Abstract

A systematic analysis of photonic bands and group index in silicon grating waveguides is performed, in order to optimize band-edge slow-light behavior in integrated structures with low losses. A combination of numerical methods and perturbation theory is adopted. It is shown that a substantial increase of slow light bandwidth is achieved when decreasing the internal width of the waveguide and the silicon thickness in the cladding region. It is also observed that a reduction of the internal width does not undermine the performance of an adiabatic taper.

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

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References

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2016 (1)

K. Qin, S. Hu, S. T. Retterer, I. I. Kravchenko, and S. M. Weiss, “Slow light mach–zehnder interferometer as label-free biosensor with scalable sensitivity,” Opt. letters 41, 753–756 (2016).
[Crossref]

2015 (2)

C. Sciancalepore, K. Hassan, T. Ferrotti, J. Harduin, H. Duprez, S. Menezo, and B. B. Bakir, “Low-loss adiabatically-tapered high-contrast gratings for slow-wave modulators on soi,” Proc. SPIE 9372, 93720G (2015).
[Crossref]

M. Minkov and V. Savona, “Wide-band slow light in compact photonic crystal coupled-cavity waveguides,” Optica. 2, 631–634 (2015).
[Crossref]

2012 (2)

2010 (2)

G. T. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
[Crossref]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

2008 (3)

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[Crossref]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (3)

R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Light. Technol. 23, 4046–4066 (2005).
[Crossref]

M. Povinelli, S. G. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13, 7145–7159 (2005).
[Crossref] [PubMed]

J. Hugonin, P. Lalanne, I. D. Villar, and I. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[Crossref]

2004 (2)

D. Gerace and L. C. Andreani, “Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs,” Phys. Rev. E 69, 056603 (2004).
[Crossref]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

1997 (1)

1996 (2)

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[Crossref]

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Andreani, L. C.

D. Gerace and L. C. Andreani, “Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs,” Phys. Rev. E 69, 056603 (2004).
[Crossref]

Baba, T.

Bakir, B. B.

C. Sciancalepore, K. Hassan, T. Ferrotti, J. Harduin, H. Duprez, S. Menezo, and B. B. Bakir, “Low-loss adiabatically-tapered high-contrast gratings for slow-wave modulators on soi,” Proc. SPIE 9372, 93720G (2015).
[Crossref]

Bendickson,

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Bloemer, M.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Bowden, C.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Boyd, R. W.

Brimont, A.

Canciamilla, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Chang-Hasnain, C. J.

R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Light. Technol. 23, 4046–4066 (2005).
[Crossref]

De La Rue, R.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Dowling, J.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Dudley, C.

Duprez, H.

C. Sciancalepore, K. Hassan, T. Ferrotti, J. Harduin, H. Duprez, S. Menezo, and B. B. Bakir, “Low-loss adiabatically-tapered high-contrast gratings for slow-wave modulators on soi,” Proc. SPIE 9372, 93720G (2015).
[Crossref]

Fan, S.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

Fedeli, J.

Ferrari, C.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Ferrotti, T.

C. Sciancalepore, K. Hassan, T. Ferrotti, J. Harduin, H. Duprez, S. Menezo, and B. B. Bakir, “Low-loss adiabatically-tapered high-contrast gratings for slow-wave modulators on soi,” Proc. SPIE 9372, 93720G (2015).
[Crossref]

Flynn, R.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Fork, R.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Gardes, F.

Gauthier, D. J.

Gerace, D.

D. Gerace and L. C. Andreani, “Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs,” Phys. Rev. E 69, 056603 (2004).
[Crossref]

Gomez-Iglesias, A.

Harduin, J.

C. Sciancalepore, K. Hassan, T. Ferrotti, J. Harduin, H. Duprez, S. Menezo, and B. B. Bakir, “Low-loss adiabatically-tapered high-contrast gratings for slow-wave modulators on soi,” Proc. SPIE 9372, 93720G (2015).
[Crossref]

Hashimoto, S.

Hassan, K.

C. Sciancalepore, K. Hassan, T. Ferrotti, J. Harduin, H. Duprez, S. Menezo, and B. B. Bakir, “Low-loss adiabatically-tapered high-contrast gratings for slow-wave modulators on soi,” Proc. SPIE 9372, 93720G (2015).
[Crossref]

Hu, S.

