Abstract

We present a closed-form approximation for estimating both the field distribution and complex propagating constant of the antiresonant reflecting optical waveguide (ARROW) based on a first-order leaky-mode analysis. The formula was obtained from a novel coupled-electric–coupled-magnetic matrix method and provides six significant figures of the real part of the propagation constant β of a SiO2/TiO2/SiO2/Si ARROW with an 8-μm core. The accuracy for the quantity of the imaginary part of β is greater than 98.4% for the TE0 mode and 99.3% for TM0. The approximate values for field components are 96.1% accurate. In addition, a slight absorption by the substrate will result in modification of the initial improper leaky-mode behavior, which grows exponentially in the substrate, yielding a proper solution.

© 2005 Optical Society of America

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  1. M. A. Duguay, Y. Kokubun, T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2–Si multiplayer structures,” Appl. Phys. Lett. 49, 13–15 (1986).
    [CrossRef]
  2. T. Sekimoto, S. Ikuta, W. Pan, S. T. Chu, Y. Kokubun, “Vertical antiresonant reflecting optical waveguide coupler for three-dimensional optical interconnects: optimum design for large tolerance, high coupling efficiency, and short coupling length,” Appl. Opt. 39, 426–430 (2000).
    [CrossRef]
  3. S. T. Chu, W. Pan, S. Sato, B. E. Little, T. Kaneko, Y. Kokubun, “ARROW-type vertical coupler filter: design and fabrication,” J. Lightwave Technol. 17, 652–658 (1999).
    [CrossRef]
  4. X. Borrise, D. Jimenez, F. Perez-Murano, A. Llobera, C. Dominguez, N. Barniol, “Characterization of antiresonant reflecting optical waveguide devices by scanning near-field optical microscopy,” J. Opt. Soc. Am. A 17, 2243–2248 (2000).
    [CrossRef]
  5. M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
    [CrossRef]
  6. R. Bernini, S. Campopiano, L. Zeni, “Design and analysis of an integrated antiresonant reflecting optical waveguide refractive-index sensor,” Appl. Opt. 41, 70–73 (2002).
    [CrossRef] [PubMed]
  7. F. Prieto, L. M. Lechuga, A. Calle, A. Llobera, C. Domínguez, “Optimized silicon antiresonant reflecting optical waveguides for sensing applications,” J. Lightwave Technol. 19, 75–83 (2001).
    [CrossRef]
  8. K. Benaissa, A. Nathan, “Silicon anti-resonant reflecting optical waveguides for sensor applications,” Sens. Actuators A 65, 33–44 (1998).
    [CrossRef]
  9. T. Hayakawa, S. Asakawa, Y. Kokubun, “ARROW-B type polarization splitter with Y-branch fabricated by a self-alignment process,” J. Lightwave Technol. 15, 1165–1170 (1997).
    [CrossRef]
  10. J. Chilwell, I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
    [CrossRef]
  11. R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
    [CrossRef]
  12. T. Baba, Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides—numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992).
    [CrossRef]
  13. W. Huang, R. M. Shubair, A. Nathan, Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. 10, 1015–1022 (1992).
    [CrossRef]
  14. J. M. Kubica, “Numerical analysis of InP/InGaAsP ARROW waveguides using transfer matrix approach,” J. Lightwave Technol. 10, 767–771 (1992).
    [CrossRef]
  15. J. C. Grant, J. C. Beal, N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. 30, 1250–1253 (1994).
    [CrossRef]
  16. B. Ray, G. W. Hanson, “Some effects of anisotropy on planar antiresonant reflecting optical waveguides,” J. Light-wave Technol. 14, 202–208 (1996).
    [CrossRef]
  17. I. Garces, F. Villuendas, J. A. Valles, C. Dominguez, M. Moreno, “Analysis of leakage properties and guiding conditions of rib antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 798–805 (1996).
    [CrossRef]
  18. B. Liu, A. Shakouri, J. E. Bowers, “Characteristic equations for different ARROW structures,” Opt. Quantum Electron. 31, 1267–1276 (1999).
    [CrossRef]
  19. R. E. Collin, Field Theory of Guided Waves (Institute of Electrical and Electronics Engineering, New York, 1991), pp. 181–184.
  20. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, Englewood Cliffs, N.J., 1991), pp. 43–45.
  21. A. K. Chu, I. J. Lan, H. W. Chang, M. H. Sheng, “Low-loss polyimide/Ta2O5/SiO2hybrid antiresonant reflecting optical waveguides at quasi-antiresonant conditions,” IEEE Photon. Technol. Lett. 14, 44–46 (2002).
    [CrossRef]
  22. A. K. Chu, H. W. Chang, M. H. Sheng, “Optical polarizer based on antiresonant reflecting optical waveguide under quasi-antiresonant conditions,” Opt. Commun. 194, 137–142 (2001).
    [CrossRef]
  23. N.-J. You, “Application of coupled E/H formulation to the design of multiple layer AR-coating for large incident angle,” M.S. thesis (National Sun Yat-sen University, Institute of Electro-optical Engineering, Kaoshiung, Taiwan, 2000).
  24. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), pp. 482–489.
  25. S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
    [CrossRef]

