Abstract

The Goos-Hänchen effects are investigated for a monochromatic Gaussian beam totally reflected by a photonic crystal with a negative effective index. By choosing an appropriate thickness for the homogeneous cladding layer, a giant negative GH lateral shift can be obtained and the totally reflected beam retains a single beam of good profile even for a very narrow incident beam. The GH lateral shift can be very sensitive to the change of the refractive index of the cladding layer, and this property can be utilized for e.g. the switching applications.

© 2006 Optical Society of America

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  1. F. Goos and H. Hänchen, "Ein neuer und fundamentaler versuch zur totalreflexion," Ann. Phys. 1, 333-346 (1947).
    [CrossRef]
  2. S. R. Seshadri, "Goos-Hänchen beam shift at total internal reflection," J. Opt. Soc. Am. A 5, 583-590 (1998).
    [CrossRef]
  3. I. Shadrivov, A. Zharov, and Y. S. Kivshar, "Giant Goos-Hanchen effect at the reflection from left-handed metamaterials," Appl. Phys. Lett. 83, 2713-2715 (2003).
    [CrossRef]
  4. I. Shadrivov, R. Ziolkowski, A. Zharov, and Y. Kivshar, "Excitation of guided waves in layered structures with negative refraction," Opt. Express 13, 481-492 (2005).
    [CrossRef] [PubMed]
  5. L. G. Wang and S. Y. Zhu, "Giant lateral shift of a light beam at the defect mode in one-dimensional photonic crystals," Opt. Lett. 31, 101-103 (2006).
    [CrossRef] [PubMed]
  6. H. M. Lai and S. W. Chan, "Large and negative Goos-Hanchen shift near the Brewster dip on reflection from weakly absorbing media," Opt. Lett. 27, 680-682 (2002)
    [CrossRef]
  7. L. Wang, H. Chen, and S. Zhu, "Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab," Opt. Lett. 30, 2936-2938 (2005).
    [CrossRef] [PubMed]
  8. D. Felbacq, A. Moreau, and R. Smaali, "Goos-Hänchen effect in the gaps of photonic crystals, " Opt. Lett. 28, 1633-1635 (2003).
    [CrossRef] [PubMed]
  9. 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] [PubMed]
  10. 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]
  11. 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]
  12. S. L. He, Z. C. Ruan, L. Chen and J. Q. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phy. Rev. B 70, 115113 (2004).
    [CrossRef]
  13. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
    [CrossRef]
  14. J. J. Chen, T. M. Grzegorczyk, B. Wu, and J A. Kong, "Role of evanescent waves in the positive and negative Goos-Hanchen shifts with left-handed material slabs," J. Appl. Phys. 98, 094905 (2005).
    [CrossRef]
  15. T. Tamir, "Leaky waves in planer optical waveguides," Nouv. Rev. Opt. 6, 273-284 (1975).
    [CrossRef]
  16. F. Schreier, M. Schmitz, and O. Bryngdahl, "Beam displacement atdiffractive structures under resonance conditions," Opt. Lett. 23, 576-578 (1998).
    [CrossRef]
  17. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
    [CrossRef]
  18. S. Enoch, E. Popov, and N. Bonod, "Analysis of the physical origin of surface modes on finite-size photonic crystals," Phys. Rev. B 72, 155101 (2005).
    [CrossRef]
  19. R. Reinisch, and M. Neviere, "Grating-enhanced nonlinear excitation of surface polaritons: An electromagnetic study," Phys. Rev. B 24, 4392-4405 (1981).
    [CrossRef]
  20. 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]

2006 (1)

2005 (4)

L. Wang, H. Chen, and S. Zhu, "Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab," Opt. Lett. 30, 2936-2938 (2005).
[CrossRef] [PubMed]

I. Shadrivov, R. Ziolkowski, A. Zharov, and Y. Kivshar, "Excitation of guided waves in layered structures with negative refraction," Opt. Express 13, 481-492 (2005).
[CrossRef] [PubMed]

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J A. Kong, "Role of evanescent waves in the positive and negative Goos-Hanchen shifts with left-handed material slabs," J. Appl. Phys. 98, 094905 (2005).
[CrossRef]

S. Enoch, E. Popov, and N. Bonod, "Analysis of the physical origin of surface modes on finite-size photonic crystals," Phys. Rev. B 72, 155101 (2005).
[CrossRef]

2004 (2)

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] [PubMed]

S. L. He, Z. C. Ruan, L. Chen and J. Q. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phy. Rev. B 70, 115113 (2004).
[CrossRef]

2003 (2)

I. Shadrivov, A. Zharov, and Y. S. Kivshar, "Giant Goos-Hanchen effect at the reflection from left-handed metamaterials," Appl. Phys. Lett. 83, 2713-2715 (2003).
[CrossRef]

D. Felbacq, A. Moreau, and R. Smaali, "Goos-Hänchen effect in the gaps of photonic crystals, " Opt. Lett. 28, 1633-1635 (2003).
[CrossRef] [PubMed]

