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Superoscillatory field features with evanescent waves
Optics Communications 356 (2015) 482–487
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Superoscillatory field features with evanescent waves Ioannis Chremmos a,n, George Fikioris b a b
Max Planck Institute for the Science of Light, D-91058 Erlangen, Germany School of Electrical and Computer Engineering, National Technical University, GR 157-73 Athens, Greece
art ic l e i nf o
a b s t r a c t
Article history: Received 28 May 2015 Received in revised form 8 August 2015 Accepted 11 August 2015
We show how to obtain optical fields possessing superoscillatory features by superposing the evanescent tails of waves undergoing total internal reflection at a plane dielectric interface. In doing so, we essentially extend the definition of superoscillations to functions expressed as a continuum of slowly decaying exponentials, while not necessarily being bandlimited in the standard (Fourier) sense. We obtain such functions by complexifying the argument of standard bandlimited superoscillatory functions with a strictly positive spectrum. Combined with our recent method for superoscillations with arbitrary polynomial shape, the present approach offers flexibility for locally shaping the evanescent field near dielectric interfaces for applications such as particle or atom trapping. & 2015 Elsevier B.V. All rights reserved.
Keywords: Superoscillations Superresolution Evanescent waves Diffraction Wave optics Near field
The phenomenon of superoscillations implies the ability of bandlimited signals to oscillate with local frequencies that are arbitrarily larger than their maximum frequency component. This counter-intuitive property has long intrigued physicists and engineers because it enables, at least theoretically, antenna or imaging systems to produce or resolve wave features much finer than their bandwidth suggests. The earliest reports of the phenomenon can therefore be found in classical works on superdirectivity [1] and subdiffraction imaging [2], as well as in information and signal theory [3,4]. The underlying idea of all these works was that extremely fine (or fast) oscillations can be obtained from appropriately weighted superpositions of slowly-varying functions whose rate of variation is limited by some fundamental bandwidth cutoff. The latter is introduced either by the operating wavelength in radiation or imaging settings or by the response time of circuits and systems in signal processing settings. Interestingly, the term superoscillations itself was coined independently in quantum mechanics in the context of weak measurements [5] and was thereafter established when referring to such faster-than-Fourier functions [6]. The interest in superoscillations and particular in superoscillatory optical imaging has recently revived with a number of promising experimental demonstrations [7,8]. Superoscillations have so far been obtained as superpositions of propagating waves in homogeneous media. From a theoretical n
Corresponding author. E-mail addresses: [email protected] (I. Chremmos), gfi[email protected] (G. Fikioris). http://dx.doi.org/10.1016/j.optcom.2015.08.029 0030-4018/& 2015 Elsevier B.V. All rights reserved.
viewpoint, this is due to their original definition within a space of bandlimited functions, namely functions whose Fourier transform has an appropriate compact support. In this context, a superoscillatory function f(x) is always expressible (through its inverse Fourier transform) as a superposition of propagating waves eikx with k being limited within a given bandwidth. From the applications viewpoint too, creating superoscillations with propagating waves has been the major focus of research with the aim of achieving optical superresolution without evanescent waves or, equivalently, subwavelength focusing of light in the far field [9,10]. In this communication we consider a different question: Is it possible to produce superoscillatory field features using as basis functions nonpropagating evanescent waves of the form e kx? Before we give an answer, the concept of superoscillations within the context of evanescent waves should first be clarified (or better, introduced). In standard superoscillations, one considers functions expressed as superpositions of propagating waves eikx over a finite range of k (bandlimited functions) which can oscillate at spatial scales s with k max s ⪡ 1, with k max being the maximum frequency in the spectrum. We use a similar concept for evanescent waves e kx. These have also a clearly defined spatial scale k 1, hence if a superposition of such waves could oscillate locally at a scale −1 , e.g. like sin (πx/s ), this would also qualify as a supers ⪡ k max oscillation, in analogy to the case of propagating waves eikx . In other words, we consider the possibility of obtaining fast oscillations (in particular, densely spaced zeros) by superposing slowly decaying exponentials instead of slowly oscillating sines and cosines. This possibility can be thought of as extending the definition
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of superoscillations to functions that are amenable to a Laplace but not necessarily to a Fourier transform (and thus may not be bandlimited in the strict sense), which may be an interesting extension both from the physics and the mathematics point of view. Specifically consider a function defined in x ≥ 0 through a continuous superposition of evanescent waves with decay constants k limited in some range [k1, k2 ] as
f (x) =
k2
∫k
F (k ) e−kx dk,
x≥0
1
(1)
