Abstract

We present a study of transverse optical forces arising in a freestanding silicon nanowire waveguide. A theoretical framework is provided for the calculation of the optical forces existing between a waveguide and a dielectric substrate. The force is evaluated using a numerical procedure based on finite-element simulations. In addition, an analytical formalism is developed which allows for a simple approximate analysis of the problem. We find that in this configuration optical forces on the order of pN can be obtained, sufficient to actuate nano-mechanical devices.

©2009 Optical Society of America

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References

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  1. S. S. Verbridge, H. G. Craighead, and J. M. Parpia, “A megahertz nanomechanical resonator with room temperature quality factor over a million,” Appl. Phys. Lett. 92, 3112–3114 (2008).
    [Crossref]
  2. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15, 17,172–17,205 (2007).
    [Crossref]
  3. H. Rokhsari, T. Kippenberg, T. Carmon, and K. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13, 5293–5301 (2005).
    [Crossref] [PubMed]
  4. B. Kemp, T. Grzegorczyk, and J. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express 13, 9280–9291 (2005).
    [Crossref] [PubMed]
  5. M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express 13, 2245–2250 (2005).
    [Crossref] [PubMed]
  6. J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A 8, 14–21 (1973).
    [Crossref]
  7. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004).
    [Crossref] [PubMed]
  8. R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express 14, 11,855–11,869 (2006).
    [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] [PubMed]
  10. M. Povinelli, S. Johnson, M. Lonèar, M. Ibanescu, E. Smythe, F. Capasso, and J. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,” Opt. Express 13, 8286–8295 (2005).
    [Crossref] [PubMed]
  11. P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nature Photonics 1, 658–665 (2007).
    [Crossref]
  12. 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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (2008).
    [Crossref]
  13. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photonics 1, 416–422 (2007).
    [Crossref]
  14. 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, 480–484 (2008).
    [Crossref] [PubMed]
  15. D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
    [Crossref] [PubMed]
  16. K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76, 061,101 (2005).
    [Crossref]
  17. J. D. Jackson, Classical electrodynamics (J.Wiley and sons, New York, 1975).
  18. F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” European Physical Journal D 46, 157–164 (2008).
    [Crossref]
  19. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
  20. A. Kumar, K. Thyagarajan, and A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Opt. Lett. 8, 63–65 (1983).
    [Crossref] [PubMed]
  21. P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. 19, 2848–2855 (1980).
    [Crossref] [PubMed]

2008 (4)

S. S. Verbridge, H. G. Craighead, and J. M. Parpia, “A megahertz nanomechanical resonator with room temperature quality factor over a million,” Appl. Phys. Lett. 92, 3112–3114 (2008).
[Crossref]

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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (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, 480–484 (2008).
[Crossref] [PubMed]

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” European Physical Journal D 46, 157–164 (2008).
[Crossref]

2007 (3)

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photonics 1, 416–422 (2007).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15, 17,172–17,205 (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,” Nature Photonics 1, 658–665 (2007).
[Crossref]

2006 (1)

R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express 14, 11,855–11,869 (2006).
[Crossref]

2005 (6)

2004 (2)

M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004).
[Crossref] [PubMed]

D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
[Crossref] [PubMed]

1983 (1)

1980 (1)

1973 (1)

J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A 8, 14–21 (1973).
[Crossref]

1969 (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Antezza, M.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” European Physical Journal D 46, 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, 480–484 (2008).
[Crossref] [PubMed]

Barnett, S. M.

R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express 14, 11,855–11,869 (2006).
[Crossref]

Budakian, R.

D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
[Crossref] [PubMed]

Capasso, F.

Carmon, T.

Carusotto, I.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” European Physical Journal D 46, 157–164 (2008).
[Crossref]

Chui, B.

D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
[Crossref] [PubMed]

Craighead, H. G.

S. S. Verbridge, H. G. Craighead, and J. M. Parpia, “A megahertz nanomechanical resonator with room temperature quality factor over a million,” Appl. Phys. Lett. 92, 3112–3114 (2008).
[Crossref]

Eichenfield, M.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photonics 1, 416–422 (2007).
[Crossref]

Ekinci, K. L.

K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76, 061,101 (2005).
[Crossref]

Ghatak, A. K.

Gordon, J. P.

J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A 8, 14–21 (1973).
[Crossref]

Grzegorczyk, T.

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, 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,” Nature Photonics 1, 658–665 (2007).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical electrodynamics (J.Wiley and sons, New York, 1975).

Joannopoulos, J.

Joannopoulos, J. D.

Johnson, S.

Johnson, S. G.

Kemp, B.

