Abstract

Several different expressions for the Goos–Hänchen shift have been given in the literature. By looking at the problem from the point of view of the conservation of energy, new and more accurate expressions can be derived. For light waves, classical optics is used; for waves associated with particles, a quantum-mechanical approach is used. A comparison is made of the results of the present work with the earlier theories.

© 1964 Optical Society of America

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References

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  1. R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), 3rd ed., p. 418.
  2. F. Goos and H. Hänchen, Ann. Physik 1, 333 (1947).
    [Crossref]
  3. F. Goos and H. Lindberg-Hänchen, Ann. Physik 5, 251 (1949).
    [Crossref]
  4. K. Artmann, Ann. Physik 2, 87 (1948).
    [Crossref]
  5. C. V. Fragstein, Ann. Physik 4, 271 (1949).
    [Crossref]
  6. J. Picht, Ann. Physik 3, 433 (1929).
    [Crossref]
  7. J. Picht, Z. Phys. 30, 905 (1929).
  8. J. Picht, Optik 12, 41 (1955).
  9. C. Schaefer and R. Pich, Ann. Physik 30, 245 (1937).
    [Crossref]
  10. K. Artmann, Ann. Physik 7, 209 (1950).
    [Crossref]
  11. K. Artmann, Ann. Physik 8, 270, 285 (1950).
    [Crossref]
  12. K. Artmann, Ann. Physik 15, 1 (1954).
    [Crossref]
  13. C. V. Fragstein and C. Schaefer, Ann. Physik 12, 84 (1953).
    [Crossref]
  14. H. Hora, Optik 17, 409 (1960).
  15. P. Debye, Ann. Physik 30, 755 (1909).
    [Crossref]
  16. H. Arzelies, Ann. Phys. (Paris) 12/1, 5 (1946).
  17. H. Arzelies, Rev. Opt. 27, 205 (1948).
  18. Born and Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 46–50.
  19. The critical angle of total rellection is denoted in this paper by ic. It should be noted that German authors usually use iG and French authors use il. Sometimes the angle of incidence is designated by α instead of i.
  20. A referee has pointed out that the experimental optical systems used in studying the Goos–Hänchen shift may not match the ideal optics assumed in the theoretical treatment because of the difficulty of removing contaminant films and of getting a good optical homogeneity, a perfect surface polish and flatness, and parallelism of the plates used.
  21. We have used j for √ − 1 in order to avoid the confusion with the symbol i used for the angle of incidence.
  22. J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Company, Inc., New York, 1960), p. 107.

1960 (1)

H. Hora, Optik 17, 409 (1960).

1955 (1)

J. Picht, Optik 12, 41 (1955).

1954 (1)

K. Artmann, Ann. Physik 15, 1 (1954).
[Crossref]

1953 (1)

C. V. Fragstein and C. Schaefer, Ann. Physik 12, 84 (1953).
[Crossref]

1950 (2)

K. Artmann, Ann. Physik 7, 209 (1950).
[Crossref]

K. Artmann, Ann. Physik 8, 270, 285 (1950).
[Crossref]

1949 (2)

F. Goos and H. Lindberg-Hänchen, Ann. Physik 5, 251 (1949).
[Crossref]

C. V. Fragstein, Ann. Physik 4, 271 (1949).
[Crossref]

1948 (2)

K. Artmann, Ann. Physik 2, 87 (1948).
[Crossref]

H. Arzelies, Rev. Opt. 27, 205 (1948).

1947 (1)

F. Goos and H. Hänchen, Ann. Physik 1, 333 (1947).
[Crossref]

1946 (1)

H. Arzelies, Ann. Phys. (Paris) 12/1, 5 (1946).

1937 (1)

C. Schaefer and R. Pich, Ann. Physik 30, 245 (1937).
[Crossref]

1929 (2)

J. Picht, Ann. Physik 3, 433 (1929).
[Crossref]

J. Picht, Z. Phys. 30, 905 (1929).

1909 (1)

P. Debye, Ann. Physik 30, 755 (1909).
[Crossref]

Artmann, K.

K. Artmann, Ann. Physik 15, 1 (1954).
[Crossref]

K. Artmann, Ann. Physik 7, 209 (1950).
[Crossref]

K. Artmann, Ann. Physik 8, 270, 285 (1950).
[Crossref]

K. Artmann, Ann. Physik 2, 87 (1948).
[Crossref]

Arzelies, H.

