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Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 29, Iss. 7 — Jul. 1, 2012
  • pp: 1407–1411
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Generalized laws of refraction that can lead to wave-optically forbidden light-ray fields

Johannes Courtial and Tomáš Tyc  »View Author Affiliations


JOSA A, Vol. 29, Issue 7, pp. 1407-1411 (2012)
http://dx.doi.org/10.1364/JOSAA.29.001407


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Abstract

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

1. INTRODUCTION

An optical interface that changes the direction of light according to a generalized law of refraction was recently demonstrated [1

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). [CrossRef]

]. 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

2. H. O. Moser and C. Rockstuhl, “3D THz metamaterials from micro/nanomanufacturing,” Laser Photon. Rev. 6, 219–244 (2012). [CrossRef]

].

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

3. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.1.2.

]) has to be the gradient of the eikonal, S, so s=S [4

4. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.1.1.

]. Taking the curl of both sides of this equation and applying the vector-calculus identity ×S=0 then yields the condition
×s=0
(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

5. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974). [CrossRef]

], which form lines of zero intensity and where the phase is undefined.) This condition limits a number of applications, including holography [6

6. C. Paterson, “Diffractive optical elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994). [CrossRef]

] and distorting mirrors [7

7. R. A. Hicks, “Controlling a ray bundle with a free-form reflector,” Opt. Lett. 33, 1672–1674 (2008). [CrossRef]

]. 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

8. 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). [CrossRef]

].

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

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). [CrossRef]

] 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

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). [CrossRef]

], 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

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). [CrossRef]

], 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 s(s) of the normalized light-ray-direction vector s 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
sx(sx,sy,sz),sy(sx,sy,sz),sz(sx,sy,sz).
(2)
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

9. A. C. Hamilton and J. Courtial, “Optical properties of a Dove-prism sheet,” J. Opt. A 10, 125302 (2008). [CrossRef]

],
sx=sx,sy=sy,sz=sz.
(3)

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, s can simply be calculated by evaluating the functions of Eq. (2). According to Eq. (1), the light-ray-direction field s can have a corresponding complex scalar wave only if its curl vanishes. The Cartesian components of this curl are
(×s)x=szysyz,
(4)
(×s)y=sxzszx,
(5)
(×s)z=syxsxy,
(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, sx/y, sy/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, sx/z, sy/z, sz/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 s that contains no z derivatives is the z component, given by Eq. (6). The corresponding condition
syxsxy=0
(7)
for the wave-optical legality of the refracted light-ray-direction field was derived in [8

8. 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). [CrossRef]

].

By applying the chain rule to the left-hand side of Eq. (7), we now bring in the functional dependence of s on s, given by Eqs. (2). Equation (7) becomes
0=sysxsxx+sysysyx+syszszxsxsxsxysxsysyysxszszy.
(8)

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 (×s)z=0, or simply
sxy=syx.
(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=sysxsxx+(sysysxsx)syx+syszszxsxsysyysxszszy.
(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” sx/sx, sy/sx, sx/sy, sy/sy, sx/sz, and sy/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:
sxsy=0,sxsz=0,
(11)
sysx=0,sysz=0,
(12)
sysysxsx=0.
(13)

We have bundled the conditions on the law derivatives together so that we can see the following. Equations (11) state that sx is neither a function of sy nor of sz, so
sx=sx(sx).
(14)
Similarly, Eqs. (12) imply that
sy=sy(sy).
(15)
Equation (13) then simply states that
dsx(sx)dsx=dsy(sy)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,
dsx(sx)dsx=N,
(17)
and so
sx=Nsx+Sx,
(18)
where Sx is a constant of integration. Similarly,
sy=Nsy+Sy,
(19)
with the same factor N. Equations (18) and (19) are the main results of this paper. They describe the most general zero-curl-preserving law of refraction.

To understand the generalized law of refraction described by Eqs. (18) and (19), we first discuss the factor N for Sx=Sy=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=sy. We can write sx and sx in terms of their respective angles with the window normal, α and α, and the length of s, s:
sx=ssinα,sx=ssinα.
(20)
When these expressions are substituted into Eq. (18), it becomes
sinα=Nsinα.
(21)
This has the form of Snell’s law.

To understand, next, the meaning of Sx and Sy, 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(Δkxx+Δkyy)]; the complex amplitude of a homogeneous plane wave with wave vector (kx,ky,kz) incident on the hologram therefore becomes, in the hologram plane,
exp[i(kxx+kyy)]exp[i(Δkxx+Δkyy)]=exp[i(kxx+kyy)],
(22)
where the transverse wavenumbers after transmission through the hologram are
kx=kx+Δkxandky=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

10. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.1.4.

