Abstract

We propose and describe a new class of optical modes consisting of superposition of three waveguide modes which can be supported by a few-mode waveguide spatially modulated by two co-spatial gratings. These supermodes bear a close, but not exact, formal analogy to the three-level quantum states involved in EIT and its attendant slow light propagation characteristics. Of particular interest is the supermode which we call the dark mode in which, in analogy with the dark state of EIT, one of the three uncoupled waveguide modes is not excited. This mode has unique dispersion characteristics that translate into a slow light propagation which possesses high bandwidth-delay product and can form the basis for a new generation of optical resonators and lasers.

© 2009 Optical Society of America

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References

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  1. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50(7), 36-42 (1997).
    [CrossRef]
  2. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397(6720), 594-598 (1999).
    [CrossRef]
  3. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409(6819), 490-493 (2001).
    [CrossRef]
  4. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A 69(6), 063804 (2004).
    [CrossRef]
  5. L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonators," Opt. Lett. 29(6), 626-628 (2004).
    [CrossRef]
  6. 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(23), 233903 (2004).
    [CrossRef]
  7. A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71(4), 043804 (2005).
    [CrossRef]
  8. Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96(12), 123901 (2006).
    [CrossRef]
  9. Q. F. Xu, J. Shakya, and M. Lipson, "Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency," Opt. Express 14(14), 6463-6468 (2006).
    [CrossRef]
  10. K. Totsuka, N. Kobayashi, and M. Tomita, "Slow light in coupled-resonator-induced transparency," Phys. Rev. Lett. 98(21), 213904 (2007).
    [CrossRef]
  11. L. Yosef Mario and M. K. Chin, "Optical buffer with higher delay-bandwidth product in a two-ring system," Opt. Express 16(3), 1796-1807 (2008).
    [CrossRef]
  12. Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, "Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism," IEEE J. Quantum Electron. 44(11), 1065-1070 (2008).
    [CrossRef]
  13. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Express, New York, 2007).
  14. E. Peral and A. Yariv, "Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering," J. Lightwave Technol. 17(5), 942-947 (1999).
    [CrossRef]
  15. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York and London, 1974).
  16. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955).
  17. X. Sun, H.-C. Liu, and A. Yariv, "Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system," Opt. Lett. 34(3), 280-282 (2009).
    [CrossRef]

2009 (1)

2008 (2)

L. Yosef Mario and M. K. Chin, "Optical buffer with higher delay-bandwidth product in a two-ring system," Opt. Express 16(3), 1796-1807 (2008).
[CrossRef]

Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, "Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism," IEEE J. Quantum Electron. 44(11), 1065-1070 (2008).
[CrossRef]

2007 (1)

K. Totsuka, N. Kobayashi, and M. Tomita, "Slow light in coupled-resonator-induced transparency," Phys. Rev. Lett. 98(21), 213904 (2007).
[CrossRef]

2006 (2)

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96(12), 123901 (2006).
[CrossRef]

Q. F. Xu, J. Shakya, and M. Lipson, "Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency," Opt. Express 14(14), 6463-6468 (2006).
[CrossRef]

2005 (1)

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

2004 (3)

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A 69(6), 063804 (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(23), 233903 (2004).
[CrossRef]

L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonators," Opt. Lett. 29(6), 626-628 (2004).
[CrossRef]

2001 (1)

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409(6819), 490-493 (2001).
[CrossRef]

1999 (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397(6720), 594-598 (1999).
[CrossRef]

E. Peral and A. Yariv, "Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering," J. Lightwave Technol. 17(5), 942-947 (1999).
[CrossRef]

1997 (1)

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50(7), 36-42 (1997).
[CrossRef]

Behroozi, C. H.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409(6819), 490-493 (2001).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397(6720), 594-598 (1999).
[CrossRef]

Boyd, R. W.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Chang, H.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Chin, M. K.

Dong, C. H.

Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, "Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism," IEEE J. Quantum Electron. 44(11), 1065-1070 (2008).
[CrossRef]

Dutton, Z.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409(6819), 490-493 (2001).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397(6720), 594-598 (1999).
[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(23), 233903 (2004).
[CrossRef]

Fan, S. H.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96(12), 123901 (2006).
[CrossRef]

Farca, G.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

Fuller, K. A.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397(6720), 594-598 (1999).
[CrossRef]

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50(7), 36-42 (1997).
[CrossRef]

Hau, L. V.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409(6819), 490-493 (2001).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397(6720), 594-598 (1999).
[CrossRef]

Ilchenko, V. S.

Jiang, X.

Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, "Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism," IEEE J. Quantum Electron. 44(11), 1065-1070 (2008).
[CrossRef]

Kobayashi, N.

K. Totsuka, N. Kobayashi, and M. Tomita, "Slow light in coupled-resonator-induced transparency," Phys. Rev. Lett. 98(21), 213904 (2007).
[CrossRef]

Lipson, M.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96(12), 123901 (2006).
[CrossRef]

Q. F. Xu, J. Shakya, and M. Lipson, "Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency," Opt. Express 14(14), 6463-6468 (2006).
[CrossRef]

Liu, C.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409(6819), 490-493 (2001).
[CrossRef]

Liu, H.-C.

Maleki, L.

Matsko, A. B.

Min, B. K.

Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, "Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism," IEEE J. Quantum Electron. 44(11), 1065-1070 (2008).
[CrossRef]

Naweed, A.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

Peral, E.

Povinelli, M. L.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96(12), 123901 (2006).
[CrossRef]

Rosenberger, A. T.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Sandhu, S.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96(12), 123901 (2006).
[CrossRef]

Savchenkov, A. A.

Shakya, J.

