Abstract

Speed manipulation of optical pulses is a very attractive research challenge enabling next-generation high-capacity all-optical communication networks. Pulses can be effectively slowed by using different integrated optical structures such as coupled-resonator waveguiding structures or photonic crystal cavities. Fast light generation by means of integrated photonic devices is currently a quite unexplored research field in spite of its crucial importance for all-optical pulse processing. In this paper, we report on the first theoretical demonstration of fast light generation in an ultra-compact double vertical stacked ring resonator coupled to a bus waveguide. Periodic coupling between the two rings leads to splitting and recombining of symmetric and anti-symmetric resonant modes. Re-established degenerate modes can form when a symmetric and an anti-symmetric mode having different resonance order exhibit the same resonance wavelength. Under degenerate mode conditions, wide wavelength ranges where the group velocity is negative or larger than the speed of light in vacuum are generated. The paper proves how this physical effect can be exploited to design fast light resonant devices. Moreover, conditions are also derived to obtain slow light operation regime.

© 2010 OSA

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2009 (8)

M. Z. Feng, W. V. Sorin, and R. S. Tucker, “Fast light and Seed of Information Transfer in the Presence of the detector Noise,” IEEE Photonics J. 1(3), 213–224 (2009).
[CrossRef]

G. S. Wiederhecker, L. Chen, A. A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[CrossRef] [PubMed]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[CrossRef]

Q. Li, Z. Zhang, J. Wang, M. Qiu, and Y. Su, “Fast light in silicon ring resonator with resonance-splitting,” Opt. Express 17(2), 933 (2009).
[CrossRef] [PubMed]

P. Chamorro-Posada and F. J. Fraile-Pelaez, “Fast and slow light in zigzag microring resonator chains,” Opt. Lett. 34(5), 626–628 (2009).
[CrossRef] [PubMed]

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Simultaneous slow and fast light effects using probe gain and pump depletion via Raman gain in atomic vapor,” Opt. Express 17(11), 8775–8780 (2009).
[CrossRef] [PubMed]

S. Rawal, R. K. Sinha, and R. M. De La Rue, “Slow light miniature devices with ultra-flattened dispersion in silicon-on-insulator photonic crystal,” Opt. Express 17(16), 13315–13325 (2009).
[CrossRef] [PubMed]

C. Ciminelli, C. E. Campanella, and M. N. Armenise, “Optimized Design of Integrated Optical Angular Velocity Sensors Based on a Passive Ring Resonator,” J. Lightwave Technol. 27(14), 2658–2666 (2009).
[CrossRef]

2008 (4)

2007 (7)

C. Ciminelli, F. Peluso, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2007).
[CrossRef]

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro–optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[CrossRef]

P. T. Rakich, M. A. Popović, M. Soljačić, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75(5), 053807 (2007).
[CrossRef]

P. K. Kondratko and S. L. Chuang, “Slow-to-fast light using absorption to gain switching in quantum-well semiconductor optical amplifier,” Opt. Express 15(16), 9963–9969 (2007).
[CrossRef] [PubMed]

H. P. Uranus, L. Zhuang, C. G. H. Roeloffzen, and H. J. W. M. Hoekstra, “Pulse advancement and delay in an integrated-optical two-port ring-resonator circuit: direct experimental observations,” Opt. Lett. 32(17), 2620–2622 (2007).
[CrossRef] [PubMed]

H. P. Uranus and H. J. W. M. Hoekstra, “Modeling of Loss-Induced Superluminal and Negative Group Velocity in Two-Port Ring-Resonator Circuits,” J. Lightwave Technol. 25(9), 2376–2384 (2007).
[CrossRef]

2006 (1)

2005 (2)

M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

2004 (1)

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92(4), 043601 (2004).
[CrossRef] [PubMed]

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

2001 (1)

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett. 86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[CrossRef]

1999 (1)

Armenise, M. N.

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[CrossRef]

Campanella, C. E.

Chamorro-Posada, P.

Chen, L.

G. S. Wiederhecker, L. Chen, A. A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Chiao, R. Y.

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92(4), 043601 (2004).
[CrossRef] [PubMed]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett. 86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

Chuang, S. L.

Ciminelli, C.

De La Rue, R. M.

Degl’Innocenti, R.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro–optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[CrossRef]

Dogariu, A.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett. 86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

Dong, P.

Feng, M. Z.

M. Z. Feng, W. V. Sorin, and R. S. Tucker, “Fast light and Seed of Information Transfer in the Presence of the detector Noise,” IEEE Photonics J. 1(3), 213–224 (2009).
[CrossRef]

Fraile-Pelaez, F. J.

