Scattering theory for stationary materials with PT symmetry
aa r X i v : . [ m a t h - ph ] N ov Scattering theory for stationary materials with PT symmetry P. A. Brand˜ao ∗ Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o, 57072-900, Brazil.
O. Korotkova † Department of Physics, University of Miami, 1320 Campo Sano Dr, Coral Gables, FL 33146, USA. (Dated: November 10, 2020)A theoretical framework is developed for scattering of scalar radiation from stationary, three-dimensional media with correlation functions of scattering potentials obeying PT -symmetry. It isillustrated that unlike in scattering from deterministic PT symmetric media, its stationary gen-eralization involves two mechanisms leading to symmetry breaking in the statistics of scatteredradiation, one stemming from the complex-valued medium realizations and the other - from thecomplex-valued degree of medium’s correlation. I. INTRODUCTION
The discovery made by Bender and Boettcher [1] onthe possibility of physical systems with complex-valuedHamiltonians to possess real-valued spectra, under condi-tions of the PT symmetry, set the basis for the explosivegrowth of non-Hermitian quantum mechanics [2]. Thisgeneralization from Hermitian to non-Hermitian quan-tum mechanics acquired a solid foundation after the in-troduction of a new definition for the inner product ofthe Hilbert space [3]. It was soon realized that PT symmetry is actually a special case of a more generalclass of pseudo-Hermitian Hamiltonians that yield realeigenvalues [4]. Electromagnetics and optics, in partic-ular, has soon benefited from such a generalization inview of the analogy between the paraxial wave equationand the time-dependent Schr¨odinger equation, with man-ifestation in the areas of wave-guiding [5], unidirectionalcrystal invisibility [6], balanced lasing and anti-lasing [7],structured light emission [8], beam dynamics in periodiclattices [9], among others.In the realm of physical optics, the PT symmetry ofa deterministic material translates to Hermiticity of itscomplex-valued index of refraction or, equivalently, scat-tering potential [10]. For static media, only spatial Her-miticity at a fixed frequency must be imposed. Physicallythis implies the perfect balance between gain and losscenters constituting the medium. Hence the symmetryin the complex-valued index of refraction must be con-sidered with precaution, for it can be geometrically sym-metric but not Hermitian, implying imbalance betweenthe gain and loss contributions. In fact, the geomet-rically symmetric (unbalanced) and the PT -symmetric(balanced) media are mutually exclusive, with the onlyexception possible when the imaginary parts of their re-fractive index, accounting for gain and loss, vanish.In this work, we pursue a set of objectives, the first ofwhich is to draw a clear distinction between a static, de- ∗ paulo.brandao@fis.ufal.br † [email protected] terministic, geometrically symmetric medium, which wewill term classic , and a PT -symmetric medium. Thesecond, and the main aim, is to introduce a class ofstochastic, stationary media obtained on taking the spa-tial correlation over the ensemble of realizations of themedium obeying PT symmetry. The detailed analysis ofthe properties of the new type of correlation functions,their mathematical modeling and their comparison withthose of classic media is then attempted. Our final goalis to develop a theoretical treatment of light scatteringfrom the new type of stationary media, and compare itwith that from classic stationary media [11]. The cen-tral quantity in the scattering theory involving generallyrandom