where F(k) is a (generally complex) amplitude function. This function remains undefined in x < 0 hence one cannot readily speak of its Fourier transform or its bandwidth. Of course, one can assume that f(x) is extended to x < 0 through its analytic continuation (simply by using Eq. (1) with negative x) in which case the function becomes unbounded as x → − ∞ and thus nonFourier transformable. Other choices lead to a Fourier transformable function, such as an even or odd reflection ( f ( − x ) = f (x ) or f ( − x ) = − f (x )) or the causal case f (x ) = 0 for x < 0. Whatever the choice, the values of f(x) in x < 0 are immaterial to our problem which is to obtain fast oscillations from a superposition of slowly decaying exponentials. In a physical setting, as for example when f(x) is the wave along the x-axis obtained after total internal reflection (TIR) at an interface at x ¼0, this implies that we do not pose any restrictions on the field or the medium on the high-index side (x < 0) other than it must produce the desired f(x) in x > 0. Hence the question is whether a function of the type defined in Eq. (1) can actually oscillate at small spatial scales, namely superoscillate. The answer is obtained by associating f(x) with the analytic continuation of a bandlimited function Φ (ξ ) (of the real variable ξ) whose Fourier transform is supported on [k1, k2 ] and equals F(k). Such a function is expressed as
Φ (ξ ) =
∫k
k2
F (k ) eikξ dk,
1
ξ real
(2)
According to the Paley–Wiener theorem [11], the complex function Φ (z ), that is obtained from Eq. (2) by complexifying the argument ξ to z = ξ + iη, is an entire function (of exponential type, namely there exists C > 0 such that |g (z )| ≤ Cek2 | z| ). Therefore, we can obtain from Eq. (2) analytic functions f(x) of a real variable x by letting z run along any smooth curve z(x) on the complex plane. Notice that, for a general path z(x), these functions will be free of singularities but not necessarily square integrable (square integrability is guaranteed by the Paley–Wiener theorem only for paths parallel to the real axis [11]). In particular, if we restrict z to the imaginary axis (z = ix ), a propagating wave eikξ turns into an evanescent wave e kx and Eq. (2) gives the f(x) of Eq. (1) in x ≥ 0, namely
f (x) = Φ (ix),
x≥0
(3)
We have therefore reached an important conclusion. A f(x) defined through Eq. (1) can be viewed as the restriction to the positive imaginary axis of an entire function Φ (z ), whose restriction to the real axis Φ (ξ ) has the Fourier transform F(k) supported on [k1, k2 ]. And by the uniqueness of an analytic continuation, the complex analytic function Φ (z ) satisfying Eq. (3) is unique. In this way, the oscillations of f(x) are mapped to oscillations of Φ (z ) along the positive imaginary axis. These oscillations can in turn be controlled by the zeros of Φ (z ). And here is where a remarkable property of entire functions applies. The zeros of entire functions Φ (z ), which are defined as the extension in the complex plane of a bandlimited function Φ (ξ ), can be placed arbitrarily close to each other no matter how small is the maximum frequency contained in Φ (ξ ) [12]. In our case of Φ (ξ ) defined as in Eq. (2), this maximum
483
frequency is k2. This property of entire functions has also been used in the past to produce standard superoscillatory functions [13,14]. One can therefore construct Φ (z ) to have a set of densely spaced zeros along the positive imaginary axis, zn = ixn , xn ≥ 0, n = 1, 2, … , to obtain a function f (x ) = Φ (ix ) having densely spaced zeros at x = xn . We have therefore shown that f(x) can indeed superoscillate in the aforementioned sense. Let us illustrate the above with an example. Consider the entire function
Φ (z ) =
(z − is )(z − 2is )(z − 3is ) sinc (z ) eiδz (z − 1)(z − 2)(z − 3)
(4)
where s ⪡ 1, δ > π and sinc (z ) = sin (πz ) /(πz ). The logic behind this Φ (z ) is, firstly, to replace the three zeros of the entire function sinc(z ) at z = 1, 2, 3 by three closely spaced zeros along the imaginary axis at z = is, 2is, 3is . The new function is also entire and of the same exponential order e π| z| with sinc(z ) hence, according to the zero-replacement theorem [14], its restriction to the real axis has the same bandwidth (2π) with sinc (ξ ). Secondly, the exponential factor eiδz of Φ (z ) becomes on the real axis the phase factor eiδξ which serves to shift the spectrum of Φ (ξ ) from [ − π , π ] to the strictly positive wavenumbers [δ − π , δ + π ]. Now, according to our previous discussion, the restriction of Φ (z ) to the positive imaginary axis, i.e.,
f (x) = Φ (ix) =
(x − s )(x − 2s )(x − 3s ) sinh (πx) −δx e , (x + i)(x + 2i)(x + 3i) πx
x≥0
(5)
is a function that is expressible as in Eq. (1) with k1 = δ − π and k2 = δ + π and with F(k) being the Fourier transform of Φ (ξ ). Moreover f(x) superoscillates at the (arbitrarily small) scale s due to the three zeros at x = s, 2s, 3s . Note also the decay of f (x ) ∼ e(π − δ ) x /x as x → + ∞. An example of this f(x) is shown in Fig. 1(a) for s ¼0.1 and δ ¼1.5π. Here s has been deliberately chosen large for illustration purposes. The corresponding complex amplitude function F(k) follows by numerically computing the Fourier transform of the Φ (ξ ) of Eq. (4) (using the inverse to Eq. (2) relation) and is shown in Fig. 1(b). The design of superoscillations using evanescent waves has therefore been reduced to the design of standard bandlimited superoscillatory functions with a strictly positive spectrum. There are two general methods available in the literature for constructing standard superoscillatory functions, apart from the zero-replacement theorem that we have just used. The first method expresses the function as a finite sum of sinc functions with coefficients that are determined by imposing a set of amplitude or derivative constraints over a finite grid of closely spaced points [15– 17]. The second method uses similar constraints but the superoscillatory function is