Abstract

Symmetry breaking is a common phenomenon in nonlinear systems, it refers to the existence of solutions that do not preserve the original symmetries of the underlying system. In nonlinear optics, symmetry breaking has been previously investigated in a number of systems, usually based on simplified model equations or temporal coupled mode theories. In this paper, we analyze the scattering of an incident plane wave by one or two circular cylinders with a Kerr nonlinearity, and show the existence of solutions that break a lateral reflection symmetry. Although symmetry breaking is a known phenomenon in nonlinear optics, it is the first time that this phenomenon was rigorously studied in simple systems with one or two circular cylinders.

© 2014 Optical Society of America

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References

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  1. N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56(2), 299–303 (1982).
  2. A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40(3), 229–232 (1982).
    [Crossref]
  3. K. Otsuka, “Pitchfork bifurcation and all-optical digital signal-processing with a coupled-element bistable system,” Opt. Lett. 14, 7274 (1989).
    [Crossref]
  4. M. Haelterman and P. Mandel, “Pitchfork bifurcation using a 2-beam nonlinear Fabry-Perot interferometer: an analytical study,” Opt. Lett. 15, 1412–1414 (1990).
    [Crossref] [PubMed]
  5. C. Paré and M. Florjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
    [Crossref] [PubMed]
  6. P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analytical results,” J. Opt. Soc. Am. B 10, 1379–1385 (1993).
    [Crossref]
  7. J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
    [Crossref]
  8. T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counter-propagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
    [Crossref] [PubMed]
  9. J. P. Torres, J. Boyce, and R.Y. Chiao, “Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability,” Phys. Rev. Lett. 83, 4293–4296 (1999).
    [Crossref]
  10. B. Maes, M. Soljačić, J. D. Joannopoulos, P. Bienstman, R. Baets, S.-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006).
    [Crossref] [PubMed]
  11. B. Maes, P. Bienstman, and R. Baets, “Symmetry breaking with coupled Fano resonances,” Opt. Express 16, 3069–3076 (2008).
    [Crossref] [PubMed]
  12. K. Huybrechts, G. Morthier, and B. Maes, “Symmetry breaking in networks of nonlinear cavities,” J. Opt. Soc. Am. B 27, 708–713 (2010).
    [Crossref]
  13. E. Bulgakov, K. Pichugin, and A. Sadreev, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
    [Crossref]
  14. E. Bulgakov and A. Sadreev, “Switching through symmetry breaking for transmission in a T-shaped photonic waveguide coupled with two identical nonlinear micro-cavities,” J. Phys. Condens. Matter 23, 315303 (2011).
    [Crossref] [PubMed]
  15. E. N. Bulgakov and A. F. Sadreev, “Symmetry breaking in photonic crystal waveguide coupled with the dipole modes of a nonlinear optical cavity,” J. Opt. Soc. Am. B 29, 2924–2928 (2012).
    [Crossref]
  16. C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
    [Crossref]
  17. V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
    [Crossref]
  18. T. Mayteevarunyoo, B. A. Malomed, and A. Reoksabutr, “Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials,” J. Mod. Opt. 58(21), 1977–1989 (2011).
    [Crossref]
  19. L. Yuan and Y. Y. Lu, “Efficient numerical method for analyzing optical bistability in photonic crystal microcavities,” Opt. Express 21(10), 11952–11964 (2013).
    [Crossref] [PubMed]
  20. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
    [Crossref] [PubMed]
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    [Crossref]
  22. L. N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, 2000.
    [Crossref]
  23. L. Yuan and Y. Y. Lu, “Analyzing second harmonic generation from arrays of cylinders using the Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 26, 587–594 (2009).
    [Crossref]

2013 (1)

2012 (2)

2011 (4)

E. Bulgakov, K. Pichugin, and A. Sadreev, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[Crossref]

E. Bulgakov and A. Sadreev, “Switching through symmetry breaking for transmission in a T-shaped photonic waveguide coupled with two identical nonlinear micro-cavities,” J. Phys. Condens. Matter 23, 315303 (2011).
[Crossref] [PubMed]

V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
[Crossref]

T. Mayteevarunyoo, B. A. Malomed, and A. Reoksabutr, “Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials,” J. Mod. Opt. 58(21), 1977–1989 (2011).
[Crossref]

2010 (1)

2009 (2)

L. Yuan and Y. Y. Lu, “Analyzing second harmonic generation from arrays of cylinders using the Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 26, 587–594 (2009).
[Crossref]

C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
[Crossref]

2008 (2)

2006 (1)

1999 (1)

J. P. Torres, J. Boyce, and R.Y. Chiao, “Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability,” Phys. Rev. Lett. 83, 4293–4296 (1999).
[Crossref]

1994 (1)

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counter-propagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[Crossref] [PubMed]

1993 (2)

P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analytical results,” J. Opt. Soc. Am. B 10, 1379–1385 (1993).
[Crossref]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[Crossref]

1990 (2)

C. Paré and M. Florjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[Crossref] [PubMed]

M. Haelterman and P. Mandel, “Pitchfork bifurcation using a 2-beam nonlinear Fabry-Perot interferometer: an analytical study,” Opt. Lett. 15, 1412–1414 (1990).
[Crossref] [PubMed]

1989 (1)

K. Otsuka, “Pitchfork bifurcation and all-optical digital signal-processing with a coupled-element bistable system,” Opt. Lett. 14, 7274 (1989).
[Crossref]

1982 (2)

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56(2), 299–303 (1982).

