Quasi-Exactly Solvable Scattering Problems, Exactness of the Born Approximation, and Broadband Unidirectional Invisibility in Two Dimensions

# Quasi-Exactly Solvable Scattering Problems, Exactness of the Born Approximation, and Broadband Unidirectional Invisibility in Two Dimensions

## Abstract

Achieving exact unidirectional invisibility in a finite frequency band has been an outstanding problem for many years. We offer a simple solution to this problem in two dimensions that is based on our solution to another more basic open problem of scattering theory, namely finding scattering potentials in two dimensions whose scattering problem is exactly solvable for energies not exceeding a critical value . This extends the notion of quasi-exact solvability to scattering theory and yields a simple condition under which the first Born approximation gives the exact expression for the scattering amplitude whenever the wavenumber for the incident wave is not greater than . Because this condition only restricts the -dependence of , we can use it to determine classes of such potentials that have certain desirable scattering features. This leads to a partial inverse scattering scheme that we employ to achieve perfect broadband unidirectional invisibility in two dimensions. We discuss an optical realization of the latter by identifying a class of two-dimensional isotropic active media that do not scatter incident TE waves with wavenumber in the range and source located at , while scattering the same waves if their source is relocated to .

Introduction.— In 1926, the year Born published his monumental work on the probabilistic interpretation of quantum mechanics born1 (), he also laid the foundations of quantum scattering theory and introduced the celebrated Born approximation born2 (). The latter proved to be an extremely powerful tool for performing scattering calculations in different areas of physics hofstadter (); koshino (); born-wolf (). The Born approximation of order corresponds to the approximation scheme in which one neglects all but the first terms in the standard series solution of the Lippmann-Schwinger equation sakurai (). Because of its central importance, Born approximation is discussed in standard textbooks on quantum mechanics sakurai (), optics born-wolf (), and scattering theory newton (). But, surprisingly, none of these address the natural problem of inquiring into potentials for which the -th order Born approximation is exact. In one dimension, these turn out to be sums of delta-function potentials pra-2012 (); bookchapter (), , with real and possibly complex parameters. In two and higher dimensions, this has been an open problem. In this letter, we offer a solution of this problem for the case in two dimensions, i.e., identify potentials in two dimensions for which the first Born approximation is exact. This may seem as a purely academic problem, but its solution proves to have far-reaching consequences; it paves the way for devising a method of engineering scattering potentials which we employ to address another outstanding problem of basic importance, namely, achieving perfect unidirectional invisibility in a tunable finite frequency band.

The study of invisible potentials has been a subject of research for many decades. In one dimension, a potential is invisible from the left (respectively right), if it does not reflect a left- (respectively right-) incident plane wave and transmits it without changing the amplitude or phase of the wave, i.e., it is transparent. Reciprocity theorem implies that the transparency cannot be unidirectional bookchapter (). A nonreal scattering potential can, however, support unidirectional reflectionlessness and invisibility lin (); invisible3-1 (); invisible3-2 (); invisible3-3 (). The principal example is

 v(x)={ze2iβxforx∈[0,L],0forx∉[0,L], (1)

where is a coupling constant, is a nonzero real parameter, and , lin (); invisible-1 (); invisible-2 (); invisible-3 (). This potential is unidirectionally invisible from the right (respectively left) for an incident wave with wavenumber , if (respectively ) and , lin (); longhi-2011 (); jones (). The latter condition is to ensure the validity of the first Born approximation. For sufficiently large values of , the first Born approximation is unreliable and the unidirectional invisibility of the potential (1) breaks down pra-2014a ().

The problem of realizing exact unidirectional invisibility is addressed in Refs. pra-2014a (); ge-2012 (); pra-2013a (); ap-2014 (); pra-2014b (); jpa-2016 (). Similarly to (1), the unidirectionally invisible potentials considered in these references display this property for a single or a discrete set of wavenumbers. In one dimension, the problem of constructing potentials that have exact unidirectional invisibility in the entire wavenumber spectrum is solved in horsley-2015 (); longhi-2015 (). See also longhi-op-2015 (); longhi-op-2016 (); jiang (); hayran ().

