Abstract

Design rules for both single-mode and polarization-independent strained silicon-on-insulator rib waveguides at the wavelength of 3.39μm are presented for the first time to our knowledge. Waveguide geometries with different parameters, such as waveguide height, rib width, etch depth, top oxide cover thickness and sidewall angle, have been studied in order to investigate and define design rules that will make devices suitable for mid-IR applications. Chebyshev bivariate interpolation with a standard deviation of less than 1% has been used to represent the zero-birefringence surface. Experimental results for the upper cladding stress level have been used to determine the influence of top oxide cover thickness and different levels of upper cladding stress on waveguide characteristics. Finally, the polarization-insensitive and single-mode locus is presented for different waveguide heights.

© 2009 Optical Society of America

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2008 (2)

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

D. Thomson, F. Y. Gardes, G. Z. Mashanovich, A. P. Knights, and G. T. Reed, “Using SiO2 carrier confinement in total internal reflection optical switches to restrict carrier diffusion in the guiding layer,” J. Lightwave Technol. 26, 1288-1294 (2008).
[CrossRef]

2007 (2)

2006 (3)

M. Graf, N. Hoyler, M. Giovannini, J. Faist, and D. Hofstetter, “InP-based quantum cascade detectors in the mid-infrared,” Appl. Phys. Lett. 88, 241118 (2006).
[CrossRef]

Q. Yang, C. Manz, W. Bronner, K. Köhler, and J. Wagner, “Room-temperature short-wavelength (λ~3.7-3.9 μm)GaInAs/AlAsSb quantum-cascade lasers,” Appl. Phys. Lett. 88, 121127 (2006).
[CrossRef]

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A 8, 840-848 (2006).
[CrossRef]

2005 (1)

2004 (1)

W. R. Headley, G. T. Reed, M. Pannicia, A. Liu, and S. Howe, “Polarization-independent optical racetrack resonators using rib waveguides in silicon-on-insulator,” Appl. Phys. Lett. 85, 5523-5525 (2004).
[CrossRef]

2003 (1)

M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40, 1615-1632 (2003).
[CrossRef]

2000 (2)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

M. Razeghi, C. Miesner, O. Röthig, K. Brunner, and G. Abstreiter, “Mid-infrared photocurrent measurements on self-assembled Ge dots in Si,” Physica E 7, 146-150 (2000).
[CrossRef]

1981 (1)

K. Okamoto, T. Hosaka, and T. Edahiro, “Stress analysis of optical fibers by a finite element method,” IEEE J. Quantum Electron. QE-17, 2123-2129 (1981).
[CrossRef]

Abstreiter, G.

M. Razeghi, C. Miesner, O. Röthig, K. Brunner, and G. Abstreiter, “Mid-infrared photocurrent measurements on self-assembled Ge dots in Si,” Physica E 7, 146-150 (2000).
[CrossRef]

Bettiol, A. A.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

Borlaug, D.

Breese, M. B. H.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

Bronner, W.

Q. Yang, C. Manz, W. Bronner, K. Köhler, and J. Wagner, “Room-temperature short-wavelength (λ~3.7-3.9 μm)GaInAs/AlAsSb quantum-cascade lasers,” Appl. Phys. Lett. 88, 121127 (2006).
[CrossRef]

Brunner, K.

M. Razeghi, C. Miesner, O. Röthig, K. Brunner, and G. Abstreiter, “Mid-infrared photocurrent measurements on self-assembled Ge dots in Si,” Physica E 7, 146-150 (2000).
[CrossRef]

Buchwald, W. R.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A 8, 840-848 (2006).
[CrossRef]

Cheben, P.

Edahiro, T.

K. Okamoto, T. Hosaka, and T. Edahiro, “Stress analysis of optical fibers by a finite element method,” IEEE J. Quantum Electron. QE-17, 2123-2129 (1981).
[CrossRef]

Emelett, S. J.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A 8, 840-848 (2006).
[CrossRef]

Faist, J.

M. Graf, N. Hoyler, M. Giovannini, J. Faist, and D. Hofstetter, “InP-based quantum cascade detectors in the mid-infrared,” Appl. Phys. Lett. 88, 241118 (2006).
[CrossRef]

Gardes, F. Y.