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40(3), 229–232 (1982).
[Crossref]

Akhmediev, N.

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[Crossref]

Akhmediev, N. N.

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56(2), 299–303 (1982).

Baets, R.

Bienstman, P.

Boyce, J.

J. P. Torres, J. Boyce, and R.Y. Chiao, “Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability,” Phys. Rev. Lett. 83, 4293–4296 (1999).
[Crossref]

Brazhnyi, V. A.

V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
[Crossref]

Bulgakov, E.

E. Bulgakov, K. Pichugin, and A. Sadreev, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[Crossref]

E. Bulgakov and A. Sadreev, “Switching through symmetry breaking for transmission in a T-shaped photonic waveguide coupled with two identical nonlinear micro-cavities,” J. Phys. Condens. Matter 23, 315303 (2011).
[Crossref] [PubMed]

Bulgakov, E. N.

Chiao, R.Y.

J. P. Torres, J. Boyce, and R.Y. Chiao, “Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability,” Phys. Rev. Lett. 83, 4293–4296 (1999).
[Crossref]

Chu, P. L.

Florjanczyk, M.

C. Paré and M. Florjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[Crossref] [PubMed]

Frantzeskakis, D. J.

C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
[Crossref]

Gorza, S.-P.

Haelterman, M.

Hu, Z.

Huybrechts, K.

Joannopoulos, J. D.

Kaplan, A. E.

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40(3), 229–232 (1982).
[Crossref]

Kevrekidis, P. G.

C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
[Crossref]

Law, K. J. H.

C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
[Crossref]

Lederer, F.

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counter-propagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[Crossref] [PubMed]

Lu, Y. Y.

Maes, B.

Malomed, B. A.

V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
[Crossref]

T. Mayteevarunyoo, B. A. Malomed, and A. Reoksabutr, “Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials,” J. Mod. Opt. 58(21), 1977–1989 (2011).
[Crossref]

C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
[Crossref]

P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analytical results,” J. Opt. Soc. Am. B 10, 1379–1385 (1993).
[Crossref]

Mandel, P.

Mayteevarunyoo, T.

T. Mayteevarunyoo, B. A. Malomed, and A. Reoksabutr, “Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials,” J. Mod. Opt. 58(21), 1977–1989 (2011).
[Crossref]

Meystre, P.

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40(3), 229–232 (1982).
[Crossref]

Morthier, G.

Otsuka, K.

K. Otsuka, “Pitchfork bifurcation and all-optical digital signal-processing with a coupled-element bistable system,” Opt. Lett. 14, 7274 (1989).
[Crossref]

Paré, C.

C. Paré and M. Florjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[Crossref] [PubMed]

Peng, G. D.

Peschel, T.

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counter-propagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[Crossref] [PubMed]

Peschel, U.

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counter-propagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[Crossref] [PubMed]

Pichugin, K.

E. Bulgakov, K. Pichugin, and A. Sadreev, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[Crossref]

Reoksabutr, A.

T. Mayteevarunyoo, B. A. Malomed, and A. Reoksabutr, “Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials,” J. Mod. Opt. 58(21), 1977–1989 (2011).
[Crossref]

Sadreev, A.

E. Bulgakov, K. Pichugin, and A. Sadreev, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[Crossref]

E. Bulgakov and A. Sadreev, “Switching through symmetry breaking for transmission in a T-shaped photonic waveguide coupled with two identical nonlinear micro-cavities,” J. Phys. Condens. Matter 23, 315303 (2011).
[Crossref] [PubMed]

Sadreev, A. F.

Soljacic, M.

Soto-Crespo, J. M.

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[Crossref]

Theocharis, G.

C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
[Crossref]

Torres, J. P.

J. P. Torres, J. Boyce, and R.Y. Chiao, “Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability,” Phys. Rev. Lett. 83, 4293–4296 (1999).
[Crossref]

Trefethen, L. N.

L. N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, 2000.
[Crossref]

Wang, C.

C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
[Crossref]

Whitaker, N.

C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
[Crossref]

Yuan, L.

