Abstract

In this paper, enhanced spatial and temporal shifts in the reflection of Gaussian wave packets from two-dimensional photonic crystal waveguides supporting above-the-light-line leaky modes are studied, for the first time to our best knowledge. Particular attention is given to two important special cases, namely, harmonic Gaussian beams and Gaussian-pulse uniform plane waves. Analytical expressions are given for enhanced spatial and temporal shifts when the stationary phase approximation holds and the incident wave excites above-the-light-line leaky modes. The enhanced spatial and temporal shifts of Gaussian wave packets are thereby related to each other via the group velocity of the excited leaky modes.

© 2012 Optical Society of America

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2010

2008

2006

2005

J. R. Thomas and A. Ishimaru, “Wave packet incident on negative-index media,” IEEE Trans. Antennas Propag. 53, 1591–1599 (2005).
[CrossRef]

2004

D. Felbacq and R. Smaâli, “Bloch modes dressed by evanescent waves and the generalized Goos–Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92, 193902 (2004).
[CrossRef]

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372–374 (2004).
[CrossRef]

2000

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos–Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
[CrossRef]

1999

1998

1986

1983

K. Yasumoto and Y. Oishi, “A new evaluation of the Goos–Hänchen shift and associated time delay,” J. Appl. Phys. 54, 2170–2176 (1983).
[CrossRef]

1977

J. Hugonin and R. Petit, “General study of displacements at total reflection,” J. Opt. 8, 73–87 (1977).
[CrossRef]

1971

1948

K. Artmann, “Berechnung der Seitenversetzung des Totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
[CrossRef]

1947

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Alishahi, F.

Anemogiannis, E.

Artmann, K.

K. Artmann, “Berechnung der Seitenversetzung des Totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
[CrossRef]

Bertoni, H. L.

Bryngdahl, O.

Fang, N.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372–374 (2004).
[CrossRef]

Felbacq, D.

D. Felbacq and R. Smaâli, “Bloch modes dressed by evanescent waves and the generalized Goos–Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92, 193902 (2004).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

He, J.

He, S.

Hesselink, L.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372–374 (2004).
[CrossRef]

Hugonin, J.

J. Hugonin and R. Petit, “General study of displacements at total reflection,” J. Opt. 8, 73–87 (1977).
[CrossRef]

Ishimaru, A.

J. R. Thomas and A. Ishimaru, “Wave packet incident on negative-index media,” IEEE Trans. Antennas Propag. 53, 1591–1599 (2005).
[CrossRef]

Jahromi, A. K.

Khavasi, A.

Khorasani, S.

Liu, Z.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372–374 (2004).
[CrossRef]

Mehrany, K.

Miri, M.

Naqavi, A.

Oishi, Y.

K. Yasumoto and Y. Oishi, “A new evaluation of the Goos–Hänchen shift and associated time delay,” J. Appl. Phys. 54, 2170–2176 (1983).
[CrossRef]

Petit, R.

J. Hugonin and R. Petit, “General study of displacements at total reflection,” J. Opt. 8, 73–87 (1977).
[CrossRef]

Rashidian, B.

Sakata, T.

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos–Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals, 1st ed. (Springer, 2001).

Schmitz, M.

Schreier, F.

Shimokawa, F.

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos–Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
[CrossRef]

Smaâli, R.

D. Felbacq and R. Smaâli, “Bloch modes dressed by evanescent waves and the generalized Goos–Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92, 193902 (2004).
[CrossRef]

Tamir, T.

Thomas, J. R.

J. R. Thomas and A. Ishimaru, “Wave packet incident on negative-index media,” IEEE Trans. Antennas Propag. 53, 1591–1599 (2005).
[CrossRef]

Togo, H.

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos–Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
[CrossRef]

Wang, L. G.

Yasumoto, K.

K. Yasumoto and Y. Oishi, “A new evaluation of the Goos–Hänchen shift and associated time delay,” J. Appl. Phys. 54, 2170–2176 (1983).
[CrossRef]

Yi, J.

Yin, X.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372–374 (2004).
[CrossRef]

Zhang, X.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372–374 (2004).
[CrossRef]

Zhu, S. Y.

Ann. Phys.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

K. Artmann, “Berechnung der Seitenversetzung des Totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
[CrossRef]

Appl. Phys. Lett.

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85, 372–374 (2004).
[CrossRef]

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos–Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
[CrossRef]

IEEE Trans. Antennas Propag.

J. R. Thomas and A. Ishimaru, “Wave packet incident on negative-index media,” IEEE Trans. Antennas Propag. 53, 1591–1599 (2005).
[CrossRef]

J. Appl. Phys.

K. Yasumoto and Y. Oishi, “A new evaluation of the Goos–Hänchen shift and associated time delay,” J. Appl. Phys. 54, 2170–2176 (1983).
[CrossRef]

J. Lightwave Technol.

J. Opt.

J. Hugonin and R. Petit, “General study of displacements at total reflection,” J. Opt. 8, 73–87 (1977).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

D. Felbacq and R. Smaâli, “Bloch modes dressed by evanescent waves and the generalized Goos–Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92, 193902 (2004).
[CrossRef]

Other

K. Sakoda, Optical Properties of Photonic Crystals, 1st ed. (Springer, 2001).

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Figures (10)

Fig. 1.
Fig. 1.

Reflection of a Gaussian wave packet upon a typical 2D-PCW.

Fig. 2.
Fig. 2.