K. Qin, S. Hu, S. T. Retterer, I. I. Kravchenko, and S. M. Weiss, “Slow light mach–zehnder interferometer as label-free biosensor with scalable sensitivity,” Opt. letters 41, 753–756 (2016).
[Crossref]

Hugonin, J.

J. Hugonin, P. Lalanne, I. D. Villar, and I. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[Crossref]

Joannopoulos, J.

Johnson, S. G.

Krauss, T. F.

Kravchenko, I. I.

K. Qin, S. Hu, S. T. Retterer, I. I. Kravchenko, and S. M. Weiss, “Slow light mach–zehnder interferometer as label-free biosensor with scalable sensitivity,” Opt. letters 41, 753–756 (2016).
[Crossref]

Ku, P.-C.

R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Light. Technol. 23, 4046–4066 (2005).
[Crossref]

Lalanne, P.

J. Hugonin, P. Lalanne, I. D. Villar, and I. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[Crossref]

Leavitt, R. P.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Ledbetter, H.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Li, J.

Li, L.

Martí, J.

Mashanovich, G.

G. T. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
[Crossref]

Matias, I.

J. Hugonin, P. Lalanne, I. D. Villar, and I. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[Crossref]

Melloni, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Menezo, S.

C. Sciancalepore, K. Hassan, T. Ferrotti, J. Harduin, H. Duprez, S. Menezo, and B. B. Bakir, “Low-loss adiabatically-tapered high-contrast gratings for slow-wave modulators on soi,” Proc. SPIE 9372, 93720G (2015).
[Crossref]

Minkov, M.

M. Minkov and V. Savona, “Wide-band slow light in compact photonic crystal coupled-cavity waveguides,” Optica. 2, 631–634 (2015).
[Crossref]

Morichetti, F.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Nguyen, H. C.

O’Faolain, L.

Povinelli, M.

Qin, K.

K. Qin, S. Hu, S. T. Retterer, I. I. Kravchenko, and S. M. Weiss, “Slow light mach–zehnder interferometer as label-free biosensor with scalable sensitivity,” Opt. letters 41, 753–756 (2016).
[Crossref]

Reed, G.

Reed, G. T.

G. T. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
[Crossref]

Reinhardt, S.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Retterer, S. T.

K. Qin, S. Hu, S. T. Retterer, I. I. Kravchenko, and S. M. Weiss, “Slow light mach–zehnder interferometer as label-free biosensor with scalable sensitivity,” Opt. letters 41, 753–756 (2016).
[Crossref]

Samarelli, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Sanchis, P.

Savona, V.

M. Minkov and V. Savona, “Wide-band slow light in compact photonic crystal coupled-cavity waveguides,” Optica. 2, 631–634 (2015).
[Crossref]

Scalora, M.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Sciancalepore, C.

C. Sciancalepore, K. Hassan, T. Ferrotti, J. Harduin, H. Duprez, S. Menezo, and B. B. Bakir, “Low-loss adiabatically-tapered high-contrast gratings for slow-wave modulators on soi,” Proc. SPIE 9372, 93720G (2015).
[Crossref]

Shi, Z.

Shinkawa, M.

Sorel, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Suh, W.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

Thomson, D.

Tocci, M.

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

Torregiani, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Tucker, R. S.

R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Light. Technol. 23, 4046–4066 (2005).
[Crossref]

Villar, I. D.

J. Hugonin, P. Lalanne, I. D. Villar, and I. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[Crossref]

Wang, Z.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

Weiss, S. M.

K. Qin, S. Hu, S. T. Retterer, I. I. Kravchenko, and S. M. Weiss, “Slow light mach–zehnder interferometer as label-free biosensor with scalable sensitivity,” Opt. letters 41, 753–756 (2016).
[Crossref]

White, T. P.

