1. INTRODUCTION

An optical interface that changes the direction of light according to a generalized law of refraction was recently demonstrated [1]. The novel aspect is that the direction change was achieved in a layer so thin that it can be treated as an interface in the effective-medium approximation. The layer consisted of a planar array of nanoscale optical resonators that imparted a spatially varying phase delay, so it was a phase hologram realized with a two-dimensional metamaterial [2].

Here we consider this law of refraction—and more general laws of refraction—in the context of a condition wave optics imposes on light-ray fields. The condition stems from the fact that, in the limit of geometrical optics, the light-ray direction $\mathbf{s}$ (defined as a unit vector in the direction of the average Poynting vector [3]) has to be the gradient of the eikonal, $S$, so $\mathbf{s}=\nabla S$ [4]. Taking the curl of both sides of this equation and applying the vector-calculus identity $\nabla \times \nabla S=0$ then yields the condition

We divide generalized laws of refraction into two categories: those that turn any incident wave-optically allowed light-ray field into another wave-optically allowed light-ray field and those that do not. We call the laws in the former category zero-curl preserving. We find that the generalized law of refraction demonstrated in [1] is one of the very few zero-curl preserving refraction laws. We speculate on the possibility of realizing, with an effective-medium interface, other generalized laws of refraction.

2. MOST GENERAL ZERO-CURL-PRESERVING LAW OF REFRACTION

We consider a planar, homogeneous surface that is surrounded by air on both sides and that changes the direction of transmitted light rays according to a generalized law of refraction. (Note that we are asking for a homogeneous change in light-ray direction and not necessarily a homogeneous surface. In the case of [1], for example, a homogeneous light-ray-direction change is achieved by a constant gradient of the phase delay introduced by the surface.) The restrictions ensure that the properties of the refracted light-ray field are due to the generalized law of refraction rather than surface shape, surface inhomogeneity [the generation of a phase singularity (vortex) in [1], for example, is due to the spatial variation of the direction change introduced by the surface], or the medium behind the surface and indeed that the direction change itself is due to the surface rather than light entering another medium. The surface does not offset light rays, and so any transmitted light ray leaves the surface from the same position where the corresponding incident light ray intersects the surface, but on the other side. Here we call such a surface a window.

We describe a generalized law of refraction in terms of the explicit dependence ${\mathbf{s}}^{\prime }(\mathbf{s})$ of the normalized light-ray-direction vector ${\mathbf{s}}^{\prime }$ immediately behind the window on the normalized light-ray-direction vector $\mathbf{s}$ immediately in front of the window. This dependence can be described in terms of the $x$, $y$, and $z$ components as the functions

Note that such a window changes the homogeneous light-ray-direction field that corresponds to any homogeneous plane wave into a new homogeneous light-ray-direction field whose direction has been changed according to the generalized law of refraction. The refracted field is thus another homogeneous plane wave and curl free. Therefore, a homogeneous plane wave is an example of a light field that is turned into another wave-optically allowed light-ray field by any window that refracts according to any generalized law of refraction. The homogeneous windows we discuss here therefore produce wave-optically forbidden light-ray fields only if the incident light-ray field is inhomogeneous.

Now consider a window in the $xy$ plane. In the plane immediately behind the window, ${\mathbf{s}}^{\prime }$ can simply be calculated by evaluating the functions of Eq. (2). According to Eq. (1), the light-ray-direction field ${\mathbf{s}}^{\prime }$ can have a corresponding complex scalar wave only if its curl vanishes. The Cartesian components of this curl are

By applying the chain rule to the left-hand side of Eq. (7), we now bring in the functional dependence of ${\mathbf{s}}^{\prime }$ on $\mathbf{s}$, given by Eqs. (2). Equation (7) becomes

From now on we consider zero-curl-preserving laws of refraction. As the incident light-ray-direction field is wave-optically allowed, it satisfies the equation ${(\nabla \times \mathbf{s})}_z=0$, or simply

The incident light-ray-direction field determines the partial derivatives $\partial s_x/\partial x$, $\partial s_y/\partial x$, $\partial s_z/\partial x$, $\partial s_y/\partial y$, and $\partial s_z/\partial y$, so we call them the “field derivatives”; the law of refraction determines the “law derivatives” $\partial s_x^{\prime }/\partial s_x$, $\partial s_y^{\prime }/\partial s_x$, $\partial s_x^{\prime }/\partial s_y$, $\partial s_y^{\prime }/\partial s_y$, $\partial s_x^{\prime }/\partial s_z$, and $\partial s_y^{\prime }/\partial s_z$. In order for a law of refraction to be zero-curl preserving, this equation has to hold for any incident light-ray field, that is, for any combination of values of the field derivatives (see Appendix A). [Note that the list of the field derivatives does not include $\partial s_x/\partial y$, as this is given by Eq. (9) and therefore not independent.] Therefore, all the terms multiplying the field derivatives have to be individually zero. This gives the following conditions on the law derivatives:

We have bundled the conditions on the law derivatives together so that we can see the following. Equations (11) state that $s_x^{\prime }$ is neither a function of $s_y$ nor of $s_z$, so

The left-hand side of Eq. (16) is dependent purely on $s_x$, while the right-hand side depends only on $s_y$. This implies that both sides have to equal a constant, $N$. Therefore,

To understand the generalized law of refraction described by Eqs. (18) and (19), we first discuss the factor $N$ for $S_x=S_y=0$. We choose the coordinate system such that the plane of incidence, which contains the window normal and the incident ray direction, is the $(x,z)$ plane, so that $s_y=0=s_y^{\prime }$. We can write $s_x$ and $s_x^{\prime }$ in terms of their respective angles with the window normal, $\alpha$ and ${\alpha }^{\prime }$, and the length of $\mathbf{s}$, $\parallel \mathbf{s}\parallel$:

