Abstract

Single-photon and correlated two-photon sources are important elements for optical information systems. Nonlinear downconversion light sources are robust and stable emitters of single photons and entangled photon pairs. However, the rate of downconverted light emission, dictated by the properties of low-symmetry nonlinear crystals, is typically very small, leading to significant constraints in device design and integration. In this Letter, we study principles of spontaneous emission control (i.e., the Purcell effect) generalized to describe the enhancement of nonlinear generation of quantum light through spontaneous parametric downconversion. We develop a theoretical framework based on eigenmode analysis to study quantum nonlinear emission in a general anisotropic, dispersive, and lossy media. Our theory provides an unprecedented insight into the emission process. We find that spontaneous parametric downconversion in a media with hyperbolic dispersion is broadband and phase-mismatch-free. We further predict a significant enhancement of the downconverted emission rate in experimentally realistic nanostructures. Our theoretical formalism and approach to Purcell enhancement of nonlinear optical processes provides a framework for description of quantum nonlinear optical phenomena in complex nanophotonic structures.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, Laser Photon. Rev. 12, 1700189 (2018).

2016 (3)

M. Z. Alam, I. de Leon, and R. W. Boyd, Science 352, 795 (2016).
[Crossref]

Q. Li, M. Davanco, and K. Srinivasan, Nat. Photonics 10, 406 (2016).
[Crossref]

A. N. Poddubny, I. V. Iorsh, and A. A. Sukhorukov, Phys. Rev. Lett. 117, 123901 (2016).
[Crossref]

2015 (6)

E. E. Narimanov and A. V. Kildishev, Nat. Photonics 9, 214 (2015).
[Crossref]

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, Nature 522, 192 (2015).
[Crossref]

Y. Li, S. Kita, P. Munoz, O. Reshef, D. I. Vulis, M. Yin, M. Loncar, and E. Mazur, Nat. Photonics 9, 738 (2015).
[Crossref]

D. Grassani, S. Azzini, M. Liscidini, M. Galli, M. J. Strain, M. Sorel, J. E. Sipe, and D. Bajoni, Optica 2, 88 (2015).
[Crossref]

C. Duncan, L. Perret, S. Palomba, M. Lapine, B. T. Kuhlmey, and C. M. de Sterke, Sci. Rep. 5, 8983 (2015).
[Crossref]

P. Lodahl, S. Mahmoodian, and S. Stobbe, Rev. Mod. Phys. 87, 347 (2015).
[Crossref]

2014 (3)

J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, Nat. Photonics 8, 104 (2014).
[Crossref]

A. M. Mahmoud and N. Engheta, Nat. Commun. 5, 5638 (2014).
[Crossref]

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, Nature 511, 65 (2014).
[Crossref]

2013 (1)

M. Fortsch, J. U. Fürst, C. Wittmann, D. Strekalov, A. Aiello, M. V. Chekhova, C. Silberhorn, G. Leuchs, and C. Marquardt, Nat. Commun. 4, 1818 (2013).
[Crossref]

2012 (4)

A. Aspuru-Guzik and P. Walther, Nat. Phys. 8, 285 (2012).
[Crossref]

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[Crossref]

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, Science 336, 205 (2012).
[Crossref]

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, J. Opt. 14, 063001 (2012).
[Crossref]

2011 (1)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref]

2007 (2)

M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, Opt. Express 15, 7479 (2007).
[Crossref]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[Crossref]

2002 (1)

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, Eur. Phys. J. D 18, 155 (2002).
[Crossref]

2001 (1)

R. Matsushima, N. Tanaka, O. Sugihara, and N. Okamoto, Chem. Lett. 30, 200 (2001).
[Crossref]

1995 (1)

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, IEEE J. Quantum Electron. 31, 769 (1995).
[Crossref]

1992 (1)

1982 (1)

A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982).

1976 (1)

M. M. Choy and R. L. Byer, Phys. Rev. B 14, 1693 (1976).
[Crossref]

Abolghasem, P.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[Crossref]

Aiello, A.