2002

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

A. K. Chu, I. J. Lan, H. W. Chang, M. H. Sheng, “Low-loss polyimide/Ta2O5/SiO2hybrid antiresonant reflecting optical waveguides at quasi-antiresonant conditions,” IEEE Photon. Technol. Lett. 14, 44–46 (2002).
[CrossRef]

R. Bernini, S. Campopiano, L. Zeni, “Design and analysis of an integrated antiresonant reflecting optical waveguide refractive-index sensor,” Appl. Opt. 41, 70–73 (2002).
[CrossRef] [PubMed]

2001

F. Prieto, L. M. Lechuga, A. Calle, A. Llobera, C. Domínguez, “Optimized silicon antiresonant reflecting optical waveguides for sensing applications,” J. Lightwave Technol. 19, 75–83 (2001).
[CrossRef]

A. K. Chu, H. W. Chang, M. H. Sheng, “Optical polarizer based on antiresonant reflecting optical waveguide under quasi-antiresonant conditions,” Opt. Commun. 194, 137–142 (2001).
[CrossRef]

2000

1999

S. T. Chu, W. Pan, S. Sato, B. E. Little, T. Kaneko, Y. Kokubun, “ARROW-type vertical coupler filter: design and fabrication,” J. Lightwave Technol. 17, 652–658 (1999).
[CrossRef]

B. Liu, A. Shakouri, J. E. Bowers, “Characteristic equations for different ARROW structures,” Opt. Quantum Electron. 31, 1267–1276 (1999).
[CrossRef]

1998

K. Benaissa, A. Nathan, “Silicon anti-resonant reflecting optical waveguides for sensor applications,” Sens. Actuators A 65, 33–44 (1998).
[CrossRef]

1997

T. Hayakawa, S. Asakawa, Y. Kokubun, “ARROW-B type polarization splitter with Y-branch fabricated by a self-alignment process,” J. Lightwave Technol. 15, 1165–1170 (1997).
[CrossRef]

1996

B. Ray, G. W. Hanson, “Some effects of anisotropy on planar antiresonant reflecting optical waveguides,” J. Light-wave Technol. 14, 202–208 (1996).
[CrossRef]

I. Garces, F. Villuendas, J. A. Valles, C. Dominguez, M. Moreno, “Analysis of leakage properties and guiding conditions of rib antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 798–805 (1996).
[CrossRef]

1995

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

1994

J. C. Grant, J. C. Beal, N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. 30, 1250–1253 (1994).
[CrossRef]

1992

T. Baba, Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides—numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992).
[CrossRef]

W. Huang, R. M. Shubair, A. Nathan, Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. 10, 1015–1022 (1992).
[CrossRef]

J. M. Kubica, “Numerical analysis of InP/InGaAsP ARROW waveguides using transfer matrix approach,” J. Lightwave Technol. 10, 767–771 (1992).
[CrossRef]

1986

M. A. Duguay, Y. Kokubun, T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2–Si multiplayer structures,” Appl. Phys. Lett. 49, 13–15 (1986).
[CrossRef]

1984

Aramburu, C.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

Asakawa, S.

T. Hayakawa, S. Asakawa, Y. Kokubun, “ARROW-B type polarization splitter with Y-branch fabricated by a self-alignment process,” J. Lightwave Technol. 15, 1165–1170 (1997).
[CrossRef]

Baba, T.

T. Baba, Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides—numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992).
[CrossRef]

Baets, R. G.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

Barniol, N.

