Abstract

A simple tapered directional coupler with a five-layered structure is employed for mode conversion between a single-order and higher-order modes. We investigate coupling from higher-order modes to a single-order field theoretically and experimentally. As a result, we confirm that the first two modes in the tapered waveguide are coupled with a single-order mode in another waveguide by computer simulation using the beam propagation method. Furthermore, we fabricated the actual device and observed the streak patterns of the first three modes.

© 1999 Optical Society of America

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  1. H. Yanagawa, T. Shimizu, S. Nakamura, I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10, 587–591 (1992).
    [CrossRef]
  2. F. Ghirardi, B. Mersali, J. Brandon, G. Herve-Gruyer, A. Carenco, “Quasi planar spot-size transformer for efficient coupling between a cleaved fiber and an InP/InGaAsP rib waveguide,” IEEE Photon. Technol. Lett. 6, 522–524 (1994).
    [CrossRef]
  3. B. Jacobs, R. Zengerle, K. Faltin, W. Weiershausen, “Vertically tapered spot size transformers fabricated by a simple masking technique,” Electron. Lett. 31, 794–796 (1995).
    [CrossRef]
  4. S. Sottini, D. Grando, L. Palchetti, E. Giorgetti, “Optical fiber-polymer guide coupling by a tapered graded index glass guide,” IEEE J. Quantum Electron. 31, 1123–1129 (1995).
    [CrossRef]
  5. R. Zengerle, H. Bruckner, H. Olzhausen, A. Kohl, “Low-loss fibre-chip coupling by buried laterally tapered InP/InGaAsP waveguide structure,” Electron. Lett. 28, 631–632 (1992).
    [CrossRef]
  6. B. Mersali, H. J. Bruckner, M. Feuillade, S. Sainson, A. Ougazzaden, A. Carenco, “Theoretical and experimental studies of a spot-size transformer with integrated waveguide for polarization insensitive optical amplifiers,” J. Lightwave Technol. 13, 1865–1871 (1995).
    [CrossRef]
  7. P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
    [CrossRef]
  8. G. H. Song, W. J. Tomlinson, “Fourier analysis and synthesis of adiabatic tapers in integrated optics,” J. Opt. Soc. Am. A 9, 1289–1300 (1992).
    [CrossRef]
  9. J. Haes, J. Willems, R. Baets, “Design of adiabatic tapers for high contrast step index waveguides,” in Linear and Nonlinear Integrated Optics, G. C. Righini, D. Yevick, eds., Proc. SPIE2212, 685–693 (1994).
    [CrossRef]
  10. P. Sewell, T. M. Benson, P. C. Kendall, T. Anada, “Tapered beam propagation,” Electron. Lett. 32, 1025–1026 (1996).
    [CrossRef]
  11. Y. Koike, T. Ishigure, E. Nihei, “High-bandwidth graded-index polymer optical fiber,” J. Lightwave Technol. 13, 1475–1489 (1995).
    [CrossRef]
  12. R. Zengerle, B. Jacobs, W. Weiershausen, K. Faltin, A. Kunz, “Efficient spot-size transformation using spatially separated tapered InP/InGaAsP twin waveguides,” J. Lightwave Technol. 14, 448–453 (1996).
    [CrossRef]
  13. D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
    [CrossRef]
  14. J. M. White, P. F. Heidrich, “Optical waveguide refractive index profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. 15, 151–155 (1976).
    [CrossRef] [PubMed]
  15. Y. H. Won, P. C. Jaussaud, G. H. Chartier, “Three-prism loss measurements of optical waveguides,” Appl. Phys. Lett. 37, 269–271 (1980).
    [CrossRef]

1996 (2)

P. Sewell, T. M. Benson, P. C. Kendall, T. Anada, “Tapered beam propagation,” Electron. Lett. 32, 1025–1026 (1996).
[CrossRef]

