Abstract

We introduce the concept of the inverse prism as the dual of the conventional prism and deduce from this duality an implementation of it based on temporal discontinuity and spatial dispersion provided by anisotropy. Moreover, we show that this inverse prism exhibits the following three unique properties: chromatic refraction birefringence, ordinary monochromatic and extraordinary polychromatic temporal refraction, and linear-to-Lissajous polarization transformation.

© 2018 Optical Society of America

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References

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  1. I. Newton, Opticks: Or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light (William and John Innys, 1721).
  2. “Uniaxial inverse prism animation,” https://goo.gl/Z771np .
  3. J. Mendonça, A. Guerreiro, and A. M. Martins, Phys. Rev. A 62, 033805 (2000).
    [Crossref]
  4. J. T. Mendonça, Theory of Photon Acceleration (CRC Press, 2000).
  5. D. K. Kalluri, Electromagnetics of Time Varying Complex Media: Frequency and Polarization Transformer, 2nd ed. (CRC Press, 2010).
  6. A. B. Shvartsburg, Phys. Usp. 48, 797 (2005).
    [Crossref]
  7. D. Kaup, A. Reiman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
    [Crossref]
  8. E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 90, 203904 (2003).
    [Crossref]
  9. E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
    [Crossref]
  10. S. Taravati and C. Caloz, IEEE Trans. Antennas Propag. 65, 442 (2017).
    [Crossref]
  11. F. R. Morgenthaler, IRE Trans. Microw. Theory Tech. 6, 167 (1958).
    [Crossref]
  12. L. Felsen and G. Whitman, IEEE Trans. Antennas Propag. 18, 242 (1970).
    [Crossref]
  13. R. L. Fante, IEEE Trans. Antennas Propag. 19, 417 (1971).
    [Crossref]
  14. Y. Xiao, D. N. Maywar, and G. P. Agrawal, J. Opt. Soc. Am. B 28, 1685 (2011).
    [Crossref]
  15. Y. Xiao, G. P. Agrawal, and D. N. Maywar, Opt. Lett. 36, 505 (2011).
    [Crossref]
  16. Y. Xiao, D. N. Maywar, and G. P. Agrawal, Opt. Lett. 39, 574 (2014).
    [Crossref]
  17. B. Plansinis, W. Donaldson, and G. Agrawal, Phys. Rev. Lett. 115, 183901 (2015).
    [Crossref]
  18. M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
    [Crossref]
  19. N. Chamanara and C. Caloz, Phys. Rev. B 94, 075413 (2016).
    [Crossref]
  20. Y. Hadad, D. Sounas, and A. Alù, Phys. Rev. B 92, 100304 (2015).
    [Crossref]
  21. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier, 2013).
  22. J. A. Kong, Theory of Electromagnetic Waves (Wiley-Interscience, 1975).
  23. M. G. Silveirinha, Phys. Rev. B 75, 115104 (2007).
    [Crossref]
  24. I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).
  25. F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’reilly, Phys. Rev. E 75, 046607 (2007).
    [Crossref]
  26. Z.-L. Deck-Léger, A. Akbarzadeh, and C. Caloz, Phys. Rev. B 97, 104305 (2017).
  27. “Supplementary material,” https://goo.gl/c9eLRz .
  28. L. Surhone, M. Timpledon, and S. Marseken, Lissajous Curve: Mathematics, Parametric Equation, Curve, Nathaniel Bowditch, Jules Antoine Lissajous, Ellipse, Parabola, Line, Circle, Radian, Rational Number (Betascript, 2010).
  29. I. Freund, Opt. Commun. 226, 351 (2003).

2017 (2)

S. Taravati and C. Caloz, IEEE Trans. Antennas Propag. 65, 442 (2017).
[Crossref]

Z.-L. Deck-Léger, A. Akbarzadeh, and C. Caloz, Phys. Rev. B 97, 104305 (2017).

2016 (1)

N. Chamanara and C. Caloz, Phys. Rev. B 94, 075413 (2016).
[Crossref]

2015 (2)

Y. Hadad, D. Sounas, and A. Alù, Phys. Rev. B 92, 100304 (2015).
[Crossref]

B. Plansinis, W. Donaldson, and G. Agrawal, Phys. Rev. Lett. 115, 183901 (2015).
[Crossref]

2014 (1)

2011 (2)

2007 (2)

M. G. Silveirinha, Phys. Rev. B 75, 115104 (2007).
[Crossref]

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’reilly, Phys. Rev. E 75, 046607 (2007).
[Crossref]

2005 (1)

