Abstract

Transformation optics aims to identify artificial materials and structures with desired electromagnetic properties by means of pertinent coordinate transformations. In general, such schemes are meant to appropriately tailor the constitutive parameters of metamaterials in order to control the trajectory of light in two and three dimensions. Here, we introduce a new class of one-dimensional optical transformations that exploits the mathematical framework of supersymmetry (SUSY). This systematic approach can be utilized to synthesize photonic configurations with identical reflection and transmission characteristics, down to the phase, for all incident angles, thus rendering them perfectly indistinguishable to an external observer. Along these lines, low-contrast dielectric arrangements can be designed to fully mimic the behavior of a given high-contrast structure that would have been otherwise beyond the reach of available materials and existing fabrication techniques. Similar strategies can also be adopted to replace negative-permittivity domains, thus averting unwanted optical losses.

© 2014 Optical Society of America

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  1. I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izvest. Akad. Nauk. 15, 309–360 (1951).
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    [Crossref]
  3. K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, 1977).
  4. P. Deift, E. Trubowitz, “Inverse scattering on the line,” Comm. Pure Appl. Math. 32, 121–251 (1979).
    [Crossref]
  5. I. Kay, H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. 27, 1503–1508 (1956).
    [Crossref]
  6. E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B185, 513–554 (1981).
  7. F. Cooper, A. Khare, U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
    [Crossref]
  8. M. M. Nieto, “Relationship between supersymmetry and the inverse method in quantum mechanics,” Phys. Lett. B 145, 208–210 (1984).
    [Crossref]
  9. A. Neveu, J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B31, 86–112 (1971).
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    [Crossref]
  11. J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref]
  12. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref]
  13. A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
    [Crossref]
  14. V. M. Shalaev, “Transforming light,” Science 322, 384–386 (2008).
    [Crossref]
  15. U. Leonhardt, T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
    [Crossref]
  16. H. Chen, C. T. Chan, P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
    [Crossref]
  17. A. Alù, M. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
  18. G. W. Milton, M. Briane, J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
    [Crossref]
  19. J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
    [Crossref]
  20. L. H. Gabrielli, J. Cardenas, C. B. Poitras, M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
    [Crossref]
  21. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
    [Crossref]
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    [Crossref]
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    [Crossref]
  27. M.-A. Miri, M. Heinrich, D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87, 043819 (2013).
  28. M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

2014 (1)

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

2013 (3)

D. Liu, L. H. Gabrielli, M. Lipson, S. G. Johnson, “Transformation inverse design,” Opt. Express 21, 14223–14243 (2013).
[Crossref]

M.-A. Miri, M. Heinrich, R. El-Ganainy, D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

M.-A. Miri, M. Heinrich, D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87, 043819 (2013).

2012 (1)

2011 (2)

B. Zhang, Y. Luo, X. Liu, G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
[Crossref]

A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref]

2010 (1)

H. Chen, C. T. Chan, P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[Crossref]

2009 (4)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[Crossref]

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

U. Leonhardt, T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[Crossref]

2008 (1)

V. M. Shalaev, “Transforming light,” Science 322, 384–386 (2008).
[Crossref]

2007 (1)

A. Alù, M. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).

2006 (3)

G. W. Milton, M. Briane, J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[Crossref]

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

1995 (1)

F. Cooper, A. Khare, U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
[Crossref]

1989 (1)

A. Khare, U. Sukhatme, “Phase-equivalent potentials obtained from supersymmetry,” J. Phys. A 22, 2847–2860 (1989).
[Crossref]

1984 (1)

M. M. Nieto, “Relationship between supersymmetry and the inverse method in quantum mechanics,” Phys. Lett. B 145, 208–210 (1984).
[Crossref]

1981 (1)

E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B185, 513–554 (1981).

1979 (1)

P. Deift, E. Trubowitz, “Inverse scattering on the line,” Comm. Pure Appl. Math. 32, 121–251 (1979).
[Crossref]

1977 (1)

1971 (1)

A. Neveu, J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B31, 86–112 (1971).

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

1956 (1)

I. Kay, H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. 27, 1503–1508 (1956).
[Crossref]

1951 (1)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izvest. Akad. Nauk. 15, 309–360 (1951).

Alù, A.

A. Alù, M. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).

Barbastathis, G.

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

Briane, M.

G. W. Milton, M. Briane, J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[Crossref]

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[Crossref]

Chadan, K.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, 1977).

Chan, C. T.

H. Chen, C. T. Chan, P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[Crossref]

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Chen, H.

