Abstract

We introduce a new method allowing rigorous electromagnetic analysis of scattering through photonic crystals comprising polygonal or round rods. For this purpose, we reformulate the C method with adaptive spatial resolution by utilizing the hybrid-spectrum connection method, permitting the use of nonidentical trapezoidal profiles. Considering polygonal rods as gratings consisting of different piecewise-differentiable surfaces, we are able to analyze the reflection and the transmittance of crystals by means of the C method. To enhance computational efficiency, we apply the recursive S-matrix approach with Redheffer’s star product to solve the transfer matrix for structures of numerous successive layers of rods.

© 2003 Optical Society of America

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  2. E. Yablonovitch, D. F. Sievenpiper, “Knitting a finer net for photons,” Nature (London) 383, 665–666 (1996).
    [CrossRef]
  3. T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelength,” Nature (London) 383, 699–702 (1996).
    [CrossRef]
  4. D. Nyyssonen, C. P. Kirk, “Optical microscope imaging of lines patterned in thick layers with variable edge geometry,” J. Opt. Soc. Am. A 5, 1270–1280 (1988).
    [CrossRef]
  5. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  6. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  7. J. Turunen, “Diffraction theory of microrelief gratings,” Chap. 2 in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).
  8. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  9. L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
    [CrossRef]
  10. G. Granet, J. Chandezon, J. P. Plumey, K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution: application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102–2108 (2001).
    [CrossRef]
  11. G. Granet, J.-P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
    [CrossRef]
  12. T. W. Preist, N. P. K. Cotter, J. R. Samples, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
    [CrossRef]
  13. L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
    [CrossRef]
  14. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  15. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  16. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  17. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
  18. E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
    [CrossRef]
  19. R. Redheffer, “Difference equations and functional equations in transmission-line theory,” Chap. 12 in Modern Mathematics for the Engineer, E. F. Beckenbach, ed. (McGraw-Hill, New York, 1961).
  20. B. Datta, A. N. Singh, History of Hindu Mathematics, 2nd ed. (Asia Publishing House, Bombay, India, 1962), pp. 75–77.
  21. P. Lalanne, E. Silberstein, “Fourier-modal method applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000).
    [CrossRef]
  22. L. Li, J. Chandezon, G. Granet, J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
    [CrossRef]
  23. T. Vallius, “Comparing the Fourier modal method FMM with the C method: analysis of conducting multilevel gratings in TM polarization,” J. Opt. Soc. Am. A 19, 1555–1562 (2002).
    [CrossRef]
  24. J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
    [CrossRef]
  25. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
    [CrossRef]
  26. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
    [CrossRef]
  27. E. Popov, B. Bozhkov, “Differential method applied for photonic crystals,” Appl. Opt. 39, 4926–4932 (2000).
    [CrossRef]

2002

2001

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

G. Granet, J. Chandezon, J. P. Plumey, K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution: application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102–2108 (2001).
[CrossRef]

2000

1999

1996

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
[CrossRef]

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

E. Yablonovitch, D. F. Sievenpiper, “Knitting a finer net for photons,” Nature (London) 383, 665–666 (1996).
[CrossRef]

T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelength,” Nature (London) 383, 699–702 (1996).
[CrossRef]

1995

1993

1988

1987

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1986

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

1982

1978

Aalto, T.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Asatryan, A. A.

Botten, L. C.

Bozhkov, B.

Brand, S.

T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelength,” Nature (London) 383, 699–702 (1996).
[CrossRef]

Chandezon, J.

Cornet, G.

Cotter, N. P. K.

Datta, B.

B. Datta, A. N. Singh, History of Hindu Mathematics, 2nd ed. (Asia Publishing House, Bombay, India, 1962), pp. 75–77.

De La Rue, R. M.

T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelength,” Nature (London) 383, 699–702 (1996).
[CrossRef]

de Sterke, C. M.

Dupuis, M. T.

