Abstract

Oblique propagation of semi-guided waves across slab waveguide structures with bent corners is investigated. A critical angle can be defined beyond which all radiation losses are suppressed. Additionally an increase of the curvature radius of the bends also leads to low-loss configurations for incidence angles below that critical angle. A combination of two bent corner systems represents a step-like structure, behaving like a Fabry-Perot interferometer, with two partial reflectors separated by the vertical height between the horizontal slabs. We numerically analyse typical high-index-contrast Si/SiO2 structures for their reflectance and transmittance properties. When increasing the curvature radius the resonant effect becomes less relevant such that full transmittance is reached with less critical conditions on the vertical distance or the incidence angle. For practical interest 3-D problems are considered, where the structures are excited by the fundamental mode of a wide, shallow rib waveguide. High transmittance levels can be observed also for these 3-D configurations depending on the width of the rib.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. Van Campenhout, P. Bienstman, and D. Van Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23(1), 401–412 (2005).
    [Crossref]
  2. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006).
    [Crossref]
  3. P. Prakash Koonath and B. Jalali, “Multilayer 3-D photonics in silicon,” Opt. Express 15(20), 12686–12691 (2007).
    [Crossref] [PubMed]
  4. P. Koonath, T. Indukuri, and B. Jalali, “Monolithic 3-D silicon photonics,” J. Lightwave Technol. 24(4), 1796–1804 (2006).
    [Crossref]
  5. N. Sherwood-Droz and M. Lipson, “Scalable 3D dense integration of photonics on bulk silicon,” Opt. Express 19(18), 17758–17765 (2011).
    [Crossref] [PubMed]
  6. R. A. Soref, E. Cortesi, F. Namavar, and L. Friedman, “Vertically integrated silicon-on-insulator waveguides,” IEEE Photonics Technol. Lett. 3(1), 22–24 (1991).
    [Crossref]
  7. J. K. Doylend, A. P. Knights, C. Brooks, and P. E. Jessop, “CMOS compatible vertical directional coupler for 3D optical circuits,” Proc. SPIE 5970, 59700G (2005).
    [Crossref]
  8. R. Sun, M. Beals, A. Pomerene, J. Cheng, C.-Y. Hong, L. Kimerling, and J. Michel, “Impedance matching vertical optical waveguide couplers for dense high index contrast circuits,” Opt. Express 16(16), 11682–11690 (2008).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  10. P. Dong and A. G. Kirk, “Compact grating coupler between vertically stacked silicon-on-insulator waveguides,” Proc. SPIE 5357, 135–142 (2004).
    [Crossref]
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    [Crossref]
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  14. T. P. Shen, R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, “Fresnel-like behavior of guided waves,” J. Opt. Soc. Am. A 4(11), 2120–2132 (1987).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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  22. C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).
  23. M. Hammer, “Oblique incidence of semi-guided waves on rectangular slab waveguide discontinuities: A vectorial QUEP solver,” Opt. Commun. 338, 447–456 (2015).
    [Crossref]
  24. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1), 37–61 (2005).
    [Crossref]
  25. E. A. J. Marcatili, “Slab-coupled waveguides,” Bell Sys. Tech. J. 53(4), 645–674 (1974).
    [Crossref]
  26. A. G. Rickman, G. T. Reed, and F. Namavar, “Silicon-on-insulator optical rib waveguide loss and mode characteristics,” J. Lightwave Technol. 12(10), 1771–1776 (1994).
    [Crossref]
  27. C. L. Chen, Foundations for Guided-Wave Optics (John Wiley and Sons, Inc., 2006).
    [Crossref]
  28. M. Hanke-Bourgeois, Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens (Vieweg+Teubner Verlag, 2008).

2017 (1)

L. Ebers, M. Hammer, and J. Förstner, “Spiral modes supported by circular dielectric tubes and tube segments,” Opt. Quantum Electron. 49(4), 176 (2017).
[Crossref]

2016 (2)

M. Hammer, A. Hildebrandt, and J. Förstner, “Full resonant transmission of semi-guided planar waves through slab waveguide steps at oblique incidence,” J. Lightwave Technol. 34(3), 997–1005 (2016).
[Crossref]

A. Hildebrandt, A. Alhaddad, M. Hammer, and J. Förstner, “Oblique incidence of semi-guided waves on step-like folds in planar dielectric slabs: Lossless vertical interconnects in 3-D integrated photonic circuits,” Proc. SPIE 9750, 975045 (2016).

