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Research Cite this article: Loran F, Mostafazadeh A. 2016 Unidirectional invisibility and non-reciprocal transmission in two and three dimensions. Proc. R. Soc. A 472: 20160250. http://dx.doi.org/10.1098/rspa.2016.0250 Received: 5 April 2016 Accepted: 10 June 2016
Subject Areas: optics, quantum physics, acoustics Keywords: scattering, reciprocity principle, non-reciprocal transmission, unidirectional invisibility, multidimensional transfer matrix Author for correspondence: Ali Mostafazadeh e-mail: [email protected]
Unidirectional invisibility and non-reciprocal transmission in two and three dimensions Farhang Loran1 and Ali Mostafazadeh2 1 Department of Physics, Isfahan University of Technology,
Isfahan 84156-83111, Iran 2 Departments of Mathematics and Physics, Koç University, 34450 Sarıyer, Istanbul, Turkey AM, 0000-0002-0739-4060 We explore the phenomenon of unidirectional invisibility in two dimensions, examine its optical realizations and discuss its three-dimensional generalization. In particular, we construct an infinite class of unidirectionally invisible optical potentials that describe the scattering of normally incident transverse electric waves by an infinite planar slab with refractive-index modulations along both the normal directions to the electric field. A by-product of this investigation is a demonstration of non-reciprocal transmission in two dimensions. To elucidate this phenomenon, we state and prove a general reciprocity theorem that applies to quantum scattering theory of real and complex potentials in two and three dimensions.
1. Introduction In one dimension, a real scattering potential has identical reflection and transmission coefficients for incoming waves from the left and right, i.e. it obeys reciprocity in reflection and transmission. In contrast, a complex scattering potential can violate reciprocity in reflection [1]. A remarkable consequence of this observation is the existence of unidirectionally reflectionless and invisible potentials [2–19]. These have recently attracted a great deal of attention, mainly because they model certain oneway optical devices with possible applications in optical circuitry. One of the most sought-after examples of a one-way element of an optical circuit is an optical diode [20]. It is generally believed that real and complex potentials are incapable of modelling an optical diode, because they are bound to respect reciprocity in transmission [21–23]. 2016 The Author(s) Published by the Royal Society. All rights reserved.
Consider the Schrödinger equation − ∇ 2 ψ(r) + v(r)ψ(r) = k2 ψ(r),
(2.1)
for a real or complex scattering potential v, where r := (x, y) and k is the wavenumber. Let us identify the x-axis with the scattering axis, and recall that for x → ±∞ the scattering solutions of (2.1) have the form 1 k dp eipy [A± (p) eiω(p)x + B± (p) e−iω(p)x ], (2.2) 2π −k where A± (p) and B± (p) are coefficient functions vanishing for p ∈ / [−k, k] and ω(p) := [26]. By definition, the transfer matrix of v is the 2 × 2 matrix operator M(p) satisfying A− (p) A+ (p) = M(p) . B+ (p) B− (p)
k2 − p2 ,
(2.3)
Similarly to its one-dimensional analogue [27], it can be expressed as a time-ordered exponential [26] ∞ −iH(x, p) dx, (2.4) M(p) := T exp −∞
where x plays the role of time, H(x, p) is the non-Hermitian effective Hamiltonian operator H(x, p) :=
1 e−iω(p)xσ 3 v(x, i∂p )K eiω(p)xσ 3 , 2ω(p)
(2.5)
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2. Transfer matrix and unidirectional invisibility in two dimension
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The same problem arises also in acoustics where designing acoustic waveguides with nonreciprocal transmission is a major area of research [24]. The search for means of achieving non-reciprocal transmission is dominated by the use of nonlinear, time-dependent or magnetic materials [20,24,25]. This work is motivated by the observation that reciprocity in reflection and transmission can both be violated within the confines of linear, stationary, isotropic, non-magnetic material provided that one employs a genuine multidimensional set-up. In particular, it is possible to construct unidirectionally invisible potentials in two and three dimensions that enjoy non-reciprocal transmission. The basic conceptual framework for this work is the multidimensional transfer matrix formulation of scattering theory that we report in reference [26]. We use it to offer a precise and convenient way of describing unidirectional reflection, unidirectional invisibility and transmission reciprocity in dimensions higher than one. The organization of this article is as follows. In §2, we review the formulation of scattering theory presented in reference [26] and use it to give a precise definition for the notions of unidirectional reflectionlessness, unidirectional transparency, unidirectional invisibility and reciprocal transmission in two dimensions. Here, we also derive some useful relations characterizing these notions. In §3, we prove a general reciprocity theorem that applies to real and complex scattering potentials in two dimensions. In §4, we construct an infinite class of unidirectionally invisible potentials in two dimensions that possess non-reciprocal transmission. In §5, we examine an optical realization of a particular example of these potentials and discuss the physical implications of their unidirectional invisibility and non-reciprocal transmission properties. In §6, we generalize the results of the preceding sections to three dimensions, in particular establishing the three-dimensional analogue of the reciprocity theorem of §3 and constructing finite-range unidirectionally invisible potentials with non-reciprocal transmission in three dimensions. Finally, in §7, we summarize our findings and present our concluding remarks.
where (r, θ) are the polar coordinates of r, and f l (θ ) is the scattering amplitude for the left-incident waves. In order to express the latter in terms of M(p), we introduce l T− (p) := B− (p)
and
l T+ (p) := A+ (p) − A− (p),
(2.7)
B+ (p) = 0,
(2.8)
and note that for a left-incident wave, A− (p) = 2π δ(p)
and
where δ stands for the Dirac delta-function. In reference [26], we show that (2.3), (2.7) and (2.8) imply l (p) = −2π M−1 T− 22 M21 δ(p), l T+ (p) = 2π (M11
(2.9)
− 1 − M12 M−1 22 M21 )δ(p)
l f l (θ ) = −(2π )−1/2 ik| cos θ|T± (k sin θ),
and
(2.10) (2.11)
where Mij are the entries of M(p) and the ± in (2.11) stands for sgn(cos θ). Similarly, we can introduce the scattering amplitude f r (θ) for a right-incident wave by expressing the asymptotic form of the corresponding scattering solution as i ikr r r −ikx e f (θ) as r → ∞. ψ (r) = e + (2.12) kr To relate f r (θ ) to M(p), we define r (p) := B− (p) − B+ (p) T−
and
r T+ (p) := A+ (p),
(2.13)
and set A− (p) = 0
and
B+ (p) = 2π δ(p).
(2.14)
These, together with (2.3), give
and
r T− (p) = −2π (1 − M−1 22 )δ(p),
(2.15)
r T+ (p) = 2π M12 M−1 22 δ(p)
(2.16)
r f r (θ ) = −(2π )−1/2 ik| cos θ|T± (k sin θ),
(2.17)
where again ± := sgn(cos θ ). l/r It is important to note that Mij and T± (p) depend on the wavenumber k. Therefore, any l/r
quantity or concept whose definition involves Mij or T± (p) is k-dependent. The particular form of the effective Hamiltonian operator (2.5) and consequently those of K and v(x, i∂p ) are dictated by the formalism developed in reference [26]. They are necessary for establishing equation (2.4), which is central to the developments reported in reference [26] and this article.
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˜ Ky ) := and v(x, ˜ Ky ) denotes the Fourier transform of v(x, y) with respect to y, i.e. v(x, ∞ −iKy y v(x, y).1 dy e −∞ The transfer matrix M(p) contains all the scattering information about v. To see this, we recall that for a left-incident plane wave with a momentum k pointing along the positive x-axis, the scattering solutions ψ l of (2.1) have the asymptotic form i ikr l l ikx e f (θ) as r → ∞, ψ (r) = e + (2.6) kr
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σ i are the Pauli matrices, K := σ 3 + iσ 2 , v(x, i∂p ) is the integral operator acting on test functions φ : [−k, k] → C according to 1 k dqv(x, ˜ p − q)φ(q), v(x, i∂p )φ(p) := 2π −k
The following is a direct consequence of (2.11) and (2.17). Theorem 2.2. Let v be as in definition 2.1. Then, v is l (p) = 0 (respectively, T r (p) = 0) for all — left- (respectively right-) reflectionless if and only if T− + p ∈ [−k, k] and l (p) = 0 (respectively T r (p) = 0) for all p ∈ [−k, k]. — left- (respectively right-) transparent if T+ −
In view of (2.9), (2.10), (2.15) and (2.16), we also have the following useful result. Theorem 2.3. Let v be as in definition 2.1. Then, v is — left-reflectionless, if and only if M21 δ(p) = 0 for all p ∈ [−k, k]. — left-invisible, if and only if M21 δ(p) = (M11 − 1)δ(p) = 0 for all p ∈ [−k, k]. — right-invisible, if and only if M12 δ(p) = (M22 − 1)δ(p) = 0 for all p ∈ [−k, k].
3. Reciprocity principle in two dimension The phrase ‘reciprocity principle’ is usually taken to be synonymous to the ‘Lorentz reciprocity theorem’ for electromagnetic fields which follows from Maxwell’s equations [20]. Here, we formulate a reciprocity principle that applies directly to the quantum scattering theory as defined by the Schrödinger equation (2.1). First, we define what we mean by reciprocity in transmission in two dimensions. Definition 3.1. A real or complex scattering potential v is said to have reciprocal transmission l (p) = T r (p). According to (2.10) and (2.15), this is for a wavenumber k, if for all p ∈ [−k, k], T+ − equivalent to −1 (3.1) (M11 − M−1 22 − M12 M22 M21 )δ(p) = 0. We can also use (2.11) and (2.17) to state it in the form f l (θ ) = f r (π − θ )
for all θ ∈ − π2 , π2 .
(3.2)
The content of definitions (2.1) and (3.1) reduce to their well-known one-dimensional analogue provided that we replace δ(p) with 1 and identify Mij with the entries of the transfer matrix in one dimension [14]. In this case, condition (3.1) for reciprocal transmission reduces to M11 − −1 M−1 22 − M12 M22 M21 = 0 which is equivalent to det M = 1. But, in one dimension, one can use Abel’s theorem for linear homogeneous ordinary differential equations [28] to show that ‘det M = 1’ holds for every real or complex scattering potential [29]. This proves that non-reciprocal transmission is forbidden in one dimension. It also suggests that in order to establish (3.1) or its violation in two (and higher) dimensions, we should seek for an analogue of Abel’s theorem for the Schrödinger equation (2.1), which is a partial differential equation.
