Abstract

The optical transmission of resonant guided-mode gratings patterned on suspended silicon nitride thin films and illuminated at normal incidence with a Gaussian beam is investigated both experimentally and theoretically. Effects due to the beam focusing and its finite size are accounted for by a phenomenological coupled-mode model whose predictions are found to be in very good agreement with the experimentally measured spectra for various grating structures and beam sizes, and which allow for a detailed analysis of the respective magnitude of these effects. These results are highly relevant for the design and optimization of such suspended structured films that are widely used for photonics, sensing, and optomechanics applications.

© 2021 Optical Society of America

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References

  • View by:

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2021 (4)

A. Parthenopoulos, A. A. Darki, B. R. Jeppesen, and A. Dantan, “Optical spatial differentiation using suspended subwavelength gratings,” Opt. Express 29, 6481–6494 (2021).
[Crossref]

K. Cheng, Y. Fan, W. Zhang, Y. Gong, S. Fei, and H. Li, “Optical realization of wave-based analog computing with metamaterials,” Appl. Sci. 11, 141 (2021).
[Crossref]

J. M. Fitzgerald, S. K. Manjeshwar, W. Wieczorek, and P. Tassin, “Cavity optomechanics with photonic bound states in the continuum,” Phys. Rev. Res. 3, 013131 (2021).
[Crossref]

A. A. Darki, A. Parthenopoulos, J. V. Nygaard, and A. Dantan, “Profilometry and stress analysis of suspended nanostructured thin films,” J. Appl. Phys. 129, 065302 (2021).
[Crossref]

2020 (4)

S. K. Manjeshwar, K. Elkhouly, J. M. Fitzgerald, M. Ekman, Y. Zhang, F. Zhang, S. M. Wang, P. Tassin, and W. Wieczorek, “Suspended photonic crystal membranes in AlGaAs heterostructures for integrated multi-element optomechanics,” Appl. Phys. Lett. 116, 264001 (2020).
[Crossref]

C. Yang, X. Wei, J. Sheng, and H. Wu, “Phonon heat transport in cavity-mediated nanomechanical resonators,” Nat. Commun. 11, 4626 (2020).
[Crossref]

L. Guillemot, T. Oksenhendler, S. Pelloquin, O. Gauthier-Lafaye, A. Monmayrant, and T. Chaneliere, “Guided-mode resonant filter external-cavity diode laser,” Laser Phys. 30, 035802 (2020).
[Crossref]

W. Yang, X. Yu, J. Zhang, and X. Deng, “Plasmonic transmitted optical differentiator based on the subwavelength gold gratings,” Opt. Lett. 45, 2295–2298 (2020).
[Crossref]

2019 (3)

X. Wei, J. Sheng, C. Yang, Y. Wu, and H. Wu, “Controllable two-membrane-in-the-middle cavity optomechanical system,” Phys. Rev. A 99, 023851 (2019).
[Crossref]

A. Cernotik, A. Dantan, and C. Genes, “Cavity quantum electrodynamics with frequency-dependent reflectors,” Phys. Rev. Lett. 122, 243601 (2019).
[Crossref]

B. Nair, A. Naesby, B. R. Jeppesen, and A. Dantan, “Suspended silicon nitride thin films with enhanced and electrically tunable reflectivity,” Phys. Scr. 94, 125013 (2019).
[Crossref]

2018 (7)

A. Naesby and A. Dantan, “Microcavities with suspended subwavelength structured mirrors,” Opt. Express 26, 29886–29894 (2018).
[Crossref]

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
[Crossref]

C. Gärtner, J. P. Moura, W. Haaxman, R. A. Norte, and S. Gröblacher, “Integrated optomechanical arrays of two high reflectivity SiN membranes,” Nano Lett. 18, 7171–7175 (2018).
[Crossref]

D. A. Bykov, L. L. Doskolovich, A. A. Morozov, V. V. Podlipnov, E. A. Bezis, P. Verma, and V. A. Soifer, “First-order optical spatial differentiator based on a guided-mode resonant grating,” Opt. Express 26, 10997–11006 (2018).
[Crossref]

Z. Dong, J. Si, X. Yu, and X. Deng, “Optical spatial differentiator based on subwavelength high-contrast gratings,” Appl. Phys. Lett. 112, 181102 (2018).
[Crossref]

