Abstract

The excitation and propagation of both scalar and vector optical waves in one-dimensional nonlinear arrays of channel waveguides was investigated both theoretically and experimentally. In this arrangement vector discrete solitons are also possible through the coexistence of two orthogonally polarized fields. At high input power levels a rapid collapse of the power back to the incidence channel occurred over a small power range for both scalar and vector inputs.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
    [CrossRef] [PubMed]
  2. J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of spatial solitons in AlGaAs wave-guides," Electron. Lett. 28, 1879-1880 (1992).
    [CrossRef]
  3. S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-47 (1973).
    [CrossRef]
  4. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled wave-guides," Opt. Lett. 13, 794-796 (1988).
    [CrossRef] [PubMed]
  5. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
    [CrossRef]
  6. J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
    [CrossRef]
  7. E. A. Ostrovskaya, N. N. Akhmediev, G. I. Stegeman, J. U. Kang, and J. S. Aitchison, "Mixed-mode spatial solitons in semiconductor waveguides," J. Opt. Soc. Am. B 14, 880-887 (1997).
    [CrossRef]
  8. R. R. Malendevich, L. Friedrich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation-related polarization instability of Kerr spatial vector solitons," J. Opt. Soc. Am. B 19, 695-702 (2002).
    [CrossRef]
  9. N. Akhmediev and J. M. Soto-Crespo, "Dynamics of soliton-like pulse-propagation in birefringent optical fibers," Phys. Rev. E 49, 5742-5754 (1994).
    [CrossRef]
  10. L. Friedrich, R. Malendevich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation related polarization instability of fast Kerr spatial solitons in slab waveguides," Opt. Commun. 186, 335-341 (2000).
    [CrossRef]
  11. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).
    [CrossRef]
  12. J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quantum Electron. 33, 341-348 (1997).
    [CrossRef]
  13. S. Droulias, K. Hizanidis, J. Meier, and D. N. Christodoulides, "X waves in nonlinear normally dispersive waveguide arrays," Opt. Express 13, 1827-1832 (2005).
    [CrossRef]

2005 (1)

2003 (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
[CrossRef]

2002 (1)

2000 (1)

L. Friedrich, R. Malendevich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation related polarization instability of fast Kerr spatial solitons in slab waveguides," Opt. Commun. 186, 335-341 (2000).
[CrossRef]

1999 (1)

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).
[CrossRef]

1998 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

1997 (2)

E. A. Ostrovskaya, N. N. Akhmediev, G. I. Stegeman, J. U. Kang, and J. S. Aitchison, "Mixed-mode spatial solitons in semiconductor waveguides," J. Opt. Soc. Am. B 14, 880-887 (1997).
[CrossRef]

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quantum Electron. 33, 341-348 (1997).
[CrossRef]

1994 (1)

N. Akhmediev and J. M. Soto-Crespo, "Dynamics of soliton-like pulse-propagation in birefringent optical fibers," Phys. Rev. E 49, 5742-5754 (1994).
[CrossRef]

1992 (1)

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of spatial solitons in AlGaAs wave-guides," Electron. Lett. 28, 1879-1880 (1992).
[CrossRef]

1988 (1)

1973 (1)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-47 (1973).
[CrossRef]

Aitchison, J. S.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
[CrossRef]

R. R. Malendevich, L. Friedrich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation-related polarization instability of Kerr spatial vector solitons," J. Opt. Soc. Am. B 19, 695-702 (2002).
[CrossRef]

L. Friedrich, R. Malendevich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation related polarization instability of fast Kerr spatial solitons in slab waveguides," Opt. Commun. 186, 335-341 (2000).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

E. A. Ostrovskaya, N. N. Akhmediev, G. I. Stegeman, J. U. Kang, and J. S. Aitchison, "Mixed-mode spatial solitons in semiconductor waveguides," J. Opt. Soc. Am. B 14, 880-887 (1997).
[CrossRef]

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quantum Electron. 33, 341-348 (1997).
[CrossRef]

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of spatial solitons in AlGaAs wave-guides," Electron. Lett. 28, 1879-1880 (1992).
[CrossRef]

Akhmediev, N.