...................................................
— left- (respectively right-) reflectionless if f l (θ) = 0 for all θ ∈ (π/2, 3π /2) (respectively, f r (θ ) = 0 for all θ ∈ (−π/2, π/2)), — left- (respectively right-) transparent if f l (θ) = 0 for all θ ∈ (−π/2, π/2) (respectively, f r (θ ) = 0 for all θ ∈ (π/2, 3π /2)), — left- (respectively right-) invisible if it is both left- (respectively right-) reflectionless and transparent, — unidirectionally reflectionless (respectively, transparent or invisible) if it is either left- or right-reflectionless (respectively, transparent or invisible), but not both.
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Definition 2.1. Let v : R2 → C be a scattering potential and k be a wavenumber. Choose a Cartesian coordinate system in which the scattering axis coincides with the x-axis. Then, v is said to be
Now, let a and b be positive real numbers, V be the rectangular region enclosed between the lines x = 0, x = a and y = ±b, and ∞ dy ψ1 (r)∂x ψ2 (r) − ψ2 (r)∂x ψ1 (r) . (3.5) j(x) := −∞
Suppose that ∂y ψj (x, y) → 0 for y → ±∞, which is consistent with (2.6) and (2.12). Then, applying the divergence theorem to J(r) and V, using (3.4), and taking the b → ∞ limit, we find that j(a) = j(0). Because a is arbitrary, this means that j(x) does not depend on x. In particular, we have3 j(−∞) = j(∞).
(3.6)
Because scattering solutions of (2.1) admit an asymptotic expression of the form (2.2), we can calculate j(±∞) whenever ψ1 and ψ2 are scattering solutions. Let Aj± and Bj± , respectively, denote the coefficient functions A± and B± appearing in the asymptotic expression (2.2) for ψj . Then, substituting this expression in (3.5), we obtain −i k dpω(p) ± (p) for x → ±∞ (3.7) j(x) = π −k and ± (p) := A1± (−p)B2± (p) − B1± (−p)A2± (p).
(3.8)
The fact that the right-hand side of (3.7) does not involve x provides an independent verification of (3.6). Next, we use the definition of the transfer matrix, namely (2.3) to express the Aj+ and Bj+ appearing in (3.8) in terms of Aj− and Bj− . Inserting the resulting formula in (3.7), we can express (3.6) in the form k dpω(p)[C1+ (−p)T ΩC2+ (p) − C1− (−p)T ΩC2− (p)] = 0, (3.9) −k
where the superscript ‘T’ stands for the transpose, Aj− (p) (3.10) Cj− (p) := and Cj+ (p) := M(p)Cj− (p), Bj− (p) 0 1 and Ω := iσ 2 = −1 0 is the standard 2 × 2 symplectic matrix. Because the choice of Cj− (p) is arbitrary, (3.9) is equivalent to ←−−−−− −−→ (3.11) M(−p)T Ω M(p) = Ω. Here, we use arrows to stress that M(−p)T acts on the test functions appearing to its left, whereas M(p) acts on those appearing to its right. For a real potential, we may choose ψ2 = ψ1∗ . If we do so, (3.3) coincides (up to an unimportant constant coefficient) with the standard probability current density corresponding to ψ1 . For a complex potential, this choice is forbidden, because ψ1∗ is no longer a solution of the Schrödinger equation (2.1). In this case, setting ψ2 = ψ1∗ gives a probability current density that is not divergence-free [30]. In contrast, (3.3) is a divergence-free (conserved) current density for both real and complex potentials provided that ψ1 and ψ2 solve (2.1).
2
Here, we assume that j(x) has finite asymptotic values for x → ±∞. For our purposes, we need to consider only j(x) for the scattering solutions ψ l/r , and for these, limx→±∞ j(x) exist.
3
...................................................
It is easy to check that whenever ψ1 and ψ2 are solutions of the Schrödinger equation (2.1), J is divergence-free2 ∇ · J(r) = 0. (3.4)
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In one dimension, the Schrödinger equation reads: −ψ (x) + v(x)ψ(x) = k2 ψ(x), and Abel’s theorem states that the Wronskian, ψ1 (x)ψ2 (x) − ψ2 (x)ψ1 (x), of any pair of solutions of this equation, ψ1 and ψ2 , is x-independent. A higher-dimensional generalization of the Wronskian is the vector field (3.3) J(r) := ψ1 (r)∇ψ2 (r) − ψ2 (r)∇ψ1 (r).
B1+ (p) = 0,
B2+ (p) = 2π δ(p),
A2− (p) = 0.
Substituting these relations in (3.8) and making use of (2.8), (2.14) and (3.7), we can write (3.6) as l r (0) = T− (0), T+
or
f l (0) = f r (π ).
(3.12)
In view of (2.11) and (2.17), this proves the following reciprocity theorem.
Theorem 3.2 (Reciprocity Principle). For every real or complex scattering potential, the forward scattering amplitude for the left- and right-incident waves coincide. Note that this theorem does not prohibit non-reciprocal transmission in the sense of definition 3.1. In particular, as we shown in §4, there are scattering potentials with different total transmission cross section from the left and right.
4. Unidirectionally invisible potentials with non-reciprocal transmission The recent interest in unidirectional invisibility is initiated by the study of the locally periodic optical potentials of the form [2–10] ⎧ ⎨z eiKx for x ∈ [0, a], (4.1) v(x) = z eiKx χa (x) = ⎩0 otherwise, where z is a coupling constant, K := 2π/a, a is a positive real parameter and χa is the characteristic function for the interval [0, a], i.e. ⎧ ⎨1 for x ∈ [0, a], χa (x) := ⎩0 otherwise. This potential is unidirectionally invisible for k = π/a provided that |z| is so small that the scattering properties of v can be determined using the first Born approximation, i.e. it displays perturbative unidirectional invisibility [15]. In optical applications, where v is related to the relative permittivity εˆ of the material according to v(x) = k2 [1 − εˆ (x)], this condition holds whenever |ˆε(x) − 1| 1. In one dimension, one can use the dynamical formulation of scattering theory developed in reference [27] to construct scattering potentials that enjoy exact unidirectional invisibility [27,31]. These have a more complicated structure than (4.1). Reference [15] provides a complete characterization of finite-range potentials that similarly to (4.1) support perturbative unidirectional invisibility. In the remainder of this section, we construct a class of perturbative unidirectionally invisible potentials in two dimensions that generalize (4.1). Consider a planar slab of optical material with relative permittivity εˆ that is located between the planes x = 0 and x = a in three-dimensional Euclidean space. Suppose that the slab has translational symmetry along the z-directions, so that εˆ depends only on x and y. Then, the interaction of this slab with the time-harmonic normally incident z-polarized transverse electric waves of the form (4.2) E(x, y, z) = E0 e−ikct ψ(x, y)ez is described by the Schrödinger equation (2.1) with the potential v(x, y) = k2 [1 − εˆ (x, y)]χa (x).
(4.3)
In equation (4.2), E0 is a non-zero complex parameter, c is the speed of light in vacuum and ez is the unit vector pointing along the positive z-axis.
...................................................
A1− (p) = 2π δ(p),
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In order to elucidate the physical meaning of equation (3.6), which is equivalent to (3.9) and (3.11), we identify ψ1 and ψ2 with scattering solutions ψ l and ψ r associated with the incident plane waves from the left and right, respectively. This corresponds to setting
l T± (p) ≈
and r (p) ≈ T±
−i ˜ v(−p ˜ ∓ , p) 2ω(p)
(4.5)
−i ˜ v(p ˜ ± , p), 2ω(p)
(4.6)
˜˜ x , Ky ) is the two-dimensional Fourier transform of v(x, y), i.e. where v(K ∞ ∞ ˜˜ x , Ky ) := dx dy e−i(Kx x+Ky y) v(x, y), v(K −∞
−∞
(4.7)
and p± := k ± ω(p) = k ± k2 − p2 . It is easy to see that equations (4.5) and (4.6) are consistent with the statement of the reciprocity principle (theorem 3.2); clearly, for p = 0, we have p− = 0, and (3.12) holds. In view of definition 2.1, equations (4.5) and (4.6) imply that v is unidirectionally invisible from the right if and only if the following conditions hold. ˜˜ ± , p) = 0 for all p ∈ [−k, k] v(p
and
˜˜ ˜˜ |v(−p − , p)| + |v(−p + , p)| = 0 for some p ∈ [−k, k].
(4.8)
In order to obtain the explicit examples of potentials of the form (4.3) that satisfy these conditions, we expand them in their Fourier series in [0, a]. This gives v(x, y) = χa (x) where cn (y) := (1/a)
∞
cn (y) einKx ,
(4.9)
n=−∞
a
−inKx v(x, y) 0 dx e
and K :=
In view of (4.7) and (4.9), ˜˜ x , Ky ) = v(K
∞
n=−∞
2π . a
eia(nK−Kx ) − 1 c˜ n (Ky ). i(nK − Kx )
(4.10)
(4.11)
Substituting this relation in (4.8), we find two complex equations for the coefficient functions c˜ n (p). These characterize perturbative right-invisible potentials v(x, y) that vanish for x ∈ / [0, a]. To obtain concrete examples of such potentials, we take a pair of distinct non-zero integers, and m, and set (4.12) cn (y) = 0 for n = 0, , m. We then solve (4.8) to express c˜ 0 , c˜ and c˜ m in terms of an arbitrary function that we denote by g˜ . This gives c˜ 0 (p) = p2 g˜ (p),
and
(4.13)
c˜ (p) =
m[ K( K − 2k) + p2 ]˜g(p) −m
(4.14)
c˜ m (p) =
[mK(mK − 2k) + p2 ]˜g(p) . m−
(4.15)
...................................................
