Abstract

We develop and experimentally verify a theoretical model for the total efficiency η0 of evanescent excitation and subsequent collection of spontaneous Raman signals by the fundamental quasi-TE and quasi-TM modes of a generic photonic channel waveguide. Single-mode silicon nitride (Si3N4) slot and strip waveguides of different dimensions are used in the experimental study. Our theoretical model is validated by the correspondence between the experimental and theoretical absolute values within the experimental errors. We extend our theoretical model to silicon-on-insulator (SOI) and titanium dioxide (TiO2) channel waveguides and study η0 as a function of index contrast, polarization of the mode and the geometry of the waveguides. We report nearly 2.5 (4 and 5) times larger η0 for the fundamental quasi-TM mode when compared to η0 for the fundamental quasi-TE mode of a typical Si3N4 (TiO2 and SOI) strip waveguide. η0 for the fundamental quasi-TE mode of a typical Si3N4, (TiO2 and SOI) slot waveguide is about 7 (22 and 90) times larger when compared to η0 for the fundamental quasi-TE mode of a strip waveguide of the similar dimensions. We attribute the observed enhancement to the higher electric field discontinuity present in high index contrast waveguides.

© 2015 Optical Society of America

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References

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2015 (2)

2014 (2)

2013 (1)

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

2011 (1)

2010 (1)

D. Liang, G. Roelkens, R. Baets, and J. E. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials (Basel) 3(3), 1782–1802 (2010).
[Crossref]

2009 (1)

2008 (1)

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta 620(1-2), 8–26 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (1)

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74(5), 051802 (2006).
[Crossref]

2005 (2)

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]

V. Mazet, C. Carteret, D. Brie, J. Idier, and B. Humbert, “Background removal from spectra by designing and minimising a non-quadratic cost function,” Chemom. Intell. Lab. Syst. 76(2), 121–133 (2005).
[Crossref]

2003 (1)

M. J. Colles and J. E. Griffiths, “Relative and absolute Raman scattering cross sections in liquids,” J. Chem. Phys. 7(56), 3384 (2003).

1997 (1)

Atwater, H. A.

Baclig, A. C.

Baets, R.

E. Ryckeboer, R. Bockstaele, M. Vanslembrouck, and R. Baets, “Glucose sensing by waveguide-based absorption spectroscopy on a silicon chip,” Biomed. Opt. Express 5(5), 1636–1648 (2014).
[Crossref] [PubMed]

A. Dhakal, A. Z. Subramanian, P. Wuytens, F. Peyskens, N. L. Thomas, and R. Baets, “Evanescent excitation and collection of spontaneous Raman spectra using silicon nitride nanophotonic waveguides,” Opt. Lett. 39(13), 4025–4028 (2014).
[Crossref] [PubMed]

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

D. Liang, G. Roelkens, R. Baets, and J. E. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials (Basel) 3(3), 1782–1802 (2010).
[Crossref]

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]

A. Dhakal, A. Z. Subramanian, N. L. Thomas, and R. Baets, “The role of index contrast in the efficiency of absorption and emission of a luminescent particle near a slab waveguide,” in 16th European Conference on Integrated Optics (ECIO, 2012), 131.

Barrios, C. A.

Beckx, S.

Bienstman, P.

Bisschop, S.

Bockstaele, R.

Bogaerts, W.

Bowers, J. E.

D. Liang, G. Roelkens, R. Baets, and J. E. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials (Basel) 3(3), 1782–1802 (2010).
[Crossref]

Brainis, E.

Brie, D.

V. Mazet, C. Carteret, D. Brie, J. Idier, and B. Humbert, “Background removal from spectra by designing and minimising a non-quadratic cost function,” Chemom. Intell. Lab. Syst. 76(2), 121–133 (2005).
[Crossref]

Briggs, R. M.

Brongersma, M. L.

Carteret, C.

V. Mazet, C. Carteret, D. Brie, J. Idier, and B. Humbert, “Background removal from spectra by designing and minimising a non-quadratic cost function,” Chemom. Intell. Lab. Syst. 76(2), 121–133 (2005).
[Crossref]

Caspers, P. J.

Casquel, R.

Choo-Smith, L.-P.

Claes, T.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Colles, M. J.