Kippenberg, T.

Kippenberg, T. J.

T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15, 17,172–17,205 (2007).
[Crossref]

Kong, J.

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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (2008).
[Crossref]

Kumar, A.

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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (2008).
[Crossref]

Li, 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, 480–484 (2008).
[Crossref] [PubMed]

Loncar, M.

Lonèar, M.

Loudon, R.

R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express 14, 11,855–11,869 (2006).
[Crossref]

Mamin, H.

D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
[Crossref] [PubMed]

Mansuripur, M.

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Michael, C. P.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photonics 1, 416–422 (2007).
[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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (2008).
[Crossref]

Painter, O.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photonics 1, 416–422 (2007).
[Crossref]

Parpia, J. M.

S. S. Verbridge, H. G. Craighead, and J. M. Parpia, “A megahertz nanomechanical resonator with room temperature quality factor over a million,” Appl. Phys. Lett. 92, 3112–3114 (2008).
[Crossref]

Perahia, R.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photonics 1, 416–422 (2007).
[Crossref]

Pernice, W. H. P.

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, 480–484 (2008).
[Crossref] [PubMed]

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,” Nature Photonics 1, 658–665 (2007).
[Crossref]

Povinelli, M.

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,” Nature Photonics 1, 658–665 (2007).
[Crossref]

Recati, A.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” European Physical Journal D 46, 157–164 (2008).
[Crossref]

Riboli, F.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” European Physical Journal D 46, 157–164 (2008).
[Crossref]

Rokhsari, H.

Roukes, M. L.

K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76, 061,101 (2005).
[Crossref]

Rugar, D.

D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
[Crossref] [PubMed]

Smythe, E.

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,” Nature Photonics 1, 658–665 (2007).
[Crossref]

Tang, H. X.

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, 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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (2008).
[Crossref]

Taylor, H. F.

Thyagarajan, K.

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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (2008).
[Crossref]

Vahala, K.

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15, 17,172–17,205 (2007).
[Crossref]

Verbridge, S. S.

S. S. Verbridge, H. G. Craighead, and J. M. Parpia, “A megahertz nanomechanical resonator with room temperature quality factor over a million,” Appl. Phys. Lett. 92, 3112–3114 (2008).
[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, 480–484 (2008).
[Crossref] [PubMed]

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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (2008).
[Crossref]

Yeh, P.

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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (2008).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. S. Verbridge, H. G. Craighead, and J. M. Parpia, “A megahertz nanomechanical resonator with room temperature quality factor over a million,” Appl. Phys. Lett. 92, 3112–3114 (2008).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

European Physical Journal D (1)

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” European Physical Journal D 46, 157–164 (2008).
[Crossref]

Nature (2)

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, 480–484 (2008).
[Crossref] [PubMed]

D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004).
[Crossref] [PubMed]

Nature Photonics (2)

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photonics 1, 416–422 (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,” Nature Photonics 1, 658–665 (2007).
[Crossref]

Opt. Express (7)

Opt. Lett. (2)

Phys. Rev. A (1)

J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A 8, 14–21 (1973).
[Crossref]

Physical Review B (Condensed Matter and Materials Physics) (1)

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,” Physical Review B (Condensed Matter and Materials Physics) 78, 165,129 (2008).
[Crossref]

Rev. Sci. Instrum. (1)

K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76, 061,101 (2005).
[Crossref]

Other (1)

J. D. Jackson, Classical electrodynamics (J.Wiley and sons, New York, 1975).

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Figures (6)

Fig. 1.
Fig. 1. Illustration of the device. a) Render view of the freestanding silicon waveguide suspended over a SiO 2 substrate. Overlaid in color is the mode pattern of a propagating wave with a wavelength of 1.55 μm. The nano-mechanical resonator is released by removing the oxide under the silicon waveguide by isotropic etching. b) A cross section view of the device in the center of the groove detailing the dimensions used in the force calculations.

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Fig. 2.
Fig. 2. Illustration of the device. a) The dispersion relation of the device in Fig 1(a) obtained from finite-element calculations. b) The optical force calculated from the dispersion relation using Eq.(3). The force is normalized to optical power and beam length and thus given in pN/μm/mW.

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Fig. 3.
Fig. 3. The optical force acting on the waveguide can be understood in analogy to the concept of a mirror charge in electro-statics. An image waveguide is assumed to be present on the other side of the substrate-air interface. The symmetric mode of the coupled waveguides is the origin of the attractive optical force.

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Fig. 4.
Fig. 4. The optical force calculated using Eq.(4) with a shielding factor of 0.65. These results agree well when the gap values are large enough, that the coupling between the waveguide and the substrate is weak.

