Abstract

This paper proposes to use slow light effects near the Brillouin zone band edge of one-dimensional gratings for reducing the size of integrated electro-optic (EO) modulators. The gratings are built within the arms of a Mach-Zehnder Interferometer (MZI) for intensity modulation. To overcome the inherent high reflection and low extinction ratio, we introduce various multi-segment grating designs. We use coupled-mode theory and derive transfer matrices to analyze the spectral transmittance and phase delay of each arm of the interferometer. Calculations show that a size-reduction of a factor of 2 or more can be achieved at λ = 1.574µm with an insertion loss of 0.17dB and an amplitude modulation extinction ratio of 18.84dB. The simulated structure is based on a Si slab-waveguide 0.2 μm thick with 30nm deep grating groves on SiO2 substrate.

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  1. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
    [CrossRef] [PubMed]
  2. Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005).
    [CrossRef]
  3. A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007).
    [CrossRef] [PubMed]
  4. E. F. Schipper, A. M. Brugman, C. Dominguez, L. M. Lechuga, R. P. H. Kooyman, and J. Greve, “The realization of an integrated Mach-Zehnder waveguide immunosensor in silicon technology,” Sens. Actuators B Chem. 40(2-3), 147–153 (1997).
    [CrossRef]
  5. B. J. Luff, J. S. Wilkinson, J. Piehler, U. Hollenbach, J. Ingenhoff, and N. Fabricius, “Integrated optical Mach–Zehnder biosensor,” J. Lightwave Technol. 16(4), 583–592 (1998).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  18. S.-L. Chuang, Physics of optoelectronic devices, (John Wiley & Sons, Inc, 1995), chap. 8.
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    [PubMed]

2010

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

2009

2007

2006

2005

M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13(18), 7145–7159 (2005).
[CrossRef] [PubMed]

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005).
[CrossRef]

2004

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

2002

1998

1997

L. Wei and J. Lit, “Phase-shifted Bragg grating filters with symmetrical structures,” J. Lightwave Technol. 15(8), 1405–1410 (1997).
[CrossRef]

E. F. Schipper, A. M. Brugman, C. Dominguez, L. M. Lechuga, R. P. H. Kooyman, and J. Greve, “The realization of an integrated Mach-Zehnder waveguide immunosensor in silicon technology,” Sens. Actuators B Chem. 40(2-3), 147–153 (1997).
[CrossRef]

1994

G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg grating and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6(8), 995–997 (1994).
[CrossRef]

1987

Agrawal, G. P.

G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg grating and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6(8), 995–997 (1994).
[CrossRef]

Brugman, A. M.

E. F. Schipper, A. M. Brugman, C. Dominguez, L. M. Lechuga, R. P. H. Kooyman, and J. Greve, “The realization of an integrated Mach-Zehnder waveguide immunosensor in silicon technology,” Sens. Actuators B Chem. 40(2-3), 147–153 (1997).
[CrossRef]

Chen, R. T.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005).
[CrossRef]

Chen, X.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005).
[CrossRef]

Chetrit, Y.

Ciftcioglu, B.

Cohen, O.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Corcoran, B.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

Deng, S.

Dominguez, C.

E. F. Schipper, A. M. Brugman, C. Dominguez, L. M. Lechuga, R. P. H. Kooyman, and J. Greve, “The realization of an integrated Mach-Zehnder waveguide immunosensor in silicon technology,” Sens. Actuators B Chem. 40(2-3), 147–153 (1997).
[CrossRef]

Ebnali-Heidari, M.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

Eggleton, B. J.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

Fabricius, N.

Fan, S.

Greve, J.

E. F. Schipper, A. M. Brugman, C. Dominguez, L. M. Lechuga, R. P. H. Kooyman, and J. Greve, “The realization of an integrated Mach-Zehnder waveguide immunosensor in silicon technology,” Sens. Actuators B Chem. 40(2-3), 147–153 (1997).
[CrossRef]

Grillet, C.

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

Gu, L.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005).
[CrossRef]

Hollenbach, U.

Huang, Z. R.

Ibanescu, M.

Ingenhoff, J.

Ippen, E.

Izhaky, N.

Jiang, W.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005).
[CrossRef]

Jiang, Y.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Jones, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Kooyman, R. P. H.

