Abstract

Nyquist sampling theorem reveals the possibility of sampling the continuous refractive index profiles of optical waveguides at periods greater than the free-space wavelength, λo. Binary encoding of these analog waveguides is investigated using the zero-order effective medium theory, while conserving the quantization of the modal spectrum implied by their boundary conditions. Both analytical and numerical approaches are developed for this analog-to-digital (A-to-D) conversion. An example is presented for the A-to-D conversion of a graded index waveguide with a hyperbolic secant profile at a sample period of 1.3λo. The results are confirmed using a beam propagation method.

© 2013 Optical Society of America

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2009 (2)

2008 (1)

G. Zhang, M. Yu, C. Tung, and G. Lo, “Quantum size effects on dielectric constants and optical absorption of ultrathin silicon films,” IEEE Electron Dev. Lett. 29, 1302–1305 (2008).

2006 (2)

A. Delage, S. Janz, B. Lamontagne, A. Bogdanov, D. Dalacu, D.-X. Xu, and K. P. Yap, “Monolithically integrated asymmetric graded and step-index couplers for microphotonic waveguides,” Opt. Express 14, 148–161 (2006).
[CrossRef]

S. Wakui, J. Nakamura, and A. Natori, “First-principles calculations of dielectric constants for ultrathin SiO2 films,” J. Vac. Sci. Technol. B 24, 1992–1996 (2006).
[CrossRef]

2005 (1)

1998 (1)

1996 (1)

Y. P. Li and C. H. Henry, “Silica-based optical integrated circuits,” IEE Proc. Optoelectron. 143, 263–280 (1996).
[CrossRef]

1992 (2)

1985 (1)

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3, 385–391 (1985).
[CrossRef]

Bardyszewski, W.

Bogdanov, A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 14, pp. 705–708.

Chiang, K. S.

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3, 385–391 (1985).
[CrossRef]

Dalacu, D.

Delage, A.

Fainman, Y.

Hadley, G. R.

G. R. Hadley, “Transparent boundary conditions for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

Haruna, M.

H. Nishihara, M. Haruna, and T. Sukara, Optical Integrated Circuits (McGraw-Hill, 1989), Chap. 2, pp. 21–25.

Henry, C. H.

Y. P. Li and C. H. Henry, “Silica-based optical integrated circuits,” IEE Proc. Optoelectron. 143, 263–280 (1996).
[CrossRef]

Janz, S.

Kim, H.-C.

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, Y. R. Triandafilov, and A. G. Nalimov, “Simulation of propagation of modes in planar gradient-index hyperbolic secant waveguide,” in 11th International Conference on Laser and Fiber-Optical Networks Modeling (LFNM), Kharkov, Ukraine, September5–8, 2011.

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, Y. R. Triandafilov, and A. G. Nalimov, “Simulation of propagation of modes in planar gradient-index hyperbolic secant waveguide,” in 11th International Conference on Laser and Fiber-Optical Networks Modeling (LFNM), Kharkov, Ukraine, September5–8, 2011.

Lamontagne, B.

Levy, U.

Li, Y. P.

Y. P. Li and C. H. Henry, “Silica-based optical integrated circuits,” IEE Proc. Optoelectron. 143, 263–280 (1996).
[CrossRef]

Lo, G.

G. Zhang, M. Yu, C. Tung, and G. Lo, “Quantum size effects on dielectric constants and optical absorption of ultrathin silicon films,” IEEE Electron Dev. Lett. 29, 1302–1305 (2008).

Molina-Fernandez, I.

Muro, K.

Nakamura, J.

S. Wakui, J. Nakamura, and A. Natori, “First-principles calculations of dielectric constants for ultrathin SiO2 films,” J. Vac. Sci. Technol. B 24, 1992–1996 (2006).
[CrossRef]

Nalimov, A. G.