R. Zengerle, B. Jacobs, W. Weiershausen, K. Faltin, A. Kunz, “Efficient spot-size transformation using spatially separated tapered InP/InGaAsP twin waveguides,” J. Lightwave Technol. 14, 448–453 (1996).
[CrossRef]

1995 (4)

Y. Koike, T. Ishigure, E. Nihei, “High-bandwidth graded-index polymer optical fiber,” J. Lightwave Technol. 13, 1475–1489 (1995).
[CrossRef]

B. Mersali, H. J. Bruckner, M. Feuillade, S. Sainson, A. Ougazzaden, A. Carenco, “Theoretical and experimental studies of a spot-size transformer with integrated waveguide for polarization insensitive optical amplifiers,” J. Lightwave Technol. 13, 1865–1871 (1995).
[CrossRef]

B. Jacobs, R. Zengerle, K. Faltin, W. Weiershausen, “Vertically tapered spot size transformers fabricated by a simple masking technique,” Electron. Lett. 31, 794–796 (1995).
[CrossRef]

S. Sottini, D. Grando, L. Palchetti, E. Giorgetti, “Optical fiber-polymer guide coupling by a tapered graded index glass guide,” IEEE J. Quantum Electron. 31, 1123–1129 (1995).
[CrossRef]

1994 (1)

F. Ghirardi, B. Mersali, J. Brandon, G. Herve-Gruyer, A. Carenco, “Quasi planar spot-size transformer for efficient coupling between a cleaved fiber and an InP/InGaAsP rib waveguide,” IEEE Photon. Technol. Lett. 6, 522–524 (1994).
[CrossRef]

1992 (3)

H. Yanagawa, T. Shimizu, S. Nakamura, I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10, 587–591 (1992).
[CrossRef]

R. Zengerle, H. Bruckner, H. Olzhausen, A. Kohl, “Low-loss fibre-chip coupling by buried laterally tapered InP/InGaAsP waveguide structure,” Electron. Lett. 28, 631–632 (1992).
[CrossRef]

G. H. Song, W. J. Tomlinson, “Fourier analysis and synthesis of adiabatic tapers in integrated optics,” J. Opt. Soc. Am. A 9, 1289–1300 (1992).
[CrossRef]

1989 (1)

D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

1987 (1)

P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[CrossRef]

1980 (1)

Y. H. Won, P. C. Jaussaud, G. H. Chartier, “Three-prism loss measurements of optical waveguides,” Appl. Phys. Lett. 37, 269–271 (1980).
[CrossRef]

1976 (1)

Anada, T.

P. Sewell, T. M. Benson, P. C. Kendall, T. Anada, “Tapered beam propagation,” Electron. Lett. 32, 1025–1026 (1996).
[CrossRef]

Baets, R.

J. Haes, J. Willems, R. Baets, “Design of adiabatic tapers for high contrast step index waveguides,” in Linear and Nonlinear Integrated Optics, G. C. Righini, D. Yevick, eds., Proc. SPIE2212, 685–693 (1994).
[CrossRef]

Benson, T. M.

P. Sewell, T. M. Benson, P. C. Kendall, T. Anada, “Tapered beam propagation,” Electron. Lett. 32, 1025–1026 (1996).
[CrossRef]

Brandon, J.

F. Ghirardi, B. Mersali, J. Brandon, G. Herve-Gruyer, A. Carenco, “Quasi planar spot-size transformer for efficient coupling between a cleaved fiber and an InP/InGaAsP rib waveguide,” IEEE Photon. Technol. Lett. 6, 522–524 (1994).
[CrossRef]

Bruckner, H.

R. Zengerle, H. Bruckner, H. Olzhausen, A. Kohl, “Low-loss fibre-chip coupling by buried laterally tapered InP/InGaAsP waveguide structure,” Electron. Lett. 28, 631–632 (1992).
[CrossRef]

Bruckner, H. J.

