Abstract

We present stable symmetry breaking solutions in a nonlinear optical cavity with dipole eigenmodes embedded into the propagation band of a directional photonic crystal waveguide for symmetric injecting condition. We demonstrate how this phenomenon can be exploited for all-optical switching of light transmission from the one side of the waveguide to another by application of input pulses. When the light injected to both sides of the waveguide has equal intensities but different phases, we reveal a wealth of new solutions.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2012

E. N. Bulgakov and A. F. Sadreev, “Giant optical vortex in photonic crystal waveguide with nonlinear optical cavity,” Phys. Rev. B 85, 165305–165306 (2012).
[CrossRef]

2011

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

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]

2009

N. Dror and B. A. Malomed, “Spontaneous symmetry breaking in coupled parametrically driven waveguides,” Phys. Rev. E 79, 016605 (2009).
[CrossRef]

2008

2007

2006

2004

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

2003

1998

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23, 1855–1857 (1998).
[CrossRef]

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking in optical dynamics of two bistable thin films,” Quantum Electron. 28, 104–107 (1998).
[CrossRef]

1997

C. F. Chien and R. V. Waterhouse, “Singular points of intensity streamlines in two-dimensional sound fields,” J. Acoust. Soc. Am. 101, 705–712 (1997).
[CrossRef]

1993

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef]

1990

1987

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef]

Babushkin, I. V.

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking in optical dynamics of two bistable thin films,” Quantum Electron. 28, 104–107 (1998).
[CrossRef]

Baets, R.

Bermel, P.

Bienstman, P.

Bravo-Abad, J.

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 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]

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]

Bulgakov, E. N.

E. N. Bulgakov and A. F. Sadreev, “Giant optical vortex in photonic crystal waveguide with nonlinear optical cavity,” Phys. Rev. B 85, 165305–165306 (2012).
[CrossRef]

Chien, C. F.

C. F. Chien and R. V. Waterhouse, “Singular points of intensity streamlines in two-dimensional sound fields,” J. Acoust. Soc. Am. 101, 705–712 (1997).
[CrossRef]

Dror, N.

N. Dror and B. A. Malomed, “Spontaneous symmetry breaking in coupled parametrically driven waveguides,” Phys. Rev. E 79, 016605 (2009).
[CrossRef]

Fan, S.

Gorza, S.-P.

Haelterman, M.

Haus, H. A.

Ikeda, K.

Joannopoulos, J.

J. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Joannopoulos, J. D.

Johnson, S. G.

Logvin, Yu. A.

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking in optical dynamics of two bistable thin films,” Quantum Electron. 28, 104–107 (1998).
[CrossRef]

Loiko, N. A.

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking in optical dynamics of two bistable thin films,” Quantum Electron. 28, 104–107 (1998).
[CrossRef]

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]

N. Dror and B. A. Malomed, “Spontaneous symmetry breaking in coupled parametrically driven waveguides,” Phys. Rev. E 79, 016605 (2009).
[CrossRef]

Mandel, P.

Manolatou, C.

Meade, R. D.

J. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Otsuka, K.

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]

Rodriguez, S. A.

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]

Sadreev, A. F.

E. N. Bulgakov and A. F. Sadreev, “Giant optical vortex in photonic crystal waveguide with nonlinear optical cavity,” Phys. Rev. B 85, 165305–165306 (2012).
[CrossRef]

Soljacic, M.

Suh, W.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

Villeneuve, P. R.

Wang, Z.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

Waterhouse, R. V.

C. F. Chien and R. V. Waterhouse, “Singular points of intensity streamlines in two-dimensional sound fields,” J. Acoust. Soc. Am. 101, 705–712 (1997).
[CrossRef]

Winn, J. N.

J. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Yanik, M. F.

IEEE J. Quantum Electron.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

J. Acoust. Soc. Am.

C. F. Chien and R. V. Waterhouse, “Singular points of intensity streamlines in two-dimensional sound fields,” J. Acoust. Soc. Am. 101, 705–712 (1997).
[CrossRef]

J. Phys. Condens. Matter

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]

Opt. Express

Opt. Lett.

