Abstract

The perturbation to the refractive index induced by a periodic electric field from two systems of interdigitated electrodes with the electrode-finger period l is analyzed for a waveguide with an electro-optically (EO) active core–cladding. It is shown that the electric field induces two superimposed transmissive refractive-index gratings with different symmetries of their cross-section distributions. One of these gratings has a constant component of an EO-induced refractive index along with its variable component with periodicity l, whereas the second grating possesses only a variable component with periodicity 2l. With the proper waveguide design, the gratings provide interaction between a guided fundamental core mode and two guided cladding modes. Through the externally applied electric potential, these gratings can be independently switched ON and OFF, or they can be activated simultaneously with electronically controlled weighting factors. Coupling coefficients of both gratings are analyzed in terms of their dependence on the electrode duty ratio and dielectric permittivities of the core and cladding. The coupled-wave equations for the superimposed gratings are written and solved. The spectral characteristics are investigated by numerical simulation. It is found that the spectral characteristics are described by a dual-dip transmission spectrum with individual electronic control of the dip depths and positions. Within the concept, a new external potential application scheme is described in which the symmetry of the cross-sectional distribution of the refractive index provides coupling only between the core mode and the cladding modes, preventing interaction of the cladding modes with each another. This simple concept opens opportunities for developing a number of tunable devices for integrated optics by use of the proposed design as a building block.

© 2002 Optical Society of America

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References

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  1. T. Erdogan, “Fiber gratings provide keys to future optical networks,” Photonics Spectra 32, 96–97 (1998).
  2. D. M. Costantini, C. A. P. Muller, S. A. Vasiliev, H. G. Limberger, R. P. Salathe, “Tunable loss filter based on metal-coated long-period filter grating,” IEEE Photon. Technol. Lett. 11, 1458–1460 (1999).
    [CrossRef]
  3. A. A. Abramov, A. Hale, R. S. Windler, T. A. Strasser, “Widely tunable long-period fiber gratings,” Electron. Lett. 35, 81–82 (1999).
    [CrossRef]
  4. Y. Joeng, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photon. Technol. Lett. 12, 519–521 (2000).
    [CrossRef]
  5. D. B. Stegall, T. Erdogan, “Dispersion control with use of long-period fiber gratings,” J. Opt. Soc. Am. A 17, 304–312 (2000).
    [CrossRef]
  6. M. Das, K. Thyagarajan, “Dispersion compensation in transmission using uniform long period fiber gratings,” Opt. Commun. 190, 159–163 (2001).
    [CrossRef]
  7. G. Levy-Yurista, A. A. Friesem, “Dual spectral filters with multi-layered grating waveguide structures,” Appl. Phys. B 72, 921–925 (2001).
    [CrossRef]
  8. J. Bao, X. Zhang, K. Chen, W. Zhou, “Spectra of dual overwritten fiber Bragg grating,” Opt. Commun. 188, 31–39 (2001).
    [CrossRef]
  9. J. Zhao, X. Shen, Y. Xia, “Beam splitting, combining and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
    [CrossRef]
  10. A. Yariv, “Frustration of Bragg reflection by cooperative dual-mode interference: a new mode of optical propagation,” Opt. Lett. 23, 1835–1836 (1998).
    [CrossRef]
  11. M. Kulishov, “Interdigitated electrode-induced phase grating with an electrically switchable and tunable period,” Appl. Opt. 38, 7356–7363 (1999).
    [CrossRef]
  12. M. Kulishov, P. Cheben, X. Daxhelet, S. Delprat, “Electro-optically induced tilted phase gratings in waveguides,” J. Opt. Soc. Am. B 18, 457–464 (2001).
    [CrossRef]
  13. E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
    [CrossRef]
  14. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Sec. 1.441.2.
  15. M. A. Hussain, S. L. Pu, “Dynamic stress intensity factor for an unbounded plate having collinear cracks,” Eng. Fract. Mech. 4, 865–876 (1972).
    [CrossRef]
  16. V. Ramaswamy, “Ray model of energy and power flow in anisotropic film waveguides,” J. Opt. Soc. Am. 64, 1313–1320 (1974).
    [CrossRef]
  17. D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972).
  18. H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, 2nd ed., T. Tamir, ed. (Springer-Verlag, New York, 1990), pp. 48–50.
  19. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  20. K. S. Lee, “Mode coupling in tilted planar waveguide gratings,” Appl. Opt. 39, 6144–6149 (2000).
    [CrossRef]
  21. D. Marcuse, “Electrooptic coupling between TE and TM modes in anisotropic slabs,” IEEE J. Quantum Electron. QE-11, 759–767 (1975).
    [CrossRef]

