Abstract

We theoretically show that coaxial waveguides composed of a metallic core, surrounded by a dielectric cylinder and clad by a metal outer layer exhibit negative refractive index modes over a broad spectral range in the visible. For narrow dielectric gaps (10 nm GaP embedded in Ag) a figure-of-merit of 18 can be achieved at λ0 = 460 nm. For larger dielectric gaps the negative index spectral range extends well below the surface plasmon resonance frequency. By fine-tuning the coaxial geometry the special case of n = −1 at a figure-of-merit of 5, or n = 0 for a decay length of 500 nm can be achieved.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2010

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef] [PubMed]

2009

R. de Waele, S. P. Burgos, A. Polman, and H. A. Atwater, “Plasmon dispersion in coaxial waveguides from single-cavity optical transmission measurements,” Nano Lett. 9, 2832–2837 (2009).
[CrossRef] [PubMed]

2008

2007

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[CrossRef]

G. Dolling, M. Wegener, and S. Linden, “Realization of a three-functional-layer negative-index photonic metamaterial,” Opt. Lett. 32, 551–553 (2007).
[CrossRef] [PubMed]

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

M. Sandtke, and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1, 573–576 (2007).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

2006

F. I. Baida, A. Belkhir, D. V. Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B 74, 205419 (2006).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

H. Miyazaki, and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96, 097401 (2006).
[CrossRef] [PubMed]

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
[CrossRef] [PubMed]

2005

J. Weeber, M. U. González, A. L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett. 87, 221101 (2005).
[CrossRef]

W. J. Fan, S. Zhang, B. Minhas, K. J. Malloy, and S. R. J. Brueck, “Enhanced infrared transmission through subwavelength coaxial metallic arrays,” Phys. Rev. Lett. 94, 033902 (2005).
[CrossRef] [PubMed]

2004

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef] [PubMed]

2003

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67, 035109 (2003).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

1999

1983

1972

P. B. Johnson, and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Alekseyev, L. V.

Alù, A.

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

Atwater, H. A.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef] [PubMed]

R. de Waele, S. P. Burgos, A. Polman, and H. A. Atwater, “Plasmon dispersion in coaxial waveguides from single-cavity optical transmission measurements,” Nano Lett. 9, 2832–2837 (2009).
[CrossRef] [PubMed]

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? a case study of three plasmonic geometries,” Opt. Express 16, 19001–19017 (2008).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Aussenegg, F. R.

A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Baida, F. I.

F. I. Baida, A. Belkhir, D. V. Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B 74, 205419 (2006).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef] [PubMed]

Baudrion, A. L.

J. Weeber, M. U. González, A. L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett. 87, 221101 (2005).
[CrossRef]

Belkhir, A.

F. I. Baida, A. Belkhir, D. V. Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B 74, 205419 (2006).
[CrossRef]

Berini, P.

Brueck, S. R. J.

W. J. Fan, S. Zhang, B. Minhas, K. J. Malloy, and S. R. J. Brueck, “Enhanced infrared transmission through subwavelength coaxial metallic arrays,” Phys. Rev. Lett. 94, 033902 (2005).
[CrossRef] [PubMed]

Burgos, S. P.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef] [PubMed]

R. de Waele, S. P. Burgos, A. Polman, and H. A. Atwater, “Plasmon dispersion in coaxial waveguides from single-cavity optical transmission measurements,” Nano Lett. 9, 2832–2837 (2009).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson, and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

de Waele, R.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef] [PubMed]

R. de Waele, S. P. Burgos, A. Polman, and H. A. Atwater, “Plasmon dispersion in coaxial waveguides from single-cavity optical transmission measurements,” Nano Lett. 9, 2832–2837 (2009).
[CrossRef] [PubMed]

Dereux, A.

J. Weeber, M. U. González, A. L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett. 87, 221101 (2005).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Dionne, J. A.

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? a case study of three plasmonic geometries,” Opt. Express 16, 19001–19017 (2008).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Dolling, G.

G. Dolling, M. Wegener, and S. Linden, “Realization of a three-functional-layer negative-index photonic metamaterial,” Opt. Lett. 32, 551–553 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef] [PubMed]

Drezet, A.

A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Engheta, N.

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

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef] [PubMed]

Fan, W. J.

W. J. Fan, S. Zhang, B. Minhas, K. J. Malloy, and S. R. J. Brueck, “Enhanced infrared transmission through subwavelength coaxial metallic arrays,” Phys. Rev. Lett. 94, 033902 (2005).
[CrossRef] [PubMed]

Genov, D.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef] [PubMed]

González, M. U.

J. Weeber, M. U. González, A. L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett. 87, 221101 (2005).
[CrossRef]

Jacob, Z.

Johnson, P. B.

P. B. Johnson, and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Krenn, J. R.

A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Kuipers, L.

E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,” Opt. Express 16, 45–57 (2008).
[CrossRef] [PubMed]

M. Sandtke, and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1, 573–576 (2007).
[CrossRef]

Kurokawa, Y.

H. Miyazaki, and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96, 097401 (2006).
[CrossRef] [PubMed]

Labeke, D. V.

F. I. Baida, A. Belkhir, D. V. Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B 74, 205419 (2006).
[CrossRef]

Lamrous, O.

F. I. Baida, A. Belkhir, D. V. Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B 74, 205419 (2006).
[CrossRef]

Leitner, A.

A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[CrossRef] [PubMed]

Linden, S.

G. Dolling, M. Wegener, and S. Linden, “Realization of a three-functional-layer negative-index photonic metamaterial,” Opt. Lett. 32, 551–553 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef] [PubMed]

Malloy, K. J.