E. F. Schipper, A. M. Brugman, C. Dominguez, L. M. Lechuga, R. P. H. Kooyman, and J. Greve, “The realization of an integrated Mach-Zehnder waveguide immunosensor in silicon technology,” Sens. Actuators B Chem. 40(2-3), 147–153 (1997).
[CrossRef]

Krauss, T. F.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D Appl. Phys. 40(9), 2666–2670 (2007).
[CrossRef]

Lechuga, L. M.

E. F. Schipper, A. M. Brugman, C. Dominguez, L. M. Lechuga, R. P. H. Kooyman, and J. Greve, “The realization of an integrated Mach-Zehnder waveguide immunosensor in silicon technology,” Sens. Actuators B Chem. 40(2-3), 147–153 (1997).
[CrossRef]

Liao, L.

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Lit, J.

L. Wei and J. Lit, “Phase-shifted Bragg grating filters with symmetrical structures,” J. Lightwave Technol. 15(8), 1405–1410 (1997).
[CrossRef]

Liu, A.

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Luff, B. J.

McDonald, J. F.

McMillan, J. F.

McNab, S. J.

Monat, C.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

Moss, D. J.

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

Nguyen, H.

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

O’Faolain, L.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

O'Faolain, L.

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

Osgood, R. M.

Paniccia, M.

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Panoiu, N. C.

Pelusi, M. D.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

Piehler, J.

Povinelli, M. L.

Pudo, D.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

Radic, S.

G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg grating and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6(8), 995–997 (1994).
[CrossRef]

Rubin, D.

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express 15(2), 660–668 (2007).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Sakuda, K.

Samara-Rubio, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Schipper, E. F.

E. F. Schipper, A. M. Brugman, C. Dominguez, L. M. Lechuga, R. P. H. Kooyman, and J. Greve, “The realization of an integrated Mach-Zehnder waveguide immunosensor in silicon technology,” Sens. Actuators B Chem. 40(2-3), 147–153 (1997).
[CrossRef]

Soljacic, M.

Vlasov, Y. A.

Wei, L.

L. Wei and J. Lit, “Phase-shifted Bragg grating filters with symmetrical structures,” J. Lightwave Technol. 15(8), 1405–1410 (1997).
[CrossRef]

White, T. P.

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

Wilkinson, J. S.

Wong, C.-W.

Yamada, M.

Yang, X.

Appl. Opt.

Appl. Phys. Lett.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, “80-micron interaction length silicon photonic crystal waveguide modulator,” Appl. Phys. Lett. 87(22), 221105 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow Light Enhanced Nonlinear Optics in Silicon Photonic Crystal Waveguides,” IEEE J. Sel. Top. Quantum Electron. 16(1), 344–356 (2010).
[CrossRef]

IEEE Photon. Technol. Lett.

G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg grating and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6(8), 995–997 (1994).
[CrossRef]

J. Lightwave Technol.

L. Wei and J. Lit, “Phase-shifted Bragg grating filters with symmetrical structures,” J. Lightwave Technol. 15(8), 1405–1410 (1997).
[CrossRef]

B. J. Luff, J. S. Wilkinson, J. Piehler, U. Hollenbach, J. Ingenhoff, and N. Fabricius, “Integrated optical Mach–Zehnder biosensor,” J. Lightwave Technol. 16(4), 583–592 (1998).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. D Appl. Phys.

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D Appl. Phys. 40(9), 2666–2670 (2007).
[CrossRef]

Nature

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

PIER

D. J. Moss, B. Corcoran, C. Monat, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, and B. J. Eggleton, “Slow-light enhanced nonlinear optics in silicon photonic crystal waveguides,” PIER 6, 273–278 (2010).

Sens. Actuators B Chem.

E. F. Schipper, A. M. Brugman, C. Dominguez, L. M. Lechuga, R. P. H. Kooyman, and J. Greve, “The realization of an integrated Mach-Zehnder waveguide immunosensor in silicon technology,” Sens. Actuators B Chem. 40(2-3), 147–153 (1997).
[CrossRef]

Other

M. Nevière, and E. Popov, Light propagation in periodic media: differential theory and design, (CRC Press, 2002).

S.-L. Chuang, Physics of optoelectronic devices, (John Wiley & Sons, Inc, 1995), chap. 8.

