Perfect lensing with phase conjugating surfaces: Towards practical realization
aa r X i v : . [ phy s i c s . op ti c s ] A ug Perfect lensing with phase conjugating surfaces:Towards practical realization
Stanislav Maslovski and Sergei Tretyakov Departmento de Engenharia Electrot´ecnica,Instituto de Telecomunica¸c˜oes - Universidade de Coimbra,P´olo II, 3030 Coimbra, Portugal Aalto University, School of Electrical Engineering,P.O. 13000, FI-00076 Aalto, Finland
E-mails: [email protected], sergei.tretyakov@aalto.fi(Dated: October 11, 2018)It is theoretically known that a pair of phase conjugating surfaces can function as a perfect lens,focusing propagating waves and enhancing evanescent waves. However, the known experimentalapproaches based on thin sheets of nonlinear materials cannot fully realize the required phase con-jugation boundary condition. In this paper we show that the ideal phase conjugating surface is inprinciple physically realizable and investigate the necessary properties of nonlinear and nonrecipro-cal particles which can be used to build a perfect lens system. The physical principle of the lensoperation is discussed in detail and directions of possible experimental realizations are outlined.
PACS numbers: 42.65.Hw, 78.67.Pt, 81.05.Xj
I. INTRODUCTION
The perfect lens is a device which focuses the field of a point source into a point, that is, the perfect lens focuses bothpropagating and evanescent fields. It is known that a planar slab of the ideal Veselago medium with the relativepermittivity and permeability both equal to − metasurface at which the incident waves refract negatively,then a parallel pair of such planar sheets will mimic the operation of the Veselago lens for the propagating planewaves. Moreover, if this metasurface supports surface modes (surface plasmon-polaritons) within a wide range of thetangential propagation factors k t > k (where k is the free space wavenumber), then also the impinging evanescent plane waves will interact resonantly with the sheets and will be tunneled through the lens with an enhanced amplitude,due to the electromagnetic coupling between the surface states excited on the sheets. Such subwavelength imagingwith linear plasmon-polariton resonant grids was theoretically predicted and confirmed experimentally in a numberof works. In 2003, we showed that two parallel sheets with phase conjugating boundary conditions for tangential fields onthe two sides of the sheets E t+ = E ∗ t − , H t+ = H ∗ t − (1)have the necessary properties of the perfect lens outlined above. In these conditions that are written for the complexamplitudes of the time-harmonic fields (symbol ∗ denotes complex conjugation operation) the indices ± indicate thefield values on the two sides of an infinitely thin phase conjugating sheet.Obviously, boundary conditions (1) cannot be realized using linear materials, and in the same paper the use ofthree-wave mixing in a nonlinear layer was proposed as an approach to realization of this effect. Phase conjugation and“time-reversal” devices were studied also earlier for other applications. It is interesting that in the same year (2003)an experimental microwave realization of a phase conjugating layer was published independently from our work. Later, the concept of perfect lensing based on two nonlinear sheets was studied theoretically in Ref. 10. Alternativeexperimental realizations of nonlinear negative refraction effect were published in Refs. 11,12.However, in known devices based on antenna arrays with mixers, or sheets of nonlinear dielectrics, or arrays ofonly electric or only magnetic particles with nonlinear insertions, the phase conjugated (“time-reversed”) productscreate waves propagating symmetrically to the both sides of the sheet, i.e., there appears a retrodirected wave FIG. 1: An ideal Veselago lens: A planar slab of a double-negative (DNG) material with the medium parameters ε = − ε and µ = − µ in free space. propagating back to the source. While the perfect lens operation can be theoretically approached even in suchsystems if the amplitudes of the nonlinear products tend to infinity in the assumption of nonphysical infinitely strongexternal pumping, the ideal phase conjugating boundary conditions (6) cannot be realized within this scenario.In this paper we discuss the physical meaning of the ideal complex-conjugation boundary conditions (1) and outlinepossible approaches for realization of such surfaces, which would potentially lead to creation of super-resolution lenses.The paper is organized as follows. In Section II we consider physical processes taking place at a phase conjugatingboundary and demonstrate that such boundary may be equivalently represented with pairs of electric and magneticsurface currents reacting (nonlinearly) to the applied magnetic and electric fields. In Section III a realization of thephase conjugating boundary with an array of bi-anisotropic inclusions is proposed and studied and the necessaryconditions on nonlinear susceptibilities of the inclusions are established. In Section IV a possible microwave design ofsuch inclusions is proposed. II. THE PHYSICAL MEANING OF THE COMPLEX-CONJUGATING BOUNDARY CONDITIONSA. Complex-conjugating boundary and perfect lens
Let us start from outlining the idea from our paper in Ref. 8. Consider an ideal Veselago lens depicted in Figure 1.Let the relative permittivity and permeability of the medium surrounding the lens be equal to 1 and the relativeparameters of the lens material to − ω , respectively. At the lens interfaces the tangentialcomponents of the fields satisfy the usual Maxwellian continuity conditions. One may notice that in this systemthe only difference between the time-harmonic [the time dependence is of the form exp(+ jωt )] field equations in theVeselago slab (region 2) ∇ × E = jωµ H , ∇ × H = − jωε E (2)and the analogous equations in the free-space regions is the sign in front of the imaginary unit. A substitution E (old) , H (old) ⇒ C E ∗ (new) , C H ∗ (new) (3)( C is an arbitrary constant; here and thereafter ∗ denotes the complex conjugation) into the field equations in region2 reformulates the problem in terms of the new field vectors in which the field equations become the same in all threeregions: ∇ × E = − jωµ H , ∇ × H = jωε E , (4)that are simply the Maxwell equations in free space. The boundary conditions on the two interfaces, however, are nomore the standard continuity conditions, but they involve complex conjugation: E t (1 , = C E ∗ t (2) , H t (1 , = C H ∗ t (2) . (5)The constant C describes the “transformation efficiency” of the nonlinear surface which transforms fields into thecomplex-conjugate state. In the known experimental realizations of phase conjugation in electrically thin layers (e.g.,Ref. 12), the efficiency has been rather small. However, by choosing a small value of C in (5), we arrive at a structurewith asymmetric properties with respect to the two sides of the surface. Indeed, complex conjugating and dividing(5) by C ∗ , we see that in this case the weak fields inside the lens should be enhanced by the surface in the same rateas they are suppressed when the surface is excited from outside. For this reason we will concentrate on the simplestchoice of C = 1, as in Ref. 8, described by the boundary conditions E t (1 , = E ∗ t (2) , H t (1 , = H ∗ t (2) . (6)In this case the complex-conjugating surface has symmetric properties with respect to its two sides, and for the ideallens operation the amplitudes of the field should not change across the sheets.Now it becomes evident that the problem involving an ideal Veselago slab is mathematically equivalent to theproblem dealing with a pair of conjugating surfaces in free space, provided that the field sources are outside of region2. Therefore, in the latter system the field solutions are the same as in the Veselago slab, and because of this thephysical phenomena taking place at the interfaces of region 2 are also the same: The propagating plane waves arerefracted negatively at the interfaces, and the evanescent modes are enhanced due to the excitation of coupled surfaceplasmon-polariton pairs. In this regard, a pair of phase conjugating planes is indistinguishable from a perfect lensproposed by Pendry. In what follows we concentrate on physical properties of such phase conjugating sheets.
