Abstract

We develop a coupled-mode theory that describes the dependence of optical gradient forces between side-coupled periodic waveguides on the longitudinal shift between the waveguides. Our approach is fully applicable to waveguides with a strong refractive-index modulation and in the regime of slow-light enhancement of optical forces, associated with the group-velocity reduction at the photonic band edge. Our method enables fast calculation of both the transverse and longitudinal forces for all longitudinal shifts, based on numerical simulations of mode profiles only at particular shift values. We perform a comparison with direct numerical simulations for photonic-crystal nanowire waveguides and demonstrate that our approach provides very accurate results for the slow-light enhanced transverse and longitudinal forces, accounting for the key features of force suppression and sign reversal at critical shift values.

© 2013 Optical Society of America

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References

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  1. M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
    [CrossRef]
  2. J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
    [CrossRef]
  3. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
    [CrossRef]
  4. M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a ‘zipper’ photonic crystal optomechanical cavity,” Opt. Express 17, 3802–3817 (2009).
    [CrossRef]
  9. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005).
    [CrossRef]
  10. P. T. Rakich, M. A. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17, 18116–18135 (2009).
    [CrossRef]
  11. Z. Wang and P. Rakich, “Response theory of optical forces in two-port photonics systems: a simplified framework for examining conservative and non-conservative forces,” Opt. Express 19, 22322–22336 (2011).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
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  20. S. Ha, A. A. Sukhorukov, and Yu. S. Kivshar, “Slow-light switching in nonlinear Bragg-grating couplers,” Opt. Lett. 32, 1429–1431 (2007).
    [CrossRef]
  21. Y. J. Tsofe and B. A. Malomed, “Quasisymmetric and asymmetric gap solitons in linearly coupled Bragg gratings with a phase shift,” Phys. Rev. E 75, 056603 (2007).
    [CrossRef]
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2012 (1)

2011 (3)

2010 (1)

J. Ma and M. L. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Lett. 97, 151102 (2010).
[CrossRef]

2009 (6)

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

P. T. Rakich, M. A. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17, 18116–18135 (2009).
[CrossRef]

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a ‘zipper’ photonic crystal optomechanical cavity,” Opt. Express 17, 3802–3817 (2009).
[CrossRef]

C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express 17, 5879–5884 (2009).
[CrossRef]

2008 (3)

2007 (2)

S. Ha, A. A. Sukhorukov, and Yu. S. Kivshar, “Slow-light switching in nonlinear Bragg-grating couplers,” Opt. Lett. 32, 1429–1431 (2007).
[CrossRef]

Y. J. Tsofe and B. A. Malomed, “Quasisymmetric and asymmetric gap solitons in linearly coupled Bragg gratings with a phase shift,” Phys. Rev. E 75, 056603 (2007).
[CrossRef]

2005 (1)

2001 (1)

1996 (1)

Baets, R.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Bagheri, M.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
[CrossRef]

Benton, C. J.

C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express 17, 5879–5884 (2009).
[CrossRef]

C. J. Benton, A. V. Gorbach, and D. V. Skryabin, “Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires,” Phys. Rev. A 78, 033818(2008).
[CrossRef]

Camacho, R.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a ‘zipper’ photonic crystal optomechanical cavity,” Opt. Express 17, 3802–3817 (2009).
[CrossRef]

Capasso, F.

Chan, J.

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a ‘zipper’ photonic crystal optomechanical cavity,” Opt. Express 17, 3802–3817 (2009).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

Chigrin, D. N.

De Vlaminck, I.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Eichenfield, M.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a ‘zipper’ photonic crystal optomechanical cavity,” Opt. Express 17, 3802–3817 (2009).
[CrossRef]

Erdogan, T.

Favuzzi, P. A.

Gorbach, A. V.

C. J. Benton, A. V. Gorbach, and D. V. Skryabin, “Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires,” Phys. Rev. A 78, 033818(2008).
[CrossRef]

Ha, S.

Ibanescu, M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Joannopoulos, J. D.

Johnson, S. G.

Kawakami, Y.

Kivshar, Y. S.

Kivshar, Yu. S.

Lagae, L.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Lavrinenko, A. V.

Li, M.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

Loncar, M.

Ma, J.

J. Ma and M. L. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Lett. 97, 151102 (2010).
[CrossRef]

Maes, B.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Malomed, B. A.

Y. J. Tsofe and B. A. Malomed, “Quasisymmetric and asymmetric gap solitons in linearly coupled Bragg gratings with a phase shift,” Phys. Rev. E 75, 056603 (2007).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Noda, S.