2002 (1)

2000 (3)

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[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]

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]

1998 (3)

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]

F. Schreier, M. Schmitz, and O. Bryngdahl, "Beam displacement atdiffractive structures under resonance conditions," Opt. Lett. 23, 576-578 (1998).
[CrossRef]

S. R. Seshadri, "Goos-Hänchen beam shift at total internal reflection," J. Opt. Soc. Am. A 5, 583-590 (1998).
[CrossRef]

1991 (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

1981 (1)

R. Reinisch, and M. Neviere, "Grating-enhanced nonlinear excitation of surface polaritons: An electromagnetic study," Phys. Rev. B 24, 4392-4405 (1981).
[CrossRef]

1975 (1)

T. Tamir, "Leaky waves in planer optical waveguides," Nouv. Rev. Opt. 6, 273-284 (1975).
[CrossRef]

1947 (1)

F. Goos and H. Hänchen, "Ein neuer und fundamentaler versuch zur totalreflexion," Ann. Phys. 1, 333-346 (1947).
[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]

Bonod, N.

S. Enoch, E. Popov, and N. Bonod, "Analysis of the physical origin of surface modes on finite-size photonic crystals," Phys. Rev. B 72, 155101 (2005).
[CrossRef]

Brommer, K. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Bryngdahl, O.

Chan, S. W.

Chen, H.

Chen, J. J.

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J A. Kong, "Role of evanescent waves in the positive and negative Goos-Hanchen shifts with left-handed material slabs," J. Appl. Phys. 98, 094905 (2005).
[CrossRef]

Chen, L.

S. L. He, Z. C. Ruan, L. Chen and J. Q. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phy. Rev. B 70, 115113 (2004).
[CrossRef]

Enoch, S.

S. Enoch, E. Popov, and N. Bonod, "Analysis of the physical origin of surface modes on finite-size photonic crystals," Phys. Rev. B 72, 155101 (2005).
[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] [PubMed]

D. Felbacq, A. Moreau, and R. Smaali, "Goos-Hänchen effect in the gaps of photonic crystals, " Opt. Lett. 28, 1633-1635 (2003).
[CrossRef] [PubMed]

Goos, F.

F. Goos and H. Hänchen, "Ein neuer und fundamentaler versuch zur totalreflexion," Ann. Phys. 1, 333-346 (1947).
[CrossRef]

Grzegorczyk, T. M.

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J A. Kong, "Role of evanescent waves in the positive and negative Goos-Hanchen shifts with left-handed material slabs," J. Appl. Phys. 98, 094905 (2005).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, "Ein neuer und fundamentaler versuch zur totalreflexion," Ann. Phys. 1, 333-346 (1947).
[CrossRef]

He, S. L.

S. L. He, Z. C. Ruan, L. Chen and J. Q. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phy. Rev. B 70, 115113 (2004).
[CrossRef]

Joannopoulos, J. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Kivshar, Y.

Kivshar, Y. S.

I. Shadrivov, A. Zharov, and Y. S. Kivshar, "Giant Goos-Hanchen effect at the reflection from left-handed metamaterials," Appl. Phys. Lett. 83, 2713-2715 (2003).
[CrossRef]

Kong, J A.

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J A. Kong, "Role of evanescent waves in the positive and negative Goos-Hanchen shifts with left-handed material slabs," J. Appl. Phys. 98, 094905 (2005).
[CrossRef]

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Lai, H. M.

H. M. Lai and S. W. Chan, "Large and negative Goos-Hanchen shift near the Brewster dip on reflection from weakly absorbing media," Opt. Lett. 27, 680-682 (2002)
[CrossRef]

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Meade, R. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Moreau, A.

Neviere, M.

R. Reinisch, and M. Neviere, "Grating-enhanced nonlinear excitation of surface polaritons: An electromagnetic study," Phys. Rev. B 24, 4392-4405 (1981).
[CrossRef]

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]

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]

Popov, E.

S. Enoch, E. Popov, and N. Bonod, "Analysis of the physical origin of surface modes on finite-size photonic crystals," Phys. Rev. B 72, 155101 (2005).
[CrossRef]

Rappe, A. M.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Reinisch, R.

R. Reinisch, and M. Neviere, "Grating-enhanced nonlinear excitation of surface polaritons: An electromagnetic study," Phys. Rev. B 24, 4392-4405 (1981).
[CrossRef]

Ruan, Z. C.

S. L. He, Z. C. Ruan, L. Chen and J. Q. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phy. Rev. B 70, 115113 (2004).
[CrossRef]

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]

Schmitz, M.

Schreier, F.

Seshadri, S. R.

Shadrivov, I.

I. Shadrivov, R. Ziolkowski, A. Zharov, and Y. Kivshar, "Excitation of guided waves in layered structures with negative refraction," Opt. Express 13, 481-492 (2005).
[CrossRef] [PubMed]

I. Shadrivov, A. Zharov, and Y. S. Kivshar, "Giant Goos-Hanchen effect at the reflection from left-handed metamaterials," Appl. Phys. Lett. 83, 2713-2715 (2003).
[CrossRef]

Shen, J. Q.