H. Chen, C. T. Chan, P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[Crossref]

Chen, H. Y.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Christodoulides, D. N.

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

M.-A. Miri, M. Heinrich, D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87, 043819 (2013).

M.-A. Miri, M. Heinrich, R. El-Ganainy, D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

Cooper, F.

F. Cooper, A. Khare, U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
[Crossref]

Deift, P.

P. Deift, E. Trubowitz, “Inverse scattering on the line,” Comm. Pure Appl. Math. 32, 121–251 (1979).
[Crossref]

El-Ganainy, R.

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

M.-A. Miri, M. Heinrich, R. El-Ganainy, D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

Engheta, N.

A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref]

A. Alù, M. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).

Gabrielli, L. H.

D. Liu, L. H. Gabrielli, M. Lipson, S. G. Johnson, “Transformation inverse design,” Opt. Express 21, 14223–14243 (2013).
[Crossref]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[Crossref]

Gel’fand, I. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izvest. Akad. Nauk. 15, 309–360 (1951).

Han, D. Z.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Heinrich, M.

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

M.-A. Miri, M. Heinrich, D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87, 043819 (2013).

M.-A. Miri, M. Heinrich, R. El-Ganainy, D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

Hong, C. S.

Johnson, S. G.

Kay, I.

I. Kay, H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. 27, 1503–1508 (1956).
[Crossref]

Khare, A.

F. Cooper, A. Khare, U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
[Crossref]

A. Khare, U. Sukhatme, “Phase-equivalent potentials obtained from supersymmetry,” J. Phys. A 22, 2847–2860 (1989).
[Crossref]

Lai, Y.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Leonhardt, U.

U. Leonhardt, T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

Levitan, B. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izvest. Akad. Nauk. 15, 309–360 (1951).

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

Lipson, M.

D. Liu, L. H. Gabrielli, M. Lipson, S. G. Johnson, “Transformation inverse design,” Opt. Express 21, 14223–14243 (2013).
[Crossref]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[Crossref]

Liu, D.

Liu, X.

B. Zhang, Y. Luo, X. Liu, G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
[Crossref]

Luo, Y.

B. Zhang, Y. Luo, X. Liu, G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
[Crossref]

Milton, G. W.

G. W. Milton, M. Briane, J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[Crossref]

Miri, M.-A.

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

M.-A. Miri, M. Heinrich, D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87, 043819 (2013).

M.-A. Miri, M. Heinrich, R. El-Ganainy, D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

Moses, H. E.

I. Kay, H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. 27, 1503–1508 (1956).
[Crossref]

Neveu, A.

A. Neveu, J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B31, 86–112 (1971).

Ng, J.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Nieto, M. M.

M. M. Nieto, “Relationship between supersymmetry and the inverse method in quantum mechanics,” Phys. Lett. B 145, 208–210 (1984).
[Crossref]

Nolte, S.

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

Pendry, J. B.

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Philbin, T. G.

U. Leonhardt, T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[Crossref]

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[Crossref]

Sabatier, P. C.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, 1977).

Salandrino, A.

A. Alù, M. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).

Schurig, D.

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Schwarz, J. H.

A. Neveu, J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B31, 86–112 (1971).

Shalaev, V. M.

V. M. Shalaev, “Transforming light,” Science 322, 384–386 (2008).
[Crossref]

Sheng, P.

H. Chen, C. T. Chan, P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[Crossref]

Silveirinha, M.

A. Alù, M. Silveirinha, A. Salandrino, N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).

Smith, D. R.

J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Stützer, S.

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

Sukhatme, U.

F. Cooper, A. Khare, U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
[Crossref]

A. Khare, U. Sukhatme, “Phase-equivalent potentials obtained from supersymmetry,” J. Phys. A 22, 2847–2860 (1989).
[Crossref]

Sun, H.

Szameit, A.

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

Trubowitz, E.

P. Deift, E. Trubowitz, “Inverse scattering on the line,” Comm. Pure Appl. Math. 32, 121–251 (1979).
[Crossref]

Vakil, A.

A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

Willis, J. R.

G. W. Milton, M. Briane, J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[Crossref]

Witten, E.

E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B185, 513–554 (1981).

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

Xiao, J. J.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Xu, H.

Yariv, A.

Yeh, P.

Yu, T.

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

Zhang, B.

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

Zhang, Z.-Q.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[Crossref]

Comm. Pure Appl. Math. (1)

P. Deift, E. Trubowitz, “Inverse scattering on the line,” Comm. Pure Appl. Math. 32, 121–251 (1979).
[Crossref]

Izvest. Akad. Nauk. (1)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izvest. Akad. Nauk. 15, 309–360 (1951).