M. Fortsch, J. U. Fürst, C. Wittmann, D. Strekalov, A. Aiello, M. V. Chekhova, C. Silberhorn, G. Leuchs, and C. Marquardt, Nat. Commun. 4, 1818 (2013).
[Crossref]

Alam, M. Z.

M. Z. Alam, I. de Leon, and R. W. Boyd, Science 352, 795 (2016).
[Crossref]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[Crossref]

Alù, A.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, Nature 511, 65 (2014).
[Crossref]

Amann, M.-C.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, Nature 511, 65 (2014).
[Crossref]

Argyropoulos, C.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, Nature 511, 65 (2014).
[Crossref]

Aspect, A.

A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982).

Aspuru-Guzik, A.

A. Aspuru-Guzik and P. Walther, Nat. Phys. 8, 285 (2012).
[Crossref]

Atwater, H. A.

R. Sokhoyan and H. A. Atwater, “Cooperative behavior of quantum dipole emitters coupled to a zero-index nanoscale waveguide,” arXiv: 1510.07071 (2015).

Azzini, S.

Bajoni, D.

Baldi, P.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, Eur. Phys. J. D 18, 155 (2002).
[Crossref]

Battle, P.

Beausoleil, R. G.

Belkin, M. A.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, Nature 511, 65 (2014).
[Crossref]

Bierlein, J. D.

Bijlani, B. J.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[Crossref]

Boehm, G.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, Nature 511, 65 (2014).
[Crossref]

Bonneau, D.

J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, Nat. Photonics 8, 104 (2014).
[Crossref]

Boyd, R. W.

M. Z. Alam, I. de Leon, and R. W. Boyd, Science 352, 795 (2016).
[Crossref]

Byer, R. L.

M. M. Choy and R. L. Byer, Phys. Rev. B 14, 1693 (1976).
[Crossref]

Chakmakjian, S. H.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, IEEE J. Quantum Electron. 31, 769 (1995).
[Crossref]

Chekhova, M. V.

M. Fortsch, J. U. Fürst, C. Wittmann, D. Strekalov, A. Aiello, M. V. Chekhova, C. Silberhorn, G. Leuchs, and C. Marquardt, Nat. Commun. 4, 1818 (2013).
[Crossref]

Chen, P.-Y.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, Nature 511, 65 (2014).
[Crossref]

Cheung, E. C.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, IEEE J. Quantum Electron. 31, 769 (1995).
[Crossref]

Choy, M. M.

M. M. Choy and R. L. Byer, Phys. Rev. B 14, 1693 (1976).
[Crossref]

Cortes, C. L.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, J. Opt. 14, 063001 (2012).
[Crossref]

Dalibard, J.

A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982).

Davanco, M.

Q. Li, M. Davanco, and K. Srinivasan, Nat. Photonics 10, 406 (2016).
[Crossref]

de Leon, I.

M. Z. Alam, I. de Leon, and R. W. Boyd, Science 352, 795 (2016).
[Crossref]

de Leon, N. P.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, Nature 522, 192 (2015).
[Crossref]

De Riedmatten, H.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, Eur. Phys. J. D 18, 155 (2002).
[Crossref]

de Sterke, C. M.

C. Duncan, L. Perret, S. Palomba, M. Lapine, B. T. Kuhlmey, and C. M. de Sterke, Sci. Rep. 5, 8983 (2015).
[Crossref]

DeMicheli, M.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, Eur. Phys. J. D 18, 155 (2002).
[Crossref]

Demmerle, F.

J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, Nature 511, 65 (2014).
[Crossref]

Devlin, R. C.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, Nature 522, 192 (2015).
[Crossref]

Dibos, A.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, Nature 522, 192 (2015).
[Crossref]

Duncan, C.

C. Duncan, L. Perret, S. Palomba, M. Lapine, B. T. Kuhlmey, and C. M. de Sterke, Sci. Rep. 5, 8983 (2015).
[Crossref]

Eisaman, M. D.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref]

Engheta, N.

