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) [CrossRef]  ] 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

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References

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  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]

2012 (2)

2011 (2)

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

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]

2010 (1)

2009 (4)

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

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]

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]

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]

2008 (3)

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

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

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

2005 (1)

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]

2003 (1)

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

2001 (1)

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

1994 (1)

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

1974 (1)

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

Aieta, F.

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]

Alonso, M. A.

Barnett, S. M.

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]

Berry, M. V.

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

Blair, M.

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]

Born, M.

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

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

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

Capasso, F.

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]

Chen, N.

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.”

Clark, L.

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]

Courtial, J.

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]

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

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]

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]

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

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]

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]

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

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

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.”

Gaburro, Z.

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]

Gal, M.

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

Genevet, P.

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]

Gibson, G.

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.”

Green, M. A.

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

Hamilton, A. C.

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]

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]

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

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]

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

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.”

Hicks, R. A.

Houston, E. A.

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]

Kats, M. A.

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]

Kirkpatrick, B. C.

Leach, J.

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]

Logean, E.

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]

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.”

Moser, H. O.

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

Nelson, J.

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

Nye, J. F.

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

Ogilvie, S.

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.”

Padgett, M.

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]

Paterson, C.

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

Reece, P. J.

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

Rockstuhl, C.

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

Roux, F. S.

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

Scharf, T.

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]

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.”

Smith, G.

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]

Sundar, B.

Tetienne, J.-P.

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]

Thomson, L. C.

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]

Tyc, T.

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

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.”

Wang, A.

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

Wolf, E.

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

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

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

Yu, N.

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]

Zambrini, R.

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]

Zhao, J.

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

J. Mod. Opt. (2)

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

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]

J. Opt. (1)

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

J. Opt. A (2)

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

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

Laser Photon. Rev. (1)

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

Nature (1)

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

New J. Phys. (2)

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

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]

Opt. Commun. (2)

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

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]

Opt. Lett. (4)

Proc. R. Soc. A (1)

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

Science (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]

Other (4)

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

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

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

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.”

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Figures (2)

Fig. 1.
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.

Download Full Size | PPT Slide | PDF

Fig. 2.
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.

Download Full Size | PPT Slide | PDF

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

∇×s=0
sx′(sx,sy,sz),sy′(sx,sy,sz),sz′(sx,sy,sz).
sx′=−sx,sy′=sy,sz′=sz.
(∇×s′)x=∂sz′∂y−∂sy′∂z,
(∇×s′)y=∂sx′∂z−∂sz′∂x,
(∇×s′)z=∂sy′∂x−∂sx′∂y,
∂sy′∂x−∂sx′∂y=0
0=∂sy′∂sx∂sx∂x+∂sy′∂sy∂sy∂x+∂sy′∂sz∂sz∂x−∂sx′∂sx∂sx∂y−∂sx′∂sy∂sy∂y−∂sx′∂sz∂sz∂y.
∂sx∂y=∂sy∂x.
0=∂sy′∂sx∂sx∂x+(∂sy′∂sy−∂sx′∂sx)∂sy∂x+∂sy′∂sz∂sz∂x−∂sx′∂sy∂sy∂y−∂sx′∂sz∂sz∂y.
∂sx′∂sy=0,∂sx′∂sz=0,
∂sy′∂sx=0,∂sy′∂sz=0,
∂sy′∂sy−∂sx′∂sx=0.
sx′=sx′(sx).
sy′=sy′(sy).
dsx′(sx)dsx=dsy′(sy)dsy.
dsx′(sx)dsx=N,
sx′=Nsx+Sx,
sy′=Nsy+Sy,
sx=∥s∥sin α,sx′=∥s∥sin α′.
sin α=N sin α′.
exp[i(kxx+kyy)]exp[i(Δkxx+Δkyy)]=exp[i(kx′x+ky′y)],
kx′=kx+Δkxandky′=ky+Δky.
k=2πλsandk′=2πλs′.
sx′=sx+λ2πΔkxandsy′=sy+λ2πΔky.
Sx=λ2πΔkx,Sy=λ2πΔky.
sx2+sy2+sz2=1.
∂∂xsx2+∂∂xsy2+∂∂xsz2=2sx∂sx∂x+2sy∂sy∂x+2sz∂sz∂x=2s·(∂∂xs).
s·(∂∂xs)=0,s·(∂∂ys)=0.

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