Abstract

A slow light waveguide made of a dielectric slab inserted in a two-dimensional photonic crystal with a negative effective refractive index is proposed and numerically studied. The waveguide may possess modes with zero group velocity, and its frequency varies with the thickness of the waveguide. A linearly tapered left-handed photonic crystal waveguide is also proposed and studied. It is shown that the so-called ‘trapped rainbow’ proposed by Tsakmakidis, Boardman, and Hess [1] is difficult to realize due to a coupling of forward- and backward-propagating modes near zero group velocity. However, different frequency components of a broadband excitation can still be separated through partial accumulation at waveguide sections of different thicknesses.

© 2008 Optical Society of America

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  1. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "??Trapped rainbow?? storage of light in metamaterials," Nature 50, 397-401 (2007).
    [CrossRef]
  2. M. D. Lukin, and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2001).
    [CrossRef] [PubMed]
  3. A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photon. News 14, 44-48 (2003).
    [CrossRef]
  4. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
    [CrossRef] [PubMed]
  5. D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101 (2004).
    [CrossRef]
  6. Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
    [CrossRef] [PubMed]
  7. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67, 057602 (2003).
  8. J. He and S. He, "Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left-handed material substrate," IEEE Microw. Wirel. Compon. Lett. 16, 96-98 (2005).
    [CrossRef]
  9. K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, "Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures," Appl. Phys. Lett. 89, 201103 (2006).
    [CrossRef]
  10. N. Garcia and M. Nieto-Vesperinas, "Left-Handed Materials Do Not Make a Perfect Lens," Phys. Rev. Lett. 88, 207403 (2002).
    [CrossRef] [PubMed]
  11. V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photon. 1, 41-48 (2007).
    [CrossRef]
  12. S. Hughes, L. Ramunno, J. F. Yong, and J. E. Sipe, "Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity," Phys. Rev. Lett. 94, 033903 (2005).
    [CrossRef] [PubMed]
  13. M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refraction-like behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
    [CrossRef]
  14. J. He, J. Yi, and S. He, "Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index," Opt. Express 14, 3024-3029 (2006).
    [CrossRef] [PubMed]
  15. E. Istrate and E. H. Sargent, "Photonic Crystal Waveguide Analysis Using Interface Boundary Conditions," IEEE J. Quantum Electron. 41, 461-467 (2005).
    [CrossRef]
  16. K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
    [CrossRef]

2007 (2)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "??Trapped rainbow?? storage of light in metamaterials," Nature 50, 397-401 (2007).
[CrossRef]

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photon. 1, 41-48 (2007).
[CrossRef]

2006 (2)

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, "Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures," Appl. Phys. Lett. 89, 201103 (2006).
[CrossRef]

J. He, J. Yi, and S. He, "Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index," Opt. Express 14, 3024-3029 (2006).
[CrossRef] [PubMed]

2005 (5)

E. Istrate and E. H. Sargent, "Photonic Crystal Waveguide Analysis Using Interface Boundary Conditions," IEEE J. Quantum Electron. 41, 461-467 (2005).
[CrossRef]

S. Hughes, L. Ramunno, J. F. Yong, and J. E. Sipe, "Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

J. He and S. He, "Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left-handed material substrate," IEEE Microw. Wirel. Compon. Lett. 16, 96-98 (2005).
[CrossRef]

2004 (1)

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101 (2004).
[CrossRef]

2003 (2)

A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photon. News 14, 44-48 (2003).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67, 057602 (2003).

2002 (1)

N. Garcia and M. Nieto-Vesperinas, "Left-Handed Materials Do Not Make a Perfect Lens," Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

2001 (1)

M. D. Lukin, and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2001).
[CrossRef] [PubMed]

2000 (1)

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refraction-like behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

1998 (1)

K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

Amemiya, K.

K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

Aryal, D. P.

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, "Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures," Appl. Phys. Lett. 89, 201103 (2006).
[CrossRef]

Baba, T.

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101 (2004).
[CrossRef]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "??Trapped rainbow?? storage of light in metamaterials," Nature 50, 397-401 (2007).
[CrossRef]

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Engelen, R. J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, "Left-Handed Materials Do Not Make a Perfect Lens," Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

Gersen, H.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Hamann, H. F.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

He, J.

J. He, J. Yi, and S. He, "Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index," Opt. Express 14, 3024-3029 (2006).
[CrossRef] [PubMed]

J. He and S. He, "Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left-handed material substrate," IEEE Microw. Wirel. Compon. Lett. 16, 96-98 (2005).
[CrossRef]

He, S.

J. He, J. Yi, and S. He, "Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index," Opt. Express 14, 3024-3029 (2006).
[CrossRef] [PubMed]

J. He and S. He, "Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left-handed material substrate," IEEE Microw. Wirel. Compon. Lett. 16, 96-98 (2005).
[CrossRef]

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "??Trapped rainbow?? storage of light in metamaterials," Nature 50, 397-401 (2007).
[CrossRef]

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, "Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures," Appl. Phys. Lett. 89, 201103 (2006).
[CrossRef]

Hughes, S.

S. Hughes, L. Ramunno, J. F. Yong, and J. E. Sipe, "Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Imamoglu, A.

M. D. Lukin, and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2001).
[CrossRef] [PubMed]

Istrate, E.

E. Istrate and E. H. Sargent, "Photonic Crystal Waveguide Analysis Using Interface Boundary Conditions," IEEE J. Quantum Electron. 41, 461-467 (2005).
[CrossRef]

Jamois, C.

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, "Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures," Appl. Phys. Lett. 89, 201103 (2006).
[CrossRef]

Karle, T. J.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67, 057602 (2003).

