Abstract

We analytically relate the giant Goos–Hänchen shift, observed at the interface of a high refractive index prism and a waveguide structure with an arbitrary refractive index profile, to the spatial resonance phenomenon. The proximity effect of the high refractive index prism on modal properties of the waveguide is discussed, and the observed shift is expressed in terms of proper and improper electromagnetic modes supported by the waveguide with no prism. The transversely increasing improper modes are shown playing an increasingly important role as the high refractive index prism comes closer to the waveguide.

© 2010 Optical Society of America

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References

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  1. T. Tamir and H. L. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971).
    [CrossRef]
  2. I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, arXiv.org, arXiv:physics/0305032v1.
  3. X. Lin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, Appl. Phys. Lett. 85, 372 (2004).
    [CrossRef]
  4. T. Okamoto, M. Yamamoto, and I. Yamaguchi, J. Opt. Soc. Am. A 17, 1880 (2000).
    [CrossRef]
  5. S. M. Vukovi, N. B. Aleksi, and D. V. Timotijevi, Eur. Phys. J. D 39, 295 (2006).
    [CrossRef]
  6. K. V. Artmann, Ann. Phys. 437, 87 (1948).
    [CrossRef]

2006 (1)

S. M. Vukovi, N. B. Aleksi, and D. V. Timotijevi, Eur. Phys. J. D 39, 295 (2006).
[CrossRef]

2004 (1)

X. Lin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, Appl. Phys. Lett. 85, 372 (2004).
[CrossRef]

2000 (1)

1971 (1)

1948 (1)

K. V. Artmann, Ann. Phys. 437, 87 (1948).
[CrossRef]

Aleksi, N. B.

S. M. Vukovi, N. B. Aleksi, and D. V. Timotijevi, Eur. Phys. J. D 39, 295 (2006).
[CrossRef]

Artmann, K. V.

K. V. Artmann, Ann. Phys. 437, 87 (1948).
[CrossRef]

Bertoni, H. L.

Fang, N.

X. Lin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, Appl. Phys. Lett. 85, 372 (2004).
[CrossRef]

Hesselink, L.

X. Lin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, Appl. Phys. Lett. 85, 372 (2004).
[CrossRef]

Kivshar, Y. S.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, arXiv.org, arXiv:physics/0305032v1.

Lin, X.

X. Lin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, Appl. Phys. Lett. 85, 372 (2004).
[CrossRef]

Liu, Z.

X. Lin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, Appl. Phys. Lett. 85, 372 (2004).
[CrossRef]

Okamoto, T.

Shadrivov, I. V.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, arXiv.org, arXiv:physics/0305032v1.

Tamir, T.

Timotijevi, D. V.

S. M. Vukovi, N. B. Aleksi, and D. V. Timotijevi, Eur. Phys. J. D 39, 295 (2006).
[CrossRef]

Vukovi, S. M.

S. M. Vukovi, N. B. Aleksi, and D. V. Timotijevi, Eur. Phys. J. D 39, 295 (2006).
[CrossRef]

Yamaguchi, I.

Yamamoto, M.

Zhang, X.

X. Lin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, Appl. Phys. Lett. 85, 372 (2004).
[CrossRef]

Zharov, A. A.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, arXiv.org, arXiv:physics/0305032v1.

Ann. Phys. (1)

K. V. Artmann, Ann. Phys. 437, 87 (1948).
[CrossRef]

Appl. Phys. Lett. (1)

X. Lin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, Appl. Phys. Lett. 85, 372 (2004).
[CrossRef]

Eur. Phys. J. D (1)

S. M. Vukovi, N. B. Aleksi, and D. V. Timotijevi, Eur. Phys. J. D 39, 295 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Other (1)

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, arXiv.org, arXiv:physics/0305032v1.

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Figures (2)

Fig. 1
Fig. 1

GHS in nanometers versus angle of incidence for the lossless structure with (curve 1) d g = 0.4 λ 0 and (2) d g = 0.55 λ 0 . Artmann’s formula (solid curve) and the proposed approximation using β p and β z given in Eqs. (14) (dashed curve).

Fig. 2
Fig. 2

GHS in nanometers versus angle of incidence for the lossy structure with (curve 1) d g = 0.4 λ 0 and (2) d g = 0.55 λ 0 . Artmann’s formula (solid curve) and the proposed approximation using β p and β z given in Eq. (14) (dashed curve).

Equations (22)

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Q T = Q × T ,
Q = [ q 11 q 12 q 21 q 22 ]
T = [ t 11 t 12 t 21 t 22 ] = [ ( 1 + f κ p κ c ) exp ( κ c d g ) ( 1 f κ p κ c ) exp ( κ c d g ) ( 1 f κ p κ c ) exp ( κ c d g ) ( 1 + f κ p κ c ) exp ( κ c d g ) ] ,
r = Q 21 Q 22 .
Q 21 ( β ) = Q 21 ( β g ) + Q 21 β | β g ( β β g ) ,
Q 22 ( β ) = Q 22 ( β g ) + Q 22 β | β g ( β β g ) .
Q 21 ( β ) = Q 21 β | β g ( β β z ) ,
Q 22 ( β ) = Q 22 β | β g ( β β p ) ,
β z = β g Q 21 ( β g ) Q 21 β | β g = β g q 21 ( β g ) q 22 ( β g ) × t 11 / t 21 1 + ( t 11 q 21 t 21 q 22 + t 11 q 21 t 21 q 22 ) | β g ,
β p = β g Q 22 ( β g ) Q 22 β | β g = β g q 21 ( β g ) q 22 ( β g ) × t 12 / t 22 1 + ( t 12 q 21 t 22 q 22 + t 12 q 21 t 22 q 22 ) | β g ,
r = Q 21 β | β g Q 22 β | β g × β β z β β p .
GHS = ϕ β = Im [ d ( ln ( r ) ) d β ] = Im [ β z β p ( β β p ) ( β β z ) ] ,
r w = q 21 ( β ) q 22 ( β ) .
Res ( β g ) = q 21 ( β g ) q 22 ( β g ) .
β g β 0 = q 21 ( β g ) q 21 ( β g ) .
t 11 t 21 = r pc 1 exp ( 2 κ c d g ) ,
t 12 t 22 = r pc exp ( 2 κ c d g ) ,
β ˜ z = β g Res ( β g ) exp ( 2 κ c ( β g ) d g ) r pc ( β g ) + Res ( β g ) / ( β g β 0 ) exp ( 2 κ c ( β g ) d g ) ,
β ˜ p = β g Res ( β g ) r pc ( β g ) exp ( 2 κ c ( β g ) d g ) 1 + r pc ( β g ) Res ( β g ) / ( β g β 0 ) exp ( 2 κ c ( β g ) d g ) .
β ˜ z = β g Res ( β g ) × r pc 1 ( β g ) exp ( 2 κ c ( β g ) d g ) ,
β ˜ p = β g Res ( β g ) × r pc ( β g ) exp ( 2 κ c ( β g ) d g ) ,
n ( x ) = [ n f 2 ( n f 2 n s 2 ) x d ] 1 / 2 ,

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