H. Arzelies, Rev. Opt. 27, 205 (1948).

H. Arzelies, Ann. Phys. (Paris) 12/1, 5 (1946).

Born,

Born and Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 46–50.

Debye, P.

P. Debye, Ann. Physik 30, 755 (1909).
[Crossref]

Fragstein, C. V.

C. V. Fragstein and C. Schaefer, Ann. Physik 12, 84 (1953).
[Crossref]

C. V. Fragstein, Ann. Physik 4, 271 (1949).
[Crossref]

Goos, F.

F. Goos and H. Lindberg-Hänchen, Ann. Physik 5, 251 (1949).
[Crossref]

F. Goos and H. Hänchen, Ann. Physik 1, 333 (1947).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, Ann. Physik 1, 333 (1947).
[Crossref]

Hora, H.

H. Hora, Optik 17, 409 (1960).

Lindberg-Hänchen, H.

F. Goos and H. Lindberg-Hänchen, Ann. Physik 5, 251 (1949).
[Crossref]

Pich, R.

C. Schaefer and R. Pich, Ann. Physik 30, 245 (1937).
[Crossref]

Picht, J.

J. Picht, Optik 12, 41 (1955).

J. Picht, Ann. Physik 3, 433 (1929).
[Crossref]

J. Picht, Z. Phys. 30, 905 (1929).

Schaefer, C.

C. V. Fragstein and C. Schaefer, Ann. Physik 12, 84 (1953).
[Crossref]

C. Schaefer and R. Pich, Ann. Physik 30, 245 (1937).
[Crossref]

Slater, J. C.

J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Company, Inc., New York, 1960), p. 107.

Wolf,

Born and Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 46–50.

Wood, R. W.

R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), 3rd ed., p. 418.

Ann. Phys. (Paris) (1)

H. Arzelies, Ann. Phys. (Paris) 12/1, 5 (1946).

Ann. Physik (11)

P. Debye, Ann. Physik 30, 755 (1909).
[Crossref]

F. Goos and H. Hänchen, Ann. Physik 1, 333 (1947).
[Crossref]

F. Goos and H. Lindberg-Hänchen, Ann. Physik 5, 251 (1949).
[Crossref]

K. Artmann, Ann. Physik 2, 87 (1948).
[Crossref]

C. V. Fragstein, Ann. Physik 4, 271 (1949).
[Crossref]

J. Picht, Ann. Physik 3, 433 (1929).
[Crossref]

C. Schaefer and R. Pich, Ann. Physik 30, 245 (1937).
[Crossref]

K. Artmann, Ann. Physik 7, 209 (1950).
[Crossref]

K. Artmann, Ann. Physik 8, 270, 285 (1950).
[Crossref]

K. Artmann, Ann. Physik 15, 1 (1954).
[Crossref]

C. V. Fragstein and C. Schaefer, Ann. Physik 12, 84 (1953).
[Crossref]

Optik (2)

H. Hora, Optik 17, 409 (1960).

J. Picht, Optik 12, 41 (1955).

Rev. Opt. (1)

H. Arzelies, Rev. Opt. 27, 205 (1948).

Z. Phys. (1)

J. Picht, Z. Phys. 30, 905 (1929).

Other (6)

Born and Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), pp. 46–50.

The critical angle of total rellection is denoted in this paper by ic. It should be noted that German authors usually use iG and French authors use il. Sometimes the angle of incidence is designated by α instead of i.

A referee has pointed out that the experimental optical systems used in studying the Goos–Hänchen shift may not match the ideal optics assumed in the theoretical treatment because of the difficulty of removing contaminant films and of getting a good optical homogeneity, a perfect surface polish and flatness, and parallelism of the plates used.

We have used j for √ − 1 in order to avoid the confusion with the symbol i used for the angle of incidence.

J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Company, Inc., New York, 1960), p. 107.

R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), 3rd ed., p. 418.

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Figures (4)

F. 1
F. 1

Total reflection. I is the incident beam. The reflected beam RI shows a behavior similar to a partially reflected beam; the reflected beam RII shows the behavior expected by Goos and Hänchen. The spacing d is the Goos–Hänchen shift.

F. 2
F. 2

Illustration of the Goos-Hänchen shift for a finite plane wave. W is the limited incident plane wave. Φ1 and Φ2 are the two fluxes of energy, the equality of which is implied by the principle of conservation of energy.