illumination and random medium is the pair-scattering matrix [12] which characterizes the change bythe medium in the correlation along two incident and twoscattered directions and which coincides, within the ac-curacy of the first Born approximation, with the Fouriertransform of the scattering potential correlation function.We derive this quantity for the general stationary PT -symmetric media and reveal its remarkable properties.In addition, a mathematical model for a Schell-like cor-relation function of the PT -symmetric medium with lin-ear phases is developed with the help of the Bochnertheorem previously used in optics in connection to fieldcorrelations [13], [14] and classic medium correlations [15]for numerical illustration of the new theory.Various aspects of stochastic light scattering from de-terministic PT -symmetric media were recently discussedin Refs. [16], [17] and [18]. The results of these stud-ies can be deduced from the general theory developedhere in the completely correlated medium limit. To ourknowledge the only example of a PT -symmetric station-ary medium had previously been considered in [19], how-ever without development of the general theory.The paper is organized as follows: the stationary PT -symmetric media and their properties are analyzed inSec. 2; the general approach for modeling of the novelmedia is developed in Sec. 3; Sec. 4 presents the generaltheory for characterization of weak scattering from suchmedia; the analytical examples relating to scattering ofSchell-like PT -symmetric media are given in Sec. 5 andthe concluding remarks are provided in Sec. 6. II. COMPLEX MATERIAL CORRELATIONFUNCTIONS WITH CLASSIC AND PT SYMMETRIES
We will confine our attention only to a (wide-sense)statistically stationary medium distributed in a three-dimensional region of space and being symmetric, in acertain sense, with respect to its geometrical center, say ~r = 0. Let us first consider the case of an unbalanced sta-tionary medium in which the realizations of the complex-valued index of refraction n ( ~r, ω ) = n r ( ~r, ω ) + in i ( ~r, ω ) (1)at position ~r and angular frequency ω , obey geometricalsymmetry relations n r ( − ~r, ω ) = n r ( ~r, ω ) , n i ( − ~r, ω ) = n i ( ~r, ω ) . (2)We will refer to such media as having classic symmetryor just classic and use them throughout the paper toset up the contrast with PT -symmetric media that wewill introduce below. The realizations of the scatteringpotential defined as [11] F ( ~r, ω ) = ( k / π )[ n ( ~r, ω ) − , (3)where k is the radiation’s wave number, then must besymmetric about the center as well: F CL ( − ~r, ω ) = F CL ( ~r, ω ) , (4)where subscript CL is used to denote classic (unbalanced)medium. See Fig. 1 (left) for visualization of real andimaginary parts of F CL .This immediately implies that the spatial correlationfunction of the scattering potential in Eq. (4) [11] C CL ( ~r , ~r , ω ) = h F ∗ CL ( ~r , ω ) F CL ( ~r , ω ) i m , (5)of the classic medium, where subscript m denotes aver-aging over medium realizations, must meet condition C CL ( − ~r , − ~r , ω ) = C CL ( ~r , ~r , ω ) , (6)i.e., must also be symmetric with respect to the medium’scenter ~r = 0. We will also refer to such stationary me-dia as classic . As we will demonstrate, due to theirinherent geometrical symmetry, such media constitutea solid comparison tool with the PT -symmetric media.Indeed, random media with spatially asymmetric, real-valued correlation functions, can also be shown to leadto broken symmetries in the statistics of scattered radi-ation [20].On the other hand, a PT -symmetric material is de-scribed by a complex refractive index whose real andimaginary parts must satisfy relations n r ( − ~r, ω ) = n r ( ~r, ω ) , n i ( − ~r, ω ) = − n i ( ~r, ω ) . (7) F R F R F I F I xxCLASSIC MEDIUM PT-SYMMETRIC MEDIUM
