Abstract

We present a rigorous analysis of the optical gradient force between coupled single-mode waveguides in three dimensions. Using eigenmode expansion we determine the optical mode patterns in the coupled system. In contrast to previous work, the sign and amplitude of the optical force is found to vary along the waveguide with a characteristic beating length. Our results establish design principles for optomechanically tunable directional couplers.

© 2009 OSA

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References

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  1. 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(22), 3042–3044 (2005).
    [CrossRef] [PubMed]
  2. A. Mizrahi and L. Schächter, “Two-slab all-optical spring,” Opt. Lett. 32(6), 692–694 (2007).
    [CrossRef] [PubMed]
  3. F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).
    [CrossRef]
  4. M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,” Opt. Express 13(20), 8286–8295 (2005).
    [CrossRef] [PubMed]
  5. P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
    [CrossRef]
  6. H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
    [CrossRef]
  7. W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
    [CrossRef] [PubMed]
  8. M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
    [CrossRef] [PubMed]
  9. M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
    [CrossRef] [PubMed]
  10. G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. 5(5), 554–557 (1993).
    [CrossRef]
  11. P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14(2), 164–166 (2002).
    [CrossRef]
  12. A. Yariv, “Quantum Electronics,” Wiley, New York (1989).
  13. A. D. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antenn. Propag. 55(6), 1495–1505 (2007).
    [CrossRef]
  14. M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
    [CrossRef]
  15. U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. 30(5), 406–408 (1994).
    [CrossRef]

2009

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[CrossRef]

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[CrossRef] [PubMed]

2008

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
[CrossRef]

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).
[CrossRef]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[CrossRef] [PubMed]

2007

A. D. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antenn. Propag. 55(6), 1495–1505 (2007).
[CrossRef]

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

A. Mizrahi and L. Schächter, “Two-slab all-optical spring,” Opt. Lett. 32(6), 692–694 (2007).
[CrossRef] [PubMed]

2005

2002

P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14(2), 164–166 (2002).
[CrossRef]

1994

U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. 30(5), 406–408 (1994).
[CrossRef]

1993

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. 5(5), 554–557 (1993).
[CrossRef]

Antezza, M.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).
[CrossRef]

Baehr-Jones, T.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[CrossRef] [PubMed]

Baets, R.

P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14(2), 164–166 (2002).
[CrossRef]

Bienstman, P.

P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14(2), 164–166 (2002).
[CrossRef]

Capasso, F.

Carusotto, I.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).
[CrossRef]

Fischer, U.

U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. 30(5), 406–408 (1994).
[CrossRef]

Hochberg, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[CrossRef] [PubMed]

Ibanescu, M.

Ippen, E. P.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kuga, T.

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
[CrossRef]

Kuramochi, E.

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
[CrossRef]

Li, M.

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[CrossRef] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[CrossRef] [PubMed]

Loncar, M.

Mizrahi, A.

Nolting, H. P.

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. 5(5), 554–557 (1993).
[CrossRef]

Notomi, M.

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
[CrossRef]

Pernice, W. H. P.

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[CrossRef]

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[CrossRef] [PubMed]

Petermann, K.

U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. 30(5), 406–408 (1994).
[CrossRef]

Popovic, M. A.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Povinelli, M. L.

Rakich, P. T.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Recati, A.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).
[CrossRef]

Riboli, F.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).
[CrossRef]

Roelens, A.

P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14(2), 164–166 (2002).
[CrossRef]

Schächter, L.

Schuppert, B.

U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. 30(5), 406–408 (1994).
[CrossRef]

Six, E.

P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14(2), 164–166 (2002).
[CrossRef]

Smythe, E. J.

Soljacic, M.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Sztefka, G.

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. 5(5), 554–557 (1993).
[CrossRef]

Tang, H. X.

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[CrossRef]

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[CrossRef] [PubMed]

Taniyama, H.

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
[CrossRef]

Torii, Y.

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
[CrossRef]

Vanwolleghem, M.