A. B. Shvartsburg, Phys. Usp. 48, 797 (2005).
[Crossref]

2004 (1)

M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
[Crossref]

2003 (3)

I. Freund, Opt. Commun. 226, 351 (2003).

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 90, 203904 (2003).
[Crossref]

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
[Crossref]

2000 (1)

J. Mendonça, A. Guerreiro, and A. M. Martins, Phys. Rev. A 62, 033805 (2000).
[Crossref]

1979 (1)

D. Kaup, A. Reiman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

1971 (1)

R. L. Fante, IEEE Trans. Antennas Propag. 19, 417 (1971).
[Crossref]

1970 (1)

L. Felsen and G. Whitman, IEEE Trans. Antennas Propag. 18, 242 (1970).
[Crossref]

1958 (1)

F. R. Morgenthaler, IRE Trans. Microw. Theory Tech. 6, 167 (1958).
[Crossref]

Agrawal, G.

B. Plansinis, W. Donaldson, and G. Agrawal, Phys. Rev. Lett. 115, 183901 (2015).
[Crossref]

Agrawal, G. P.

Akbarzadeh, A.

Z.-L. Deck-Léger, A. Akbarzadeh, and C. Caloz, Phys. Rev. B 97, 104305 (2017).

Alù, A.

Y. Hadad, D. Sounas, and A. Alù, Phys. Rev. B 92, 100304 (2015).
[Crossref]

Amann, A.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’reilly, Phys. Rev. E 75, 046607 (2007).
[Crossref]

Bers, A.

D. Kaup, A. Reiman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

Biancalana, F.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’reilly, Phys. Rev. E 75, 046607 (2007).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier, 2013).

Caloz, C.

S. Taravati and C. Caloz, IEEE Trans. Antennas Propag. 65, 442 (2017).
[Crossref]

Z.-L. Deck-Léger, A. Akbarzadeh, and C. Caloz, Phys. Rev. B 97, 104305 (2017).

N. Chamanara and C. Caloz, Phys. Rev. B 94, 075413 (2016).
[Crossref]

Chamanara, N.

N. Chamanara and C. Caloz, Phys. Rev. B 94, 075413 (2016).
[Crossref]

Deck-Léger, Z.-L.

Z.-L. Deck-Léger, A. Akbarzadeh, and C. Caloz, Phys. Rev. B 97, 104305 (2017).

Donaldson, W.

B. Plansinis, W. Donaldson, and G. Agrawal, Phys. Rev. Lett. 115, 183901 (2015).
[Crossref]

Fan, S.

M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
[Crossref]

Fante, R. L.

R. L. Fante, IEEE Trans. Antennas Propag. 19, 417 (1971).
[Crossref]

Felsen, L.

L. Felsen and G. Whitman, IEEE Trans. Antennas Propag. 18, 242 (1970).
[Crossref]

Freund, I.

I. Freund, Opt. Commun. 226, 351 (2003).

Guerreiro, A.

J. Mendonça, A. Guerreiro, and A. M. Martins, Phys. Rev. A 62, 033805 (2000).
[Crossref]

Hadad, Y.

Y. Hadad, D. Sounas, and A. Alù, Phys. Rev. B 92, 100304 (2015).
[Crossref]

Joannopoulos, J. D.

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
[Crossref]

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 90, 203904 (2003).
[Crossref]

Kalluri, D. K.

D. K. Kalluri, Electromagnetics of Time Varying Complex Media: Frequency and Polarization Transformer, 2nd ed. (CRC Press, 2010).

Kaup, D.

D. Kaup, A. Reiman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

Kong, J. A.

J. A. Kong, Theory of Electromagnetic Waves (Wiley-Interscience, 1975).

Lindell, I. V.

I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Marseken, S.

L. Surhone, M. Timpledon, and S. Marseken, Lissajous Curve: Mathematics, Parametric Equation, Curve, Nathaniel Bowditch, Jules Antoine Lissajous, Ellipse, Parabola, Line, Circle, Radian, Rational Number (Betascript, 2010).

Martins, A. M.

J. Mendonça, A. Guerreiro, and A. M. Martins, Phys. Rev. A 62, 033805 (2000).
[Crossref]

Maywar, D. N.

Mendonça, J.

J. Mendonça, A. Guerreiro, and A. M. Martins, Phys. Rev. A 62, 033805 (2000).
[Crossref]

Mendonça, J. T.

J. T. Mendonça, Theory of Photon Acceleration (CRC Press, 2000).

Morgenthaler, F. R.

F. R. Morgenthaler, IRE Trans. Microw. Theory Tech. 6, 167 (1958).
[Crossref]

Newton, I.