2001 (5)

M. Das, K. Thyagarajan, “Dispersion compensation in transmission using uniform long period fiber gratings,” Opt. Commun. 190, 159–163 (2001).
[CrossRef]

G. Levy-Yurista, A. A. Friesem, “Dual spectral filters with multi-layered grating waveguide structures,” Appl. Phys. B 72, 921–925 (2001).
[CrossRef]

J. Bao, X. Zhang, K. Chen, W. Zhou, “Spectra of dual overwritten fiber Bragg grating,” Opt. Commun. 188, 31–39 (2001).
[CrossRef]

J. Zhao, X. Shen, Y. Xia, “Beam splitting, combining and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[CrossRef]

M. Kulishov, P. Cheben, X. Daxhelet, S. Delprat, “Electro-optically induced tilted phase gratings in waveguides,” J. Opt. Soc. Am. B 18, 457–464 (2001).
[CrossRef]

2000 (3)

K. S. Lee, “Mode coupling in tilted planar waveguide gratings,” Appl. Opt. 39, 6144–6149 (2000).
[CrossRef]

D. B. Stegall, T. Erdogan, “Dispersion control with use of long-period fiber gratings,” J. Opt. Soc. Am. A 17, 304–312 (2000).
[CrossRef]

Y. Joeng, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photon. Technol. Lett. 12, 519–521 (2000).
[CrossRef]

1999 (3)

D. M. Costantini, C. A. P. Muller, S. A. Vasiliev, H. G. Limberger, R. P. Salathe, “Tunable loss filter based on metal-coated long-period filter grating,” IEEE Photon. Technol. Lett. 11, 1458–1460 (1999).
[CrossRef]

A. A. Abramov, A. Hale, R. S. Windler, T. A. Strasser, “Widely tunable long-period fiber gratings,” Electron. Lett. 35, 81–82 (1999).
[CrossRef]

M. Kulishov, “Interdigitated electrode-induced phase grating with an electrically switchable and tunable period,” Appl. Opt. 38, 7356–7363 (1999).
[CrossRef]

1998 (2)

A. Yariv, “Frustration of Bragg reflection by cooperative dual-mode interference: a new mode of optical propagation,” Opt. Lett. 23, 1835–1836 (1998).
[CrossRef]

T. Erdogan, “Fiber gratings provide keys to future optical networks,” Photonics Spectra 32, 96–97 (1998).

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

1987 (1)

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

1975 (1)

D. Marcuse, “Electrooptic coupling between TE and TM modes in anisotropic slabs,” IEEE J. Quantum Electron. QE-11, 759–767 (1975).
[CrossRef]

1974 (1)

1972 (1)

M. A. Hussain, S. L. Pu, “Dynamic stress intensity factor for an unbounded plate having collinear cracks,” Eng. Fract. Mech. 4, 865–876 (1972).
[CrossRef]

Abramov, A. A.

A. A. Abramov, A. Hale, R. S. Windler, T. A. Strasser, “Widely tunable long-period fiber gratings,” Electron. Lett. 35, 81–82 (1999).
[CrossRef]

Bao, J.

J. Bao, X. Zhang, K. Chen, W. Zhou, “Spectra of dual overwritten fiber Bragg grating,” Opt. Commun. 188, 31–39 (2001).
[CrossRef]

Cheben, P.

Chen, K.

J. Bao, X. Zhang, K. Chen, W. Zhou, “Spectra of dual overwritten fiber Bragg grating,” Opt. Commun. 188, 31–39 (2001).
[CrossRef]

Choi, S.

Y. Joeng, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photon. Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Costantini, D. M.

D. M. Costantini, C. A. P. Muller, S. A. Vasiliev, H. G. Limberger, R. P. Salathe, “Tunable loss filter based on metal-coated long-period filter grating,” IEEE Photon. Technol. Lett. 11, 1458–1460 (1999).
[CrossRef]

Das, M.

M. Das, K. Thyagarajan, “Dispersion compensation in transmission using uniform long period fiber gratings,” Opt. Commun. 190, 159–163 (2001).
[CrossRef]

Daxhelet, X.

Delprat, S.

Erdogan, T.