−∞
where I is the 2 × 2 identity matrix, and ‘≈’ stands for the first Born approximation. Substituting (2.5) in this relation to determine Mij and using these in (2.9), (2.10), (2.15) and (2.16) give
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In what follows, we consider situations where |ˆε(x, y) − 1| is so small that the first Born approximation provides a reliable description of the behaviour of the system. In this case, we can compute the transfer matrix (2.4), using the formula ∞ dx H(x, p), (4.4) M(p) ≈ I − i
z
8
x
s a b
Figure 1. Schematic view of an optically active wire of rectangular cross section with a screen placed to its right at x = d. (Online version in colour.)
Evaluating the inverse Fourier transform of these relations and using the result together with (4.12) in (4.9), we find ei Kx − m eimKx mK i Kx imKx ( K − 2k) e g(y) , − (mK − 2k) e v(x, y) = χa (x) − 1 + g (y) + −m −m (4.16) where g is the inverse Fourier transform of g˜ . Equation (4.16) gives a right-invisible potential for each choice of k, , m and g, provided that g and g be such that (4.16) defines a well-behaved scattering potential and that the first Born approximation is justified. Typical examples are g(y) = g0 e−y
2
/2b2
(4.17)
and g(y) = g0 b−4 y2 (y − b)2 χb (y),
(4.18)
where g0 is a possibly complex non-zero coupling constant, b is a positive real parameter, and |g0 | is sufficiently small. Note that (4.18) corresponds to a finite-range right-invisible potential whose support is the rectangular region: [0, a] × [0, b]. It describes an optically active wire of rectangular cross section that is aligned along the z-axis, as shown in figure 1. l (p). In order to make sure that the potentials (4.16) are not invisible from the left, we examine T± First, we substitute (4.13)–(4.15) in (4.11) and make use of (4.10) and (4.12) to derive ˜˜ x , Ky ) = v(K
mK2 (1 − e−iaKx )[K2y + (Kx − 2k)Kx ]˜g(Ky ) iKx (Kx − K)(Kx − mK)
.
This, together with (4.5), implies l T± (p) ≈
2 mK2 k(1 − eiap∓ ) g˜ (p). ω(p)(p∓ + K)(p∓ + mK)
(4.19)
Equation (4.19) provides an explicit demonstration of non-reciprocity in transmission, because for l (p) = 0, whereas T r (p) ≈ 0 for all p ∈ [−k, k]. generic choices of g and p, we have T+ −
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s
d
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y
Using (4.19) in (2.11), we also find
(4.20)
According to this relation and the fact that f r (θ ) ≈ 0, the potential (4.16) is unidirectionally invisible from the right, but it is neither left-reflectionless nor left-transparent.
5. Optical manifestation of unidirectional invisibility and non-reciprocal transmission We can identify the unidirectionally invisible potentials of the form (4.16) with optical potentials describing the interaction of the linearly polarized electromagnetic waves (4.2) with an optically active wire of rectangular cross section as depicted in figure 1. In this section, we explore the implications of the non-reciprocal transmission property of these potentials for the transmitted power associated with the left- and right-incident waves. This requires the computation of the Poynting vector for the corresponding scattered waves. First, we recall that for a charge-free, non-magnetic, isotropic medium, the time-averaged energy density and Poynting vector are respectively given by
u = 14 Re(E · D∗ + B · H∗ )
and
S := 12 Re(E × H∗ ),
(5.1)
where E and H are the electric and magnetic fields, D = ε0 εˆ E, B = μ0 H, and ε0 and μ0 are the permittivity and permeability of the vacuum, respectively. For time-harmonic waves, Maxwell’s √ equations imply that H = (1/ik) ε0 /μ0 ∇ × E. Substituting this relation in (5.1) and making use of (4.2), we have
u =
ε0 |E0 |2 Re[ˆε(r)]|ψ(r)|2 + k−2 |∇ψ(r)|2 4
(5.2)
|E0 |2 Im ψ(r)∗ ∇ψ(r) , 2μ0 ck
(5.3)
and
S =
where ‘Re’ and ‘Im’, respectively, denote the real and imaginary part of their argument. Now, consider the time-averaged energy density and Poynting vector for the scattered waves ψ l/r , which we denote by ul/r and Sl/r , respectively. We can use the latter to compute the reflected and transmitted power to the left (x = −∞) and right (x = ∞). It is not difficult to see that the contribution of the scattering potential to these quantities, i.e. the difference between their value in the presence and absence of the potential, is proportional to l/r
P± := ±
∞ −∞
dy ex · ( Sl/r − S∅ )
,
(5.4)
x=±∞
where ex is the unit vector pointing along the positive x-axis, and the subscript ‘∅’ means that the l , corresponding quantity is computed in the absence of the potential, i.e. for v = 0. Note that P− l r r l P+ , P− and P+ , respectively, correspond to the reflected power for ψ , transmitted power for ψ l , transmitted power for ψ r and reflected power for ψ r .
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√ − 2i mK2 k[1 − eiak(1−cos θ) ]˜g(k sin θ) . π [k(1 − cos θ ) + K][k(1 − cos θ) + mK]
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f l (θ ) ≈ √
9
l/r
implies that the total reflected and transmitted power for ψ l are quadratic functions of the strength of the potential.4 Next, we examine a more realistic situation where we intend to determine the power transmitted to a finite screen placed at a large but finite distance from the wire. To this end, we introduce the dimensionless parameters uˆ :=
u − u∅
u∅
and Sˆ :=
S − S∅ . | S∅ |
(5.5)
In the light of the right invisibility of the wire, these vanish for a right-incident wave. For a scattering solution corresponding to a left-incident wave, they have the following asymptotic expression (in the limit r → ∞): r , (5.6) uˆ ≈ (1 + cos θ)ξ (r, θ ) and Sˆ ≈ ξ (r, θ) ex + r where
−ikx
ξ (r, θ) := Re[e
l
ψ ] − 1 = Re
i ikr(1−cos θ) l e f (θ) . kr
We can use (5.6) to compute the effect of the wire on the power transmitted to a screen determined by x = d, |y| ≤ s/2 and |z| ≤ s/2, where d and s are real parameters, and d is much larger than the side lengths of the wire (figure 1). Because the screen is parallel to the y–z plane, the difference between the time-averaged power transmitted to the screen in the presence and absence of the wire is proportional to 1 s/2 1 s/2 dy Sˆ · ex ≈ dy uˆ , Pˆ := s −s/2 s −s/2 x=d
x=d
where the second relation follows from (5.6). Figure 2 shows the plots of Pˆ as a function of s/a for the cases where m = − = 1, g is given by (4.18) with b = a, d = 100a and k = 2π/a, 4π/a, 8π/a and 12π/a. For a left-incident wave, the transmitted power exceeds its vacuum value provided that the screen, which is placed to the right of the wire, is sufficiently large. The opposite is the case for a right-incident wave that does not get affected by the wire. This is a clear manifestation of the non-reciprocal transmission property This means that if we scale the potential as v → αv for a sufficiently small real number α so that the first Born approximation remains valid, the total reflected and transmitted power for ψ l scale by a factor of α 2 .
4
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For an optical wire described by an optical potential of the form (4.16), f r (θ) ≈ 0 for all θ ∈ r ≈ 0, whereas (−π/2, 3π /2), and f l (0) ≈ 0 by virtue of the reciprocity principle. Therefore, P± l P± > 0. This, in particular, shows that the wire displays non-reciprocal transmission, because l = P r . P+ − l and P l are respectively proportional to the total It is interesting to observe that P− 3π /2+ π/2 reflection and transmission cross section, i.e. π/2 dθ|f l (θ)|2 and −π/2 dθ|f l (θ)|2 . This, in turn,
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We can use (2.2), (2.7), (2.8), (2.11), (2.13), (2.14), (2.17), (5.3) and (5.4) to compute P± . The result is |E0 |2 3π /2 l P− = dθ|f l (θ )|2 , 2μ0 ck π/2 π/2 √ |E0 |2 l l 2 l dθ|f (θ)| − 8π Im[f (0)] , P+ = 2μ0 ck −π /2 3π /2 √ |E0 |2 r r 2 r dθ|f (θ )| − 8π Im[f (0)] , P− = 2μ0 ck π/2 |E0 |2 π/2 r P+ = dθ|f r (θ )|2 . 2μ0 ck −π/2
s/a
0
5
10
15
20
25
15
20
25
DPˆ
10
DPˆ
0
0 s/a
Figure 2. Graphs of Pˆ as a function of s/a for k = 2π/a (thin solid navy curve in the top panel), k = 4π/a (dashed red curve in the top panel), k = 8π/a (dotted black curve in the bottom panel) and k = 12π/a (thick solid purple curve in the bottom panel). (Online version in colour.)
of the system. As seen from figure 2, Pˆ tends to zero as s → 0. This isin conformity with the l ˆ reciprocity principle (theorem 3.2), because lims→0 P = 2ξ (x, 0) = 2Re i/kxf (0) and f l (0) = f r (π ) ≈ 0. The above calculation of the transmitted power to a finite screen reveals its linear dependence on the strength of the optical potential. This is in contrast to the total transmitted power which has a quadratic dependence on the strength of the potential. In view of the fact that we consider a weak optical potential, so that the first Born approximation is reliable, this shows that the set-up involving a finite screen is more desirable for displaying the non-reciprocal transmission property of the system.5
6. Generalization to three dimension The results we have reported in the preceding sections admit a straightforward generalization to three dimensions. Here, we provide a brief summary of the results in this direction. We begin by listing some convenient conventions and notation. In what follows, v is a real or complex scattering potential defined on R3 . We use a Cartesian coordinate system in which the scattering axis corresponds to the z-axis. Following [26], we employ over-arrows to denote two-dimensional vectors obtained by projecting three-dimensional vectors onto the x–y plane. For example, ρ := (x, y) and p := (px , py ), whereas r := (x, y, z) and 2 := ∂x2 + ∂y2 . p := (px , py , pz ). Similarly, we employ the notation: ∂p := (∂px , ∂py ) and ∇
(a) Transfer matrix in three dimension In order to define an appropriate notion of a transfer matrix in three dimensions, we pursue the approach of §2 with the role of x, p and [−k, k] respectively, played by z, p and Dk := {p ∈ R2 | |p| ≤ k}. In particular, the scattering solutions of the Schrödinger equation tend to 1 2 ip·ρ iω(p)z −iω(p)z A , (6.1) d p e ( p ) e + B ( p ) e ± ± (2π )2 Dk 5 The observation of this effect, using an infinite screen, would involve measuring the total transmitted power which is much smaller.