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Fig. 5.
Fig. 5. Comparison of the approximate solution for the effective index and the numerical solution of the dispersion Eq. for the effective index. The approximation of neff with the exponential decay from Eq.(27) is very close to the direct solution of Eq.(12).

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Fig. 6.
Fig. 6. The calculated force in comparison with the effective index method. When the effective width is used in the calculations (red markers), both methods are in good agreement. Strong deviations are found when the actual width of 110 nm is used in the analytical expressions.

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Equations (38)

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

F = − 1 ω ∂ ω ∂ g U
F = 1 n eff ∂ n eff ∂ g U
F n = n g c n eff ∂ n eff ∂ g
F coupled = α n g c n eff ∂ n eff ∂ g
E ( r , t ) = E ( y ) e i ( ωt − βz )
H ( r , t ) = H ( y ) e i ( ωt − βz )
ψ = { 1 , − β ωμ i ωμ } T E x
k s = β 2 − k 2 n s 2
k a = β 2 − k 2 n a 2
k c = k 2 n c 2 − β 2
E x = A { a 1 e k s y y < 0 a 2 e − k a y + a 3 e k a y 0 ≤ y < g cos ( kc ( y − g ) + ϕ ) g ≤ y < g + h a 4 e − k a ( y − g − h ) y ≥ g + h
k c h = tan − 1 ( k a k c ) + tan − 1 ( k a 2 sinh ( k a g ) + k a k s cosh ( k a g ) k s k c sinh ( k a g ) + k c k a cosh ( k a g ) ) − m π
a 1 = 2 a 3 k a k a + k s a 2 = a 3 k a − k s k a + k s
a 3 = k a + k s 2 M a 4 = k c k a 2 + k s 2
M = ( k a k c ) 2 [ k a sinh ( k a g ) + k s cosh ( k a g ) ] 2 + [ k s sinh ( k a g ) + k a cosh ( k a g ) ] 2
β ( g ) = β 0 + β 1 e − σg
k a ≈ α a + β 0 β 1 α a e − σg
k s ≈ α s + β 0 β 1 α s e − σg
k c ≈ α c − β 0 β 1 α c e − σg
α a = β 0 2 − k 2 n a 2
α s = β 0 2 − k 2 n s 2
α c = k 2 n s 2 − β 0 2
tan ( k c h ) = k a k c 1 + Q 1 − ( k a k c ) 2 Q
Q = k a sinh ( k a g ) + k s cosh ( k a g ) k s sinh ( k a g ) + k a cosh ( k a g )
tan ( k c h ) ≈ tan ( α c ) ≡ P
( P − h α c β 0 β 1 ) ( 1 − ( k a k c ) 2 k s k a ) − k a k c ( 1 + k s k a ) ( 1 + P h α c β 0 β 1 ) = 0
[ P h α c 2 ( 1 α s + 1 α a ) − h α c ( 1 α c 2 + 1 α a α s ) ] β 0 3 β 1 3
+ [ P ( 1 α c 2 + 1 α a α s ) + h α c ( 2 + α s α a + α a α s )
+ 1 α c + ( 1 α s + 1 α a ) − Ph α c ( α c α s + α c α a − α s α c − α a α c ) ] β 0 2 β 1 2
− [ P ( 2 + α s α a + α a α s ) + h α c ( α c 2 − α s α a ) + Ph ( α a + α s ) + ( α c α s + α x α a − α s α c − α a α c ) ] β 0 β 1
+ [ P ( α c 2 − α s α a ) − α c ( α s + α a ) ] = 0
σ = 2 k a In ( 2 ) − In { P ( k a − k s ) ( k a 2 + k c 2 ) ( k a + k s ) ( k a 2 − k c 2 ) P + 2 k a k c }
n eff ( g ) = k ( β 0 + β 1 e − σg ) = n 0 + n 1 e − σg
T yy = − ε 0 4 { ∣ E x ∣ 2 + c 2 μ 2 ( ∣ H z ∣ 2 − ∣ H y ∣ 2 ) }
= − ε 0 4 { ∣ E x ∣ 2 ( 1 − β 2 k 2 ) + 1 k 2 ∣ ∂ y E x ∣ 2 }
T yy = − ε 0 4 ∣ E x ∣ 2 ( 1 − n a 2 )
F optical = − T yy = ε 0 M 2 A 2 ( k 2 − β 2 ) ( n s 2 − n a 2 )
w eff = h ∫ A wg ( E y wg ) 2 dxdy ∫ A wg ( E y analytical ) 2 dxdy

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