D. B. Stegall, T. Erdogan, “Dispersion control with use of long-period fiber gratings,” J. Opt. Soc. Am. A 17, 304–312 (2000).
[CrossRef]

T. Erdogan, “Fiber gratings provide keys to future optical networks,” Photonics Spectra 32, 96–97 (1998).

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Friesem, A. A.

G. Levy-Yurista, A. A. Friesem, “Dual spectral filters with multi-layered grating waveguide structures,” Appl. Phys. B 72, 921–925 (2001).
[CrossRef]

Gaylord, T. K.

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

Glytsis, E. N.

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Sec. 1.441.2.

Hale, A.

A. A. Abramov, A. Hale, R. S. Windler, T. A. Strasser, “Widely tunable long-period fiber gratings,” Electron. Lett. 35, 81–82 (1999).
[CrossRef]

Hussain, M. A.

M. A. Hussain, S. L. Pu, “Dynamic stress intensity factor for an unbounded plate having collinear cracks,” Eng. Fract. Mech. 4, 865–876 (1972).
[CrossRef]

Joeng, Y.

Y. Joeng, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photon. Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, 2nd ed., T. Tamir, ed. (Springer-Verlag, New York, 1990), pp. 48–50.

Kulishov, M.

Lee, B.

Y. Joeng, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photon. Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Lee, K. S.

Levy-Yurista, G.

G. Levy-Yurista, A. A. Friesem, “Dual spectral filters with multi-layered grating waveguide structures,” Appl. Phys. B 72, 921–925 (2001).
[CrossRef]

Limberger, H. G.

D. M. Costantini, C. A. P. Muller, S. A. Vasiliev, H. G. Limberger, R. P. Salathe, “Tunable loss filter based on metal-coated long-period filter grating,” IEEE Photon. Technol. Lett. 11, 1458–1460 (1999).
[CrossRef]

Marcuse, D.

D. Marcuse, “Electrooptic coupling between TE and TM modes in anisotropic slabs,” IEEE J. Quantum Electron. QE-11, 759–767 (1975).
[CrossRef]

D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972).

Moharam, M. G.

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

Muller, C. A. P.

D. M. Costantini, C. A. P. Muller, S. A. Vasiliev, H. G. Limberger, R. P. Salathe, “Tunable loss filter based on metal-coated long-period filter grating,” IEEE Photon. Technol. Lett. 11, 1458–1460 (1999).
[CrossRef]

Oh, K.

Y. Joeng, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photon. Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Pu, S. L.

M. A. Hussain, S. L. Pu, “Dynamic stress intensity factor for an unbounded plate having collinear cracks,” Eng. Fract. Mech. 4, 865–876 (1972).
[CrossRef]

Ramaswamy, V.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Sec. 1.441.2.

Salathe, R. P.

D. M. Costantini, C. A. P. Muller, S. A. Vasiliev, H. G. Limberger, R. P. Salathe, “Tunable loss filter based on metal-coated long-period filter grating,” IEEE Photon. Technol. Lett. 11, 1458–1460 (1999).
[CrossRef]

Seo, H. S.

Y. Joeng, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photon. Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Shen, X.

J. Zhao, X. Shen, Y. Xia, “Beam splitting, combining and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[CrossRef]

Stegall, D. B.

Strasser, T. A.

A. A. Abramov, A. Hale, R. S. Windler, T. A. Strasser, “Widely tunable long-period fiber gratings,” Electron. Lett. 35, 81–82 (1999).
[CrossRef]

Thyagarajan, K.

M. Das, K. Thyagarajan, “Dispersion compensation in transmission using uniform long period fiber gratings,” Opt. Commun. 190, 159–163 (2001).
[CrossRef]

Vasiliev, S. A.

D. M. Costantini, C. A. P. Muller, S. A. Vasiliev, H. G. Limberger, R. P. Salathe, “Tunable loss filter based on metal-coated long-period filter grating,” IEEE Photon. Technol. Lett. 11, 1458–1460 (1999).
[CrossRef]

Windler, R. S.

A. A. Abramov, A. Hale, R. S. Windler, T. A. Strasser, “Widely tunable long-period fiber gratings,” Electron. Lett. 35, 81–82 (1999).
[CrossRef]

Xia, Y.

J. Zhao, X. Shen, Y. Xia, “Beam splitting, combining and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[CrossRef]

Yang, B.

Y. Joeng, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photon. Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Yariv, A.