B. Plane-wave propagation across the phase conjugating sheet
Let us consider a single phase conjugating surface located at z = 0. We decompose the tangential electric andmagnetic fields into plane waves at both sides of the surface: E t ( x, y ) (cid:12)(cid:12)(cid:12) z = ± = 1(2 π ) Z Z E t ( k x , k y ) (cid:12)(cid:12)(cid:12) z = ± e − j ( k x x + k y y ) dk x dk y , (7) H t ( x, y ) (cid:12)(cid:12)(cid:12) z = ± = 1(2 π ) Z Z H t ( k x , k y ) (cid:12)(cid:12)(cid:12) z = ± e − j ( k x x + k y y ) dk x dk y . (8)It is easy to see that the boundary conditions (6) require that the plane-wave components satisfy E t ( k x , k y ) (cid:12)(cid:12)(cid:12) z =+0 = E ∗ t ( − k x , − k y ) (cid:12)(cid:12)(cid:12) z = − , (9) H t ( k x , k y ) (cid:12)(cid:12)(cid:12) z =+0 = H ∗ t ( − k x , − k y ) (cid:12)(cid:12)(cid:12) z = − . (10)From these relations we immediately realize that the propagating modes refract negatively at the conjugating interfacedue to the change in the sign of the tangential component of the wave vector k t = k x x + k y y . This is very differentfrom the case of the same refraction at an interface with a double-negative medium where the normal component ofthe wave vector changes sign.What about the evanescent modes? It can be shown that they are at resonance with the phase conjugating surfaceso that the strong (theoretically infinite) reflection takes place at the surface, although the field transformation in thesheets (the complex conjugate operation) does not amplify the fields. The details are given in Ref. 8; here we will justtry to convince the reader with the following simple physical argument.Let us consider a single plane-wave component interacting with a single phase conjugating surface in free space.The plane wave is incident from the half space z <
0. On both sides of the phase conjugating sheet the tangentialfields satisfy the usual relation between the fields in a free-space plane wave (the following equations are valid for TMor TE waves separately): E t ( k x , k y ) (cid:12)(cid:12)(cid:12) z = ± = − Z TM , TE z × H t ( k x , k y ) (cid:12)(cid:12)(cid:12) z = ± . (11)Here the free-space wave impedances for TM- and TE-polarized waves read Z TM = η s − k k , Z TE = η q − k k , (12)where k = ω/c is the free-space wavenumber, η = p µ /ε , and k = k x + k y . But on the other hand the tangentialfields satisfy also the boundary condition (6). Together with the impedance relation (11) this leads to E t ( k x , k y ) (cid:12)(cid:12)(cid:12) z = − = E ∗ t ( − k x , − k y ) (cid:12)(cid:12)(cid:12) z =+0 = − Z ∗ TM , TE z × H ∗ t ( − k x , − k y ) (cid:12)(cid:12)(cid:12) z =+0 = − Z ∗ TM , TE z × H t ( k x , k y ) (cid:12)(cid:12)(cid:12) z = − . (13)Thus, a wave incident on the sheet from one side “sees” the surface impedance which equals to the complex conjugateof the wave impedance in free space. The reflection coefficient reads R TM , TE = Z ∗ TM , TE − Z TM , TE Z ∗ TM , TE + Z TM , TE . (14)If the wave is a propagating wave, that is, | k t | ≡ k t ≡ q k x + k y < k ≡ ω/c , then its wave impedance is areal number, and the reflection coefficients equal zero. This proves that the propagating modes experience negativerefraction without any reflection.If the wave is evanescent, that is, k t ≡ q k x + k y > k , the wave impedances (11) are purely imaginary and thereflection coefficient is infinite. The transmission coefficient is also infinite due to the boundary condition (5). Thisis obviously a resonant condition which can be also understood as a condition for existence of a surface mode. Notethat this resonant condition holds for all values of the tangential wave number k t > k , which is the condition forperfect lensing of all evanescent field components.The physical reason of such a resonance is rather simple. Consider, for instance, the waves of TM polarization.The characteristic impedance of an evanescent TM wave (12) can be written as Z TM = − jη α/k , where α is thedecay factor: α = p k − k . We conclude that a TM evanescent wave has capacitive characteristic impedance. Dueto the conjugating interface, the same impedance of the matching TM wave behind the interface (at z = +0) is seenas inductive in front of the interface (at z = − were the first to identify and explainthe resonance of the same nature that happens at the border of a double-positive and a double-negative material.Therefore, when an incident evanescent wave excites a conjugating surface, it resonantly excites a surface modethat matches its transversal propagation factor and this results in very strong (theoretically infinite) reflected andtransmitted waves at the surface. The strong reflection was also found to be the key to sub-wavelength imaging in apair of parametrically pumped nonlinear nonmagnetic sheets studied in Ref. 10 The above discussion shows that ina metasurface realizing the boundary conditions (6) the reflection and transmission coefficients for evanescent modestend to infinity due to a high-quality resonance (theoretically, with an infinite quality factor), while, in Ref. 10, thesurface itself parametrically amplifies the fields. C. Equivalent surface currents on the phase conjugating sheet
