Abstract

We design compact (a few wavelength long) and efficient (>99%) injectors for coupling light into slow Bloch modes of periodic thin film stacks and of periodic slab waveguides. The study includes the derivation of closed-form expressions for the injection efficiency as a function of the group-velocity of injected light, and the proof that 100% coupling efficiencies for arbitrary small group velocities is possible with an injector length scaling as log(c/vg). The trade-off between the injector bandwidth and the group velocity of the injected light is also considered.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (July 1997).
    [CrossRef]
  2. J. Khurgin, "Expanding the bandwidth of slow-light photonic devices based on coupled resonators," Opt. Lett. 30, 2778-2780 (2005).
    [CrossRef] [PubMed]
  3. J. Poon, L. Zhu, G. DeRose, and A. Yariv, "Transmission and group delay of microring coupled-resonator optical waveguides," Opt. Lett. 31, 456-458 (2006).
    [CrossRef] [PubMed]
  4. M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004).
    [CrossRef] [PubMed]
  5. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi and T. Tanabe "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
    [CrossRef] [PubMed]
  6. A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004).
    [CrossRef]
  7. D. Mori and T. Baba, "Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide," Opt. Express 13, 9398-9408 (2005).
    [CrossRef] [PubMed]
  8. T. F. Krauss, "Photonic Crystals shine on," Phys. World 32-36, (February 2006).
  9. A. Melloni, F. Morichetti, and M. Martinelli, "Linear and Nonlinear propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365-379 (2003).
    [CrossRef]
  10. G. Lenz, B. J. Eggleton, C. K. Madsen, and R.E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electronics 37, 525-532 (2001).
    [CrossRef]
  11. Y. A. Vlasov and S. J. McNab, "Coupling into the slow light mode in slab-type photonic crystal waveguides," Opt. Lett. 31, 50-52 (2006).
    [CrossRef] [PubMed]
  12. M. Povinelli, S. Johnson, and J. Joannopoulos, "Slow-light, band-edge waveguides for tuneable time delays," Opt. Express 13, 7145-7159 (2005).
    [CrossRef] [PubMed]
  13. J. P. Hugonin and P. Lalanne, "Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization," J. Opt. Soc. Am. A. 22, 1844-1849 (2005).
    [CrossRef]
  14. P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York 1988).
  15. J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure," Phys. Rev. E 53, 4107-4121 (1996).
    [CrossRef]
  16. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall International, London, 1984).
  17. K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, Berlin 2001) Chap. 11.
  18. B. Momeni and A. Adibi, "Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal," Appl. Phys. Lett. 87, 171104 (2005).
    [CrossRef]
  19. L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (J. Wiley and Sons, New York, 1995).
  20. C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, "Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities," Opt. Exp. 13, 245-255 (2005).
    [CrossRef]
  21. P. Lalanne and J. P. Hugonin, "Bloch-wave engineering for high Q’s, small V’s microcavities," IEEE J. Quantum Electron. 39, 1430-1438 (2003).
    [CrossRef]
  22. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM J. Optim. 9,112-147 (1998).
    [CrossRef]
  23. D. Gerace and L. C. Andreani, "Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs," Photonics and Nanostructures-fundamentals and applications 3, 120-128 (2005).
    [CrossRef]
  24. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005).
    [CrossRef]
  25. M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, "Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth," Opt. Express 15, 219-226 (2007).
    [CrossRef] [PubMed]
  26. H. A. Macleod, Thin-film optical filters, (Adam Hilger LTD, London 1969).
  27. R. E. Collin, Field theory of guided waves, (London, Section 9 1960).

2007

2006

2005

J. P. Hugonin and P. Lalanne, "Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization," J. Opt. Soc. Am. A. 22, 1844-1849 (2005).
[CrossRef]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005).
[CrossRef]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi and T. Tanabe "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

M. Povinelli, S. Johnson, and J. Joannopoulos, "Slow-light, band-edge waveguides for tuneable time delays," Opt. Express 13, 7145-7159 (2005).
[CrossRef] [PubMed]

J. Khurgin, "Expanding the bandwidth of slow-light photonic devices based on coupled resonators," Opt. Lett. 30, 2778-2780 (2005).
[CrossRef] [PubMed]

D. Mori and T. Baba, "Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide," Opt. Express 13, 9398-9408 (2005).
[CrossRef] [PubMed]

B. Momeni and A. Adibi, "Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal," Appl. Phys. Lett. 87, 171104 (2005).
[CrossRef]

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, "Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities," Opt. Exp. 13, 245-255 (2005).
[CrossRef]

2004

M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004).
[CrossRef]

2003

A. Melloni, F. Morichetti, and M. Martinelli, "Linear and Nonlinear propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365-379 (2003).
[CrossRef]

P. Lalanne and J. P. Hugonin, "Bloch-wave engineering for high Q’s, small V’s microcavities," IEEE J. Quantum Electron. 39, 1430-1438 (2003).
[CrossRef]

2001

G. Lenz, B. J. Eggleton, C. K. Madsen, and R.E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electronics 37, 525-532 (2001).
[CrossRef]

1998

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM J. Optim. 9,112-147 (1998).
[CrossRef]

1997

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (July 1997).
[CrossRef]

1996

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure," Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Adibi, A.