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photon. Rev. 12, 1800017 (2018).
[Crossref]

J. P. Moura, R. A. Norte, J. Guo, C. Schäfermeier, and S. Gröblacher, “Centimeter-scale suspended photonic crystal mirrors,” Opt. Express 26, 1895–1909 (2018).
[Crossref]

2017 (3)

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deleglise, “High-finesse Fabry–Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light Sci. Appl. 6, e16190 (2017).
[Crossref]

H. Xu, U. Kemiktarak, J. Fan, S. Ragole, J. Lawall, and J. M. Taylor, “Observation of optomechanical buckling transitions,” Nat. Commun. 8, 14481 (2017).
[Crossref]

B. Nair, A. Naesby, and A. Dantan, “Optomechanical characterization of silicon nitride membrane arrays,” Opt. Lett. 42, 1341–1344 (2017).
[Crossref]

2016 (3)

S. Bernard, C. Reinhardt, V. Dumont, Y.-A. Peter, and J. C. Sankey, “Precision resonance tuning and design of SiN photonic crystal reflectors,” Opt. Lett. 41, 5624–5627 (2016).
[Crossref]

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref]

C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
[Crossref]

2015 (3)

2014 (2)

U. Kemiktarak, M. Durand, M. Metcalfe, and J. Lawall, “Mode competition and anomalous cooling in a multimode phonon laser,” Phys. Rev. Lett. 113, 030802 (2014).
[Crossref]

A. Xuereb, C. Genes, G. Pupillo, M. Paternostro, and A. Dantan, “Reconfigurable long-range phonon dynamics in optomechanical arrays,” Phys. Rev. Lett. 112, 133603 (2014).
[Crossref]

2013 (1)

2012 (6)

A. Xuereb, C. Genes, and A. Dantan, “Strong coupling and long-range collective interactions in optomechanical arrays,” Phys. Rev. Lett. 109, 223601 (2012).
[Crossref]

U. Kemiktarak, M. Durand, M. Metcalfe, and J. Lawall, “Cavity optomechanics with sub-wavelength grating mirrors,” New J. Phys. 14, 125010 (2012).
[Crossref]

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photon. 4, 379–440 (2012).
[Crossref]

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2012).
[Crossref]

U. Kemiktarak, M. Metcalfe, M. Durand, and J. Lawall, “Mechanically compliant grating reflectors for optomechanics,” Appl. Phys. Lett. 100, 061124 (2012).
[Crossref]

C. H. Bui, J. Zheng, S. W. Hoch, L. Y. T. Lee, J. G. E. Harris, and C. W. Wong, “High-reflectivity, high-Q micromechanical membranes via guided resonances for enhanced optomechanical coupling,” Appl. Phys. Lett. 100, 021110 (2012).
[Crossref]

2009 (1)

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichoimetric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

2008 (2)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

E. Kenyon, M. W. Cresswell, H. J. Patrick, and T. A. Germer, “Modeling the effects of finite size gratings on scatterometry measurements,” Proc. SPIE 6922, 69223P (2008).
[Crossref]

2006 (1)

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[Crossref]

2004 (3)

2003 (4)

S. T. Thurman and G. M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. 42, 3225–3233 (2003).
[Crossref]

E. Bonnet, X. Letartre, A. Cachard, A. V. Tishchenko, and O. Parriaux, “High resonant reflection of a confined free space beam by a high contrast segmented waveguide,” Opt. Quantum Electron. 35, 1025–1036 (2003).
[Crossref]

A. V. Tishchenko, M. Hamdoun, and O. Parriaux, “Two-dimensional coupled mode equation for grating waveguide excitation by a focused beam,” Opt. Quantum Electron. 35, 475–491 (2003).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

2002 (1)

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

2001 (2)

2000 (2)

1998 (1)

S. Glasberg, A. Sharon, D. Rosenblatt, and A. A. Friesem, “Spectral shifts and line-shapes asymmetries in the resonant response of grating waveguide structures,” Opt. Commun. 145, 291–299 (1998).
[Crossref]

1997 (2)

S. M. Loktev, N. M. Lyndin, O. Parriaux, V. A. Sychugov, and A. V. Tishchenko, “Reflection of a finite light beam from a finite waveguide grating,” Quantum Electron. 27, 447–451 (1997).
[Crossref]

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[Crossref]

1995 (2)

J. Saarinen and E. Noponen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[Crossref]

J. C. Brazas and L. Li, “Analysis of input-grating couplers having finite lengths,” Appl. Opt. 34, 3786–3792 (1995).
[Crossref]

1993 (2)

R. Magnusson and S. S. Wang, “Optical waveguide-grating filters,” Proc. SPIE 2108, 380–390 (1993).
[Crossref]

S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
[Crossref]

Atwater, H. A.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2012).
[Crossref]

Basset, G.