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0
as z → ±∞, and the transfer matrix M(p) is defined by (2.3) with p changed to p. Again, we can show that it satisfies (2.4) for an effective Hamiltonian operator of the form
provided that we replace x with z [26]. Next, we recall that in three dimensions the scattering solutions of the Schrödinger equation admit the asymptotic expression e±ikz +
eikr l/r f (ϑ, ϕ) r
for r → ∞,
where + and − correspond to the left- and right-incident waves, (r, ϑ, ϕ) are the spherical coordinates of r with ϑ denoting the polar angle, and f l/r (ϑ, ϕ) is the scattering amplitude. This turns out to satisfy [26] f l/r (ϑ, ϕ) = −
ik| cos ϑ| l/r T± (k sin ϑ cos ϕ, k sin ϑ sin ϕ), 2π
l/r
where ± := sgn(cos ϑ) and T± are given by (2.9), (2.10), (2.15) and (2.16) with δ(p) replaced l/r
by 2π δ(px )δ(py ). Because the entries of the transfer matrix determine T± , it contains all the information about the scattering features of the potential.
(b) Reciprocity principle in three dimension The argument we use in §3 to establish the reciprocity principle in two dimensions can be easily generalized to three dimensions. Here, we pursue an alternative approach that also has a twodimensional analogue. Let ψ1 and ψ2 be any pair of scattering solutions of the Schrödinger equation (2.1) in three dimensions, and d2 p ψ˜ 1 (−p, z)∂z ψ˜ 2 (p, z) − ψ˜ 2 (p, z)∂z ψ˜ 1 (−p, z) , j(z) := 2 Dk 4π ∞ ∞ −i(xpx +ypy ) ψ(x, y, z) is the Fourier transform of ψ(x, ˜ y, z) over x ˜ p, z) := where ψ( −∞ dx −∞ dy e and y. Clearly, ψ˜ j satisfy the Fourier transformed Schrödinger equation
˜ p, z) + v(i∂p , z)ψ( ˜ p, z), ˜ p, z) = ω(p)2 ψ( − ∂z2 ψ(
(6.2)
with ω(p) := k2 − p2 . With the help of (6.2), we can easily show that ∂z j(z) = 0. Therefore, j(z) is actually z-independent. In particular, j(−∞) = j(+∞).
(6.3)
Next, we use the asymptotic form (6.1) of the scattering solutions to compute j(z) for z → ±∞. This gives d2 p [−iω(p) ± (p)], (6.4) j(±∞) = 2 Dk 2π where ± (p) is defined by (3.8) with p changes to p. Imposing (6.3) and making use of (6.4), we are led to the three-dimensional analogues of (3.11) and (3.12) which are equivalent to f l (0, ϕ) = f r (π , ϕ). This argument establishes the validity of theorem 3.2 (reciprocity principle) in three dimensions.
(c) Unidirectional invisibility in three dimension The content of definitions 2.1 and 3.1 applies for scattering potentials in three dimensions provided that we respectively use p and Dk in place of p and [−k, k]. In particular, reciprocal l ( r ( p) = T− p) for all p ∈ Dk or equivalently f l (ϑ, ϕ) = f r (π − ϑ, ϕ) for transmission means that T+
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1 e−iω(p)zσ 3 v(i∂p , z) K eiω(p)zσ 3 , 2ω(p)
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H(z, p) :=
12
and r (p) ≈ T±
−i ˜ v( ˜ p, −p∓ ) 2ω(p)
(6.5)
−i ˜ v( ˜ p, p± ), 2ω(p)
(6.6)
˜˜ K, ˜˜ x , Ky , Kz ) = 3 d3 r e−iK·r v(r) is the three-dimensional Fourier transform Kz ) := v(K where v( R of v(r), and p± := k ± ω(p). In particular, for left-invisible and right-invisible potentials, we respectively have ˜˜ p, −p± ) ≈ 0 v( (6.7) and
˜˜ p, p± ) ≈ 0. v(
(6.8)
Repeating the construction of §4, we can use (6.8) to construct right-invisible potentials of the form ei Kx − m eimKx 2 v(x, y, z) = χc (z) − 1 + ∇ g(x, y) −m mK i Kx imKx ( K − 2k) e g(x, y) , − (mK − 2k) e + −m where c is a length scale, and m are distinct non-zero integers, K := 2π/c, and g is an arbitrary well-behaved function with sufficiently rapid asymptotic decay rate. For example, let a and b be positive real parameters, g0 be a non-zero real or complex number, and g(x, y) = g0 a−4 b−4 x2 y2 (x − a)2 (y − b)2 χa (x)χb (y). Then, for each choice of and m, v(x, y, z) is a right-invisible potential that vanishes outside the (rectangular) cube defined by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c. It is easy to check that it is neither left-reflectionless nor left-transparent. Therefore, it is a finite-range unidirectionally invisible potential with non-reciprocal transmission.
7. Concluding remarks For a scattering potential in one dimension, the reciprocity in transmission is a consequence of the x-independence of the Wronskian of the left- and right-incident scattering solutions of the Schrödinger equation. In this article, we have introduced a similar conserved quantity involving solutions of the Schrödinger equation in two and three dimensions and used it to prove a reciprocity theorem that applies for arbitrary real and complex scattering potentials in these dimensions. This theorem states that the forward scattering amplitude for left- and right-incident waves coincide. The condition for having reciprocal transmission in higher dimensions is much stronger. It requires the invariance of the scattering amplitude under the parity transformation that flips the orientation of the scattering axis. In two dimensions, it takes the form f l (θ) = f r (π − θ), whereas the reciprocity principle only demands f l (0) = f r (π ). In particular, it is possible to have potentials whose total transmission cross section to the left differs from that to the right. An extreme example is a unidirectionally invisible potential that is not transparent from both the directions. We have given a precise definition for the concept
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l (p) ≈ T±
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all ϑ ∈ [0, π/2) and ϕ ∈ [0, 2π ). This is much stronger a condition than the one imposed by the reciprocity principle. In particular, it is possible to construct left-invisible (right-invisible) scattering potentials with non-trivial transmission from the right (left). We can follow the approach of §4 to construct weak unidirectionally invisible potentials that admit a reliable description using the first Born approximation. Similarly to the case of two dimensions, we find that in three dimensions the first Born approximation yields the following analogues of equations (4.5) and (4.6):
paper. F.L. worked out the material on Reciprocity Principle in two and three dimensions in consultation with A.M. A.M. worked out the material on unidirectional invisibility in two and three dimensions in consultation with F.L. Each author has actively contributed to the improvement of the results initially obtained by the other. A.M. wrote the paper. Both authors gave final approval for publication. Competing interests. We declare we have no competing interests. ˙ Funding. This work was supported by the Scientific and Technological Research Council of Turkey (TÜBITAK) in the framework of the project no: 112T951 and by Turkish Academy of Sciences (TÜBA).
Acknowledgements. We are indebted to Aref Mostafazadeh for his help in preparing figure 1.
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Authors’ contributions. F.L. and A.M. conceived of the basic ideas and conceptual developments reported in this
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of unidirectional invisibility in two and three dimensions and devised a general method of constructing explicit examples of perturbative unidirectionally invisible potentials. These are the multidimensional generalizations of the complex exponential potentials (4.1) in one dimension whose study initiated the overwhelming recent interest in unidirectional invisibility. We have constructed an infinite family of unidirectionally invisible potentials in two dimensions that admit simple optical realizations. These include as a special case a right-invisible potential modelling an optical wire with a rectangular cross section that is neither reflectionless nor transparent from the left. For this model, we have explored the behaviour of the total transmitted and reflected power as well as the power transmitted to a finite screen placed at finite distance from the wire. We have also constructed similar unidirectionally invisible potentials in three dimensions. We expect our results to pave the way for a systematic study of the phenomenon of unidirectional invisibility in realistic multidimensional systems. They should be of particular interest for devising optical and acoustic devices displaying non-reciprocal transmission.