V. V. Kotlyar, A. A. Kovalev, Y. R. Triandafilov, and A. G. Nalimov, “Simulation of propagation of modes in planar gradient-index hyperbolic secant waveguide,” in 11th International Conference on Laser and Fiber-Optical Networks Modeling (LFNM), Kharkov, Ukraine, September5–8, 2011.

Natori, A.

S. Wakui, J. Nakamura, and A. Natori, “First-principles calculations of dielectric constants for ultrathin SiO2 films,” J. Vac. Sci. Technol. B 24, 1992–1996 (2006).
[CrossRef]

Nezhad, M.

Nishihara, H.

H. Nishihara, M. Haruna, and T. Sukara, Optical Integrated Circuits (McGraw-Hill, 1989), Chap. 2, pp. 21–25.

Ortega-Monux, A.

Pang, L.

Pollock, C. R.

C. R. Pollock, Fundamentals of Optoelectronics, 1st ed. (Irwin, 1994), Chap. 9, pp. 243–270.

Ramadan, T.

Schwartz, M.

M. Schwartz, Information Transmission, Modulation, and Noise, 4th ed. (McGraw-Hill, 1990), Chap. 3, pp. 96–100.

Shiraishi, K.

Sukara, T.

H. Nishihara, M. Haruna, and T. Sukara, Optical Integrated Circuits (McGraw-Hill, 1989), Chap. 2, pp. 21–25.

Triandafilov, Y. R.

V. V. Kotlyar, A. A. Kovalev, Y. R. Triandafilov, and A. G. Nalimov, “Simulation of propagation of modes in planar gradient-index hyperbolic secant waveguide,” in 11th International Conference on Laser and Fiber-Optical Networks Modeling (LFNM), Kharkov, Ukraine, September5–8, 2011.

Tsai, C.-H.

Tung, C.

G. Zhang, M. Yu, C. Tung, and G. Lo, “Quantum size effects on dielectric constants and optical absorption of ultrathin silicon films,” IEEE Electron Dev. Lett. 29, 1302–1305 (2008).

Wakui, S.

S. Wakui, J. Nakamura, and A. Natori, “First-principles calculations of dielectric constants for ultrathin SiO2 films,” J. Vac. Sci. Technol. B 24, 1992–1996 (2006).
[CrossRef]

Wanguemert-Perez, J. G.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 14, pp. 705–708.

Xu, D.-X.

Yap, K. P.

Yevick, D.

Yu, M.

G. Zhang, M. Yu, C. Tung, and G. Lo, “Quantum size effects on dielectric constants and optical absorption of ultrathin silicon films,” IEEE Electron Dev. Lett. 29, 1302–1305 (2008).

Zhang, G.

G. Zhang, M. Yu, C. Tung, and G. Lo, “Quantum size effects on dielectric constants and optical absorption of ultrathin silicon films,” IEEE Electron Dev. Lett. 29, 1302–1305 (2008).

IEE Proc. Optoelectron. (1)

Y. P. Li and C. H. Henry, “Silica-based optical integrated circuits,” IEE Proc. Optoelectron. 143, 263–280 (1996).
[CrossRef]

IEEE Electron Dev. Lett. (1)

G. Zhang, M. Yu, C. Tung, and G. Lo, “Quantum size effects on dielectric constants and optical absorption of ultrathin silicon films,” IEEE Electron Dev. Lett. 29, 1302–1305 (2008).

IEEE J. Quantum Electron. (1)

G. R. Hadley, “Transparent boundary conditions for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

J. Lightwave Technol. (4)

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

J. Vac. Sci. Technol. B (1)

S. Wakui, J. Nakamura, and A. Natori, “First-principles calculations of dielectric constants for ultrathin SiO2 films,” J. Vac. Sci. Technol. B 24, 1992–1996 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (5)

H. Nishihara, M. Haruna, and T. Sukara, Optical Integrated Circuits (McGraw-Hill, 1989), Chap. 2, pp. 21–25.

V. V. Kotlyar, A. A. Kovalev, Y. R. Triandafilov, and A. G. Nalimov, “Simulation of propagation of modes in planar gradient-index hyperbolic secant waveguide,” in 11th International Conference on Laser and Fiber-Optical Networks Modeling (LFNM), Kharkov, Ukraine, September5–8, 2011.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 14, pp. 705–708.

