Abstract

A fundamental challenge has plagued computer-generated volumetric holography since its inception: design methods are available only in the perturbative limit, but this poses serious limitations on efficiency and the amount of multiplexing achievable. Given the recent progress in highly tailorable artificial media, such as metamaterials, the need for general and robust design techniques grows. We present a method based on the electromagnetic variational principle that applies to media that can be described as collections of point dipoles, as most metamaterials are. We demonstrate its efficacy by designing highly efficient, non-perturbative, multiplexing devices.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

2017 (1)

L. Pulido-Mancera, P. T. Bowen, M. F. Imani, N. Kundtz, and D. Smith, “Polarizability extraction of complementary metamaterial elements in waveguides for aperture modeling,” Phys. Rev. B 96, 235402 (2017).
[Crossref]

2015 (1)

2013 (2)

2010 (2)

D. R. Smith, “Analytic expressions for the constitutive parameters of magnetoelectric metamaterials,” Phys. Rev. E 81, 036605 (2010).
[Crossref]

T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics 4, 188 (2010).
[Crossref]

2007 (1)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41 (2007).
[Crossref]

2006 (1)

2005 (2)

2002 (2)

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

N. Garcia, E. V. Ponizovskaya, and J. Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett. 80, 1120–1122 (2002).
[Crossref]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

2000 (2)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933 (2000).
[Crossref] [PubMed]

1994 (1)

1993 (1)

D. Brady and D. Psaltis, “Information capacity of 3-D holographic data storage,” Opt. Quantum Electron. 25, S597–S610 (1993).
[Crossref]

1992 (2)

D. Brady and D. Psaltis, “Control of volume holograms,” J. Opt. Soc. Am. A 9, 1167–1182 (1992).
[Crossref]

A. Lakhtakia, “General theory of the Purcell-Pennypacker scattering approach and its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
[Crossref]

1991 (2)

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

1980 (1)

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microw. Theory Techn. 28, 878–886 (1980).
[Crossref]

1979 (1)

1974 (1)

J. E. Sipe and J. Van Kranendonk, “Macroscopic electromagnetic theory of resonant dielectrics,” Phys. Rev. A 9, 1806 (1974).
[Crossref]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1969 (1)

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[Crossref]

1968 (1)

1967 (1)

1964 (1)

1963 (1)

1962 (1)

1950 (1)

B. A. Lippmann and J. Schwinger, “Variational principles for scattering processes. I,” Phys. Rev. 79, 469 (1950).
[Crossref]

1926 (1)

M. Born, “Quantenmechanik der Stoßvorgänge,” Zeitschrift für Physik A 38, 803–827 (1926).
[Crossref]

Bergmann, R. B.

Bløtekjaer, K.

Born, M.

M. Born, “Quantenmechanik der Stoßvorgänge,” Zeitschrift für Physik A 38, 803–827 (1926).
[Crossref]

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light(Elsevier, 2013).

Bowen, P. T.

L. Pulido-Mancera, P. T. Bowen, M. F. Imani, N. Kundtz, and D. Smith, “Polarizability extraction of complementary metamaterial elements in waveguides for aperture modeling,” Phys. Rev. B 96, 235402 (2017).
[Crossref]

P. T. Bowen, “Metamaterials analysis, modeling, and design in the point dipole approximation,” Ph.D. thesis, Duke University, Department of Electrical and Computer Engineering (2017).

Brady, D.

D. Brady and D. Psaltis, “Information capacity of 3-D holographic data storage,” Opt. Quantum Electron. 25, S597–S610 (1993).
[Crossref]

D. Brady and D. Psaltis, “Control of volume holograms,” J. Opt. Soc. Am. A 9, 1167–1182 (1992).
[Crossref]

Brown, B. R.

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[Crossref]

Cai, W.

Chen, C. H.

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microw. Theory Techn. 28, 878–886 (1980).
[Crossref]

Chettiar, U. K.