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15. Mostafazadeh A. 2014 Transfer matrices as non-unitary S-matrices, multimode unidirectional invisibility, and perturbative inverse scattering. Phys. Rev. A 89, 012709. (doi:10.1103/ PhysRevA.89.012709) 16. Mostafazadeh A. 2015 Perturbative unidirectional invisibility. Phys. Rev. A 92, 023831. (doi:10.1103/PhysRevA.92.023831) 17. Horsley SAR, Artoni M, La Rocca GC. 2015 Spatial Kramers–Kronig relations and the reflection of waves. Nat. Photon. 9, 436–439. (doi:10.1038/nphoton.2015.106) 18. Longhi S. 2015 Wave reflection in dielectric media obeying spatial Kramers–Kronig relations. Eur. Phys. Lett. 112, 64001. (doi:10.1209/0295-5075/112/64001) 19. Longhi S. 2015 Half-spectral unidirectional invisibility in non-Hermitian periodic optical structures. Opt. Lett. 40, 5694–5697. (doi:10.1364/OL.40.005694) 20. Jalas D et al. 2013 What is and what is not an optical isolator. Nat. Photon. 7, 579–582. (doi:10.1038/nphoton.2013.185) 21. Ahmed Z. 2001 Schrödinger transmission through one-dimensional complex potentials. Phys. Rev. A 64, 042716. (doi:10.1103/PhysRevA.64.042716) 22. Mostafazadeh A. 2009 Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102, 220402. (doi:10.1103/PhysRevLett.102.220402) ˜ 23. Sánchez-Soto LL, Monzóna JJ, Barriuso AG, Carinena JF. 2012 The transfer matrix: a geometrical perspective. Phys. Rep. 513, 191–227. (doi:10.1016/j.physrep.2011.10.002) 24. Fleury R, Sounas D, Haberman MR, Alú A. 2015 Nonreciprocal acoustics. Acoust. Today 11, 14–21. 25. Aplet LJ, Carson JW. 1964 A Faraday effect optical isolator. Appl. Opt. 3, 544–545. (doi:10.1364/ AO.3.000544) 26. Loran F, Mostafazadeh A. 2016 Transfer matrix formulation of scattering theory in two and three dimensions. Phys. Rev. A 93, 042707. (doi:10.1103/PhysRevA.93.042707) 27. Mostafazadeh A. 2014 A dynamical formulation of one-dimensional scattering theory and its applications in optics. Ann. Phys. (N.Y.) 341, 77–85. (doi:10.1016/j.aop.2013.11.008) 28. Boyce WE, DiPrima RC. 2005 Elementary differential equations and boundary value problems. Hoboken, NJ: Wiley. 29. Mostafazadeh A, Mehri-Dehnavi H. 2009 Spectral singularities, biorthonormal systems, and a two-parameter family of complex point interactions. J. Phys. A 42, 125303. (doi:10.1088/1751-8113/42/12/125303) 30. Bethe H. 1940 A. A continuum theory of the compound nucleus. Phys. Rev. 57, 1125. (doi:10.1103/PhysRev.57.1125) 31. Mostafazadeh A. 2014 Unidirectionally invisible potentials as local building blocks of all scattering potentials. Phys. Rev. A 90, 023833. (doi:10.1103/PhysRevA.90.023833)
Research Cite this article: Loran F, Mostafazadeh A. 2016 Unidirectional invisibility and non-reciprocal transmission in two and three dimensions. Proc. R. Soc. A 472: 20160250. http://dx.doi.org/10.1098/rspa.2016.0250 Received: 5 April 2016 Accepted: 10 June 2016
Subject Areas: optics, quantum physics, acoustics Keywords: scattering, reciprocity principle, non-reciprocal transmission, unidirectional invisibility, multidimensional transfer matrix Author for correspondence: Ali Mostafazadeh e-mail: [email protected]
Unidirectional invisibility and non-reciprocal transmission in two and three dimensions Farhang Loran1 and Ali Mostafazadeh2 1 Department of Physics, Isfahan University of Technology,
Isfahan 84156-83111, Iran 2 Departments of Mathematics and Physics, Koç University, 34450 Sarıyer, Istanbul, Turkey AM, 0000-0002-0739-4060 We explore the phenomenon of unidirectional invisibility in two dimensions, examine its optical realizations and discuss its three-dimensional generalization. In particular, we construct an infinite class of unidirectionally invisible optical potentials that describe the scattering of normally incident transverse electric waves by an infinite planar slab with refractive-index modulations along both the normal directions to the electric field. A by-product of this investigation is a demonstration of non-reciprocal transmission in two dimensions. To elucidate this phenomenon, we state and prove a general reciprocity theorem that applies to quantum scattering theory of real and complex potentials in two and three dimensions.
1. Introduction In one dimension, a real scattering potential has identical reflection and transmission coefficients for incoming waves from the left and right, i.e. it obeys reciprocity in reflection and transmission. In contrast, a complex scattering potential can violate reciprocity in reflection [1]. A remarkable consequence of this observation is the existence of unidirectionally reflectionless and invisible potentials [2–19]. These have recently attracted a great deal of attention, mainly because they model certain oneway optical devices with possible applications in optical circuitry. One of the most sought-after examples of a one-way element of an optical circuit is an optical diode [20]. It is generally believed that real and complex potentials are incapable of modelling an optical diode, because they are bound to respect reciprocity in transmission [21–23]. 2016 The Author(s) Published by the Royal Society. All rights reserved.
Consider the Schrödinger equation − ∇ 2 ψ(r) + v(r)ψ(r) = k2 ψ(r),
(2.1)
for a real or complex scattering potential v, where r := (x, y) and k is the wavenumber. Let us identify the x-axis with the scattering axis, and recall that for x → ±∞ the scattering solutions of (2.1) have the form 1 k dp eipy [A± (p) eiω(p)x + B± (p) e−iω(p)x ], (2.2) 2π −k where A± (p) and B± (p) are coefficient functions vanishing for p ∈ / [−k, k] and ω(p) := [26]. By definition, the transfer matrix of v is the 2 × 2 matrix operator M(p) satisfying A− (p) A+ (p) = M(p) . B+ (p) B− (p)
k2 − p2 ,
(2.3)
Similarly to its one-dimensional analogue [27], it can be expressed as a time-ordered exponential [26] ∞ −iH(x, p) dx, (2.4) M(p) := T exp −∞
where x plays the role of time, H(x, p) is the non-Hermitian effective Hamiltonian operator H(x, p) :=
1 e−iω(p)xσ 3 v(x, i∂p )K eiω(p)xσ 3 , 2ω(p)
(2.5)
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2. Transfer matrix and unidirectional invisibility in two dimension
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The same problem arises also in acoustics where designing acoustic waveguides with nonreciprocal transmission is a major area of research [24]. The search for means of achieving non-reciprocal transmission is dominated by the use of nonlinear, time-dependent or magnetic materials [20,24,25]. This work is motivated by the observation that reciprocity in reflection and transmission can both be violated within the confines of linear, stationary, isotropic, non-magnetic material provided that one employs a genuine multidimensional set-up. In particular, it is possible to construct unidirectionally invisible potentials in two and three dimensions that enjoy non-reciprocal transmission. The basic conceptual framework for this work is the multidimensional transfer matrix formulation of scattering theory that we report in reference [26]. We use it to offer a precise and convenient way of describing unidirectional reflection, unidirectional invisibility and transmission reciprocity in dimensions higher than one. The organization of this article is as follows. In §2, we review the formulation of scattering theory presented in reference [26] and use it to give a precise definition for the notions of unidirectional reflectionlessness, unidirectional transparency, unidirectional invisibility and reciprocal transmission in two dimensions. Here, we also derive some useful relations characterizing these notions. In §3, we prove a general reciprocity theorem that applies to real and complex scattering potentials in two dimensions. In §4, we construct an infinite class of unidirectionally invisible potentials in two dimensions that possess non-reciprocal transmission. In §5, we examine an optical realization of a particular example of these potentials and discuss the physical implications of their unidirectional invisibility and non-reciprocal transmission properties. In §6, we generalize the results of the preceding sections to three dimensions, in particular establishing the three-dimensional analogue of the reciprocity theorem of §3 and constructing finite-range unidirectionally invisible potentials with non-reciprocal transmission in three dimensions. Finally, in §7, we summarize our findings and present our concluding remarks.
where (r, θ) are the polar coordinates of r, and f l (θ ) is the scattering amplitude for the left-incident waves. In order to express the latter in terms of M(p), we introduce l T− (p) := B− (p)
and
l T+ (p) := A+ (p) − A− (p),
(2.7)
B+ (p) = 0,
(2.8)
and note that for a left-incident wave, A− (p) = 2π δ(p)
and
where δ stands for the Dirac delta-function. In reference [26], we show that (2.3), (2.7) and (2.8) imply l (p) = −2π M−1 T− 22 M21 δ(p), l T+ (p) = 2π (M11
(2.9)
− 1 − M12 M−1 22 M21 )δ(p)
l f l (θ ) = −(2π )−1/2 ik| cos θ|T± (k sin θ),
and
(2.10) (2.11)
where Mij are the entries of M(p) and the ± in (2.11) stands for sgn(cos θ). Similarly, we can introduce the scattering amplitude f r (θ) for a right-incident wave by expressing the asymptotic form of the corresponding scattering solution as i ikr r r −ikx e f (θ) as r → ∞. ψ (r) = e + (2.12) kr To relate f r (θ ) to M(p), we define r (p) := B− (p) − B+ (p) T−
and
r T+ (p) := A+ (p),
(2.13)
and set A− (p) = 0
and
B+ (p) = 2π δ(p).
(2.14)
These, together with (2.3), give
and
r T− (p) = −2π (1 − M−1 22 )δ(p),
(2.15)
r T+ (p) = 2π M12 M−1 22 δ(p)
(2.16)
r f r (θ ) = −(2π )−1/2 ik| cos θ|T± (k sin θ),
(2.17)
where again ± := sgn(cos θ ). l/r It is important to note that Mij and T± (p) depend on the wavenumber k. Therefore, any l/r
quantity or concept whose definition involves Mij or T± (p) is k-dependent. The particular form of the effective Hamiltonian operator (2.5) and consequently those of K and v(x, i∂p ) are dictated by the formalism developed in reference [26]. They are necessary for establishing equation (2.4), which is central to the developments reported in reference [26] and this article.
1
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˜ Ky ) := and v(x, ˜ Ky ) denotes the Fourier transform of v(x, y) with respect to y, i.e. v(x, ∞ −iKy y v(x, y).1 dy e −∞ The transfer matrix M(p) contains all the scattering information about v. To see this, we recall that for a left-incident plane wave with a momentum k pointing along the positive x-axis, the scattering solutions ψ l of (2.1) have the asymptotic form i ikr l l ikx e f (θ) as r → ∞, ψ (r) = e + (2.6) kr
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σ i are the Pauli matrices, K := σ 3 + iσ 2 , v(x, i∂p ) is the integral operator acting on test functions φ : [−k, k] → C according to 1 k dqv(x, ˜ p − q)φ(q), v(x, i∂p )φ(p) := 2π −k
The following is a direct consequence of (2.11) and (2.17). Theorem 2.2. Let v be as in definition 2.1. Then, v is l (p) = 0 (respectively, T r (p) = 0) for all — left- (respectively right-) reflectionless if and only if T− + p ∈ [−k, k] and l (p) = 0 (respectively T r (p) = 0) for all p ∈ [−k, k]. — left- (respectively right-) transparent if T+ −
In view of (2.9), (2.10), (2.15) and (2.16), we also have the following useful result. Theorem 2.3. Let v be as in definition 2.1. Then, v is — left-reflectionless, if and only if M21 δ(p) = 0 for all p ∈ [−k, k]. — left-invisible, if and only if M21 δ(p) = (M11 − 1)δ(p) = 0 for all p ∈ [−k, k]. — right-invisible, if and only if M12 δ(p) = (M22 − 1)δ(p) = 0 for all p ∈ [−k, k].