J. Mod. Opt. (1)

T. Mayteevarunyoo, B. A. Malomed, and A. Reoksabutr, “Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials,” J. Mod. Opt. 58(21), 1977–1989 (2011).
[Crossref]

J. Opt. Soc. Am. B (5)

J. Phys. Condens. Matter (1)

E. Bulgakov and A. Sadreev, “Switching through symmetry breaking for transmission in a T-shaped photonic waveguide coupled with two identical nonlinear micro-cavities,” J. Phys. Condens. Matter 23, 315303 (2011).
[Crossref] [PubMed]

Opt. Commun. (1)

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40(3), 229–232 (1982).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

K. Otsuka, “Pitchfork bifurcation and all-optical digital signal-processing with a coupled-element bistable system,” Opt. Lett. 14, 7274 (1989).
[Crossref]

M. Haelterman and P. Mandel, “Pitchfork bifurcation using a 2-beam nonlinear Fabry-Perot interferometer: an analytical study,” Opt. Lett. 15, 1412–1414 (1990).
[Crossref] [PubMed]

Phys. Rev. A (3)

C. Paré and M. Florjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[Crossref] [PubMed]

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counter-propagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[Crossref] [PubMed]

V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
[Crossref]

Phys. Rev. B (1)

E. Bulgakov, K. Pichugin, and A. Sadreev, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[Crossref]

Phys. Rev. E (2)

C. Wang, G. Theocharis, P. G. Kevrekidis, N. Whitaker, K. J. H. Law, D. J. Frantzeskakis, and B. A. Malomed, “Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential,” Phys. Rev. E 80, 046611 (2009).
[Crossref]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[Crossref]

Phys. Rev. Lett. (1)

J. P. Torres, J. Boyce, and R.Y. Chiao, “Bilateral symmetry breaking in a nonlinear Fabry-Perot cavity exhibiting optical tristability,” Phys. Rev. Lett. 83, 4293–4296 (1999).
[Crossref]

Sov. Phys. JETP (1)

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56(2), 299–303 (1982).

Other (1)

L. N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, 2000.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 One or two nonlinear cylinders parallel to the z axis and illuminated by a plane wave propagating in the x direction.
Fig. 2
Fig. 2 Case I: (a) normalized scattered power P vs. the intensity of the incident wave I, where the blue and red curves represent symmetric and asymmetric solutions, respectively; (b) difference between the normalized scattered powers of the asymmetric and symmetric solutions; (c) the asymmetry measure η for the asymmetric solutions.
Fig. 3
Fig. 3 Electric field distributions for case I with I = 40I0. (a) and (b): real and imaginary parts of the symmetric solution; (c) and (d): real and imaginary parts of an asymmetric solution.
Fig. 4
Fig. 4 Case II: (a) normalized scattered power P vs. intensity I of the incident wave, where blue and red curves represent the symmetric and asymmetric solutions; (b) difference between the normalized scattered powers of the asymmetric and symmetric solutions; (c) the measure of asymmetry η.
Fig. 5
Fig. 5 Electric field distributions for case II with I = 30I0. (a) and (b): real and imaginary parts of the symmetric solution; (c) and (d): real and imaginary parts of an asymmetric solution.

Equations (12)

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

2 u x 2 + 2 u y 2 + k 0 2 ( n 2 + γ | u | 2 ) u = 0 ,
u ( i ) ( x , y ) = A e i k 0 n 0 x ,
lim r r [ u ( s ) r i k 0 n 0 u ( s ) ] = 0 ,
u ν = Λ u + A f on Ω ,
u ( s ) ( x , y ) = m = c m H m ( 1 ) ( k 0 n 0 r ) e i m θ , r > a ,
Λ e i m θ = λ m e i m θ , λ m = k 0 n 0 H m ( 1 ) ( k 0 n 0 a ) H m ( 1 ) ( k 0 n 0 a ) ,
u ( s ) ( x , y ) = l = 1 2 m = c m ( l ) H m ( 1 ) ( k 0 n 0 r ( l ) ) e i m θ ( l ) ,
u ( j + 1 ) + k 0 2 γ | u ( j ) | 2 u ( j + 1 ) = 0 ,
u ( j + 1 ) + k 0 2 γ { 2 | u ( j ) | 2 u ( j + 1 ) + [ u ( j ) ] 2 u ¯ ( j + 1 ) } = 2 k 0 2 γ | u ( j ) | 2 u ( j ) .
u ( j + 1 ) + 2 k 0 2 γ | u ( j ) | 2 u ( j + 1 ) = k 0 2 γ | u ( j ) | 2 u ( j ) .
r p = a cos ( p π 2 N 1 ) , θ q = π ( 2 q 1 ) M ,
η = Ω | u ( x , y ) u ( x , y ) | 2 d x d y 4 Ω | u ( x , y ) | 2 d x d y

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