Yanik, M. F.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

J. Light. Technol. (1)

R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Light. Technol. 23, 4046–4066 (2005).
[Crossref]

J. Opt. (1)

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Photonics (3)

G. T. Reed, G. Mashanovich, F. Gardes, and D. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
[Crossref]

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Opt. letters (1)

K. Qin, S. Hu, S. T. Retterer, I. I. Kravchenko, and S. M. Weiss, “Slow light mach–zehnder interferometer as label-free biosensor with scalable sensitivity,” Opt. letters 41, 753–756 (2016).
[Crossref]

Opt. Quantum Electron. (1)

J. Hugonin, P. Lalanne, I. D. Villar, and I. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[Crossref]

Optica. (1)

M. Minkov and V. Savona, “Wide-band slow light in compact photonic crystal coupled-cavity waveguides,” Optica. 2, 631–634 (2015).
[Crossref]

Phys. Rev. E (2)

M. Scalora, R. Flynn, S. Reinhardt, R. Fork, M. Bloemer, M. Tocci, C. Bowden, H. Ledbetter, Bendickson, J. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078 (1996).
[Crossref]

D. Gerace and L. C. Andreani, “Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs,” Phys. Rev. E 69, 056603 (2004).
[Crossref]

Phys. Rev. Lett. (1)

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

Proc. SPIE (1)

C. Sciancalepore, K. Hassan, T. Ferrotti, J. Harduin, H. Duprez, S. Menezo, and B. B. Bakir, “Low-loss adiabatically-tapered high-contrast gratings for slow-wave modulators on soi,” Proc. SPIE 9372, 93720G (2015).
[Crossref]

Supplementary Material (1)

NameDescription
» Data File 1       This dataset contains data about all the simulated structure. We provide all the geometrical parameters, W1, W2, t, and a (the period needed to have for each structure the lower band gap at a wavelength of 1300 nm). We also provide the fit parameters

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Figures (9)

Fig. 1
Fig. 1 Schematic cross section (a), top view (b) and 3D view (c) of the grating waveguide structure analyzed in this work. All the relevant parameters are defined: orange represents Si and yellow represents SiO2 materials (missing in 3D view). The black banded region in the 2D view shows the borders to which coordinate transformations are applied (see Sec. 2).
Fig. 2
Fig. 2 (a) Complete band dispersion calculated for the same structure as in Ref. [15]; (b) Close-up of the fundamental TE mode near the band gap. Both full numerical solution and the corresponding fit using only the red square points are shown. Parameters of the structure are: W1 = 400 nm, W2 = 800 nm, t = 150 nm, a= 213.2 nm, d2/a=0.5; Fit parameters are: Ω0 = 1.6559 · 10−1, n = 3.3429, U = 2.3945 · 10−2.
Fig. 3
Fig. 3 (a) Band structure and (b) corresponding group index for different widths of the internal section, W1, (the corresponding period is given in the legend). Fixed parameters are: h = 310 nm, t = 150 nm, W2 = 800 nm, d2/a=0.5.
Fig. 4
Fig. 4 Contour plot showing the slow light bandwidth for ng > 20 as a function of W1 and W2. The other parameters are fixed as h = 310 nm, t = 150 nm, d2/a=0.5.
Fig. 5
Fig. 5 Slow light bandwidth for ng > 10 as a function of the ratio d2/a for some selected sets of parameters.
Fig. 6
Fig. 6 Contour plot showing the slow light bandwidth for ng > 10 as a function of W1 and W2 for d2/a = 0.5 and different values of t: (a) 150 nm, (b) 100 nm, and (c) 50 nm.
Fig. 7
Fig. 7 Evolution of the group index versus wavelength as function of the cladding silicon thickness, t, for the structure with W1 = 100 nm, W2 = 800 nm and d2/a = 0.5. A bandwidth of about 10 nm for ng > 10 is evident for the simulated structure with t = 50 nm.
Fig. 8
Fig. 8 Performances of 150 periods long adiabatic tapers for t = 150 nm, W2 = 800 nm and different values of W1, respectively 100 and 400 nm: (a) group index as a function of wavelength, (b) transmission as a function of wavelength, (c) transmission as a function of group index. An example of a 20 periods long tapering structure is shown in the inset.
Fig. 9
Fig. 9 (a) Schematic representation of the Mach-Zehnder interferometer, (b) effective group index of the slow light arm and (c) transmission at the output 1 channel of the interferometer.

Equations (4)

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Ω ( δ , Ω 0 , n , U ) = { Ω 0 2 + [ δ 2 + 4 δ 2 ( 1 δ 2 ) 2 U 2 4 n 2 ] } 1 2
S BS = 1 2 [ 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 ]
S W ( ω ) = [ e i k L 0 0 e i k L ]
n g = c k ω = c L ϕ ω

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