C. R. Pollock, Fundamentals of Optoelectronics, 1st ed. (Irwin, 1994), Chap. 9, pp. 243–270.

M. Schwartz, Information Transmission, Modulation, and Noise, 4th ed. (McGraw-Hill, 1990), Chap. 3, pp. 96–100.

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of a graded-index analog waveguide (left), which is converted to a digital waveguide (right). The optical confinement is along the x axis while the modal propagation is along the z axis.

Fig. 2.
Fig. 2.

Schematic of the feasibility region (bold solid boundaries) of the analytical A-to-D conversion rules of analog multimode waveguides in the nAND space. The dashed and dotted lines show the bounds whose crossings define this space. The overlap between the vertical solid lines at nA,min and nA,max, inside the feasible region, identifies the range of ND (shaded area) for the A-to-D conversion.

Fig. 3.
Fig. 3.

Schematic of (a) truncated parabolic and (b) sinusoidal refractive index profiles with discontinuities at the core-cladding interface (top), which are converted to binary profiles (bottom) using the analytical A-to-D conversion rules in Eq. (3). Due to the invariance of w, nmin, and nmax in these profiles, the same refractive index pulses are used in the binary encoding process.

Fig. 4.
Fig. 4.

Effective index quantization error, q, versus number of samples, M, for the A-to-D conversion of analog waveguides with (a) truncated parabolic and (b) sinusoidal refractive-index profiles at λo=1.55μm. Both guides have w=4.0μm, nmin=2.63, nmax=2.67, and n2=1.44. The solid lines correspond to the ideal layer thickness. The dashed, dash-dot, and dotted lines correspond to precision in layers thickness of 0.1, 0.2, and 0.5 nm, respectively.

Fig. 5.
Fig. 5.

Schematic of parabolic (solid) and hyperbolic-secant (dashed) refractive index profiles, which are continuous at the core-cladding interface.

Fig. 6.
Fig. 6.

A-to-D conversion map in the fmaxnmax space at λo=1.55μm and different number of samples of (a) 16, (b) 18, (c) 25, and (d) 40 for an analog waveguide with w=4.0μm, n2=1.44, and a parabolic refractive-index profile that is continuous at the core-cladding interface. The forbidden space is shown in gray.

Fig. 7.
Fig. 7.

A-to-D conversion map in the fmaxw space at λo=1.55μm and M=14 for an analog waveguide with HS profile that is binary encoded by refractive index pulses of ND=0.17 and n2=1.44. The forbidden space is shown in gray.

Fig. 8.
Fig. 8.

Widening of an A-to-D conversion window (white) in the fmaxw space as the number of samples increases for an analog waveguide with HS profile at λo=1.55μm.

Fig. 9.
Fig. 9.

Quantization errors in modal effective index, q, and in modal profile, Q, versus the normalized sample period Λ/λo, for the A-to-D conversion of a HS waveguide of w=28μm and n2=1.44 at λo=1.55μm. The linearly interpolated points correspond to Λ/λo=0.65, 0.90, 1.13, 1.29, and 2.26.

Tables (1)

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Table 1. Effective Indexes of the TE Modes of Analog and Digital Waveguides Corresponding to the HS Profile Given in the Text and Their Absolute Difference Error

Equations (6)

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Λλo/2ND,
ǧmin/(1fmax)<Λ<ǧmax/(1fmin),
nA,maxNDnA,min2/η(N1),
nA,max2/η(N0.5)NDnA,min2/η(N1),
nA,max2/η(N0.5)NDη+η2+nA,min2,
(nmax2nmin2)/(nmax2n22)1/(2N1),

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