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photon. Rev. 12, 1800017 (2018).
[Crossref]

Bawaj, M.

P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
[Crossref]

Bendickson, J. M.

Bernard, S.

Bezis, E. A.

Bonnet, E.

E. Bonnet, X. Letartre, A. Cachard, A. V. Tishchenko, and O. Parriaux, “High resonant reflection of a confined free space beam by a high contrast segmented waveguide,” Opt. Quantum Electron. 35, 1025–1036 (2003).
[Crossref]

Bourassa, A.

C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise SiN trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
[Crossref]

Boye, R. R.

Braive, R.

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deleglise, “High-finesse Fabry–Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light Sci. Appl. 6, e16190 (2017).
[Crossref]

Brazas, J. C.

Briant, T.

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Adv. Opt. Photon. (1)

Appl. Opt. (5)

Appl. Phys. Lett. (4)

Z. Dong, J. Si, X. Yu, and X. Deng, “Optical spatial differentiator based on subwavelength high-contrast gratings,” Appl. Phys. Lett. 112, 181102 (2018).
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U. Kemiktarak, M. Metcalfe, M. Durand, and J. Lawall, “Mechanically compliant grating reflectors for optomechanics,” Appl. Phys. Lett. 100, 061124 (2012).
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C. H. Bui, J. Zheng, S. W. Hoch, L. Y. T. Lee, J. G. E. Harris, and C. W. Wong, “High-reflectivity, high-Q micromechanical membranes via guided resonances for enhanced optomechanical coupling,” Appl. Phys. Lett. 100, 021110 (2012).
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S. K. Manjeshwar, K. Elkhouly, J. M. Fitzgerald, M. Ekman, Y. Zhang, F. Zhang, S. M. Wang, P. Tassin, and W. Wieczorek, “Suspended photonic crystal membranes in AlGaAs heterostructures for integrated multi-element optomechanics,” Appl. Phys. Lett. 116, 264001 (2020).
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Appl. Sci. (1)

K. Cheng, Y. Fan, W. Zhang, Y. Gong, S. Fei, and H. Li, “Optical realization of wave-based analog computing with metamaterials,” Appl. Sci. 11, 141 (2021).
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IEEE J. Quantum Electron. (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
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J. Appl. Phys. (1)

A. A. Darki, A. Parthenopoulos, J. V. Nygaard, and A. Dantan, “Profilometry and stress analysis of suspended nanostructured thin films,” J. Appl. Phys. 129, 065302 (2021).
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J. Opt. Soc. Am. A (5)

Laser Photon. Rev. (1)

G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent advances in resonant waveguide gratings,” Laser Photon. Rev. 12, 1800017 (2018).
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Laser Phys. (1)

L. Guillemot, T. Oksenhendler, S. Pelloquin, O. Gauthier-Lafaye, A. Monmayrant, and T. Chaneliere, “Guided-mode resonant filter external-cavity diode laser,” Laser Phys. 30, 035802 (2020).
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Light Sci. Appl. (1)

X. Chen, C. Chardin, K. Makles, C. Caër, S. Chua, R. Braive, I. Robert-Philip, T. Briant, P.-F. Cohadon, A. Heidmann, T. Jacqmin, and S. Deleglise, “High-finesse Fabry–Perot cavities with bidimensional Si3N4 photonic-crystal slabs,” Light Sci. Appl. 6, e16190 (2017).
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Nano Lett. (1)

C. Gärtner, J. P. Moura, W. Haaxman, R. A. Norte, and S. Gröblacher, “Integrated optomechanical arrays of two high reflectivity SiN membranes,” Nano Lett. 18, 7171–7175 (2018).
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Nat. Commun. (2)

