Abstract

The evanescent tails of a guiding mode as well as its first and second derivatives were measured by a modified end-fire coupling method. The effective index of the waveguide can be obtained by simultaneously fitting these three fields using single parameter. Combined with an inverse calculation algorithm, the fields with fitted evanescent tails showed great improvement in the refractive index profiling of the optical waveguide, especially at the substrate region. Single-mode optical fibers and planar waveguides of proton-exchanged (PE) and titanium-indiffusion (Ti:LiNbO3) on lithium niobate substrates with different refractive index profiles were measured for the demonstration.

©2012 Optical Society of America

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References

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    [Crossref]
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  22. M. N. Armenise, “Fabrication techniques of lithium niobate waveguides,” IEE Proc. 135, 85–91 (1988).
  23. C.-C. Wei, P.-K. Wei, and W. Fann, “Direct measurements of the true vibrational amplitudes in shear force microscopy,” Appl. Phys. Lett. 67(26), 3835–3837 (1995).
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    [Crossref]
  25. I. Fatadin, D. Ives, and M. Wicks, “Accurate magnified near-field measurement of optical waveguides using a calibrated CCD camera,” J. Lightwave Technol. 24(12), 5067–5074 (2006).
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    [Crossref]
  27. Y. Tomita, M. Sugimoto, and K. Eda, “Direct bonding of LiNbO3 single crystals for optical waveguides,” Appl. Phys. Lett. 66(12), 1484–1485 (1995).
    [Crossref]
  28. G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. 31(7), 1054–1058 (2009).
    [Crossref]

2012 (1)

L. Wang and B.-X. Xiang, “Planar waveguides in magnesium doped stoichiometric LiNbO3 crystals formed by MeV oxygen ion implantations,” Nucl. Instrum. Meth. Phys. Res. Sect. B 272, 121–124 (2012).
[Crossref]

2011 (2)

Y. Dattner and O. Yadid-Pecht, “Analysis of the effective refractive index of silicon waveguides through the constructive and destructive interference in a Mach-Zehnder interferometer,” IEEE Photonics J. 3(6), 1123–1132 (2011).
[Crossref]

W.-S. Tsai, S.-C. Piao, and P.-K. Wei, “Refractive index measurement of optical waveguides using modified end-fire coupling method,” Opt. Lett. 36(11), 2008–2010 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (2)

S. Barai and A. Sharma, “Inverse algorithm with built-in spatial filter to obtain the 2-D refractive index profile of optical waveguides from the propagating mode near-field profile,” J. Lightwave Technol. 27(11), 1514–1521 (2009).
[Crossref]

G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. 31(7), 1054–1058 (2009).
[Crossref]

2008 (1)

X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008).
[Crossref]

2007 (1)

W.-S. Tsai, W.-S. Wang, and P.-K. Wei, “Two-dimensional refractive index profiling by using differential near-field scanning optical microscopy,” Appl. Phys. Lett. 91(6), 061123 (2007).
[Crossref]

2006 (1)

2002 (1)

A. L. Campillo, J. W. P. Hsu, C. A. White, and C. D. W. Jones, “Direct measurement of the guided modes in LiNbO3 waveguides,” Appl. Phys. Lett. 80(13), 2239–2241 (2002).
[Crossref]

1999 (1)

D. P. Tsai, C. W. Yang, S.-Z. Lo, and H. E. Jackson, “Imaging local index variations in an optical waveguide using a tapping mode near-field scanning optical microscope,” Appl. Phys. Lett. 75(8), 1039–1041 (1999).
[Crossref]

1998 (1)

1996 (3)

G. L. Yip, P. C. Noutsios, and L. Chen, “Improved propagation-mode near-field method for refractive-index profiling of optical waveguides,” Appl. Opt. 35(12), 2060–2068 (1996).
[Crossref] [PubMed]

I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured nearfield intenstity,” J. Lightwave Technol. 14(3), 423–428 (1996).
[Crossref]

D. Brooks and S. Ruschin, “Improved near-field method for refractive index measurement of optical waveguides,” IEEE Photon. Technol. Lett. 8(2), 254–256 (1996).
[Crossref]

1995 (3)

F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995).
[Crossref]

Y. Tomita, M. Sugimoto, and K. Eda, “Direct bonding of LiNbO3 single crystals for optical waveguides,” Appl. Phys. Lett. 66(12), 1484–1485 (1995).
[Crossref]

C.-C. Wei, P.-K. Wei, and W. Fann, “Direct measurements of the true vibrational amplitudes in shear force microscopy,” Appl. Phys. Lett. 67(26), 3835–3837 (1995).
[Crossref]

1994 (1)

P. K. Wei and W. S. Wang, “A TE-TM mode splitter on lithium niobate using Ti, Ni, and MgO diffusions,” IEEE Photon. Technol. Lett. 6(2), 245–248 (1994).
[Crossref]

1990 (1)

J. Helms, J. Schmidtchen, B. Schüppert, and K. Petermann, “Error analysis for refractive-index profile determination from near-field measurements,” J. Lightwave Technol. 8(5), 625–633 (1990).
[Crossref]

1988 (1)

M. N. Armenise, “Fabrication techniques of lithium niobate waveguides,” IEE Proc. 135, 85–91 (1988).

1986 (1)

P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” J. Mod. Opt. 33, 127–143 (1986).