I. Newton, Opticks: Or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light (William and John Innys, 1721).

O’reilly, E. P.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’reilly, Phys. Rev. E 75, 046607 (2007).
[Crossref]

Plansinis, B.

B. Plansinis, W. Donaldson, and G. Agrawal, Phys. Rev. Lett. 115, 183901 (2015).
[Crossref]

Reed, E. J.

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
[Crossref]

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 90, 203904 (2003).
[Crossref]

Reiman, A.

D. Kaup, A. Reiman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

Shvartsburg, A. B.

A. B. Shvartsburg, Phys. Usp. 48, 797 (2005).
[Crossref]

Sihvola, A.

I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Silveirinha, M. G.

M. G. Silveirinha, Phys. Rev. B 75, 115104 (2007).
[Crossref]

Soljacic, M.

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 90, 203904 (2003).
[Crossref]

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
[Crossref]

Sounas, D.

Y. Hadad, D. Sounas, and A. Alù, Phys. Rev. B 92, 100304 (2015).
[Crossref]

Surhone, L.

L. Surhone, M. Timpledon, and S. Marseken, Lissajous Curve: Mathematics, Parametric Equation, Curve, Nathaniel Bowditch, Jules Antoine Lissajous, Ellipse, Parabola, Line, Circle, Radian, Rational Number (Betascript, 2010).

Taravati, S.

S. Taravati and C. Caloz, IEEE Trans. Antennas Propag. 65, 442 (2017).
[Crossref]

Timpledon, M.

L. Surhone, M. Timpledon, and S. Marseken, Lissajous Curve: Mathematics, Parametric Equation, Curve, Nathaniel Bowditch, Jules Antoine Lissajous, Ellipse, Parabola, Line, Circle, Radian, Rational Number (Betascript, 2010).

Tretyakov, S.

I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Uskov, A. V.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’reilly, Phys. Rev. E 75, 046607 (2007).
[Crossref]

Viitanen, A.

I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Whitman, G.

L. Felsen and G. Whitman, IEEE Trans. Antennas Propag. 18, 242 (1970).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier, 2013).

Xiao, Y.

Yanik, M. F.

M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
[Crossref]

IEEE Trans. Antennas Propag. (3)

L. Felsen and G. Whitman, IEEE Trans. Antennas Propag. 18, 242 (1970).
[Crossref]

R. L. Fante, IEEE Trans. Antennas Propag. 19, 417 (1971).
[Crossref]

S. Taravati and C. Caloz, IEEE Trans. Antennas Propag. 65, 442 (2017).
[Crossref]

IRE Trans. Microw. Theory Tech. (1)

F. R. Morgenthaler, IRE Trans. Microw. Theory Tech. 6, 167 (1958).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

I. Freund, Opt. Commun. 226, 351 (2003).

Opt. Lett. (2)

Phys. Rev. A (1)

J. Mendonça, A. Guerreiro, and A. M. Martins, Phys. Rev. A 62, 033805 (2000).
[Crossref]

Phys. Rev. B (4)

N. Chamanara and C. Caloz, Phys. Rev. B 94, 075413 (2016).
[Crossref]

Y. Hadad, D. Sounas, and A. Alù, Phys. Rev. B 92, 100304 (2015).
[Crossref]

Z.-L. Deck-Léger, A. Akbarzadeh, and C. Caloz, Phys. Rev. B 97, 104305 (2017).

M. G. Silveirinha, Phys. Rev. B 75, 115104 (2007).
[Crossref]

Phys. Rev. E (1)

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’reilly, Phys. Rev. E 75, 046607 (2007).
[Crossref]

Phys. Rev. Lett. (4)

B. Plansinis, W. Donaldson, and G. Agrawal, Phys. Rev. Lett. 115, 183901 (2015).
[Crossref]

M. F. Yanik and S. Fan, Phys. Rev. Lett. 93, 173903 (2004).
[Crossref]

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 90, 203904 (2003).
[Crossref]

E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
[Crossref]

Phys. Usp. (1)

A. B. Shvartsburg, Phys. Usp. 48, 797 (2005).
[Crossref]

Rev. Mod. Phys. (1)

D. Kaup, A. Reiman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

Other (9)

J. T. Mendonça, Theory of Photon Acceleration (CRC Press, 2000).

D. K. Kalluri, Electromagnetics of Time Varying Complex Media: Frequency and Polarization Transformer, 2nd ed. (CRC Press, 2010).

I. Newton, Opticks: Or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light (William and John Innys, 1721).

“Uniaxial inverse prism animation,” https://goo.gl/Z771np .

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier, 2013).