Gaylord, T. K.

Granet, G.

G. Granet, J. Chandezon, J. P. Plumey, K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution: application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102–2108 (2001).
[CrossRef]

L. Li, J. Chandezon, G. Granet, J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
[CrossRef]

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, J.-P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Grann, E. B.

Heimala, P.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Kirk, C. P.

Knop, K.

Krauss, T. H.

T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelength,” Nature (London) 383, 699–702 (1996).
[CrossRef]

Kuittinen, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Lalanne, P.

Leppihalme, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Li, L.

Mashev, L.

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

Maystre, D.

McPhedran, R. C.

Moharam, M. G.

Nicorovici, N. A.

Nyyssonen, D.

Plumey, J. P.

Plumey, J.-P.

L. Li, J. Chandezon, G. Granet, J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
[CrossRef]

G. Granet, J.-P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Pommet, D. A.

Popov, E.

E. Popov, B. Bozhkov, “Differential method applied for photonic crystals,” Appl. Opt. 39, 4926–4932 (2000).
[CrossRef]

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

Preist, T. W.

Raniriharinosy, K.

Redheffer, R.

R. Redheffer, “Difference equations and functional equations in transmission-line theory,” Chap. 12 in Modern Mathematics for the Engineer, E. F. Beckenbach, ed. (McGraw-Hill, New York, 1961).

Robinson, P. A.

Samples, J. R.

Sievenpiper, D. F.

E. Yablonovitch, D. F. Sievenpiper, “Knitting a finer net for photons,” Nature (London) 383, 665–666 (1996).
[CrossRef]

Silberstein, E.

Singh, A. N.

B. Datta, A. N. Singh, History of Hindu Mathematics, 2nd ed. (Asia Publishing House, Bombay, India, 1962), pp. 75–77.

Tervo, J.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Turunen, J.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

J. Turunen, “Diffraction theory of microrelief gratings,” Chap. 2 in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

Vahimaa, P.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Vallius, T.

Yablonovitch, E.

E. Yablonovitch, D. F. Sievenpiper, “Knitting a finer net for photons,” Nature (London) 383, 665–666 (1996).
[CrossRef]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

T. W. Preist, N. P. K. Cotter, J. R. Samples, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
[CrossRef]

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
[CrossRef]

G. Granet, J. Chandezon, J. P. Plumey, K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution: application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102–2108 (2001).
[CrossRef]

D. Nyyssonen, C. P. Kirk, “Optical microscope imaging of lines patterned in thick layers with variable edge geometry,” J. Opt. Soc. Am. A 5, 1270–1280 (1988).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

T. Vallius, “Comparing the Fourier modal method FMM with the C method: analysis of conducting multilevel gratings in TM polarization,” J. Opt. Soc. Am. A 19, 1555–1562 (2002).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[CrossRef]

Nature (London)

E. Yablonovitch, D. F. Sievenpiper, “Knitting a finer net for photons,” Nature (London) 383, 665–666 (1996).
[CrossRef]

T. H. Krauss, R. M. De La Rue, S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelength,” Nature (London) 383, 699–702 (1996).
[CrossRef]

Opt. Acta

E. Popov, L. Mashev, “Convergence of Rayleigh–Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

Opt. Commun.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Pure Appl. Opt.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, J.-P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Other

J. Turunen, “Diffraction theory of microrelief gratings,” Chap. 2 in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

R. Redheffer, “Difference equations and functional equations in transmission-line theory,” Chap. 12 in Modern Mathematics for the Engineer, E. F. Beckenbach, ed. (McGraw-Hill, New York, 1961).

B. Datta, A. N. Singh, History of Hindu Mathematics, 2nd ed. (Asia Publishing House, Bombay, India, 1962), pp. 75–77.

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Figures (6)

Fig. 1
Fig. 1

General trapezoidal multilayer grating with nonidentical profiles.

