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# THIRD

## SERIES' VOL. 5, NO. 4

15 February 1972
Calculation and Experimental Proof of the Transverse Shift Induced
by
Total Internal Reflection of a Circularly Polarized Light Beam
Christian Imbert
Institut d'Optique, Faculte des Sciences, 91-0rsay, France
(Received 20 July 1970)
Wiegrefe, Fedorov, Costa de Beauregard, and Schilling have discussed the transverse en-
ergy flux existing in total reQection of an elliptically polarized light beam, the latter two pro-
posing formulas for the transverse shift of the reQected beam. We have calculated the trans-
. verse shift by an energy-flux-conservation argument similar to Kristoffel's and to Renard's
in their deduction of the longitudinal Goos-Hanchen shift, thus obtaining a formula different
from those of the previous authors. We have also tested experimentally the existence of the
transverse shift, in the optimal case of circular polarization and quasilimit total reQection,
by using two slightly different multiplying procedures. Our measurements definitely vindi-
cate our own formula for the transverse shift against both Costa de Beauregard's and Schil-
ling's. The relevance of our results in connection with noncollinearity of velocity and momen-
tum of the spinning photon inside the evanescent wave is very briefly discussed.
I. INTRODUCTION
It is now well known that the Poynting vector in-
side an inhomogeneous plane wave (that is, a for-
mally plane wave with a complex propagation vec-
tor) is, in general, oblique on the phase planes. It
seems that Boguslawski' was the first to call atten-
tion to this point in 1912 for the case of two inter-
fering plane waves propagating in an absorbing
medium. Soon after, Wiegrefe' considered in
1914-1916the much more interesting case of
Fresnel's evanescent wave, where the real and
imaginary parts of the propagation vector are or-
thogonal to each other and there is no energy
ab-
sorption. He calculated the Poynting vector inside
the evanescent wave generated by an incident plane
wave of arbitrary linear polarization, thus display-
ing, in general, a nonzero component of this vector
normal to the incidence plane. It is surprising that
he did not mention that this remarkable transverse
energy flux turns out to be considerably stronger
when the incident beam is elliptically polarized,
and is indeed maximal (given the incidence angle)
when the evanescent wave is circularly polarized.
Rose and Wiegrefe' have attempted to prove ex-
perimentally the existence of this transverse ener-
gy
flux, but, in the light of subsequent work, it is
clear today that their approach was not the best.
In 1929 a remarkable phenomenon associated with
the classical, longitudinal, energy flux inside
Fresnel's evanescent wave was predicted theoreti-
cally on the basis of a stationary-phase argument
by
Picht': a parallel shift of the reflected light ray
inside the incidence plane, as if the photons per-
formed some tunneling inside the evanescent wave
before coming back in the medium of higher index.
This longitudinal shift of the reflected pencil was
indeed proved experimentally in 1947 by Goos and
Hanchen.
'
Later, Acloque and Guillemet6 and
Osterberg and Smith' demonstrated in a very strik-
ing fashion the tunneling of the photons inside the
evanescent wave.
In 1955 Fedorov' called attention to Wiegrefe's
papers and to the fact that an elliptic polarization
Copyright 1972 by the American Physical Society.
V88 CHRISTIAN IMBERT
of the incident beam entails a much larger trans-
verse energy flux. He produced a compact calcu-
lation of the Poynting vector inside Fresnel's eva-
nescent wave and, in the general case, announced
that a lateral shift of the reflected beam should be
associated with the transverse energy flux, just as
the Goos-Hanchen shift is associated with the lon-
gitudinal energy flux. However, he produced no
calculation of the expected transverse shift.
In 1964-1965 Costa de Beauregard, not aware of
the previous writings, approached the subject with
a different motivation: noncollinearity of velocity
and momentum of spinning particles,
'
which he dis-
cussed for propagating electron" or photon"
waves. In the total reflection of an incident pure
plane wave, translational invariance in the direc-
tion orthogonal to the incidence plane entails that
the photon's momentum has a zero component in
this direction, while, as previously said, in gen-
eral the energy flux has a nonzero component. A
Fourier analysis" clearly shows that this phenom-
enon is not an artifact inherent in an idealized sit-
uation, and that it is still present in the more re-
alistic case of a bundle of waves.
In 1965 Schilling" produced a synthetic calcula-
tion of the longitudinal and lateral shifts of a total-
ly reflected beam by using a stationary-phase ar-
gument. A peculiarity of his approach is that the
field quantities inside the evanescent wave do not
enter his calculations. As for the final result, he
added to the Noether type" formula for the Goos-
Hanchen shift a formula for the transverse shift
which, in the light of our own theoretical and ex-
perimental work, is of the correct sign and order