J. A. Kong, Theory of Electromagnetic Waves (Wiley-Interscience, 1975).

“Supplementary material,” https://goo.gl/c9eLRz .

L. Surhone, M. Timpledon, and S. Marseken, Lissajous Curve: Mathematics, Parametric Equation, Curve, Nathaniel Bowditch, Jules Antoine Lissajous, Ellipse, Parabola, Line, Circle, Radian, Rational Number (Betascript, 2010).

I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

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Figures (4)

Fig. 1.
Fig. 1. Comparison of the conventional prism and the inverse prism, formed by the refractive indices n 1 and n 2 with n 2 > n 1 . (a) Conventional prism, decomposing white light into its constituent colors, using spatial discontinuity [ n = n ( r ) ] and temporal dispersion [ n 2 = n 2 ( ω ) ]. (b) Corresponding mapping from temporal frequencies ( ω ) to spatial frequencies ( k ). (c) Inverse prism, transforming multidirectional light into direction-dependent new colors, using temporal discontinuity [ n = n ( t ) ] and spatial dispersion [ n 2 = n 2 ( k ) ] (animation available in Ref. [2]). (d) Corresponding mapping from spatial to temporal frequencies.
Fig. 2.
Fig. 2. Implementation of the inverse prism in Figs. 1(c) and 1(d), as the time-varying medium described by Eq. (1). (a) Space–time representation of the inverse prism, whose anisotropy, here uniaxial, induces birefringence, with inset showing the x z space, the s , p polarizations, and the angle of wave propagation θ . (b) Circular (isotropic) and elliptical (anisotropic) isofrequency curves corresponding to Eq. (3) and Eq. (6), respectively, for n 1 < n < n .
Fig. 3.
Fig. 3. Temporal frequency birefringence of the uniaxial inverse prism of Fig. 2. In a hybrid wave, the s -polarized and p -polarized parts experience ordinary (monochromatic) and extraordinary (polychromatic) temporal refraction, respectively.
Fig. 4.
Fig. 4. Lissajous polarization cycle of the inverse prism of Fig. 2 for two different sets of parameters. (a)  E y = 2 , E z = 1 , ω 1 / c = 10 , n 1 = 1 , n = 5 , n = 2 , ϕ y = π / 8 , and ϕ z = 0 . (b)  E y = 10 , E z = 20 , ω 1 / c = 16 , n 1 = 1 , n = 8 , n = 4 , ϕ y = π / 18 , and ϕ z = π / 6 .

Equations (17)

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n ( t ) = n 1 u ( t 0 t ) + n ¯ ¯ 2 u ( t t 0 ) ,
k 1 = n 1 ω 1 c = k x 2 + k z 2 or ω 1 = c n 1 k x 2 + k z 2 ,
k s = n ω s c = ± k x 2 + k z 2       or       ω s = ± c n k x 2 + k z 2 ,
T s = n 1 ( n + n 1 ) 2 n 2 ,
R s = n 1 ( n n 1 ) 2 n 2 .
n 2 n 2 k 0 2 = n 2 n 2 ( ω p c ) 2 = n 2 k x 2 + n 2 k z 2 ,
ω p = ω = ± ( c n ) 2 k x 2 + ( c n ) 2 k z 2 ,
T p = n 1 2 n 4 k z 2 + n 4 k x 2 n 2 n 2 k x 2 + k z 2 ( ω 1 + ω p 2 ω p ) ,
R p = n 1 2 n 4 k z 2 + n 4 k x 2 n 2 n 2 k x 2 + k z 2 ( ω 1 ω p 2 ω p ) .
n 1 ω 1 = ± n ω s .
n 1 ω 1 = ± n n n 2 cos 2 ( θ ) + n 2 sin 2 ( θ ) ω p = n ( k ) ω p ,
E 1 ( x , t ) = e y E y cos ( k x x ω 1 t ) + e z E z cos ( k x x ω 1 t ) .
E 2 + ( x , t ) = e y T s E y cos [ k x x ω s ( t t 0 ) ω 1 t 0 ] + e z T p E z cos [ k x x ω p ( t t 0 ) ω 1 t 0 ] ,
E 2 + ( x , t ) = e y A cos ( ω s t + ϕ y ) + e z B cos ( ω p t + ϕ z ) ,
A = T s E y , B = T p E z ,
ϕ y = k x x ( ω s ω 1 ) t 0 , ϕ z = k x x ( ω p ω 1 ) t 0 .
E 2 y + ( t ) = A cos ( ω s t + ϕ y ) , E 2 z + ( t ) = B cos ( ω p t + ϕ z ) ,

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