J. Appl. Phys. (1)

I. Kay, H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. 27, 1503–1508 (1956).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

A. Khare, U. Sukhatme, “Phase-equivalent potentials obtained from supersymmetry,” J. Phys. A 22, 2847–2860 (1989).
[Crossref]

Nat. Commun. (1)

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).

Nat. Mater. (2)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref]

H. Chen, C. T. Chan, P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[Crossref]

Nat. Photonics (1)

L. H. Gabrielli, J. Cardenas, C. B. Poitras, M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[Crossref]

New J. Phys. (1)

G. W. Milton, M. Briane, J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[Crossref]

Nucl. Phys. (2)

E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B185, 513–554 (1981).

A. Neveu, J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B31, 86–112 (1971).

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

Opt. Express (1)

Phys. Lett. B (1)

M. M. Nieto, “Relationship between supersymmetry and the inverse method in quantum mechanics,” Phys. Lett. B 145, 208–210 (1984).
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Supplementary Material (1)

» Supplement 1: PDF (1978 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Schematic overview of the different SUSY optical transformations. Starting from a given fundamental structure ϵ, supersymmetric partners ϵp can be constructed. Whereas the broken SUSY system ϵp(br) preserves all bound modes, unbroken SUSY (ϵp(ub)) removes the fundamental mode. Regardless, in both cases, the intensity reflection and transmission coefficients of the superpartners are identical to those of the fundamental system. In order to maintain the full complex scattering characteristics, a family ϵf of isophase structures can be synthesized. Finally, a hierarchical sequence of higher-order superpartners ϵp,2N(ub) may be utilized to obtain a scattering-equivalent structure, which requires a substantially lower refractive index contrast than that involved in the original system ϵ.

Fig. 2.
Fig. 2.

Relative permittivity distributions of the original and the transformed potentials. (a) The fundamental system has a step-like profile ϵ(X)=1+exp[(X/5)8]. (b) Superpartner in the unbroken SUSY regime. (c) Superpartner in the broken SUSY case. (d) Phase-equivalent structures. (e) Scattering geometry. (f–h) Superpotentials W corresponding to panels (b–d). (j) Identical reflectivity R (solid line) and transmittivity T (dashed line) corresponding to Figs. 1(a)1(d). (k–m) Relative phases of the reflection (ΔΦr, solid line) and transmission (ΔΦt, dashed) coefficients of the structures in (b–d) compared to the fundamental system (a) as a function of the incident angle θ. The scattering characteristics were evaluated by means of the differential transfer matrix method [23].

Fig. 3.
Fig. 3.

Reflection/transmission characteristics of structures obtained by SUSY transformations depicted in Fig. 2 as functions of wavelength λ and angle of incidence θ. (a–c) Intensity difference in transmission. (d–f) Relative phases in reflection and (g–j) relative phases in transmission. The dashed lines follow the phase jumps of π, which originate from the interaction with guided modes in the fundamental structure and unbroken-SUSY partner. Top row, unbroken SUSY; middle row, broken SUSY; bottom row, isophase case (C=0.5).

Fig. 4.
Fig. 4.

(a) Hypothetical high-contrast dielectric layer arrangement that supports N=9 guided modes. (b) Hierarchical sequence of partner structures obtained through iterative SUSY transformations. (c) Despite the general trend toward lower-contrast configurations, each intermediate step inherits the reflectivity and transmittivity of the fundamental system (a). (d) The resulting low-contrast structure is free of bound states and faithfully mimics the intensity-scattering characteristics of the original high-contrast configuration for all angles of incidence. Note that the transverse coordinate x scales in units of λ0/2π.

Fig. 5.
Fig. 5.

(a) Metal–dielectric grating arrangement comprising five layers of negative electrical permittivity (red sections). (b) An entirely dielectric superpartner grating constructed in the broken SUSY regime, using the respective superpotential (c). (d) Despite the absence of any metallic regions, the equivalent structure exhibits identical reflectivities/transmittivities.

Tables (1)

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Table 1. Reflection and Transmission Coefficients for the Different SUSY Transformationsa

Equations (6)

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Hψ(X)=Ωψ(X).
ϵ(X)=+WW2+α,
ϵp(X)=WW2+α.
W=Xln(ψ0)
Wf(X;C)=W+Xln(C+Xψ02(X)dX).
ϵf(X;C)=ϵ(X)+2XXln(C+Xψ02(X)dX),

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