One may notice that the boundary conditions (6) imply discontinuity of both tangential electric and magnetic fieldsacross the phase conjugating plane. The jumps of the fields can be expressed as follows: E t ( x, y ) (cid:12)(cid:12)(cid:12) z =+0 − E t ( x, y ) (cid:12)(cid:12)(cid:12) z = − = − j Im[ E t ( x, y )] (cid:12)(cid:12)(cid:12) z = − , (15) H t ( x, y ) (cid:12)(cid:12)(cid:12) z =+0 − H t ( x, y ) (cid:12)(cid:12)(cid:12) z = − = − j Im[ H t ( x, y )] (cid:12)(cid:12)(cid:12) z = − . (16)These jumps are related to the equivalent magnetic and electric surface currents that exist on the surface: z × J m ( x, y ) = E t ( x, y ) (cid:12)(cid:12)(cid:12) z =+0 − E t ( x, y ) (cid:12)(cid:12)(cid:12) z = − = − j Im[ E t ( x, y )] (cid:12)(cid:12)(cid:12) z = − = 2 j Im[ E t ( x, y )] (cid:12)(cid:12)(cid:12) z =+0 , (17) − z × J e ( x, y ) = H t ( x, y ) (cid:12)(cid:12)(cid:12) z =+0 − H t ( x, y ) (cid:12)(cid:12)(cid:12) z = − = − j Im[ H t ( x, y )] (cid:12)(cid:12)(cid:12) z = − = 2 j Im[ H t ( x, y )] (cid:12)(cid:12)(cid:12) z =+0 . (18)Let us stress already at this point that the relations (17)–(18) are the only physical conditions one has to satisfyin a subwavelength imaging device based on phase conjugating sheets. Nothing more is required! Essentially, theserelations tell us that it is enough in practice to realize a metasurface that reacts with certain magnetic and electricsurface currents to the imaginary parts of the tangential electric and magnetic fields at a given side of the surface.This also shows that there is no need for extra-strong pumping or super-efficient nonlinear conversion as in Ref. 10.Moreover, from (17)–(18) we observe that the induced electric current should be proportional to the magnetic field onthe sheet (more precisely, to its imaginary part). Likewise, the induced magnetic current is proportional to the electric field. This is quite different from the known approaches based on layers of nonlinear dielectrics or magnetics. Let us decompose each of these surface currents into a sum of two currents: J e = J (1)e + J (2)e , J m = J (1)m + J (2)m ,where − z × J (1)e = − H t (cid:12)(cid:12)(cid:12) z = − , z × J (1)m = − E t (cid:12)(cid:12)(cid:12) z = − , (19) − z × J (2)e = H t (cid:12)(cid:12)(cid:12) z =+0 , z × J (2)m = E t (cid:12)(cid:12)(cid:12) z =+0 . (20)One can see that the pair of the surface currents J (1)e , J (1)m is essentially an equivalent Huygens source defined at theplane z = −
0. In the half-space z > z < z > J (1)e , J (1)m when concerned with the half-space z > J (2)e , J (2)m when concernedwith the half-space z <
0. These currents form a Huygens source defined at z = +0 plane. In the half-space z < z > J (1)e , J (1)m plays another role when concerned with the half-space z < − z × J (1)e = − H ∗ t (cid:12)(cid:12)(cid:12) z =+0 , z × J (1)m = − E ∗ t (cid:12)(cid:12)(cid:12) z =+0 , (21)from which it is evident that these currents can be identified also as an equivalent source located at the plane z = +0that produces at z < z >
0. Respectively, J (2)e , J (2)m produce the conjugated field in the region z > • The pair J (1)e , J (1)m cancels the field incident from z < z > z > z < • The pair J (2)e , J (2)m cancels the field incident from z > z < z < z > III. PHASE CONJUGATING SURFACE AS AN ARRAY OF BI-ANISOTROPIC NONLINEARINCLUSIONS
From the above results we see that the ideal phase conjugating surface should respond to the fields with both electricand magnetic polarization. Dependence of the induced electric current on the magnetic field and vice versa suggeststhat the structure should have some magneto-electric coupling. In this section we investigate if it is possible to realizethe ideal phase conjugating boundary conditions (6) with a planar array of nonlinear bi-anisotropic particles.
A. General requirements on susceptibilities of inclusions
Let us first find out how the total induced electric and magnetic surface current densities depend on the incident electric and magnetic fields in the array plane. To do that, we consider an isolated phase conjugating surface in thefield of a single
TM (or TE) polarized plane electromagnetic wave (propagating or evanescent). Taking into accountthe conjugating boundary condition (6) we may formally write the total tangential electric and magnetic fields onboth sides of the surface as E t ( x, y ) (cid:12)(cid:12)(cid:12) z = − = (1 + R TM , TE ) E inct ( x, y ) (cid:12)(cid:12)(cid:12) z =0 , (22) E t ( x, y ) (cid:12)(cid:12)(cid:12) z =+0 = (1 + R ∗ TM , TE ) (cid:0) E inct ( x, y ) (cid:1) ∗ (cid:12)(cid:12)(cid:12) z =0 , (23) H t ( x, y ) (cid:12)(cid:12)(cid:12) z = − = (1 − R TM , TE ) H inct ( x, y ) (cid:12)(cid:12)(cid:12) z =0 , (24) H t ( x, y ) (cid:12)(cid:12)(cid:12) z =+0 = (1 − R ∗ TM , TE ) (cid:0) H inct ( x, y ) (cid:1) ∗ (cid:12)(cid:12)(cid:12) z =0 , (25)where the reflection coefficients R TM , TE are given by (14), from which we notice that R ∗ TM , TE = − R TM , TE . The aboveexpressions hold for both propagating and evanescent waves incident from the half space z < − z × J m ( x, y ) = 2 j Im( E inct ( x, y )) (cid:12)(cid:12)(cid:12) z =0 + 2 R TM , TE Re( E inct ( x, y )) (cid:12)(cid:12)(cid:12) z =0 , (26) z × J e ( x, y ) = 2 j Im( H inct ( x, y )) (cid:12)(cid:12)(cid:12) z =0 − R TM , TE Re( H inct ( x, y )) (cid:12)(cid:12)(cid:12) z =0 . (27)The addends on the right-hand side of (26)–(27) that are proportional to the imaginary part of the incident fieldare relevant for the propagating waves (for these waves R TM , TE = 0), while for the evanescent waves the addendsproportional to the real part of the field are of the most importance, because | R TM , TE | → ∞ for these modes. It isinstructive to compare these observations with the discussion in Sec. II C. From (17)–(18) it follows that in termsof the total tangential fields at a given side of the phase conjugating sheet, the conditions for both propagating andevanescent waves are the same: Eqs. (17)–(18) do not distinguish these waves. Physically, this is because the locationswhere one must measure the fields and where one must create the surface currents are not at the same point , if onewants to approach a design directly suggested by these equations. Indeed, the mathematical form of Eqs. (17)–(18)demands that the field values must be taken at a point slightly displaced off the surface z = 0. On the contrary, inan array of particles (considered here as point objects) the equivalent surface currents depend only on the fields inthe array plane and, as we will see soon, the required reaction to this field happens to be different for the two typesof waves.A related observation is that in a realistic structure, e.g., an array of nonlinear polarizable bi-anisotropic inclusions,it is not the incident field, but the local field E loct , H loct , that excites each and every inclusion in the