Dispersion diagram of the above-the-light-line modes supported by the 2D-PCW shown in the inset of the figure. The hatched region and the dashed–dotted line indicate the photonic bands of the 2D-PC and the light line, respectively.

Fig. 3.
Fig. 3.

(a) Spatial shift for a harmonic Gaussian beam with W0=100λ0 (under the stationary phase approximation): spatial shift from Artmann’s formula (dashed line), 2/βi (dotted line), and the center-of-gravity shift calculated by using Eq. (3) (solid line) (b) Spatial shift for a harmonic Gaussian beam with W0=5λ0 (when the stationary phase approximation is not valid): the center-of-gravity shift calculated by using Eq. (3) (solid line) and the FDTD simulation (circles).

Fig. 4.
Fig. 4.

(a) Fourier transform of the incident wave packet with T0=103×(2π/ω0), ω0=0.35(2πc/a), θ=34.7°, and the spatial width of W0=100λ0. (b) Fourier transform of the incident wave packet with T0=103×(2π/ω0), ω0=0.35(2πc/a), θ=34.7°, and the spatial width of W0=5λ0.

Fig. 5.
Fig. 5.

FDTD simulation of reflection of a Gaussian beam from the interface of a 2D-PC, which supports a defect mode with positive group velocity that is in near resonance with the incident beam, at the frequency of ωn=0.35 and the incident angle of θi=34.7°.

Fig. 6.
Fig. 6.

(a) The overall temporal delay of the Gaussian-pulse uniform plane wave with T0=200×(2π/ω0) when the stationary phase approximation is valid: the temporal counterpart of Artmann’s formula, ΔtGP=φR/ω(φR/kx)/vp (solid line), the proposed approximation ΔtGP=2/ωi2/βivp (circles), and the rigorous center-of-gravity shift (dashed). (b) The overall temporal delay of the Gaussian-pulse uniform plane wave with T0=5×(2π/ω0) when the stationary phase approximation is not valid: the rigorous center-of-gravity shift (solid line) and the FDTD results (circles).

Fig. 7.
Fig. 7.

(a) Fourier transform of the incident wave packet with W0=103λ0, ω0=0.35(2πc/a), θ=34.7°, and temporal width of T0=200×(2π/ω0). (b) Fourier transform of the incident wave packet with W0=103λ0, ω0=0.35(2πc/a), θ=34.7°, and temporal width of T0=5×(2π/ω0).

Fig. 8.
Fig. 8.

Group velocity of the excited leaky mode when the incident Gaussian wave packets are in resonance with the structure: the well-known analytical formula vg=ω/kx (solid line), the proposed approximation vg=ΔxWP/ΔtWPωi/βi under the stationary phase approximation (dotted line), and the proposed approximation vg=ΔxWP/ΔtWP when the stationary phase approximation is not valid (circles).

Fig. 9.
Fig. 9.

Dispersion diagram of the leaky mode with negative group velocity for the 2D-PCW shown in the inset of the figure. The hatched region and the dashed–dotted line indicate the photonic bands of the 2D-PC and the light line, respectively.

Fig. 10.
Fig. 10.

FDTD simulation of reflection of a Gaussian beam from the interface of a 2D-PC that supports a defect mode with negative group velocity that is in near resonance with incident beam, at the frequency of ωn=0.43 and incident angle of θi=42.1°.

Tables (1)

Tables Icon

Table 1. Negative Spatial Shift for the Structure Shown in Fig. 9, Inset

Equations (19)

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Ei(x,y=0,t)=Re{ψi(x,t)};ψi(x,t)=exp[iω0(txsinθc)]×exp[(txsinθc)2T02(xcosθ)2W02],
Er(x,0,t)=Re{ψr(x,t)};ψr(x,t)=1/(2π)2++exp[iφR(kx,ω)]Ψi(kx,ω)exp[i(ωtkxx)]dkxdω.
ΔxGB=Ai*exp[iφR(kx,ω0)]ddkx(exp[iφR(kx,ω0)])AidkxAi*Aidkx,
Ai(kx)=12π+Ei(x,y=0)exp(ikxx)dkx,
Er(x,0,t)=exp([(xΔxGB)cosθ]2W02)×cos(ω0[t(xΔxGB)sinθc]),
ΔxGB=φRkx|kx=ω0/csinθ.
ρ(kx,ω0)r(kx)(kxkz)(kxkp).
ΔxGBIm{kzkp(kxkp)(kxkz)}|kx=βr.
ΔxGB2βi.
ΔtGP=+Ai*exp[iφR(kx0,ω)]ddω(exp[iφR(kx0,ω)])Aidω+Ai*Aidω,
Ai(ω)=12π+Ei(x=0,y=0,t)exp(iωt)dt,
Er(x,0,t)=exp([(tΔt)(xΔx)sinθc]2T02)×cos(ω0[(tΔt)(xΔx)sinθc]),
Er(x,0,t)=exp([(tΔtGP)xsinθc]2T02)×cos(ω0[(tΔtGP)xsinθc]).
ΔtGP=ΔtΔxsinθc=ΔtΔxvp.
ΔtGP=2ωiΔxvp=2ωi2vpβi.
Er(x,0,t)=exp([(tΔtWP)(xΔxWP)sinθc]2/T02[(xΔxWP)cosθ]2/W02)cos(ω0[(tΔtWP)(xΔxWP)sinθc]),
ΔtWP=ΔtGP+ΔxGBvp.
ΔxWP=ΔxGB.
ΔxWP=vgΔtWP.

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