Giovannini, M.

M. Graf, N. Hoyler, M. Giovannini, J. Faist, and D. Hofstetter, “InP-based quantum cascade detectors in the mid-infrared,” Appl. Phys. Lett. 88, 241118 (2006).
[CrossRef]

Goodier, J. N.

S. Timoshenko and J. N. Goodier, Theory of Elasticity, 2nd ed. (McGraw-Hill, 1951).

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Graf, M.

M. Graf, N. Hoyler, M. Giovannini, J. Faist, and D. Hofstetter, “InP-based quantum cascade detectors in the mid-infrared,” Appl. Phys. Lett. 88, 241118 (2006).
[CrossRef]

Hawkins, G. J.

G. J. Hawkins, “Spectral characteristics of infrared optical materials and filters,” Ph.D. thesis (University of Reading, 1998).

Headley, W. R.

W. R. Headley, G. T. Reed, M. Pannicia, A. Liu, and S. Howe, “Polarization-independent optical racetrack resonators using rib waveguides in silicon-on-insulator,” Appl. Phys. Lett. 85, 5523-5525 (2004).
[CrossRef]

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Hoffmann, R. W.

R. W. Hoffmann, in Measurement Techniques for Thin Films, B.Schwartz and N.Schwartz, eds. (Electrochemical Society, 1967), p. 312.

Hofstetter, D.

M. Graf, N. Hoyler, M. Giovannini, J. Faist, and D. Hofstetter, “InP-based quantum cascade detectors in the mid-infrared,” Appl. Phys. Lett. 88, 241118 (2006).
[CrossRef]

Hosaka, T.

K. Okamoto, T. Hosaka, and T. Edahiro, “Stress analysis of optical fibers by a finite element method,” IEEE J. Quantum Electron. QE-17, 2123-2129 (1981).
[CrossRef]

Howe, S.

W. R. Headley, G. T. Reed, M. Pannicia, A. Liu, and S. Howe, “Polarization-independent optical racetrack resonators using rib waveguides in silicon-on-insulator,” Appl. Phys. Lett. 85, 5523-5525 (2004).
[CrossRef]

Hoyler, N.

M. Graf, N. Hoyler, M. Giovannini, J. Faist, and D. Hofstetter, “InP-based quantum cascade detectors in the mid-infrared,” Appl. Phys. Lett. 88, 241118 (2006).
[CrossRef]

Huang, M.

M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40, 1615-1632 (2003).
[CrossRef]

Jalali, B.

Janz, S.

Kawano, K.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell's Equation and the Schrödinger Equation (Wiley, 2001).
[PubMed]

Kitoh, T.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell's Equation and the Schrödinger Equation (Wiley, 2001).
[PubMed]

Knights, A. P.

Köhler, K.

Q. Yang, C. Manz, W. Bronner, K. Köhler, and J. Wagner, “Room-temperature short-wavelength (λ~3.7-3.9 μm)GaInAs/AlAsSb quantum-cascade lasers,” Appl. Phys. Lett. 88, 121127 (2006).
[CrossRef]

Lamontagne, B.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 2nd ed. (Pergamon, 1970).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 2nd ed. (Pergamon, 1970).

Liu, A.

W. R. Headley, G. T. Reed, M. Pannicia, A. Liu, and S. Howe, “Polarization-independent optical racetrack resonators using rib waveguides in silicon-on-insulator,” Appl. Phys. Lett. 85, 5523-5525 (2004).
[CrossRef]

Manz, C.

Q. Yang, C. Manz, W. Bronner, K. Köhler, and J. Wagner, “Room-temperature short-wavelength (λ~3.7-3.9 μm)GaInAs/AlAsSb quantum-cascade lasers,” Appl. Phys. Lett. 88, 121127 (2006).
[CrossRef]

Mashanovich, G. Z.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

D. Thomson, F. Y. Gardes, G. Z. Mashanovich, A. P. Knights, and G. T. Reed, “Using SiO2 carrier confinement in total internal reflection optical switches to restrict carrier diffusion in the guiding layer,” J. Lightwave Technol. 26, 1288-1294 (2008).
[CrossRef]

M. M. Milošević, P. S. Matavulj, B. D. Timotijević, G. T. Reed, and G. Z. Mashanovich, “Design rules for single-mode and polarization independent silicon-on-insulator rib waveguides using stress engineering,” J. Lightwave Technol. 26, 1840-1846 (2007).