3. Reciprocity principle in two dimension The phrase ‘reciprocity principle’ is usually taken to be synonymous to the ‘Lorentz reciprocity theorem’ for electromagnetic fields which follows from Maxwell’s equations [20]. Here, we formulate a reciprocity principle that applies directly to the quantum scattering theory as defined by the Schrödinger equation (2.1). First, we define what we mean by reciprocity in transmission in two dimensions. Definition 3.1. A real or complex scattering potential v is said to have reciprocal transmission l (p) = T r (p). According to (2.10) and (2.15), this is for a wavenumber k, if for all p ∈ [−k, k], T+ − equivalent to −1 (3.1) (M11 − M−1 22 − M12 M22 M21 )δ(p) = 0. We can also use (2.11) and (2.17) to state it in the form f l (θ ) = f r (π − θ )
for all θ ∈ − π2 , π2 .
(3.2)
The content of definitions (2.1) and (3.1) reduce to their well-known one-dimensional analogue provided that we replace δ(p) with 1 and identify Mij with the entries of the transfer matrix in one dimension [14]. In this case, condition (3.1) for reciprocal transmission reduces to M11 − −1 M−1 22 − M12 M22 M21 = 0 which is equivalent to det M = 1. But, in one dimension, one can use Abel’s theorem for linear homogeneous ordinary differential equations [28] to show that ‘det M = 1’ holds for every real or complex scattering potential [29]. This proves that non-reciprocal transmission is forbidden in one dimension. It also suggests that in order to establish (3.1) or its violation in two (and higher) dimensions, we should seek for an analogue of Abel’s theorem for the Schrödinger equation (2.1), which is a partial differential equation.
...................................................
— left- (respectively right-) reflectionless if f l (θ) = 0 for all θ ∈ (π/2, 3π /2) (respectively, f r (θ ) = 0 for all θ ∈ (−π/2, π/2)), — left- (respectively right-) transparent if f l (θ) = 0 for all θ ∈ (−π/2, π/2) (respectively, f r (θ ) = 0 for all θ ∈ (π/2, 3π /2)), — left- (respectively right-) invisible if it is both left- (respectively right-) reflectionless and transparent, — unidirectionally reflectionless (respectively, transparent or invisible) if it is either left- or right-reflectionless (respectively, transparent or invisible), but not both.
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Definition 2.1. Let v : R2 → C be a scattering potential and k be a wavenumber. Choose a Cartesian coordinate system in which the scattering axis coincides with the x-axis. Then, v is said to be
Now, let a and b be positive real numbers, V be the rectangular region enclosed between the lines x = 0, x = a and y = ±b, and ∞ dy ψ1 (r)∂x ψ2 (r) − ψ2 (r)∂x ψ1 (r) . (3.5) j(x) := −∞
Suppose that ∂y ψj (x, y) → 0 for y → ±∞, which is consistent with (2.6) and (2.12). Then, applying the divergence theorem to J(r) and V, using (3.4), and taking the b → ∞ limit, we find that j(a) = j(0). Because a is arbitrary, this means that j(x) does not depend on x. In particular, we have3 j(−∞) = j(∞).
(3.6)
Because scattering solutions of (2.1) admit an asymptotic expression of the form (2.2), we can calculate j(±∞) whenever ψ1 and ψ2 are scattering solutions. Let Aj± and Bj± , respectively, denote the coefficient functions A± and B± appearing in the asymptotic expression (2.2) for ψj . Then, substituting this expression in (3.5), we obtain −i k dpω(p) ± (p) for x → ±∞ (3.7) j(x) = π −k and ± (p) := A1± (−p)B2± (p) − B1± (−p)A2± (p).
(3.8)
The fact that the right-hand side of (3.7) does not involve x provides an independent verification of (3.6). Next, we use the definition of the transfer matrix, namely (2.3) to express the Aj+ and Bj+ appearing in (3.8) in terms of Aj− and Bj− . Inserting the resulting formula in (3.7), we can express (3.6) in the form k dpω(p)[C1+ (−p)T ΩC2+ (p) − C1− (−p)T ΩC2− (p)] = 0, (3.9) −k
where the superscript ‘T’ stands for the transpose, Aj− (p) (3.10) Cj− (p) := and Cj+ (p) := M(p)Cj− (p), Bj− (p) 0 1 and Ω := iσ 2 = −1 0 is the standard 2 × 2 symplectic matrix. Because the choice of Cj− (p) is arbitrary, (3.9) is equivalent to ←−−−−− −−→ (3.11) M(−p)T Ω M(p) = Ω. Here, we use arrows to stress that M(−p)T acts on the test functions appearing to its left, whereas M(p) acts on those appearing to its right. For a real potential, we may choose ψ2 = ψ1∗ . If we do so, (3.3) coincides (up to an unimportant constant coefficient) with the standard probability current density corresponding to ψ1 . For a complex potential, this choice is forbidden, because ψ1∗ is no longer a solution of the Schrödinger equation (2.1). In this case, setting ψ2 = ψ1∗ gives a probability current density that is not divergence-free [30]. In contrast, (3.3) is a divergence-free (conserved) current density for both real and complex potentials provided that ψ1 and ψ2 solve (2.1).
2
Here, we assume that j(x) has finite asymptotic values for x → ±∞. For our purposes, we need to consider only j(x) for the scattering solutions ψ l/r , and for these, limx→±∞ j(x) exist.
3
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It is easy to check that whenever ψ1 and ψ2 are solutions of the Schrödinger equation (2.1), J is divergence-free2 ∇ · J(r) = 0. (3.4)
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In one dimension, the Schrödinger equation reads: −ψ (x) + v(x)ψ(x) = k2 ψ(x), and Abel’s theorem states that the Wronskian, ψ1 (x)ψ2 (x) − ψ2 (x)ψ1 (x), of any pair of solutions of this equation, ψ1 and ψ2 , is x-independent. A higher-dimensional generalization of the Wronskian is the vector field (3.3) J(r) := ψ1 (r)∇ψ2 (r) − ψ2 (r)∇ψ1 (r).
B1+ (p) = 0,
B2+ (p) = 2π δ(p),
A2− (p) = 0.
Substituting these relations in (3.8) and making use of (2.8), (2.14) and (3.7), we can write (3.6) as l r (0) = T− (0), T+
or
f l (0) = f r (π ).
(3.12)
In view of (2.11) and (2.17), this proves the following reciprocity theorem.
Theorem 3.2 (Reciprocity Principle). For every real or complex scattering potential, the forward scattering amplitude for the left- and right-incident waves coincide. Note that this theorem does not prohibit non-reciprocal transmission in the sense of definition 3.1. In particular, as we shown in §4, there are scattering potentials with different total transmission cross section from the left and right.
4. Unidirectionally invisible potentials with non-reciprocal transmission The recent interest in unidirectional invisibility is initiated by the study of the locally periodic optical potentials of the form [2–10] ⎧ ⎨z eiKx for x ∈ [0, a], (4.1) v(x) = z eiKx χa (x) = ⎩0 otherwise, where z is a coupling constant, K := 2π/a, a is a positive real parameter and χa is the characteristic function for the interval [0, a], i.e. ⎧ ⎨1 for x ∈ [0, a], χa (x) := ⎩0 otherwise. This potential is unidirectionally invisible for k = π/a provided that |z| is so small that the scattering properties of v can be determined using the first Born approximation, i.e. it displays perturbative unidirectional invisibility [15]. In optical applications, where v is related to the relative permittivity εˆ of the material according to v(x) = k2 [1 − εˆ (x)], this condition holds whenever |ˆε(x) − 1| 1. In one dimension, one can use the dynamical formulation of scattering theory developed in reference [27] to construct scattering potentials that enjoy exact unidirectional invisibility [27,31]. These have a more complicated structure than (4.1). Reference [15] provides a complete characterization of finite-range potentials that similarly to (4.1) support perturbative unidirectional invisibility. In the remainder of this section, we construct a class of perturbative unidirectionally invisible potentials in two dimensions that generalize (4.1). Consider a planar slab of optical material with relative permittivity εˆ that is located between the planes x = 0 and x = a in three-dimensional Euclidean space. Suppose that the slab has translational symmetry along the z-directions, so that εˆ depends only on x and y. Then, the interaction of this slab with the time-harmonic normally incident z-polarized transverse electric waves of the form (4.2) E(x, y, z) = E0 e−ikct ψ(x, y)ez is described by the Schrödinger equation (2.1) with the potential v(x, y) = k2 [1 − εˆ (x, y)]χa (x).
(4.3)
In equation (4.2), E0 is a non-zero complex parameter, c is the speed of light in vacuum and ez is the unit vector pointing along the positive z-axis.
...................................................
A1− (p) = 2π δ(p),
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In order to elucidate the physical meaning of equation (3.6), which is equivalent to (3.9) and (3.11), we identify ψ1 and ψ2 with scattering solutions ψ l and ψ r associated with the incident plane waves from the left and right, respectively. This corresponds to setting
l T± (p) ≈
and r (p) ≈ T±
−i ˜ v(−p ˜ ∓ , p) 2ω(p)
(4.5)
−i ˜ v(p ˜ ± , p), 2ω(p)
(4.6)
˜˜ x , Ky ) is the two-dimensional Fourier transform of v(x, y), i.e. where v(K ∞ ∞ ˜˜ x , Ky ) := dx dy e−i(Kx x+Ky y) v(x, y), v(K −∞
−∞
(4.7)
and p± := k ± ω(p) = k ± k2 − p2 . It is easy to see that equations (4.5) and (4.6) are consistent with the statement of the reciprocity principle (theorem 3.2); clearly, for p = 0, we have p− = 0, and (3.12) holds. In view of definition 2.1, equations (4.5) and (4.6) imply that v is unidirectionally invisible from the right if and only if the following conditions hold. ˜˜ ± , p) = 0 for all p ∈ [−k, k] v(p
and
˜˜ ˜˜ |v(−p − , p)| + |v(−p + , p)| = 0 for some p ∈ [−k, k].
