Abstract

Characteristics of grating-assisted coupling between two parallel waveguides are analyzed. The influence of the grating parameters, such as groove depth, duty cycle, and refractive indices is considered. Chirped and parallel gratings as well as gratings with sinusoidal envelope periodicity are also addressed. The analysis is based on a unified coupled-mode formalism, with the transfer-matrix method as a general solution technique. It is shown how to modify the grating parameters to provide a specific spectral response (reflectivity and transmission coefficients). As an example, two parallel gratings are used to obtain a similar response to a single grating of double length. The location of the grating between the two waveguides as well as the light-wave injection direction are very important. The presented methods and effects may be useful for design and analysis in the fields of optical communications, sensing, and processing.

© 1999 Optical Society of America

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References

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  1. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  2. D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
    [CrossRef]
  3. S. S. Orlov, A. Yariv, S. V. Essen, “Coupled mode analysis of fiber-optic add-drop filters for dense wavelength-division multiplexing,” Opt. Lett. 22, 688–690 (1997).
    [CrossRef] [PubMed]
  4. I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum. Electron. 28, 93–135 (1996).
    [CrossRef]
  5. D. G. Hall, ed., Coupled-Mode Theory in Guided-Wave Optics Vol. MS 84 ofSPIE Milestone Series (SPIE, Bellingham, Wash., 1993).
  6. A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum. Electron. 34, 1109–1116 (1998).
    [CrossRef]
  7. N. Izhaky, A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A 16, 1303–1311 (1999).
    [CrossRef]
  8. A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
    [CrossRef]
  9. H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), pp. 43–50.
  10. T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum. Electron. 24, 1507–1518 (1988).
    [CrossRef]
  11. W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U Press, Cambridge, UK, 1986), pp. 578–588.

1999 (1)

1998 (1)

A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum. Electron. 34, 1109–1116 (1998).
[CrossRef]

1997 (1)

1996 (1)

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum. Electron. 28, 93–135 (1996).
[CrossRef]

1991 (1)

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

1988 (1)

T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum. Electron. 24, 1507–1518 (1988).
[CrossRef]

1985 (1)

A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

1972 (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Bennion, I.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum. Electron. 28, 93–135 (1996).
[CrossRef]

Doran, N. J.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum. Electron. 28, 93–135 (1996).
[CrossRef]

Essen, S. V.

Flannery, B.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U Press, Cambridge, UK, 1986), pp. 578–588.

Glinski, J.

T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum. Electron. 24, 1507–1518 (1988).
[CrossRef]

Hall, D. G.

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

Hardy, A.

N. Izhaky, A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A 16, 1303–1311 (1999).
[CrossRef]

A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum. Electron. 34, 1109–1116 (1998).
[CrossRef]

A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

Izhaky, N.

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), pp. 43–50.

Makino, T.

T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum. Electron. 24, 1507–1518 (1988).
[CrossRef]

Orlov, S. S.

Press, W.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U Press, Cambridge, UK, 1986), pp. 578–588.

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Streifer, W.

A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

Sugden, K.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum. Electron. 28, 93–135 (1996).
[CrossRef]

Teudolsky, S.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U Press, Cambridge, UK, 1986), pp. 578–588.

Vetterling, W.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U Press, Cambridge, UK, 1986), pp. 578–588.

Williams, J. A. R.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum. Electron. 28, 93–135 (1996).
[CrossRef]

Yariv, A.

Zhang, L.

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum. Electron. 28, 93–135 (1996).
[CrossRef]

IEEE J. Quantum. Electron. (2)

A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum. Electron. 34, 1109–1116 (1998).
[CrossRef]

T. Makino, J. Glinski, “Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers,” IEEE J. Quantum. Electron. 24, 1507–1518 (1988).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Lightwave Technol. (1)

A. Hardy, W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

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

Opt. Lett. (1)

Opt. Quantum. Electron. (1)

I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, N. J. Doran, “UV-written in-fibre Bragg gratings,” Opt. Quantum. Electron. 28, 93–135 (1996).
[CrossRef]

Prog. Opt. (1)

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

Other (3)

D. G. Hall, ed., Coupled-Mode Theory in Guided-Wave Optics Vol. MS 84 ofSPIE Milestone Series (SPIE, Bellingham, Wash., 1993).

