Abstract

A two-dimensional periodic sub-wavelength array of vertical dielectric cylinders on a glass substrate is studied numerically using three different electromagnetic approaches. It is shown that such structure can present a narrow-band spectral resonance characterized by large angular tolerances and 100% maximum in reflection. In particular, in a two-nanometer spectral bandwidth the reflectivity stays above 90% within angles of incidence exceeding 10 degrees for unpolarized light. Bloch modal analysis shows that these properties are due to the excitation of a hybrid mode that is created in the structure by a guided-like mode and a localized cavity mode. The first one is due to the collective effect of the array, while the second one comes from the mode(s) of a single step-index fiber.

© 2016 Optical Society of America

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References

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    [Crossref]
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2015 (4)

A. Taghizadeh, J. Mørk, and I.-S. Chung, “Ultracompact resonator with high quality-factor based on a hybrid grating structure,” Opt. Express 23(11), 14913–14921 (2015).
[Crossref] [PubMed]

Y. Wang, D. Stellinga, A. B. Klemm, C. P. Reardon, and T. F. Krauss, “Tunable optical tilters based on Silicon Nitride high contrast gratings,” IEEE J. Sel. Top. Quantum Electron. 21, 2700706 (2015).

P. Qiao, L. Zhu, W. C. Chew, and C. J. Chang-Hasnain, “Theory and design of two-dimensional high-contrast-grating phased arrays,” Opt. Express 23(19), 24508–24524 (2015).
[Crossref] [PubMed]

D. A. Bekele, G. C. M. R. C. I.-S. Park, R. Malureanu, and I.-S. Chung, “Polarization-independent wideband high-index-contrast grating mirror,” IEEE Photonics Technol. Lett. 27(16), 1733–1736 (2015).
[Crossref]

2014 (1)

2012 (3)

2010 (3)

2008 (1)

2007 (1)

Ph. Boyer, G. Renversez, E. Popov, and M. Nevère, “Improved differential method for microstructured optical fibers,” J. Opt. A, Pure Appl. Opt. 9(7), 728–740 (2007).
[Crossref]

2006 (3)

G. Renversez, P. Boyer, and A. Sagrini, “Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling,” Opt. Express 14(12), 5682–5687 (2006).
[Crossref] [PubMed]

L. Chen, M. C. Y. Huang, C. F. R. Mateus, C. J. Chang-Hasnain, and Y. Suzuki, “Fabrication and design of an integrable subwavelength ultrabroadband dielectric mirror,” Appl. Phys. Lett. 88(3), 031102 (2006).
[Crossref]

P. Lalanne, J.-P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: a Coupled Bloch-mode insight,” J. Light. Tech. 24(6), 2442–2449 (2006).
[Crossref]

2005 (1)

2004 (2)

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE. Phot. Tech. Lett. 16(2), 518–520 (2004).
[Crossref]

A. Nicolet, S. Guenneau, C. Guezaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math. 168(1-2), 321–329 (2004).
[Crossref]

2003 (1)

2002 (1)

2001 (1)

1998 (1)

1997 (2)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997).
[Crossref]

1986 (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical Study of the Anomalies of Coated Dielectric Gratings,” Opt. Acta (Lond.) 33(5), 607–619 (1986).
[Crossref]

Bekele, D. A.

D. A. Bekele, G. C. M. R. C. I.-S. Park, R. Malureanu, and I.-S. Chung, “Polarization-independent wideband high-index-contrast grating mirror,” IEEE Photonics Technol. Lett. 27(16), 1733–1736 (2015).
[Crossref]

Belharet, D.

Boyer, P.

Boyer, Ph.

Ph. Boyer, G. Renversez, E. Popov, and M. Nevère, “Improved differential method for microstructured optical fibers,” J. Opt. A, Pure Appl. Opt. 9(7), 728–740 (2007).
[Crossref]

Buet, X.

Chang-Hasnain, C. J.