Denisyuk, Y. N.

Y. N. Denisyuk, “On the reflection of optical properties of an object in a wave field of light scattered by it,” Dokl. Akad. Nauk SSSR pp. 1275–1278 (1962).

Drachev, V. P.

Draine, B. T.

Enoch, S.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Falldorf, C.

Flatau, P. J.

Garcia, N.

N. Garcia, E. V. Ponizovskaya, and J. Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett. 80, 1120–1122 (2002).
[Crossref]

Geissbuehler, M.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gerke, T. D.

T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics 4, 188 (2010).
[Crossref]

Goodman, J. J.

Guérin, N.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Gülses, A. A.

Heine, J.

Hong, J. H.

Imani, M. F.

L. Pulido-Mancera, P. T. Bowen, M. F. Imani, N. Kundtz, and D. Smith, “Polarizability extraction of complementary metamaterial elements in waveguides for aperture modeling,” Phys. Rev. B 96, 235402 (2017).
[Crossref]

Jameson, A.

A. Jameson, Computational Fluid Dynamics Review(John Wiley & Sons, 1995), chap. Optimum aerodynamic design using control theory, pp. 495–528.

Jenkins, B. K.

Kamau, E. N.

Kildishev, A. V.

Kivshar, Y. S.

Kroll, N.

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933 (2000).
[Crossref] [PubMed]

Kundtz, N.

L. Pulido-Mancera, P. T. Bowen, M. F. Imani, N. Kundtz, and D. Smith, “Polarizability extraction of complementary metamaterial elements in waveguides for aperture modeling,” Phys. Rev. B 96, 235402 (2017).
[Crossref]

Lakhtakia, A.

A. Lakhtakia, “General theory of the Purcell-Pennypacker scattering approach and its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media (Pergamon Press, 1984), chap. 8, pp. 39–40.

Lasser, T.

Leith, E. N.

Lien, C.-D.

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microw. Theory Techn. 28, 878–886 (1980).
[Crossref]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media (Pergamon Press, 1984), chap. 8, pp. 39–40.

Lippmann, B. A.

B. A. Lippmann and J. Schwinger, “Variational principles for scattering processes. I,” Phys. Rev. 79, 469 (1950).
[Crossref]

Lohmann, A. W.

Marks, D. L.

Morrison, S. K.

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Paris, D. P.

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Piestun, R.

T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics 4, 188 (2010).
[Crossref]

Ponizovskaya, E. V.

N. Garcia, E. V. Ponizovskaya, and J. Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett. 80, 1120–1122 (2002).
[Crossref]

Psaltis, D.

D. Brady and D. Psaltis, “Information capacity of 3-D holographic data storage,” Opt. Quantum Electron. 25, S597–S610 (1993).
[Crossref]

D. Brady and D. Psaltis, “Control of volume holograms,” J. Opt. Soc. Am. A 9, 1167–1182 (1992).
[Crossref]

Pulido-Mancera, L.

L. Pulido-Mancera, P. T. Bowen, M. F. Imani, N. Kundtz, and D. Smith, “Polarizability extraction of complementary metamaterial elements in waveguides for aperture modeling,” Phys. Rev. B 96, 235402 (2017).
[Crossref]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Russer, P.

K. F. Warnick and P. Russer, “Green’s theorem in electromagnetic field theory,” in “Proceedings of the European Microwave Association,” , vol. 12 (2006), 141–146.

Sabouroux, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics(Addison-Wesley, 1994).

Sarychev, A. K.

Saxena, R.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Schwinger, J.

B. A. Lippmann and J. Schwinger, “Variational principles for scattering processes. I,” Phys. Rev. 79, 469 (1950).
[Crossref]

Shadrivov, I. V.

Shalaev, V. M.

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Sihvola, A.

J. Venermo and A. Sihvola, “Dielectric polarizability of circular cylinder,” J. Electrostat. 63, 101–117 (2005).
[Crossref]

Sipe, J. E.