M. M. Milošević, P. S. Matavulj, and G. Z. Mashanovich, “Stress-induced characteristics of silicon-on-insulator rib waveguides,” in Proceedings 15th Telecommunication Forum--TELFOR (2007), pp. 401-404.

M. M. Milošević, P. S. Matavulj, and G. Z. Mashanovich, “Interpolation of the zero-birefringent surface by use of Chebyshev polynomial,” presented at the 12th Serbian Mathematical Congress, Novi Sad, Serbia, Aug. 28-Sept. 2, 2008.

Matavulj, P.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

Matavulj, P. S.

M. M. Milošević, P. S. Matavulj, B. D. Timotijević, G. T. Reed, and G. Z. Mashanovich, “Design rules for single-mode and polarization independent silicon-on-insulator rib waveguides using stress engineering,” J. Lightwave Technol. 26, 1840-1846 (2007).

M. M. Milošević, P. S. Matavulj, and G. Z. Mashanovich, “Stress-induced characteristics of silicon-on-insulator rib waveguides,” in Proceedings 15th Telecommunication Forum--TELFOR (2007), pp. 401-404.

M. M. Milošević, P. S. Matavulj, and G. Z. Mashanovich, “Interpolation of the zero-birefringent surface by use of Chebyshev polynomial,” presented at the 12th Serbian Mathematical Congress, Novi Sad, Serbia, Aug. 28-Sept. 2, 2008.

Miesner, C.

M. Razeghi, C. Miesner, O. Röthig, K. Brunner, and G. Abstreiter, “Mid-infrared photocurrent measurements on self-assembled Ge dots in Si,” Physica E 7, 146-150 (2000).
[CrossRef]

Miloševic, M.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

Miloševic, M. M.

M. M. Milošević, P. S. Matavulj, B. D. Timotijević, G. T. Reed, and G. Z. Mashanovich, “Design rules for single-mode and polarization independent silicon-on-insulator rib waveguides using stress engineering,” J. Lightwave Technol. 26, 1840-1846 (2007).

M. M. Milošević, P. S. Matavulj, and G. Z. Mashanovich, “Stress-induced characteristics of silicon-on-insulator rib waveguides,” in Proceedings 15th Telecommunication Forum--TELFOR (2007), pp. 401-404.

M. M. Milošević, P. S. Matavulj, and G. Z. Mashanovich, “Interpolation of the zero-birefringent surface by use of Chebyshev polynomial,” presented at the 12th Serbian Mathematical Congress, Novi Sad, Serbia, Aug. 28-Sept. 2, 2008.

Okamoto, K.

K. Okamoto, T. Hosaka, and T. Edahiro, “Stress analysis of optical fibers by a finite element method,” IEEE J. Quantum Electron. QE-17, 2123-2129 (1981).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985), vol. 1.

Pannicia, M.

W. R. Headley, G. T. Reed, M. Pannicia, A. Liu, and S. Howe, “Polarization-independent optical racetrack resonators using rib waveguides in silicon-on-insulator,” Appl. Phys. Lett. 85, 5523-5525 (2004).
[CrossRef]

Picard, M.-J.

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Raghunathan, V.

Razeghi, M.

M. Razeghi, C. Miesner, O. Röthig, K. Brunner, and G. Abstreiter, “Mid-infrared photocurrent measurements on self-assembled Ge dots in Si,” Physica E 7, 146-150 (2000).
[CrossRef]

Reed, G. T.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

D. Thomson, F. Y. Gardes, G. Z. Mashanovich, A. P. Knights, and G. T. Reed, “Using SiO2 carrier confinement in total internal reflection optical switches to restrict carrier diffusion in the guiding layer,” J. Lightwave Technol. 26, 1288-1294 (2008).
[CrossRef]

M. M. Milošević, P. S. Matavulj, B. D. Timotijević, G. T. Reed, and G. Z. Mashanovich, “Design rules for single-mode and polarization independent silicon-on-insulator rib waveguides using stress engineering,” J. Lightwave Technol. 26, 1840-1846 (2007).