M. J. Colles and J. E. Griffiths, “Relative and absolute Raman scattering cross sections in liquids,” J. Chem. Phys. 7(56), 3384 (2003).

de Ridder, R. M.

Deshpande, P.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Dhakal, A.

A. Dhakal, A. Z. Subramanian, P. Wuytens, F. Peyskens, N. L. Thomas, and R. Baets, “Evanescent excitation and collection of spontaneous Raman spectra using silicon nitride nanophotonic waveguides,” Opt. Lett. 39(13), 4025–4028 (2014).
[Crossref] [PubMed]

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

A. Dhakal, A. Z. Subramanian, N. L. Thomas, and R. Baets, “The role of index contrast in the efficiency of absorption and emission of a luminescent particle near a slab waveguide,” in 16th European Conference on Integrated Optics (ECIO, 2012), 131.

Driessen, A.

Du Bois, B.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Dumon, P.

Fan, X.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta 620(1-2), 8–26 (2008).
[Crossref] [PubMed]

Griffiths, J. E.

M. J. Colles and J. E. Griffiths, “Relative and absolute Raman scattering cross sections in liquids,” J. Chem. Phys. 7(56), 3384 (2003).

Griol, A.

Guille, A.

Gylfason, K. B.

Häyrinen, M.

Helin, P.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Hens, Z.

Holgado, M.

Honkanen, S.

Humbert, B.

V. Mazet, C. Carteret, D. Brie, J. Idier, and B. Humbert, “Background removal from spectra by designing and minimising a non-quadratic cost function,” Chemom. Intell. Lab. Syst. 76(2), 121–133 (2005).
[Crossref]

Idier, J.

V. Mazet, C. Carteret, D. Brie, J. Idier, and B. Humbert, “Background removal from spectra by designing and minimising a non-quadratic cost function,” Chemom. Intell. Lab. Syst. 76(2), 121–133 (2005).
[Crossref]

Ismail, N.

Jansen, R.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Jun, Y. C.

Kalkman, J.

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74(5), 051802 (2006).
[Crossref]

Kippenberg, T. J.

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74(5), 051802 (2006).
[Crossref]

Kuittinen, M.

Leyssens, K.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Liang, D.

D. Liang, G. Roelkens, R. Baets, and J. E. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials (Basel) 3(3), 1782–1802 (2010).
[Crossref]

Luyssaert, B.

Mazet, V.

V. Mazet, C. Carteret, D. Brie, J. Idier, and B. Humbert, “Background removal from spectra by designing and minimising a non-quadratic cost function,” Chemom. Intell. Lab. Syst. 76(2), 121–133 (2005).
[Crossref]

Neutens, P.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Novotny, L.

Peyskens, F.

A. Dhakal, A. Z. Subramanian, P. Wuytens, F. Peyskens, N. L. Thomas, and R. Baets, “Evanescent excitation and collection of spontaneous Raman spectra using silicon nitride nanophotonic waveguides,” Opt. Lett. 39(13), 4025–4028 (2014).
[Crossref] [PubMed]

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Pollnau, M.

Polman, A.

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74(5), 051802 (2006).
[Crossref]

Puppels, G. J.

Roelkens, G.

D. Liang, G. Roelkens, R. Baets, and J. E. Bowers, “Hybrid integrated platforms for silicon photonics,” Materials (Basel) 3(3), 1782–1802 (2010).
[Crossref]

Rottenberg, X.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Roussey, M.

Ryckeboer, E.

Sánchez, B.

Säynätjoki, A.

Selvaraja, S.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Severi, S.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Shopova, S. I.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta 620(1-2), 8–26 (2008).
[Crossref] [PubMed]

Sohlström, H.

Subramanian, A. Z.

A. Dhakal, A. Z. Subramanian, P. Wuytens, F. Peyskens, N. L. Thomas, and R. Baets, “Evanescent excitation and collection of spontaneous Raman spectra using silicon nitride nanophotonic waveguides,” Opt. Lett. 39(13), 4025–4028 (2014).
[Crossref] [PubMed]

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

A. Dhakal, A. Z. Subramanian, N. L. Thomas, and R. Baets, “The role of index contrast in the efficiency of absorption and emission of a luminescent particle near a slab waveguide,” in 16th European Conference on Integrated Optics (ECIO, 2012), 131.

