Abstract

We present a design of a one dimensional dielectric waveguide that can trap a broadband light pulse with different frequency component stored at different positions, effectively forming a “trapped rainbow” [Nature 450, 397 (2007)]. The spectrum of the rainbow covers the whole visible range. To do this, we first show that the dispersion of a SiO2 waveguide with a Si grating placed on top can be engineered by the design parameter of the grating. Specifically, guided modes with zero group velocity(frozen modes) can be realized. Negative Goos-Hänchen shift along the surface of the grating is responsible for such a dispersion control. The frequency of the frozen mode is tuned by changing the lateral feature parameters (period and duty cycle) of the grating. By tuning the grating feature point by point along the waveguide, a light pulse can be trapped with different frequency components frozen at different positions, so that a “rainbow” is formed. The device is expected to have extremely low ohmic loss because only dielectric materials are used. A planar geometry also promises much reduced fabrication difficulty.

© 2015 Optical Society of America

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References

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  1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
    [Crossref]
  2. P. S. Ma, Y. E. Kwon, and Y. Y. Kim, “Wave dispersion tailoring in an elastic waveguide by phononic crystals,” Appl. Phys. Lett. 103, 151901 (2013).
    [Crossref]
  3. 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]
  4. S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
    [Crossref]
  5. P. Colman, S. Combriè, G. Lehoucq, and A. De Rossi, “Control of dispersion in photonic crystal waveguides using group symmetry theory,” Opt. Express 20, 13108–13114 (2012).
    [Crossref] [PubMed]
  6. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
    [Crossref] [PubMed]
  7. E. Kirby, J. Hamm, T. Pickering, K. Tsakmakidis, and O. Hess, “Evanescent gain for slow and stopped light in negative refractive index heterostructures,” Phys. Rev. B 84, 041103(R) (2011).
    [Crossref]
  8. Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108, 5169–5173 (2011).
    [Crossref] [PubMed]
  9. Q. Gan and F. J. Bartoli, “Surface dispersion engineering of planar plasmonic chirped grating for complete visible rainbow trapping,” Appl. Phys. Lett. 98, 251103 (2011).
    [Crossref]
  10. V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).
    [Crossref]
  11. F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Berlin) 436, 333–346 (1947).
    [Crossref]
  12. T. Tamir and E. Garmire, Integrated Optics (Springer-Verlag, 1979).
  13. R. Yang, W. Zhu, and J. Li, “Giant positive and negative goos-hänchen shift on dielectric gratings caused by guided mode resonance,” Opt. Express 22, 2043–2050 (2014).
    [Crossref] [PubMed]
  14. R. Renard, “Total reflection: A new evaluation of the Goos-Hänchen shift,” J. Opt. Soc. Am. 54, 1190–1196 (1964).
    [Crossref]
  15. http://www.fourmilab.ch/documents/specrend/ .
  16. D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photon. 4, 466–470 (2010).
    [Crossref]

2014 (1)

2013 (1)

P. S. Ma, Y. E. Kwon, and Y. Y. Kim, “Wave dispersion tailoring in an elastic waveguide by phononic crystals,” Appl. Phys. Lett. 103, 151901 (2013).
[Crossref]

2012 (1)

2011 (3)

E. Kirby, J. Hamm, T. Pickering, K. Tsakmakidis, and O. Hess, “Evanescent gain for slow and stopped light in negative refractive index heterostructures,” Phys. Rev. B 84, 041103(R) (2011).
[Crossref]

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108, 5169–5173 (2011).
[Crossref] [PubMed]

Q. Gan and F. J. Bartoli, “Surface dispersion engineering of planar plasmonic chirped grating for complete visible rainbow trapping,” Appl. Phys. Lett. 98, 251103 (2011).
[Crossref]

2010 (3)

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).
[Crossref]

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[Crossref]

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photon. 4, 466–470 (2010).
[Crossref]

2008 (1)

2007 (1)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

1964 (1)

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Berlin) 436, 333–346 (1947).
[Crossref]

Bartoli, F. J.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108, 5169–5173 (2011).
[Crossref] [PubMed]

Q. Gan and F. J. Bartoli, “Surface dispersion engineering of planar plasmonic chirped grating for complete visible rainbow trapping,” Appl. Phys. Lett. 98, 251103 (2011).
[Crossref]

Beausoleil, R.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photon. 4, 466–470 (2010).
[Crossref]

Beggs, D. M.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[Crossref]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

Colman, P.