FIG. 1. Real and imaginary parts of typical classic (left) and PT -symmetric (right) media’s scattering potentials, along asingle Cartesian axis, say x . Then the scattering potential of such a medium is also PT -symmetric and satisfies condition F ∗ P T ( − ~r, ω ) = F P T ( ~r, ω ) . (8)In view of Eq. (8) the two-point correlation function ofthe PT -symmetric scatterer C P T ( ~r , ~r , ω ) = h F ∗ P T ( ~r , ω ) F P T ( ~r , ω ) i m , (9)must satisfy relation C ∗ P T ( − ~r , − ~r , ω ) = C P T ( ~r , ~r , ω ) . (10)We will term the media with correlation in Eq. (10) PT - symmetric stationary media .In order to avoid possible misconceptions we will nowoutline some other similarities and differences betweenthe two media types, based on comparison between Eqs.(6) and (10). First, switching the order of spatial argu-ments in any complex-valued correlation function yields C ( ~r , ~r , ω ) = C ∗ ( ~r , ~r , ω ) which implies with the use ofEqs. (6) and (10) that C CL ( ~r , ~r , ω ) = C ∗ CL ( − ~r , − ~r , ω ) ,C P T ( ~r , ~r , ω ) = C P T ( − ~r , − ~r , ω ) . (11)Second, on defining the measure of the strength of scat-tering potential for the two media by the expressions I CL ( ~r, ω ) = C CL ( ~r, ~r, ω ) , I P T ( ~r, ω ) = C P T ( ~r, ~r, ω ) , (12)we find that they are given by the same formula: I α ( ~r, ω ) = h| F ( ~r, ω ) | i m ( α = CL, P T ) , (13)being, due to Eq. (6) and (10), the real-valued and evenfunctions I α ( − ~r, ω ) = I α ( ~r, ω ) . (14)In order to highlight the distinctive feature of station-ary PT -symmetric media we introduce a new quantity, anti-strength of scattering potential , which is an analog ofa cross-correlation function used sometimes in connectionto optical fields (c.f. Ref. [21]) N P T ( ~r, ω ) = C P T ( − ~r, ~r, ω ) = h F ( ~r, ω ) i m , (15)and note that Im[ N P T ( ~r, ω )] = ( k / π ) h n r n i ( n r − n i − i relates to the amount of asymmetry that a PT -symmetric medium introduces on scattering, whileIm[ N P T (0 , ω )] = 0. For classic media this quantity canalso be defined but coincides with I CL .More generally, the degrees of potential correlation de-fined at any points ~r and ~r by expressions µ α ( ~r , ~r , ω ) = C α ( ~r , ~r , ω ) p I ( ~r , ω ) p I ( ~r , ω ) , ( α = CL, P T )(16)are seen to satisfy the same relations as C CL and C P T ,respectively, but their values are bound to the unit circleof the complex plane. Equation (16) has a particularlysimple form at symmetric points: µ α ( − ~r, ~r, ω ) = N α ( ~r, ω ) I ( ~r, ω ) , ( α = CL, P T ) . (17)This immediately implies that while a classic medium isfully correlated (for such points), viz., µ CL ( − ~r, ~r, ω ) = 1 , (18)the PT -symmetric medium can have any, generallycomplex-valued, correlation state: µ P T ( − ~r, ~r, ω ) = h F ( ~r, ω ) i m h| F ( ~r, ω ) | i m . (19) III. MODELING OF PT -SYMMETRIC MEDIACORRELATION FUNCTIONS The Bochner theorem of functional analysis was em-ployed in Ref. [15] for modeling of novel three-dimensional scattering media (see also [13], [14] for itsapplication to stationary radiation). We will now ex-plore the application of this idea specifically to the PT -symmetric stationary media. For C P T ( ~r , ~r , ω ) to rep-resent a genuine correlation function, it is sufficient towrite it as C P T ( ~r , ~r , ω ) = Z p ( ~v, ω ) H ∗ P T ( ~r , ~v, ω ) H P T ( ~r , ~v, ω ) d v, (20)where p ( ~v, ω ) ≥ H P T ( ~r, ~v, ω ) is