Beal, J. C.

J. C. Grant, J. C. Beal, N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. 30, 1250–1253 (1994).
[CrossRef]

Benaissa, K.

K. Benaissa, A. Nathan, “Silicon anti-resonant reflecting optical waveguides for sensor applications,” Sens. Actuators A 65, 33–44 (1998).
[CrossRef]

Bernini, R.

Borrise, X.

Bowers, J. E.

B. Liu, A. Shakouri, J. E. Bowers, “Characteristic equations for different ARROW structures,” Opt. Quantum Electron. 31, 1267–1276 (1999).
[CrossRef]

Calle, A.

Campopiano, S.

Chang, H. W.

A. K. Chu, I. J. Lan, H. W. Chang, M. H. Sheng, “Low-loss polyimide/Ta2O5/SiO2hybrid antiresonant reflecting optical waveguides at quasi-antiresonant conditions,” IEEE Photon. Technol. Lett. 14, 44–46 (2002).
[CrossRef]

A. K. Chu, H. W. Chang, M. H. Sheng, “Optical polarizer based on antiresonant reflecting optical waveguide under quasi-antiresonant conditions,” Opt. Commun. 194, 137–142 (2001).
[CrossRef]

Chilwell, J.

Chow, Y. L.

W. Huang, R. M. Shubair, A. Nathan, Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. 10, 1015–1022 (1992).
[CrossRef]

Chu, A. K.

A. K. Chu, I. J. Lan, H. W. Chang, M. H. Sheng, “Low-loss polyimide/Ta2O5/SiO2hybrid antiresonant reflecting optical waveguides at quasi-antiresonant conditions,” IEEE Photon. Technol. Lett. 14, 44–46 (2002).
[CrossRef]

A. K. Chu, H. W. Chang, M. H. Sheng, “Optical polarizer based on antiresonant reflecting optical waveguide under quasi-antiresonant conditions,” Opt. Commun. 194, 137–142 (2001).
[CrossRef]

Chu, S. T.

Chung, Y.

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Coldren, L. A.

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (Institute of Electrical and Electronics Engineering, New York, 1991), pp. 181–184.

Dagli, N.

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

De Mesel, K.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

Dominguez, C.

X. Borrise, D. Jimenez, F. Perez-Murano, A. Llobera, C. Dominguez, N. Barniol, “Characterization of antiresonant reflecting optical waveguide devices by scanning near-field optical microscopy,” J. Opt. Soc. Am. A 17, 2243–2248 (2000).
[CrossRef]

I. Garces, F. Villuendas, J. A. Valles, C. Dominguez, M. Moreno, “Analysis of leakage properties and guiding conditions of rib antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 798–805 (1996).
[CrossRef]

Domínguez, C.

Duguay, M. A.

M. A. Duguay, Y. Kokubun, T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2–Si multiplayer structures,” Appl. Phys. Lett. 49, 13–15 (1986).
[CrossRef]

Frenette, N. J. P.

J. C. Grant, J. C. Beal, N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. 30, 1250–1253 (1994).
[CrossRef]

Fuentes, D.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

Galarza, M.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

Garces, I.

I. Garces, F. Villuendas, J. A. Valles, C. Dominguez, M. Moreno, “Analysis of leakage properties and guiding conditions of rib antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 798–805 (1996).
[CrossRef]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Grant, J. C.

J. C. Grant, J. C. Beal, N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. 30, 1250–1253 (1994).
[CrossRef]

Hanson, G. W.

B. Ray, G. W. Hanson, “Some effects of anisotropy on planar antiresonant reflecting optical waveguides,” J. Light-wave Technol. 14, 202–208 (1996).
[CrossRef]

Hayakawa, T.

T. Hayakawa, S. Asakawa, Y. Kokubun, “ARROW-B type polarization splitter with Y-branch fabricated by a self-alignment process,” J. Lightwave Technol. 15, 1165–1170 (1997).
[CrossRef]

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Hodgkinson, I.

Huang, W.

W. Huang, R. M. Shubair, A. Nathan, Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. 10, 1015–1022 (1992).
[CrossRef]

Ikuta, S.

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, Englewood Cliffs, N.J., 1991), pp. 43–45.

Jimenez, D.

Kaneko, T.

Koch, T. L.

