We use the stationary phase method to determine the paths of optical beams that propagate through a dielectric block. In the presence of partial internal reflection, we recover the geometrical result obtained by using Snell's law. For total internal reflection, the stationary phase method overreaches Snell's law, predicting the Goos-Hänchen shift.

1.
M.
Born
and
E.
Wolf
,
Principles of Optics
(
Cambridge U.P.
,
Cambridge
,
1999
).
2.
B. E. A.
Saleh
and
M. C.
Teich
,
Fundamentals of Photonics
(
Wiley & Sons
,
New Jersey
,
2007
).
3.
M.
Kryjevskaia
,
M. R.
Stetzer
, and
P. R. L.
Heron
, “
Student understanding of wave behavior at a boundary: The relationships among wavelength, propagation speed, and frequency
,”
Am. J. Phys.
80
,
339
347
(
2012
).
4.
R.
Heller
, “
On the teaching of the Snell-Descartes law of refraction
,”
Am. J. Phys.
16
,
356
357
(
1948
).
5.
J. W.
Shirley
, “
An early experimental determination of Snell's law
,”
Am. J. Phys.
19
,
507
508
(
1951
).
6.
C. V.
Bertsch
and
B. A.
Greenbaum
, “
New apparatus for Snell's law
,”
Am. J. Phys.
26
,
340
(
1958
).
7.
H. E.
Bates
, “
An analogue model for teaching reflection and refraction of waves
,”
Am. J. Phys.
48
,
275
277
(
1980
).
8.
D.
Drosdoff
and
A.
Widom
, “
Snell's law from an elementary particle viewpoint
,”
Am. J. Phys.
73
,
973
975
(
2005
).
9.
J. J.
Lynch
, “
Snell's law with large blocks
,”
Phys. Teach.
45
,
180
182
(
2007
).
10.
L. I.
Shiff
,
Quantum Mechanics
(
McGraw-Hill
,
New York
,
1955
).
11.
C.
Cohen-Tannoudji
,
B.
Diu
, and
F.
Laloë
,
Quantum Mechanics
(
Wiley
,
Paris
,
1977
).
12.
S.
Longhi
, “
Quantum-optical analogies using photonic structures
,”
Laser Photon Rev.
3
,
243
261
(
2009
).
13.
X.
Chen
,
X. J.
Lu
,
Y.
Ban
, and
C. F.
Li
, “
Electronic analogy of the Goos-Hänchen effect: A review
,”
J. Opt.
15
,
033001-1
12
(
2013
).
14.
H. F.
Meiners
, “
Optical analog of quantum-mechanical barrier penetration
,”
Am. J. Phys.
33
,
xviii
(
1965
).
15.
P. L.
Garrido
,
S.
Goldstein
,
J.
Lukkarinen
, and
R.
Tumulka
, “
Paradoxical reflection in quantum mechanics
,”
Am. J. Phys.
79
,
1218
1231
(
2011
).
16.
G.
Zhu
and
C.
Singh
, “
Surveying students understanding of quantum mechanics in one spatial dimension
,”
Am. J. Phys.
80
,
252
259
(
2012
).
17.
S. De
Leo
and
P.
Rotelli
, “
Localized beams and dielectric barriers
,”
J. Opt. A
10
,
115001-1
5
(
2008
).
18.
S. De
Leo
and
P.
Rotelli
, “
Laser interacting with a dielectric block
,”
Eur. Phys. J. D
61
,
481
488
(
2011
).
19.
S. De
Leo
and
P.
Rotelli
, “
Resonant laser tunneling
,”
Eur. Phys. J. D
65
,
563
570
(
2011
).
20.
M.
Selmke
and
F.
Cichos
, “
Photonic Rutherford scattering: A classical and quantum mechanical analogy in ray and wave optics
,”
Am. J. Phys.
81
,
405
413
(
2013
).
21.
E.
Wigner
, “
Lower limit for the energy derivative of the scattering phase shift
,”
Phys. Rev.
98
,
145
147
(
1955
).
22.
R. B.
Dingle
,
Asymptotic Expansions: Their Derivation and Interpretation
(
Academic Press
,
London
1973
).
23.
N.
Bleistein
and
R.
Handelsman
,
Asymptotic Expansions of Integrals
(
Dover
,
New York
,
1975
).
24.
D. C.
Look
, “
Novel demonstration of total internal reflection
,”
Am. J. Phys.
49
,
794
(
1981
).
25.
E.
Richard
and
V.
Keuren
, “
Refractive index measurement using total internal reflection
,”
Am. J. Phys.
73
,
611
615
(
2005
).
26.
F.
Goos
and
H.
Hänchen
, “
Ein neuer und fundamentaler Versuch zur totalreflexion
,”
Ann. Phys.
436
,
333
346
(
1947
).
27.
S. R.
Seshadri
, “
Goos-Hänchen beam shift at total internal reflection
,”
J. Opt. Soc. Am. A
5
,
583
585
(
1988
).
28.
A.
Aiello
, “
Goos-Hänchen and Imbert-Federov shifts: A novel perspective
,”
New J. Phys.
14
,
013058-1
12
(
2012
).
29.
K. Y.
Bliokh
and
A.
Aiello
, “
Goos-Hänchen and Imbert-Fedorov beam shifts: An overview
,”
J. Opt.
15
,
014001-1
16
(
2013
).
30.
M.
Andrews
, “
The evolution of free wave packets
,”
Am. J. Phys.
76
,
1102
1107
(
2008
).
31.
F.
Mooney
, “
Snell's law equivalent to the conservation of tangential momentum
,”
Am. J. Phys.
19
,
385
(
1951
).
32.
F. P.
Zanella
,
D. V.
Magalhes
,
M. M.
Oliveira
,
R. F.
Bianchi
, and
L.
Misoguti
, “
Frustrated total internal reflection: A simple application and demonstration
,”
Am. J. Phys.
71
,
494
496
(
2003
).
33.
K.
Yasumoto
and
Y.
Oishi
, “
A new evaluation of the Goos-Hänchen shift and associated time delay
,”
J. Appl. Phys.
54
,
2170
2176
(
1983
).
34.
W.
van Dijk
and
K. A.
Kiers
, “
Time delay in simple one dimensional systems
,”
Am. J. Phys.
60
,
520
527
(
1992
).
35.
L. de la
Torre
, “
Wave packet distortion and time delay
,”
Am. J. Phys.
65
,
123
125
(
1997
).
36.
C.
Bahrim
and
W. T.
Hsu
, “
Precise measurements of the refractive indices for dielectrics using an improved Brewster angle method
,”
Am. J. Phys.
77
,
337
344
(
2009
).
37.
N. J.
Harrick
, “
Study of physics and chemistry of surfaces from frustrated total internal reflections
,”
Phys. Rev. Lett.
4
,
224
226
(
1960
).
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