F. 3
F. 3

Illustration of the system of axes used in calculating the shift for a light beam of width L, incident at angle i. I is the incident wave, R the reflected wave. S is the surface of separation between the two media and P is the plane of incidence.

F. 4
F. 4

Illustration of the coordinate axes used for the quantum-mechanical treatment of the shift in the case of a beam of particles. The potential V in region I (x<0) is taken less than potential V′ in region II (x>0). The set of axes ( OX , OY ) is chosen such that is OX parallel to the direction of incidence D. OY is perpendicular to OX .

Tables (1)

Tables Icon

Table I Numerical comparison between experimental results and several theories. The quantities k and k are related to the observed quantities d and d by Eq. (12).

Equations (59)

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M 2 2 = 64 π 2 D 1 2 K 1 μ 1 · cos 2 i ( 2 sin 2 i K μ ) μ 2 cos 2 i + sin 2 i K μ .
M 1 = [ 4 π / ( K 1 μ 1 ) 1 2 ] D 1 .
Φ 2 = L 32 π 2 ( μ 2 K 2 ) 1 2 1 γ 2 2 1 + γ 2 2 λ 2 γ 2 M 2 2 ,
Φ 1 = ( L d / 8 π ) ( μ 1 / K 1 ) 1 2 M 1 2 ,
γ 2 2 = 1 n 2 / sin 2 i , sin 2 i > n 2 = K μ , V 1 = 1 / ( K 1 μ 1 ) 1 2 , V 2 = 1 / ( K 2 μ 2 ) 1 2 ,
d = d = 1 π · μ n sin i cos 2 i μ 2 cos 2 i + sin 2 i n 2 · λ 2 ( sin 2 i n 2 ) 1 2 .
d = 1 π · μ sin i cos 2 i μ 2 cos 2 i + sin 2 i n 2 · λ 1 ( sin 2 i n 2 ) 1 2 .
d ( sin i / π ) [ λ 1 / ( sin 2 i n 2 ) 1 2 ] ,
d ( 1 / π n 1 ) [ n 2 λ 1 / ( sin 2 i n 2 ) 1 2 ] .
d = 1 π · K sin i cos 2 i K 2 cos 2 i + sin 2 i n 2 · λ 1 ( sin 2 i n 2 ) 1 2 .
d ( sin i / π n 2 ) [ λ 1 / ( sin 2 i n 2 ) 1 2 ] ,
d ( 1 / n 2 ) ( 1 / π n 1 ) [ n 2 λ 1 / ( sin 2 i n 2 ) 1 2 ] .
d = π 1 [ λ 1 / ( 1 n 2 ) 1 2 ]
d = ( π n 2 ) 1 [ λ 1 / ( 1 n 2 ) 1 2 ] .
d ( , ) = k ( , ) n 2 [ λ 1 / ( sin 2 i n 2 ) 1 2 ] ,
k = 1 π n 1 , k = 1 n 2 · 1 π n 1 = 1 n · 1 π n 2 ,
k = sin i π n 2 , k = 1 n 2 · sin i π n 2 ,
k = 1 π · cos 2 i 1 n 2 · sin i n 2 , k = 1 π · n 2 cos 2 i n 4 cos 2 i + sin 2 i n 2 · sin i n 2 ,
( 2 / 2 m ) 2 ψ + υ ψ = i ( ψ / t ) ,
sin i c = [ ( E V ) / ( E V ) ] 1 2 .
U 1 = F exp [ j ( 2 π / h ) ( x p x + y p y ) ] .
U 2 = G exp [ j ( 2 π / h ) ( x p x + y p y ) ] .
U = H exp [ j ( 2 π / h ) ( x p x + y p y ) ] .
p x = [ 2 m ( E V ) ] 1 2 cos i ,
p x = { 2 m [ ( E V ) ( E V ) sin 2 i ] } 1 2 ,
p y = [ 2 m ( E V ) ] 1 2 ( sin i ) .