Zhang, X.

J. Bao, X. Zhang, K. Chen, W. Zhou, “Spectra of dual overwritten fiber Bragg grating,” Opt. Commun. 188, 31–39 (2001).
[CrossRef]

Zhao, J.

J. Zhao, X. Shen, Y. Xia, “Beam splitting, combining and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[CrossRef]

Zhou, W.

J. Bao, X. Zhang, K. Chen, W. Zhou, “Spectra of dual overwritten fiber Bragg grating,” Opt. Commun. 188, 31–39 (2001).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

G. Levy-Yurista, A. A. Friesem, “Dual spectral filters with multi-layered grating waveguide structures,” Appl. Phys. B 72, 921–925 (2001).
[CrossRef]

Electron. Lett. (1)

A. A. Abramov, A. Hale, R. S. Windler, T. A. Strasser, “Widely tunable long-period fiber gratings,” Electron. Lett. 35, 81–82 (1999).
[CrossRef]

Eng. Fract. Mech. (1)

M. A. Hussain, S. L. Pu, “Dynamic stress intensity factor for an unbounded plate having collinear cracks,” Eng. Fract. Mech. 4, 865–876 (1972).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Marcuse, “Electrooptic coupling between TE and TM modes in anisotropic slabs,” IEEE J. Quantum Electron. QE-11, 759–767 (1975).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

Y. Joeng, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh, “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photon. Technol. Lett. 12, 519–521 (2000).
[CrossRef]

D. M. Costantini, C. A. P. Muller, S. A. Vasiliev, H. G. Limberger, R. P. Salathe, “Tunable loss filter based on metal-coated long-period filter grating,” IEEE Photon. Technol. Lett. 11, 1458–1460 (1999).
[CrossRef]

J. Lightwave Technol. (2)

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

Opt. Commun. (2)

J. Bao, X. Zhang, K. Chen, W. Zhou, “Spectra of dual overwritten fiber Bragg grating,” Opt. Commun. 188, 31–39 (2001).
[CrossRef]

M. Das, K. Thyagarajan, “Dispersion compensation in transmission using uniform long period fiber gratings,” Opt. Commun. 190, 159–163 (2001).
[CrossRef]

Opt. Laser Technol. (1)

J. Zhao, X. Shen, Y. Xia, “Beam splitting, combining and cross coupling through multiple superimposed volume-index gratings,” Opt. Laser Technol. 33, 23–28 (2001).
[CrossRef]

Opt. Lett. (1)

Photonics Spectra (1)

T. Erdogan, “Fiber gratings provide keys to future optical networks,” Photonics Spectra 32, 96–97 (1998).

Other (3)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Sec. 1.441.2.

D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972).

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, 2nd ed., T. Tamir, ed. (Springer-Verlag, New York, 1990), pp. 48–50.

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Figures (17)

Fig. 1
Fig. 1

(a) Geometry of the EO-induced waveguide gratings with interdigitated electrode structure. The electrostatic problem solution for this geometry can be presented as a linear combination of the Laplace’s equation solutions for the structures shown in (b) and (c).

Fig. 2
Fig. 2

Boundary conditions for the normalized electric potential ϕb(4)(x, h+d)/V0 and the normalized charge density ρ1(x)h/V0 for (a) and (c) narrow electrodes (a/l=0.2) and (b) and (d) wide electrode (a/l=0.6) for the following parameter set: 2h=0.0745l, d=1.376h, (s)=3.5, xx(cl)=zzcl=4, xx(co)=8.4, zz(co)=5.8, and N=40.

Fig. 3
Fig. 3

Boundary conditions for the normalized electric potential ϕc(4)(x, h+d)/V0 and the normalized charge density ρ2(x)h/V0 for (a) and (c) narrow electrodes (a/l=0.2) and (b) and (d) wide electrode (a/l=0.6) for the following parameter set: 2h=0.0745l, d=1.376h, (s)=3.5, xx(cl)=zz(cl)=4; xx(co)=8.4, zz(co)=5.8, and N=40.

Fig. 4
Fig. 4

Three-dimensional distribution of the normal component of the normalized electrode electric field Ez/(V0/h) in the waveguide core for different values of ΔV voltages. (a) ΔV=0, (b) ΔV=-1.6V0, and (c) ΔV=-2V0 for the following parameter set: a/l=0.25, 2h=0.0745l, d=1.376h, (s)=3.5, xx(cl)=zz(cl)=4, xx(co)=8.4, zz(co)=5.8, and N=40.