W. R. Headley, G. T. Reed, M. Pannicia, A. Liu, and S. Howe, “Polarization-independent optical racetrack resonators using rib waveguides in silicon-on-insulator,” Appl. Phys. Lett. 85, 5523-5525 (2004).
[CrossRef]

Rice, R. R.

Röthig, O.

M. Razeghi, C. Miesner, O. Röthig, K. Brunner, and G. Abstreiter, “Mid-infrared photocurrent measurements on self-assembled Ge dots in Si,” Physica E 7, 146-150 (2000).
[CrossRef]

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Soref, R. A.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A 8, 840-848 (2006).
[CrossRef]

Stankovic, S.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

Tarr, N. G.

Teo, E. J.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

Thomson, D.

Timoshenko, S.

S. Timoshenko and J. N. Goodier, Theory of Elasticity, 2nd ed. (McGraw-Hill, 1951).

Timotijevic, B.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

Timotijevic, B. D.

Wagner, J.

Q. Yang, C. Manz, W. Bronner, K. Köhler, and J. Wagner, “Room-temperature short-wavelength (λ~3.7-3.9 μm)GaInAs/AlAsSb quantum-cascade lasers,” Appl. Phys. Lett. 88, 121127 (2006).
[CrossRef]

Xu, D.-X.

Yang, P. Y.

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
[CrossRef]

Yang, Q.

Q. Yang, C. Manz, W. Bronner, K. Köhler, and J. Wagner, “Room-temperature short-wavelength (λ~3.7-3.9 μm)GaInAs/AlAsSb quantum-cascade lasers,” Appl. Phys. Lett. 88, 121127 (2006).
[CrossRef]

Ye, W. N.

Appl. Phys. Lett. (3)

M. Graf, N. Hoyler, M. Giovannini, J. Faist, and D. Hofstetter, “InP-based quantum cascade detectors in the mid-infrared,” Appl. Phys. Lett. 88, 241118 (2006).
[CrossRef]

Q. Yang, C. Manz, W. Bronner, K. Köhler, and J. Wagner, “Room-temperature short-wavelength (λ~3.7-3.9 μm)GaInAs/AlAsSb quantum-cascade lasers,” Appl. Phys. Lett. 88, 121127 (2006).
[CrossRef]

W. R. Headley, G. T. Reed, M. Pannicia, A. Liu, and S. Howe, “Polarization-independent optical racetrack resonators using rib waveguides in silicon-on-insulator,” Appl. Phys. Lett. 85, 5523-5525 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Okamoto, T. Hosaka, and T. Edahiro, “Stress analysis of optical fibers by a finite element method,” IEEE J. Quantum Electron. QE-17, 2123-2129 (1981).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Int. J. Solids Struct. (1)

M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40, 1615-1632 (2003).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. A (1)

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A 8, 840-848 (2006).
[CrossRef]

Opt. Express (1)

Physica E (1)

M. Razeghi, C. Miesner, O. Röthig, K. Brunner, and G. Abstreiter, “Mid-infrared photocurrent measurements on self-assembled Ge dots in Si,” Physica E 7, 146-150 (2000).
[CrossRef]

Semicond. Sci. Technol. (1)

G. Z. Mashanovich, M. Milošević, P. Matavulj, S. Stanković, B. Timotijević, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23, 064002 (2008).
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M. M. Milošević, P. S. Matavulj, and G. Z. Mashanovich, “Stress-induced characteristics of silicon-on-insulator rib waveguides,” in Proceedings 15th Telecommunication Forum--TELFOR (2007), pp. 401-404.

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Figures (9)

Fig. 1
Fig. 1

Modeled structure: t is the top oxide layer thickness, H is rib height, D ( D = H h ) is etch depth, W is waveguide rib width, and θ is the rib sidewall angle. Inset, scanning electron microscope structure.