Phys. Rev. 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]

Phys. Rev. B

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. N. Bulgakov and A. F. Sadreev, “Giant optical vortex in photonic crystal waveguide with nonlinear optical cavity,” Phys. Rev. B 85, 165305–165306 (2012).
[CrossRef]

Phys. Rev. E

N. Dror and B. A. Malomed, “Spontaneous symmetry breaking in coupled parametrically driven waveguides,” Phys. Rev. E 79, 016605 (2009).
[CrossRef]

Phys. Rev. Lett.

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef]

Quantum Electron.

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking in optical dynamics of two bistable thin films,” Quantum Electron. 28, 104–107 (1998).
[CrossRef]

Other

J. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

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Figures (4)

Fig. 1.
Fig. 1.

Two cavity dipole TM eigenmodes (space profiles of the electric field directed parallel to the rods) with the eigenfrequencies ω1a/2πc=0.371 and ω2a/2πc=0.367 in the two-dimensional square lattice PhC consisting of GaAs dielectric rods with radius 0.18a and dielectric constant ϵ=11.56, where a=0.5μm is the lattice unit. These rods are shown by black open circles. The defect shown by open white circle has the radius 0.288a and ϵ0=12. It is shifted relative to the waveguide center line by the distance 0.2a.

Fig. 2.
Fig. 2.

Frequency behavior of (a) intensities of dipole modes and (b) transmissions to the left TL and to the right TR for light injection with Ein=0.08, θ=0 onto both sides of the waveguide. In (a), blue lines show the intensity of even dipole mode I1=|A1|2, while red lines show the intensity of odd dipole mode I2=|A2|2. The parameters are given in the beginning of Section 3. In (b) and (c), red lines show TR, blue solid lines show TL. (c) Transmissions as dependent on the input amplitude Ein for ω=0.361. In (a)–(c), dashed lines show the symmetry-preserving solution, while solid lines show the SB solution. The thicker lines mark stable solutions. (d) Time dependence of the transmissions to the left (blue lines) and to the right (red lines), which follow the impulses of the input light. The first and second impulses have amplitudes 2 and 5 and durations 150 and 200, respectively (are not shown). In (b) and (c), only stable solutions are presented.

Fig. 3.
Fig. 3.

Absolute value of light amplitude (electric field) and optical streamlines (white lines) in the PhC waveguide with single nonlinear defect shown by gray open circle for ωa/2πc=0.36, P=10W/a, θ=0.

Fig. 4.
Fig. 4.

(a) Intensities of dipole modes (red for I1 and blue for I2), (b) and (c) transmissions to the left (blue lines) and to the right (red lines), and (d) ratio of maximal value of the Poynting vector power current in the interior of the defect rod to the input Poynting vector for Ein=0.064, ωa/2πc=0.361 as a function of the phase difference of the light inputs. In (a), blue lines show the intensity of even dipole mode, red lines show the intensity of odd dipole mode. Thicker lines mark the stable domains of the solutions.

Equations (8)

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iA1˙=[ω1iγ1]A1+iγ1Eineiωt(1+eiθ),iA2˙=[ω2iγ2]A2+iγ2Eineiωt(1eiθ),
δϵ(r⃗)=n0cn2|E(r⃗)|24πn0cn2|A1E1(r⃗)+A2E2(r⃗)|24π,
iA1˙=[ω1+V11iγ1]A1+V12A2+iγ1Eineiωt(1+eiθ),iA1˙=[ω2+V22iγ2]A2+V21A1+iγ2Eineiωt(1eiθ),
m|V|n=(ωm+ωn)4Nmd2r⃗δϵ(r⃗)Em(r⃗)En(r⃗),
Nm=d2r⃗ϵ(r⃗)Em2(r⃗)=a2cn2,
[ωω1+λ11I1+λ12I2+iγ1]A1+2λ12Re(A1*A2)A2=iγ1Ein(1+eiθ),2λ12Re(A1*A2)A1+[ωω2+λ22I2+λ12I1+iγ2]A2=iγ2Ein(1eiθ),
λmn=(ωm+ωn)n0c2n2216πa2σEm2(x,y)En2(x,y)d2r⃗,
tL=γ1A1+γ2A2Ein,tR=γ1A1γ2A2Eineiθ,

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