W. J. Fan, S. Zhang, B. Minhas, K. J. Malloy, and S. R. J. Brueck, “Enhanced infrared transmission through subwavelength coaxial metallic arrays,” Phys. Rev. Lett. 94, 033902 (2005).
[CrossRef] [PubMed]

Meier, M.

Minhas, B.

W. J. Fan, S. Zhang, B. Minhas, K. J. Malloy, and S. R. J. Brueck, “Enhanced infrared transmission through subwavelength coaxial metallic arrays,” Phys. Rev. Lett. 94, 033902 (2005).
[CrossRef] [PubMed]

Miyazaki, H.

H. Miyazaki, and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96, 097401 (2006).
[CrossRef] [PubMed]

Narimanov, E.

Ozbay, E.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Polman, A.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative index metamaterial at visible frequencies,” Nat. Mater. 9, 407–412 (2010).
[CrossRef] [PubMed]

R. de Waele, S. P. Burgos, A. Polman, and H. A. Atwater, “Plasmon dispersion in coaxial waveguides from single-cavity optical transmission measurements,” Nano Lett. 9, 2832–2837 (2009).
[CrossRef] [PubMed]

E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,” Opt. Express 16, 45–57 (2008).
[CrossRef] [PubMed]

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? a case study of three plasmonic geometries,” Opt. Express 16, 19001–19017 (2008).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Salandrino, A.

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

Sandtke, M.

M. Sandtke, and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1, 573–576 (2007).
[CrossRef]

Shalaev, V. M.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[CrossRef]

Shvets, G.

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67, 035109 (2003).
[CrossRef]

Silveirinha, M. G.

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

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006).
[CrossRef] [PubMed]

Stockman, M. I.

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Ulin-Avila, E.

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Figures (6)

Fig. 1.
Fig. 1.

Coaxial plasmon waveguide geometry and numerical mode solving method. (a) Schematic cross-section of a coaxial waveguide with the definition of the cylindrical polar coordinates, r, ϕ and z. The metallic inner core and outer cladding separate a dielectric channel. A schematic wave propagating in the waveguide in the direction of positive z is also indicated. (b) Argument θ of the determinant, det[M(k)], plotted in the complex k-plane for a Ag/Si/Ag waveguide with 75 nm inner core diameter and 10-nm-wide dielectric channel at ω = 3×1015 rad/s. By cycling around the closed loop indicated by the dashed square the net number of discontinuities in θ is determined. Zero positions are indicated by the white circles.

Fig. 2.
Fig. 2.

Dispersion relations of the three lowest-order modes of a coaxial waveguide with 75-nm-diameter Ag core, 25-nm wide Si channel, and infinite outer Ag cladding. Radial frequency is plotted versus propagation constant k′ (a), attenuation constant k″ (b), and figure-of-merit k′/k″ (c). The Ag/Si surface plasmon resonance frequency ωSP = 3.15×1015 rad/s (λ0 = 598 nm) is indicated by the horizontal line. Panel (a) shows two modes with positive index (blue dashed curve and green dotted curve) and one mode with a negative index below a frequency of ~ 3.8×1015 rad/s (red drawn curve). The insets in (a) show the Hy field distribution in the transverse plane of the waveguide at 2.8×1015 rad/s for the positive-index mode (blue dashed dispersion curve) and at 3.6×1015 rad/s for the negative-index mode.

Fig. 3.
Fig. 3.

Dispersion relations for negative-index coaxial waveguides with Ag core and cladding and Si dielectric channel, (a): ω(k′); (b): ω(k″). The inner core diameter is fixed at 75 nm and the Si-channel thickness w is 10 nm, 30 nm, and 70 nm. Positive-index modes [as shown in Fig. 2(a)] are not shown in the figure. The bold sections of the dispersion curves indicate the spectral range over which the negative-index mode is dominant, i.e. has lower loss than the positive index modes. The frequency where the red and green dispersion curves cross k′ = 0 is indicated by the star-symbols.

Fig. 4.
Fig. 4.

Dispersion relations for coaxial waveguides with Ag core and cladding and 70-nm-wide Si dielectric channel, (a): ω(k′); (b): ω(k″). The inner core diameter, dcore, is 45 nm (blue curves), 75 nm (green curves) and 100 nm (red curves). Only modes with negative index are plotted. Bold lines indicate the spectral range where the mode is dominant over the positive-index mode.

Fig. 5.
Fig. 5.

Dispersion relations for coaxial waveguides with 75-nm-diameter metal core, 25-nm-wide dielectric channel and infinite metal cladding, (a): ω(k′); (b): FOM k′/k″. The type of metal in the core and cladding, as well as the dielectric material was varied. The frequency axes are normalized to the corresponding surface plasmon resonance frequency ωSP. Blue curves are for a Si channel surrounded by Ag (ωSP = 3.15×1015 rad/s), green curves for GaP in Ag (ωSP = 3.49×1015 rad/s), red curves for SiO2 in Ag (ωSP = 5.24× 1015 rad/s) and purple curves for Si in Au (ωSP = 2.77×1015 rad/s). Bold curves indicate the spectral range where the negative-index mode is dominant.

Fig. 6.
Fig. 6.

Magnetic field images of a coaxial waveguide with 75 nm Ag core diameter, 25-nm-wide GaP-filled ring and infinite Ag cladding at a frequency of 3.75×1015 rad/s. In (a) we plot the Hy field distribution on the boundary of the Ag cladding with the dielectric channel. Note that the phase-fronts in the waveguide are in general not perpendicular to the optical axis (z-axis). In (b-d) we plot the polar magnetic field components in the transverse plane. The amplitude of the fields plotted in the figure has the same order of magnitude in all of the four panels.

Equations (4)

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κ i 2 ε i ω 2 c 2 k 2
det [ M ( k ) ] = 0
det [ M ( k ) ] = det [ M ( k ) ] e i θ
L = 1 2 k .

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