a complex-valuedfunction and the integration extends over the three-dimensional space of vector ~v . Equation (10) imposessome restrictions on the possible forms that H P T ( ~r, ~v, ω )can assume: Z p ( ~v, ω ) H P T ( − ~r , ~v, ω ) H ∗ P T ( − ~r , ~v, ω ) d v = Z p ( ~v, ω ) H ∗ P T ( ~r , ~v, ω ) H P T ( ~r , ~v, ω ) d v. (21)For this relation to be satisfied it is sufficient that H P T ( − ~r , ~v, ω ) H ∗ P T ( − ~r , ~v, ω )= H ∗ P T ( ~r , ~v, ω ) H P T ( ~r , ~v, ω ) , (22) or H ∗ P T ( − ~r, ~v, ω ) = H P T ( ~r, ~v, ω ) , (23)implying that H P T ( ~r, ~v, ω ) is PT -symmetric. It is inter-esting to compare Eq. (23) with that for classic media.On writing C CL ( ~r , ~r , ω ) = Z p ( ~v, ω ) H ∗ CL ( ~r , ~v, ω ) H CL ( ~r , ~v, ω ) d v, (24)we find that in view of the first of Eqs. (11), no addi-tional condition is imposed on H CL : indeed, it can beany complex-valued function. We stress that condition(23) is only sufficient, in general H P T may have a moregeneral form.For Schell-like scatterers the correlation class takes theform of the three-dimensional Fourier transform kernel H ( ~r, ~v, ω ) = a ( ~r, ω ) exp( − πi~r · ~v ) , (25)where a ( ~r, ω ) must satisfy a ∗ ( − ~r, ω ) = a ( ~r, ω ) providedEq. (23) holds. Then, Eq. (20) simplifies as C P T ( ~r , ~r , ω ) = a ∗ ( ~r , ω ) a ( ~r , ω ) × Z p ( ~v, ω ) exp[ − πi~v · ( ~r − ~r )] d v = a ∗ ( ~r , ω ) a ( ~r , ω ) g ( ~r d , ω ) ⊗ g ( ~r d , ω )= p I P T ( ~r , ω ) p I P T ( ~r , ω ) × exp[ − iψ ( ~r , ω )] exp[ iψ ( ~r , ω )] × µ P T ( ~r d , ω ) , (26)where p I F ( ~r, ω ) = | a ( ~r, ω ) | , ψ ( ~r, ω ) = arg[ a ( ~r, ω )], ⊗ denotes the 3D convolution, ~r d = ~r − ~r , µ P T ( ~r d , ω ) = g ( ~r d , ω ) ⊗ g ( ~r d , ω ) and g ( ~r d , ω ) is given by integral g ( ~r d , ω ) = Z p p ( ~v ) exp( − πi~v · ~r d ) d v. (27)There are two fundamental ways in which C P T in Eq.(26) can possess nontrivial phase: function ψ can be non-trivial or µ P T can be complex-valued. This results ina qualitatively different far-field spectral density distri-butions. In the former case an originally deterministic PT -symmetric medium is randomized, resulting in par-tial “blurring” of the scattered spectral density, as com-pared with that for µ P T = 1. In the latter case themedium is not PT -symmetric and if µ P T = 1 it wouldscatter to spectral density symmetric about the z -axis.However, if | µ P T | → PT -symmetry like effects in which Fourier transform of | µ P T | determines the scattered spectral density profileand arg( µ P T ) determines the off-axis shift. Both condi-tions can hold simultaneously, leading to a much morecomplex scattering outcome. In particular, two mecha-nisms can annihilate the asymmetries produced by themindividually.
IV. SCATTERING THEORY FOR STATIONARY PT -SYMMETRIC MEDIA Within the validity of the first Born approximation,the cross-spectral density W s ( r ˆ s , r ˆ s , ω ) of radiationscattered to the far zone of a stationary medium is givenby integral ([11], p. 120) W s ( r ˆ s ,r ˆ s , ω ) = 1 r Z V Z V W i ( ~r , ~r , ω ) C ( ~r , ~r , ω ) × exp[ − ik (ˆ s · ~r − ˆ s · ~r )] d r d r , (28)where W i ( ~r , ~r , ω ) is the cross-spectral density of theincident field. We can express Eq. (28) as W s ( r ˆ s ,r ˆ s , ω )= 1 r f W i ( ~r , ~r , ω ) ⊛ e C ( ~r , ~r , ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( − k ˆ s ,k ˆ s ,ω ) , (29)where tilde stands for three-dimensional Fourier trans-form and ⊛ denotes convolution in six dimensions. Thisexpression indicates that e C plays the crucial part in char-acterizing the redistribution of energy along various inci-dent and scattered