Download Generalized laws of refraction that can lead to

J. Courtial and T. Tyc
Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A
1407
Generalized laws of refraction that can lead
to wave-optically forbidden light-ray fields
Johannes Courtial1,* and Tomáš Tyc2
2
1
SUPA, School of Physics & Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
Institute of Theoretical Physics & Astrophysics, Masaryk University, 61137 Brno, Czech Republic
*Corresponding author: [email protected]
Received January 11, 2012; revised April 25, 2012; accepted May 14, 2012;
posted May 15, 2012 (Doc. ID 161368); published June 22, 2012
The recent demonstration of a metamaterial phase hologram so thin that it can be classified as an interface in the
effective-medium approximation [Science 334, 333 (2011)] has dramatically increased interest in generalized laws
of refraction. Based on the fact that scalar wave optics allows only certain light-ray fields, we divide generalized
laws of refraction into two categories. When applied to a planar cross section through any allowed light-ray field,
the laws in the first category always result in a cross section through an allowed light-ray field again, whereas the
laws in the second category can result in a cross section through a forbidden light-ray field. © 2012 Optical
Society of America
OCIS codes: 160.1245, 240.3990.
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 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
s ∇S [4]. Taking the curl of both sides of this equation and
applying the vector-calculus identity ∇ × ∇S 0 then yields
the condition
∇×s0
1
on the light-ray field. Equation (1) states that the curl of any
physical light-ray-direction field has to be zero. (Note that this
condition only holds almost everywhere; it does not hold at
phase singularities [5], which form lines of zero intensity
and where the phase is undefined.) This condition limits a
number of applications, including holography [6] and distorting mirrors [7]. It has previously been realized that specific
generalized laws of refraction can lead to wave-optically
forbidden light-ray fields, i.e., light-ray fields that violate
Eq. (1) [8].
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
1084-7529/12/071407-05$15.00/0
of refraction demonstrated in [1] is one of the very few zerocurl preserving refraction laws. We speculate on the possibility of realizing, with an effective-medium interface, other
generalized laws of refraction.
2. MOST GENERAL ZERO-CURLPRESERVING 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 lightray-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 s0 s of the normalized light-ray-direction
vector s0 immediately behind the window on the normalized
light-ray-direction vector s immediately in front of the window. This dependence can be described in terms of the x,
y, and z components as the functions
s0x sx ; sy ; sz ;
s0y sx ; sy ; sz ;
© 2012 Optical Society of America
s0z sx ; sy ; sz :
(2)
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J. Opt. Soc. Am. A / Vol. 29, No. 7 / July 2012
J. Courtial and T. Tyc
For example, in the case of a window that flips the x component of the light-ray direction without altering its y and z components [9],
s0x −sx ;
s0y sy ;
s0z sz :
(3)
Note that such a window changes the homogeneous lightray-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, s0 can simply be calculated
by evaluating the functions of Eq. (2). According to Eq. (1),
the light-ray-direction field s0 can have a corresponding complex scalar wave only if its curl vanishes. The Cartesian components of this curl are
∇ × s0 x ∂s0z ∂s0y
−
;
∂y ∂z
(4)
∇ × s0 y ∂s0x ∂s0z
−
;
∂z ∂x
(5)
∇ × s0 z ∂s0y ∂s0x
−
;
∂x ∂y
(6)
and all of these components have to vanish for the refracted
light-ray field to be allowed wave optically. The partial derivatives with respect to x and y, ∂s0x ∕ ∂y, ∂s0y ∕ ∂x, etc., can easily
be calculated, as the field in the plane immediately in front of
the window and the ray-direction mapping performed by the
window together fully determine the field in the plane immediately behind the window. In contrast, the partial derivatives
with respect to z, ∂s0x ∕ ∂z, ∂s0y ∕ ∂z, ∂s0z ∕ ∂z, depend on the way
the field propagates, which is determined by wave optics.
Therefore, there is no obvious way in which to calculate
the right-hand sides of Eqs. (4) and (5), as they contain z derivatives: we need to know the wave to know how it propagates, but that wave might not exist in the first place. The only
component of the curl of s0 that contains no z derivatives is the
z component, given by Eq. (6). The corresponding condition
∂s0y ∂s0x
−
0
∂x ∂y
(7)
for the wave-optical legality of the refracted light-ray-direction
field was derived in [8].