B. Mersali, H. J. Bruckner, M. Feuillade, S. Sainson, A. Ougazzaden, A. Carenco, “Theoretical and experimental studies of a spot-size transformer with integrated waveguide for polarization insensitive optical amplifiers,” J. Lightwave Technol. 13, 1865–1871 (1995).
[CrossRef]

Carenco, A.

B. Mersali, H. J. Bruckner, M. Feuillade, S. Sainson, A. Ougazzaden, A. Carenco, “Theoretical and experimental studies of a spot-size transformer with integrated waveguide for polarization insensitive optical amplifiers,” J. Lightwave Technol. 13, 1865–1871 (1995).
[CrossRef]

F. Ghirardi, B. Mersali, J. Brandon, G. Herve-Gruyer, A. Carenco, “Quasi planar spot-size transformer for efficient coupling between a cleaved fiber and an InP/InGaAsP rib waveguide,” IEEE Photon. Technol. Lett. 6, 522–524 (1994).
[CrossRef]

Chartier, G. H.

Y. H. Won, P. C. Jaussaud, G. H. Chartier, “Three-prism loss measurements of optical waveguides,” Appl. Phys. Lett. 37, 269–271 (1980).
[CrossRef]

Faltin, K.

R. Zengerle, B. Jacobs, W. Weiershausen, K. Faltin, A. Kunz, “Efficient spot-size transformation using spatially separated tapered InP/InGaAsP twin waveguides,” J. Lightwave Technol. 14, 448–453 (1996).
[CrossRef]

B. Jacobs, R. Zengerle, K. Faltin, W. Weiershausen, “Vertically tapered spot size transformers fabricated by a simple masking technique,” Electron. Lett. 31, 794–796 (1995).
[CrossRef]

Feuillade, M.

B. Mersali, H. J. Bruckner, M. Feuillade, S. Sainson, A. Ougazzaden, A. Carenco, “Theoretical and experimental studies of a spot-size transformer with integrated waveguide for polarization insensitive optical amplifiers,” J. Lightwave Technol. 13, 1865–1871 (1995).
[CrossRef]

Ghirardi, F.

F. Ghirardi, B. Mersali, J. Brandon, G. Herve-Gruyer, A. Carenco, “Quasi planar spot-size transformer for efficient coupling between a cleaved fiber and an InP/InGaAsP rib waveguide,” IEEE Photon. Technol. Lett. 6, 522–524 (1994).
[CrossRef]

Giorgetti, E.

S. Sottini, D. Grando, L. Palchetti, E. Giorgetti, “Optical fiber-polymer guide coupling by a tapered graded index glass guide,” IEEE J. Quantum Electron. 31, 1123–1129 (1995).
[CrossRef]

Grando, D.

S. Sottini, D. Grando, L. Palchetti, E. Giorgetti, “Optical fiber-polymer guide coupling by a tapered graded index glass guide,” IEEE J. Quantum Electron. 31, 1123–1129 (1995).
[CrossRef]

Haes, J.

J. Haes, J. Willems, R. Baets, “Design of adiabatic tapers for high contrast step index waveguides,” in Linear and Nonlinear Integrated Optics, G. C. Righini, D. Yevick, eds., Proc. SPIE2212, 685–693 (1994).
[CrossRef]

Heidrich, P. F.

Hermansson, B.

D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

Herve-Gruyer, G.

F. Ghirardi, B. Mersali, J. Brandon, G. Herve-Gruyer, A. Carenco, “Quasi planar spot-size transformer for efficient coupling between a cleaved fiber and an InP/InGaAsP rib waveguide,” IEEE Photon. Technol. Lett. 6, 522–524 (1994).
[CrossRef]

Ishigure, T.

Y. Koike, T. Ishigure, E. Nihei, “High-bandwidth graded-index polymer optical fiber,” J. Lightwave Technol. 13, 1475–1489 (1995).
[CrossRef]

Jacobs, B.