Fig. 5
Fig. 5

First core mode TE1 (solid curve); first cladding mode TE2 (dashed curve); and second cladding mode TE3 (dotted curve) distributions across the waveguide cross section.

Fig. 6
Fig. 6

Coupling coefficients (a) κijh/(V0r13) and (b) σiih/(V0r13) in 1/m units as a function of ΔV/V0 [(a) κ12, solid curve; κ13, dotted curve; and κ23, dashed curve; (b) σ11, solid curve; σ22, dashed curve; σ33, dotted curve] for the following waveguide and electrode parameters: 2h/l=0.0745, d/h=1.376, (s)=3.5, xx(cl)=zz(cl)=5, xx(co)=zz(co)=5.8, a/l=0.5, V0=50 V, and r13=30 pm/V.

Fig. 7
Fig. 7

Coupling coefficients (a) κijh/(V0r13) and (b) σiih/(V0r13) in 1/m units as a function of the electrode duty ratio for ΔV/V0=-1.75 [(a) κ12, solid curve; κ13, dotted curve; κ23, dashed curve; (b) σ11, solid curve; σ22, dashed curve; σ33, dotted curve] for the same waveguide and electrode parameters as in Fig. 6.

Fig. 8
Fig. 8

Coupling coefficients (a) and (c) κijh/(V0r13) and (b) and (d) σiih/(V0r13) in 1/m units as a function of the dielectric permittivity of the (a) and (b) cladding xx(cl)=zz(cl) and the dielectric permittivity of the (c) and (d) core zz(co) for the fixed xx(co) for a/l=0.5; ΔV/V0=-1.75. The rest of the parameters are the same as in Fig. 6 [(a) κ12, solid curve; κ13, dotted curve; κ23, dashed curve; (b) σ11, solid curve, σ22; dashed curve, and σ33; dotted curve].

Fig. 9
Fig. 9

Coupling coefficients (a) κijh/(V0r13) and (b) σiih/(V0r13) in 1/m units as a function of ΔV/V0 [(a) κ12, solid curve; κ13, dotted curve; κ23, dashed curve; (b) σ11, solid curve; σ22, dashed curve; σ33, dotted curve] for the following waveguide and electrode parameters: 2h/l=0.0745, d/h=1.376, (s)=3.5, xx(cl)=zz(cl)=4, xx(co)=5.8, zz(co)=4, a/l=0.8, V0=50 V, and r13=30 pm/V.

Fig. 10
Fig. 10

(a) Electric potential application scheme for the EO-induced waveguide gratings that provides coupling between the core mode and the cladding modes with the same symmetry as the core mode. The electrostatic problem solution for this geometry can be presented as a linear combination of the Laplace’s equation solutions for the structures (b) and (c).

Fig. 11
Fig. 11

Transmission spectra of waveguide grating (a) for both b and c activated gratings (ΔV/V0=-1.77, V0=57 V) (b) individually activated grating b (ΔV=0, V0=6.5 V) and (c) individually activated grating c (ΔV/V0=-2; V0=50 V) and the same waveguide parameter set as for Fig. 6.

Fig. 12
Fig. 12

Double-dip transmission spectra when both b and c activated gratings with ΔV/V0=-1.22; (a) V0=40 V, (b) V0=30 V, and (c) V0=20 V and the same waveguide parameter set as for Fig. 9.

Fig. 13
Fig. 13

Transmission spectra of (a) individually activated b grating (ΔV=0; solid curve, V0=20 V; dotted curve, V0=10 V; dashed curve, V0=5 V) and (b) individually activated c grating (ΔV/V0=-2; solid curve, V0=50 V; dotted curve, V0=25 V; dashed curve, V0=12.5 V) for the same waveguide parameter set as for Fig. 6.

Fig. 14
Fig. 14

(a) Electric potential application scheme for the EO-induced waveguide gratings that provides zero coupling between cladding modes with the corresponding potential distributions for each partial grating at (b) ΔV=0, and (c) ΔV/V0=-2. (d) The mode distributions involved in the interaction and (e) the momentum diagram in which β2-β3 coupling is eliminated due to zero overlapping integral for these mode interactions.