(4.8)
In order to obtain the explicit examples of potentials of the form (4.3) that satisfy these conditions, we expand them in their Fourier series in [0, a]. This gives v(x, y) = χa (x) where cn (y) := (1/a)
∞
cn (y) einKx ,
(4.9)
n=−∞
a
−inKx v(x, y) 0 dx e
and K :=
In view of (4.7) and (4.9), ˜˜ x , Ky ) = v(K
∞
n=−∞
2π . a
eia(nK−Kx ) − 1 c˜ n (Ky ). i(nK − Kx )
(4.10)
(4.11)
Substituting this relation in (4.8), we find two complex equations for the coefficient functions c˜ n (p). These characterize perturbative right-invisible potentials v(x, y) that vanish for x ∈ / [0, a]. To obtain concrete examples of such potentials, we take a pair of distinct non-zero integers, and m, and set (4.12) cn (y) = 0 for n = 0, , m. We then solve (4.8) to express c˜ 0 , c˜ and c˜ m in terms of an arbitrary function that we denote by g˜ . This gives c˜ 0 (p) = p2 g˜ (p),
and
(4.13)
c˜ (p) =
m[ K( K − 2k) + p2 ]˜g(p) −m
(4.14)
c˜ m (p) =
[mK(mK − 2k) + p2 ]˜g(p) . m−
(4.15)
...................................................
−∞
where I is the 2 × 2 identity matrix, and ‘≈’ stands for the first Born approximation. Substituting (2.5) in this relation to determine Mij and using these in (2.9), (2.10), (2.15) and (2.16) give
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In what follows, we consider situations where |ˆε(x, y) − 1| is so small that the first Born approximation provides a reliable description of the behaviour of the system. In this case, we can compute the transfer matrix (2.4), using the formula ∞ dx H(x, p), (4.4) M(p) ≈ I − i
z
8
x
s a b
Figure 1. Schematic view of an optically active wire of rectangular cross section with a screen placed to its right at x = d. (Online version in colour.)
Evaluating the inverse Fourier transform of these relations and using the result together with (4.12) in (4.9), we find ei Kx − m eimKx mK i Kx imKx ( K − 2k) e g(y) , − (mK − 2k) e v(x, y) = χa (x) − 1 + g (y) + −m −m (4.16) where g is the inverse Fourier transform of g˜ . Equation (4.16) gives a right-invisible potential for each choice of k, , m and g, provided that g and g be such that (4.16) defines a well-behaved scattering potential and that the first Born approximation is justified. Typical examples are g(y) = g0 e−y
2
/2b2
(4.17)
and g(y) = g0 b−4 y2 (y − b)2 χb (y),
(4.18)
where g0 is a possibly complex non-zero coupling constant, b is a positive real parameter, and |g0 | is sufficiently small. Note that (4.18) corresponds to a finite-range right-invisible potential whose support is the rectangular region: [0, a] × [0, b]. It describes an optically active wire of rectangular cross section that is aligned along the z-axis, as shown in figure 1. l (p). In order to make sure that the potentials (4.16) are not invisible from the left, we examine T± First, we substitute (4.13)–(4.15) in (4.11) and make use of (4.10) and (4.12) to derive ˜˜ x , Ky ) = v(K
mK2 (1 − e−iaKx )[K2y + (Kx − 2k)Kx ]˜g(Ky ) iKx (Kx − K)(Kx − mK)
.
This, together with (4.5), implies l T± (p) ≈
2 mK2 k(1 − eiap∓ ) g˜ (p). ω(p)(p∓ + K)(p∓ + mK)
(4.19)
Equation (4.19) provides an explicit demonstration of non-reciprocity in transmission, because for l (p) = 0, whereas T r (p) ≈ 0 for all p ∈ [−k, k]. generic choices of g and p, we have T+ −
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s
d
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y
Using (4.19) in (2.11), we also find
(4.20)
According to this relation and the fact that f r (θ ) ≈ 0, the potential (4.16) is unidirectionally invisible from the right, but it is neither left-reflectionless nor left-transparent.
5. Optical manifestation of unidirectional invisibility and non-reciprocal transmission We can identify the unidirectionally invisible potentials of the form (4.16) with optical potentials describing the interaction of the linearly polarized electromagnetic waves (4.2) with an optically active wire of rectangular cross section as depicted in figure 1. In this section, we explore the implications of the non-reciprocal transmission property of these potentials for the transmitted power associated with the left- and right-incident waves. This requires the computation of the Poynting vector for the corresponding scattered waves. First, we recall that for a charge-free, non-magnetic, isotropic medium, the time-averaged energy density and Poynting vector are respectively given by
u = 14 Re(E · D∗ + B · H∗ )
and
S := 12 Re(E × H∗ ),
(5.1)
where E and H are the electric and magnetic fields, D = ε0 εˆ E, B = μ0 H, and ε0 and μ0 are the permittivity and permeability of the vacuum, respectively. For time-harmonic waves, Maxwell’s √ equations imply that H = (1/ik) ε0 /μ0 ∇ × E. Substituting this relation in (5.1) and making use of (4.2), we have
u =
ε0 |E0 |2 Re[ˆε(r)]|ψ(r)|2 + k−2 |∇ψ(r)|2 4
(5.2)
|E0 |2 Im ψ(r)∗ ∇ψ(r) , 2μ0 ck
(5.3)
and
S =
where ‘Re’ and ‘Im’, respectively, denote the real and imaginary part of their argument. Now, consider the time-averaged energy density and Poynting vector for the scattered waves ψ l/r , which we denote by ul/r and Sl/r , respectively. We can use the latter to compute the reflected and transmitted power to the left (x = −∞) and right (x = ∞). It is not difficult to see that the contribution of the scattering potential to these quantities, i.e. the difference between their value in the presence and absence of the potential, is proportional to l/r
P± := ±
∞ −∞
dy ex · ( Sl/r − S∅ )
,
(5.4)
x=±∞
where ex is the unit vector pointing along the positive x-axis, and the subscript ‘∅’ means that the l , corresponding quantity is computed in the absence of the potential, i.e. for v = 0. Note that P− l r r l P+ , P− and P+ , respectively, correspond to the reflected power for ψ , transmitted power for ψ l , transmitted power for ψ r and reflected power for ψ r .
...................................................
√ − 2i mK2 k[1 − eiak(1−cos θ) ]˜g(k sin θ) . π [k(1 − cos θ ) + K][k(1 − cos θ) + mK]
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f l (θ ) ≈ √
9
l/r
implies that the total reflected and transmitted power for ψ l are quadratic functions of the strength of the potential.4 Next, we examine a more realistic situation where we intend to determine the power transmitted to a finite screen placed at a large but finite distance from the wire. To this end, we introduce the dimensionless parameters uˆ :=
u − u∅
u∅
and Sˆ :=
S − S∅ . | S∅ |
(5.5)
In the light of the right invisibility of the wire, these vanish for a right-incident wave. For a scattering solution corresponding to a left-incident wave, they have the following asymptotic expression (in the limit r → ∞): r , (5.6) uˆ ≈ (1 + cos θ)ξ (r, θ ) and Sˆ ≈ ξ (r, θ) ex + r where
−ikx
ξ (r, θ) := Re[e
l
ψ ] − 1 = Re
i ikr(1−cos θ) l e f (θ) . kr
We can use (5.6) to compute the effect of the wire on the power transmitted to a screen determined by x = d, |y| ≤ s/2 and |z| ≤ s/2, where d and s are real parameters, and d is much larger than the side lengths of the wire (figure 1). Because the screen is parallel to the y–z plane, the difference between the time-averaged power transmitted to the screen in the presence and absence of the wire is proportional to 1 s/2 1 s/2 dy Sˆ · ex ≈ dy uˆ , Pˆ := s −s/2 s −s/2 x=d
x=d
where the second relation follows from (5.6). Figure 2 shows the plots of Pˆ as a function of s/a for the cases where m = − = 1, g is given by (4.18) with b = a, d = 100a and k = 2π/a, 4π/a, 8π/a and 12π/a. For a left-incident wave, the transmitted power exceeds its vacuum value provided that the screen, which is placed to the right of the wire, is sufficiently large. The opposite is the case for a right-incident wave that does not get affected by the wire. This is a clear manifestation of the non-reciprocal transmission property This means that if we scale the potential as v → αv for a sufficiently small real number α so that the first Born approximation remains valid, the total reflected and transmitted power for ψ l scale by a factor of α 2 .