directions. See also [12] for more gen-eral expressions for W s involving e C for the intermediatescattered field, relating to the pair-scattering matrix. Ex-pression (29) substantially simplifies under the assump-tion that the incident field is a polychromatic plane wavepropagating along direction ˆ s , and, hence, having thecross-spectral density of the form W i ( ~r , ~r , ω ) = S i ( ω ) exp[ ik ˆ s · ( ~r − ~r )] , (30)where S i ( ω ) is the (position-independent) spectral den-sity. After substituting from Eq. (30) into (29) we obtain W s ( r ˆ s , r ˆ s , ω ) = S i ( ω ) r e C ( − ~K , ~K , ω ) , (31)where ~K = k (ˆ s − ˆ s ) and ~K = k (ˆ s − ˆ s ) (32)are the momentum transfer vectors characterizing scat-tering from incident direction ˆ s to outgoing direction ˆ s j , j = 1 ,
2. We note that in cases when W i involves morethan one direction, the momentum transfer vectors havemore general form: ~K = k (ˆ s − ˆ s ′ ) and ~K = k (ˆ s − ˆ s ′ ),i.e., they depend on two incident and two scattered di-rections.Let us now analyze ˜ C ( − ~K , ~K , ω ) in detail:˜ C ( − ~K , ~K , ω ) = h ˜ F r ( − ~K , ω ) ˜ F r ( ~K , ω ) i m + h ˜ F i ( − ~K , ω ) ˜ F i ( ~K , ω ) i m + i (cid:2) h ˜ F r ( − ~K , ω ) ˜ F i ( ~K , ω ) i m − h ˜ F i ( − ~K , ω ) ˜ F r ( ~K , ω ) i m (cid:3) . (33) For classic medium both F r ( ~r, ω ) and F i ( ~r, ω ) are real-valued and even, hence both ˜ F r ( ~K, ω ) and ˜ F i ( ~K, ω )are real valued and even, ˜ F r ( − ~K, ω ) = ˜ F r ( ~K, ω )[ ˜ F i ( − ~K, ω ) = ˜ F i ( ~K, ω )]. This implies that˜ C CL ( − ~K , ~K , ω ) = ˜ C ∗ CL ( − ~K , ~K , ω ) . (34)Since all Fourier transforms entering ˜ C CL are real func-tions, it must always be complex-valued if ~K = ~K , un-less F i ( ~r, ω ) = 0. For PT -symmetric medium F r ( ~r, ω )[ F i ( ~r, ω )] is even [odd], hence we have ˜ F r ( − ~K, ω ) =˜ F r ( ~K, ω ) [ ˜ F i ( − ~K, ω ) = − ˜ F i ( ~K, ω )], and ˜ F r ( ~K, ω ) is real-valued, while ˜ F i ( ~K, ω ) is purely imaginary. This implies˜ C P T ( − ~K , ~K , ω ) = ˜ C P T ( − ~K , ~K , ω ) . (35)Further, since ˜ F i ( ~K, ω ) is a purely imaginary function,Eq. (35) implies that ˜ C P T must always be real-valued,even for ~K = ~K and F i ( ~r, ω ) = 0.Along the same scattered direction ˆ s = ˆ s = ˆ s , ~K = ~K = ~K , and, hence, the spectral density becomes S s ( r ˆ s, ω ) = S i ( ω ) r ˜ C ( − ~K, ~K, ω )= S i ( ω ) r ˜ N ( ~K, ω ) , (36)where˜ N ( ~K, ω ) = ˜ C ( − ~K, ~K, ω )= h ˜ F r ( − ~K, ω ) ˜ F r ( ~K, ω ) i m + h ˜ F i ( − ~K, ω ) ˜ F i ( ~K, ω ) i m + i [ h ˜ F r ( − ~K, ω ) ˜ F i ( ~K, ω ) i m − h ˜ F i ( − ~K, ω ) ˜ F r ( ~K, ω ) i m ] . (37)In particular,˜ N CL ( ~K, ω ) = h ˜ F r ( ~K, ω ) i m + h ˜ F i ( ~K, ω ) i m = h| ˜ F ( ~K, ω ) | i m , (38)being the Fourier transform of I CL ( ~r, ω ) in Eq. (13) and˜ N P T ( ~K, ω ) = h [ ˜ F r ( ~K, ω ) + i ˜ F i ( ~K, ω )] i m = h ˜ F ( ~K, ω ) i m , (39)being the Fourier transform of N P T ( ~r, ω ) in Eq. (15).Both ˜ N CL ( ~K, ω ) and ˜ N P T ( ~K, ω ) are obviously real-valued. By definition, the spectral degree of coherenceof the scattered field is µ s ( r ˆ s , r ˆ s , ω ) = W s ( − ~K , ~K , ω ) q W s ( − ~K , ~K , ω ) q W s ( − ~K , ~K , ω )(40)and in view of Eq. (31) it generally yields µ s ( r ˆ s , r ˆ s , ω ) = ˜ C ( − ~K , ~K , ω ) q ˜ N ( ~K , ω ) q ˜ N ( ~K , ω ) . (41)To summarize the results of this section: (I) for genericvectors ~K and ~K , ˜ C P T must be real-valued while ˜ C CL must be complex-valued; (II) as compared with ˜ N CL ,˜ N P T contains an additional term being the correlationfunction of the real and imaginary parts of the scatteringpotential.