N. Akhmediev and J. M. Soto-Crespo, "Dynamics of soliton-like pulse-propagation in birefringent optical fibers," Phys. Rev. E 49, 5742-5754 (1994).
[CrossRef]

Akhmediev, N. N.

Alhemyari, K.

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of spatial solitons in AlGaAs wave-guides," Electron. Lett. 28, 1879-1880 (1992).
[CrossRef]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Christodoulides, D.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
[CrossRef]

Christodoulides, D. N.

Droulias, S.

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Friedrich, L.

R. R. Malendevich, L. Friedrich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation-related polarization instability of Kerr spatial vector solitons," J. Opt. Soc. Am. B 19, 695-702 (2002).
[CrossRef]

L. Friedrich, R. Malendevich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation related polarization instability of fast Kerr spatial solitons in slab waveguides," Opt. Commun. 186, 335-341 (2000).
[CrossRef]

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-47 (1973).
[CrossRef]

Garvin, H. L.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-47 (1973).
[CrossRef]

Grant, R. S.

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of spatial solitons in AlGaAs wave-guides," Electron. Lett. 28, 1879-1880 (1992).
[CrossRef]

Hizanidis, K.

Hudock, J.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
[CrossRef]

Hunsperger, R. G.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-47 (1973).
[CrossRef]

Hutchings, D. C.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quantum Electron. 33, 341-348 (1997).
[CrossRef]

Ironside, C. N.

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of spatial solitons in AlGaAs wave-guides," Electron. Lett. 28, 1879-1880 (1992).
[CrossRef]

Joseph, R. I.

Kang, J. U.

E. A. Ostrovskaya, N. N. Akhmediev, G. I. Stegeman, J. U. Kang, and J. S. Aitchison, "Mixed-mode spatial solitons in semiconductor waveguides," J. Opt. Soc. Am. B 14, 880-887 (1997).
[CrossRef]

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quantum Electron. 33, 341-348 (1997).
[CrossRef]

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

Malendevich, R.

L. Friedrich, R. Malendevich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation related polarization instability of fast Kerr spatial solitons in slab waveguides," Opt. Commun. 186, 335-341 (2000).
[CrossRef]

Malendevich, R. R.

Meier, J.

S. Droulias, K. Hizanidis, J. Meier, and D. N. Christodoulides, "X waves in nonlinear normally dispersive waveguide arrays," Opt. Express 13, 1827-1832 (2005).
[CrossRef]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
[CrossRef]

Morandotti, R.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Ostrovskaya, E. A.

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).
[CrossRef]

Sibbett, W.

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of spatial solitons in AlGaAs wave-guides," Electron. Lett. 28, 1879-1880 (1992).
[CrossRef]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Somekh, S.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-47 (1973).
[CrossRef]

Soto-Crespo, J. M.

R. R. Malendevich, L. Friedrich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation-related polarization instability of Kerr spatial vector solitons," J. Opt. Soc. Am. B 19, 695-702 (2002).
[CrossRef]

L. Friedrich, R. Malendevich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation related polarization instability of fast Kerr spatial solitons in slab waveguides," Opt. Commun. 186, 335-341 (2000).
[CrossRef]

N. Akhmediev and J. M. Soto-Crespo, "Dynamics of soliton-like pulse-propagation in birefringent optical fibers," Phys. Rev. E 49, 5742-5754 (1994).
[CrossRef]

Stegeman, G.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
[CrossRef]

Stegeman, G. I.