2015 (1)

M. Hammer, “Oblique incidence of semi-guided waves on rectangular slab waveguide discontinuities: A vectorial QUEP solver,” Opt. Commun. 338, 447–456 (2015).
[Crossref]

2013 (2)

2011 (1)

2008 (1)

2007 (1)

2006 (2)

P. Koonath, T. Indukuri, and B. Jalali, “Monolithic 3-D silicon photonics,” J. Lightwave Technol. 24(4), 1796–1804 (2006).
[Crossref]

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006).
[Crossref]

2005 (3)

W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. Van Campenhout, P. Bienstman, and D. Van Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23(1), 401–412 (2005).
[Crossref]

J. K. Doylend, A. P. Knights, C. Brooks, and P. E. Jessop, “CMOS compatible vertical directional coupler for 3D optical circuits,” Proc. SPIE 5970, 59700G (2005).
[Crossref]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1), 37–61 (2005).
[Crossref]

2004 (1)

P. Dong and A. G. Kirk, “Compact grating coupler between vertically stacked silicon-on-insulator waveguides,” Proc. SPIE 5357, 135–142 (2004).
[Crossref]

2003 (1)

D. N. Chien, K. Tanaka, and M. Tanaka, “Guided wave equivalents of Snell’s and Brewster’s laws”, Opt. Commun. 225(4), 319–329 (2003).
[Crossref]

1994 (2)

S. Misawa, M. Aoki, S. Fujita, A. Takaura, T. Kihara, K. Yokomori, and H. Funato, “Focusing waveguide mirror with a tapered edge,” Appl. Opt. 33(16), 3365–3370 (1994).
[Crossref] [PubMed]

A. G. Rickman, G. T. Reed, and F. Namavar, “Silicon-on-insulator optical rib waveguide loss and mode characteristics,” J. Lightwave Technol. 12(10), 1771–1776 (1994).
[Crossref]

1991 (1)

R. A. Soref, E. Cortesi, F. Namavar, and L. Friedman, “Vertically integrated silicon-on-insulator waveguides,” IEEE Photonics Technol. Lett. 3(1), 22–24 (1991).
[Crossref]

1990 (1)

W. Biehlig and U. Langbein, “Three-dimensional step discontinuities in planar waveguides: Angular-spectrum representation of guided wavefields and generalized matrix-operator formalism,” Opt. Quantum Electron. 22(4), 319–333 (1990).
[Crossref]

1987 (1)

1974 (1)

E. A. J. Marcatili, “Slab-coupled waveguides,” Bell Sys. Tech. J. 53(4), 645–674 (1974).
[Crossref]

Ahn, D.

Alhaddad, A.

A. Hildebrandt, A. Alhaddad, M. Hammer, and J. Förstner, “Oblique incidence of semi-guided waves on step-like folds in planar dielectric slabs: Lossless vertical interconnects in 3-D integrated photonic circuits,” Proc. SPIE 9750, 975045 (2016).

Aoki, M.

Baets, R.

Bauters, J. F.

Beals, M.

Beckx, S.

Bessette, J. T.

Biehlig, W.

W. Biehlig and U. Langbein, “Three-dimensional step discontinuities in planar waveguides: Angular-spectrum representation of guided wavefields and generalized matrix-operator formalism,” Opt. Quantum Electron. 22(4), 319–333 (1990).
[Crossref]

Bienstman, P.

Bogaerts, W.

Bowers, J. E.

Brooks, C.

J. K. Doylend, A. P. Knights, C. Brooks, and P. E. Jessop, “CMOS compatible vertical directional coupler for 3D optical circuits,” Proc. SPIE 5970, 59700G (2005).
[Crossref]

Chen, A.

Chen, C. L.

C. L. Chen, Foundations for Guided-Wave Optics (John Wiley and Sons, Inc., 2006).
[Crossref]

Cheng, J.

Chien, D. N.

D. N. Chien, K. Tanaka, and M. Tanaka, “Guided wave equivalents of Snell’s and Brewster’s laws”, Opt. Commun. 225(4), 319–329 (2003).
[Crossref]

Cortesi, E.

R. A. Soref, E. Cortesi, F. Namavar, and L. Friedman, “Vertically integrated silicon-on-insulator waveguides,” IEEE Photonics Technol. Lett. 3(1), 22–24 (1991).
[Crossref]

Ctyroký, J.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1), 37–61 (2005).
[Crossref]

Davenport, M. L.

Dong, P.

P. Dong and A. G. Kirk, “Compact grating coupler between vertically stacked silicon-on-insulator waveguides,” Proc. SPIE 5357, 135–142 (2004).
[Crossref]

Doylend, J. K.