G F = cos i ( sin 2 i c sin 2 i ) 1 2 cos i + ( sin 2 i c sin 2 i ) 1 2 ,
H F = 2 cos i cos i + ( sin 2 i c sin 2 i ) 1 2 .
U 1 = F exp [ j ( 2 π / h ) ( x p x + y p y ) ] .
U 2 = G exp [ j ( 2 π / h ) ( x p x + y p y ) ] .
U = H exp [ ( 2 π / h ) x p x ] exp [ j ( 2 π / h ) y p y ] .
p x = [ 2 m ( E V ) ] 1 2 cos i ,
p x = { 2 m [ ( E V ) sin 2 i ( E V ) ] } 1 2 ,
p y = [ 2 m ( E V ) ] 1 2 ( sin i ) .
H / F = 2 cos i / [ cos i + j ( sin 2 i sin 2 i c ) 1 2 ] ,
G / F = exp ( j χ ) ,
χ = 2 arctan [ ( sin 2 i sin 2 i c ) / ( 1 sin 2 i ) ] 1 2 .
ψ 0 = c { [ 2 m ( E V ) ] / a } 1 2 × i 0 + i 0 exp { j 2 π h ( x cos i y sin i ) [ 2 m ( E V ) ] 1 2 } d i .
ψ 0 R = c [ 2 m ( E V ) a ] 1 2 i 0 + i 0 exp ( j χ ) · exp { j ( 2 π / h ) ( x cos i + y sin i ) [ 2 m ( E V ) ] 1 2 } d i .
χ ( sin i ) = ν = 0 1 ν ! [ ν χ ( sin i ) ( sin i ) ν ] i 0 ( sin i sin i 0 ) ν ,
χ = χ ( sin i 0 ) + [ χ ( sin i ) ( sin i ) ] i 0 ( sin i sin i 0 ) + η ,
R = exp j { χ ( sin i 0 ) [ χ ( sin i ) ( sin i ) ] i 0 sin i 0 } ,
ψ 0 R = R C [ 2 m ( E V ) a ] 1 2 i 0 + i 0 exp ( j η ) × exp [ j 2 π h ( x cos i + { y + | χ ( sin i ) ( sin i ) | i 0 × 1 [ 2 m ( E V ) ] 1 2 · h 2 π } sin i ) [ 2 m ( E V ) ] 1 2 d i ] .
ψ 0 R ( x , y ) = R ψ 0 { x ; y + h 2 π [ 2 m ( E V ) ] 1 2 | χ ( sin i ) ( sin i ) | i 0 } .
d y = h 2 π [ 2 m ( E V ) ] 1 2 [ χ ( sin i ) ( sin i ) ] i 0 = h 2 π [ 2 m ( E V ) ] 1 2 [ χ i ] i 0 1 cos i 0 .
d = d y cos i 0 = h 2 π [ 2 m ( E V ) ] 1 2 [ χ i ] i 0 ,
( χ i ) i 0 = 2 sin i 0 ( sin 2 i 0 sin 2 i c ) 1 2 ,
h / [ 2 m ( E V ) ] 1 2 = λ ( de Broglie ) .
d = sin i 0 π · λ ( sin 2 i 0 sin 2 i c ) 1 2 ,
J = j ( / 2 m ) [ ψ * ψ ψ ψ * ] .
ψ = H exp ( 2 π h { 2 m [ ( E V ) sin 2 i ( E V ) ] } 1 2 x ) × exp { j 2 π h [ 2 m [ ( E V ) ] 1 2 · y ( sin i ) } × exp [ j 2 π E t h ] .
Φ 2 = 0 J y ( ψ ) d x ,
Φ 2 = ( sin i ) H * H × [ 2 ( E V ) / m ] 1 2 2 { ( 2 m / 2 ) [ ( E V ) sin 2 i ( E V ) ] } 1 2 ,
Φ 2 = ( sin i ) H * H 2 m ( sin 2 i sin 2 i c ) 1 2 .
ψ 1 = F exp { j ( 2 π / h ) [ 2 m ( E V ) ] 1 2 X } × exp [ j ( 2 π E t / h ) ] .
Φ 1 = F * F [ 2 ( E V ) / m ] 1 2 d .
d = ( sin i ) h 4 π [ 2 m ( E V ) ] 1 2 · H * H F * F · 1 ( sin 2 i sin 2 i c ) 1 2 .
H * H / F * F = 4 cos 2 i / ( cos 2 i + sin 2 i sin 2 i c ) .
d = sin i π · cos 2 i cos 2 i + sin 2 i sin 2 i c · λ ( sin 2 i sin 2 i c ) 1 2 ,