Abstract

An extremely simple derivation of the Goos-Hänchen shift is presented for total internal reflection at a plane interface between two semiinfinite dielectric media, as well as for optical waveguides of plane and circular cross section. The derivation is based on energy considerations, requires knowledge of Fresnel's equation only, and shows explicitly that the shift is due to the flow of energy across the dielectric boundary.

© 1976 Optical Society of America

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References

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  1. H. K. V. Lotsch, Optik 32, 116, 189 (1970);Optik 32, 299, 553 (1971).
  2. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic Press, New York, 1972), p. 74.
  3. B. R. Horowitz, T. Tamir, J. Opt. Soc. Am. 61, 586 (1971).
    [CrossRef]
  4. A. W. Snyder, J. D. Love, IEEE Trans. Microwave Theory Tech. MTT23, Jan. (1975);follows from Eq. (16) with n1i= 0.
  5. H. Kogelnik, H. P. Weber, J. Opt. Soc. Am. 64, 174 (1974).
    [CrossRef]
  6. A. W. Snyder, Appl. Phys. 4, 273 (1974); see Sect. 9.1.
    [CrossRef]
  7. R. Sammut, A. W. Snyder, Appl. Opt.15, in press (1976).
    [PubMed]

1975 (1)

A. W. Snyder, J. D. Love, IEEE Trans. Microwave Theory Tech. MTT23, Jan. (1975);follows from Eq. (16) with n1i= 0.

1974 (2)

H. Kogelnik, H. P. Weber, J. Opt. Soc. Am. 64, 174 (1974).
[CrossRef]

A. W. Snyder, Appl. Phys. 4, 273 (1974); see Sect. 9.1.
[CrossRef]

1971 (1)

1970 (1)

H. K. V. Lotsch, Optik 32, 116, 189 (1970);Optik 32, 299, 553 (1971).

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic Press, New York, 1972), p. 74.

Horowitz, B. R.

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic Press, New York, 1972), p. 74.

Kogelnik, H.

Lotsch, H. K. V.

H. K. V. Lotsch, Optik 32, 116, 189 (1970);Optik 32, 299, 553 (1971).

Love, J. D.

A. W. Snyder, J. D. Love, IEEE Trans. Microwave Theory Tech. MTT23, Jan. (1975);follows from Eq. (16) with n1i= 0.

Sammut, R.

R. Sammut, A. W. Snyder, Appl. Opt.15, in press (1976).
[PubMed]

Snyder, A. W.

A. W. Snyder, J. D. Love, IEEE Trans. Microwave Theory Tech. MTT23, Jan. (1975);follows from Eq. (16) with n1i= 0.

A. W. Snyder, Appl. Phys. 4, 273 (1974); see Sect. 9.1.
[CrossRef]

R. Sammut, A. W. Snyder, Appl. Opt.15, in press (1976).
[PubMed]

Tamir, T.

Weber, H. P.

Appl. Phys. (1)

A. W. Snyder, Appl. Phys. 4, 273 (1974); see Sect. 9.1.
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, J. D. Love, IEEE Trans. Microwave Theory Tech. MTT23, Jan. (1975);follows from Eq. (16) with n1i= 0.

J. Opt. Soc. Am. (2)

Optik (1)

H. K. V. Lotsch, Optik 32, 116, 189 (1970);Optik 32, 299, 553 (1971).

Other (2)

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic Press, New York, 1972), p. 74.

R. Sammut, A. W. Snyder, Appl. Opt.15, in press (1976).
[PubMed]

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Figures (2)

Fig. 1
Fig. 1

Total internal reflection (α > αc) from a plane interface exhibiting the Goos-Hänchen shift s. The arrows represent light rays.

Fig. 2
Fig. 2

Model used to illustrate the trajectory of a ray undergoing total internal reflection within a dielectric slab. The dotted line is the geometric continuation of the ray into medium 2 that would exist if n1 = n2. N(s) is the number of reflections in a specified length.

Equations (24)

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s E = ( 2 k 1 ) tan α c ( sin 2 α sin 2 α c ) 1 / 2 ,
s H = s E / sin 2 α c ,
k 1 = 2 π n 1 / λ ,
l tot = z / sin α
= ( z 1 + s ) N ( s ) / sin α
= ( l + s / sin α ) N ( s ) ,
γ = l ξ N ( s ) ,
ξ = 2 k 1 n 1 i n 1 r ,
γ = ( l ξ + T ) N ( o ) ,
N ( o ) = l tot / l
= ( 1 + s / l sin α ) N ( s ) .
T = 1 power of reflected wave power of incident wave .
T E T H sin 2 α c
2 k 1 ( sin 2 α c cos α c ) ξ ( sin 2 α sin 2 α c ) 1 / 2 ,
s = T l sin α / ( l ξ + T ) .
s = T sin α / ξ ,
k 1 d ( sin 2 α sin 2 α c ) 1 / 2 1 .
η ( θ z , α ) = power of the mode in the core total guided power of the mode
1 2 sin 2 θ z k 1 d cos 2 α c ( sin 2 α sin 2 α c ) 1 / 2 ,
γ z ξ η ( θ z , α ) ,
s = [ l P c 1 P co ] sin α ,
l = d cos α / sin 2 θ z .
P c 1 P co = 1 η ( θ z , α ) η ( θ z , α ) .
η = 1 + T / l ξ .

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