H. Xu, U. Kemiktarak, J. Fan, S. Ragole, J. Lawall, and J. M. Taylor, “Observation of optomechanical buckling transitions,” Nat. Commun. 8, 14481 (2017).
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C. Yang, X. Wei, J. Sheng, and H. Wu, “Phonon heat transport in cavity-mediated nanomechanical resonators,” Nat. Commun. 11, 4626 (2020).
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Nature (2)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
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P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2012).
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New J. Phys. (2)

U. Kemiktarak, M. Durand, M. Metcalfe, and J. Lawall, “Cavity optomechanics with sub-wavelength grating mirrors,” New J. Phys. 14, 125010 (2012).
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P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Malossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New J. Phys. 20, 083024 (2018).
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Opt. Commun. (1)

S. Glasberg, A. Sharon, D. Rosenblatt, and A. A. Friesem, “Spectral shifts and line-shapes asymmetries in the resonant response of grating waveguide structures,” Opt. Commun. 145, 291–299 (1998).
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Opt. Eng. (1)

J. Saarinen and E. Noponen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
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Opt. Express (7)

Opt. Lett. (3)

Opt. Quantum Electron. (2)

E. Bonnet, X. Letartre, A. Cachard, A. V. Tishchenko, and O. Parriaux, “High resonant reflection of a confined free space beam by a high contrast segmented waveguide,” Opt. Quantum Electron. 35, 1025–1036 (2003).
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Phys. Rev. A (1)

X. Wei, J. Sheng, C. Yang, Y. Wu, and H. Wu, “Controllable two-membrane-in-the-middle cavity optomechanical system,” Phys. Rev. A 99, 023851 (2019).
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Phys. Rev. B (2)

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
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S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
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A. Xuereb, C. Genes, and A. Dantan, “Strong coupling and long-range collective interactions in optomechanical arrays,” Phys. Rev. Lett. 109, 223601 (2012).
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Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (12)

Fig. 1.
Fig. 1. Illustration of the situation considered: a suspended thin film patterned with a subwavelength grating is illuminated at normal incidence by a Gaussian beam of linearly polarized, monochromatic light focused on the grating. Both the beam collimation and its finite size affect the resonant interaction of the incident light with a guided mode in the grating.
Fig. 2.
Fig. 2. (a) Microscope top-view picture of sample C showing a 200 µm square SWG (green) patterned on a 500 µm square suspended ${\rm{S}}{{\rm{i}}_3}{{\rm{N}}_4}$ film (white). (b) Zoom-in on the top-right corner of the patterned area. (c) Result of an AFM scan of the SWG. (d) Suspended SWG geometry considered: a Gaussian beam (waist ${w_0}$, divergence angle ${\theta _D} = \lambda /\pi {w_0}$) impinges at normal incidence on a suspended subwavelength grating consisting of periodic trapezoidal fingers (period $\Lambda$, finger depth $d$, top and mean finger widths ${w_t}$ and ${w_m}$) on a slab waveguide with thickness $t - d$. The incident light is linearly polarized in either the $x$ direction (TM) or the $y$ direction (TE). (e) Schematic of the setup used for the optical characterization of the SWGs (see text for details). DL, diode laser; ${{\rm{L}}_1}$, ${{\rm{L}}_2}$, lenses; BS, 50:50 beam splitter; HWP, half-wave plate; D, achromatic doublet; ${{\rm{P}}_i}$, ${{\rm{P}}_t}$, photodiodes.
Fig. 3.
Fig. 3. Measured normalized transmission spectra of samples A–E for different incoming beam waists ${w_0}$.
Fig. 4.
Fig. 4. Top, AR-coated waveguide illuminated at oblique incidence. Bottom, suspended SWG illuminated at normal incidence.
Fig. 5.
Fig. 5. Normalized transmission spectra for sample A. Black dots, experimental data for a waist ${w_0} = 100 \; \unicode{x00B5}{\rm m}$. Black plain line, RCWA predictions for normally incident plane wave. Black dashed line, result of a fit of the experimental data with the coupled-mode model for normally incident plane wave. Gray plain line, RCWA predictions for plane wave with incidence angle 0.5°. Gray dashed line, result of a fit of the experimental data with the coupled-mode model plane wave with incidence angle 0.5°.
Fig. 6.
Fig. 6. Collimation effects. Top, RCWA predicted transmission spectra for sample A based on the experimentally determined grating parameters and a Gaussian angular average for different waists ${w_0} = 20$, 30, 43, 67, and 100 µm. Bottom, corresponding coupled-mode model predicted spectra for ${t_d} = 0.87$, ${\lambda _0} = 941.8\;{\rm{nm}}$, ${\lambda _1} = 943.2\;{\rm{nm}}$, ${\delta _\lambda} = \lambda _0^2/(2\pi)\delta = 2.9\;{\rm{nm}}$, ${\lambda _2} = 925.3\;{\rm{nm}}$, and $\nu = 2.9 \times {10^{- 5}} \;{{\rm{nm}}^{- 2}}$.
Fig. 7.
Fig. 7. Finite-size effects. Normalized transmission spectra of sample A predicted by the waveguide model for the same waists as in Fig. 6.
Fig. 8.
Fig. 8. Comparison between the experimental transmission spectra of sample A (top) and those resulting from a fit with the phenomenological model including both collimation and finite-size effects (bottom).
Fig. 9.
Fig. 9. Sample B. Simulated transmission spectra including collimation effects only (top left), finite-size effects only (top right), and both effects (bottom left), as well as the corresponding experimental spectra (bottom right).
Fig. 10.
Fig. 10. Sample C. Simulated transmission spectra including collimation effects only (top left), finite-size effects only (top right), and both effects (bottom left), as well as the corresponding experimental spectra (bottom right).
Fig. 11.
Fig. 11. Sample D. Simulated transmission spectra including collimation effects only (top left), finite-size effects only (top right), and both effects (bottom left), as well as the corresponding experimental spectra (bottom right).
Fig. 12.
Fig. 12. Sample E. Simulated transmission spectra including collimation effects only (top left), finite-size effects only (top right), and both effects (bottom left), as well as the corresponding experimental spectra (bottom right).