1985 (1)

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3(2), 385–391 (1985).
[Crossref]

1983 (1)

A. Yi-Yan, “Index instabilities in protonexchanged LiNbO3 waveguides,” Appl. Phys. Lett. 42(8), 633–635 (1983).
[Crossref]

1976 (1)

1974 (1)

Armenise, M. N.

M. N. Armenise, “Fabrication techniques of lithium niobate waveguides,” IEE Proc. 135, 85–91 (1988).

Barai, S.

Bosso, S.

F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995).
[Crossref]

Brooks, D.

D. Brooks and S. Ruschin, “Improved near-field method for refractive index measurement of optical waveguides,” IEEE Photon. Technol. Lett. 8(2), 254–256 (1996).
[Crossref]

Caccavale, F.

F. Caccavale, F. Segato, I. Mansour, and M. Gianesin, “A finite differences method for the reconstruction of refractive index profiles from near-field measurements,” J. Lightwave Technol. 16(7), 1348–1353 (1998).
[Crossref]

I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured nearfield intenstity,” J. Lightwave Technol. 14(3), 423–428 (1996).
[Crossref]

F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995).
[Crossref]

Campillo, A. L.

A. L. Campillo, J. W. P. Hsu, C. A. White, and C. D. W. Jones, “Direct measurement of the guided modes in LiNbO3 waveguides,” Appl. Phys. Lett. 80(13), 2239–2241 (2002).
[Crossref]

Chakraborty, P.

F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995).
[Crossref]

Chandler, P. J.

P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” J. Mod. Opt. 33, 127–143 (1986).

Chen, F.

X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008).
[Crossref]

Chen, L.

Chiang, K. S.

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3(2), 385–391 (1985).
[Crossref]

Corsini, R.

F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995).
[Crossref]

Dattner, Y.

Y. Dattner and O. Yadid-Pecht, “Analysis of the effective refractive index of silicon waveguides through the constructive and destructive interference in a Mach-Zehnder interferometer,” IEEE Photonics J. 3(6), 1123–1132 (2011).
[Crossref]

Eda, K.

Y. Tomita, M. Sugimoto, and K. Eda, “Direct bonding of LiNbO3 single crystals for optical waveguides,” Appl. Phys. Lett. 66(12), 1484–1485 (1995).
[Crossref]

Fann, W.

C.-C. Wei, P.-K. Wei, and W. Fann, “Direct measurements of the true vibrational amplitudes in shear force microscopy,” Appl. Phys. Lett. 67(26), 3835–3837 (1995).
[Crossref]

Fatadin, I.

Gianello, G.

F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995).
[Crossref]

Gianesin, M.

Guarino, A.

G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. 31(7), 1054–1058 (2009).
[Crossref]

Günter, P.

G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. 31(7), 1054–1058 (2009).
[Crossref]

Hajfler, J.

G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. 31(7), 1054–1058 (2009).
[Crossref]

Heidrich, P. F.

Helms, J.

J. Helms, J. Schmidtchen, B. Schüppert, and K. Petermann, “Error analysis for refractive-index profile determination from near-field measurements,” J. Lightwave Technol. 8(5), 625–633 (1990).
[Crossref]

Hsu, J. W. P.

A. L. Campillo, J. W. P. Hsu, C. A. White, and C. D. W. Jones, “Direct measurement of the guided modes in LiNbO3 waveguides,” Appl. Phys. Lett. 80(13), 2239–2241 (2002).
[Crossref]

Ives, D.

Jackson, H. E.

D. P. Tsai, C. W. Yang, S.-Z. Lo, and H. E. Jackson, “Imaging local index variations in an optical waveguide using a tapping mode near-field scanning optical microscope,” Appl. Phys. Lett. 75(8), 1039–1041 (1999).
[Crossref]

Jones, C. D. W.

A. L. Campillo, J. W. P. Hsu, C. A. White, and C. D. W. Jones, “Direct measurement of the guided modes in LiNbO3 waveguides,” Appl. Phys. Lett. 80(13), 2239–2241 (2002).
[Crossref]

Koechlin, M.