Fig. 15
Fig. 15

Transmission spectrum for the structure with uniform potential distribution on the electrodes V0=28 V, ΔV/V0=-1.77, and L=35 mm (dashed curve) and the structure with the same length, in which one half of the length is under V0=34 V, ΔV/V0=-1.61 and the second half is under V0=27 V; ΔV/V0=-2. The waveguide parameters are set as for Fig. 6.

Fig. 16
Fig. 16

(a) Electric potential application scheme for the EO-induced waveguide gratings that provides a zero constant component of the electric field, and as a result dc coupling coefficients are always zero with the corresponding potential distributions for each partial grating at (b) ΔV=0 and (c) ΔV/V0=-2.

Fig. 17
Fig. 17

Transmission spectra for the structure with parameters the same as in Fig. 15 except that σ11=σ22=σ33=0 for both curves.

Equations (100)

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xx(i)2ϕ(i)x2+zz(i)2ϕ(i)z2=0,
ϕ(i)(x, z)=1+ΔV2V0ϕb(i)(x, z)+ΔV2V0ϕc(i)(x, z).
ϕ(2)(x, h)=ϕ(3)(x, h),
ϕ(3)(x,-h)=ϕ(4)(x,-h),
zz(cl)ϕ(2)(x, h)z=zz(co)ϕ1(3)(x, h)z,
zz(co)ϕ(3)(x,-h)z=zz(cl)ϕ(4)(x,-h)z.
ϕb(1)(x, z)=V0B0h+zz(co)zz(cl)B0d+n=1Cn exp(-nkz)cos(nkx),
ϕb(2)(x, z)=V0B0h+zz(co)zz(cl)B0(z-h)+n=1En×sinh(nkhδ3)cosh[nkδ2(z-h)]+zz(co)δ3zz(cl)δ2cosh(nkhδ3)sinh[nkδ2(z-h)]×cos(nkx),
ϕb(3)(x, z)=V0B0z+n=1En sinh(nkδ3z)cos(nkx),
ϕb(4)(x, z)=V0-B0h+zz(co)zz(cl)B0(z+h)+n=1En×sinh(nkhδ3)cosh[nkδ2(z+h)]-zz(co)δ3zz(cl)δ2cosh(nkhδ3)sinh[nkδ2(z+h)]×cos(nkx),
ϕb(5)(x, z)=V0-B0h-zz(3)zz(2)B0d-n=1Cn exp(nkz)cos(nkx),
ϕb(1)(x, h+d)=ϕb(2)(x, h+d),
ϕb(2)(x, h+d)=V0,|x-ml|a/2,
(s)ϕb(1)(x, h+d)z=zz(cl)ϕb(2)(x, h+d)z,a/2<|x-ml|l/2,
B0h+dzz(co)zz(cl)+n=1EnFn cos(nkx)=1,
|x-ml|a/2, 
zz(co)B0h+n=1nkhEn[(s)Fn+δ2zz(cl)Gn]cos(nkx)=0,
a/2<|x-ml|l/2,
Fn=sinh(nkhδ3)cosh(nkδ2d)+zz(co)δ3zz(cl)δ2cosh(nkhδ3)×sinh(nkδ2d),
Gn=sinh(nkhδ3)sinh(nkδ2d)+zz(co)δ3zz(cl)δ2cosh(nkhδ3)×cosh(nkδ2d),
B0h+dzz(co)zz(cl)+n=1En*Fn(s)Fn+δ2zz(cl)Gncos(nkx)=1,
|x-ml|a/2,
zz(co)B0h+n=1nkhEn* cos(nkx)=0,
a/2<|x-ml|l/2,
En*=En[(s)Fn+δ2zz(cl)Gn].