Oskooi, A.

Painter, O.

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a ‘zipper’ photonic crystal optomechanical cavity,” Opt. Express 17, 3802–3817 (2009).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

Pelinovsky, D. E.

Pernice, W. H. P.

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

Pernice, W. P. H.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
[CrossRef]

Poot, M.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
[CrossRef]

Popovic, M. A.

Povinelli, M. L.

J. Ma and M. L. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Lett. 97, 151102 (2010).
[CrossRef]

M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005).
[CrossRef]

Rakich, P.

Rakich, P. T.

Roels, J.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Sipe, J. E.

Skryabin, D. V.

C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express 17, 5879–5884 (2009).
[CrossRef]

C. J. Benton, A. V. Gorbach, and D. V. Skryabin, “Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires,” Phys. Rev. A 78, 033818(2008).
[CrossRef]

Smythe, E. J.

Sukhorukov, A. A.

Sun, Y.

Tang, H. X.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

Tsofe, Y. J.

Y. J. Tsofe and B. A. Malomed, “Quasisymmetric and asymmetric gap solitons in linearly coupled Bragg gratings with a phase shift,” Phys. Rev. E 75, 056603 (2007).
[CrossRef]

Vahala, K. J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

Van Thourhout, D.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Wang, Z.

White, T. P.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Yeh, P.

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, 1988).

Appl. Phys. Lett. (1)

J. Ma and M. L. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Lett. 97, 151102 (2010).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

Nat. Nanotechnol. (2)

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6, 726–732 (2011).
[CrossRef]

Nat. Photonics (1)

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

Nature (1)

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

Opt. Express (5)

Opt. Lett. (4)

Phys. Rev. A (1)

C. J. Benton, A. V. Gorbach, and D. V. Skryabin, “Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires,” Phys. Rev. A 78, 033818(2008).
[CrossRef]

Phys. Rev. E (1)

Y. J. Tsofe and B. A. Malomed, “Quasisymmetric and asymmetric gap solitons in linearly coupled Bragg gratings with a phase shift,” Phys. Rev. E 75, 056603 (2007).
[CrossRef]

Other (4)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

J. B. Khurgin and R. S. Tucker, eds., Slow Light: Science and Applications (Taylor & Francis, 2009).

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, 1988).

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

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Figures (4)

Fig. 1.
Fig. 1.

(a) 3D sketch of the longitudinally shifted side-coupled photonic-crystal waveguides. The waveguides are identical, and each is patterned with a regular array of rectangular airholes of period a. (b) Top view of the waveguide geometry.

Fig. 2.
Fig. 2.

(a) Dispersion relation for the TM bands near the band edge for a half-period longitudinal shift (ϕ=π), calculated using a fully vectorial plane-wave method. Labels show the bandgap center frequency (ω0), lower band edge frequency (ωe), second-lowest band’s frequency at the edge of the first Brillouin zone (ωb), and bandgap width (Δω). (b)–(g) Comparison between plane-wave simulation results (circles) and CMT prediction (solid curves) for the second lowest band [shown in red in (a)] at different longitudinal shifts ϕ: (b) 0, (c) 0.2π, (d) 0.4π, (e) 0.6π, (f) 0.8π, and (g) π. For all the plots, the structure geometrical parameters are w=7a/6, d=a/6, wx=0.445a, wy=2a/3; the waveguide height is h=2a/3; and the refractive index of the waveguides is 2.

Fig. 3.
Fig. 3.

Normalized amplitudes of (a) even-forward, (b) odd-forward, (c) even-backward, and (d) odd-backward mode at the lower band edge (Ke, ωe) as a function of the longitudinal shift Δx. The curves were calculated using the CMT model with coefficients C0=0.1048/a, ρ=0.2129/a, and n0=1.7511.

Fig. 4.
Fig. 4.

Comparison of the optical forces acting on the waveguides as a function of the longitudinal shift, calculated using CMT (solid curves) and full numerical simulations (circles). (a) Longitudinal force. (b) Transverse force. Shown are the force values per unit length per unit energy density.