structure. Thelatter has a contribution from the secondary field of the induced currents. We may therefore write for the signals atthe frequency of the incident wave: E loct ( x, y ) = E inct ( x, y ) + β ee · J e ( x, y ) , (28) H loct ( x, y ) = H inct ( x, y ) + β mm · J m ( x, y ) , (29)where β ee , mm are the so-called interaction dyadics. For arbitrary distributed currents these dyadics are understoodas operators acting on the currents. However, for the following it is enough to consider the currents of the form J e , m ( x, y ) = J e , m k t exp( − j k t · r ) + J e , m − k t exp( j k t · r ). In this case, in order for (28)–(29) to hold in a simple dyadic sense,the interaction dyadics must satisfy β ee , mm ≡ β ee , mm ( k t ) = β ee , mm ( − k t ), i.e., the lattice (not the particles!) musthave a center of symmetry. There are no cross-terms in (28)–(29) because the tangential magnetic (electric) field ofan array of tangential electric (magnetic) dipoles vanishes in the plane of the array.Additionally, the induced electric and magnetic currents must be sensitive to the phase of the external field, becausethe conjugating boundary reacts differently to the real and imaginary parts of the tangential electric and magneticfields. Therefore, the inclusions must react differently to the corresponding components of the local fields. Based onthe above discussion we may write J e = α reee · Re( E loct ) + jα imee · Im( E loct ) + α reem · Re( H loct ) + jα imem · Im( H loct ) , (30) J m = α reme · Re( E loct ) + jα imme · Im( E loct ) + α remm · Re( H loct ) + jα immm · Im( H loct ) , (31)where α re , imee , mm , me , em are the dyadic polarizabilities to the real and imaginary components of the local fields.Substituting (31)–(30) into (28)–(29) we obtain E inct = ( I t − β ee · α reee ) · Re( E loct ) + j ( I t − β ee · α imee ) · Im( E loct ) − β ee · α reem · Re( H loct ) − jβ ee · α imem · Im( H loct ) , (32) H inct = ( I t − β mm · α remm ) · Re( H loct ) + j ( I t − β mm · α immm ) · Im( H loct ) − β mm · α reme · Re( E loct ) − jβ mm · α imme · Im( E loct ) , (33)where I t is the unit dyadic in the plane of the array. Respectively,Re( E inct ) = Re( I t − β ee · α reee ) · Re( E loct ) + Im( β ee · α imee ) · Im( E loct ) − Re( β ee · α reem ) · Re( H loct ) + Im( β ee · α imem ) · Im( H loct ) , (34)Im( E inct ) = − Im( β ee · α reee ) · Re( E loct ) + Re( I t − β ee · α imee ) · Im( E loct ) − Im( β ee · α reem ) · Re( H loct ) − Re( β ee · α imem ) · Im( H loct ) , (35)Re( H inct ) = Re( I t − β mm · α remm ) · Re( H loct ) + Im( β mm · α immm ) · Im( H loct ) − Re( β mm · α reme ) · Re( E loct ) + Im( β mm · α imme ) · Im( E loct ) , (36)Im( H inct ) = − Im( β mm · α remm ) · Re( H loct ) + Re( I t − β mm · α immm ) · Im( H loct ) − Im( β mm · α reme ) · Re( E loct ) − Re( β mm · α imme ) · Im( E loct ) . (37)These expressions can be substituted into (26)–(27) from which one obtains a set of dyadic relations for the polar-izabilities assuming that the four components of the local fields Re( E loct ( x, y )), Re( H loct ( x, y )), Im( E loct ( x, y )), andIm( H loct ( x, y )) are independent. Doing so we obtain the following relations: z × α reme = 2 j Im( β ee · α reee ) − R · Re( I t − β ee · α reee ) , (38) j z × α imme = − j Re( I t − β ee · α imee ) − R · Im( β ee · α imee ) , (39) z × α remm = 2 j Im( β ee · α reem ) + 2 R · Re( β ee · α reem ) , (40) j z × α immm = 2 j Re( β ee · α imem ) − R · Im( β ee · α imem ) , (41) z × α reee = − j Im( β mm · α reme ) + 2 R · Re( β mm · α reme ) , (42) j z × α imee = − j Re( β mm · α imme ) − R · Im( β mm · α imme ) , (43) z × α reem = − j Im( β mm · α remm ) − R · Re( I t − β mm · α remm ) , (44) j z × α imem = 2 j Re( I t − β mm · α immm ) − R · Im( β mm · α immm ) , (45)where R is the dyadic reflection coefficient defined in terms of R TM , TE as R = R TE z z ×× k t k t k + R TM k t k t k (46)(for the definition of the double cross product and other dyadic algebra rules see, e.g., Ref. 14). One may notice thatbecause R in the above relations is either zero or purely imaginary: ( R ) ∗ = − R , it follows that Re( α reee , mm , em , me ) =Im( α imee , mm , em , me ) = 0. Therefore, the equations (38)–(45) can be also written as z × α reme = 2 h Re( β ee ) + jR · Im( β ee ) i · α reee − R, (47) z × α imme = 2 h Re( β ee ) + jR · Im( β ee ) i · α imee − I t , (48) z × α remm = 2 h Re( β ee ) + jR · Im( β ee ) i · α reem , (49) z × α immm = 2 h Re( β ee ) + jR · Im( β ee ) i · α imem , (50) z × α reee = − h Re( β mm ) − jR · Im( β mm ) i · α reme , (51) z × α imee = − h Re( β mm ) − jR · Im( β mm ) i · α imme , (52) z × α reem = − h Re( β mm ) − jR · Im( β mm ) i · α remm − R, (53) z × α imem = − h Re( β mm ) − jR · Im( β mm ) i · α immm + 2 I t . (54)Let us consider first the propagating part of the spectrum. For such waves, R = 0, and the above relations simplify.Also, we can write β ee = η β , and β mm = η − β , where β is the dimensionless interaction dyadic which, by duality,is the same for the electric and magnetic currents as they are due to the electric and magnetic dipole moments thatbelong to the same particles in the array. The solution of the system of dyadic equations (47)–(54) in the case of R = 0 is α reee , mm , em , me = 0 , (55) α imme = − α imem = 2 (cid:20) I t + 4 (cid:16) z × Re( β ) (cid:17) (cid:21) − · ( z × I t ) , (56) η α imee = η − α immm = 4 z × Re( β ) · (cid:20) I t + 4 (cid:16) z × Re( β ) (cid:17) (cid:21) − · ( z × I t ) . (57)It can be shown that the real part of the interaction dyadic β for a planar array verifiesRe( β ) = −