n=1En* cos(nkx)=1-B0h+dzz(co)zz(cl)[zz(cl)δ2+(s)]+n=1RnEn* cos(nkx),
|x-ml|a/2,
Rn=1-[(s)+zz(cl)δ2]Fn, limn Fn=[(s)+δzz(cl)]-1,limn Rn=0.
ρ1(x)=zz(co)B0h+n=1nkhEn* cos(nkx).
zz(co)B0h=2l0a/2ρ1(ξ)dξ,
En*=2πnh0a/2ρ1(ξ)cos(nkξ)dξ.
-2πh0a/2ρ1(ξ)n=1cos(nkx)cos(nkξ)ndξ=1-B0h+dzz(co)zz(cl)[zz(cl)δ2+(s)]+2πhn=1Rnncos(nkx)0a/2ρ1(ξ)cos(nkξ)dξ, 0xa/2.
n=1cos(nkx)cos(nkξ)n=-12ln[|cos(kx)-cos(kξ)|].
cos(kx)=cos2(ka/4)+sin2(ka/4)cos(kζ),
cos(kξ)=cos2(ka/4)+sin2(ka/4)cos(kη),
ρ1[ξ(ζ)]dξdζ=m=0am cos(mkζ).
ln[|cos(kx)-cos(kξ)|]=lnsin2ka4-2p=1cos(pkζ)cos(pkη)p,
cos(nkx)=s=1nbs(n) cos(skη),
-a0 lnsin2ka4+m=1ammcos(nkη)=1-B01+dhzz(co)zz(cl)[zz(cl)δ2+(s)]kh+m=0ns=0asn=1Rnnbm(n)bs(n)cos(mkη), 0ζl/2.
a0zz(cl)δ2+(s)zz(co)1+dhzz(co)zz(cl)kh-lnsin2ka4-2n=1Rnn[b0(n)]2=[zz(cl)δ2+(s)]kh+s=1asn=1Rnnb0(n)bs(n).
m=1n=1Rnnbm(n)bs(n)-δmsmam=-2a0n=1Rnnb0(n)bs(n),
m=1AZms+AZmAZsΔ-δmsmam=-2AZsΔ[zz(cl)δ2+(s)]kh,
AZms=n=1Rnnbm(n)bs(n),AZm=n=1Rnnbm(n)b0(n),
Δ=[zz(cl)+(s)]zz(co)kh1+dhzz(co)zz(cl)-lnsin2ka4-2n=1Rnn[b0(n)]2.
nkhEn=2a0b0(n)+m=1nambm(n)(s)Fn+δ2zz(cl)Gn.
Ex(i)(x, z)=-ϕ(i)(x, z)x,Ez(i)(x, z)=-ϕ(i)(x, z)z.
ϕc(1)(x, z)=V0n=0An exp[-(n+1/2)kz]×cos[(n+1/2)kx],
ϕc(2)(x, z)=V0n=0Dncosh[(n+1/2)khδ3]×cosh[(n+1/2)kδ2(z-h)]+zz(co)δ3zz(cl)δ2sinh[(n+1/2)khδ3]×sinh[(n+1/2)kδ2(z-h)]×cos[(n+1/2)kx],
ϕc(3)(x, z)=V0n=0Dn sinh[(n+1/2)δ3kz]×cos[(n+1/2)kx],
ϕc(4)(x, z)=V0n=0Dncosh[(n+1/2)khδ3]×cosh[(n+1/2)kδ2(z+h)]-zz(co)δ3zz(cl)δ2sinh[(n+1/2)khδ3]×sinh[(n+1/2)kδ2(z+h)]×cos[(n+1/2)kx],
ϕc(5)(x, z)=V0n=0An exp[(n+1/2)kz]×cos[(n+1/2)kx].
n=0Dn*Tn cos[(n+1/2)kx]=1,
0<xa/2,
n=0(n+1/2)Dn* cos[(n+1/2)kx]=0,
a/2<x l/2,
Dn*=Dn[(s)Mn+zz(cl)δ2Nn],
Tn=Mn(s)Mn+zz(cl)δ2Nn,
Nn=cosh[(n+1/2)khδ3]×sinh[(n+1/2)kδ2d]+zz(co)δ3zz(cl)δ2sinh[(n+1/2)khδ3]×cosh[(n+1/2)kδ2d],
Mn=cosh[(n+1/2)khδ3]×cosh[(n+1/2)kδ2d]+zz(co)δ3zz(cl)δ2×sinh[(n+1/2)khδ3]×cosh[(n+1/2)kδ2d].
Dk*=Pk[cos(πa/l)](k+1/2)K[cos(πa/2l)]-n=0Dn*(Tn-1)αnk,
αnk=(k+1/2)0πa/lPn(cos ξ)Pk(cos ξ)sin ξdξ.
Ez(3)(x, z)=ΔV2V0Ecz(3)(x, z)+1+ΔV2V0Ebz(3)(x, z),
Ebz(3)(x, z)=-V0hB0h+n=1nkhδ3En cosh(nδ3kz)cos(nkx),