G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. 31(7), 1054–1058 (2009).
[Crossref]

Lama, F. L.

P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” J. Mod. Opt. 33, 127–143 (1986).

Liu, H.

X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008).
[Crossref]

Liu, X.

X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008).
[Crossref]

Lo, S.-Z.

D. P. Tsai, C. W. Yang, S.-Z. Lo, and H. E. Jackson, “Imaging local index variations in an optical waveguide using a tapping mode near-field scanning optical microscope,” Appl. Phys. Lett. 75(8), 1039–1041 (1999).
[Crossref]

Lu, F.

X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008).
[Crossref]

Mansour, I.

F. Caccavale, F. Segato, I. Mansour, and M. Gianesin, “A finite differences method for the reconstruction of refractive index profiles from near-field measurements,” J. Lightwave Technol. 16(7), 1348–1353 (1998).
[Crossref]

I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured nearfield intenstity,” J. Lightwave Technol. 14(3), 423–428 (1996).
[Crossref]

F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995).
[Crossref]

Martin, W. E.

Mussi, G.

F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995).
[Crossref]

Noutsios, P. C.

Oven, R.

Petermann, K.

J. Helms, J. Schmidtchen, B. Schüppert, and K. Petermann, “Error analysis for refractive-index profile determination from near-field measurements,” J. Lightwave Technol. 8(5), 625–633 (1990).
[Crossref]

Piao, S.-C.

Poberaj, G.

G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. 31(7), 1054–1058 (2009).
[Crossref]

Quaranta, A.

F. Caccavale, P. Chakraborty, A. Quaranta, I. Mansour, G. Gianello, S. Bosso, R. Corsini, and G. Mussi, “Secondary-ion-mass spectrometry and near-field studies of Ti:LiNbO3 optical waveguides,” J. Appl. Phys. 78(9), 5345–5350 (1995).
[Crossref]

Ruschin, S.

D. Brooks and S. Ruschin, “Improved near-field method for refractive index measurement of optical waveguides,” IEEE Photon. Technol. Lett. 8(2), 254–256 (1996).
[Crossref]

Schmidtchen, J.

J. Helms, J. Schmidtchen, B. Schüppert, and K. Petermann, “Error analysis for refractive-index profile determination from near-field measurements,” J. Lightwave Technol. 8(5), 625–633 (1990).
[Crossref]

Schüppert, B.

J. Helms, J. Schmidtchen, B. Schüppert, and K. Petermann, “Error analysis for refractive-index profile determination from near-field measurements,” J. Lightwave Technol. 8(5), 625–633 (1990).
[Crossref]

Segato, F.

Sharma, A.

Sugimoto, M.

Y. Tomita, M. Sugimoto, and K. Eda, “Direct bonding of LiNbO3 single crystals for optical waveguides,” Appl. Phys. Lett. 66(12), 1484–1485 (1995).
[Crossref]

Sulser, F.

G. Poberaj, M. Koechlin, F. Sulser, A. Guarino, J. Hajfler, and P. Günter, “Ion-sliced lithium niobate thin films for active photonic devices,” Opt. Mater. 31(7), 1054–1058 (2009).
[Crossref]

Tan, Y.

X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008).
[Crossref]

Tomita, Y.

Y. Tomita, M. Sugimoto, and K. Eda, “Direct bonding of LiNbO3 single crystals for optical waveguides,” Appl. Phys. Lett. 66(12), 1484–1485 (1995).
[Crossref]

Tsai, D. P.

D. P. Tsai, C. W. Yang, S.-Z. Lo, and H. E. Jackson, “Imaging local index variations in an optical waveguide using a tapping mode near-field scanning optical microscope,” Appl. Phys. Lett. 75(8), 1039–1041 (1999).
[Crossref]

Tsai, W.-S.

W.-S. Tsai, S.-C. Piao, and P.-K. Wei, “Refractive index measurement of optical waveguides using modified end-fire coupling method,” Opt. Lett. 36(11), 2008–2010 (2011).
[Crossref] [PubMed]

W.-S. Tsai, W.-S. Wang, and P.-K. Wei, “Two-dimensional refractive index profiling by using differential near-field scanning optical microscopy,” Appl. Phys. Lett. 91(6), 061123 (2007).
[Crossref]

Wang, L.

L. Wang and B.-X. Xiang, “Planar waveguides in magnesium doped stoichiometric LiNbO3 crystals formed by MeV oxygen ion implantations,” Nucl. Instrum. Meth. Phys. Res. Sect. B 272, 121–124 (2012).
[Crossref]

X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008).
[Crossref]

X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008).
[Crossref]

Wang, W. S.