J. F. Bauters, M. L. Davenport, M. J. R. Heck, J. K. Doylend, A. Chen, A. W. Fang, and J. E. Bowers, “Silicon on ultra-low-loss waveguide photonic integration platform,” Opt. Express 21(1), 544–555 (2013).
[Crossref] [PubMed]

J. K. Doylend, A. P. Knights, C. Brooks, and P. E. Jessop, “CMOS compatible vertical directional coupler for 3D optical circuits,” Proc. SPIE 5970, 59700G (2005).
[Crossref]

Dumon, P.

Ebers, L.

L. Ebers, M. Hammer, and J. Förstner, “Spiral modes supported by circular dielectric tubes and tube segments,” Opt. Quantum Electron. 49(4), 176 (2017).
[Crossref]

L. Ebers, “Oblique semi-guided waves: Modeling quasi-2-D waveguide optics with COMSOL,” Master Thesis, Paderborn University, 2017.

Fang, A. W.

Förstner, J.

L. Ebers, M. Hammer, and J. Förstner, “Spiral modes supported by circular dielectric tubes and tube segments,” Opt. Quantum Electron. 49(4), 176 (2017).
[Crossref]

M. Hammer, A. Hildebrandt, and J. Förstner, “Full resonant transmission of semi-guided planar waves through slab waveguide steps at oblique incidence,” J. Lightwave Technol. 34(3), 997–1005 (2016).
[Crossref]

A. Hildebrandt, A. Alhaddad, M. Hammer, and J. Förstner, “Oblique incidence of semi-guided waves on step-like folds in planar dielectric slabs: Lossless vertical interconnects in 3-D integrated photonic circuits,” Proc. SPIE 9750, 975045 (2016).

Friedman, L.

R. A. Soref, E. Cortesi, F. Namavar, and L. Friedman, “Vertically integrated silicon-on-insulator waveguides,” IEEE Photonics Technol. Lett. 3(1), 22–24 (1991).
[Crossref]

Fujita, S.

Funato, H.

Hammer, M.

L. Ebers, M. Hammer, and J. Förstner, “Spiral modes supported by circular dielectric tubes and tube segments,” Opt. Quantum Electron. 49(4), 176 (2017).
[Crossref]

A. Hildebrandt, A. Alhaddad, M. Hammer, and J. Förstner, “Oblique incidence of semi-guided waves on step-like folds in planar dielectric slabs: Lossless vertical interconnects in 3-D integrated photonic circuits,” Proc. SPIE 9750, 975045 (2016).

M. Hammer, A. Hildebrandt, and J. Förstner, “Full resonant transmission of semi-guided planar waves through slab waveguide steps at oblique incidence,” J. Lightwave Technol. 34(3), 997–1005 (2016).
[Crossref]

M. Hammer, “Oblique incidence of semi-guided waves on rectangular slab waveguide discontinuities: A vectorial QUEP solver,” Opt. Commun. 338, 447–456 (2015).
[Crossref]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1), 37–61 (2005).
[Crossref]

Hanke-Bourgeois, M.

M. Hanke-Bourgeois, Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens (Vieweg+Teubner Verlag, 2008).

Heck, M. J. R.

Hildebrandt, A.

M. Hammer, A. Hildebrandt, and J. Förstner, “Full resonant transmission of semi-guided planar waves through slab waveguide steps at oblique incidence,” J. Lightwave Technol. 34(3), 997–1005 (2016).
[Crossref]

A. Hildebrandt, A. Alhaddad, M. Hammer, and J. Förstner, “Oblique incidence of semi-guided waves on step-like folds in planar dielectric slabs: Lossless vertical interconnects in 3-D integrated photonic circuits,” Proc. SPIE 9750, 975045 (2016).

Hiremath, K. R.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1), 37–61 (2005).
[Crossref]

Hong, C.-Y.

Indukuri, T.

Jalali, B.

Jessop, P. E.

J. K. Doylend, A. P. Knights, C. Brooks, and P. E. Jessop, “CMOS compatible vertical directional coupler for 3D optical circuits,” Proc. SPIE 5970, 59700G (2005).
[Crossref]

Kihara, T.

Kimerling, L.

Kirk, A. G.

P. Dong and A. G. Kirk, “Compact grating coupler between vertically stacked silicon-on-insulator waveguides,” Proc. SPIE 5357, 135–142 (2004).
[Crossref]

Knights, A. P.

J. K. Doylend, A. P. Knights, C. Brooks, and P. E. Jessop, “CMOS compatible vertical directional coupler for 3D optical circuits,” Proc. SPIE 5970, 59700G (2005).
[Crossref]

Koonath, P.

Langbein, U.