12 cos θ z z ×× k t k t k − cos θ k t k t k + k A π I t , (58)where θ is the angle of incidence: cos θ = p − k /k , and A is the unit cell area. This result holds for arrays witharbitrary unit cell geometries, provided that the arrays do not produce higher-order diffraction lobes.It is quite interesting that the imaginary part of the interaction constant that contains the information aboutthe microstructure of the array has completely disappeared from the above solution. The imaginary part of theinteraction constant does not contribute in this case because the induced currents J e and J m are always imaginary,and the respective additions to the interaction field j Im( β ee ) · J e and j Im( β mm ) · J m are real-valued, to which theparticles do not react. Thus, the interaction of the particles in the array is irrelevant in the considered case, and eachparticle radiates effectively as in free space.We may substitute (58) into (56)–(57), taking into account that (cid:16) z × Re( β ) (cid:17) = " − − (cid:18) k A π (cid:19) + k A π (cid:18) cos θ + 1cos θ (cid:19) I t , (59)and obtain α imme = − α imem = 6 πk A (cid:18) cos θ + 1cos θ − k A π (cid:19) − ( z × I t ) , (60) η α imee = η − α immm = − πk A (cid:18) cos θ + 1cos θ − k A π (cid:19) − ( z z ×× Re( β )) . (61)For practical purposes, considering the phase conjugation of paraxial beams in dense arrays, we may approximate theabove relations as α imme = − α imem ≈ πk A ( z × I t ) , (62) η α imee = η − α immm ≈ πk A I t . (63)Because of the form of the relations (22)–(25), the obtained exact solutions (56)–(57) and their approximations (62)–(63) are valid for arbitrary plane waves incident from the half space z <
0. The solution for the case of incidence fromthe half space z > z with − z in (56)–(57) and (62)–(63), which changes signs of α imem , me .From the above results we see that the particle must be “invisible” for the real-valued electric and magnetic fields,while the polarizabilities of the particle to the imaginary-valued fields must be such that the electric and magneticcurrents form Huygens pairs that absorb the incident wave and produce the phase conjugated wave.For the evanescent waves | R TM , TE | → ∞ , therefore, it is convenient to multiply the equations (47)–(54) by R − from the left. Then, in the limit R − → η α reee = η − α remm = − j h Im( β ) i − , (64) α imee , mm = α re , imem , me = 0 . (65)The same solution can be also obtained with a more accurate treatment. Let us introduce the notations C e =Re( β ee ) + jR · Im( β ee ) and C m = Re( β mm ) − jR · Im( β mm ). Then, in these notations, we may, for example, write thesolution for α reee as α reee = 4 h I t + 4( z × C m ) · ( z × C e ) i − · ( z × C m ) · ( z × R )= (cid:20) I t + 14 ( z × C e ) − · ( z × C m ) − (cid:21) − · ( z × C e ) − · ( z × R )= C − · R + O (cid:0) R − (cid:1) . (66)Next, C − · R = − j h Im( β ee ) i − · (cid:20) I t − jR − · Re( β ee ) · (cid:16) Im( β ee ) (cid:17) − (cid:21) − = − j h Im( β ee ) i − + O (cid:0) R − (cid:1) , (67)which leads to (64).From (64)–(65) we conclude that to conjugate the evanescent part of the spectrum the inclusions must be “invisible”to the imaginary part of the electric and magnetic fields. The inclusions do not have to be bi-anisotropic in this case.The particles are purely reactive and their reactance should compensate the reactance due to particle interactions,creating a resonant structure.From a physical point of view, condition (64) can be understood as a condition for a surface polariton resonanceat the array surface. Indeed, for particles with the polarizabilities (64)–(65) we may write J e = α reee · Re( E loc ). Fromthe other hand, E loc = E inc + β ee · J e . Therefore, J e = α reee · Re( E inc + β ee · J e ) = α reee · Re( E inc ) + jα reee · Im( β ee ) · J e , (68)because both α reee and J e are purely imaginary. From here J e = h I t − jα reee · Im( β ee ) i − · α reee · Re( E inc ) , (69)and we see that J e → ∞ when condition (64) is fulfilled. A similar resonance is responsible for the enhancement ofthe evanescent waves in a pair of linear plasmon-polariton resonant grids studied in previous works. B. Electromagnetic properties of the particles forming phase conjugating sheets
Although the principle of operation of the field-conjugating perfect lens is the nonlinear operation of complexconjugation of electromagnetic fields, it is interesting to observe that the particles which perform this operation arecharacterized by linear polarizabilities with respect to the real or imaginary parts of the fields. The nonlinear natureof the particles is thus only in their selective sensitivity to either real or imaginary parts of the complex amplitude ofthe local fields.Considering the particle response to the real or imaginary field components separately, we may apply the theoryof usual linear bi-anisotropic particles. For the particles which react to the imaginary parts of the field (the particles0excited by the propagating part of the spatial spectrum), we rewrite relations (30), (31), and (62) in terms of theinduced electric and magnetic dipole moments of individual particles p e , m and the local fields: p e = (cid:18) − j πǫ k (cid:19) j Im( E loct ) + η (cid:18) j πǫ k (cid:19) z × j Im( H loct )= a ee · j Im( E loct ) + η a em · j Im( H loct ) , (70) p m = (cid:18) − j πµ k (cid:19) j Im( H loct ) − η (cid:18) j πµ k (cid:19) z × j Im( E loct )= a mm · j Im( H loct ) + 1 η a me · j Im( E loct ) (71)(the surface current densities are related to the dipole moments of individual particles as J e , m = jω p e , m /A ).These relations show that the particles reacting to the propagating part of the spectrum are bi-anisotropic andnonreciprocal. The magnetoelectric coupling is due to nonreciprocity only (no magnetoelectric coupling due toreciprocal spatial dispersion effects), because the coupling dyadics satisfy a em = a T me . (72)Furthermore, because these dyadics are antisymmetric ( a em = − a T em , a me = − a T me ), materials formed by particles ofthis type belong to the class of moving media. The polarizabilities of lossless bi-anisotropic particles satisfy the following conditions (e.g., Ref. 18): a ee = a † ee , a mm = a † mm , a em = a † me , (73)where † denotes the Hermitian conjugation operation. Obviously, the inclusions with the polarizabilities (70) and (71)have the opposite property of being purely passive or active (there is no stored electromagnetic energy in their nearfields), because