S. L. He, Z. C. Ruan, L. Chen and J. Q. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phy. Rev. B 70, 115113 (2004).
[CrossRef]

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]

Smaali, R.

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] [PubMed]

Tamir, T.

T. Tamir, "Leaky waves in planer optical waveguides," Nouv. Rev. Opt. 6, 273-284 (1975).
[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]

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]

Wang, L.

Wang, L. G.

Wu, B.

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J A. Kong, "Role of evanescent waves in the positive and negative Goos-Hanchen shifts with left-handed material slabs," J. Appl. Phys. 98, 094905 (2005).
[CrossRef]

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Zharov, A.

I. Shadrivov, R. Ziolkowski, A. Zharov, and Y. Kivshar, "Excitation of guided waves in layered structures with negative refraction," Opt. Express 13, 481-492 (2005).
[CrossRef] [PubMed]

I. Shadrivov, A. Zharov, and Y. S. Kivshar, "Giant Goos-Hanchen effect at the reflection from left-handed metamaterials," Appl. Phys. Lett. 83, 2713-2715 (2003).
[CrossRef]

Zhu, S.

Zhu, S. Y.

Ziolkowski, R.

Ann. Phys. (1)

F. Goos and H. Hänchen, "Ein neuer und fundamentaler versuch zur totalreflexion," Ann. Phys. 1, 333-346 (1947).
[CrossRef]

Appl. Phys. Lett. (2)

I. Shadrivov, A. Zharov, and Y. S. Kivshar, "Giant Goos-Hanchen effect at the reflection from left-handed metamaterials," Appl. Phys. Lett. 83, 2713-2715 (2003).
[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]

J. Appl. Phys. (1)

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J A. Kong, "Role of evanescent waves in the positive and negative Goos-Hanchen shifts with left-handed material slabs," J. Appl. Phys. 98, 094905 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nouv. Rev. Opt. (1)

T. Tamir, "Leaky waves in planer optical waveguides," Nouv. Rev. Opt. 6, 273-284 (1975).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phy. Rev. B (1)

S. L. He, Z. C. Ruan, L. Chen and J. Q. Shen, "Focusing properties of a photonic crystal slab with negative refraction," Phy. Rev. B 70, 115113 (2004).
[CrossRef]

Phys. Rev. B (5)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

S. Enoch, E. Popov, and N. Bonod, "Analysis of the physical origin of surface modes on finite-size photonic crystals," Phys. Rev. B 72, 155101 (2005).
[CrossRef]

R. Reinisch, and M. Neviere, "Grating-enhanced nonlinear excitation of surface polaritons: An electromagnetic study," Phys. Rev. B 24, 4392-4405 (1981).
[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]

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]

Phys. Rev. E (1)

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

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] [PubMed]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Schematic diagram of the photonic crystal structure considered in this paper. The insert shows the effective index of the PC as the frequency increases. (b) The region of total reflection in the frequency range of negative refraction when d =0. The shadow region corresponds to the total reflection region calculated by a layer-KKR method, the dashed line corresponds to total reflection boundary estimated by kx = sin(θi )∣neff ∣/nAir and the light line corresponds to kx = 0)/c.

Fig. 2.
Fig. 2.

Goos-Hänchen lateral shifts as the mean incident angle increases at several different frequencies when the incident beam is totally reflected by the PC. No cladding layer over the PC for this case. The beam waist is fixed to w = 25a. The minimal incident angle for each curve is determined by the total internal reflection condition.

Fig. 3.
Fig. 3.

(a) The GH shift as the thickness d of the cladding layer increases. (b) The width of the reflected beam as d increases. The parameters for the incident Gaussian beam are chosen as ω = 0.335(2πc/a), w = 25a and θi = 45°. The insets show the profiles of the field intensity for the reflected beams.

Fig. 4.
Fig. 4.

(a) Schematic diagram for multi-reflection of light in the PC structure. (b) The GH shift as the cladding thickness d increases when the refraction index of the cladding layer nclad =2.0. The incident Gassian beam is the same as used in Fig. 3.

Fig. 5.
Fig. 5.

FDTD simulation for the distribution of the electric fields of a Gaussian beam reflected from a PC structure with cladding thickness d=0.113a.

Fig. 6.
Fig. 6.

Goos-Hänchen lateral shift as the refractive index of the cladding layer varies slightly. The thickness of the cladding layer is d = 0.75a. The insets show the corresponding profiles of the field intensity of the reflected beams at three different values of the refractive index.

Equations (3)

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E y i x z = A k x w exp i ( x k x z k 0 2 k x 2 ) d k x
G r = x E y r x 0 2 dx E y r x 0 2 dx
φ i , A + φ i , B + 2 d k n 2 k i , x 2 = 2 , m = 0 1,2⋯ i = 1,0

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