4
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For an optical wire described by an optical potential of the form (4.16), f r (θ) ≈ 0 for all θ ∈ r ≈ 0, whereas (−π/2, 3π /2), and f l (0) ≈ 0 by virtue of the reciprocity principle. Therefore, P± l P± > 0. This, in particular, shows that the wire displays non-reciprocal transmission, because l = P r . P+ − l and P l are respectively proportional to the total It is interesting to observe that P− 3π /2+ π/2 reflection and transmission cross section, i.e. π/2 dθ|f l (θ)|2 and −π/2 dθ|f l (θ)|2 . This, in turn,
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We can use (2.2), (2.7), (2.8), (2.11), (2.13), (2.14), (2.17), (5.3) and (5.4) to compute P± . The result is |E0 |2 3π /2 l P− = dθ|f l (θ )|2 , 2μ0 ck π/2 π/2 √ |E0 |2 l l 2 l dθ|f (θ)| − 8π Im[f (0)] , P+ = 2μ0 ck −π /2 3π /2 √ |E0 |2 r r 2 r dθ|f (θ )| − 8π Im[f (0)] , P− = 2μ0 ck π/2 |E0 |2 π/2 r P+ = dθ|f r (θ )|2 . 2μ0 ck −π/2
s/a
0
5
10
15
20
25
15
20
25
DPˆ
10
DPˆ
0
0 s/a
Figure 2. Graphs of Pˆ as a function of s/a for k = 2π/a (thin solid navy curve in the top panel), k = 4π/a (dashed red curve in the top panel), k = 8π/a (dotted black curve in the bottom panel) and k = 12π/a (thick solid purple curve in the bottom panel). (Online version in colour.)
of the system. As seen from figure 2, Pˆ tends to zero as s → 0. This isin conformity with the l ˆ reciprocity principle (theorem 3.2), because lims→0 P = 2ξ (x, 0) = 2Re i/kxf (0) and f l (0) = f r (π ) ≈ 0. The above calculation of the transmitted power to a finite screen reveals its linear dependence on the strength of the optical potential. This is in contrast to the total transmitted power which has a quadratic dependence on the strength of the potential. In view of the fact that we consider a weak optical potential, so that the first Born approximation is reliable, this shows that the set-up involving a finite screen is more desirable for displaying the non-reciprocal transmission property of the system.5
6. Generalization to three dimension The results we have reported in the preceding sections admit a straightforward generalization to three dimensions. Here, we provide a brief summary of the results in this direction. We begin by listing some convenient conventions and notation. In what follows, v is a real or complex scattering potential defined on R3 . We use a Cartesian coordinate system in which the scattering axis corresponds to the z-axis. Following [26], we employ over-arrows to denote two-dimensional vectors obtained by projecting three-dimensional vectors onto the x–y plane. For example, ρ := (x, y) and p := (px , py ), whereas r := (x, y, z) and 2 := ∂x2 + ∂y2 . p := (px , py , pz ). Similarly, we employ the notation: ∂p := (∂px , ∂py ) and ∇
(a) Transfer matrix in three dimension In order to define an appropriate notion of a transfer matrix in three dimensions, we pursue the approach of §2 with the role of x, p and [−k, k] respectively, played by z, p and Dk := {p ∈ R2 | |p| ≤ k}. In particular, the scattering solutions of the Schrödinger equation tend to 1 2 ip·ρ iω(p)z −iω(p)z A , (6.1) d p e ( p ) e + B ( p ) e ± ± (2π )2 Dk 5 The observation of this effect, using an infinite screen, would involve measuring the total transmitted power which is much smaller.
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0
as z → ±∞, and the transfer matrix M(p) is defined by (2.3) with p changed to p. Again, we can show that it satisfies (2.4) for an effective Hamiltonian operator of the form
provided that we replace x with z [26]. Next, we recall that in three dimensions the scattering solutions of the Schrödinger equation admit the asymptotic expression e±ikz +
eikr l/r f (ϑ, ϕ) r
for r → ∞,
where + and − correspond to the left- and right-incident waves, (r, ϑ, ϕ) are the spherical coordinates of r with ϑ denoting the polar angle, and f l/r (ϑ, ϕ) is the scattering amplitude. This turns out to satisfy [26] f l/r (ϑ, ϕ) = −
ik| cos ϑ| l/r T± (k sin ϑ cos ϕ, k sin ϑ sin ϕ), 2π
l/r
where ± := sgn(cos ϑ) and T± are given by (2.9), (2.10), (2.15) and (2.16) with δ(p) replaced l/r
by 2π δ(px )δ(py ). Because the entries of the transfer matrix determine T± , it contains all the information about the scattering features of the potential.
(b) Reciprocity principle in three dimension The argument we use in §3 to establish the reciprocity principle in two dimensions can be easily generalized to three dimensions. Here, we pursue an alternative approach that also has a twodimensional analogue. Let ψ1 and ψ2 be any pair of scattering solutions of the Schrödinger equation (2.1) in three dimensions, and d2 p ψ˜ 1 (−p, z)∂z ψ˜ 2 (p, z) − ψ˜ 2 (p, z)∂z ψ˜ 1 (−p, z) , j(z) := 2 Dk 4π ∞ ∞ −i(xpx +ypy ) ψ(x, y, z) is the Fourier transform of ψ(x, ˜ y, z) over x ˜ p, z) := where ψ( −∞ dx −∞ dy e and y. Clearly, ψ˜ j satisfy the Fourier transformed Schrödinger equation
˜ p, z) + v(i∂p , z)ψ( ˜ p, z), ˜ p, z) = ω(p)2 ψ( − ∂z2 ψ(
(6.2)
with ω(p) := k2 − p2 . With the help of (6.2), we can easily show that ∂z j(z) = 0. Therefore, j(z) is actually z-independent. In particular, j(−∞) = j(+∞).
(6.3)
Next, we use the asymptotic form (6.1) of the scattering solutions to compute j(z) for z → ±∞. This gives d2 p [−iω(p) ± (p)], (6.4) j(±∞) = 2 Dk 2π where ± (p) is defined by (3.8) with p changes to p. Imposing (6.3) and making use of (6.4), we are led to the three-dimensional analogues of (3.11) and (3.12) which are equivalent to f l (0, ϕ) = f r (π , ϕ). This argument establishes the validity of theorem 3.2 (reciprocity principle) in three dimensions.
(c) Unidirectional invisibility in three dimension The content of definitions 2.1 and 3.1 applies for scattering potentials in three dimensions provided that we respectively use p and Dk in place of p and [−k, k]. In particular, reciprocal l ( r ( p) = T− p) for all p ∈ Dk or equivalently f l (ϑ, ϕ) = f r (π − ϑ, ϕ) for transmission means that T+
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1 e−iω(p)zσ 3 v(i∂p , z) K eiω(p)zσ 3 , 2ω(p)
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H(z, p) :=
12
and r (p) ≈ T±
−i ˜ v( ˜ p, −p∓ ) 2ω(p)
(6.5)
−i ˜ v( ˜ p, p± ), 2ω(p)
(6.6)
˜˜ K, ˜˜ x , Ky , Kz ) = 3 d3 r e−iK·r v(r) is the three-dimensional Fourier transform Kz ) := v(K where v( R of v(r), and p± := k ± ω(p). In particular, for left-invisible and right-invisible potentials, we respectively have ˜˜ p, −p± ) ≈ 0 v( (6.7) and
˜˜ p, p± ) ≈ 0. v(
(6.8)
Repeating the construction of §4, we can use (6.8) to construct right-invisible potentials of the form ei Kx − m eimKx 2 v(x, y, z) = χc (z) − 1 + ∇ g(x, y) −m mK i Kx imKx ( K − 2k) e g(x, y) , − (mK − 2k) e + −m where c is a length scale, and m are distinct non-zero integers, K := 2π/c, and g is an arbitrary well-behaved function with sufficiently rapid asymptotic decay rate. For example, let a and b be positive real parameters, g0 be a non-zero real or complex number, and g(x, y) = g0 a−4 b−4 x2 y2 (x − a)2 (y − b)2 χa (x)χb (y). Then, for each choice of and m, v(x, y, z) is a right-invisible potential that vanishes outside the (rectangular) cube defined by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c. It is easy to check that it is neither left-reflectionless nor left-transparent. Therefore, it is a finite-range unidirectionally invisible potential with non-reciprocal transmission.
7. Concluding remarks For a scattering potential in one dimension, the reciprocity in transmission is a consequence of the x-independence of the Wronskian of the left- and right-incident scattering solutions of the Schrödinger equation. In this article, we have introduced a similar conserved quantity involving solutions of the Schrödinger equation in two and three dimensions and used it to prove a reciprocity theorem that applies for arbitrary real and complex scattering potentials in these dimensions. This theorem states that the forward scattering amplitude for left- and right-incident waves coincide. The condition for having reciprocal transmission in higher dimensions is much stronger. It requires the invariance of the scattering amplitude under the parity transformation that flips the orientation of the scattering axis. In two dimensions, it takes the form f l (θ) = f r (π − θ), whereas the reciprocity principle only demands f l (0) = f r (π ). In particular, it is possible to have potentials whose total transmission cross section to the left differs from that to the right. An extreme example is a unidirectionally invisible potential that is not transparent from both the directions. We have given a precise definition for the concept
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l (p) ≈ T±
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all ϑ ∈ [0, π/2) and ϕ ∈ [0, 2π ). This is much stronger a condition than the one imposed by the reciprocity principle. In particular, it is possible to construct left-invisible (right-invisible) scattering potentials with non-trivial transmission from the right (left). We can follow the approach of §4 to construct weak unidirectionally invisible potentials that admit a reliable description using the first Born approximation. Similarly to the case of two dimensions, we find that in three dimensions the first Born approximation yields the following analogues of equations (4.5) and (4.6):
paper. F.L. worked out the material on Reciprocity Principle in two and three dimensions in consultation with A.M. A.M. worked out the material on unidirectional invisibility in two and three dimensions in consultation with F.L. Each author has actively contributed to the improvement of the results initially obtained by the other. A.M. wrote the paper. Both authors gave final approval for publication. Competing interests. We declare we have no competing interests. ˙ Funding. This work was supported by the Scientific and Technological Research Council of Turkey (TÜBITAK) in the framework of the project no: 112T951 and by Turkish Academy of Sciences (TÜBA).
Acknowledgements. We are indebted to Aref Mostafazadeh for his help in preparing figure 1.
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Authors’ contributions. F.L. and A.M. conceived of the basic ideas and conceptual developments reported in this
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of unidirectional invisibility in two and three dimensions and devised a general method of constructing explicit examples of perturbative unidirectionally invisible potentials. These are the multidimensional generalizations of the complex exponential potentials (4.1) in one dimension whose study initiated the overwhelming recent interest in unidirectional invisibility. We have constructed an infinite family of unidirectionally invisible potentials in two dimensions that admit simple optical realizations. These include as a special case a right-invisible potential modelling an optical wire with a rectangular cross section that is neither reflectionless nor transparent from the left. For this model, we have explored the behaviour of the total transmitted and reflected power as well as the power transmitted to a finite screen placed at finite distance from the wire. We have also constructed similar unidirectionally invisible potentials in three dimensions. We expect our results to pave the way for a systematic study of the phenomenon of unidirectional invisibility in realistic multidimensional systems. They should be of particular interest for devising optical and acoustic devices displaying non-reciprocal transmission.
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