B. Momeni and A. Adibi, "Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal," Appl. Phys. Lett. 87, 171104 (2005).
[CrossRef]

Baba, T.

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure," Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

DeRose, G.

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure," Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R.E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electronics 37, 525-532 (2001).
[CrossRef]

Eich, M.

A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004).
[CrossRef]

Engelen, R. J. P.

Fan, S.

M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

Harris, S. E.

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (July 1997).
[CrossRef]

Hughes, S.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Hugonin, J. P.

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, "Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities," Opt. Exp. 13, 245-255 (2005).
[CrossRef]

J. P. Hugonin and P. Lalanne, "Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization," J. Opt. Soc. Am. A. 22, 1844-1849 (2005).
[CrossRef]

P. Lalanne and J. P. Hugonin, "Bloch-wave engineering for high Q’s, small V’s microcavities," IEEE J. Quantum Electron. 39, 1430-1438 (2003).
[CrossRef]

Joannopoulos, J.

Johnson, S.

Khurgin, J.

Kira, G.

Krauss, T. F.

Kuipers, L.

Kuramochi, E.

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi and T. Tanabe "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM J. Optim. 9,112-147 (1998).
[CrossRef]

Lalanne, P.

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, "Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities," Opt. Exp. 13, 245-255 (2005).
[CrossRef]

J. P. Hugonin and P. Lalanne, "Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization," J. Opt. Soc. Am. A. 22, 1844-1849 (2005).
[CrossRef]

P. Lalanne and J. P. Hugonin, "Bloch-wave engineering for high Q’s, small V’s microcavities," IEEE J. Quantum Electron. 39, 1430-1438 (2003).
[CrossRef]

Lecamp, G.

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, "Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities," Opt. Exp. 13, 245-255 (2005).
[CrossRef]

Lenz, G.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R.E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electronics 37, 525-532 (2001).
[CrossRef]

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R.E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electronics 37, 525-532 (2001).
[CrossRef]

Martinelli, M.

A. Melloni, F. Morichetti, and M. Martinelli, "Linear and Nonlinear propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365-379 (2003).
[CrossRef]

McNab, S. J.

Melloni, A.

A. Melloni, F. Morichetti, and M. Martinelli, "Linear and Nonlinear propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365-379 (2003).
[CrossRef]

Michaeli, A.

Mitsugi, S.

Momeni, B.

B. Momeni and A. Adibi, "Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal," Appl. Phys. Lett. 87, 171104 (2005).
[CrossRef]

Mori, D.

Morichetti, F.

A. Melloni, F. Morichetti, and M. Martinelli, "Linear and Nonlinear propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365-379 (2003).
[CrossRef]

Notomi, M.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005).
[CrossRef]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi and T. Tanabe "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

Petrov, A. Yu.

A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004).
[CrossRef]

Poon, J.

Povinelli, M.

Ramunno, L.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM J. Optim. 9,112-147 (1998).
[CrossRef]

Salib, M.

Sauvan, C.

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, "Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities," Opt. Exp. 13, 245-255 (2005).
[CrossRef]

Scalora, M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure," Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Settle, M. D.

Shinya, A.

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi and T. Tanabe "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Slusher, R.E.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R.E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electronics 37, 525-532 (2001).
[CrossRef]

Tanabe, T.

Vlasov, Y. A.

Watanabe, T.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM J. Optim. 9,112-147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM J. Optim. 9,112-147 (1998).
[CrossRef]

Yanik, M. F.

M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

Yariv, A.

Zhu, L.

Appl. Phys. Lett.

B. Momeni and A. Adibi, "Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal," Appl. Phys. Lett. 87, 171104 (2005).
[CrossRef]

A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004).
[CrossRef]

IEEE J. Quantum Electron.

P. Lalanne and J. P. Hugonin, "Bloch-wave engineering for high Q’s, small V’s microcavities," IEEE J. Quantum Electron. 39, 1430-1438 (2003).
[CrossRef]

IEEE J. Quantum Electronics

G. Lenz, B. J. Eggleton, C. K. Madsen, and R.E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electronics 37, 525-532 (2001).
[CrossRef]

J. Opt. Soc. Am. A.

J. P. Hugonin and P. Lalanne, "Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization," J. Opt. Soc. Am. A. 22, 1844-1849 (2005).
[CrossRef]

Opt. Exp.