P. Qiao, L. Zhu, W. C. Chew, and C. J. Chang-Hasnain, “Theory and design of two-dimensional high-contrast-grating phased arrays,” Opt. Express 23(19), 24508–24524 (2015).
[Crossref] [PubMed]

V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18(16), 16973–16988 (2010).
[Crossref] [PubMed]

L. Chen, M. C. Y. Huang, C. F. R. Mateus, C. J. Chang-Hasnain, and Y. Suzuki, “Fabrication and design of an integrable subwavelength ultrabroadband dielectric mirror,” Appl. Phys. Lett. 88(3), 031102 (2006).
[Crossref]

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE. Phot. Tech. Lett. 16(2), 518–520 (2004).
[Crossref]

Chavel, P.

P. Lalanne, J.-P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: a Coupled Bloch-mode insight,” J. Light. Tech. 24(6), 2442–2449 (2006).
[Crossref]

Chen, L.

L. Chen, M. C. Y. Huang, C. F. R. Mateus, C. J. Chang-Hasnain, and Y. Suzuki, “Fabrication and design of an integrable subwavelength ultrabroadband dielectric mirror,” Appl. Phys. Lett. 88(3), 031102 (2006).
[Crossref]

Chew, W. C.

Chung, I.-S.

D. A. Bekele, G. C. M. R. C. I.-S. Park, R. Malureanu, and I.-S. Chung, “Polarization-independent wideband high-index-contrast grating mirror,” IEEE Photonics Technol. Lett. 27(16), 1733–1736 (2015).
[Crossref]

A. Taghizadeh, J. Mørk, and I.-S. Chung, “Ultracompact resonator with high quality-factor based on a hybrid grating structure,” Opt. Express 23(11), 14913–14921 (2015).
[Crossref] [PubMed]

Commandré, M.

Coutaz, J.-L.

Daran, E.

Demésy, G.

Deng, Y.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE. Phot. Tech. Lett. 16(2), 518–520 (2004).
[Crossref]

Fehrembach, A.-L.

A.-L. Fehrembach, F. Lemarchand, A. Talneau, and A. Sentenac, “High Q polarization independent guided-mode resonance filter with “doubly periodic” etched Ta2O5 bidimensional grating,” IEEE J. Light. Tech. 28, 2037–2044 (2010).

A. Sentenac and A.-L. Fehrembach, “Angular tolerant resonant grating filters under oblique incidence,” J. Opt. Soc. Am. A 22(3), 475–480 (2005).
[Crossref] [PubMed]

A.-L. Fehrembach, D. Maystre, and A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19(6), 1136–1144 (2002).
[Crossref] [PubMed]

Friesem, A. A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

Garet, F.

Gauthier-Lafaye, O.

Giovannini, H.

Guenneau, S.

A. Nicolet, S. Guenneau, C. Guezaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math. 168(1-2), 321–329 (2004).
[Crossref]

Guezaine, C.

A. Nicolet, S. Guenneau, C. Guezaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math. 168(1-2), 321–329 (2004).
[Crossref]

Hatanaka, K.

Huang, M. C. Y.

L. Chen, M. C. Y. Huang, C. F. R. Mateus, C. J. Chang-Hasnain, and Y. Suzuki, “Fabrication and design of an integrable subwavelength ultrabroadband dielectric mirror,” Appl. Phys. Lett. 88(3), 031102 (2006).
[Crossref]

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE. Phot. Tech. Lett. 16(2), 518–520 (2004).
[Crossref]

Hugonin, J.-P.

P. Lalanne, J.-P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: a Coupled Bloch-mode insight,” J. Light. Tech. 24(6), 2442–2449 (2006).
[Crossref]

Inoue, J.

Kämpfe, T.

Karagodsky, V.

Kintaka, K.

Klemm, A. B.

Y. Wang, D. Stellinga, A. B. Klemm, C. P. Reardon, and T. F. Krauss, “Tunable optical tilters based on Silicon Nitride high contrast gratings,” IEEE J. Sel. Top. Quantum Electron. 21, 2700706 (2015).

Krauss, T. F.

Y. Wang, D. Stellinga, A. B. Klemm, C. P. Reardon, and T. F. Krauss, “Tunable optical tilters based on Silicon Nitride high contrast gratings,” IEEE J. Sel. Top. Quantum Electron. 21, 2700706 (2015).

Lalanne, P.