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), pp. 43–50.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U Press, Cambridge, UK, 1986), pp. 578–588.

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Figures (9)

Fig. 1
Fig. 1

Schematic illustration of the structure for grating-assisted coupling with two parallel single-mode waveguides, a rectangular grating between them, and the four involved waves.

Fig. 2
Fig. 2

Squared amplitudes as functions of the groove depth (Δh). (a) u a +(L)/u a +(0) and u a -(0)/u a +(0). (b) u b -(0)/u a +(0) and u b +(L)/u a +(0).

Fig. 3
Fig. 3

Squared amplitudes as functions of the duty cycle (s/Λ). (a) u a +(L)/u a +(0) and u a -(0)/u a +(0). (b) u b -(0)/u a +(0) and u b +(L)/u a +(0).

Fig. 4
Fig. 4

Squared amplitudes as functions of n g1 at n g2 = n 1 (segmented waveguide). (a) u a +(L)/u a +(0) and u a -(0)/u a +(0). (b) u b -(0)/u a +(0) and u b +(L)/u a +(0).

Fig. 5
Fig. 5

Spectral values of the squared amplitudes in the case of a negatively step-chirped grating of α = -10-6 µm, Λmin = 128.17 nm, Λmax = 130.49 nm, and L = 300 µm. (a) u a +(L)/u a +(0) and u a -(0)/u a +(0) as functions of wavelength. (b) u b -(0)/u a +(0) and u b +(L)/u a +(0) as functions of wavelength.

Fig. 6
Fig. 6

Spectral values of the squared amplitudes in the case of two cascaded gratings. The first (0 ≤ z ≤ 150 µm) planned for λ0 = 0.9 µm (Λ1 = 130.49 nm) and the second (150 < z ≤ 300 µm) planned for λ0 = 0.8932 µm (Λ2 = 129.49 nm). (a) u a +(L)/u a +(0) and u a -(0)/u a +(0) as functions of wavelength. (b) u b -(0)/u a +(0) and u b +(L)/u a +(0) as functions of wavelength.

Fig. 7
Fig. 7

Spectral values of the squared amplitudes in the case of two parallel gratings (L = 150 µm) of periods as those in Fig. 6. The groove depths are Δh 1 = 96 nm and Δh 2 = 22 nm (near to waveguide a). (a) u a +(L)/u a +(0) and u a -(0)/u a +(0) as functions of wavelength. (b) u b -(0)/u a +(0) and u b +(L)/u a +(0) as functions of wavelength.

Fig. 8
Fig. 8

Normalized forward propagating power P +(z)/P +(0) and normalized backward propagating power P -(z)/P +(0) in the whole structure for sinusoidal envelope periodicity. The grating is in the middle between the two waveguides, and γ = 10-10 µm, N = 25, and L = 300 µm.

Fig. 9
Fig. 9

Spectral values of the squared amplitudes in the case of sinusoidal envelope periodicity. The parameters are the same as those in Fig. 8. (a) u a +(L)/u a +(0) and u a -(0)/u a +(0) as functions of wavelength. (b) u b -(0)/u a +(0) and u b +(L)/u a +(0) as functions of wavelength.

Tables (1)

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Table 1 Structure Parameters (defaults)

Equations (5)

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

Etx, y, z=ua+z+ua-zEtax, y+[ub+z+ub-zEtbx, y,
Htx, y, z=ua+z-ua-zHtax, y+ub+z-ub-zHtbx, y,
dUzdz=iMzUz,
βbλ0±βaλ0=2πΛ,
ua+0=1, ub+0=ua-L=ub-L=0.

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