Gondarenko, A. A.

G. S. Wiederhecker, L. Chen, A. A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Gopal, V.

M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75(5), 053807 (2007).
[CrossRef]

Guarino, A.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro–optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[CrossRef]

Günter, P.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro–optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[CrossRef]

Hemmer, P. R.

Hickmann, J. M.

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92(4), 043601 (2004).
[CrossRef] [PubMed]

Hoekstra, H. J. W. M.

Ippen, E. P.

P. T. Rakich, M. A. Popović, M. Soljačić, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Kondratko, P. K.

Kuzmich, A.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett. 86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

Lee, R. K.

Li, Q.

Lin, Q.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[CrossRef]

Lipson, M.

G. S. Wiederhecker, L. Chen, A. A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[CrossRef] [PubMed]

S. Manipatruni, P. Dong, Q. Xu, and M. Lipson, “Tunable superluminal propagation on a silicon microchip,” Opt. Lett. 33(24), 2928–2930 (2008).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Manipatruni, S.

McCormick, C. F.

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92(4), 043601 (2004).
[CrossRef] [PubMed]

Messall, M.

M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75(5), 053807 (2007).
[CrossRef]

Milonni, P. W.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett. 86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

Painter, O.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[CrossRef]

Pati, G. S.

Peluso, F.

C. Ciminelli, F. Peluso, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2007).
[CrossRef]

Poberaj, G.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro–optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[CrossRef]

Popescu, S.

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92(4), 043601 (2004).
[CrossRef] [PubMed]

Popovic, M. A.

P. T. Rakich, M. A. Popović, M. Soljačić, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Qiu, M.

Rakich, P. T.

P. T. Rakich, M. A. Popović, M. Soljačić, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Rawal, S.

Rezzonico, D.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro–optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[CrossRef]

Roeloffzen, C. G. H.

Rosenberg, J.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[CrossRef]

Salit, K.

G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Simultaneous slow and fast light effects using probe gain and pump depletion via Raman gain in atomic vapor,” Opt. Express 17(11), 8775–8780 (2009).
[CrossRef] [PubMed]

M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75(5), 053807 (2007).
[CrossRef]

Salit, M.

Scherer, A.

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Shahriar, M. S.

Sinha, R. K.

Soljacic, M.

P. T. Rakich, M. A. Popović, M. Soljačić, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Solli, D. R.

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92(4), 043601 (2004).
[CrossRef] [PubMed]

Sorin, W. V.

M. Z. Feng, W. V. Sorin, and R. S. Tucker, “Fast light and Seed of Information Transfer in the Presence of the detector Noise,” IEEE Photonics J. 1(3), 213–224 (2009).
[CrossRef]

Su, H.

Su, Y.

Sumetsky, M.

Thévenaz, L.

L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2(8), 474–481 (2008).
[CrossRef]

Tripathi, R.

M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75(5), 053807 (2007).
[CrossRef]

Tseng, S.

Tucker, R. S.

M. Z. Feng, W. V. Sorin, and R. S. Tucker, “Fast light and Seed of Information Transfer in the Presence of the detector Noise,” IEEE Photonics J. 1(3), 213–224 (2009).
[CrossRef]

Uranus, H. P.

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

Wang, J.

Wang, L. J.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett. 86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

Wiederhecker, G. S.

G. S. Wiederhecker, L. Chen, A. A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Xu, Q.

S. Manipatruni, P. Dong, Q. Xu, and M. Lipson, “Tunable superluminal propagation on a silicon microchip,” Opt. Lett. 33(24), 2928–2930 (2008).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Xu, Y.

Yariv, A.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[CrossRef]

Yum, H. N.

Zhang, Z.

Zhuang, L.

Electron. Lett. (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[CrossRef]

IEEE Photonics J. (1)

M. Z. Feng, W. V. Sorin, and R. S. Tucker, “Fast light and Seed of Information Transfer in the Presence of the detector Noise,” IEEE Photonics J. 1(3), 213–224 (2009).
[CrossRef]

J. Lightwave Technol. (2)

Nat. Photonics (5)

P. T. Rakich, M. A. Popović, M. Soljačić, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[CrossRef]

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro–optically tunable microring resonators in lithium niobate,” Nat. Photonics 1(7), 407–410 (2007).
[CrossRef]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[CrossRef]

L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2(8), 474–481 (2008).
[CrossRef]

Nature (3)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

G. S. Wiederhecker, L. Chen, A. A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Opt. Express (7)

Opt. Lett. (4)

Phys. Rev. A (1)