P. Lalanne, J.-P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: a Coupled Bloch-mode insight,” J. Light. Tech. 24(6), 2442–2449 (2006).
[Crossref]

Lemarchand, F.

A.-L. Fehrembach, F. Lemarchand, A. Talneau, and A. Sentenac, “High Q polarization independent guided-mode resonance filter with “doubly periodic” etched Ta2O5 bidimensional grating,” IEEE J. Light. Tech. 28, 2037–2044 (2010).

F. Lemarchand, A. Sentenac, and H. Giovannini, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23(15), 1149–1151 (1998).
[Crossref] [PubMed]

Li, L.

Lozes-Dupuy, F.

Magnusson, R.

Majima, T.

Malureanu, R.

D. A. Bekele, G. C. M. R. C. I.-S. Park, R. Malureanu, and I.-S. Chung, “Polarization-independent wideband high-index-contrast grating mirror,” IEEE Photonics Technol. Lett. 27(16), 1733–1736 (2015).
[Crossref]

Mashev, L.

E. Popov, L. Mashev, and D. Maystre, “Theoretical Study of the Anomalies of Coated Dielectric Gratings,” Opt. Acta (Lond.) 33(5), 607–619 (1986).
[Crossref]

Mateus, C. F. R.

L. Chen, M. C. Y. Huang, C. F. R. Mateus, C. J. Chang-Hasnain, and Y. Suzuki, “Fabrication and design of an integrable subwavelength ultrabroadband dielectric mirror,” Appl. Phys. Lett. 88(3), 031102 (2006).
[Crossref]

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE. Phot. Tech. Lett. 16(2), 518–520 (2004).
[Crossref]

Maystre, D.

A.-L. Fehrembach, D. Maystre, and A. Sentenac, “Phenomenological theory of filtering by resonant dielectric gratings,” J. Opt. Soc. Am. A 19(6), 1136–1144 (2002).
[Crossref] [PubMed]

E. Popov, L. Mashev, and D. Maystre, “Theoretical Study of the Anomalies of Coated Dielectric Gratings,” Opt. Acta (Lond.) 33(5), 607–619 (1986).
[Crossref]

Monmayrant, A.

Mørk, J.

Neureuther, A. R.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE. Phot. Tech. Lett. 16(2), 518–520 (2004).
[Crossref]

Nevère, M.

Ph. Boyer, G. Renversez, E. Popov, and M. Nevère, “Improved differential method for microstructured optical fibers,” J. Opt. A, Pure Appl. Opt. 9(7), 728–740 (2007).
[Crossref]

Nevière, M.

Nicolet, A.

G. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “All-purpose finite element formulation for arbitrarily shaped crossed-gratings embedded in a multilayered stack,” J. Opt. Soc. Am. A 27(4), 878–889 (2010).
[Crossref] [PubMed]

A. Nicolet, S. Guenneau, C. Guezaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math. 168(1-2), 321–329 (2004).
[Crossref]

Nishii, J.

Park, G. C. M. R. C. I.-S.

D. A. Bekele, G. C. M. R. C. I.-S. Park, R. Malureanu, and I.-S. Chung, “Polarization-independent wideband high-index-contrast grating mirror,” IEEE Photonics Technol. Lett. 27(16), 1733–1736 (2015).
[Crossref]

Parriaux, O.

Popov, E.

Ph. Boyer, G. Renversez, E. Popov, and M. Nevère, “Improved differential method for microstructured optical fibers,” J. Opt. A, Pure Appl. Opt. 9(7), 728–740 (2007).
[Crossref]

E. Popov and M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18(11), 2886–2894 (2001).
[Crossref] [PubMed]

E. Popov, L. Mashev, and D. Maystre, “Theoretical Study of the Anomalies of Coated Dielectric Gratings,” Opt. Acta (Lond.) 33(5), 607–619 (1986).
[Crossref]

Qiao, P.

Reardon, C. P.

Y. Wang, D. Stellinga, A. B. Klemm, C. P. Reardon, and T. F. Krauss, “Tunable optical tilters based on Silicon Nitride high contrast gratings,” IEEE J. Sel. Top. Quantum Electron. 21, 2700706 (2015).