of magnitude, but is I/cos'i times too small (i
de-
noting the incidence angle).
Our" calculation of the new lateral shift essen-
tially uses the transverse energy flux inside Fres-
nel's evanescent wave; it consists of an energy-
conservation argument similar to the one used by
Kristoffel" and by
Renard" in their deductions of
the longitudinal Goos-Hanchen shift. Apart from
being compact and simple, it yields a value for the
new transverse shift that is not only different from
those given by- the previous authors,
""
but is also
unambiguously supported by our measurements.
"
Ricard" has also produced a calculation of the
longitudinal and transverse shifts of the reflected
beam which, though not based on the energy-flux-
conservation argument, yields exactly the same
results.
In our measurements it is clear that we could not
use the Goos-Hanchen' multiplying procedure, be-
cause then the transverse displacements would be
opposite at the alternating total reflections. There
are two obvious ways out of this difficulty. One is
to reverse the circular polarization of the incident
beam between two alternating total reflections; this
is possible (as explained below) by using a prism
the section of which is an isosceles triangle and
the two sides of which receive a metallic reflect-
ing coating. The other solution is to use additive
rather than alternating successive reflections, as
is possible inside a prism with a regular polygonal
section. These two techniques have been used with
equal success and complete consistency of the re-
sults.
The new transverse shift is much smaller than
the Goos-HKnchen longitudinal shift: At its maxi-
mum, that is, for circular polarization and for an
incidence angle very near the limiting angle of to-
tal reflection, it goes to a limit that is finite and
of the order of half a wavelength; let us recall that
the longitudinal shift goes in principle to, infinity
when the incidence angle approaches its limiting
value, and that it was in fact of some ten wave-
lengths in Goos and Hanchen's experiments. For
these reasons our experiment is a priori more
delicate than Goos and Hanchen's and additional
refinements are needed.
The first of these is that the circular polariza-
tion state of the beam must be preserved in sign
and in magnitude at each of the successive total
reflections. Fresne1.
's
formulas show that this will
be the case provided that each total reflection oc-
curs extremely near the limiting case. This, in
turn, requires high precision in the realization of
both of our multiplying prisms.
Contrary to Goos and Hanchen, it would be diffi-
cult for us to mark a zero point outside our beam.
Fortunately this difficulty can be appropriately by-
passed by marking our beam by a rectilinear ob-
ject and illuminating one half of it by right and the
other by left circularly polarized light: This will
double the effect.
Finally, since the effect we are looking for is of
the order of the wavelength, it is important that it
is not hidden inside a diffraction pattern. For this
reason our rectilinear object was a phase object.
"
II. CALCULATION OF THE LONGITUDINAL
AND TRANSVERSE SHIFTS OF THE REFLECTED
BEAM IN TOTAL REFLECTION
We will base our demonstration on the classical
Fresnel-Maxwell formulas for total reflection oc-
curring on the plane z =0
separating the vacuum
z &0 from a noriabsorbing homogeneous medium
z
~
0 with real permittivity c and permeability
p,
=
1, and thus of admittance
I'=Dc
and index of
refraction n =
cv e . It is noteworthy that every-
thing significant for us occurs in the evanescent
wave, that is, in vacuo; it is thus clear that we are
dealing with properties of the electromagnetic field
CALCULATION AND EXPERIMENTAL PROOF. . . V89
the reflection and transmission coefficients. Be-
tween them and the angular frequency cu one has
the formulas
glass
k,
=
k=n u&/c, k,
=
(d/c,
k,
.
n,
.
=k, n, ,
or nn, =n, .
(2)
vacuum
or air
Sy
FIG. 1. Transverse component of the Peynting vector
above and below the reflecting plane.
Total reflection occurs when c((
&
a(
=
1/n (that is,
nt
=no(, &1).
,
so that c(t is imaginary with the ex-
pression j(n(2
1)'/'.
The transmitted wave is then,
according to Fresnel's theory, a formal plane wave
with a complex propagation vector, the real k' and
imaginary 4,
'
parts of which are orthogonal to each
other. In this case the reflection and transmission
coefficients assume the values
in itself, the presence of matter merely creating
the appropriate boundary conditions.
The well-known Fresnel-Maxwell formulas for
the incident, i, reflected, r,
and transmitted (in
vacuum), f, plane waves are, apart from phase
factors
p,
E'=(r;Ei'i Ei
-&
Eij)
If
'
=
(-
Yr(Ei i
YE
ii t
Y&
(Ei ) t
E" =
(-yr, iE',y1E1,
-o.
;riiElj ),
H"
=(Yy;r(Ej, Yt'iiEll, Yet;t' E'),
E'
=(rt&iiEii &1E1, -&t&iiEii)
( yt~lE' ~IIEII ' +t~. JEJ.}.
with the symbols II and a denoting linear polariza-
tion parallel and perpendicular to the incidence
plane
y
=
0; n, P
=-
0, and
y
the projecting cosines
of the propagation vectors k-, and the
r's
and
v's
y(&(
-
J(&( &( )
2 2 1/2
2
+j((2
2
(2
2)1/2
2y;~;
7
y+
2
+j((22+2)1
/2
2y
y+j((22+2)1
/2
entailing
Using (1), (2), (3), (4) and the definition of the
Poynting vector,
S=