V. APPLICATION TO SCHELL-MODELSCATTERERS WITH LINEAR ANDQUADRATIC PHASES
Let us assume that C P T of the scatterer is given byEq. (26) where a ( ~r, ω ) and µ ( ~r d , ω ) are selected as a ( ~r, ω ) = I exp (cid:18) − r a (cid:19) exp( − i~α · ~r ) ,µ ( ~r d , ω ) = exp (cid:20) − ( ~r − ~r ) d (cid:21) exp[ i~β · ( ~r − ~r )] , (42)where I is a constant, ~α , ~β are real vectors, the formerrelating to the non-Hermiticity of the material’s realiza-tions and the latter characterizes non-Hermiticity of thematerial’s correlation function. Further, a is related tothe dimensions of the scatterer and d is the correlationlength. In the case of linear phases the effects of ~α and ~β can be combined: ~γ = ~α + ~β. (43)Note that if ~α = − ~β then C P T is symmetric and real val-ued and will result in axially symmetric scattered spec-tral density, even though the realizations of the materialare PT -symmetric. Such simple phase cancellation is notpossible in cases when the phase functions are non-linear.Then using Eqs. (42) in Eq. (26) and applying three-dimensional Fourier transform of C P T ( ~r , ~r , ω ) yields˜ C P T ( − ~K , ~K , ω ) = I (2 π ) / a d (2 a + d ) / exp − a ~K · ~K d /a ! × exp " − ( a + d )( ~K + ~K ) /
22 + d /a × exp " − d ~γ · ( ~K + ~K − ~γ )2 + d /a . (44)We confirm, based on the results of the previous section,that ˜ C P T in (44) is real-valued. Also, in the limit d →∞ , the scattering results pertinent to deterministic PT -symmetric media can be deduced. Indeed, it is impliedby Eq. (44) that as d → ∞ ˜ C P T ( − ~K , ~K , d → ∞ , ω ) = I (2 π ) / a × exp (cid:26) − a h ~K + ~K − ~γ · ( ~K + ~K − ~γ ) i(cid:27) , (45) as expected. Expression (44) can be also written in formevidently demonstrating its non-Hermitian character:˜ C P T ( − ~K , ~K , ω ) = ˜ C P T ( − ~K , ~K , ~γ = 0 , ω ) × exp " ~γ · ( ~K + ~K − ~γ )2 /d + 1 /a , (46)where ˜ C ~γ =0 ( − ~K , ~K , ω ) is the correlation function for aHermitian scatterer with ~γ = 0, given explicitly by˜ C P T ( − ~K , ~K , ~γ = 0 , ω ) = I (2 π ) / a d (2 a + d ) / × exp " − ( a + d )( ~K + ~K ) / − a ~K · ~K d /a . (47)In particular, case ~α = 0 and ~β = 0 can be associatedwith the situation when the scatterer is deterministic and PT -symmetric and is randomized by a real-valued corre-lation function. On the other hand, if ~β = 0 but ~α = 0can be thought as a classic deterministic medium, but itstill produces PT -symmetry like effects. In principle, onecan have a match ~α = − ~β or ~γ = 0. The resulting fieldcannot be distinguished from a classic field, even thoughit would scatter from PT -symmetric medium. This is avery special case of the discussion after Eq. (26) for bothlinear phases. It appears impossible to arrange for sucha match if at least one of the phases is not linear.The spectral density produced on scattering of theplane wave to the far-zone of the PT -symmetric mediumcan be deduced from Eq. (44) as S sP T ( r ˆ s, ω ) = I S i ( ω ) r (2 π ) / a d (2 a + d ) / × exp (cid:18) − γ σ (cid:19) exp − ~K + 2 ~γ · ~Kσ ! , (48)where σ = 2 /d + 1 /a . Figures 2 and 3 give numericalexamples of S sP T in Eq. (48) for several values of a~γ =( aγ x , aγ y , aγ z ) depending on polar angle φ and azimuthalangle θ of the spherical coordinate system, defined as: s x = sin θ cos φ, s y = sin θ sin φ, s z = cos θ. (49)Consider now an example relating to classic medium.The C CL of the classic scatterer directly correspondingto Eq. (42) would have the form a ( ~r, ω ) = I exp (cid:18) − r a (cid:19) exp[ − i ( α x | x | + α y | y | + α z | z | )] ,µ ( ~r d , ω ) = exp (cid:20) − r d d (cid:21) exp[ i ( β x | x d | + β y | y d | + β z | z d | )] . (50)However, the Fourier transforms of a and µ do not leadto simple analytic equations for ˜ C CL . Instead we set a ( ~r, ω ) = I exp (cid:18) − r a (cid:19) exp( − iαr ) (51) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIG. 2. Contour plot of the position-dependent term in thespectral density (48) of the scattered radiation field for (a) a~γ = (0 , , a~γ = (0 . , ,
0) and (c) a~γ = (1 , , a~γ = (0 , . ,
0) and (e) a~γ = (0 , , ka =1, d/a = 1 and incident direction ˆ s = ˆ z . while keeping µ CL as in Eq. (42) with ~β = 0. Such classicmedium with quadratic phase yields the complex-valued˜ C CL of the form:˜ C CL ( − ~K , ~K , ω ) = I (cid:18) πc (cid:19) / (cid:18) c + c c + c (cid:19) − / × exp " − ( a + d )( ~K + ~K )4 + 8 α a d + 2 d /a × exp " iαa d ( ~K − ~K )4 + 8 α a d + 2 d /a × exp " a ~K · ~K α a d + 2 d /a , (52)where c = 4 α a d + 2 a + d , c = 1 /d + 1 /a , c = − α , c = 2 a d α and c = α + a d . Equation(52) clearly satisfies the condition ˜ C CL ( − ~K , ~K , ω ) =˜ C ∗ CL ( − ~K , ~K , ω ), as expected. Also, ˜ C CL is a complex- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIG. 3. Contour plot of the position-dependent term in thespectral density (48) of the scattered radiation field for (a) d/a = 0 .
1, (b) d/a = 0 . d/a = 1. Parameters used: ka = 1, a~γ = (1 , ,
1) and incident direction ˆ s = ˆ z . valued function but ˜ C P T ( − ~K, ~K, ω ) is real-valued. Thespectral density produced on scattering from the classicmedium, obtained from (52), is given by S sCL ( r ˆ s, ω ) = I S i ( ω ) r (cid:18) πc (cid:19) / (cid:18) c + c c + c (cid:19) − / × exp − ~K α a + 2 /d + 1 /a ! , (53)and it depends only on the angle between ˆ s and ˆ s .This is in contrast to the spectral density for the PT -symmetric medium where it depends not only on ~K butalso on ~γ · ~K [see Eq. (48)].In closing this section, let us analyze the spectral de-gree of coherence µ s ( r ˆ s , r ˆ s , ω ) for the radiation fieldscattered by classic and PT -symmetric media. Direct π x yz ss s θ θ s-s sin n^ ^^ ^ ^ ^ θ FIG. 4. Geometry for the symmetrical scattered directionswhere ˆ s is the direction of the incident field and θ is theangle between ˆ s and ˆ s . FIG. 5. Spectral degree of coherence µ sPT,CL ( θ ) for radia-tion scattered by PT -symmetric (continuous lines) and classic(dashed lines) materials. Three values of d/a are considered:0.1, 1 and 3. Parameters: ka = 1 and α/k = 2. substitution of Eqs. (44) and (52) into Eq. (41) gives µ sCL ( r ˆ s , r ˆ s , ω ) = exp (cid:20) iαa d ( K − K )4 + 8 α a d + 2 d /a (cid:21) × exp " − a ( ~K − ~K ) α a d + 2 d /a , (54)and µ sP T (ˆ s , ˆ s , ω ) = exp " − a ( ~K − ~K ) d /a . (55)It is usually the case where a pair of symmetrical direc- tions are chosen to highlight the statistical properties ofthe scattered radiation, for example (see Fig. 4)ˆ s = ˆ s, ˆ s = ˆ s − n sin θ, (56)The symmetry suggested by these vectors depends onlyon the angle between ˆ s and ˆ s and is independent of thecoordinate system. In the particular, basis-dependent,situation where ˆ s = ˆ z , angle θ is the azimuthal angle inspherical polar coordinates. For this choice, µ sCL ( θ, ω ) = exp (cid:18) − a k sin θ d / a + 2 α a d (cid:19) ,µ sP T ( θ, ω ) = exp (cid:18) − a k sin θ d / a (cid:19) . (57)We thus found, for these particular symmetrical direc-tions, that the spectral degree of coherence for the PT -symmetric scatterers depends only on its geometricalproperties while for the classic medium it depends on thephase α [see Eq. (51)]. Figure 5 shows how µ sCL,P T ( θ, ω ),given by (57), vary as a function of θ for several valuesof d/a . VI. SUMMARY