M. A. Duguay, Y. Kokubun, T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2–Si multiplayer structures,” Appl. Phys. Lett. 49, 13–15 (1986).
[CrossRef]

Kokubun, Y.

T. Sekimoto, S. Ikuta, W. Pan, S. T. Chu, Y. Kokubun, “Vertical antiresonant reflecting optical waveguide coupler for three-dimensional optical interconnects: optimum design for large tolerance, high coupling efficiency, and short coupling length,” Appl. Opt. 39, 426–430 (2000).
[CrossRef]

S. T. Chu, W. Pan, S. Sato, B. E. Little, T. Kaneko, Y. Kokubun, “ARROW-type vertical coupler filter: design and fabrication,” J. Lightwave Technol. 17, 652–658 (1999).
[CrossRef]

T. Hayakawa, S. Asakawa, Y. Kokubun, “ARROW-B type polarization splitter with Y-branch fabricated by a self-alignment process,” J. Lightwave Technol. 15, 1165–1170 (1997).
[CrossRef]

T. Baba, Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides—numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992).
[CrossRef]

M. A. Duguay, Y. Kokubun, T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2–Si multiplayer structures,” Appl. Phys. Lett. 49, 13–15 (1986).
[CrossRef]

Kubica, J. M.

J. M. Kubica, “Numerical analysis of InP/InGaAsP ARROW waveguides using transfer matrix approach,” J. Lightwave Technol. 10, 767–771 (1992).
[CrossRef]

Lan, I. J.

A. K. Chu, I. J. Lan, H. W. Chang, M. H. Sheng, “Low-loss polyimide/Ta2O5/SiO2hybrid antiresonant reflecting optical waveguides at quasi-antiresonant conditions,” IEEE Photon. Technol. Lett. 14, 44–46 (2002).
[CrossRef]

Lechuga, L. M.

Lee, S.-L.

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Little, B. E.

Liu, B.

B. Liu, A. Shakouri, J. E. Bowers, “Characteristic equations for different ARROW structures,” Opt. Quantum Electron. 31, 1267–1276 (1999).
[CrossRef]

Llobera, A.

Lopez-Amo, M.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

Moerman, I.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

Moreno, M.

I. Garces, F. Villuendas, J. A. Valles, C. Dominguez, M. Moreno, “Analysis of leakage properties and guiding conditions of rib antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 798–805 (1996).
[CrossRef]

Nathan, A.

K. Benaissa, A. Nathan, “Silicon anti-resonant reflecting optical waveguides for sensor applications,” Sens. Actuators A 65, 33–44 (1998).
[CrossRef]

W. Huang, R. M. Shubair, A. Nathan, Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. 10, 1015–1022 (1992).
[CrossRef]

Pan, W.

Perez-Murano, F.

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Prieto, F.

Ray, B.

B. Ray, G. W. Hanson, “Some effects of anisotropy on planar antiresonant reflecting optical waveguides,” J. Light-wave Technol. 14, 202–208 (1996).
[CrossRef]

Sato, S.

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Sekimoto, T.

Shakouri, A.

B. Liu, A. Shakouri, J. E. Bowers, “Characteristic equations for different ARROW structures,” Opt. Quantum Electron. 31, 1267–1276 (1999).
[CrossRef]

Sheng, M. H.

A. K. Chu, I. J. Lan, H. W. Chang, M. H. Sheng, “Low-loss polyimide/Ta2O5/SiO2hybrid antiresonant reflecting optical waveguides at quasi-antiresonant conditions,” IEEE Photon. Technol. Lett. 14, 44–46 (2002).
[CrossRef]

A. K. Chu, H. W. Chang, M. H. Sheng, “Optical polarizer based on antiresonant reflecting optical waveguide under quasi-antiresonant conditions,” Opt. Commun. 194, 137–142 (2001).
[CrossRef]

Shubair, R. M.

W. Huang, R. M. Shubair, A. Nathan, Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. 10, 1015–1022 (1992).
[CrossRef]

Valles, J. A.

I. Garces, F. Villuendas, J. A. Valles, C. Dominguez, M. Moreno, “Analysis of leakage properties and guiding conditions of rib antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 798–805 (1996).
[CrossRef]

Van Daele, P.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

Verstuyft, S.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

Villuendas, F.

I. Garces, F. Villuendas, J. A. Valles, C. Dominguez, M. Moreno, “Analysis of leakage properties and guiding conditions of rib antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 798–805 (1996).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), pp. 482–489.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), pp. 482–489.