Ref. prsa-2016 () generalizes the notion of unidirectional invisibility at isolated values of the wavenumber to two and three dimensions and examines particular examples of potentials that display this property perturbatively, i.e., similarly to (1) the potential is so weak that the first Born approximation is relible. A more recent development is the construction of a class of scattering potentials in two dimensions that display exact broadband omnidirectional invisibility ol-2017 (); these do not scatter incident plane waves with an arbitrary incidence angle provided that their wavenumber does not exceed a prescribed value . The broadband invisibility achieved in ol-2017 () is bidirectional in the sense that the scattering amplitude vanishes for regardless of the position of the source of the incident wave. The problem of realizing unidirectional invisibility in an extended frequency band, which we address in this letter, has been a well-known open problem for many years. This is a much more important problem, because its solution would allow for the design of material with broadband nonreciprocal functionalities.

Potential scattering in two dimensions.— Consider a two-dimensional scattering setup where the scatterer is described by a possibly complex-valued potential . The source of the incident wave, which is considered to be a plane wave, can be placed at or . We use the terms “left-incident” and “right-incident waves” to refer to these cases, respectively. In general, the wave vector of the incident wave makes an angle with the -axis, i.e., , where is the unit vector along the -axis with . We use to label for the left/right-incident waves. Clearly,

 0

By definition, if is a scattering potential, the solutions of the Schrödinger equation,

 [−∇2+v(x,y)]ψ(x,y)=k2ψ(x,y), (3)

tend to plane waves at spatial infinities. In particular, (3) admits the so-called “scattering solutions,” , that satisfy as , where , are the polar coordinates of , and is the scattering amplitude for the left/right-incident waves adhikari-86 (). The latter stores the scattering properties of the potential. Therefore its determination is the main objective of the scattering theory.

It is not difficult to show that the first Born approximation yields prsa-2016 ():

 fl/r(θ)=−~~v(\smallk(cosθ−cosθl/r0),k(sinθ−sinθl/r0))2√2π, (4)

where is the two-dimensional Fourier transform of . We wish to find conditions under which (4) gives the exact expression for the scattering amplitudes of the potential. Our main technical tool for achieving this purpose is the transfer-matrix formulation of potential scattering in two dimensions pra-2016 (). We summarize its basic ingredients in the sequel.
1. The transfer matrix of a scattering potential is a matrix with operator entries acting in the function space

 Fk:={ξ:R→C|ξ(p)=0 for |p|≥k}.

2. We can express as the time-ordered exponential of a non-Hermitian effective matrix Hamiltonian with operator entries acting in ;

 M= Texp{−i∫∞−∞dxH(x)}:=I−i∫∞−∞dxH(x)+ (−i)2∫∞−∞dx2∫x2−∞dx1H(x2)H(x1)+⋯, (5)

where is the time-ordering operation with playing the role of time, and is the identity operator for the space of two-component state vectors with components belonging to . The entries of are defined by

 Hij(x)ξ(p):=ϵie−iϵiϖ(p)x2ϖ(p)v(x,i∂p)[eiϵjϖ(p)xξ(p)], (6)

where , , is the linear operator acting in according to

 v(x,i∂p)ξ(p):=12π∫k−kdq~v(x,p−q)ξ(q), (7)

and is the Fourier transform of with respect to .
3. The scattering amplitudes are given by

 fl/r(θ)=−ik|cosθ|√2π×{Tl/r−(ksinθ)for cosθ<0,Tl/r+(ksinθ)for cosθ>0, (8)

where are the elements of fulfilling

 M22Tl−(p)=−2πM21δ(p−pl0), (9) Tl+(p)=M12Tl−(p)+2π(M11−I)δ(p−pl0), (10) M22Tr−(p)=−2π(M22−I)δ(p−pr0), (11) Tr+(p)=M12[Tr−(p)+2πδ(p−pr0)], (12)

and is the identity operator acting in .

Because is in general an integral operator, (9) and (11) are linear integral equations. According to (8), we can solve the scattering problem for the potential provided that we determine the transfer matrix and solve (9) and (11). Refs. pra-2016 (); pra-2017 (); jpa-2018 () offer details of the application of this scheme for solving concrete scattering problems.

Exact solvability below a prescribed energy.— The transfer matrix , the functions , and the scattering amplitude depend on the incidence angle and the wavenumber . A critical observation underlying the present study is that under a fairly simple condition on the potential, the Dyson series for the transfer matrix (5) truncates for wavenumbers not exceeding a critical value , i.e., energies . Furthermore, the same condition allows for an explicit solution of the integral equations yielding .