R. Zengerle, B. Jacobs, W. Weiershausen, K. Faltin, A. Kunz, “Efficient spot-size transformation using spatially separated tapered InP/InGaAsP twin waveguides,” J. Lightwave Technol. 14, 448–453 (1996).
[CrossRef]

B. Jacobs, R. Zengerle, K. Faltin, W. Weiershausen, “Vertically tapered spot size transformers fabricated by a simple masking technique,” Electron. Lett. 31, 794–796 (1995).
[CrossRef]

Jaussaud, P. C.

Y. H. Won, P. C. Jaussaud, G. H. Chartier, “Three-prism loss measurements of optical waveguides,” Appl. Phys. Lett. 37, 269–271 (1980).
[CrossRef]

Kendall, P. C.

P. Sewell, T. M. Benson, P. C. Kendall, T. Anada, “Tapered beam propagation,” Electron. Lett. 32, 1025–1026 (1996).
[CrossRef]

Kohl, A.

R. Zengerle, H. Bruckner, H. Olzhausen, A. Kohl, “Low-loss fibre-chip coupling by buried laterally tapered InP/InGaAsP waveguide structure,” Electron. Lett. 28, 631–632 (1992).
[CrossRef]

Koike, Y.

Y. Koike, T. Ishigure, E. Nihei, “High-bandwidth graded-index polymer optical fiber,” J. Lightwave Technol. 13, 1475–1489 (1995).
[CrossRef]

Kunz, A.

R. Zengerle, B. Jacobs, W. Weiershausen, K. Faltin, A. Kunz, “Efficient spot-size transformation using spatially separated tapered InP/InGaAsP twin waveguides,” J. Lightwave Technol. 14, 448–453 (1996).
[CrossRef]

Mersali, B.

B. Mersali, H. J. Bruckner, M. Feuillade, S. Sainson, A. Ougazzaden, A. Carenco, “Theoretical and experimental studies of a spot-size transformer with integrated waveguide for polarization insensitive optical amplifiers,” J. Lightwave Technol. 13, 1865–1871 (1995).
[CrossRef]

F. Ghirardi, B. Mersali, J. Brandon, G. Herve-Gruyer, A. Carenco, “Quasi planar spot-size transformer for efficient coupling between a cleaved fiber and an InP/InGaAsP rib waveguide,” IEEE Photon. Technol. Lett. 6, 522–524 (1994).
[CrossRef]

Nakamura, S.

H. Yanagawa, T. Shimizu, S. Nakamura, I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10, 587–591 (1992).
[CrossRef]

Nihei, E.

Y. Koike, T. Ishigure, E. Nihei, “High-bandwidth graded-index polymer optical fiber,” J. Lightwave Technol. 13, 1475–1489 (1995).
[CrossRef]

Ohyama, I.

H. Yanagawa, T. Shimizu, S. Nakamura, I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10, 587–591 (1992).
[CrossRef]

Olzhausen, H.

R. Zengerle, H. Bruckner, H. Olzhausen, A. Kohl, “Low-loss fibre-chip coupling by buried laterally tapered InP/InGaAsP waveguide structure,” Electron. Lett. 28, 631–632 (1992).
[CrossRef]

Ougazzaden, A.

B. Mersali, H. J. Bruckner, M. Feuillade, S. Sainson, A. Ougazzaden, A. Carenco, “Theoretical and experimental studies of a spot-size transformer with integrated waveguide for polarization insensitive optical amplifiers,” J. Lightwave Technol. 13, 1865–1871 (1995).
[CrossRef]

Palchetti, L.

S. Sottini, D. Grando, L. Palchetti, E. Giorgetti, “Optical fiber-polymer guide coupling by a tapered graded index glass guide,” IEEE J. Quantum Electron. 31, 1123–1129 (1995).
[CrossRef]

Ramaswamy, R. V.

P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[CrossRef]

Sainson, S.

B. Mersali, H. J. Bruckner, M. Feuillade, S. Sainson, A. Ougazzaden, A. Carenco, “Theoretical and experimental studies of a spot-size transformer with integrated waveguide for polarization insensitive optical amplifiers,” J. Lightwave Technol. 13, 1865–1871 (1995).
[CrossRef]

Sewell, P.