Combriè, S.

De Rossi, A.

Ding, Y. J.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108, 5169–5173 (2011).
[Crossref] [PubMed]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Fattal, D.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photon. 4, 466–470 (2010).
[Crossref]

Fiorentino, M.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photon. 4, 466–470 (2010).
[Crossref]

Gan, Q.

Q. Gan and F. J. Bartoli, “Surface dispersion engineering of planar plasmonic chirped grating for complete visible rainbow trapping,” Appl. Phys. Lett. 98, 251103 (2011).
[Crossref]

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108, 5169–5173 (2011).
[Crossref] [PubMed]

Gao, Y.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108, 5169–5173 (2011).
[Crossref] [PubMed]

Garmire, E.

T. Tamir and E. Garmire, Integrated Optics (Springer-Verlag, 1979).

Gomez-Iglesias, A.

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Berlin) 436, 333–346 (1947).
[Crossref]

Hamm, J.

E. Kirby, J. Hamm, T. Pickering, K. Tsakmakidis, and O. Hess, “Evanescent gain for slow and stopped light in negative refractive index heterostructures,” Phys. Rev. B 84, 041103(R) (2011).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Berlin) 436, 333–346 (1947).
[Crossref]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Hess, O.

E. Kirby, J. Hamm, T. Pickering, K. Tsakmakidis, and O. Hess, “Evanescent gain for slow and stopped light in negative refractive index heterostructures,” Phys. Rev. B 84, 041103(R) (2011).
[Crossref]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

Kildishev, A. V.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).
[Crossref]

Kim, Y. Y.

P. S. Ma, Y. E. Kwon, and Y. Y. Kim, “Wave dispersion tailoring in an elastic waveguide by phononic crystals,” Appl. Phys. Lett. 103, 151901 (2013).
[Crossref]

Kirby, E.

E. Kirby, J. Hamm, T. Pickering, K. Tsakmakidis, and O. Hess, “Evanescent gain for slow and stopped light in negative refractive index heterostructures,” Phys. Rev. B 84, 041103(R) (2011).
[Crossref]

Krauss, T. F.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[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]

Kwon, Y. E.

P. S. Ma, Y. E. Kwon, and Y. Y. Kim, “Wave dispersion tailoring in an elastic waveguide by phononic crystals,” Appl. Phys. Lett. 103, 151901 (2013).
[Crossref]

Lehoucq, G.

Li, J.

Ma, P. S.

P. S. Ma, Y. E. Kwon, and Y. Y. Kim, “Wave dispersion tailoring in an elastic waveguide by phononic crystals,” Appl. Phys. Lett. 103, 151901 (2013).
[Crossref]

Melloni, A.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[Crossref]

O’Faolain, L.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[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]

Peng, Z.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photon. 4, 466–470 (2010).
[Crossref]

Pickering, T.

E. Kirby, J. Hamm, T. Pickering, K. Tsakmakidis, and O. Hess, “Evanescent gain for slow and stopped light in negative refractive index heterostructures,” Phys. Rev. B 84, 041103(R) (2011).
[Crossref]

Renard, R.

Schulz, S. A.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[Crossref]

Shalaev, V. M.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).
[Crossref]

Smolyaninov, I. I.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).
[Crossref]

Smolyaninova, V. N.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).
[Crossref]

Tamir, T.

T. Tamir and E. Garmire, Integrated Optics (Springer-Verlag, 1979).

Tsakmakidis, K.

E. Kirby, J. Hamm, T. Pickering, K. Tsakmakidis, and O. Hess, “Evanescent gain for slow and stopped light in negative refractive index heterostructures,” Phys. Rev. B 84, 041103(R) (2011).
[Crossref]

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

Vezenov, D.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108, 5169–5173 (2011).
[Crossref] [PubMed]

Wagner, K.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108, 5169–5173 (2011).
[Crossref] [PubMed]

White, T. P.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[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]

Yang, R.

Zhu, W.