J. E. Sipe and J. Van Kranendonk, “Macroscopic electromagnetic theory of resonant dielectrics,” Phys. Rev. A 9, 1806 (1974).
[Crossref]

J. Van Kranendonk and J. E. Sipe, “Foundations of the macroscopic electromagnetic theory of dielectric media,” in “Progress in Optics,” , vol. XV (Elsevier, 1977), pp. 245–350.
[Crossref]

Smith, D.

L. Pulido-Mancera, P. T. Bowen, M. F. Imani, N. Kundtz, and D. Smith, “Polarizability extraction of complementary metamaterial elements in waveguides for aperture modeling,” Phys. Rev. B 96, 235402 (2017).
[Crossref]

Smith, D. R.

D. L. Marks and D. R. Smith, “Inverse scattering with a non self-adjoint variational formulation,” Opt. Express 26, 7655–7671 (2018).
[Crossref] [PubMed]

D. L. Marks and D. R. Smith, “Linear solutions to metamaterial volume hologram design using a variational approach,” J. Opt. Soc. Am. A 35, 567–576 (2018).
[Crossref]

D. R. Smith, “Analytic expressions for the constitutive parameters of magnetoelectric metamaterials,” Phys. Rev. E 81, 036605 (2010).
[Crossref]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933 (2000).
[Crossref] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Tayeb, G.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Upatnieks, J.

van Heerden, P. J.

Van Kranendonk, J.

J. E. Sipe and J. Van Kranendonk, “Macroscopic electromagnetic theory of resonant dielectrics,” Phys. Rev. A 9, 1806 (1974).
[Crossref]

J. Van Kranendonk and J. E. Sipe, “Foundations of the macroscopic electromagnetic theory of dielectric media,” in “Progress in Optics,” , vol. XV (Elsevier, 1977), pp. 245–350.
[Crossref]

Venermo, J.

J. Venermo and A. Sihvola, “Dielectric polarizability of circular cylinder,” J. Electrostat. 63, 101–117 (2005).
[Crossref]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Vincent, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Warnick, K. F.

K. F. Warnick and P. Russer, “Green’s theorem in electromagnetic field theory,” in “Proceedings of the European Microwave Association,” , vol. 12 (2006), 141–146.

Wolf, E.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light(Elsevier, 2013).

Xiao, J. Q.

N. Garcia, E. V. Ponizovskaya, and J. Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett. 80, 1120–1122 (2002).
[Crossref]

Yuan, H.-K.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

N. Garcia, E. V. Ponizovskaya, and J. Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett. 80, 1120–1122 (2002).
[Crossref]

Astrophys. J. (3)

A. Lakhtakia, “General theory of the Purcell-Pennypacker scattering approach and its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
[Crossref]

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

IBM J. Res. Dev. (1)

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[Crossref]

IEEE Trans. Microw. Theory Techn. (1)

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Supplementary Material (3)

NameDescription
» Visualization 1       Animation of simplex device design. Left: map of intrinsic electric polarizabilities. Right: overlap of field intensities for electric field and its adjoint.
» Visualization 2       Animation of triplex device design. Top left: map of intrinsic electric polarizabilities. Remaining panels: overlap of field intensities for electric field and respective adjoint couples.
» Visualization 3       Animation of triplex device design with random inclusion. Top left: map of intrinsic electric polarizabilities. The inclusion is highlighted by the solid white line box. Remaining panels: overlap of field intensities for electric field and respective

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Figures (11)