expressed as a Fourier integral and a minimum-energy solution is sought using variational techniques [18,19]. These methods have a discrete logic in the sense that the superoscillatory function is constrained over a discrete set of points. This however implies some lack of control over the actual shape of the superoscillatory curve across the interval that contains these points (superoscillatory interval). Towards a continuous design of superoscillations, we have recently proposed a simple method for creating superoscillations that mimic a given polynomial with arbitrarily high precision within some finite interval [20]. Such superoscillatory functions are obtained as the product of the target polynomial with a bandlimited envelope function whose Fourier transform has at least N 1 continuous derivatives and a Nth derivative of bounded variation, with N being the order of the polynomial. For example consider the entire function
⎛ z⎞ Φ (z ) = − (iz + s )(iz + 2s )(iz + 3s ) sinc 4 ⎜ ⎟ eiδz ⎝ 4⎠
(6)
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a
b
c
d
Fig. 1. Magnitude of the f(x) (in log scale) of (a) Eq. (5) and (c) Eq. (7) for s ¼ 0.1, δ ¼1.5π. (b, d) Real part (blue curve), imaginary part (red curve) and magnitude (dashed curve) of the corresponding amplitude function F(k). The vertical lines at the discontinuity points in (d) are merely to guide the eye. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
and its restriction to the real axis Φ (ξ ). The Fourier transform of the envelope function sinc 4 (ξ/4) is supported on [ − π , π ] and it is C2 ( − ∞ , ∞), namely twice continuously differentiable along the entire real line, and with a bounded (although not continuous) 3rd derivative [20]. Using standard properties of Fourier analysis it follows that the product of this envelope with any 3rd order polynomial is also a bandlimited (and square integrable) function [20]. Again, the exponential factor eiδξ , δ > π , serves to shift the support of the Fourier transform to [δ − π , δ + π ]. Setting z = ix , x ≥ 0, in Eq. (6) we obtain
⎛ sinh (πx/4) ⎞4 f (x) = (x − s )(x − 2s )(x − 3s ) ⎜ ⎟ e − δx ⎝ πx/4 ⎠
(7)
which is a function expressible as in Eq. (1). An example of this function and its corresponding amplitude function is shown in Fig. 1(c) and (d) respectively. Let us now use the above in a simple 2D refraction scenario that involves evanescent waves produced by the TIR of a y-polarized optical beam at a dielectric interface between two half-spaces with wave numbers κ1 (in x < 0) and κ2 (x ≥ 0), κ1 > κ 2. The incident beam in x < 0 is chosen so that the evanescent wave in the positive x half-plane (z = 0, x ≥ 0) equals the f(x) of Eq. (1). Then the electric field on both sides of the interface is written as
⎞ ⎛ ⎜ Ey (x ≥ 0, z )⎟ = ⎜ Ey (x < 0, z )⎟ ⎠ ⎝ where β =
∫k
k2
1
⎞ ⎛ e−kx F (k ) ⎜⎜ T−1 (k )[eiax + R k e−iax]⎟⎟ eiβz dk, ⎠ ⎝
( )
(8)
κ22 + k2 (propagation constant parallel to the inter-
face), a = κ12 − κ22 − k2 (propagation constant normal to the interface) and T = 2a/(a + ik ), R = (a − ik ) /(a + ik ) are the Fresnel reflection and transmission coefficients, respectively. An example
is shown in Fig. 2 using the amplitude function F(k) of Fig. 1(d). Fig. 2(a) shows the incident field propagating freely in a medium with κ1 = 12. The incident field is obtained by using for all x the lower formula of Eq. (8) with R ¼0 and is composed of a bundle of plane waves at angles ranging from 49.6° to 84.5° with respect to the x-axis. In Fig. 2(b) this field undergoes TIR at an interface to a medium with κ 2 = 9, which implies a critical TIR angle θc = sin−1(κ 2/κ1) = 48.6°. The superoscillatory behaviour is revealed in Fig. 2(c) which zooms around the (0, 0) point, as well as in Fig. 2 (e) which plots the field along the x-axis. For x ≥ 0 the function of Fig. 1(c) is obviously reproduced. Finally Fig. 1(d) shows the phase of the electric field around the superoscillatory region. What is here interesting is that phase singularities are observed around the zeros of the field. Indeed, there is a e−iϕ vortex phase singularity around the zeros at (x, z ) = (s, 0) and (2s, 0) and a e+iϕ singularity around the third zero at (x, z ) = (3s, 0), with ϕ indicating the polar angle around the position of the corresponding zero. It is so far known that such vortex singularities occur at the zeros of twodimensional fields obtained from the superposition of plane waves [21]. The rapid variation of the phase around the zero implies a large local wavenumber and hence a superoscillation. We here see that similar vortices occur in superpositions of evanescent waves like the ones considered in present work. In the previous example note that the choice of δ ¼ 1.5π was quite arbitrary. From the examples of Eqs. (5) and (7) it was clear that we must have δ > π so that Φ (ξ ) has a positive spectrum. There could be however some practical limitation. A very large δ means that the field f(x) decays very fast which, after Eq. (8), implies that a very large index difference is required between the two media (a must remain real for large values of k). Notice that the examples of Eqs. (4) and (6) suggest a similarity between the two methods used to construct the entire function Φ (z ) from which the superoscillatory f(x) made of exponentials is
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a
c
485
b
d
e
Fig. 2. (a) Magnitude (|Ey |) of the incident field of Eq. (8) freely propagating in a medium with κ 1 = 12 (distances and wavenumbers in dimensionless units). The amplitude function F(k) is that of Fig. 1. (b) |Ey | in the presence of an interface (white line) to a medium with κ 2 = 9 . (c) log10|Ey | and (d) phase (arg Ey ) in the presence of the interface and in a region around the origin. (e) Field magnitude along the x-axis in logarithmic scale.