Ann. Phys. (Berlin) (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Berlin) 436, 333–346 (1947).
[Crossref]

Appl. Phys. Lett. (3)

P. S. Ma, Y. E. Kwon, and Y. Y. Kim, “Wave dispersion tailoring in an elastic waveguide by phononic crystals,” Appl. Phys. Lett. 103, 151901 (2013).
[Crossref]

Q. Gan and F. J. Bartoli, “Surface dispersion engineering of planar plasmonic chirped grating for complete visible rainbow trapping,” Appl. Phys. Lett. 98, 251103 (2011).
[Crossref]

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).
[Crossref]

J. Opt. (1)

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[Crossref]

J. Opt. Soc. Am. (1)

Nature (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

Nature Photon. (1)

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photon. 4, 466–470 (2010).
[Crossref]

Opt. Express (3)

Phys. Rev. B (1)

E. Kirby, J. Hamm, T. Pickering, K. Tsakmakidis, and O. Hess, “Evanescent gain for slow and stopped light in negative refractive index heterostructures,” Phys. Rev. B 84, 041103(R) (2011).
[Crossref]

Proc. Natl. Acad. Sci. U.S.A. (1)

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108, 5169–5173 (2011).
[Crossref] [PubMed]

Other (2)

T. Tamir and E. Garmire, Integrated Optics (Springer-Verlag, 1979).

http://www.fourmilab.ch/documents/specrend/ .

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Figures (5)

Fig. 1
Fig. 1 Energy flow and Goos-Hänchen shift in a planar waveguide. The incidence is colored in red, Goos-Hänchen shift zs in blue, while the forward displacement zw in orange. The rays travelling inside the waveguide are in black.
Fig. 2
Fig. 2 (a): Schematic of an infinite thick substrate decorated by a dielectric grating. I and R stand for incidence and reflected beam respectively. zs is the Goos-Hänchen shift. Duty cycle is defined as Γ = a/Λ. (b):Reflection phase vs Incidence angle θ of different interfaces when incidence coming from the SiO2 substrate. Operating free space wavelength is 1.5μm. GratingI parameters: ΛI = 0.53μm, tI = 0.097μm, ΓI = 0.65. GratingII parameters:ΛII = 0.43μm, tII = 0.11μmII = 0.93.
Fig. 3
Fig. 3 (a): The dispersion property of waveguide with grating parameters: t = 0.04μm,Λ = 0.16μm, Γ = 0.6, εh = 10.24, εl = εc = 1; waveguide parameters: h = 0.2μm, εs = 2.09. The upper and lower dashed lines are light lines of the free space and the waveguide. The blue circles stands for the FDTD result while red cross stands for the plane wave approximation analysis. Solid violet line shows the dispersion of single grating laying on a SiO2 substrate, while soild orange line shows the dispersion of bare waveguide. (b): The instantaneous field distribution of the frozen mode in three periods of the waveguide.
Fig. 4
Fig. 4 The dispersion of two types of waveguide in the difference of Period = 0.16μm(Blue circle) and Period = 0.17μm(Red triangle).
Fig. 5
Fig. 5 (a): Top: The schematic of the rainbow-trapping device composed of multiple waveguide segments. Bottom: Full wave analysis of the trapped rainbow obtain by FDTD simulation and Fast Fourier Transform. For the waveguide, t = 0.2μm. For the grating, Γ = 0.6, t = 0.04μm, period range is designed from 0.13 → 0.26μm. The Gaussian pulse enters from the left of the structure in the slab waveguide(not shown in the plot). Both figures demonstrate the field distribution in the frequency domain(vertical axis) and the spatial position(horizontal axis). (b): Rainbow trapping by waveguide of continuously varying parameters. The period varies from 0.130μm to 0.267μm gradually. The other design parameters and the excitation is the same as (a).

Equations (5)

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

z s = ϕ k x
2 n h k 0 cos θ + ϕ 1 ( θ , k 0 ) + ϕ 2 ( θ , k 0 ) = 2 m π
2 h n 2 k 0 2 k x 2 + ϕ 1 ( k x , k 0 ) + ϕ 2 ( k x , k 0 ) = 2 m π
2 n h Δ k 0 1 k x 2 / ( n k 0 ) 2 2 h k x n 2 k 0 2 k x 2 Δ k x + j = 1 , 2 ( ϕ j k 0 Δ k 0 + ϕ j k x Δ k x ) = 0
( 2 n h 1 k x 2 / ( n k 0 ) 2 + j = 1 , 2 ϕ j k 0 ) k 0 k x = 2 h k x n 2 k 0 2 k x 2 + j = 1 , 2 ϕ j k x

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