Consider a scattering potential such that

 ~v(x,Ky)=0  for  Ky≤α, (13)

and let . Then in view of (7) and the fact that , we have

 v(x,i∂p)ξ(p)=12π∫k+pαdq~v(x,q)ξ(p−q). (14)

For the cases where , we can use this equation to show that for all . This together with (6) imply

 Hij(x2)Hi′j′(x1)ξ(p)=0  for  k≤α. (15)

Hence vanishes, and (5) implies

 M=I−i∫∞−∞dxH(x)  for  k≤α. (16)

Next, we use (15) and (16) to show that

 (Mij−δijI)(Mi′j′−δi′j′I)=0  for  k≤α, (17)

where denotes the Kronecker delta symbol, and stands for the zero operator acting in . This in turn allows for a straightforward calculation of . To see this, we express (9) and (11) in the form

 Tl−(p) = −(M22−1)Tl−(p)−2πM21δ(p−pl0), (18) Tr−(p) = −(M22−1)[Tr−(p)+2πδ(p−pr0)]. (19)

Similarly to (10) and (12), the right-hand sides of these equations consist of terms obtained by applying operators of the form to functions belonging to . In light of (17), this implies . Using this relation in (10), (12), (18), and (19), we find

 Tl−(p) = −2πM21δ(p−pl0), (20) Tl+(p) = 2π(M11−I)δ(p−pl0), (21) Tr−(p) = −2π(M22−1)δ(p−pr0), (22) Tr+(p) = 2πM12δ(p−pr0). (23)

To obtain more explicit formulas for , we introduce and use (6), (14), and (16) to show that . Substituting this equation in (20) – (23) and using the result together with the identities, and , in (8), we obtain (4). This proves the following theorem.
Theorem 1 Let be a scattering potential satisfying (13). Then the first Born approximation provides the exact solution of its scattering problem whenever the wavenumber of the incident wave does not exceed .

This result is reminiscent of the notion of a quasi-exactly solvable potential turbiner (). The time-independent Schrödinger equation for such a potential can be solved to determine finitely many low-lying bound state energies and the corresponding eigenfunctions. The potentials fulfilling (13) may also be viewed as quasi-exactly solvable, because their scattering problem is exactly solvable for energies .

Perfect broadband unidirectional invisibility.— Consider the scattering of left- and right-incident waves with wavenumber by a potential satisfying (13), and suppose that . Then (4) holds, and (2) implies

 −2α≤−2k≤k(cosθ−cosθl0)≤k≤α, (24) −α<−k≤k(cosθ−cosθr0)<2k≤2α. (25)

According to (4) and (24), for all possible values of and provided that for . In other words this is a sufficient condition for the left-invisibility of whenever . For scattering potentials of physical interest, this condition is equivalent to

 ~v(Kx,y)=0  for  Kx∈[−2α,α), (26)

where is the Fourier transform of with respect to . Similarly, we can use (25) to obtain the following sufficient condition for the right-invisibility of the potential for .

 ~v(Kx,y)=0  for  Kx∈(−α,2α]. (27)

Because (26) and (27) do not coincide, there is a range of values of for which only one of these conditions holds. This suggests the possibility of achieving broadband unidirectional invisibility.

Suppose that (13) and (27) hold. Then (4) applies, and the potential is right-invisible for all and . To ensure that it is not left-invisible, we must determine values of within the interval and ranges of values of and for which , i.e., . Conditions (13) and (27) violate this inequality unless and . We have used these relations together with (2) to show that

 −π2<θl0<0, π2≤θ<π, α√2

and , where . Similarly, we find the following necessary conditions for unidirectional left-invisibility of : and

 π<θr0≤3π2, 0<θ≤π2, α√2

The following theorem summarizes our results on sufficient conditions for broadband unidirectional invisibility.
Theorem 2. A scattering potential is unidirectionally right- (respectively left-) invisible for wavenumbers , if it satisfies (13), (27), and

 ~v(Kx,y)≠0  for  Kx∈[−2α,−α], (28)

(respectively (13), (26), and for .)

Next, we recall that under multiplication of a function by a phase factor its Fourier transform changes to . This allows for expressing conditions appearing in the statement of Theorems 1 and 2 in terms of functions with vanishing Fourier transform with respect to or on the negative or positive - or -axes footnote1 (). To be more specific, let and be scattering potentials satisfying

 ~w±(Kx,y)=0  respectively for  ±Kx≤0, (29) ~w±(x,Ky)=0  for  Ky≤0, (30) vl(x,y)=eiαy[γe−i2αxw−(x,y)+eiαxw+(x,y)], (31) vr(x,y)=eiαy[e−iαxw−(x,y)+γe2iαxw+(x,y)], (32)

where . Then, according to Theorem 2, for , is unidirectionally left-invisible, if