You, N.-J.

N.-J. You, “Application of coupled E/H formulation to the design of multiple layer AR-coating for large incident angle,” M.S. thesis (National Sun Yat-sen University, Institute of Electro-optical Engineering, Kaoshiung, Taiwan, 2000).

Zeni, L.

Appl. Opt.

Appl. Phys. Lett.

M. A. Duguay, Y. Kokubun, T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2–Si multiplayer structures,” Appl. Phys. Lett. 49, 13–15 (1986).
[CrossRef]

IEEE J. Quantum Electron.

J. C. Grant, J. C. Beal, N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. 30, 1250–1253 (1994).
[CrossRef]

T. Baba, Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides—numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992).
[CrossRef]

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

M. Galarza, K. De Mesel, S. Verstuyft, D. Fuentes, C. Aramburu, M. Lopez-Amo, I. Moerman, P. Van Daele, R. G. Baets, “Mode-expanded 1.55 μm InP–InGaAsP Fabry–Perot lasers using ARROW waveguides for efficient fiber coupling,” IEEE J. Sel. Top. Quantum Electron. 8, 1389–1398 (2002).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Typical five-layered ARROW structure and the corresponding index profile labeled with the symbols used throughout this paper.

Fig. 2
Fig. 2

Radiation loss coefficient at a wavelength of 1.3 μm for TE0 and TM0 modes in the air/SiO2/TiO2/SiO2/Si ARROW. Solid curves, lines results from the CE–CH method, considered numerically exact. Circles, closed-form data calculated from approximation (19); crosses, obtained by the mode analysis method.12 We can clearly see that the two approximation data sets are virtually on top of each other and are both highly accurate.

Fig. 3
Fig. 3

Radiation loss coefficient for TE0 and TM0 modes in the air/SiO2/Si/SiO2/Si ARROW; repeats the results of Fig. 2 except that the first cladding is replaced by a higher-index material. Comparing the two figures, we see that a bigger index contrast between the first and the second claddings (TiO2/SiO2 versus Si/SiO2) will lead to a larger radiation loss difference between the two polarizations.

Fig. 4
Fig. 4

Radiation loss coefficient for an air/SiO2/Si laminated waveguide for various core thicknesses at a wavelength of 1.3 μm. Data are calculated with the CE–CH method. A low-loss waveguide (< 1 dB/cm) for both TE0 and TM0 modes can be obtained with this simple waveguide by use of a thick core (greater than 36 μm).

Fig. 5
Fig. 5

Plotted are the real and the imaginary field distributions of the TE0 mode of the ARROW with an 8-μm core and a TiO2 first cladding (n3 = 2.27) at λ = 1.3 μm. Solid curves, obtained by solution of the CE–CH matrix. Crosses, layer interfaces estimated from closed-form formulas. Only parts of the tails are shown to prevent data from overlapping.

Fig. 6
Fig. 6

Profiles for the TM0 mode. All parameters are the same as in Fig. 5. Note the tenfold y-scale difference in the imaginary data between the two plots. We can clearly see a much higher field penetration for the TM0 mode. Our closed-form formula works as well in this case.

Fig. 7
Fig. 7

Profiles for the TE0 mode with a 4-μm core. All other parameters remain unchanged. The effectiveness of the ARROW will be reduced if an insufficient core thickness (reduced by half) is used because of increased field penetration (more than double) into the substrate.

Fig. 8
Fig. 8

Calculated field distribution in the ARROW with an absorptive substrate (n5 = 3.5 − 0.3i) for the TE0 mode. All other parameters are the same as in Fig. 5. We see that as the absorbing power of the substrate increases as it gradually converts the improper mode into a proper one, making the field in the substrate diminish exponentially.

Fig. 9
Fig. 9

Same calculation as in Fig. 8 for the TM0 mode. As indicated in Table 2, the complex wave number q5 remains essentially the same for both TE0 and TM0 modes and for small and great core thickness when the substrate becomes absorptive.

Fig. 10
Fig. 10

(N + 1)-layered waveguide with index ni and thickness di for layer i; xi is the position of the interface between layer i and i + 1; Ei is the y-component electric field on xis. The first and the last boundaries x0 and xN+1 are free to be positioned at any location or at infinity. All CE–CH formulas in this paper assume two half-spaces in the superstrate and substrate.