Sun, Y.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta 620(1-2), 8–26 (2008).
[Crossref] [PubMed]

Suter, J. D.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta 620(1-2), 8–26 (2008).
[Crossref] [PubMed]

Taillaert, D.

Thomas, N. L.

A. Dhakal, A. Z. Subramanian, P. Wuytens, F. Peyskens, N. L. Thomas, and R. Baets, “Evanescent excitation and collection of spontaneous Raman spectra using silicon nitride nanophotonic waveguides,” Opt. Lett. 39(13), 4025–4028 (2014).
[Crossref] [PubMed]

A. Dhakal, A. Z. Subramanian, N. L. Thomas, and R. Baets, “The role of index contrast in the efficiency of absorption and emission of a luminescent particle near a slab waveguide,” in 16th European Conference on Integrated Optics (ECIO, 2012), 131.

Vahala, K. J.

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74(5), 051802 (2006).
[Crossref]

Van Campenhout, J.

Van Dorpe, P.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. Du Bois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-loss singlemode PECVD silicon nitride photonic wire waveguides for 532–900 nm wavelength window fabricated within a CMOS pilot line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Van Thourhout, D.

Vanslembrouck, M.

White, I. M.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta 620(1-2), 8–26 (2008).
[Crossref] [PubMed]

Wiaux, V.

Wörhoff, K.

Wuytens, P.

Zhu, H.

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Figures (9)

Fig. 1
Fig. 1 Schematic of a dielectric a) strip and b) slot waveguides defined by a core of higher refractive index (Si3N4, TiO2 or silicon) patterned on a lower index bottom cladding (typically silicon dioxide). The scattering particles are assumed to be embedded with a uniform density in the upper cladding (colorless region). A particle at position r0 is shown for both cases for theoretical analysis in the Appendix.
Fig. 2
Fig. 2 Typical evanescently measured Raman spectra (normalized to 819/cm peak from slot waveguide) of IPA after background subtraction. The case shown is for excitation and collection by a TE-mode, pump power of 40 mW, 8 ± 1 dB coupling losses per facet and integration time of 5 s. The black spectrum is obtained using a w = 700 nm slot waveguide with s = 150 nm. The blue spectrum is obtained by using 700 nm wide Si3N4 strip waveguides and is shifted vertically for clarity (with blue axes). We observe an enhancement of the signal by more than a factor six as expected theoretically for TE modes.
Fig. 3
Fig. 3 Experimental data from the Raman peak of IPA at 819 cm−1 a) strip waveguides, w = 550 nm, b) strip waveguides, w = 700 nm, c) slot waveguides, w = 700 nm, s = 150 nm d) slot waveguides, w = 800 nm, s = 150 nm. Diamonds: experimental values. Lines: LSE fit of ζ (L) as per Eq. (1). Black diamond and black dashed lines: TM polarized excitation and collection. Blue diamond and blue solid lines: TE polarized excitation and collection.
Fig. 4
Fig. 4 The theoretical and experimental values of η0 obtained for a) TE modes and b) TM modes of Si3N4 waveguides. The markers represent the estimated experimental values obtained from the LSE fit of the experimental data of Fig. 3 with Eq. (1). The error bars represent the measurement and estimation errors. Lines represent the theoretical curves obtained from Eq. (A.11). The red solid lines are the theoretical curve for slot waveguides with s = 150 nm. The blue and black dashed lines are the theoretical curves for TE and TM polarizations respectively for strip waveguides. Circle: TE polarization, strip waveguides. Left handed triangles: TM polarization, strip waveguides. Right handed triangles: TM polarization, slot waveguides. Square: TE polarization, slot waveguides.
Fig. 5
Fig. 5 Theoretical conversion efficiency curves for strip waveguides as a function of total waveguide width w for different material systems and the corresponding excitation wavelengths. Blue, red and magenta colors are respectively for SOI, TiO2 and Si3N4 cores. Solid lines: TE polarized excitation and collection. Dashed lines: TM polarized excitation and collection. Only the curve segments corresponding to single mode operation from the cut-off width are shown. Note that y-axis in log scale.
Fig. 6
Fig. 6 Theoretical conversion efficiency curves for slot waveguides for a) Si3N4 core with λ0 = 785 nm, b) TiO2 core with λ0 = 1064 nm c) silicon core with λ0 = 1550 nm. The solid lines: TE polarized excitation and collection. Dashed lines: TM polarized excitation and collection. Green, magenta and red lines are respectively for slot width of 150 nm, 50 nm and 10 nm. Only the curve segments corresponding to single mode operation from the cut-off width are shown.
Fig. 7
Fig. 7 Plot of the modal field of the TiO2 slot waveguides for a) TE polarization for s = 150 nm, w = 750 nm, λ0 = 785 nm, b) TM polarization for s = 150 nm, w = 750 nm, λ0 = 785 nm, c) TE polarization, s = 150 nm, w = 750 nm, λ0 = 1064 nm d) TM polarization for s = 150 nm, w = 750 nm, λ0 = 1064 nm, e) TE polarization for s = 50 nm, w = 650 nm, λ0 = 1064 nm f) TM polarization for s = 50 nm, w = 650 nm, λ0 = 1064 nm. The arrow shows the electric field vector at a given location.
Fig. 8
Fig. 8 Theoretical conversion efficiency curves for TiO2 a) strip b) slot waveguides with λ0 = 785 nm (dashed lines) and λ0 = 1064 nm (solid lines). Black and blue curves: strip waveguides. Green: slot waveguides, s = 150 nm. Magenta: slot waveguide, s = 50 nm. The values of η0 for TM polarization for all the slot widths (data not shown) do not differ significantly from the values for strip waveguides. Only the curve segments corresponding to single mode operation from the cut-off width are shown.
Fig. 9
Fig. 9 The conversion efficiency η0 of slot waveguides with s = 100 nm as a function of total waveguide width w for different material systems and corresponding excitation wavelengths. Note that y-axis in log scale.