P. K. Wei and W. S. Wang, “A TE-TM mode splitter on lithium niobate using Ti, Ni, and MgO diffusions,” IEEE Photon. Technol. Lett. 6(2), 245–248 (1994).
[Crossref]

Wang, W.-S.

W.-S. Tsai, W.-S. Wang, and P.-K. Wei, “Two-dimensional refractive index profiling by using differential near-field scanning optical microscopy,” Appl. Phys. Lett. 91(6), 061123 (2007).
[Crossref]

Wei, C.-C.

C.-C. Wei, P.-K. Wei, and W. Fann, “Direct measurements of the true vibrational amplitudes in shear force microscopy,” Appl. Phys. Lett. 67(26), 3835–3837 (1995).
[Crossref]

Wei, P. K.

P. K. Wei and W. S. Wang, “A TE-TM mode splitter on lithium niobate using Ti, Ni, and MgO diffusions,” IEEE Photon. Technol. Lett. 6(2), 245–248 (1994).
[Crossref]

Wei, P.-K.

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[Crossref]

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[Crossref]

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[Crossref]

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X. Liu, F. Lu, F. Chen, Y. Tan, R. Zhang, H. Liu, L. Wang, and L. Wang, “Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O2+ ion implantation into lithium niobate,” Opt. Commun. 281(6), 1529–1533 (2008).
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Figures (9)

Fig. 1
Fig. 1 (a) The slab waveguide. (b) Theoretical calculation of optical fields with analytical solutions. The arrow shows the beginning of the evanescent wave determined from the peak value of Iyy. I, I' and I” indicate the measured fields for the fitting.

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Fig. 2
Fig. 2 Experimental setup of the modified end-fire coupling method.

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Fig. 3
Fig. 3 Fourier spectrum of the recorded sequence.

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Fig. 4
Fig. 4 Measured intensities of the single mode fiber extracted from vibrating harmonic frequencies: (a) Intensity, I. (b) First-order differential field, Ix. (c) Second-order differential field, Ixx. Measured intensities with evanescent tails fitted with Bessel functions: (d) I, (e) Ix, (f) Ixx.

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Fig. 5
Fig. 5 One-dimensional profiles of (a) I (b) Ix (c) Ixx. (d) Three-dimensional colored surface of the reconstructed index profile of the single mode fiber Δn. (e) One-dimensional cross-sectional plot of Δn(x).

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Fig. 6
Fig. 6 Measured intensities of PE on LiNbO3 planar waveguide extracted from vibrating harmonic frequencies. (a) Intensity, I. (b) First-order differential field, Iy. (c) Second-order differential field, Iyy. Measured (red lines) profiles and measured fields fitted with exponential functions (blue lines) in y direction: (d) I (e) Iy (f) Iyy.

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Fig. 7
Fig. 7 Measured intensities of Ti:LiNbO3 planar waveguide extracted from vibrating harmonic frequencies. (a) Intensity, I. (b) First-order differential field, Iy. (c) Second-order differential field, Iyy. Measured (red lines) profiles and measured fields fitted with exponential functions (blue lines) in y direction: (d) I (e) Iy (f) Iyy.

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Fig. 8
Fig. 8 Measured effective indexes with prism coupler of (a) Ti:LiNbO3 and (b) PE on LiNbO3 single mode planar waveguides.

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Fig. 9
Fig. 9 Reconstructed refractive index profile in y direction, Δn(y), of PE (green line) and Ti:LiNbO3 (blue line) single mode planar waveguides.

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Equations (6)

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

Δn≅− 1 4 k 0 2 n s ∇ 2 E E +Δ n eff ,
Δn≈− 1 4 n s k 0 2 [ I ″ I − 1 2 ( I ′ I ) 2 ]+Δ n eff ,
{ I(y)= A 2 exp( −2αy ) I y = I ′ ( y )=−2α A 2 exp( −2αy ) I yy = I ″ ( y )=4 α 2 A 2 exp( −2αy ) , 0≤y≤∞,
Δn≈− 1 4 n s k 0 2 [ 4 α 2 − 1 2 ( −2α ) 2 ]+Δ n eff =− α 2 2 n s k 0 2 +Δ n eff =− k 0 2 ( n eff 2 − n s 2 ) 2 n s k 0 2 +Δ n eff ≈− k 0 2 ( 2 n s Δ n eff ) 2 n s k 0 2 +Δ n eff =0,
{ E y = A 0 ( J 0 ( pr ) J 0 ( pa ) ) r≤a E y = A 0 ( K 0 ( qr ) K 0 ( qa ) ) r≥a ,
I[ y(t) ]≈I[ y(0) ]+Δy I y [ y(0) ]sinωt+ 1 4 Δ y 2 I yy [ y(0) ]cos2ωt,

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