M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75(5), 053807 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92(4), 043601 (2004).
[CrossRef] [PubMed]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett. 86(18), 3925–3929 (2001).
[CrossRef] [PubMed]

Proc. SPIE (1)

C. Ciminelli, F. Peluso, and M. N. Armenise, “A new integrated optical angular velocity sensor,” Proc. SPIE 5728, 93–100 (2007).
[CrossRef]

Other (4)

K. J. Vahala, Optical microcavities (World Scientific Publishing, Singapore, 2004).

J. Heebner, R. Grover, and T. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, New York, 2007).

P. W. Milonni, Fast light, slow light and left-handed light (Institute of Physics, 2004).

S. L. Chuang, Physics of Optoelectronic Devices (Wiley-Interscience Publication, New York, 1995).

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Figures (9)

Fig. 1
Fig. 1

Two vertically-stacked ring resonators coupled to a straight waveguide.(a) Top-view. (b) Cross-section.

Fig. 2
Fig. 2

Spectral properties of the optical cavity including the two vertically-stacked rings (● crossing between lines relevant to adjacent resonant orders, ● crossing between lines relevant to non adjacent resonant orders).

Fig. 3
Fig. 3

Spectral response of the proposed optical cavity for both weak and strong power transfer between the bus waveguide and the bottom ring. (a) Contour plot of the spectral response for h = 0.1. (b) spectral response for different values of κ and for h = 0.1. (c) Contour plot of the spectral response for h = 0.7. (d) Spectral response for different values of κ and for h = 0.7.

Fig. 4
Fig. 4

Spectral response for h = 0.7 and α and κ values maximizing the resonance depth.

Fig. 5
Fig. 5

Resonator phase behavior near the crossing point of two resonance lines having different symmetries. (a) h = 0.1. (b) h = 0.7.

Fig. 6
Fig. 6

Group index dependence on λ for h = 0.1 and h = 0.7 (κ = 0.1 μm−1).

Fig. 7
Fig. 7

Dependence on λ of n g / λ for h = 0.1 and h = 0.7 (κ = 0.1 μm−1).

Fig. 8
Fig. 8

Group index dependence on λ for three values of κ and for h = 0.1.

Fig. 9
Fig. 9

ng versus κ for different values of λ ( = 1.55148 µm, 1.55 µm, 1.551 µm, 1.552 µm).

Equations (14)

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{ d a 1 d ζ = i β a 1 + i κ a 2 d a 2 d ζ = i κ a 1 + i β a 2
{ a 1 ( ζ ) = a 1 ( 0 ) cos ( κ ζ ) exp ( i β ζ α 2 ζ ) + i a 2 ( 0 ) sin ( κ ζ ) exp ( i β ζ α 2 ζ ) a 2 ( ζ ) = i a 1 ( 0 ) sin ( κ ζ ) exp ( i β ζ α 2 ζ ) + a 2 ( 0 ) cos ( κ ζ ) exp ( i β ζ α 2 ζ )
a 2 ( L ) = a 2 ( 0 )
[ E t h r a 1 ( 0 ) ] = [ τ i h i h τ ] [ E i n a 1 ( L ) ]
a 1 ( L ) = a 1 ( 0 ) i h E i n τ
a 1 ( L ) = E i n h i e i β L [ e i β L cos ( κ L ) e L α 2 ] e L α + τ e 2 i β L ( 1 + τ ) e i β L + L α 2 cos ( κ L )
E t h r = τ E i n + i h a 1 ( L )
E t h r E i n = τ + h 2 e i β L [ e i β L cos ( κ L ) e L α 2 ] e L α + τ e 2 i β L ( 1 + τ ) e i β L + L α 2 cos ( κ L )
T ( λ ) = | τ + h 2 e i β L [ e i β L cos ( κ L ) e L α 2 ] e L α + τ e 2 i β L ( 1 + τ ) e i β L + L α 2 cos ( κ L ) | 2
Δ λ ( κ ) = κ λ m 2 π n e f f
Φ = arc tan Im ( E t h r / E i n ) Re ( E t h r / E i n ) = = arc tan e L α ( cos κ L ) e L α 2 ( 1 + τ ) ( cos y + sin y ) + τ ( cos 2 y sin 2 y ) e L α ( cos κ L ) e L α 2 ( 1 + τ ) ( cos y sin y ) + τ ( cos 2 y + sin 2 y )
n g = c v g = c ( β e f f ω ) 1
β e f f ω = 1 L Φ y y ω
n g = c ( β e f f ω ) 1 = c ( c / n e f f ) ( Φ y ) 1 = n e f f Φ y

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