Q. F. Xu, J. Shakya, and M. Lipson, "Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency," Opt. Express 14(14), 6463-6468 (2006).
[CrossRef]

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96(12), 123901 (2006).
[CrossRef]

Shopova, S. I.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

Smith, D. D.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A 69(6), 063804 (2004).
[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(23), 233903 (2004).
[CrossRef]

Sun, X.

Tomita, M.

K. Totsuka, N. Kobayashi, and M. Tomita, "Slow light in coupled-resonator-induced transparency," Phys. Rev. Lett. 98(21), 213904 (2007).
[CrossRef]

Totsuka, K.

K. Totsuka, N. Kobayashi, and M. Tomita, "Slow light in coupled-resonator-induced transparency," Phys. Rev. Lett. 98(21), 213904 (2007).
[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(23), 233903 (2004).
[CrossRef]

Xiao, Y. F.

Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, "Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism," IEEE J. Quantum Electron. 44(11), 1065-1070 (2008).
[CrossRef]

Xu, Q. F.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96(12), 123901 (2006).
[CrossRef]

Q. F. Xu, J. Shakya, and M. Lipson, "Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency," Opt. Express 14(14), 6463-6468 (2006).
[CrossRef]

Yang, L.

Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, "Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism," IEEE J. Quantum Electron. 44(11), 1065-1070 (2008).
[CrossRef]

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(23), 233903 (2004).
[CrossRef]

Yariv, A.

Yosef Mario, L.

IEEE J. Quantum Electron. (1)

Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, "Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism," IEEE J. Quantum Electron. 44(11), 1065-1070 (2008).
[CrossRef]

J. Lightwave Technol. (1)

Nature (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397(6720), 594-598 (1999).
[CrossRef]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409(6819), 490-493 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (2)

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71(4), 043804 (2005).
[CrossRef]

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A 69(6), 063804 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

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(23), 233903 (2004).
[CrossRef]

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96(12), 123901 (2006).
[CrossRef]

K. Totsuka, N. Kobayashi, and M. Tomita, "Slow light in coupled-resonator-induced transparency," Phys. Rev. Lett. 98(21), 213904 (2007).
[CrossRef]

Phys. Today (1)

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50(7), 36-42 (1997).
[CrossRef]

Other (3)

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Express, New York, 2007).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York and London, 1974).

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955).

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Figures (6)

Fig. 1.
Fig. 1.

Four configurations of the directions of the three modes. The black grating (/) couples modes A and B, while the red grating (\) couples mode B and C. The gratings are short-period or long-period depending on the directions of the connected modes.

Fig. 2.
Fig. 2.

(a) Band structures of a GIT waveguide. na =1.5, nb =2.5, nc =2. κab =90/m, κbc=100/m. Dash lines are the band structure without grating perturbation. (b) Zoom-in figure of the bending region. The red dashed curve is the dark mode.

Fig. 3.
Fig. 3.

(a) Transmission spectrum of a uniform structure. κab =900/m, κbc =1,000/m, and L=Lmin =1.44 cm. The refractive indices are na =1.45, nb =1.425, and nc =1.4. (b) Transmission spectrum in a narrower span of the same structure in (a). (c) Transmission spectrum of a uniform structure with L=6L min . (d) Transmission spectrum of a uniform structure with L=6 Lmin and periodic inversion of κab . (e) Group delay of the structure in (d).

Fig. 4.
Fig. 4.

Energy distribution of a uniform structure with L=2Lmin and an input A(0)=1. κab =900/m, κbc =1,000/m, and L=2.88 cm. The refractive indices are na =1.45, nb =1.425, and nc =1.4.

Fig. 5.
Fig. 5.

(a) κab and κbc in a uniform structure with periodic inversion. (b) κab and κbc in an adiabatic structure. Lad and Luni are the lengths of the adiabatic and uniform region, respectively.

Fig. 6.
Fig. 6.

(a) Transmission and (b) group delay of an adiabatic structure. Lad =3 cm, Luni =4 cm, κbc =4,000/m, and α max=0.9. The refractive indices are na =1.45, nb =1.425, and nc =1.4.

Equations (20)

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

dadz=jβaa+κabexp(j2πΛabz)b
dbdz=jβbb+κba+exp(j2πΛabz)a+κbcexp(j2πΛcbz)c,
dcdz=jβcc+κcbexp(j2πΛcbz)b
ddz[ABC]=[jδaκab0κbajδbκbc0κcbjδc]·[ABC],
ddt=Ψ=[jΔωj2Ωp0j2Ωp0j2Ωc0j2Ωc0]|Ψ,
ddz[ABC]=[jδaκab0κabjδbjδbc0jκbcjδc]·[ABC]K· [ABC] ,
1κab2+κbc2 [κbc0jκab] , eigenvalue=0 ;
[±κbc2κab2κbcjκab],eigenvalues=± j κbc2κab2 ± j Δ β0 .
vg,dark=1α2na+α2nc.c,
A(z)=11α2[1α2cos(Δβ0z)]
B(z)=α1α2sin(Δβ0z),
C(z)=jα1α2[1cos(Δβ0z)]
Q=βaL1+2α2(1α2)2,
vdark=11α2[10jα],vbright1,2=12(1α2)[1α21jα] .
x(z)=i=dark,brightai(z)vi(z)exp[j0zβi(z)dz].
i=dark,bright[(zai)vi+ai(zvi)]exp[j0zβi(z)dz]=0 .
dabright1/dz=ubright1T(zvdark)exp[j0zβbright1(z)dz],
abright1(z)=2jκbc(1α2)32αzsin(βbrigth12z)exp[jβbright12z]
dαdz κbc4 (1α2)32 ε .
α(z)=sin1 (tan(az)),

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