they satisfy the opposite conditions: a ee = − a † ee , a mm = − a † mm , a em = − a † me . (74)The power extracted from the local fields by one pair of the particles reads P = 12 Re { J ∗ e · E loc + J ∗ m · H loc } A . (75)Assuming paraxial wave propagation, we can use the polarizability expressions (62) and (63) and assume that theelectric and magnetic local fields are related by the free-space wave impedance η . In this approximation we find, forthe case of plane wave incidence from the half space z < P = 12 3 πk η [Im( E loc )] = 6 πk η [Im( E loc )] = 6 πη k [Im( H loc )] . (76)As is clear from this result, each of the polarizability components in (62) and (63) brings equal contributions tothe extracted power. Noting that the induced dipole moments of ideal absorption-free dipole scatters read (at theresonance) p e0 = (cid:18) − j πǫ k (cid:19) E loc , p m0 = (cid:18) − j πµ k (cid:19) H loc , (77)we see that a pair of such ideal dipoles would extract from the fields exactly the same amount of power as our phaseconjugating particles (when the complex amplitude of the local field is purely imaginary at the point of the particle).Thus, we can conclude that the particles described by (62) and (63) actually do not absorb power. They act asideal absorption-free particles, which receive power from the incident field and re-radiate the same amount of power,creating phase conjugated waves of the same intensity as the incident propagating waves.It is easy to check that the same particles with the polarizabilities (70)–(71) do not react to the plane waves incidentfrom the half space z >
0. Indeed, the sign of the magnetic field in these waves is opposite, therefore, the contributionsdue to E loc and H loc to the electric dipole moment p e and the magnetic dipole moment p m compensate each other inEqs. (70)–(71), so that p e = p m = 0 under such excitation. As was mentioned in Section III A, to conjugate the wavesincident from the half space z >
0, one must change the signs of the magnetoelectric coupling terms in (70)–(71).Physically, this requires another array of inclusions with slightly different topology (more details in the next section).Fortunately, the particles in the two arrays do not interact, so that in practice it is possible to combine the two typesof particles in a single plane, for example, in a chess board-like structure.1
IV. DESIGN OF THE PHASE CONJUGATING BI-ANISOTROPIC INCLUSIONS AT MICROWAVESA. Case of propagating waves
To approach the design of nonlinear phase conjugating inclusions at microwaves one may start from the ideas behindthe well-known omega particle.
An omega particle is a combination of a short dipole antenna and a small loopantenna. The particle is planar, so that both the dipole and the loop may be printed on a single side of a printed-circuit board. In the most common variant of this linear and reciprocal inclusion the dipole is directly connectedto the loop. The nonlinearity and nonreciprocity, thus, can be achieved if one inserts a nonlinear and nonreciprocalfour-pole network between the two antennas.To identify what kind of network one may need, let us first briefly analyze how the linear omega particle reactsto the local electric and magnetic fields. We consider the case when the particle lies in the xz -plane with the dipoleantenna oriented along the x -axis. In this geometry, the dipole reacts to the x -component of the electric field, E loc x ,and the loop reacts to the y -component of the magnetic field, H loc y .The electromotive force (EMF) induced by the local field in the dipole can be written as E dip = h dip E loc x , (78)where h dip is the effective height of the dipole antenna. For a short dipole of the total length 2 l , the effective heightis h dip = l . Respectively, the EMF induced in the loop by the magnetic field reads E loop = − jωµ SH loc y , (79)where S = πr is the area of the loop. Under a normal plane wave incidence, H loc y = ± E loc x /η (the two signs are forthe two possible directions of incidence), therefore, E loop = ∓ j ( k S ) E loc x = ∓ jh loop E loc x , (80)where we have introduced the effective height of a small loop antenna h loop = k S . One may notice that the EMFinduced in the loop is in quadrature with respect to the EMF in the dipole. Therefore, when the two antennas aredirectly connected, these EMFs add up, but never fully compensate (or fully complement) each other. In fact, in themost interesting case when h loop = h dip = l , the total EMF induced in the antennas is E tot = (1 ∓ j ) lE loc x . (81)Respectively, the induced current at the point where the dipole connects to the loop is I dip = (1 ∓ j ) lE loc x Z dip + Z loop , (82)where Z dip is the input impedance of the dipole and Z loop is the input impedance of the loop. The induced electricand magnetic dipole moments are proportional to this current: p e ,x = I dip ljω = − j (1 ∓ j ) l E loc x ω ( Z dip + Z loop ) , (83) p m ,y = µ I dip S = η (1 ∓ j ) l E loc x ω ( Z dip + Z loop ) , (84)where we use the fact that S = l/k if h loop = h dip = l . When the particle is at resonance, Z dip + Z loop = 2 R rad + R loss ,where R rad = η k l π is the radiation resistance of a short dipole antenna (when h loop = h dip both antennas havethe same radiation resistance) and R loss corresponds to the ohmic loss in metal, which we may neglect. From theserelations we see that the induced dipole moments in the linear omega particle are in quadrature.However, from (70)–(71) it follows, first, that in the nonlinear particle which we want to design the contributionsdue to E loc and H loc in the expressions for both dipole moments must be in phase (for the wave incident from z < ∓ j in(83)–(84) is replaced with ±
1, and η in (84) with − jη . As will be seen in a few moments, this can be achieved withmicrowave nonlinear circuits known as balanced modulators (BM).2 FIG. 2: (a) Equivalent circuit of an idealized balanced modulator (BM). (b) A network composed of two balanced modulatorsconnected through a low-pass filter (LPF).
