of a closed cavity共or equivalently along two planes z⫽z₁ and z⫽z₂) is equivalent to time re- versal at all points inside the cavity 共or in the strip z₁⬍z

⬍z₂). This assertion is a consequence of the following re- sult: two fields defined inside the strip z₁⬍z⬍z₂ that are complex conjugates in the two planes z⫽z₁ and z⫽z₂ are complex conjugates at all points within the strip z₁⬍z⬍z₂. Therefore, they are time reversed from each other in the cavity. This result holds for fields containing evanescent components. It can be derived by extending the discussion in Ref. 关29兴 to a situation involving phase conjugation along two planes z⫽z₁ and z⫽z₂. Consequently, phase conjuga- tion on a closed surface 共or along two planes兲 may be a practical way to achieve complete time reversal of a field.

The second part of the answer must take into account the presence of sources inside the cavity in the direct experi- ment. In theory, reversing time leads automatically to the transformation of all primary sources into sinks. Therefore, to achieve time reversal experimentally, the field on the sur- face of the cavity has to be time reversed, and the sources have to be transformed into sinks. This is probably the great- est experimental challenge. This is also the necessary condi- tion to obtain complete time reversal 共i.e., with evanescent waves included兲and achieve, for example, time-reversed fo- cusing below the diffraction limit. The necessity of replacing sources by sinks in the time-reversed situation can be under- stood as follows. In the direct situation, a subwavelength source radiates a localized field whose angular spectrum con-

tains evanescent waves. In the reversed situation, the sink is equivalent to a source with opposite phase. This localized source also radiates evanescent waves which allow the time- reversed field to focus below the diffraction limit.

VI. CONCLUSION

In summary, we have derived general properties of the S matrix of fields containing evanescent components. In par- ticular, we have shown that energy 共or probability兲conser- vation leads to relationships that extend the well-known uni- tarity condition of the asymptotic S matrix. Using the partitioned S matrix, we have shown that these relationships lead to extended Stokes relations. We have also obtained different relationships as a consequence of time-reversal in- variance. On this basis, we have discussed the link between unitarity, time-reversal symmetry, and reciprocity. With the increasing interest in techniques based on measurement and control of evanescent waves, we think that this work should find broad applications. In particular, we have briefly dis- cussed its implications in time reversal of scattered fields by phase conjugation.

ACKNOWLEDGMENTS

We wish to thank M. Fink, A. Garcia Martin, and J. L.

Thomas for helpful discussions. This work was supported by the French-Spanish Integrated Program PICASSO. M.N.-V.

acknowledges the support of Fundacio´n Ramo´n Areces.

APPENDIX: GENERALIZED STOKES RELATIONS FOR FIELDS CONTAINING EVANESCENT WAVES In this Appendix, we summarize the relations that are obtained by inserting the coefficients of the partitioned S matrix Eq. 共4兲 into relations 共12兲–共14兲. The first four rela- tions are the generalized Stokes relationships obtained in Ref. 关25兴. The other relations are extensions of the Stokes relationships to fields containing evanescent components.

We use the notations␥

⬘

^⫽␥^(K

⬘

^{) and}␥

⬙

^⫽␥^(K

⬙

^).

1. Relations involving homogeneous waves only Relations valid for K⭐k and K

⬙

^⭐k:

冕

K⬘^⭐kd²K

⬘

关␳共K,K

⬘

兲␶*_共K

⬙

^,K

⬘

兲⫹t共K,K

⬘

兲r*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽0, 共A1兲

冕

^K_⬘⭐k

d²K

⬘

关␳共K,K

⬘

兲␳*_共K

⬙

^,K

⬘

兲⫹t共K,K

⬘

兲t*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽␦共K⫺K

⬙

兲, 共A2兲

冕

K⬘^⭐kd²K

⬘

关r共K,K

⬘

兲r*_共K

⬙

^,K

⬘

兲⫹␶共K,K

⬘

兲␶*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽␦共K⫺K

⬙

兲, 共A3兲

冕

^K_⬘⭐k

d²K

⬘

关r共K,K

⬘

兲t*_共K

⬙

^,K

⬘

兲⫹␶共K,K

⬘

兲␳*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽0. 共A4兲

2. Relations involving conversion of homogeneous to evanescent waves

Relations valid for K⭐k and K

⬙

⬎k:

冕

^K_⬘⭐k

d²K

⬘

关␳共K,K

⬘

兲␶*_共K

⬙

^,K

⬘

兲⫹t共K,K

⬘

兲r*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽t共K,K

⬙

兲, 共A5兲

冕

^K_⬘⭐k

d²K

⬘

关␳共K,K

⬘

兲␳*_共K

⬙

^,K

⬘

兲⫹t共K,K

⬘

兲t*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽␳共K,K

⬙

兲, 共A6兲

冕

^K_⬘⭐k

d²K

⬘

关r共K,K

⬘

兲r*_共K

⬙

^,K

⬘

兲⫹␶共K,K

⬘

兲␶*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽r共K,K

⬙

兲. 共A7兲

冕

^K_⬘⭐k

d²K

⬘

关r共K,K

⬘

兲t*_共K

⬙

^,K

⬘

兲⫹␶共K,K

⬘

兲␳*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽␶共K,K

⬙

兲. 共A8兲

3. Relations involving conversion of evanescent to evanescent waves

Relations valid for K⬎k and K

⬙

⬎k:

冕

^K_⬘⭐k

d²K

⬘

关␳共K,K

⬘

兲␶*_共K

⬙

^,K

⬘

兲⫹t共K,K

⬘

兲r*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽t共K,K

⬙

兲⫺t*_共⫺K,⫺K

⬙

兲, 共A9兲

冕

K⬘^⭐kd²K

⬘

关␳共K,K

⬘

兲␳*_共K

⬙

^,K

⬘

^兲⫹t共K,K

⬘

兲t*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽␳共K,K

⬙

兲⫺␳*_共⫺K,⫺K

⬙

兲, 共A10兲

冕

K⬘^⭐kd²K

⬘

关r共K,K

⬘

兲r*_共K

⬙

^,K

⬘

兲⫹␶共K,K

⬘

兲␶*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽r共K,K

⬙

^{兲⫺r*共⫺K,⫺K}

⬙

兲, 共A11兲

冕

K⬘^⭐kd²K

⬘

关r共K,K

⬘

兲t*_共K

⬙

^,K

⬘

兲⫹␶共K,K

⬘

兲␳*_共K

⬙

^,K

⬘

兲兴␥

⬙

␥

⬘

⫽␶共K,K

⬙

兲⫺␶*_共⫺K,⫺K

⬙

兲. 共A12兲

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