derived. In both cases, Φ (z ) has a polynomial factor whose complex zeros are responsible for the superoscillatory behaviour of f (x). Notice, however, that in the second case the polynomial need not replace the zeros of an entire function but is directly multiplied with it. The only requirement is that the Fourier transform of the restriction of this envelope function to the real ξ-axis, e (ξ ), is sufficiently smooth. Stated otherwise, we only need a square-integrable e (ξ ) that is bandlimited in, say, [ − π , π ] (Paley–Wiener space PW ( − π , π )), and decays sufficiently fast as |ξ| → ∞ so that its product with a polynomial of given order belongs to the same Paley–Wiener space. In our example of Eq. (7), e (ξ ) = sinc 4 (ξ/4) decays as |ξ|−4 which allows us to multiply it with a 3rd polynomial and still obtain a square-integrable bandlimited function. This alternative explanation follows directly from the smoothness-decay property of Fourier transforms [22]. Most importantly, our method can be applied entirely in the real line without referring to the analytic continuation of the envelope or needing to know its complex zeros. For example, one may start with a envelope e (ξ ) whose Fourier transform is a bump function, i.e. has a finite support, say [ − π , π ], and is C ∞ ( − ∞ , ∞), namely infinitely differentiable along the entire real line. Due to the smoothness (infinite differentiability) of its Fourier transform, such an envelope can be multiplied by a polynomial pN (ξ ) of arbitrarily high order N to obtain a standard superoscillatory function Φ (ξ ) = e (ξ ) pN (ξ ) with any number of oscillations. The corresponding superoscillatory function made of evanescent waves follows as f (x ) = Φ (ix ) e−δx with δ > π . As an example consider the function
⎛ N ⎞ (ix/s )n ⎟ − δx f (x) = ⎜⎜ ∑ ⎟ e (ix) e , ⎝ n = 0 n! ⎠
x≥0 (9)
where
e (ξ ) =
π
∫−π
E (k ) eikξ dk,
ξ real
(10)
is the inverse Fourier transform of the bump function 2 2 E (k ) = e−1/ (π − k ) for |k| < π and E (k ) = 0 for |k| ≥ π . In Eq. (9) one recognizes the truncated Taylor expansion of eix / s , hence for sufficiently large N and s ⪡ 1 the function f(x) of Eq. (9) behaves close to x ¼0 like the propagating wave Ceix / s . This sounds quite counter-intuitive for a function composed of evanescent waves e kx. From Eq. (9) it is obvious that, in order to obtain f(x), we only need to compute e(ix) (by setting ξ = ix in Eq. (10)) and not the entire extension of e (ξ ) in the complex plane or its complex zeros. An example of the function of Eq. (9) is given in Fig. 3 for s¼ 0.001 and N ¼ 20. The oscillations of f(x) close to the origin are shown in Fig. 3(a), while Fig. 3(b) gives an expanded view of |f (x )| over a broad range of x. Notable is the difference in the amplitude of the superoscillations compared to the amplitude of the function in the region of “normal” variation. We have shown in [20] that this difference is in the order of s N which in our example gives ∼1058 taking also into account a e−δ factor. This rough estimate is quite close to the 53 orders of magnitude difference observed between Fig. 3(a) and (b). The last example suggests that one can use this approach to shape the evanescent field near an interface in a desired fashion. For
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a
b
Fig. 3. (a) Real part (blue line), imaginary part (red line) and magnitude (dashed line) of the f(x) of Eq. (9) for s ¼0.001, N ¼20 and δ = 1.5π for 0 < x < 0.01. (b) Magnitude of the same function for 0 < x < 25. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
a
b
d
e
c
f
Fig. 4. (a) Magnitude (|Ey |) of the incident field of Eq. (8) freely propagating in a medium with κ 1 = 12. (b) |Ey | in the presence of an interface (white line) to a medium with κ 2 = 9 . (c) The amplitude function F (k ) = E′(k − 1.5π ) − 0.2E (k − 1.5π ) where E(k) is the bump function defined below Eq. (10). (d) |Ey | and (e) phase (arg Ey ) in the presence of the interface and in a region around the origin. (f) Field magnitude along the x-axis in logarithmic scale.
example one may wish to realize an optical trap for particles or atoms with a potential that approximates a harmonic oscillator in the direction normal to the interface. Such a potential requires the intensity |E (x, z )|2 of the field to follow a parabolic law in x so that the component of the gradient force normal to the interface ( Fx ∝ ∂|E|2 /∂x ) varies linearly with the distance from the centre of the trap [23]. This field can be obtained very simply by multiplying an envelope like the one of Eq. (9) with a linear polynomial x − x0 , where x0 is the distance of the centre of the trap from the interface, namely f (x ) = (x − x0 ) e (ix ) e−δx . Notice in this example that we are still applying the logic of Eq. (9) even though we are not interested in superoscillations but in a single zero (hence the use of a polynomial with order 1). The corresponding amplitude function F(k) is also simple to obtain. This f(x) follows from the complexification of
Φ (ξ ) = ( − iξ − x0 ) e (ξ ) eiδξ ,which is immediately Fourier-transformed to give F (k ) = E′(k − δ ) − x0 E (k − δ ),where E(k) is the Fourier transform of e (ξ ) and E′(k ) its first derivative. Fig. 4 shows a diffraction scenario similar to Fig. 2 for a beam with this spectrum F(k) (plotted in Fig. 4(c)). Notice also in the phase map of Fig. 4(e) the e−iϕ vortex singularity occurring at the zero at (x, z ) = (0.2, 0). As a final comment, note that the concepts presented in this work can be extended to functions that decay with different, nonexponential laws, such as inverse powers. The problem could be formulated, for example, as producing superoscillations with spatial scale s ⪡ b2 by superposing functions of the form (x/b)−n , with b ∈ [b1, b2 ] acting like the continuous parameter. Although similar in its formulation, the analytical treatment of this problem is quite different from the exponential case (cannot be treated
I. Chremmos, G. Fikioris / Optics Communications 356 (2015) 482–487
with Fourier transforms) and is left to a follow-up work. In conclusion, we have introduced superoscillatory-type optical fields that are obtained as superpositions of evanescent waves near a plane dielectric interface. In this context we have extended the concept (or space) of superoscillatory functions from functions that are expressed as continuous superpositions of propagating waves e±ikx to functions expressed as continuous superpositions of exponential waves e kx, with 0 ≤ k ≤ k max in both cases. For the evanescent-wave case, a superoscillation was physically defined as an oscillation at a spatial scale s that is much smaller than the −1 minimum length scale (k max ) of the constituent exponentials. The main idea reported in this work is that such functions can be obtained by complexifying the argument of standard bandlimited superoscillatory functions with a strictly positive spectrum. We have furthermore combined this idea with our method for continuously engineered polynomial-shaped superoscillations to produce evanescent waves with a desired shape. Superoscillatory evanescent waves may be useful for locally shaping the evanescent field near interfaces for applications such as particle or atom trapping. From a broader perspective, the idea presented in this communication extends the applications of optical superoscillations from the far to the near field regime.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Superoscillatory field features with evanescent waves Ioannis Chremmos a,n, George Fikioris b a b
Max Planck Institute for the Science of Light, D-91058 Erlangen, Germany School of Electrical and Computer Engineering, National Technical University, GR 157-73 Athens, Greece
art ic l e i nf o
a b s t r a c t
Article history: Received 28 May 2015 Received in revised form 8 August 2015 Accepted 11 August 2015
We show how to obtain optical fields possessing superoscillatory features by superposing the evanescent tails of waves undergoing total internal reflection at a plane dielectric interface. In doing so, we essentially extend the definition of superoscillations to functions expressed as a continuum of slowly decaying exponentials, while not necessarily being bandlimited in the standard (Fourier) sense. We obtain such functions by complexifying the argument of standard bandlimited superoscillatory functions with a strictly positive spectrum. Combined with our recent method for superoscillations with arbitrary polynomial shape, the present approach offers flexibility for locally shaping the evanescent field near dielectric interfaces for applications such as particle or atom trapping. & 2015 Elsevier B.V. All rights reserved.