Renversez, G.

Ph. Boyer, G. Renversez, E. Popov, and M. Nevère, “Improved differential method for microstructured optical fibers,” J. Opt. A, Pure Appl. Opt. 9(7), 728–740 (2007).
[Crossref]

G. Renversez, P. Boyer, and A. Sagrini, “Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling,” Opt. Express 14(12), 5682–5687 (2006).
[Crossref] [PubMed]

Rosenblatt, D.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

Sagrini, A.

Sedgwick, F. G.

Sentenac, A.

Sharon, A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

Shokooh-Saremi, M.

Stellinga, D.

Y. Wang, D. Stellinga, A. B. Klemm, C. P. Reardon, and T. F. Krauss, “Tunable optical tilters based on Silicon Nitride high contrast gratings,” IEEE J. Sel. Top. Quantum Electron. 21, 2700706 (2015).

Suzuki, Y.

L. Chen, M. C. Y. Huang, C. F. R. Mateus, C. J. Chang-Hasnain, and Y. Suzuki, “Fabrication and design of an integrable subwavelength ultrabroadband dielectric mirror,” Appl. Phys. Lett. 88(3), 031102 (2006).
[Crossref]

Taghizadeh, A.

Talneau, A.

A.-L. Fehrembach, F. Lemarchand, A. Talneau, and A. Sentenac, “High Q polarization independent guided-mode resonance filter with “doubly periodic” etched Ta2O5 bidimensional grating,” IEEE J. Light. Tech. 28, 2037–2044 (2010).

Ura, S.

Wang, Y.

Y. Wang, D. Stellinga, A. B. Klemm, C. P. Reardon, and T. F. Krauss, “Tunable optical tilters based on Silicon Nitride high contrast gratings,” IEEE J. Sel. Top. Quantum Electron. 21, 2700706 (2015).

Zhu, L.

Zolla, F.

G. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “All-purpose finite element formulation for arbitrarily shaped crossed-gratings embedded in a multilayered stack,” J. Opt. Soc. Am. A 27(4), 878–889 (2010).
[Crossref] [PubMed]

A. Nicolet, S. Guenneau, C. Guezaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math. 168(1-2), 321–329 (2004).
[Crossref]

Appl. Phys. Lett. (1)

L. Chen, M. C. Y. Huang, C. F. R. Mateus, C. J. Chang-Hasnain, and Y. Suzuki, “Fabrication and design of an integrable subwavelength ultrabroadband dielectric mirror,” Appl. Phys. Lett. 88(3), 031102 (2006).
[Crossref]

IEEE J. Light. Tech. (1)

A.-L. Fehrembach, F. Lemarchand, A. Talneau, and A. Sentenac, “High Q polarization independent guided-mode resonance filter with “doubly periodic” etched Ta2O5 bidimensional grating,” IEEE J. Light. Tech. 28, 2037–2044 (2010).

IEEE J. Quantum Electron. (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Wang, D. Stellinga, A. B. Klemm, C. P. Reardon, and T. F. Krauss, “Tunable optical tilters based on Silicon Nitride high contrast gratings,” IEEE J. Sel. Top. Quantum Electron. 21, 2700706 (2015).

IEEE Photonics Technol. Lett. (1)

D. A. Bekele, G. C. M. R. C. I.-S. Park, R. Malureanu, and I.-S. Chung, “Polarization-independent wideband high-index-contrast grating mirror,” IEEE Photonics Technol. Lett. 27(16), 1733–1736 (2015).
[Crossref]

IEEE. Phot. Tech. Lett. (1)

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE. Phot. Tech. Lett. 16(2), 518–520 (2004).
[Crossref]

J. Comput. Appl. Math. (1)

A. Nicolet, S. Guenneau, C. Guezaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math. 168(1-2), 321–329 (2004).
[Crossref]

J. Light. Tech. (1)

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Figures (11)