M. V. Klein, and T. E. Furtak, Optics, (John Wiley & Sons, Inc, 1986), chap.5.
[PubMed]

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Figures (10)

Fig. 1
Fig. 1

(a) A 1D grating incorporated in a MZI configuration, (b) symmetric slab waveguide with 1D grating on one surface, and (c) illustration of steady-state fields.

Fig. 2
Fig. 2

Transmittance for (a) a 20-period rectangular grating for three groove depths, and (b) a grating with 30nm-deep grooves but having different numbers of periods.

Fig. 3
Fig. 3

Dispersion relations for rectangular grating grooves of different depths.

Fig. 4
Fig. 4

Dispersion relation of (a) βCreal (b) |βCimg | for rectangular grating (solid) and triangular grating (dash-dot) at ∆n = 0, and rectangular grating (dash) at ∆n = −0.01.

Fig. 5
Fig. 5

(a) Δβ for a waveguide slab with and without gratings (b) Lπ and length reduction ratio for a slab waveguide with rectangular grating grooves 30nm deep, as compared to a planar slab waveguide.

Fig. 6
Fig. 6

Illustration of a multi-segment grating waveguide.

Fig. 7
Fig. 7

(a) Scattering matrix (b) Transmission matrix.

Fig. 8
Fig. 8

Tr and Tm dependence on D for M = 2.

Fig. 9
Fig. 9

T 1 verse grating length in the MZI.

Fig. 10
Fig. 10

Tr and Tm vs. D for M = 2 at L = 156Λ.

Tables (1)

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Table 1 Summary of Different Multi-segment Designs

Equations (11)

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{ Δ ε r ( x , z ) = p = d p ( x ) e i p 2 π Λ z , E y ( x , z ) = m = A m ( z ) E y ( m ) ( x ) e i β m z ,
β C = π Λ ± [ ( β 1 ( ω ) ) ( π Λ ) ] 2 | K | 2 ,
| K | = ω 4 | d ( x ) | E y ( s ) ( x ) | 2 d x | .
| K | = { | ω ε 0 4 π ( n 2 2 n 1 2 ) d 2 a d 2 | E y ( 1 ) ( x ) | 2 d x |  ,                            (rectangular) | ω ε 0 4 π ( n 2 2 n 1 2 ) d / 2 a d / 2 cos [ π 2 a ( x d 2 ) ] | E y ( 1 ) | 2 d x |    .     (triangular)
| t ( L ) | 2 = 1 ( δ β | K | ) 2 ( δ β | K | ) 2 ( sinh ( S L ) ) 2 + ( 1 ( δ β | K | ) 2 ) ( cosh ( S L ) ) 2 .
{ S 11 = b 1 a 1 | a 2 = 0 = B ( 0 ) e i π Λ 0 A ( 0 ) e i π Λ 0 | B ( L ) = 0 = K * sinh ( S L ) Δ β sinh ( S L ) + i S cosh ( S L ) , S 21 = b 2 a 1 | a 2 = 0 = A ( L ) e i π Λ L A ( 0 ) e i π Λ 0 | B ( L ) = 0 = i S Δ β sinh ( S L ) + i S cosh ( S L ) e i π Λ L , S 22 = b 2 a 2 | a 1 = 0 = A ( L ) e i π Λ L B ( L ) e i π Λ L | A ( 0 ) = 0 = K sinh ( S L ) Δ β sinh ( S L ) + i S cosh ( S L ) e i 2 π Λ L , S 12 = b 1 a 2 | a 1 = 0 = B ( 0 ) e i π Λ 0 B ( L ) e i π Λ L | A ( 0 ) = 0 = i S Δ β sinh ( S L ) + i S cosh ( S L ) e i π Λ L .
T g = 1 S 21 [ 1 S 22 S 11 S 12 S 21 S 11 S 22 ] .
T f = [ e i β s D 0 0 e i β s D ] .
T = ( T g T f ) ( T g T f ) ( T g T f ) ( M 1 )   terms T g ,
r = T 21 T 11    and     t = 1 T 11 ,
T m , r = T 1 2 1 + R 1 2 2 R 1 cos δ ,

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