Keywords: Superoscillations Superresolution Evanescent waves Diffraction Wave optics Near field
The phenomenon of superoscillations implies the ability of bandlimited signals to oscillate with local frequencies that are arbitrarily larger than their maximum frequency component. This counter-intuitive property has long intrigued physicists and engineers because it enables, at least theoretically, antenna or imaging systems to produce or resolve wave features much finer than their bandwidth suggests. The earliest reports of the phenomenon can therefore be found in classical works on superdirectivity [1] and subdiffraction imaging [2], as well as in information and signal theory [3,4]. The underlying idea of all these works was that extremely fine (or fast) oscillations can be obtained from appropriately weighted superpositions of slowly-varying functions whose rate of variation is limited by some fundamental bandwidth cutoff. The latter is introduced either by the operating wavelength in radiation or imaging settings or by the response time of circuits and systems in signal processing settings. Interestingly, the term superoscillations itself was coined independently in quantum mechanics in the context of weak measurements [5] and was thereafter established when referring to such faster-than-Fourier functions [6]. The interest in superoscillations and particular in superoscillatory optical imaging has recently revived with a number of promising experimental demonstrations [7,8]. Superoscillations have so far been obtained as superpositions of propagating waves in homogeneous media. From a theoretical n
Corresponding author. E-mail addresses: [email protected] (I. Chremmos), gfi[email protected] (G. Fikioris). http://dx.doi.org/10.1016/j.optcom.2015.08.029 0030-4018/& 2015 Elsevier B.V. All rights reserved.
viewpoint, this is due to their original definition within a space of bandlimited functions, namely functions whose Fourier transform has an appropriate compact support. In this context, a superoscillatory function f(x) is always expressible (through its inverse Fourier transform) as a superposition of propagating waves eikx with k being limited within a given bandwidth. From the applications viewpoint too, creating superoscillations with propagating waves has been the major focus of research with the aim of achieving optical superresolution without evanescent waves or, equivalently, subwavelength focusing of light in the far field [9,10]. In this communication we consider a different question: Is it possible to produce superoscillatory field features using as basis functions nonpropagating evanescent waves of the form e kx? Before we give an answer, the concept of superoscillations within the context of evanescent waves should first be clarified (or better, introduced). In standard superoscillations, one considers functions expressed as superpositions of propagating waves eikx over a finite range of k (bandlimited functions) which can oscillate at spatial scales s with k max s ⪡ 1, with k max being the maximum frequency in the spectrum. We use a similar concept for evanescent waves e kx. These have also a clearly defined spatial scale k 1, hence if a superposition of such waves could oscillate locally at a scale −1 , e.g. like sin (πx/s ), this would also qualify as a supers ⪡ k max oscillation, in analogy to the case of propagating waves eikx . In other words, we consider the possibility of obtaining fast oscillations (in particular, densely spaced zeros) by superposing slowly decaying exponentials instead of slowly oscillating sines and cosines. This possibility can be thought of as extending the definition
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of superoscillations to functions that are amenable to a Laplace but not necessarily to a Fourier transform (and thus may not be bandlimited in the strict sense), which may be an interesting extension both from the physics and the mathematics point of view. Specifically consider a function defined in x ≥ 0 through a continuous superposition of evanescent waves with decay constants k limited in some range [k1, k2 ] as
f (x) =
k2
∫k
F (k ) e−kx dk,
x≥0
1
(1)
where F(k) is a (generally complex) amplitude function. This function remains undefined in x < 0 hence one cannot readily speak of its Fourier transform or its bandwidth. Of course, one can assume that f(x) is extended to x < 0 through its analytic continuation (simply by using Eq. (1) with negative x) in which case the function becomes unbounded as x → − ∞ and thus nonFourier transformable. Other choices lead to a Fourier transformable function, such as an even or odd reflection ( f ( − x ) = f (x ) or f ( − x ) = − f (x )) or the causal case f (x ) = 0 for x < 0. Whatever the choice, the values of f(x) in x < 0 are immaterial to our problem which is to obtain fast oscillations from a superposition of slowly decaying exponentials. In a physical setting, as for example when f(x) is the wave along the x-axis obtained after total internal reflection (TIR) at an interface at x ¼0, this implies that we do not pose any restrictions on the field or the medium on the high-index side (x < 0) other than it must produce the desired f(x) in x > 0. Hence the question is whether a function of the type defined in Eq. (1) can actually oscillate at small spatial scales, namely superoscillate. The answer is obtained by associating f(x) with the analytic continuation of a bandlimited function Φ (ξ ) (of the real variable ξ) whose Fourier transform is supported on [k1, k2 ] and equals F(k). Such a function is expressed as