R. R. Malendevich, L. Friedrich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation-related polarization instability of Kerr spatial vector solitons," J. Opt. Soc. Am. B 19, 695-702 (2002).
[CrossRef]

L. Friedrich, R. Malendevich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation related polarization instability of fast Kerr spatial solitons in slab waveguides," Opt. Commun. 186, 335-341 (2000).
[CrossRef]

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quantum Electron. 33, 341-348 (1997).
[CrossRef]

E. A. Ostrovskaya, N. N. Akhmediev, G. I. Stegeman, J. U. Kang, and J. S. Aitchison, "Mixed-mode spatial solitons in semiconductor waveguides," J. Opt. Soc. Am. B 14, 880-887 (1997).
[CrossRef]

Villeneuve, A.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quantum Electron. 33, 341-348 (1997).
[CrossRef]

Yariv, A.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-47 (1973).
[CrossRef]

Appl. Phys. Lett. (1)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-47 (1973).
[CrossRef]

Electron. Lett. (1)

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of spatial solitons in AlGaAs wave-guides," Electron. Lett. 28, 1879-1880 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quantum Electron. 33, 341-348 (1997).
[CrossRef]

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

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

L. Friedrich, R. Malendevich, G. I. Stegeman, J. M. Soto-Crespo, N. N. Akhmediev, and J. S. Aitchison, "Radiation related polarization instability of fast Kerr spatial solitons in slab waveguides," Opt. Commun. 186, 335-341 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

N. Akhmediev and J. M. Soto-Crespo, "Dynamics of soliton-like pulse-propagation in birefringent optical fibers," Phys. Rev. E 49, 5742-5754 (1994).
[CrossRef]

Phys. Rev. Lett. (3)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete vector solitons in Kerr nonlinear waveguide arrays," Phys. Rev. Lett. 91, 143907 1-4 (2003).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of discrete solitons in optical waveguide arrays," Phys. Rev. Lett. 83, 2726-2729 (1999).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

(a) Sample design. (b) Electron microscope image of a sample.

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

(a) Recorded intensity at the input and the output surface of the sample. (b) Linear propagation of light along the array for single-channel excitation.

Fig. 4
Fig. 4

Output energy distribution for single-channel excitation and increasing input power.

Fig. 5
Fig. 5

Ratio between total power and power contained in the excitation channel for increasing input power. Circles are measured values; the solid curve is a numerical simulation.

Fig. 6
Fig. 6

Spatial energy distribution for excitation of a wider beam with increasing TE power. (a) Experimental result, (b) numerical simulation.

Fig. 7
Fig. 7

Spatial output energy distribution when (a) TE and TM beam are separately excited and (b) for both beams excited together. In all cases, the peak power was 650 W in each polarization.

Fig. 8
Fig. 8

Ratio between total power and power contained in the excitation channel for increasing TM power. TE power was fixed at 450 W , and both components were excited in phase.

Fig. 9
Fig. 9

Dependence of (top) TE and (bottom) TM output beam width on the TM input power and the phase difference between TE and TM wave.

Fig. 10
Fig. 10

Dependency of the TE output power on the TM input power and the phase difference between TE and TM wave. TE input power was 450 W .

Equations (6)

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

i d a n TE d ζ + a n TE + κ ( a n + 1 TE + a n 1 TE ) + ( a n TE 2 + A b n TM 2 ) a n TE
+ B ( b n TM ) 2 ( a n TE ) * = 0 ,
i d b n TM d ζ b n TM + κ ( b n + 1 TM + b n 1 TM ) + ( b n TM 2 + A a n TE 2 ) b n TM
+ B ( a n TE ) 2 ( b n TM ) * = 0 ,
J = n ( a n TE 2 + b n TM 2 ) ,
H = n [ γ ( a n TE a n 1 TE 2 + b n TM + b n 1 TM 2 ) a n TE 2 ( 1 + 2 γ ) + b n TM 2 ( 1 2 γ ) 1 2 ( a n TE 4 + b n TM 4 ) A a n TE b n TM 2 1 2 B ( a n TE 2 b n TE * 2 + a n TE * 2 b n TM 2 ) ] .

Metrics