Tables (2)

Tables Icon

Table 1 Numerical Results, Propagation Constants β (rad/μm) (Fig. 2)

Tables Icon

Table 2 Numerical Results for β and q5 for Absorptive Substrates and q5 for Nonabsorptive Substrates

Equations (37)

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E y ( x ) = E 1 exp [ - j q 1 ( x - d 2 ) ] ( superstrate ) , E y ( x ) = sin [ q 2 ( d 2 - x ) ] sin ( q 2 d 2 ) E 2 + sin [ q 2 x ] sin ( q 2 d 2 ) E 1 ( core ) , E y ( x ) = sin [ q 3 ( - x ) ] sin ( q 3 d 3 ) E 3 + sin [ q 3 ( x + d 3 ) ] sin ( q 3 d 3 ) E 2 ( first cladding ) , E y ( x ) = sin [ q 4 ( - d 3 - x ) ] sin ( q 4 d 4 ) E 4 + sin [ q 4 ( x + d 3 + d 4 ) ] sin ( q 4 d 4 ) E 3 ( second cladding ) , E y ( x ) = E 4 exp [ j q 5 ( x + d 3 + d 4 ) ] ( substrate ) ,
A E ( β ) E = 0 ,
[ α 1 + q 2 cot ( q 2 d 2 ) - q 2 csc ( q 2 d 2 ) 0 0 - q 2 csc ( q 2 d 2 ) q 2 cot ( q 2 d 2 ) + q 3 cot ( q 3 d 3 ) - q 3 csc ( q 3 d 3 ) 0 0 - q 3 csc ( q 3 d 3 ) q 3 cot ( q 3 d 3 ) + q 4 cot ( q 4 d 4 ) - q 4 csc ( q 4 d 4 ) 0 0 - q 4 csc ( q 4 d 4 ) q 4 cot ( q 4 d 4 ) + j q 5 ] ,
E = [ E 1 E 2 E 3 E 4 ] .
A H ( β ) H = 0 .
[ α 1 ɛ 1 + q 2 cot ( q 2 d 2 ) ɛ 2 - q 2 csc ( q 2 d 2 ) ɛ 2 0 0 - q 2 csc ( q 2 d 2 ) ɛ 2 q 2 cot ( q 2 d 2 ) ɛ 2 + q 3 cot ( q 3 d 3 ) ɛ 3 - q 3 csc ( q 3 d 3 ) ɛ 3 0 0 - q 3 csc ( q 3 d 3 ) ɛ 3 q 3 cot ( q 3 d 3 ) ɛ 3 + q 4 cot ( q 4 d 4 ) ɛ 4 - q 4 csc ( q 4 d 4 ) ɛ 4 0 0 - q 4 csc ( q 4 d 4 ) ɛ 4 q 4 cot ( q 4 d 4 ) ɛ 4 + j q 5 ɛ 5 ] ,
H = [ H 1 H 2 H 3 H 4 ] .
β ˜ ν = [ ( k 0 n 2 ) 2 - ( ν + 1 d 2 π ) 2 ] 1 / 2 , ν = 0 ,             1 ,             2 ,             3 ,             .
d 4 = ( 2 N + 1 ) π 2 q ˜ 4 = λ 4 n 4 [ 1 - ( n 2 n 4 ) 2 + ( ν + 1 2 n 4 d 2 λ ) 2 ] - 1 / 2 ( 2 N + 1 ) ,
d 3 = λ 4 n 3 [ 1 - ( n 2 n 3 ) 2 + ( ν + 1 2 n 3 d 2 λ ) 2 ] - 1 / 2 ( 2 M + 1 ) ,
q i = [ k 0 2 ( n i 2 - n 2 2 ) + ( π - δ d 2 ) 2 ] 1 / 2 , i = 1 ,             3 ,             4 ,             5.
q i q ˜ i [ 1 - π δ / ( q ˜ i 2 d 2 2 ) ] ,             i = 1 ,             3 ,             4 ,             5.