Equations (13)

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ζ(L) P col (L) P tx = 1 2 ρσ η 0 [ e ΔαL 1 Δα ]
Δ tot = Δ ( P tx ) 2 +Δ (σ) 2 +Δ ( P col ) 2 +( 1 R 2 )
| d 0 ( r 0 , ω s ) | 2 = λ 0 4 k ν σ( ω p , ω s ) | E ( r 0 , ω p ) | 2
| E ( r 0 , ω p ) | 2 = P pump | e m ( r 0 , ω p ) | 2 v g ( ω p ) ε 0 ε( r , ω p ) | e m ( r , ω p ) | 2 d r = P pump ε 0 v g ( ω p ) A ˜ eff ( r 0 , ω p )
A ˜ eff ( r 0 , ω p ) ε( r , ω p ) | e m ( r , ω p ) | 2 d r | e m ( r 0 , ω p ) | 2
P 0 = ω 4 | d 0 | 2 12π ε 0 c 3 = 4 π 3 3 k ν ε 0 2 σ( ω p , ω s ) n g ( ω p ) A ˜ eff ( r 0 , ω p ) P pump
P wg ( r 0 , ω s )= P 0 3 4π n g ( ω s ) n( ω s ) λ s 2 A ˜ eff ( r 0 , ω s )
P wg ( r 0 , ω s ) P pump = π 2 ε 0 2 k ν n g ( ω p ) n g ( ω s ) n( ω s ) λ s 2 σ( ω p , ω s ) A ˜ eff ( r 0 , ω p ) A ˜ eff ( r 0 , ω s ) = Λ wg ( r 0 , ω p , ω s )σ( ω p , ω s )
Λ wg ( r 0 , ω p ω s ω)= 1 n(ω) k ν ( π ε 0 n g (ω) λ 0 A ˜ eff ( r 0 , ω p ) ) 2
P wg ( r 0 , ω s ) P pump σ( ω p , ω s )ρ V Λ wg ( r 0 , ω p , ω s )d r
η 0 (dz, ω p , ω s ) d dz V Λ wg ( r 0 , ω p , ω s )d r = S Λ wg ( r 0 , ω p , ω s )d r
d P wg ( ω s ) P pump = η 0 (dz, ω p , ω s )ρσ( ω p , ω s )dz
η 0 ( ω p , ω s )= up.clad Λ wg ( r 0 , ω p , ω s )d r

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