A. M. Mahmoud and N. Engheta, Nat. Commun. 5, 5638 (2014).
[Crossref]

Ezaki, M.

J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, Nat. Photonics 8, 104 (2014).
[Crossref]

Fan, J.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, Rev. Sci. Instrum. 82, 071101 (2011).
[Crossref]

Fiorentino, M.

Fortsch, M.

M. Fortsch, J. U. Fürst, C. Wittmann, D. Strekalov, A. Aiello, M. V. Chekhova, C. Silberhorn, G. Leuchs, and C. Marquardt, Nat. Commun. 4, 1818 (2013).
[Crossref]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[Crossref]

Fürst, J. U.

M. Fortsch, J. U. Fürst, C. Wittmann, D. Strekalov, A. Aiello, M. V. Chekhova, C. Silberhorn, G. Leuchs, and C. Marquardt, Nat. Commun. 4, 1818 (2013).
[Crossref]

Galli, M.

Ginzburg, P.

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, Laser Photon. Rev. 12, 1700189 (2018).

Gisin, N.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, Eur. Phys. J. D 18, 155 (2002).
[Crossref]

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[Crossref]

Grassani, D.

Hadfield, R. H.

J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, Nat. Photonics 8, 104 (2014).
[Crossref]

Helmy, A. S.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[Crossref]

High, A. A.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, Nature 522, 192 (2015).
[Crossref]

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[Crossref]

Horn, R.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[Crossref]

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[Crossref]

Iizuka, N.

J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, Nat. Photonics 8, 104 (2014).
[Crossref]

Iorsh, I. V.

A. N. Poddubny, I. V. Iorsh, and A. A. Sukhorukov, Phys. Rev. Lett. 117, 123901 (2016).
[Crossref]

Jacob, Z.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, Science 336, 205 (2012).
[Crossref]

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, J. Opt. 14, 063001 (2012).
[Crossref]

Kang, D.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, Phys. Rev. Lett. 108, 153605 (2012).
[Crossref]

Kildishev, A. V.

E. E. Narimanov and A. V. Kildishev, Nat. Photonics 9, 214 (2015).
[Crossref]

Kita, S.

Y. Li, S. Kita, P. Munoz, O. Reshef, D. I. Vulis, M. Yin, M. Loncar, and E. Mazur, Nat. Photonics 9, 738 (2015).
[Crossref]

Koch, K.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, IEEE J. Quantum Electron. 31, 769 (1995).
[Crossref]

Krasavin, A. V.

G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, Laser Photon. Rev. 12, 1700189 (2018).

Kretzschmar, I.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, Science 336, 205 (2012).
[Crossref]

Krishnamoorthy, H. N. S.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, Science 336, 205 (2012).
[Crossref]

Kuhlmey, B. T.

C. Duncan, L. Perret, S. Palomba, M. Lapine, B. T. Kuhlmey, and C. M. de Sterke, Sci. Rep. 5, 8983 (2015).
[Crossref]

Lapine, M.

C. Duncan, L. Perret, S. Palomba, M. Lapine, B. T. Kuhlmey, and C. M. de Sterke, Sci. Rep. 5, 8983 (2015).
[Crossref]

Lee, J.

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M. Fortsch, J. U. Fürst, C. Wittmann, D. Strekalov, A. Aiello, M. V. Chekhova, C. Silberhorn, G. Leuchs, and C. Marquardt, Nat. Commun. 4, 1818 (2013).
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G. Marino, P. Segovia, A. V. Krasavin, P. Ginzburg, N. Olivier, G. A. Wurtz, and A. V. Zayats, Laser Photon. Rev. 12, 1700189 (2018).