Φ (ξ ) =
∫k
k2
F (k ) eikξ dk,
1
ξ real
(2)
According to the Paley–Wiener theorem [11], the complex function Φ (z ), that is obtained from Eq. (2) by complexifying the argument ξ to z = ξ + iη, is an entire function (of exponential type, namely there exists C > 0 such that |g (z )| ≤ Cek2 | z| ). Therefore, we can obtain from Eq. (2) analytic functions f(x) of a real variable x by letting z run along any smooth curve z(x) on the complex plane. Notice that, for a general path z(x), these functions will be free of singularities but not necessarily square integrable (square integrability is guaranteed by the Paley–Wiener theorem only for paths parallel to the real axis [11]). In particular, if we restrict z to the imaginary axis (z = ix ), a propagating wave eikξ turns into an evanescent wave e kx and Eq. (2) gives the f(x) of Eq. (1) in x ≥ 0, namely
f (x) = Φ (ix),
x≥0
(3)
We have therefore reached an important conclusion. A f(x) defined through Eq. (1) can be viewed as the restriction to the positive imaginary axis of an entire function Φ (z ), whose restriction to the real axis Φ (ξ ) has the Fourier transform F(k) supported on [k1, k2 ]. And by the uniqueness of an analytic continuation, the complex analytic function Φ (z ) satisfying Eq. (3) is unique. In this way, the oscillations of f(x) are mapped to oscillations of Φ (z ) along the positive imaginary axis. These oscillations can in turn be controlled by the zeros of Φ (z ). And here is where a remarkable property of entire functions applies. The zeros of entire functions Φ (z ), which are defined as the extension in the complex plane of a bandlimited function Φ (ξ ), can be placed arbitrarily close to each other no matter how small is the maximum frequency contained in Φ (ξ ) [12]. In our case of Φ (ξ ) defined as in Eq. (2), this maximum
483
frequency is k2. This property of entire functions has also been used in the past to produce standard superoscillatory functions [13,14]. One can therefore construct Φ (z ) to have a set of densely spaced zeros along the positive imaginary axis, zn = ixn , xn ≥ 0, n = 1, 2, … , to obtain a function f (x ) = Φ (ix ) having densely spaced zeros at x = xn . We have therefore shown that f(x) can indeed superoscillate in the aforementioned sense. Let us illustrate the above with an example. Consider the entire function
Φ (z ) =
(z − is )(z − 2is )(z − 3is ) sinc (z ) eiδz (z − 1)(z − 2)(z − 3)
(4)
where s ⪡ 1, δ > π and sinc (z ) = sin (πz ) /(πz ). The logic behind this Φ (z ) is, firstly, to replace the three zeros of the entire function sinc(z ) at z = 1, 2, 3 by three closely spaced zeros along the imaginary axis at z = is, 2is, 3is . The new function is also entire and of the same exponential order e π| z| with sinc(z ) hence, according to the zero-replacement theorem [14], its restriction to the real axis has the same bandwidth (2π) with sinc (ξ ). Secondly, the exponential factor eiδz of Φ (z ) becomes on the real axis the phase factor eiδξ which serves to shift the spectrum of Φ (ξ ) from [ − π , π ] to the strictly positive wavenumbers [δ − π , δ + π ]. Now, according to our previous discussion, the restriction of Φ (z ) to the positive imaginary axis, i.e.,
f (x) = Φ (ix) =
(x − s )(x − 2s )(x − 3s ) sinh (πx) −δx e , (x + i)(x + 2i)(x + 3i) πx
x≥0
(5)
is a function that is expressible as in Eq. (1) with k1 = δ − π and k2 = δ + π and with F(k) being the Fourier transform of Φ (ξ ). Moreover f(x) superoscillates at the (arbitrarily small) scale s due to the three zeros at x = s, 2s, 3s . Note also the decay of f (x ) ∼ e(π − δ ) x /x as x → + ∞. An example of this f(x) is shown in Fig. 1(a) for s ¼0.1 and δ ¼1.5π. Here s has been deliberately chosen large for illustration purposes. The corresponding complex amplitude function F(k) follows by numerically computing the Fourier transform of the Φ (ξ ) of Eq. (4) (using the inverse to Eq. (2) relation) and is shown in Fig. 1(b). The design of superoscillations using evanescent waves has therefore been reduced to the design of standard bandlimited superoscillatory functions with a strictly positive spectrum. There are two general methods available in the literature for constructing standard superoscillatory functions, apart from the zero-replacement theorem that we have just used. The first method expresses the function as a finite sum of sinc functions with coefficients that are determined by imposing a set of amplitude or derivative constraints over a finite grid of closely spaced points [15– 17]. The second method uses similar constraints but the superoscillatory function is expressed as a Fourier integral and a minimum-energy solution is sought using variational techniques [18,19]. These methods have a discrete logic in the sense that the superoscillatory function is constrained over a discrete set of points. This however implies some lack of control over the actual shape of the superoscillatory curve across the interval that contains these points (superoscillatory interval). Towards a continuous design of superoscillations, we have recently proposed a simple method for creating superoscillations that mimic a given polynomial with arbitrarily high precision within some finite interval [20]. Such superoscillatory functions are obtained as the product of the target polynomial with a bandlimited envelope function whose Fourier transform has at least N 1 continuous derivatives and a Nth derivative of bounded variation, with N being the order of the polynomial. For example consider the entire function