Fig. 2
Fig. 2

Stress field distribution in a SOI rib waveguide. (a) Stress field σ x , (b) stress field σ y and corresponding electric field profiles: (c) quasi-TE mode, (d) quasi-TM mode. H = 2.95 μ m , D = 1.7 μ m , W = 2.5 μ m , t = 1 μ m , θ = 90 ° , σ film = 290 MPa , λ = 3.39 μ m .

Fig. 3
Fig. 3

ZBC as a function of waveguide rib width and etch depth for different values of top oxide thickness at an operating wavelength of λ = 3.39 μ m . The waveguide height is equal to H = 2.95 μ m , while the upper cladding stress is equal to σ film = 290 MPa ( θ = 90 ° ) .

Fig. 4
Fig. 4

Zero-birefringent surface for an operating wavelength of λ = 3.39 μ m , waveguide height of H = 2.95 μ m and cladding stress of σ film = 290 MPa ( θ = 90 ° ) .

Fig. 5
Fig. 5

ZBC as a function of waveguide rib width and etch depth for different values of rib sidewall angle. λ = 3.39 μ m , H = 2.95 μ m , t = 1 μ m , σ film = 290 MPa .

Fig. 6
Fig. 6

ZBC as a function of waveguide rib width and etch depth for different values of top oxide thickness for waveguide height of (a) H = 2.5 μ m and (b) H = 3.5 μ m . λ = 3.39 μ m , θ = 90 ° , σ film = 290 MPa .

Fig. 7
Fig. 7

ZBC as a function of waveguide rib width and etch depth for different values of upper cladding stress levels. λ = 3.39 μ m , H = 2.95 μ m , t = 1 μ m , θ = 90 ° .

Fig. 8
Fig. 8

SMC and ZBC as a function of waveguide rib width and etch depth for waveguide height of (a) H = 2.5 μ m , (b) H = 2.95 μ m , (c) H = 3.5 μ m . λ = 3.39 μ m , t = 1 μ m , θ = 90 ° , σ film = 290 MPa .

Fig. 9
Fig. 9

Waveguide geometries that exhibit both single-mode and polarization-independence conditions at three different waveguide heights. λ = 3.39 μ m , t = 1 μ m , θ = 90 ° , σ film = 290 MPa .

Tables (1)

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Table 1 Experimental Results for Cladding Stress Measurements and Corresponding Standard Deviation of Measured Data a

Equations (12)

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Δ N x = N x N 0 = C 1 σ x C 2 ( σ y + σ z ) ,
Δ N y = N y N 0 = C 1 σ y C 2 ( σ x + σ z ) ,
( C 1 C 2 ) = N 0 3 2 E ( 1 2 ν ν 1 ν ) ( p 11 p 12 ) .
( ε x ε y ε z ) = 1 E ( 1 ν ν ν 1 ν ν ν 1 ) ( σ x σ y σ z ) + ( α Δ T α Δ T α Δ T ) ,
( σ x σ y σ z ) = E ( 1 + ν ) ( 1 2 ν ) ( 1 ν ν ν ν 1 ν ν ν ν 1 ν ) ( ε x ε y ε z ) α E Δ T 1 2 ν ( 1 1 1 ) .
σ film = E s h s 6 ( 1 ν s ) R t d ,
D = a + b T 1 ( t ) + c T 1 ( W ) + d T 2 ( t ) + e T 1 ( t ) T 1 ( W ) + f T 2 ( W ) + g T 3 ( t ) + h T 2 ( t ) T 1 ( W ) + i T 1 ( t ) T 2 ( W ) + j T 3 ( W ) + k T 4 ( t ) + l T 3 ( t ) T 1 ( W ) + m T 2 ( t ) T 2 ( W ) + n T 1 ( t ) T 3 ( W ) + o T 4 ( W ) ,
T n ( x ) = 1 4 π i ( 1 t 2 ) t n 1 1 2 t x + t 2 d t ,
W [ μ m ] = ( 0.3058 + 0.6142 θ [ rad ] ) λ [ μ m ] .
D = a + c H + e ln W + g H 2 + i ( ln W ) 2 + k H ln W 1 + b H + d ln W + f H 2 + h ( ln W ) 2 + j H ln W ,
W min = 1.21982 + 0.66379 H 0.00292 H 2 ,
W max = 2.72265 1.23862 H + 0.35506 H 2 .

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