A balanced modulator is a three-port device such that there are two ports that may both serve as input and output(the ports are exchangeable due to the symmetry of the circuit; that is one of the reasons why the circuit is called balanced modulator), and the third port to which a voltage from a local oscillator is applied. The function that theBM performs is a multiplication of the signal applied to one of its input ports and the signal of the local oscillator.One may represent an idealized BM with an equivalent circuit shown in Fig. 2(a). The controllable current sourcesin the circuit depend on the instantaneous voltages at the ports as follows: i ( t ) = K u ( t ) u lo ( t ) , (85) i ( t ) = K u ( t ) u lo ( t ) , (86)where u lo ( t ) is the voltage at the local oscillator port and u , ( t ) and i , ( t ) are the voltages at the other two ports.We need power conserving BMs which do not absorb or store power that is delivered to ports 1 and 2, hence, u ( t ) i ( t ) + u ( t ) i ( t ) = 0. From here, K = − K . The input and output resistances of the BM shown in the figurewith dashed lines are assumed to be very large and are not taken into account.Consider now the network depicted in Fig. 2(b). In this network we have connected two BMs through a low-passfilter (LPF). One may imagine this LPF as a Π-type CLC -filter with an inductor in a series branch and two capacitorsin the parallel branches. For us, however, the only thing that is important here is that this LPF freely passes throughthe direct-current (DC) component and blocks all high-frequency components (the DC path through the filter is shownwith dashed lines).The whole network is designed to operate with signals at the frequency ω = 2 πf = 2 π/T , therefore, we mayrepresent the voltages u , ( t ) as u , ( t ) = U re1 , cos ωt − U im1 , sin ωt. (87)We apply the local oscillator signal at the frequency ω and the phase ϕ = π/ u lo , = − sin ωt , andanother signal at the same frequency and ϕ = 0 to the second BM: u lo , = cos ωt . Therefore, we may write for theDC currents i ′ , (here h . . . i T denotes averaging over a period): i ′ = K (cid:10) ( U re1 cos ωt − U im1 sin ωt )( − sin ωt ) (cid:11) T = 12 K U im1 , (88) i ′ = K (cid:10) ( U re2 cos ωt − U im2 sin ωt ) cos ωt (cid:11) T = 12 K U re2 . (89)But as is dictated by the topology of the network, i ′ = − i ′ , therefore, U im1 = U re2 . (90)Next, we express the high-frequency currents at the ports 1 and 2: i ( t ) = ( K u ′ )( − sin ωt ) = − I im1 sin ωt, (91) i ( t ) = ( K u ′ ) cos ωt = I re2 cos ωt, (92)where I im1 = K u ′ and I re2 = K u ′ . But again, from the topology of the network, the DC voltages satisfy u ′ = u ′ , therefore, I im1 = − I re2 . (93)3 FIG. 3: The topology of the two types of phase conjugating bi-anisotropic inclusions for the phase conjugating surface operatingat the frequency f . Each particle consists of a short dipole antenna with the impedance Z dip and a small loop antenna withthe impedance Z loop . The two antennas are interconnected through a nonlinear and nonreciprocal network composed of twobalanced modulators (BM) and a low-pass filter (LPF). The mixers are fed by a local oscillator with two signal outputs inquadrature ϕ = 0 ◦ and ϕ = 90 ◦ . See the main text for further explanations. Notice also that always I re1 = I im2 = 0.Thus, from (90) and (93) it is evident that the considered network operates, essentially, as a “connector” betweenthe imaginary current and voltage at the first port, and the real current and voltage at the second port. The networkalso enforces zero real current in the first port and zero imaginary current in the second port. This is exactly whatwe need in the design of the phase conjugating particles, and the corresponding topologies including the antennas areshown in Fig. 3.In these designs we connect the dipole and loop antennas to the BM-based network discussed above. The electricalsize of the circuit is negligible, and both antennas are excited by the same local field. Let us analyze the operation ofthe variant shown at the top of Fig. 3. First of all, it is evident that when placed in an arbitrary local field, the EMFsin both antennas are still given by (78) and (79). However, contrary to what happens in a linear omega particle, thereal part of E dip and the imaginary part of E loop will not be able to excite any current in the antennas, because thesecomponents are blocked by the BMs. Therefore, the relevant parts of the EMFs in the two antennas are (as above,we let h dip = h loop = l ) E imdip = l Im( E loc x ) , (94) E reloop = η l Im( H loc y ) . (95)Analogously, the voltage drops on the reactive parts X dip = Im( Z dip ) and X loop = Im( Z loop ) of the input impedancesof the dipole and the loop have no effect as well, as they are in quadrature with respect to the current flowing throughthem. Thus, the only relevant part of the input impedance is the radiation resistance.From the topology of the network, I imdip = − I im1 and I reloop = I re2 , therefore, I imdip = I reloop . Next, because of (90), theEMFs (94) and (95) and the rest of the equivalent circuits of the two antennas (only two R rad remain) appear to beessentially connected in series, hence, I imdip = I reloop = l [Im( E loc x ) + η Im( H loc y )]2 R rad . (96)The complex amplitudes of the electric and magnetic dipole moments, therefore, read p e ,x = l ( jI imdip ) jω = l ωR rad [Im( E loc x ) + η Im( H loc y )] , (97) p m ,y = µ I reloop S = η l ωR rad [Im( E loc x ) + η Im( H loc y )] , (98)4 FIG. 4: A nonlinear single-port network used as a load for a short electric dipole. The network is composed of the sameelements as in Fig. 2, with an additional operational amplifier working in the current-to-voltage conversion mode. which is a particular case of (70)–(71) for the considered polarization.As can be readily verified, the second design variant shown at the bottom of Fig. 3 has the opposite signs of themagnetoelectric interaction terms, and, thus, must be used to conjugate the plane waves incident from the half space z > B. Case of evanescent waves