[ α 1 + q 2 cos ( π - δ e ) sin ( π - δ e ) - q 2 sin ( π - δ e ) 0 0 - q 2 sin ( π - δ e ) q 2 cos ( π - δ e ) sin ( π - δ e ) + q 3 cos ( π 2 - a δ e ) sin ( π 2 - a δ e ) - q 3 sin ( π 2 - a δ e ) 0 0 - q 3 sin ( π 2 - a δ e ) q 3 cos ( π 2 - a δ e ) sin ( π 2 - a δ e ) + q 2 cos ( π - δ e 2 ) sin ( π - δ e 2 ) - q 2 sin ( π - δ e 2 ) 0 0 - q 2 sin ( π - δ e 2 ) q 2 · cos ( π - δ e 2 ) sin ( π - δ e 2 ) + j q 5 ] .
| ( - α 1 d 2 δ e + q 2 d 2 ) q 2 d 2 0 0 q 2 d 2 q 2 d 2 - q 3 d 2 a δ e 2 q 3 d 2 δ e 0 0 q 3 d 2 - q 3 d 2 a δ e - q 2 d 2 δ e / 2 q 2 d 2 0 0 q 2 d 2 - q 2 d 2 δ e / 2 - j q 5 d 2 | .
[ - α ˜ 1 d 2 δ e + q 2 d 2 q 2 d 2 0 0 q 2 d 2 q 2 d 2 q ˜ 3 d 2 δ e 0 0 q ˜ 3 d 2 - q 2 d 2 δ e 2 q 2 d 2 0 0 q 2 d 2 - j q ˜ 5 d 2 ] .
- j π 2 α ˜ 1 q ˜ 5 d 2 2 δ e 2 2 + π 3 α ˜ 1 d 2 δ e - 3 π 2 α ˜ 1 d 2 δ e 2 - j α ˜ 1 q ˜ 3 2 q ˜ 5 d 2 4 δ e 2 + j π q ˜ 3 2 q ˜ 5 d 2 3 δ e - j q ˜ 3 2 q ˜ 5 d 2 3 δ e 2 = 0.
δ e = 0 or δ e = - j π 3 α ˜ 1 d 2 + π q ˜ 3 2 q ˜ 5 d 2 3 π 2 α ˜ 1 q ˜ 5 d 2 2 2 + α ˜ 1 q ˜ 3 2 q ˜ 5 d 2 4 + q ˜ 3 2 q ˜ 5 d 2 3 - j 3 π 2 α ˜ 1 d 2 .
[ - d 2 α 1 δ m ɛ 1 + π - δ m ɛ 2 π - δ m ɛ 2 0 0 π - δ m ɛ 2 π - δ m ɛ 2 d 2 q 3 δ m ɛ 3 0 0 d 2 q 3 ɛ 3 - q 3 d 2 a δ m ɛ 3 - ( π - δ m ) δ m 2 ɛ 2 π - δ m ɛ 2 0 0 π - δ m ɛ 2 - ( π - δ m ) δ m 2 ɛ 2 - j d 2 q 5 ɛ 5 ] .
[ - d 2 α ˜ 1 δ m ɛ 1 + π - δ m ɛ 2 π - δ m ɛ 2 0 0 π - δ m ɛ 2 π - δ m ɛ 2 d 2 q ˜ 3 δ m ɛ 3 0 0 d 2 q ˜ 3 ɛ 3 - π δ m 2 ɛ 2 π - δ m ɛ 2 0 0 π - δ m ɛ 2 - j d 2 q ˜ 5 ɛ 5 ] .
δ m = - j 2 ɛ 3 2 ɛ 5 π 3 α ˜ 1 d 2 + 2 ɛ 1 ɛ 2 2 π q ˜ 3 2 q ˜ 5 d 2 3 ɛ 2 ɛ 3 2 π 2 α ˜ 1 q ˜ 5 d 2 2 + 2 ɛ 2 3 α ˜ 1 q ˜ 3 2 q ˜ 5 d 2 4 + 2 ɛ 1 ɛ 2 2 q ˜ 3 2 q ˜ 5 d 2 3 - j 6 ɛ 3 2 ɛ 5 π 2 α ˜ 1 d 2 ,
α Im ( 2 π δ - δ 2 2 d 2 2 k 0 n 2 ) Im ( λ δ 2 d 2 2 n 2 ) .
E 3 = - α ˜ 1 q ˜ 3 ,             H 3 = - n 3 2 α ˜ 1 n 1 2 q ˜ 3 .
E 4 = - α ˜ 1 q 2 e j q ˜ 3 q ˜ 5 = j α ˜ 1 ( π - δ e ) q ˜ 3 q ˜ 5 d 2 , H 4 = j n 3 2 n 5 2 α ˜ 1 q 2 m n 1 2 n 2 2 q ˜ 3 q ˜ 5 = j n 3 2 n 5 2 α ˜ 1 ( π - δ m ) n 1 2 n 2 2 q ˜ 3 q ˜ 5 d 2 .
E 2 = π - δ e q ˜ 3 d 2 ( δ e 2 E 3 - E 4 ) , H 2 = n 3 2 ( π - δ m ) n 2 2 q ˜ 3 d 2 ( δ m 2 H 3 - H 4 ) .