P. Sewell, T. M. Benson, P. C. Kendall, T. Anada, “Tapered beam propagation,” Electron. Lett. 32, 1025–1026 (1996).
[CrossRef]

Shimizu, T.

H. Yanagawa, T. Shimizu, S. Nakamura, I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10, 587–591 (1992).
[CrossRef]

Song, G. H.

Sottini, S.

S. Sottini, D. Grando, L. Palchetti, E. Giorgetti, “Optical fiber-polymer guide coupling by a tapered graded index glass guide,” IEEE J. Quantum Electron. 31, 1123–1129 (1995).
[CrossRef]

Suchoski, P. G.

P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[CrossRef]

Tomlinson, W. J.

Weiershausen, W.

R. Zengerle, B. Jacobs, W. Weiershausen, K. Faltin, A. Kunz, “Efficient spot-size transformation using spatially separated tapered InP/InGaAsP twin waveguides,” J. Lightwave Technol. 14, 448–453 (1996).
[CrossRef]

B. Jacobs, R. Zengerle, K. Faltin, W. Weiershausen, “Vertically tapered spot size transformers fabricated by a simple masking technique,” Electron. Lett. 31, 794–796 (1995).
[CrossRef]

White, J. M.

Willems, J.

J. Haes, J. Willems, R. Baets, “Design of adiabatic tapers for high contrast step index waveguides,” in Linear and Nonlinear Integrated Optics, G. C. Righini, D. Yevick, eds., Proc. SPIE2212, 685–693 (1994).
[CrossRef]

Won, Y. H.

Y. H. Won, P. C. Jaussaud, G. H. Chartier, “Three-prism loss measurements of optical waveguides,” Appl. Phys. Lett. 37, 269–271 (1980).
[CrossRef]

Yanagawa, H.

H. Yanagawa, T. Shimizu, S. Nakamura, I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10, 587–591 (1992).
[CrossRef]

Yevick, D.

D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

Zengerle, R.

R. Zengerle, B. Jacobs, W. Weiershausen, K. Faltin, A. Kunz, “Efficient spot-size transformation using spatially separated tapered InP/InGaAsP twin waveguides,” J. Lightwave Technol. 14, 448–453 (1996).
[CrossRef]

B. Jacobs, R. Zengerle, K. Faltin, W. Weiershausen, “Vertically tapered spot size transformers fabricated by a simple masking technique,” Electron. Lett. 31, 794–796 (1995).
[CrossRef]

R. Zengerle, H. Bruckner, H. Olzhausen, A. Kohl, “Low-loss fibre-chip coupling by buried laterally tapered InP/InGaAsP waveguide structure,” Electron. Lett. 28, 631–632 (1992).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. H. Won, P. C. Jaussaud, G. H. Chartier, “Three-prism loss measurements of optical waveguides,” Appl. Phys. Lett. 37, 269–271 (1980).
[CrossRef]

Electron. Lett. (3)

P. Sewell, T. M. Benson, P. C. Kendall, T. Anada, “Tapered beam propagation,” Electron. Lett. 32, 1025–1026 (1996).
[CrossRef]

B. Jacobs, R. Zengerle, K. Faltin, W. Weiershausen, “Vertically tapered spot size transformers fabricated by a simple masking technique,” Electron. Lett. 31, 794–796 (1995).
[CrossRef]

R. Zengerle, H. Bruckner, H. Olzhausen, A. Kohl, “Low-loss fibre-chip coupling by buried laterally tapered InP/InGaAsP waveguide structure,” Electron. Lett. 28, 631–632 (1992).
[CrossRef]

IEEE J. Quantum Electron. (3)

P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[CrossRef]

S. Sottini, D. Grando, L. Palchetti, E. Giorgetti, “Optical fiber-polymer guide coupling by a tapered graded index glass guide,” IEEE J. Quantum Electron. 31, 1123–1129 (1995).
[CrossRef]