W. Biehlig and U. Langbein, “Three-dimensional step discontinuities in planar waveguides: Angular-spectrum representation of guided wavefields and generalized matrix-operator formalism,” Opt. Quantum Electron. 22(4), 319–333 (1990).
[Crossref]

Lipson, M.

Luyssaert, B.

Maradudin, A. A.

Marcatili, E. A. J.

E. A. J. Marcatili, “Slab-coupled waveguides,” Bell Sys. Tech. J. 53(4), 645–674 (1974).
[Crossref]

Michel, J.

Misawa, S.

Namavar, F.

A. G. Rickman, G. T. Reed, and F. Namavar, “Silicon-on-insulator optical rib waveguide loss and mode characteristics,” J. Lightwave Technol. 12(10), 1771–1776 (1994).
[Crossref]

R. A. Soref, E. Cortesi, F. Namavar, and L. Friedman, “Vertically integrated silicon-on-insulator waveguides,” IEEE Photonics Technol. Lett. 3(1), 22–24 (1991).
[Crossref]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2000).

Pomerene, A.

Prakash Koonath, P.

Prkna, L.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1), 37–61 (2005).
[Crossref]

Reed, G. T.

A. G. Rickman, G. T. Reed, and F. Namavar, “Silicon-on-insulator optical rib waveguide loss and mode characteristics,” J. Lightwave Technol. 12(10), 1771–1776 (1994).
[Crossref]

Rickman, A. G.

A. G. Rickman, G. T. Reed, and F. Namavar, “Silicon-on-insulator optical rib waveguide loss and mode characteristics,” J. Lightwave Technol. 12(10), 1771–1776 (1994).
[Crossref]

Shen, T. P.

Sherwood-Droz, N.

Soref, R.

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006).
[Crossref]

Soref, R. A.

R. A. Soref, E. Cortesi, F. Namavar, and L. Friedman, “Vertically integrated silicon-on-insulator waveguides,” IEEE Photonics Technol. Lett. 3(1), 22–24 (1991).
[Crossref]

Stegeman, G. I.

Stoffer, R.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1), 37–61 (2005).
[Crossref]

Sun, R.

Taillaert, D.

Takaura, A.

Tanaka, K.

D. N. Chien, K. Tanaka, and M. Tanaka, “Guided wave equivalents of Snell’s and Brewster’s laws”, Opt. Commun. 225(4), 319–329 (2003).
[Crossref]

Tanaka, M.

D. N. Chien, K. Tanaka, and M. Tanaka, “Guided wave equivalents of Snell’s and Brewster’s laws”, Opt. Commun. 225(4), 319–329 (2003).
[Crossref]

Van Campenhout, J.

Van Thourhout, D.

Vassallo, C.

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Wallis, R. F.

Wiaux, V.

Yokomori, K.

Appl. Opt. (1)

Bell Sys. Tech. J. (1)

E. A. J. Marcatili, “Slab-coupled waveguides,” Bell Sys. Tech. J. 53(4), 645–674 (1974).
[Crossref]

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

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006).
[Crossref]

IEEE Photonics Technol. Lett. (1)

R. A. Soref, E. Cortesi, F. Namavar, and L. Friedman, “Vertically integrated silicon-on-insulator waveguides,” IEEE Photonics Technol. Lett. 3(1), 22–24 (1991).
[Crossref]

J. Lightwave Technol. (4)

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

Opt. Commun. (2)

D. N. Chien, K. Tanaka, and M. Tanaka, “Guided wave equivalents of Snell’s and Brewster’s laws”, Opt. Commun. 225(4), 319–329 (2003).
[Crossref]

M. Hammer, “Oblique incidence of semi-guided waves on rectangular slab waveguide discontinuities: A vectorial QUEP solver,” Opt. Commun. 338, 447–456 (2015).
[Crossref]

Opt. Express (5)

Opt. Quantum Electron. (3)

W. Biehlig and U. Langbein, “Three-dimensional step discontinuities in planar waveguides: Angular-spectrum representation of guided wavefields and generalized matrix-operator formalism,” Opt. Quantum Electron. 22(4), 319–333 (1990).
[Crossref]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1), 37–61 (2005).
[Crossref]

L. Ebers, M. Hammer, and J. Förstner, “Spiral modes supported by circular dielectric tubes and tube segments,” Opt. Quantum Electron. 49(4), 176 (2017).
[Crossref]

Proc. SPIE (3)

A. Hildebrandt, A. Alhaddad, M. Hammer, and J. Förstner, “Oblique incidence of semi-guided waves on step-like folds in planar dielectric slabs: Lossless vertical interconnects in 3-D integrated photonic circuits,” Proc. SPIE 9750, 975045 (2016).