⎛ z⎞ Φ (z ) = − (iz + s )(iz + 2s )(iz + 3s ) sinc 4 ⎜ ⎟ eiδz ⎝ 4⎠
(6)
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a
b
c
d
Fig. 1. Magnitude of the f(x) (in log scale) of (a) Eq. (5) and (c) Eq. (7) for s ¼ 0.1, δ ¼1.5π. (b, d) Real part (blue curve), imaginary part (red curve) and magnitude (dashed curve) of the corresponding amplitude function F(k). The vertical lines at the discontinuity points in (d) are merely to guide the eye. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
and its restriction to the real axis Φ (ξ ). The Fourier transform of the envelope function sinc 4 (ξ/4) is supported on [ − π , π ] and it is C2 ( − ∞ , ∞), namely twice continuously differentiable along the entire real line, and with a bounded (although not continuous) 3rd derivative [20]. Using standard properties of Fourier analysis it follows that the product of this envelope with any 3rd order polynomial is also a bandlimited (and square integrable) function [20]. Again, the exponential factor eiδξ , δ > π , serves to shift the support of the Fourier transform to [δ − π , δ + π ]. Setting z = ix , x ≥ 0, in Eq. (6) we obtain
⎛ sinh (πx/4) ⎞4 f (x) = (x − s )(x − 2s )(x − 3s ) ⎜ ⎟ e − δx ⎝ πx/4 ⎠
(7)
which is a function expressible as in Eq. (1). An example of this function and its corresponding amplitude function is shown in Fig. 1(c) and (d) respectively. Let us now use the above in a simple 2D refraction scenario that involves evanescent waves produced by the TIR of a y-polarized optical beam at a dielectric interface between two half-spaces with wave numbers κ1 (in x < 0) and κ2 (x ≥ 0), κ1 > κ 2. The incident beam in x < 0 is chosen so that the evanescent wave in the positive x half-plane (z = 0, x ≥ 0) equals the f(x) of Eq. (1). Then the electric field on both sides of the interface is written as
⎞ ⎛ ⎜ Ey (x ≥ 0, z )⎟ = ⎜ Ey (x < 0, z )⎟ ⎠ ⎝ where β =
∫k
k2
1
⎞ ⎛ e−kx F (k ) ⎜⎜ T−1 (k )[eiax + R k e−iax]⎟⎟ eiβz dk, ⎠ ⎝
( )
(8)
κ22 + k2 (propagation constant parallel to the inter-
face), a = κ12 − κ22 − k2 (propagation constant normal to the interface) and T = 2a/(a + ik ), R = (a − ik ) /(a + ik ) are the Fresnel reflection and transmission coefficients, respectively. An example
is shown in Fig. 2 using the amplitude function F(k) of Fig. 1(d). Fig. 2(a) shows the incident field propagating freely in a medium with κ1 = 12. The incident field is obtained by using for all x the lower formula of Eq. (8) with R ¼0 and is composed of a bundle of plane waves at angles ranging from 49.6° to 84.5° with respect to the x-axis. In Fig. 2(b) this field undergoes TIR at an interface to a medium with κ 2 = 9, which implies a critical TIR angle θc = sin−1(κ 2/κ1) = 48.6°. The superoscillatory behaviour is revealed in Fig. 2(c) which zooms around the (0, 0) point, as well as in Fig. 2 (e) which plots the field along the x-axis. For x ≥ 0 the function of Fig. 1(c) is obviously reproduced. Finally Fig. 1(d) shows the phase of the electric field around the superoscillatory region. What is here interesting is that phase singularities are observed around the zeros of the field. Indeed, there is a e−iϕ vortex phase singularity around the zeros at (x, z ) = (s, 0) and (2s, 0) and a e+iϕ singularity around the third zero at (x, z ) = (3s, 0), with ϕ indicating the polar angle around the position of the corresponding zero. It is so far known that such vortex singularities occur at the zeros of twodimensional fields obtained from the superposition of plane waves [21]. The rapid variation of the phase around the zero implies a large local wavenumber and hence a superoscillation. We here see that similar vortices occur in superpositions of evanescent waves like the ones considered in present work. In the previous example note that the choice of δ ¼ 1.5π was quite arbitrary. From the examples of Eqs. (5) and (7) it was clear that we must have δ > π so that Φ (ξ ) has a positive spectrum. There could be however some practical limitation. A very large δ means that the field f(x) decays very fast which, after Eq. (8), implies that a very large index difference is required between the two media (a must remain real for large values of k). Notice that the examples of Eqs. (4) and (6) suggest a similarity between the two methods used to construct the entire function Φ (z ) from which the superoscillatory f(x) made of exponentials is
I. Chremmos, G. Fikioris / Optics Communications 356 (2015) 482–487
a
c
485
b
d
e
Fig. 2. (a) Magnitude (|Ey |) of the incident field of Eq. (8) freely propagating in a medium with κ 1 = 12 (distances and wavenumbers in dimensionless units). The amplitude function F(k) is that of Fig. 1. (b) |Ey | in the presence of an interface (white line) to a medium with κ 2 = 9 . (c) log10|Ey | and (d) phase (arg Ey ) in the presence of the interface and in a region around the origin. (e) Field magnitude along the x-axis in logarithmic scale.