Z i = η i cos θ i = η 0 cos θ i n i for TM mode , Z i = η i cos θ i = η 0 n i cos θ i for TE mode ,
r TE = Z II - Z I Z II + Z I = n I cos θ I - n II cos θ II n I cos θ I + n II cos θ II - ( 1 - 2 n I cos θ I n II cos θ II ) , r TM = Z II - Z I Z II + Z I = n I cos θ II - n II cos θ I n I cos θ II + n II cos θ I 1 - 2 n II cos θ I n I cos θ II .
T TE = 1 - r TE 2 4 n I cos θ I n II cos θ II , T TM = 1 - r TM 2 4 n II cos θ I n I cos θ II .
T TM / T TE n II 2 / n I 2 .
Z eff TM = n 4 4 n 3 4 n 3 2 - ( n 2 sin θ 2 ) 2 n 4 2 - ( n 2 sin θ 2 ) 2 cos θ 5 n 5 ,
Z eff TE = n 4 2 - ( n 2 sin θ 2 ) 2 n 3 2 - ( n 2 sin θ 2 ) 2 1 n 5 cos θ 5 .
T TM T TE n 3 4 n 2 4 n 5 2 n 2 2 .
E y ( i ) ( x , z ) = { A i exp [ - j q i ( x - x i - 1 ) ] + B i exp [ + j q i × ( x - x i - 1 ) ] } exp ( - j β z ) ,
E i = A i exp [ - j q i ( x i - x i - 1 ) ] + B i exp [ + j q i ( x i - x i - 1 ) ] = A i + 1 + B i + 1 .
E y ( x ) = E 1 exp [ + j q 1 ( x - x 1 ) ] ( for layer 1 ) , E y ( x ) = sin q i ( x i - x ) sin q i d i E i - 1 + sin q i ( x - x i - 1 ) sin q i d i E i ( for layers 2 ~ N ) , E y ( x ) = E N exp [ - j q N + 1 ( x - x N ) ] ( for layer N + 1 ) ,
H z ( x ) = 1 j ω μ 1 j q 1 E 1 exp [ j q 1 ( x - x 1 ) ] ( for layer 1 ) , H z ( x ) = 1 j ω μ i [ - q i cos q i ( x i - x ) sin q i d i E i - 1 + q i cos q i ( x - x i - 1 ) sin q i d i E i ] ( for layer 2 ~ N ) , H z ( x ) = 1 j ω μ N + 1 ( - j q N + 1 ) E N exp [ - j q N + 1 ( x - x N ) ] ( for layer N + 1 ) .
[ a 11 a 12 0 0 0 0 a 21 a 22 a 23 0 0 0 a 32 0 0 a i , i - 1 a i , i a i , i + 1 0 0 a N - 2 , N - 1 0 0 0 a N - 1 , N - 2 a N - 1 , N - 1 a N - 1 , N 0 0 0 0 a N , N - 1 a N , N ] [ E 1 E 2 E i E N - 1 E N ] = [ 0 0 0 0 0 ] .
a 11 = j q 1 μ 1 + q 2 cot ( q 2 d 2 ) μ 2 , a 12 = - q 2 csc ( q 2 d 2 ) μ 2 , a i , i - 1 = - q i csc ( q i d i ) μ i , a i , i = q i cot ( q i d i ) μ i + q i + 1 cot ( q i + 1 d i + 1 ) μ i + 1 , a i , i + 1 = - q i + 1 csc ( q i + 1 d i + 1 ) μ i + 1 , a N , N - 1 = - q N csc ( q N d N ) μ N , a N , N = q N cot ( q N d N ) μ N + j q N + 1 μ N + 1 , i = 2 , 3 , , N - 1.

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