To phase-conjugate the evanescent waves, the particles must react to the real parts of electric and magnetic fields,as follows from (64)–(65). In this case the inclusions are simple electric and magnetic dipoles without magneto-electricinteraction. Therefore, as the base for our design we may choose the loaded dipole and loop antennas. In what follows,we consider in detail the case of a linear electric dipole oriented along the x -axis (the magnetic dipole along the y -axismay be considered in a dual manner).When a short loaded dipole is placed in an electric field, an EMF is induced in the dipole, with the value givenby (78). Respectively, the current induced in the dipole is I dip = lE loc x Z dip + Z load , (99)where Z load is the impedance of a bulk load connected to the dipole. The induced electric dipole moment of theloaded dipole reads p e ,x = I dip ljω = − j l E loc x ω ( Z dip + Z load ) . (100)It is easy to verify that if we chose the load so that Z load = − Z dip + jη l A Im( β ) , (101)then the condition for the surface electric current polarizability (64) in an array of such particles will be satisfied,with an exception that the linear particles will react to both real and imaginary parts of the electric field.With the use of BMs we may get rid of the reaction to the imaginary part of the field and also find a simple wayto realize the necessary loading (101). Consider the network shown in Fig. 4. This network is similar to the ones weconsidered in Sec. IV A. It is composed of a pair of BMs and a LPF, with an additional element in the middle that isan operational amplifier working in the current-to-voltage conversion mode. We pump the first BM at the frequency f with the phase ϕ = 0, and the second one with the phase ϕ = ± π/ u ′ = 0, and, therefore, i = 0. Next,assuming that the voltage at the input of the circuit is u = U re1 cos ωt − U im1 sin ωt , ω = 2 πf = 2 π/T , we obtain i ′ = K (cid:10) ( U re1 cos ωt − U im1 sin ωt ) cos ωt (cid:11) T = 12 K U re1 . (102)Hence, the DC voltage at the output of the operational amplifier is u ′ = i ′ R = 12 K RU re1 , (103)5 FIG. 5: The topology of the two particles loaded with nonlinear circuits designed to operate with the evanescent part of thespatial spectrum. Top: a short electric dipole loaded with a nonlinear load. Bottom: a small magnetic loop loaded with anonlinear load. where R is the resistor in the feedback loop of the operational amplifier. Therefore, i ( t ) = ∓ K u ′ sin ωt = ∓ K RU re1 sin ωt = − I im2 sin ωt, (104)where I im2 = ± K RU re1 /
2. This current is the input current of the whole network, because the input current of thefirst BM equals zero. Thus, we have designed a circuit in which a real input voltage induces an imaginary current,i.e., the circuit behaves almost as a usual reactance (the plus sign in the expression for the current corresponds to thecapacitance and the minus sign — to the inductance), with the difference that the loading circuit is not sensitive atall to the imaginary input voltage. This is exactly what we need to realize the nonlinear particles reacting only to thereal part of the electric field. Indeed, to realize the required loading one has to choose the parameters of the circuitso that ± K R = − X dip + η l A Im( β ) . (105)It is interesting to note that there is no need to compensate the real part of the dipole impedance Z dip (the radiationresistance), because the circuit reacts only to the real part of the input voltage, and the additional voltage drop onthe radiation resistance U R = jI im2 R rad is purely imaginary.An example topology of an electric dipole particle with the nonlinear active load is shown in Fig. 5, top, and thesame for the magnetic dipole particle in Fig. 5, bottom (notice the difference in phases of the local oscillator signalsin both schematics). We would like to stress that the operational amplifier seen in this schematic works with a DCsignal. Thus, its role is not to amplify the evanescent fields of an incoming wave, but just to provide the necessary reactance of the loading circuit in order to tune the structure to a resonance. In turn, the evanescent modes in thisstructure are enhanced because of this resonance. V. CONCLUSIONS
In this paper, the concept of perfect lensing with a pair of phase conjugating surfaces introduced by us earlier, i.e.,a possibility to achieve optical resolution well below the wavelength limit without using double-negative materials,has been further developed. Working as a planar lens, a pair of phase conjugating sheets is able to focus propagatingmodes of a source due to the negative refraction at the interfaces and, in the same time, enhance the evanescent modesdue to surface plasmon-polariton resonances, i.e., it provides sub-wavelength resolution imaging indistinguishable fromthe perfect lens proposed by Pendry, while not suffering from its known drawbacks.6We have investigated in detail the physics of operation of nonlinear sheets with the boundary conditions of the form E t ( ω ) | = E t ( ω ) ∗ | , H t ( ω ) | = H t ( ω ) ∗ | and have demonstrated that they are, in principle, physically realizable withdevices imposing the necessary relations between the fields and the equivalent electric and magnetic surface currentsat the phase conjugating boundary. Namely, we have shown that the mentioned surface currents must form Huygenssources that radiate towards a given side of the boundary, negating the fields incident from the other side and creatingcomplex-conjugated fields in the corresponding half space.As a possible realization of such sheets, we have proposed and considered in this paper arrays of nonlinear andnonreciprocal bi-anisotropic inclusions reacting differently to the propagating and evanescent plane waves. At mi-crowaves, the considered design makes use of balanced modulators (a type of mixers) to provide for the requirednonlinearity and nonreciprocity of the circuit. At optics, a design utilizing similar principles may become feasible infuture as the field of optical nanocircuits develops further.As a final note we would like to mention that in such arrays the enhancement of the evanescent waves is due toa high-quality surface mode resonance, as in the grids of passive resonant inclusions considered in Ref. 3. This is incontrast to Ref. 10 where the phase conjugating surface must itself parametrically amplify the fields which requiresan unphysically high conversion efficiency. J. Pendry, Negative refraction makes a perfect lens,
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