To understand, next, the meaning of $S_x$ and $S_y$, we compare Eqs. (18) and (19) to the equations for the transverse wave-vector components behind the phase hologram of a thin, transparent wedge. (Note that, locally, any phase hologram is of such a form at all points other than on lines or points where it introduces phase discontinuities.) The effect of such a phase hologram on a light beam is a multiplication of the beam’s complex amplitude in the hologram plane (the $z$ plane) by a factor $\exp [i(\mathrm{\Delta }{k}_{x}x+\mathrm{\Delta }{k}_{y}y)]$; the complex amplitude of a homogeneous plane wave with wave vector $(k_x,k_y,k_z)$ incident on the hologram therefore becomes, in the hologram plane,

We can translate the wave-vector change for a plane wave described by Eq. (23) into a corresponding direction change. For a plane wave (and, locally, any physical light field behaves as a plane wave [10] at all points other than phase discontinuities), the propagation direction is that of the $\mathbf{k}$ vector, namely

3. REALIZING OTHER GENERALIZED LAWS OF REFRACTION

Snell’s law of refraction is, of course, realized at every interface between isotropic media with different refractive indices [Fig. 1(a)]. In this case, the factor $N$ in Eq. (21) is the ratio of the refractive indices. But Snell’s law is zero-curl preserving, irrespective of the refractive indices on either side of the interface, so our condition does not forbid interfaces such as a window with air on both sides that nevertheless refracts according to Snell’s law [Eq. (21)] with $N\ne 1$. Inclined phase fronts would be discontinuous across such an interface [Fig. 1(b)]. If such an interface existed, it would help solve the important energy-related problem of coupling of light into and out of silicon [11].

 

Fig. 1. Refraction according to Snell’s law. (a) This happens naturally at the interface between different refractive indices, $n$ and $n^{\prime }$, in which case the phase fronts (thick lines) line up. (b) Same direction-change law, realized with a hypothetical interface (horizontal line) with the same refractive index, $n$, on either side. Note that the phase fronts do not line up at the interface.

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One can imagine even more exotic and intriguing generalized laws of refraction, namely the ones that do not preserve zero curl. Such laws of refraction have so far only been realized in compromised form, namely accompanied by a nonzero and inhomogeneous ray offset, and so these current realizations are too imperfect for the considerations from Section 2 to apply. Nevertheless, they currently represent the closest there exists to windows that refract according to generalized laws of refraction not preserving zero curl. Moreover, they have led to interesting concepts that would equally apply to perfect realizations of the same generalized laws of refraction. These realizations use so-called METAmaTerial fOr raYs (METATOYs) [12–14] (Fig. 2). The compromise used by METATOYs is pixelation, i.e., piecewise redirection of the phase front. At the border between neighboring pixels, the phase is discontinuous (even if the illuminating wave is a homogeneous plane wave); in a sense, METATOYs concentrate each pixel’s curl into phase singularities along the pixel’s edge [15]. Provided the pixels (i.e., phase-front pieces) are too small to be resolved by an observer and at the same time large compared to the wavelength [8], this compromise can be as unnoticeable as a computer monitor’s pixelation; METATOYs then appear to refract light according to exotic generalized laws of refraction. The generalized laws of refraction that have been realized include Snell’s law but with sines replaced by tangents, resulting in imaging that stretches the longitudinal direction but not the transverse directions [13,16]; flipping of one transverse ray-direction component, like in Eq. (3) [9,12]; and rotation of the light-ray direction by an arbitrary, but fixed, angle $\alpha$ around the local window normal [14,17] (Fig. 2), which can be described as Snell’s law of refraction between media whose complex ray-optical refractive indices differ by a factor $\exp (i\alpha )$ [18] and which leads to the concept of imaging between complex object and image distances [19].

 

Fig. 2. Example of the view through a window that performs generalized refraction that does not preserve the curl of the incident light-ray field. Here a Rubik’s cube is seen through a window that rotates the light-ray direction around the local window normal; see [14] for details.

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We are not aware of any fundamental reason that forbids the perfect (no accompanying ray offset, etc.) realization of such exotic generalized laws of refraction. When an interface that (perfectly) realizes such generalized refraction is illuminated with a light-ray field whose curl remains zero after refraction, then the interface should simply refract. What would happen if such an interface is illuminated by an incoming field that would, according to the interface’s generalized law of refraction, be turned into an outgoing field that is wave-optically forbidden? We speculate that the incoming field might undergo total internal reflection (TIR). In Snell’s law of refraction, TIR occurs whenever the refracted field becomes evanescent, which happens when the transverse part of the wave vector becomes greater than the wavenumber. Something similar might happen when the refracted light-ray field is wave-optically forbidden—after all, if it existed, the wave corresponding to a light-ray field with nonzero curl would have nonzero vortex-charge density [8], and in a sense all optical vortices are, at their core, evanescent [20,21].

4. CONCLUSIONS

Generalized laws of refraction that do not leave light-ray fields curl free can lead to wave-optically forbidden light-ray fields. Surprisingly few generalized laws of refraction are zero-curl preserving, i.e., never change a light-ray field from being wave-optically allowed to forbidden; these are Snell’s law and phase-hologram refraction (and combinations of these). Remarkably, with the recent demonstration of a phase-hologram interface [1], all of these have now been realized in the form of interfaces.

It is interesting to speculate whether or not it might be possible to realize other generalized laws of refraction. If this was possible, then these would have the potential to solve important problems and realize new optical-design ideas.