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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Purcell enhancement of quantum nonlinear light generation with metamaterials. (a) An excited two-level system in free space decays by spontaneous emission. This process may be enhanced by modifying the emitter coupling to the photonic environment, (b). (c) Light may also be spontaneously emitted within a nonlinear crystal, when pump photons spontaneously fission, creating quantum-correlated signal-idler photon pairs. Nonlinear generation depends on the density of optical states, ρreg, and the strength of light–material interaction H^int, which also depends on phase matching between the pump, signal, and idler waves (Δk)—a condition hard to meet in bulk nonlinear materials. (d) Poling of a crystal minimizes the phase mismatch (Δk0) for a spectrally narrow operation range, but in general the density of optical modes is not significantly changed (ρpoledρreg). (e) Modification of the density of optical states in a resonator or a waveguide enhances emission and modifies the phase-matching conditions, but the high quality factor limits the operational frequency range. (f) Conversely, metamaterials may enable nonresonant, broadband, phase-mismatch-free Purcell enhancement of spontaneous nonlinear light emission.
Fig. 2.
Fig. 2. (a) Map of possible isofrequency surfaces (Vk) for a uniaxial crystal. Elliptical and hyperbolic dispersion regimes may be accessed by controlling the signs of ordinary, ϵ, and extraordinary, ϵ, permittivities. Notably, a semi-infinite number of optical modes are available in materials with hyperbolic dispersion (ρVk). (b) Spectral diagram of the downconversion process treated here. The metamaterial is pumped in an elliptical dispersion regime, whereas spontaneous downconversion of the signal-idler pairs occurs in the hyperbolic dispersion regime. Color gradient depicts the expected fluorescence probability.
Fig. 3.
Fig. 3. (a) Phase-matching diagram and corresponding photon emission pattern for an isotropic medium. For this closed isofrequency manifold, phase matching is possible only for a limited range of pump wavevectors (|kp|2|ks|). The dashed line denotes a condition beyond which phase matching is not possible. The photon emission is conical in this case. (b,c) Schematic of phase matching in layer- and wire-like hyperbolic metamaterials for pump propagation along the metamaterial axis. Open hyperbolic isofrequency surfaces for both of the hyperbolic cases imply pump wavevector-independent phase-mismatch-free operation. The light emission is conical, similar to the isotropic case in panel (a). (d) A phase-mismatch-free condition is also obtained for pump propagating along the x axis of the hyperbolic layer and wire metamaterials, respectively. However, in this case signal and idler waves encompass an infinitely large state space enabling a significant Purcell-like enhancement of nonlinear luminescence. Panel (d) shows the expected emission patterns in red for this pump propagation configuration.
Fig. 4.
Fig. 4. (a) Calculated spectral power density of emitted signal photons for a wire-like hyperbolic LiNbO3–Ag metamaterial for different pump wavelengths. The dashed curve shows expected emission from a bulk LiNbO3 crystal of similar thickness at λp=350  nm. (b) Spectral power density of emitted signal photons for a layer-hyperbolic metamaterial. The dashed curve shows the expected emission from a bulk LiNbO3 at λp=500  nm, for comparison. (c) Single-photon luminescence rate for frequency-degenerate downconversion (i.e., 2λp=λs=λi) as a function of a pump wavelength for layered hyperbolic and wire-like hyperbolic LiNbO3–Ag metamaterials. (d) Signal-photon spectral power density at the frequency-degenerate downconversion wavelength for a layered hyperbolic metamaterial as function of pump wavelength and propagation length. (e) Calculated signal photon spectral power density at the frequency-degenerate downconversion wavelength as a function of the pump wavelength for different propagation lengths for a layer-hyperbolic 70 nm GaP–30 nm Ag metamaterial. Inset shows the calculated luminescence for a 400 nm pump wavelength. (f) Signal photon emission map for a layer-hyperbolic GaP-Ag crystal at 400 nm pump wavelength after 500 nm of propagation.

Equations (1)

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dPsdλs=πc3L2λs4λiPpϵ0np×d2kscks2cki2ksωkiωN(ks)|1eiΔkLiΔkL|2eγ(ks)L,

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