] at all points other than phase discontinuities), the propagation direction is that of the k vector, namely
k=2πλsandk=2πλs.
(24)
Substitution of the wave-vector components into Eqs. (23) gives
sx=sx+λ2πΔkxandsy=sy+λ2πΔky.
(25)
Equations (25) are, of course, Eqs. (18) and (19) with N=1 and
Sx=λ2πΔkx,Sy=λ2πΔky.
(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

Fig. 1. Refraction according to Snell’s law. (a) This happens naturally at the interface between different refractive indices, n and n, 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.

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

12. 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). [CrossRef]

14

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.”

] (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

15. J. Courtial and T. Tyc, “METATOYs and optical vortices,” J. Opt. 13, 115704 (2011). [CrossRef]

]. 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

8. 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). [CrossRef]

], 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

13. 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). [CrossRef]

,16

16. J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008). [CrossRef]

]; flipping of one transverse ray-direction component, like in Eq. (3) [9

9. A. C. Hamilton and J. Courtial, “Optical properties of a Dove-prism sheet,” J. Opt. A 10, 125302 (2008). [CrossRef]

,12

12. 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). [CrossRef]

]; and rotation of the light-ray direction by an arbitrary, but fixed, angle α around the local window normal [14

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.”

,17

17. A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, “Local light-ray rotation,” J. Opt. A 11, 085705 (2009). [CrossRef]

] (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(iα) [18

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). [CrossRef]

] and which leads to the concept of imaging between complex object and image distances [19

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). [CrossRef]

].

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.

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

8. 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). [CrossRef]

], and in a sense all optical vortices are, at their core, evanescent [20

20. F. S. Roux, “Optical vortex density limitation,” Opt. Commun. 223, 31–37 (2003). [CrossRef]

,21

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). [CrossRef]

].

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

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). [CrossRef]

], 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

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:
sx2+sy2+sz2=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
xsx2+xsy2+xsz2=2sxsxx+2sysyx+2szszx=2s·(xs).
(A2)
A similar argument holds for the y derivatives, so the direction s has to satisfy the equations
s·(xs)=0,s·(ys)=0.
(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). [CrossRef]

2.

H. O. Moser and C. Rockstuhl, “3D THz metamaterials from micro/nanomanufacturing,” Laser Photon. Rev. 6, 219–244 (2012). [CrossRef]

3.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.1.2.

4.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.1.1.

5.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974). [CrossRef]

6.

C. Paterson, “Diffractive optical elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994). [CrossRef]

7.

R. A. Hicks, “Controlling a ray bundle with a free-form reflector,” Opt. Lett. 33, 1672–1674 (2008). [CrossRef]

8.

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). [CrossRef]

9.

A. C. Hamilton and J. Courtial, “Optical properties of a Dove-prism sheet,” J. Opt. A 10, 125302 (2008). [CrossRef]

10.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.1.4.

11.

M. A. Green, J. Zhao, A. Wang, P. J. Reece, and M. Gal, “Efficient silicon light-emitting diodes,” Nature 412, 805–808 (2001). [CrossRef]

12.

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). [CrossRef]

13.

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). [CrossRef]

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). [CrossRef]

16.

J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008). [CrossRef]

17.

A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, “Local light-ray rotation,” J. Opt. A 11, 085705 (2009). [CrossRef]

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). [CrossRef]

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). [CrossRef]

20.

F. S. Roux, “Optical vortex density limitation,” Opt. Commun. 223, 31–37 (2003). [CrossRef]

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). [CrossRef]

OCIS Codes
(160.1245) Materials : Artificially engineered materials
(240.3990) Optics at surfaces : Micro-optical devices

ToC Category:
Materials

History
Original Manuscript: January 11, 2012
Revised Manuscript: April 25, 2012
Manuscript Accepted: May 14, 2012
Published: June 22, 2012

Virtual Issues
July 17, 2012 Spotlight on Optics

Citation
Johannes Courtial and Tomáš Tyc, "Generalized laws of refraction that can lead to wave-optically forbidden light-ray fields," J. Opt. Soc. Am. A 29, 1407-1411 (2012)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-29-7-1407


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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). [CrossRef]
  2. H. O. Moser and C. Rockstuhl, “3D THz metamaterials from micro/nanomanufacturing,” Laser Photon. Rev. 6, 219–244 (2012). [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.1.2.
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.1.1.
  5. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974). [CrossRef]
  6. C. Paterson, “Diffractive optical elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994). [CrossRef]
  7. R. A. Hicks, “Controlling a ray bundle with a free-form reflector,” Opt. Lett. 33, 1672–1674 (2008). [CrossRef]
  8. 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). [CrossRef]
  9. A. C. Hamilton and J. Courtial, “Optical properties of a Dove-prism sheet,” J. Opt. A 10, 125302 (2008). [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.1.4.
  11. M. A. Green, J. Zhao, A. Wang, P. J. Reece, and M. Gal, “Efficient silicon light-emitting diodes,” Nature 412, 805–808 (2001). [CrossRef]
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