By applying the chain rule to the left-hand side of Eq. (7),
we now bring in the functional dependence of s0 on s, given by
Eqs. (2). Equation (7) becomes
0
∂s0y ∂sx ∂s0y ∂sy ∂s0y ∂sz ∂s0x ∂sx ∂s0x ∂sy ∂s0x ∂sz
−
−
−
:
∂sx ∂x ∂sy ∂x ∂sz ∂x ∂sx ∂y ∂sy ∂y ∂sz ∂y
(8)
From now on we consider zero-curl-preserving laws of
refraction. As the incident light-ray-direction field is waveoptically allowed, it satisfies the equation ∇ × sz 0, or
simply
∂sx ∂sy
:
∂y
∂x
(9)
For the outgoing light-ray-direction field to be wave-optically
allowed, both sides of Eq. (8) have to be zero. Substituting
Eq. (9) into Eq. (8) gives
0
0
∂s0y ∂sx
∂sy ∂s0x ∂sy ∂s0y ∂sz ∂s0x ∂sy ∂s0x ∂sz
−
−
−
:
∂sx ∂x
∂sy ∂sx ∂x ∂sz ∂x ∂sy ∂y ∂sz ∂y
(10)
The incident light-ray-direction field determines the partial
derivatives ∂sx ∕ ∂x, ∂sy ∕ ∂x, ∂sz ∕ ∂x, ∂sy ∕ ∂y, and ∂sz ∕ ∂y, so we
call them the “field derivatives”; the law of refraction determines the “law derivatives” ∂s0x ∕ ∂sx , ∂s0y ∕ ∂sx , ∂s0x ∕ ∂sy , ∂s0y ∕ ∂sy ,
∂s0x ∕ ∂sz , and ∂s0y ∕ ∂sz . 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 ∂sx ∕ ∂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:
∂s0x
0;
∂sy
∂s0x
0;
∂sz
(11)
∂s0y
0;
∂sx
∂s0y
0;
∂sz
(12)
∂s0y ∂s0x
−
0:
∂sy ∂sx
(13)
We have bundled the conditions on the law derivatives together so that we can see the following. Equations (11) state
that s0x is neither a function of sy nor of sz , so
s0x s0x sx :
(14)
Similarly, Eqs. (12) imply that
s0y s0y sy :
(15)
Equation (13) then simply states that
ds0x sx ds0y sy :
dsx
dsy
(16)
The left-hand side of Eq. (16) is dependent purely on sx ,
while the right-hand side depends only on sy . This implies that
both sides have to equal a constant, N. Therefore,
J. Courtial and T. Tyc
Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A
ds0x sx N;
dsx
(17)
Substitution of the wave-vector components into Eqs. (23)
gives
s0x sx and so
s0x Nsx S x ;
(18)
where S x is a constant of integration. Similarly,
s0y Nsy S y ;
(19)
with the same factor N. Equations (18) and (19) are the main
results of this paper. They describe the most general zero-curlpreserving law of refraction.
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
sy 0 s0y . We can write sx and s0x in terms of their respective angles with the window normal, α and α0 , and the length
of s, ∥s∥:
s0x ∥s∥ sin α0 :
sx ∥s∥ sin α;
(20)
When these expressions are substituted into Eq. (18), it
becomes
sin α N sin α0 :
(21)
This has the form of Snell’s law.
To understand, next, the meaning of S x and S y , we compare
Eqs. (18) and (19) to the equations for the transverse wavevector 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 expiΔkx x Δky y; the complex amplitude of a
homogeneous plane wave with wave vector kx ; ky ; kz incident on the hologram therefore becomes, in the hologram
plane,
expikx x ky y expiΔkx x Δky y expik0x x k0y y;
(22)
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λ
Δkx
2π
and s0y sy λ
Δky :
2π
(25)
Equations (25) are, of course, Eqs. (18) and (19) with N 1
and
Sx λ
Δkx ;
2π
Sy λ
Δky :
2π
(26)
Equations (18) and (19) therefore describe a combination of
the light-ray-direction changes due to Snell’s law and due to
the addition of a phase wedge.
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 ≠ 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].
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].
where the transverse wavenumbers after transmission
through the hologram are
k0x kx Δkx
and
k0y ky Δky :
(23)
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 k vector,
namely
k
2π
s
λ
and k0 2π 0
s:
λ
(24)
Fig. 1. (Color online) Refraction according to Snell’s law. (a) This
happens naturally at the interface between different refractive indices, n and n0 , 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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J. Opt. Soc. Am. A / Vol. 29, No. 7 / July 2012
J. Courtial and T. Tyc
Remarkably, with the recent demonstration of a phasehologram 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.
APPENDIX A
Fig. 2. (Color online) 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.
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 α 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 expiα [18] and which leads to the concept
of imaging between complex object and image distances [19].
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 waveoptically 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).
We have stated above that any combination of field derivatives can occur. Here we justify this assertion.