Fig. 1
Fig. 1 Prescribed input (incoming from the left) and output (outgoing rightward) waves for the simplex device. The black dots represent the dipole locations.
Fig. 2
Fig. 2 Simplex device design. Left: map of intrinsic electric polarizabilities. ΔA denotes the volume (area) of the λ/4 × λ/4 unit cell. Right: overlap of field intensities for E and Ea. The iteration steps are animated in Visualization 1.
Fig. 3
Fig. 3 Average value of gradient for simplex device (left y-axis, blue continuous concave line) and generalized reaction (right y-axis, orange continuous convex line) as a function of number of design cycle iterations.
Fig. 4
Fig. 4 Operation of simplex device. Total fields (real part, left and imaginary part, right) are shown.
Fig. 5
Fig. 5 Prescribed input (incoming from the left) and output (outgoing rightward) waves for the triplex device. The black dots represent the dipole locations.
Fig. 6
Fig. 6 Triplex device design. Top left: map of intrinsic electric polarizabilities. ΔA denotes the volume (area) of the λ/4 × λ/4 unit cell. Remaining panels: overlap of field intensities for E and Ea couples. The iteration steps are animated in Visualization 2.
Fig. 7
Fig. 7 Average value of gradient for triplex design (left y-axis, blue continuous concave line) and generalized reaction (right y-axis, orange continuous convex line) as a function of number of design cycle iterations.
Fig. 8
Fig. 8 Operation of triplex device. Total fields (real part, left and imaginary part, right) are shown.
Fig. 9
Fig. 9 Triplex device design with random inclusion. Top left: map of intrinsic electric polarizabilities. ΔA denotes the volume (area) of the λ/4 × λ/4 unit cell. The inclusion is highlighted by the solid white line box. Remaining panels: overlap of field intensities for E and Ea couples. The iteration steps are animated in Visualization 3.
Fig. 10
Fig. 10 Average value of gradient for triplex device with random inclusion (left y-axis, blue continuous concave line) and generalized reaction (right y-axis, orange continuous convex line) as a function of number of design cycle iterations.
Fig. 11
Fig. 11 Operation of triplex device with random inclusion. Total fields (real part, left and imaginary part, right) are shown.

Equations (29)

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E = E inc + E scatt .
M ^ E = E inc ,
j = 1 N M i j E j = E inc , i i [ 1 , N ] , with
M i j δ i j G i j α j ,
p i α i E i .
L ^ E = s
L ^ a E a = s a .
F E a , L ^ E s a , E E a , s
L ^ diag ( α ) M ^ with s diag ( α ) E inc .
L ^ a = diag ( α ) ( I ¯ ¯ G ¯ ¯ diag ( α ) ) = diag ( α ) * ( I ¯ ¯ G ¯ ¯ * diag ( α ) * ) = L ^ * ,
L ^ E = s
L ^ * E a = s a or
L ^ E a * = s a * ,
( I ¯ ¯ G ¯ ¯ diag ( α ) ) E = E inc
( I ¯ ¯ G ¯ ¯ diag ( α ) ) E a * = E inc a * .
F T n = 1 N w w n ( E n a , L ^ E n s n a , E n E n a , s n ) ,
P ϵ 0 V n = 1 N w w n ( E n a , s n + s n a , E n ) ,
S = i ω u J E ,
η I Re ( i E a , E ) E , E E a , E a .
η i j I Re ( i E j a , E i ) E i , E i E j a , E j a
L ^ ' G ¯ ¯ 1 M ^ .
δ F T = n = 1 N w w n E n a diag ( δ α ) E n ,
δ F T = j = 1 N F T α j δ α j with
F T α j = n = 1 N w w n E n , j a * E n , j .
α F T = n = 1 N w w n E n a * E n ,
α ( p + 1 ) = α ( p ) γ E m 2 α F T ( p ) ,
α = Δ A ϵ 0 ( n 2 1 ) ,
[ R G B ] = [ 1 0 0.5 0.5 0 1 ] [ | E n | | E n a | ] .
η ¯ ¯ = ( 0.9915 0.2201 0.0098 44.5790 0.0260 111.1839 0.0038 127.1493 0.9957 0.0363 0.0060 22.2258 0.0311 130.5633 0.0074 26.2793 0.9939 0.1623 ) .

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