Fig. 1
Fig. 1 Schematical presentation of a two-dimensional array of dielectric cylinders.
Fig. 2
Fig. 2 Reflectivity of the dielectric grating in normal incidence as a function of the cylinder diameter and height given in microns. Period d = 0.5 µm, λ = 0.865 µm. (a) logarithmic scale over a large interval of diameters; (b) zoom of the upper right corner.
Fig. 3
Fig. 3 (a). Spectral dependence of the reflectivity in normal incidence for d = 0.5 µm, h = 1 µm, Φ = 0.4333 µm; (b) Angular dependence of the reflectivity for λ = 0.865 µm, incident polarization along the x-axis. The other parameters as in Fig. 3.
Fig. 4
Fig. 4 (a). The dependence of the mode effective index neff = γp/k0 equal to the normalized propagation constant γ in z-direction as a function of the cylinder diameter. d = 0.5 µm, λ = 0.865 µm. The red line corresponds to a mode that cannot be excited with linear incident polarization in normal incidence; (b) the same as in (a) but for d = 5 µm. In addition, the squares stay for the modes that can propagate along a single infinitely long fiber.
Fig. 5
Fig. 5 The map of the modulus of the electric field of mode 3 as a function of x and y, Φ = 0.4333 µm and λ = 0.865 µm, and normal incidence. All coordinates are given in microns.
Fig. 6
Fig. 6 Variation of neff as a function of the period d for a fixed cylinder diameter Φ = 0.4333 µm and λ = 0.865 µm, and normal incidence.
Fig. 7
Fig. 7 x-y maps of the modulus of electric field of mode 1 (b, e), mode 2 (c, f), and the total field (a,d) for two grating periods, d = 2 µm (a-c) and 0.5 µm (d-f) for a fixed cylinder diameter 0.4333 µm, wavelength 0.865 µm and polarization along x-direction. The modal fields calculated for a 2D array of rods infinitely long in z does not depend on z-direction, while the total field, which is a solution of the 3D diffraction problem is calculated at mid-height z = 0.5 µm with h = 1 µm. Black circles represent the cylinder boundaries.
Fig. 8
Fig. 8 Fourier spectrum of the x-components of the electric field of mode 1 and mode 2, corresponding to Fig. 7 for two different period values. Vertical axis represent the values of the modulus of the Fourier components, numbered on the horizontal axis.
Fig. 9
Fig. 9 Vertical (d-f) and horizontal (a-c) distribution of electric field, calculated by taking into account all existing modes (a, d), mode 1 only (c, f), or mode 2 alone (b, e). The vertical maps are calculated at y = 0, and the horizontal maps at z = 0.5. The cylinder boundaries are presented by black circles or lines.
Fig. 10
Fig. 10 The y-z field maps calculated at x = 0 and taking into account different modal contributions: (a) mode 2 alone; (b) mode 1 alone; (d) mode 1 and mode 2; (c) all modal components
Fig. 11
Fig. 11 Spectral dependence of the reflectivity corresponding to Fig. 3(a) and obtained by suppressing all other modes than mode 1 (blue), or mode 2 (red), or both of them (black). The squares are obtained by using the complete modal set.

Tables (1)

Tables Icon

Table 1 The values of the real part of the normalized mode propagation constant for the different modes of a system with d = 0.5 µm, Φ = 0.4333 µm, λ = 0.865 µm, together with the modulus of the up- and downwards traveling mode amplitudes b for h = 1 µm.

Equations (7)

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d dz F=MF.
( E x E y H x H y )= m,n,p e i k 0 ( αmx+ β n y ) V mn,p e i γ p z b p ,
( E x E y H x H y )= p ( E x,p (x,y) E y,p (x,y) H x,p (x,y) H y,p (x,y) ) e i γ p z b p ,
( E x,p (x,y) E y,p (x,y) H x,p (x,y) H y,p (x,y) )= m,n,p e i k 0 ( αmx+ β n y ) V mn,p
n eff = ε ¯ r (λ/d) 2 ,
t Fano = t nonresonant + t Lorentz = t nonresonant + c λ λ p = t nonresonant λ λ z λ λ p
t Fano = c 1 λ λ 1 p + c 2 λ λ 2 p =( c 1 + c 2 ) λ λ z ( λ λ 1 p )( λ λ 2 p ) λ z = c 1 λ 2 p + c 2 λ 1 p c 1 + c 2 ,

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