Equations (34)

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(ExEyEzHxHyHz)(x,y,z;ω)=±(ExEyEzHxHyHz)(x,y,z;ω),
{E,H}(x+a,y,z;ω)={E,H}(x,y,z;ω)exp(iKx),
{E,H}(x,y,z;K)=exp[i(KKb)x][A(f)(K){E(f),H(f)}(x,y,z)exp(iKbx)+A(b)(K){E(b),H(b)}(x,y,z)exp(iKbx)],
(Ex(b)Ey(b)Ez(b)Hx(b)Hy(b)Hz(b))(x,y,z)=(Ex(f)Ey(f)Ez(f)Hx(f)Hy(f)+Hz(f))(x,y,z).
{E,H}(x,y,z;t)=A(f)(x,t){E(f),H(f)}(x,y,z)exp(iKbx)+A(b)(x,t){E(b),H(b)}(x,y,z)exp(iKbx).
(ExEyEzHxHyHz)=(Ex(e)Ex(e)Ex(o)Ex(o)Ey(e)Ey(e)Ey(o)Ey(o)Ez(e)Ez(e)Ez(o)Ez(o)Hx(e)Hx(e)Hx(o)Hx(o)Hy(e)Hy(e)Hy(o)Hy(o)Hz(e)Hz(e)Hz(o)Hz(o))(A(ef)eiKbxA(eb)eiKbxA(of)eiKbxA(ob)eiKbx),
icu1t+in0u1x+ρw1+C0u2=0,
icw1tin0w1x+ρ*u1+C0w2=0,
icu2t+in0u2x+ρeiϕw2+C0u1=0,
icw2tin0w2x+ρ*eiϕu2+C0w1=0,
ϕ=Δx2π/a.
{A(ef),A(eb)}=({u,w}1+{u,w}2)/2,{A(of),A(ob)}=({u,w}1{u,w}2)/2.
u1,2(x,t)=U1,2ei(KKb)xi(ωω0)t,w1,2(x,t)=W1,2ei(KKb)xi(ωω0)t,
(ωω0)2c2=(Kπ/a)2n02+C02+|ρ|2±2C0(Kπ/a)2n02+|ρ|2cos2(ϕ/2).
(ωω0)2c2=[(Kπ/a)/n0±C0]2+|ρ|2.
Ke=π/aC0n0,ωe=ω0±|ρ|c.
ρ=Δω/(2c),
K=Kb=π/a,ωb=ω0cC02+|ρ|2,
C0=1c(ωbω0)2|Δω|24.
n0=(π/aKe)/C0.
Fα=βSβTαββ^dSβ,
Fα=SyTαyy^dSy|y=0.
Txy=ϵ(Ex*Ey+ExEy*)+μ(Hx*Hy+HxHy*),
Tyy=ϵ(|Ey|2|Ex|2|Ez|2)+μ(|Hy|2|Hx|2|Hz|2).
fx=(|A(ef)|2|A(eb)|2)fx(e)+(|A(of)|2|A(ob)|2)fx(o)+[(A(ef)A(of)*A(eb)A(ob)*)fx(oe)+c.c.],
fx(e)=Sy[ϵ(Ex(e)*Ey(e)+Ex(e)Ey(e)*)+μ(Hx(e)*Hy(e)+Hx(e)Hy(e)*)]y^dSy|y=0,
fx(o)=Sy[ϵ(Ex(o)*Ey(o)+Ex(o)Ey(o)*)+μ(Hx(o)*Hy(o)+Hx(o)Hy(o)*)]y^dSy|y=0,
fx(oe)=Sy[ϵ(Ex(e)Ey(o)*+Ex(o)*Ey(e))+μ(Hx(e)Hy(o)*+Hx(o)*Hy(e))]y^dSy|y=0.
fy=(|A(ef)|2+|A(eb)|2)fy(e)+(|A(of)|2+|A(ob)|2)fy(o)+[(A(ef)A(of)*A(eb)A(ob)*+c.c.)fy(oe)+c.c.],
fy(e)=Sy[ϵ(|Ey(e)|2|Ex(e)|2|Ez(e)|2)+μ(|Hy(e)|2|Hx(e)|2|Hz(e)|2)]y^dSy,
fy(o)=Sy[ϵ(|Ey(o)|2|Ex(o)|2|Ez(o)|2)+μ(|Hy(o)|2|Hx(o)|2|Hz(o)|2)]y^dSy|y=0,
fy(oe)=Sy[ϵ(Ey(e)Ey(o)*Ex(e)Ex(o)*Ez(e)Ez(o)*)+μ(Hy(e)Hy(o)*Hx(e)Hx(o)*Hz(e)Hz(o)*)]y^dSy|y=0.
fx=(A(ef)A(of)*A(eb)A(ob)*)fx(oe)+c.c.
fy=(|A(ef)|2+|A(eb)|2)fy(e)+(|A(of)|2+|A(ob)|2)fy(o).

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