 ~w+(Kx,y)≠0 for Kx∈[0,α], (33)

and is unidirectionally right-invisible, if

 ~w−(Kx,y)≠0 for Kx∈[−α,0]. (34)

We can construct specific examples for using functions of the form

 gj(x):=(x/Lj+i)−nj−1,

where , and and are respectively positive real numbers and positive integers. It is easy to check that for . This in turn implies that the function satisfies for . In view of these properties of and , we can show that the following choices for fulfill (29), (30), (33), and (34).

 w−(x,y)=z−¯g1(x)g2(y), w+(x,y)=z+g3(x)g4(y), (35)

where are nonzero real or complex coupling constants. Substituting (35) in (31) and (32), we therefore find scattering potentials that are unidirectionally left- and right-invisible for . It is important to note that this is true for arbitrary choices of and , and that linear combinations of potentials of this form (with different choices for and ) posses the same unidirectional invisibility property.

Optical realization.— Complex scattering potentials in two dimensions may be used to model the scattering of transverse electric (TE) waves by the inhomogeneities of an effectively two-dimensional nonmagnetic isotropic medium , born-wolf (); ol-2017 (). Let label the permittivity profile of and suppose that as , tends to a constant value . For a TE wave propagating in , we can express its electric field in the form , where is a constant amplitude, labels time, is the unit vector along the -axis, , and are respectively the permittivity and the speed of light in vacuum, and is the wavenumber. We can use Maxwell’s equations to identify with a solution of the Schrödinger equation (3) for the potential

 v(x,y)=k2[1−^ε(x,y)], (36)

where is the relative permittivity of .

In order for the medium to display perfect broadband unidirectional invisibility for wavenumbers , it suffices to select a reasonable value for , identify the left-hand side of (36) with one of the unidirectionally invisible potentials of the preceding section, and solve this equation for .

For example, consider the of (35) with and given by , and let and . Substituting this choice for in (32), setting , and equating the resulting potential with the left-hand side of (36), we find

 ^ε(x,y)=1+z−eiα(y−x)[(x/a−i)(y/b+i)]2. (37)

By construction, the first Born approximation provides an exact description of the scattering of TE waves by the optical medium possessing this relative permittivity profile provided that their wavenumber does not exceed . Substituting (37) in (36) and making use of (4), we find the following expression for the scattering amplitudes.

 fl/r(θ)=π√2πz−a2b2k4X(ak,cl/r)X(bk,sl/r), (38)

where , , and . Because, , for we have . This implies that . Therefore, for all , i.e., the medium is right-invisible. Because for there are ranges of values of within the interval for which , the right-invisibility of the medium is unidirectional. Fig. 1 provides a graphical demonstration of this behavior for , , and .

As expected, the medium described by the premittivity profile (37) has a nonzero scattering amplitude for left-incident waves with incidence angle provided that their wavelength lies between 500.0 nm and 707.1 nm.

Concluding remarks.— Quasi-exactly solvable scattering potentials are potentials whose scattering problem is exactly solvable for wavenumbers not exceeding a prescribed value . In this letter we have employed the transfer-matrix formulation of the scattering theory to identify a class of quasi-exactly solvable potentials in two dimensions. The first Born approximation provides the exact expression for the scattering amplitudes of these potentials for . In one dimension, the only potential whose scattering problem is exactly solvable by the first Born approximation is the delta-function potential bookchapter (); pra-2012 (). This result does not extend to two dimensions; the delta-function potential in two dimensions is not exactly solvable by the first Born approximation pra-2016 (); jpa-2018 (); manuel ().

Condition (13) that ensures the exactness of the first Born approximation for wavenumbers does not restrict the -dependence of the potential . Because, according to (4), such a potential may be obtained by performing the inverse Fourier transform of the scattering amplitudes , we can fix the -dependence of the potential by demanding that has a certain desirable behavior for . This yields an extremely effective partial inverse scattering prescription which we have presently employed for achieving perfect broadband unidirectional invisibility. The latter admits a straightforward optical realization involving certain two-dimensional nonmagnetic isotropic media with regions of gain and loss. We expect a two-dimensional analog of the setup employed in Ref. jiang () to allow for an experimental detection of this effect.

The generalization of our results to three dimensions does not pose any major difficulty. In particular, one can pursue the above-mentioned idea of partial inverse scattering in three dimensions. The progress in this direction can have interesting applications in acoustics.

Acknowledgements.— We are indebted to the Turkish Academy of Sciences (TÜBA) for providing the financial support for FL’s visit to Koç University during which we have carried out a major part of the work reported here.

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