Tables (2)

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Table 1. Geometrical Parameters of the SWGs and Incident Light Polarization

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Table 2. Analysis Results for Samples A–E

Equations (26)

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E ~ i n ( k x ) = d x E i n ( x ) e i k x x ,
E t r ( x ) = 1 2 π d k x t ( k x ) E ~ i n ( k x ) e i k x x .
E i n ( x ) = E 0 e x 2 / w 0 2 ,
E ~ i n ( k x ) = E 0 π w 0 e ( k x w 0 / 2 ) 2 .
T = d k x | t ( k x ) E ~ i n ( k x ) | 2 d k x | E ~ i n ( k x ) | 2 .
t C M ( k x ) = t d ( k k 0 ) ( k k 2 ) ν k x 2 ( k k 1 i δ ) ( k k 2 ) ν k x 2 ,
ν θ = ν k 0 k 2 k 0 .
r w a v ( ) = η e i ϕ n = 0 ( 1 η ) n e i n ϕ = η e i ϕ 1 ( 1 η ) e i ϕ ,
β = k sin θ 0 ± 2 π Λ .
ϕ = 2 κ ( t d ) + 2 ϕ c + 2 ϕ s = 2 m π ,
κ = n 2 k 2 β 2
ϕ s = tan 1 ( n 2 ρ γ κ ) ,
ϕ c = tan 1 [ n 2 ρ γ e κ tanh [ tanh 1 ( γ γ e ) + γ e d ] ] ,
γ = β 2 k 2 ,
γ e = β 2 n e 2 k 2 ,
ρ = 0 a n d n e = n 2 f + 1 f
ρ = 1 a n d n e = 1 / f / n 2 + 1 f
η = δ λ 0 2 2 π | d ϕ d λ | k = k 0 .
r w a v ( N ) = η e i ϕ n = 0 N ( 1 η ) n e i n ϕ = η e i ϕ 1 ( 1 η ) e i ϕ [ 1 ( 1 η ) N e i N ϕ ] = ( 1 α ) r w a v ( ) ,
α = ( 1 η ) N e i N ϕ .
l = | d ϕ d β | k = k 0 .
t = t d + a ( k k 1 ) i δ t d k k 0 k k 1 i δ ,
t f i n = ( 1 α ) t + α t d = t d + ( 1 α ) a k k 1 + i δ = t d k k 0 + α Δ k k 1 i δ ,
α ( 1 η ) w 0 / l = e w 0 / A ,
A = l ln ( 1 η ) .
t C M , f i n ( k x ) = t d ( k k 0 + α Δ ) ( k k 2 ) ν k x 2 ( k k 1 i δ ) ( k k 2 ) ν k x 2 .

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