D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

F. Ghirardi, B. Mersali, J. Brandon, G. Herve-Gruyer, A. Carenco, “Quasi planar spot-size transformer for efficient coupling between a cleaved fiber and an InP/InGaAsP rib waveguide,” IEEE Photon. Technol. Lett. 6, 522–524 (1994).
[CrossRef]

J. Lightwave Technol. (4)

H. Yanagawa, T. Shimizu, S. Nakamura, I. Ohyama, “Index-and-dimensional taper and its application to photonic devices,” J. Lightwave Technol. 10, 587–591 (1992).
[CrossRef]

B. Mersali, H. J. Bruckner, M. Feuillade, S. Sainson, A. Ougazzaden, A. Carenco, “Theoretical and experimental studies of a spot-size transformer with integrated waveguide for polarization insensitive optical amplifiers,” J. Lightwave Technol. 13, 1865–1871 (1995).
[CrossRef]

Y. Koike, T. Ishigure, E. Nihei, “High-bandwidth graded-index polymer optical fiber,” J. Lightwave Technol. 13, 1475–1489 (1995).
[CrossRef]

R. Zengerle, B. Jacobs, W. Weiershausen, K. Faltin, A. Kunz, “Efficient spot-size transformation using spatially separated tapered InP/InGaAsP twin waveguides,” J. Lightwave Technol. 14, 448–453 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

Other (1)

J. Haes, J. Willems, R. Baets, “Design of adiabatic tapers for high contrast step index waveguides,” in Linear and Nonlinear Integrated Optics, G. C. Righini, D. Yevick, eds., Proc. SPIE2212, 685–693 (1994).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Sketch of a five-layer tapered coupler with (a) a first-order mode and (b) a second-order mode.

Fig. 2
Fig. 2

Taper length dependence of η for mode conversion from a first-order mode in the tapered layer to a second-order mode in the top layer.

Fig. 3
Fig. 3

Intensity distribution between the tapered and the top layers with (a) a second-order mode in the tapered layer, (b) a first-order mode in the tapered layer, (c) a first-order mode both in the tapered layer and in the top layer.

Fig. 4
Fig. 4

Tapered length dependence of η for the case shown in Fig. 3(c).

Fig. 5
Fig. 5

Dependence of z on χ12 and χ21.

Fig. 6
Fig. 6

Scheme of the fabrication system for a tapered waveguide by use of the ion-exchange method.

Fig. 7
Fig. 7

Dependence of d s on n p . With (a) the K+ - Na+ exchange and (b) the Ag+ - Na+ exchange. Open circles, 2-h soaking time; filled circles, 3-h soaking time.

Fig. 8
Fig. 8

Dependence of the index change from the surface index of BZS4 on Ag+.

Fig. 9
Fig. 9

Mode dispersion curve of the first-order and the second-order modes in the tapered layer and of the first-order mode in the top layer.

Fig. 10
Fig. 10

Electric field of the first-order modes in the tapered layer and in the top layer.

Fig. 11
Fig. 11

Actual device parameters obtained with the experiment.

Fig. 12
Fig. 12

Experimental setup with two types of waveguide with (a) a three-layered structure and (b) a five-layered structure.

Fig. 13
Fig. 13

Streak patterns of three layers with (a) a first-order mode, (b) a second-order mode, (c) a third-order mode.

Fig. 14
Fig. 14

Streak patterns of five layers with (a) a first-order mode, (b) a second-order mode, (c) a third-order mode.

Equations (4)

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

Ez, t=AzE1 exp-jωt+β1z+BzE2 exp-jωt+β2z,
dAdz+jχ12A=0, dBdz+jχ21B=0,
χpq=ω0--Nx, y2-Nqx, y2Ep·Eqdxdy--uzEp×Hp+Ep×Hpdxdy,
η=- EinyxEoutyxdx2,

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