Ecz(3)(x, z)=-V0hn=0(n+1/2)δ3khDn sinh×[(n+1/2)δ3kz]cos[(n+1/2)kx].
Hz(i)=-jωμ0Ey(i)x=-βωμ0Ey(i),Hx(i)=jωμ0Ey(i)z.
Ey=Zeexp[-α(z-h-d)],<zh+d,cos[k2(z-h-d)]-αk2sin[k2(z-h-d)],h+d<z<h,Ae cos(k3z),|z|h,cos[k2(z+h+d)]+αk2sin[k2(z+h+d)],-h<z-(h+d),exp[α(z+h+d)],-(h+d)<z-,
Ey=Zoexp[-α(z-h-d)],<zh+d,cos[k2(z-h-d)]-αk2sin[k2(z-h-d)],h+d<z<h,Ao sin(k3z),|z|h,-cos[k2(z+h+d)]-αk2sin[k2(z+h+d)],-h<z-(h+d),-exp[α(z+h+d)],-(h+d)<z-.
α=(β2-ns2k02)1/2,k2=(ncl2k02-β2)1/2, k3=(nco2k02-β2)1/2,
Ae=cos(k2d)+α/k2 sin(k2d)cos(k3h),
Ao=cos(k2d)+α/k2 sin(k2d)sin(k3h).
tan(k3h)=αk3-k2k3tan(k2d)1+αk2tan(k2d).
cot(k3h)=-αk3-k2k3tan(k2d)1+αk2tan(k2d)
P=-12-+EyHz*dz=1W/m,
Zo,e=(μ0c)/βAo,e22k3h±sin(2k3h)4k3+αk22+(k22-α2)sin(2k2d)-4αk2 cos2(k2d)+2(k22+α2)k2d4k23+12α1/2,
Δnco,cl(TE)=-nco,cl3r13(co, cl)2Ex(x, z),
Δnco,cl(TM)=-nco,cl3r33(co,cl)2Ez(x, z),
da1dx=jσ11a1+jκ12a2 exp(-jΔα12x)+jκ13a3 exp(-jΔα13x),
da2dx=jσ22a2+jκ12a1 exp(-jΔα12x)+jκ23a3 exp(-jΔα23x),
da3dx=jσ33a3+jκ13a1 exp(-jΔα13x)+jκ23a2 exp(-jΔα23x),
db1dx=jκ12b2 exp(-jΔβ12x)+jκ13b3 exp(-jΔβ13x),
db2dx=jκ12b1 exp(-jΔβ12x)+jκ23b3 exp(-jΔβ23x),
db3dx=jκ13b1 exp(-jΔβ13x)+jκ23b2 exp(-jΔβ23x),
b1=a1 exp(-jσ11x),b2=a2 exp(-jσ22x),
b3=a3 exp(-jσ33x),Δβ12=Δα12-σ11+σ22,
Δβ13=Δα13-σ11+σ33,Δβ23=Δα23-σ22+σ33.
dRdx=-jΔβ12+Δβ132R-jκ12S-jκ13P,
dSdx=-jΔβ12-Δβ132S-jκ12R-jκ23P,
dPdx=+jΔβ12-Δβ132P-jκ13S-jκ23S,
R(z)=b1 exp[-j(Δβ12+Δβ13)x/2],
S(z)=b2 exp[-j(Δβ13-Δβ12)x/2],
P(z)=b3 exp[-j(Δβ12-Δβ13)x/2].
κij=ω4-Δ(z)Eyi(z)Eyj*(z)dz,
σ11=β12μ0k0cV0B01+ΔV2V0r31nco4Z1A1×-h+h cos2(k31z)dz,
σ22=β22μ0k0cV0B01+ΔV2V0r31nco4Z2A2×-h+h sin2(k32z)dz,
σ33=β32μ0k0cV0B01+ΔV2V0r31nco4Z3A3×-h+h cos2(k33z)dz,
κ12=β1β22μ0k0cΔV4δ3kD1r31nco4Z1Z2A1A2×-h+h cos(k31z)sin(k32z)sinh(δ3kz/2)dz,
κ13=β1β32μ0k0cV01+ΔV2V0δ3kE1r31nco4Z1Z3A1A3×-h+h cos(k31z)cos(k33z)cosh(δ3kz)dz,
κ23=β2β32μ0k0cΔV4δ3kD1r31nco4Z2Z3A2A3×-h+h cos(k33z)sin(k32z)sinh(δ3kz/2)dz.
rij=00r3100r3100r330r150r1500000,
Δ=r31Ez0r15Ex0r31Ez0r15Ex0r33Ez.

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