P. Dong and A. G. Kirk, “Compact grating coupler between vertically stacked silicon-on-insulator waveguides,” Proc. SPIE 5357, 135–142 (2004).
[Crossref]

J. K. Doylend, A. P. Knights, C. Brooks, and P. E. Jessop, “CMOS compatible vertical directional coupler for 3D optical circuits,” Proc. SPIE 5970, 59700G (2005).
[Crossref]

Other (6)

COMSOL Multiphysics, numerical finite element software, version 5.2. https://www.comsol.de/comsol-multiphysics .

L. Ebers, “Oblique semi-guided waves: Modeling quasi-2-D waveguide optics with COMSOL,” Master Thesis, Paderborn University, 2017.

K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2000).

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

C. L. Chen, Foundations for Guided-Wave Optics (John Wiley and Sons, Inc., 2006).
[Crossref]

M. Hanke-Bourgeois, Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens (Vieweg+Teubner Verlag, 2008).

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Figures (10)

Fig. 1
Fig. 1 Bent corner (a) and bent step (b) configuration; oblique wave propagation with incidence angle φ. Images show incoming, reflected (R) and transmitted (T) waves.
Fig. 2
Fig. 2 Cross section of the bent corner (a) and bent step (b) configuration with refractive index (b) configuration with refractive index nf = 3.4 in the core and nc = 1.45 in the claddings, thickness d = 0.25 μm and vacuum wavelength λ0 = 1.55 μm. Incidence angle φ, vertical step height h and curvature radius r are variable parameters.
Fig. 3
Fig. 3 Transmittances T TE 0, T TM 0 and reflectances R TE 0, R TM 0 of the bent corner structure depending on the incidence angle φ for different curvature radii r.
Fig. 4
Fig. 4 Absolute electric field |E| for the bent corner structure for different radii and angles of incidence with maximal TE transmittance (upper row) and maximal TM transmittance (bottom row).
Fig. 5
Fig. 5 Transmittances T TE 0, T TM 0 and reflectances R TE 0, R TM 0 of the bent step structure depending on the vertical distance h for specified incidence angles φ and curvature radii r from the corner structure in Section 2.1. Vertical lines indicate heights with maximal transmittance that are selected for the plots of Figs. 67.
Fig. 6
Fig. 6 Transmittances T TE 0, T TM 0 and reflectances R TE 0, R TM 0 of the bent step for different incidence angles φ around the primary incidence angle φ0 for specific radii r and heights h.
Fig. 7
Fig. 7 Absolute electric field |E| for bent step structures for different radii, heights and angles of incidence with maximal TE transmittance from Fig. 5.
Fig. 8
Fig. 8 Connection of a rib and slab waveguide (a) of thickness d with refractive indices nf, nc in rotated coordinates (x′, y, z′). In (b) the view from above for incoming and outgoing waves under primary incidence angle φ0 is shown. The ribs have a width of l and an etch depth of d′ = 0.01µm; other parameters are as given for Fig. 2.
Fig. 9
Fig. 9 Electric field |E| for incidence of laterally guided modes for the bent step structure for different radii r, primary angles φ0 and heights h as used before in Fig. 7 for rib widths (a) l = 40µm, (b) l = 60µm, (c) l = 10µm, (d) l = 40µm (e) l = 10µm, (f) l = 10µm.
Fig. 10
Fig. 10 TE0 transmittances for the 3-D (blue line) bent step structure (config. (C1)–(C6)) and sharp step structure (config. (S1)–(S2)) from [12], for incoming and outgoing rib of width l, depending on the height h for different incidence angles and radii. The red line refers to the corresponding 2-D results from Section 2.1, with laterally unlimited incoming waves.

Equations (7)

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( E H ) ( x , y , z ) = A w ( k z ) { Ψ 0 ( k z , y ) e i k x ( k z ) ( x x 0 ) + ρ } e i k z ( z z 0 ) d k z .
Φ 0 ( y , z ) = A j w j ( k z ) Ψ j ( k z , y ) e i k z ( z z 0 ) d k z ,
( Ψ , Φ ) y : = 1 4 ( E 1 y * H 2 z E 1 z * H 2 y + H 1 z * E 2 y H 1 y * E 2 z ) d y
w 0 ( k z ) = 1 2 π A ( Ψ 0 , Φ 0 ) y e i k z ( z z 0 ) d z .
Ψ out ( y , z ) = j u j Φ j ( y , z ) .
( Φ , Ψ ) y , z : = 1 4 ( E 1 y * H 2 z E 1 z * H 2 y + H 1 z * E 2 y H 1 y * E 2 z ) d y d z ,
u 0 = ( Φ 0 , Ψ out ) y , z