We have introduced a class of random, stationary me-dia whose second-order spatial correlation functions ofscattering potential obey the conditions of PT -symmetryand have derived their major properties. In such me-dia the balance between gain and loss centers can beachieved by requiring it from the individual realizationsof the scattering potential (or, alternatively, the indexof refraction). Also, individual potential’s realizationscan be passive (no gain or loss present) but the correla-tion function might possess a phase term leading to PT -symmetric like effects. In order to illustrate the distinc-tive nature of the introduced media we have made a closecomparison of their properties with those of “classic” me-dia, i.e., geometrically symmetric media with unbalancedgain and loss. Then we have applied the Bochner theo-rem of functional analysis for analytical modeling of thegenuine PT -symmetric correlation functions, and consid-ered in detail a particular but very important subclass ofsuch media with uniform (Schell-like) correlations.Further we have developed a theory of light scatteringfrom the PT -symmetric media and applied it to an ex-ample of a plane wave interacting with Gaussian Schell-model media with linear phase functions. Such an exam-ple provides the required insight into the nature of light-medium interactions while offering unsurpassed simplic-ity. In particular, it illustrates the fact that both typesof phase terms (of potential’s realizations and of correla-tion function) responsible for asymmetric intensity scat-tering can be present, the combination either enhancingthe produced asymmetry or suppressing (annihilating) it. [1] C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” PhysicalReview Letters 80, 5243 (1998).[2] C. M. Bender, PT Symmetry: In Quantum and ClassicalPhysics (World Scientific, Singapore, 2018).[3] C. M. Bender, D. C. Brody, and H. F. Jones, “Complexextension of quantum mechanics”, Physical Review Let-ters 89, 270401 (2002).[4] A. Mostafazadeh, “Pseudo-Hermiticity versus PT sym-metry: the necessary condition for the reality of thespectrum of a non-Hermitian Hamiltonian,” Journal ofMathematical Physics 43, 205 (2002).[5] C. E. R¨uter, K. G. Markis, El-Ganainy, D. N. R.Christodoulides, M. Segev, and D. Kip, “Observation ofPT symmetry in optics,” Nature Physics 6, 192 (2010).[6] L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B.Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer,“Experimental demonstration of a unidirectional reflec-tionless parity-time metamaterial at optical frequencies,”Nature Materials 12, 108 (2013).[7] Z. J. Wong, Y.-L. Xu, J. Kim, K. O’Brien, Y. Wang, L.Feng and X. Zhang, “Lasing and anti-lasing in a singlecavity”, Nature Photonics 10, 796 (2016).[8] P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N.M. Litchinitser, and L. Feng, “Lasing and anti-lasing ina single cavity,” Science 353, 464 (2016).[9] K. G. Makris, R. El-Ganainy, D. Christodoulides, and Z.H. Musslimani, “Beam dynamics in PT symmetric opti-cal lattices,” Physical Review Letters 100, 103904 (2008).[10] S. Longhi, “Parity-time symmetry meets photonics: Anewtwist in non-Hermitian optics,” Europhysics Letters120, 64001 (2017). [11] E. Wolf,