Fig. 2
Fig. 2

Geometry of the diffraction problem. The cylinders are divided into two continuous trapezoidal profiles.

Fig. 3
Fig. 3

Two photonic crystals with refractive indices n0=1 and n1=1.5. (a) d=0.9λ, ΔH=1λ, and the diameter of the square is 0.5λ. (b) d=1λ, ΔH=1.5λ, and the diameters of the hexagons are 0.5λ and 0.2λ. In structure (a) we have 20 layers of squares, whereas in structure (b) the number of layers of hexagonal rods is 50.

Fig. 4
Fig. 4

Sum of the transmitted and reflected energies in TE, (solid curves) and TM (dashed curves) polarizations for the grating comprising tilted squares with incident angles (a) θ=5° and (b) θ=20° and the grating comprising hexagons with (c) θ=5° and (d) θ=20°.

Fig. 5
Fig. 5

Convergence of diffraction orders -3 (solid curves), -2 (dashed curves), and -1 (dotted curves) for the conducting structure in (a) TE polarization and (b) TM polarization.

Fig. 6
Fig. 6

Reflectance of round periodic cylinders in (a) TE polarization and (b) TM polarization obtained by using polygonal approximation. The number of corners in the polygons is 10 (dotted curves), 14 (dashed curves), and 18 (solid curves).

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

2x2 H(x, z)+2z2 H(x, z)+k2ϵˆr,pH(x, z)=0,
x=F(u),
z=aq[F(u)]+vq.
dxdu=f(u),
daq(u)du=hq(u),
x=1f(u) u-1f(u) hq(u) v,
z=vq.
1f hq 1f hq+f 2vq2 H(u, vq)-1f hq 1f u+u 1f hq×vq H(u, vq)+1f u 1f u H(u, vq)+k2ϵˆr,pH(u, vq)=0.
H(p, q)(u, vq)=mHm exp{i[αmu+γ(p, q)vq]},
H(p, q)(u, vq)=1i vq H(p, q)(u, vq)
-αf-1α+fk2np200IH(p, q)H(p, q)=γ-hqf-1α-αF-1hqhqf-1hq+fI0H(p, q)H(p, q),
LA-1LBH(p, q)H(p, q)=1γ H(p, q)H(p, q).
xl(u)=a1+a2u+a32π sin2π u-ul-1ul-ul-1,
a1=ulxl-1-ul-1xlul-ul-1,
a2=xl-xl-1ul-ul-1,
a3=G(ul-ul-1)-(xl-xl-1)
H±(p, q)(u, vq)=nmPn(p, q)± exp[1γn(p, q)±vq]Hmn(p, q)± exp(iαmu),
H˜±(p, q)(u, vr)=nm exp(iαmu)H˜mn(p, q)± exp[iγn(p, q)±vr]Pn(p, q)±
H˜n±(p, q)=Ln(p, q)±Hn±(p, q)
[Ln(p, q)±]ml=Lm-l,n(p, q)±=1d 0d exp{iγn(p, q)±×[ar(u)-aq(u)]-i2π(m-l)u/d}du,
H(p)(x, z)=H+(p, p)[u, vp(z)]+H-(p, p+1)[u, vp+1(z)].
H+(p, p)[u, vp(z)]+H˜-(p, p+1)[u, vp(z)]=H˜+(p-1, p-1)[u, vp(z)]+H-(p-1, p)[u, vp(z)],
G±(p, q)=1ϵˆr,p [hqf-1αH±(p, q)-(f+hqf-1hq)H±(p, q)Γ±(p, q)],
S(p)=H+(p, p)-H-(p-1, p)G+(p, p)-G-(p-1, p)-1H˜+(p-1, p-1)-H˜-(p, p+1)G˜+(p-1, p-1)-G˜-(p, p+1);
ηRm=Re(γm1/γ01)|Tm|2,
ηTm=C Re(γmP/γ01)|Rm|2,

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