derived. In both cases, Φ (z ) has a polynomial factor whose complex zeros are responsible for the superoscillatory behaviour of f (x). Notice, however, that in the second case the polynomial need not replace the zeros of an entire function but is directly multiplied with it. The only requirement is that the Fourier transform of the restriction of this envelope function to the real ξ-axis, e (ξ ), is sufficiently smooth. Stated otherwise, we only need a square-integrable e (ξ ) that is bandlimited in, say, [ − π , π ] (Paley–Wiener space PW ( − π , π )), and decays sufficiently fast as |ξ| → ∞ so that its product with a polynomial of given order belongs to the same Paley–Wiener space. In our example of Eq. (7), e (ξ ) = sinc 4 (ξ/4) decays as |ξ|−4 which allows us to multiply it with a 3rd polynomial and still obtain a square-integrable bandlimited function. This alternative explanation follows directly from the smoothness-decay property of Fourier transforms [22]. Most importantly, our method can be applied entirely in the real line without referring to the analytic continuation of the envelope or needing to know its complex zeros. For example, one may start with a envelope e (ξ ) whose Fourier transform is a bump function, i.e. has a finite support, say [ − π , π ], and is C ∞ ( − ∞ , ∞), namely infinitely differentiable along the entire real line. Due to the smoothness (infinite differentiability) of its Fourier transform, such an envelope can be multiplied by a polynomial pN (ξ ) of arbitrarily high order N to obtain a standard superoscillatory function Φ (ξ ) = e (ξ ) pN (ξ ) with any number of oscillations. The corresponding superoscillatory function made of evanescent waves follows as f (x ) = Φ (ix ) e−δx with δ > π . As an example consider the function
⎛ N ⎞ (ix/s )n ⎟ − δx f (x) = ⎜⎜ ∑ ⎟ e (ix) e , ⎝ n = 0 n! ⎠
x≥0 (9)
where
e (ξ ) =
π
∫−π
E (k ) eikξ dk,
ξ real
(10)
is the inverse Fourier transform of the bump function 2 2 E (k ) = e−1/ (π − k ) for |k| < π and E (k ) = 0 for |k| ≥ π . In Eq. (9) one recognizes the truncated Taylor expansion of eix / s , hence for sufficiently large N and s ⪡ 1 the function f(x) of Eq. (9) behaves close to x ¼0 like the propagating wave Ceix / s . This sounds quite counter-intuitive for a function composed of evanescent waves e kx. From Eq. (9) it is obvious that, in order to obtain f(x), we only need to compute e(ix) (by setting ξ = ix in Eq. (10)) and not the entire extension of e (ξ ) in the complex plane or its complex zeros. An example of the function of Eq. (9) is given in Fig. 3 for s¼ 0.001 and N ¼ 20. The oscillations of f(x) close to the origin are shown in Fig. 3(a), while Fig. 3(b) gives an expanded view of |f (x )| over a broad range of x. Notable is the difference in the amplitude of the superoscillations compared to the amplitude of the function in the region of “normal” variation. We have shown in [20] that this difference is in the order of s N which in our example gives ∼1058 taking also into account a e−δ factor. This rough estimate is quite close to the 53 orders of magnitude difference observed between Fig. 3(a) and (b). The last example suggests that one can use this approach to shape the evanescent field near an interface in a desired fashion. For
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a
b
Fig. 3. (a) Real part (blue line), imaginary part (red line) and magnitude (dashed line) of the f(x) of Eq. (9) for s ¼0.001, N ¼20 and δ = 1.5π for 0 < x < 0.01. (b) Magnitude of the same function for 0 < x < 25. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
a
b
d
e
c
f
Fig. 4. (a) Magnitude (|Ey |) of the incident field of Eq. (8) freely propagating in a medium with κ 1 = 12. (b) |Ey | in the presence of an interface (white line) to a medium with κ 2 = 9 . (c) The amplitude function F (k ) = E′(k − 1.5π ) − 0.2E (k − 1.5π ) where E(k) is the bump function defined below Eq. (10). (d) |Ey | and (e) phase (arg Ey ) in the presence of the interface and in a region around the origin. (f) Field magnitude along the x-axis in logarithmic scale.
example one may wish to realize an optical trap for particles or atoms with a potential that approximates a harmonic oscillator in the direction normal to the interface. Such a potential requires the intensity |E (x, z )|2 of the field to follow a parabolic law in x so that the component of the gradient force normal to the interface ( Fx ∝ ∂|E|2 /∂x ) varies linearly with the distance from the centre of the trap [23]. This field can be obtained very simply by multiplying an envelope like the one of Eq. (9) with a linear polynomial x − x0 , where x0 is the distance of the centre of the trap from the interface, namely f (x ) = (x − x0 ) e (ix ) e−δx . Notice in this example that we are still applying the logic of Eq. (9) even though we are not interested in superoscillations but in a single zero (hence the use of a polynomial with order 1). The corresponding amplitude function F(k) is also simple to obtain. This f(x) follows from the complexification of
Φ (ξ ) = ( − iξ − x0 ) e (ξ ) eiδξ ,which is immediately Fourier-transformed to give F (k ) = E′(k − δ ) − x0 E (k − δ ),where E(k) is the Fourier transform of e (ξ ) and E′(k ) its first derivative. Fig. 4 shows a diffraction scenario similar to Fig. 2 for a beam with this spectrum F(k) (plotted in Fig. 4(c)). Notice also in the phase map of Fig. 4(e) the e−iϕ vortex singularity occurring at the zero at (x, z ) = (0.2, 0). As a final comment, note that the concepts presented in this work can be extended to functions that decay with different, nonexponential laws, such as inverse powers. The problem could be formulated, for example, as producing superoscillations with spatial scale s ⪡ b2 by superposing functions of the form (x/b)−n , with b ∈ [b1, b2 ] acting like the continuous parameter. Although similar in its formulation, the analytical treatment of this problem is quite different from the exponential case (cannot be treated
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with Fourier transforms) and is left to a follow-up work. In conclusion, we have introduced superoscillatory-type optical fields that are obtained as superpositions of evanescent waves near a plane dielectric interface. In this context we have extended the concept (or space) of superoscillatory functions from functions that are expressed as continuous superpositions of propagating waves e±ikx to functions expressed as continuous superpositions of exponential waves e kx, with 0 ≤ k ≤ k max in both cases. For the evanescent-wave case, a superoscillation was physically defined as an oscillation at a spatial scale s that is much smaller than the −1 minimum length scale (k max ) of the constituent exponentials. The main idea reported in this work is that such functions can be obtained by complexifying the argument of standard bandlimited superoscillatory functions with a strictly positive spectrum. We have furthermore combined this idea with our method for continuously engineered polynomial-shaped superoscillations to produce evanescent waves with a desired shape. Superoscillatory evanescent waves may be useful for locally shaping the evanescent field near interfaces for applications such as particle or atom trapping. From a broader perspective, the idea presented in this communication extends the applications of optical superoscillations from the far to the near field regime.
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