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, "Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities," Opt. Exp. 13, 245-255 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

A. Melloni, F. Morichetti, and M. Martinelli, "Linear and Nonlinear propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365-379 (2003).
[CrossRef]

Phys. Rev. B

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005).
[CrossRef]

Phys. Rev. E

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure," Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Phys. Rev. Lett.

M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

Phys. Today

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (July 1997).
[CrossRef]

SIAM J. Optim.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM J. Optim. 9,112-147 (1998).
[CrossRef]

Other

D. Gerace and L. C. Andreani, "Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs," Photonics and Nanostructures-fundamentals and applications 3, 120-128 (2005).
[CrossRef]

L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (J. Wiley and Sons, New York, 1995).

H. A. Macleod, Thin-film optical filters, (Adam Hilger LTD, London 1969).

R. E. Collin, Field theory of guided waves, (London, Section 9 1960).

T. F. Krauss, "Photonic Crystals shine on," Phys. World 32-36, (February 2006).

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall International, London, 1984).

K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, Berlin 2001) Chap. 11.

P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Impedance mismatch problem. (a): Scattering at an interface between a uniform medium (refractive index n) and a semi-infinite bi-layer periodic stack (refractive indices nH and nL). (b) and (c): Injection efficiency T as a function of the group-velocity of the periodic-stack Bloch mode for different fill factors (0.1<f<0.9). Colormaps and white curves are exact numerical results obtained with the 2×2 transfer-matrix method in [14], and the superimposed solid-red curves are obtained using the approximate closed-form expressions. (b): data obtained in the vicinity of the valence band edge of a weak-modulation (nH=1.5 and nL=1.4) stack for n=nH. (c): data obtained in the vicinity of the conduction band edge of a strong-modulation stack (nH=3.495 and nL=1) for n=nL.

Fig. 2.
Fig. 2.

A slow-mode injector at a single frequency is basically a mirror. (a) Definition of the injector parameters : length L , interface S1 between the coupler and the uniform medium and interface S2 between the coupler and the periodic stack supporting a slow Bloch mode. (b) Perfect injector at a single frequency for a plane wave incident from the uniform medium. (c) Reciprocal problem. The incident illumination is the reciprocal Bloch mode propagating towards the negative z-direction. (d) General solution of the synthesis problem : the injector is composed of a mirror and of a phase plate.

Fig. 3.
Fig. 3.

Coupling efficiency into the slow-mode of a periodic stack (nH, nL) as a function of the group velocity (log scale) of the slow mode. Different injectors are considered. They are all composed of Bragg mirrors (nH, nL) with an increasing number of repeated pairs, m=1, …5. The results are obtained in the vicinity of the valence band edge of the periodic stack. The horizontal arrows indicate the bandwidths in GHz of the different injectors for T=0.8 (-1dB).

Fig. 4.
Fig. 4.

Broadband injection from a planar waveguide to a periodic waveguide near the valence-band edge (λ=1.65 μm). (a) Injector geometry optimized for coupling at vg/c≈0.01. From left to right, the injector slit- and ridge-widths are 84, 143,127, 158, 174 nm and 239, 213, 173, 166, 166 nm, respectively. The superimposed red curve represents the squared modulus of the transverse electric field at optimal coupling. (b) Performance of the injector predicted by fully vectorial computational results for the radiation loss L (dashed curve), the modal reflection R (blue circles) and the transmission T (actually 1-T is shown with a solid black curve). All quantities are displayed in a log scale. Inset: Injection efficiency T as a function of vg with (red) and without (blue) injector. The maximum injection efficiency is as large as 0.999. In the absence of injector, this efficiency is only 0.25%.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

B + ( z = z 0 + p a ) > = A H P + > + B H P > ,
0 < u < 1 ,
1 < u < 0 ,
B ( z = z 0 + p a ) > = A ' H P + > + B ' H P > ,
r = [ ( n n H ) + u ( n + n H ) exp ( ) ] [ ( n + n H ) + u ( n n H ) exp ( ) ] ,
( v g c ) [ α ( 1 + u 2 ) + 2 β u ] = 2 n H ( 1 u 2 ) ,
u VB = 1 ( α + β VB 2 n H ) v g c + O ( v g c ) 2 , and
u CB = 1 + ( α β CB 2 n H ) v g c + O ( v g c ) 2 ,
ϕ ( 0 ) πf n H [ 1 ± ε 1 ( 2 ε 0 ) ] ε 0 1 2 ,
T VB = ( α + β VB n H ) v g c , T CB = ( α β CB n H ) v g c .
L = log ( c v g ) ,

Metrics