We have already taken into account the condition the incident field has to satisfy to be wave-optically allowed [Eq. (9)],
which means that Eq. (10) does not contain any terms proportional to ∂sy ∕ ∂x. The other condition that might restrict the
possible values of the derivatives listed above is the normalization of the vector s:
s2x s2y s2z 1:
(A1)
Differentiating both sides of Eq. (A1) with respect to x gives
zero for the right-hand side, and for the left-hand side
∂sy
∂ 2
∂
∂
∂s
∂s
s s2 s2 2sx x 2sy
2sz z
∂x x ∂x y ∂x z
∂x
∂x
∂x
∂
2s ·
s :
∂x
(A2)
A similar argument holds for the y derivatives, so the direction
s has to satisfy the equations
s·
∂
s 0;
∂x
s·
∂
s 0:
∂y
(A3)
For a given vector ∂s ∕ ∂x, possible vectors s that satisfy the left
equation are those that are perpendicular to ∂s ∕ ∂x. Similarly,
solutions to the right equation are perpendicular to the vector
∂s ∕ ∂y [note that the length of the x component of the vector
∂s ∕ ∂y is restricted by Eq. (9), but the vector can still point in
any direction]. Vectors s that satisfy both equations are those
that point in the directions where the plane perpendicular to
∂s ∕ ∂x and the plane perpendicular to ∂s ∕ ∂y (both planes pass
through the origin) intersect. Such directions always exist, so
it is indeed true that any combination of ∂sx ∕ ∂x, ∂sy ∕ ∂x,
∂sz ∕ ∂x, ∂sy ∕ ∂y, and ∂sz ∕ ∂y can occur.
ACKNOWLEDGMENTS
Many thanks to Martin Šarbort for very helpful discussions.
T. T. acknowledges support of the Quest for Ultimate Electromagnetics using Spatial Transformations (QUEST) programme grant of the Engineering and Physical Sciences
Research Council (EPSRC).
REFERENCES
1. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F.
Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science
334, 333–337 (2011).
2. H. O. Moser and C. Rockstuhl, “3D THz metamaterials from micro/nanomanufacturing,” Laser Photon. Rev. 6, 219–244 (2012).
3. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980),
Chap. 3.1.2.
J. Courtial and T. Tyc
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
M. Born and E. Wolf, Principles of Optics (Pergamon, 1980),
Chap. 3.1.1.
J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R.
Soc. A 336, 165–190 (1974).
C. Paterson, “Diffractive optical elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994).
R. A. Hicks, “Controlling a ray bundle with a free-form reflector,”
Opt. Lett. 33, 1672–1674 (2008).
A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray
optics without wave-optical analog in the ray-optics limit,” New
J. Phys. 11, 013042 (2009).
A. C. Hamilton and J. Courtial, “Optical properties of a
Dove-prism sheet,” J. Opt. A 10, 125302 (2008).
M. Born and E. Wolf, Principles of Optics (Pergamon, 1980),
Chap. 3.1.4.
M. A. Green, J. Zhao, A. Wang, P. J. Reece, and M. Gal, “Efficient
silicon light-emitting diodes,” Nature 412, 805–808 (2001).
M. Blair, L. Clark, E. A. Houston, G. Smith, J. Leach, A. C.
Hamilton, and J. Courtial, “Experimental demonstration of a
light-ray-direction-flipping METATOY based on confocal lenticular arrays,” Opt. Commun. 282, 4299–4302 (2009).
J. Courtial, B. C. Kirkpatrick, E. Logean, and T. Scharf, “Experimental demonstration of ray-optical refraction with confocal
lenslet arrays,” Opt. Lett. 35, 4060–4062 (2010).
Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A
1411
14. J. Courtial, N. Chen, S. Ogilvie, B. C. Kirkpatrick, A. C. Hamilton,
G. Gibson, T. Tyc, E. Logean, and T. Scharf are preparing a
manuscript to be called “Experimental demonstration of
windows representing complex ray-optical refractive-index
interfaces.”
15. J. Courtial and T. Tyc, “METATOYs and optical vortices,” J. Opt.
13, 115704 (2011).
16. J. Courtial, “Ray-optical refraction with confocal lenslet arrays,”
New J. Phys. 10, 083033 (2008).
17. A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, “Local lightray rotation,” J. Opt. A 11, 085705 (2009).
18. B. Sundar, A. C. Hamilton, and J. Courtial, “Fermat’s principle
and the formal equivalence of local light-ray rotation and refraction at the interface between homogeneous media,” Opt. Lett.
34, 374–376 (2009).
19. J. Courtial, B. C. Kirkpatrick, and M. A. Alonso, “Imaging
with complex ray-optical refractive-index interfaces between
complex object and image distances,” Opt. Lett. 37, 701–703
(2012).
20. F. S. Roux, “Optical vortex density limitation,” Opt. Commun.
223, 31–37 (2003).
21. R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett,
“Momentum paradox in a vortex core,” J. Mod. Opt. 52,
1135–1144 (2005).