Klaedtke, A.

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, "Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures," Appl. Phys. Lett. 89, 201103 (2006).
[CrossRef]

Korterik, J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Krauss, T. F.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Kuipers, L.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Lukin, M. D.

M. D. Lukin, and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2001).
[CrossRef] [PubMed]

Martinelli, M.

A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photon. News 14, 44-48 (2003).
[CrossRef]

McNab, S. J.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Melloni, A.

A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photon. News 14, 44-48 (2003).
[CrossRef]

Mori, D.

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101 (2004).
[CrossRef]

Morichetti, F.

A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photon. News 14, 44-48 (2003).
[CrossRef]

Nieto-Vesperinas, M.

N. Garcia and M. Nieto-Vesperinas, "Left-Handed Materials Do Not Make a Perfect Lens," Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

Notomi, M.

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refraction-like behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

O'Boyle, M.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Ohtaka, K.

K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

Ramunno, L.

S. Hughes, L. Ramunno, J. F. Yong, and J. E. Sipe, "Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Sargent, E. H.

E. Istrate and E. H. Sargent, "Photonic Crystal Waveguide Analysis Using Interface Boundary Conditions," IEEE J. Quantum Electron. 41, 461-467 (2005).
[CrossRef]

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67, 057602 (2003).

Shalaev, V. M.

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photon. 1, 41-48 (2007).
[CrossRef]

Sipe, J. E.

S. Hughes, L. Ramunno, J. F. Yong, and J. E. Sipe, "Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67, 057602 (2003).

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "??Trapped rainbow?? storage of light in metamaterials," Nature 50, 397-401 (2007).
[CrossRef]

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, "Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures," Appl. Phys. Lett. 89, 201103 (2006).
[CrossRef]

Ueta, T.

K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

van Hulst, N. F.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Vlasov, Y. A.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Yi, J.

J. He, J. Yi, and S. He, "Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index," Opt. Express 14, 3024-3029 (2006).
[CrossRef] [PubMed]

Yong, J. F.

S. Hughes, L. Ramunno, J. F. Yong, and J. E. Sipe, "Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101 (2004).
[CrossRef]

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, "Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures," Appl. Phys. Lett. 89, 201103 (2006).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. Istrate and E. H. Sargent, "Photonic Crystal Waveguide Analysis Using Interface Boundary Conditions," IEEE J. Quantum Electron. 41, 461-467 (2005).
[CrossRef]

IEEE Microw. Wirel. Compon. Lett. (1)

J. He and S. He, "Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left-handed material substrate," IEEE Microw. Wirel. Compon. Lett. 16, 96-98 (2005).
[CrossRef]

Nat. Photon. (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photon. 1, 41-48 (2007).
[CrossRef]

Nature (3)

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "??Trapped rainbow?? storage of light in metamaterials," Nature 50, 397-401 (2007).
[CrossRef]

M. D. Lukin, and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2001).
[CrossRef] [PubMed]

Opt. Express (1)

J. He, J. Yi, and S. He, "Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index," Opt. Express 14, 3024-3029 (2006).
[CrossRef] [PubMed]

Opt. Photon. News (1)

A. Melloni, F. Morichetti, and M. Martinelli, "Optical slow wave structures," Opt. Photon. News 14, 44-48 (2003).
[CrossRef]

Phys. Rev. B (2)

K. Ohtaka, T. Ueta, and K. Amemiya, "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys. Rev. B 57, 2550-2568 (1998).
[CrossRef]

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refraction-like behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

Phys. Rev. E (1)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67, 057602 (2003).

Phys. Rev. Lett. (3)

N. Garcia and M. Nieto-Vesperinas, "Left-Handed Materials Do Not Make a Perfect Lens," Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in Photonic Crystal Waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

S. Hughes, L. Ramunno, J. F. Yong, and J. E. Sipe, "Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the proposed left-handed photonic crystal waveguide.

Fig. 2.
Fig. 2.

(a). Dispersion curves for oscillatory modes in LHPC waveguides of different thickness d. The total thickness of the waveguide is fixed as d+2w =2.464a. The two dotted lines correspond to the light line in the PC and the light line in the core medium. Backward modes and forward modes of A2 are denoted by Mb and M f , respectively. Circle point PL indicates the low cut-off frequency of mode A2, and asterisk point PH indicates the high cut-off frequency of backward mode on A2. (b)-(d) Distributions of electric field for modes A2, A1 and A0, respectively, when d=1.9a.

Fig. 3
Fig. 3

(a) Group velocities for modes A2 for the four waveguides in Fig. 2(a). Circle point PL corresponds to the low cut-off frequency. Modes at PL have a zero group velocity, and are denoted by Ms. Backward modes and forward modes are denoted by Mb and M f , respectively. (b)-(d) Light intensity patterns of modes (b) Mb, (c) M f and (d) Ms for the LHPC waveguide with d= 1.9a.

Fig. 4.
Fig. 4.

(a). Schematic diagram of a tapered LHPC structure with length L=31a. The width of the dielectric core is d=1.8a at entrance P a and d=2.0a at exit P b . The light is launched from the uniform input waveguide W1. (b) Spatial distributions of light intensity for a pulse propagating in the tapered LHPC structure. Time interval for every two frames is Δt=40a/c.

Fig. 5.
Fig. 5.

Spatial field distributions in the tapered structure for light launched from W1 at normalized frequency of (a) f=0.312(2πc/a), (b) f=0.314(2πc/a), (c) f=0.316(2πc/a) and (d) f=0.318(2πc/a).

Equations (1)

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r 1 r 2 e 2 i κ x d = 1 ,

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