P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14(2), 164–166 (2002).
[CrossRef]

Xiong, C.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[CrossRef] [PubMed]

Yaghjian, A. D.

A. D. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antenn. Propag. 55(6), 1495–1505 (2007).
[CrossRef]

Yamamoto, T.

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
[CrossRef]

Yoshikawa, Y.

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
[CrossRef]

Zinke, T.

U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. 30(5), 406–408 (1994).
[CrossRef]

Electron. Lett.

U. Fischer, T. Zinke, B. Schuppert, and K. Petermann, “Singlemode optical switches based on SOI waveguides with large cross-section,” Electron. Lett. 30(5), 406–408 (1994).
[CrossRef]

Eur. Phys. J. D

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46(1), 157–164 (2008).
[CrossRef]

IEEE Photon. Technol. Lett.

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. 5(5), 554–557 (1993).
[CrossRef]

P. Bienstman, E. Six, A. Roelens, M. Vanwolleghem, and R. Baets, “Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers,” IEEE Photon. Technol. Lett. 14(2), 164–166 (2002).
[CrossRef]

IEEE Trans. Antenn. Propag.

A. D. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antenn. Propag. 55(6), 1495–1505 (2007).
[CrossRef]

Nat. Nanotechnol.

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[CrossRef] [PubMed]

Nat. Photonics

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[CrossRef]

Nature

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. B

H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008).
[CrossRef]

Other

A. Yariv, “Quantum Electronics,” Wiley, New York (1989).

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Figures (5)

Fig. 1
Fig. 1

(a) The transfer matrix description of the directional coupler. The coupler is excited from the left side with mode amplitudes L and R in the left and right waveguides, respectively. (b) The geometric dimensions of the waveguides that form the directional coupler.

Fig. 2
Fig. 2

The different excitation cases used in the determination of the force distribution on coupled beams. The Left mode is excited, when only the left waveguide carries its fundamental mode, and the equivalent holds for the Right mode. The Even/Odd modes are the eigenmodes of the coupled waveguide system. The Symmetric mode arises from equal excitation of both the left and the right waveguide modes. The Anti-symmetric mode results from equal excitation of the Left and Right waveguide modes with a phase difference of π.

Fig. 3
Fig. 3

Shown is a plot of the field distribution of a directional coupler of 10μm length along the propagation direction. We illustrate the profiles for normalized modes with a phase difference at the input side of 0 and π (0° and 180°). Because both the even and odd mode of the coupled waveguides are excited, a beating pattern with a beating length of 6.4μm is present.

Fig. 4
Fig. 4

The different optical force contributions to the net optical force for two coupled freestanding waveguides of cross-section 220x300nm2. (a) The attractive force resulting from the excitation of the even mode and the force resulting from the odd mode. This force component is attractive for gaps smaller than 130nm and repulsive otherwise. (b) The beating force term due to the excitation of a superposition of even and odd modes.

Fig. 5
Fig. 5

(a) Tunability of the net optical force. By varying the input phase from 0 to π, the sign of the force can be switched from negative to positive. (b) Comparison of the efficiency of a directional coupler using for single and double waveguide excitation.

Equations (6)

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

L=aE+bOR=cE+dO}(LR)=(abcd)(EO)=M(EO)
a=LE*dxdyb=LO*dxdyc=RE*dxdyd=RO*dxdy
F(z)=weFe+woFo+cos(Δβz)wbFb
Fe=(a2+c2+2accos(Φ0))CRe{ε0(EEn^)EE*ε02(EEEE*)n^+μ0(EHn^)EH*μ02(EHEH*)n^}dl
Fo=(b2+d2+2bdcos(Φ0))CRe{ε02(OEn^)OE*ε02(OEOE*)n^+μ02(OHn^)OH*μ02(OHOH*)n^}dl
Fb=4cos(Δβz)(ab+cd+2(ad+bc)cos(Φ0))CRe{ε02(OEn^)EE*ε04(OEEE*)n^+μ02(OHn^)EH*μ04(OHEH*)n^}dl

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