Abstract

We derive a formal description of local light-ray rotation in terms of complex refractive indices. We show that Fermat’s principle holds, and we derive an extended Snell’s law. The change in the angle of a light ray with respect to the normal of a refractive index interface is described by the modulus of the refractive index ratio; the rotation around the interface normal is described by the argument of the refractive index ratio.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
    [CrossRef] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  3. A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. (to be published).
  4. J. Courtial and J. Nelson, New J. Phys. 10, 023028 (2008).
    [CrossRef]
  5. J. Courtial, New J. Phys. 10, 083033 (2008).
    [CrossRef]
  6. A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, arXiv:0809.2646v2 (2008).
  7. F. G. Smith and J. H. Thomson, Optics, 2nd ed. (Wiley, 1988), Chap. 6.1.
  8. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), p. 613.
  9. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.3.2.
  10. J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
    [CrossRef]

2008 (3)

J. Courtial and J. Nelson, New J. Phys. 10, 023028 (2008).
[CrossRef]

J. Courtial, New J. Phys. 10, 083033 (2008).
[CrossRef]

A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, arXiv:0809.2646v2 (2008).

2006 (1)

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

2004 (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
[CrossRef]

1988 (1)

F. G. Smith and J. H. Thomson, Optics, 2nd ed. (Wiley, 1988), Chap. 6.1.

1980 (2)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), p. 613.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.3.2.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), p. 613.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.3.2.

Courtial, J.

A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, arXiv:0809.2646v2 (2008).

J. Courtial, New J. Phys. 10, 083033 (2008).
[CrossRef]

J. Courtial and J. Nelson, New J. Phys. 10, 023028 (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. (to be published).

Hamilton, A. C.

A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, arXiv:0809.2646v2 (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. (to be published).

Nelson, J.

J. Courtial and J. Nelson, New J. Phys. 10, 023028 (2008).
[CrossRef]

A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, arXiv:0809.2646v2 (2008).

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
[CrossRef]

Smith, F. G.

F. G. Smith and J. H. Thomson, Optics, 2nd ed. (Wiley, 1988), Chap. 6.1.

Sundar, B.

A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, arXiv:0809.2646v2 (2008).

Thomson, J. H.

F. G. Smith and J. H. Thomson, Optics, 2nd ed. (Wiley, 1988), Chap. 6.1.

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), p. 613.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.3.2.

New J. Phys. (3)

A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. (to be published).

J. Courtial and J. Nelson, New J. Phys. 10, 023028 (2008).
[CrossRef]

J. Courtial, New J. Phys. 10, 083033 (2008).
[CrossRef]

Phys. Today (1)

J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
[CrossRef]

Science (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

Other (4)

A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, arXiv:0809.2646v2 (2008).

F. G. Smith and J. H. Thomson, Optics, 2nd ed. (Wiley, 1988), Chap. 6.1.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), p. 613.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 3.3.2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Geometry of refraction at a planar interface between two media with different refractive indices, n 1 and n 2 . (a) Light ray travels from a point A in front of the interface to a point P on the interface and then to a point B behind the interface. The geometrical distance between A and P is d 1 and that between P and B is d 2 . (b) Light-ray direction can be represented by two angles, θ and ϕ, which, respectively, represent the angle with respect to the interface normal (the z axis) and the angle of the projection into the interface plane with respect to the x axis. Alternatively, the light-ray direction can be described by the projection of the normalized direction vector into the interface plane. With a complex plane in the interface plane as shown, this projection can then be described by a single complex number, c.

Fig. 2
Fig. 2

Plots of the complex numbers c 1 and c 2 representing various types of refraction. (a) and (b) Examples of standard refraction ( n 1 n 2 real and positive). In (a) n 1 n 2 > 1 , and in (b) n 1 n 2 < 1 . (c) Complex refractive index ratio n 1 n 2 = exp ( i α ) leads to local light-ray rotation through an angle α (here α = 90 ° ). (d) Light-ray rotation through 180° is equivalent to negative refraction with a refractive index ratio n 1 n 2 = 1 .

Equations (19)

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

Δ = n 1 d 1 + n 2 d 2
= n 1 x 1 2 + y 1 2 + z 1 2 + n 2 x 2 2 + y 2 2 + z 2 2 ,
X = x 1 + x 2 , Y = y 1 + y 2 ,
Δ = n 1 x 1 2 + y 1 2 + z 1 2 + n 2 ( X x 1 ) 2 + ( Y y 1 ) 2 + z 2 2 .
Δ x 1 = 0 , Δ y 1 = 0 .
n 1 x 1 r 1 n 2 x 2 r 2 = 0 , n 1 y 1 r 1 n 2 y 2 r 2 = 0 ,
x j r j = sin θ j cos ϕ j , y j r j = sin θ j sin ϕ j .
n 1 sin θ 1 exp ( i ϕ 1 ) = n 2 sin θ 2 exp ( i ϕ 2 ) .
c = sin θ exp ( i ϕ ) .
n 1 c 1 = n 2 c 2 ,
n 1 n 2 = c 2 c 1 .
n 1 sin θ 1 = n 2 sin θ 2 .
n 1 n 2 = exp ( i α )
c 2 c 1 = exp ( i α ) .
n 1 n 2 = n 1 n 2 exp [ i arg ( n 1 n 2 ) ]
sin θ 2 exp ( i ϕ 2 ) sin θ 1 exp ( i ϕ 1 ) = sin θ 2 exp [ i ( ϕ 1 + α ) ] sin θ 1 exp ( i ϕ 1 ) = sin θ 2 sin θ 1 exp ( i α ) ,
n 1 n 2 = sin θ 2 sin θ 1 ,
arg ( n 1 n 2 ) = α .
c = sin ( θ ) exp ( i ϕ ) = sin θ exp [ i ( ϕ + 180 ° ) ] .

Metrics