, '(E*
x H+ Ex H*},
we find, in the evanescent wave,
(4)
S'
=(c(,/2o(, ) exp[2k;((2

c(t')' 'zj(7(. 11E(~E( +Tii TiiEii Eii (j/&()(o.'c((
} (1'1~iiE'*E(I

c.c.
),
0) (7)
and inside the medium, where the incident and reflected wave are superposed,
S' = 2Y(c(;(E'*E' (1+r*r +N*x, +Nr*)+EIl*EIl(1+rilrii+N*rii+Nr, *l), E1*EIi(N"rii Nr*)+c c
,
0-), . .(8)
with
N
~P*"P' =
exp[j(k*,

k, ) rI . (9)
Thus, either in the medium or in the vacuum,
there is an energy flux in the x direction, which
was of course expected, and no energy flux in the
z direction, which is also very natural. The im-
portant point for us is that there is in general a
nonzero energy flux in the
y
direction normal to
the incidence plane; Fig. 1 shows the z dependence
of the Scomponent, in the quasilimiting case,
for a left circularly polarized incident beam (right
circular polarization would yield the symmetrical
curve}.
(2 (}
x
gr j x
(10)
Now we recall Kristoffel's" and Renard's" meth-
od for calculating the longitudinal Goos-Hanchen'
shift. According to formulas (7},
where a common
real exponential factor is present, there is a net
energy flux inside the evanescent wave through the
semiplanes x=cte, 2 &0. Assuming (Fig. 2) that,
per unit length in the
y
direction, this flux
J
St
dz
is equal to the incident flux coming inside a prism
of oblique thickness
L,
and of course also to the
outgoing flux through a prism of the same thickness
L,
one obtains
V90 CHRISTIAN IMB ERT
C'
b
f
s
&X
FIG. 2. Kristoffel*s and Renard's reasoning for de-
ducing the formula of the Goos-85nchen longitudinal
shift through conservation of the energy flux.
Ly
FIG. 3. Our reasoning for deducing the formula of the
new transverse shift through conservation of energy flux
whence
ai(7&%&Ei Et.
+ 7
II TIIEll Ell )
x
2i (a
2
a. 2)1/2(1
a
2)1/2(EikEt
+
EitEt
)
This expression, first obtained by Kristoffel, is
in excellent agreement with the measured value of
the Goos-Hanchen longitudinal shift.
It is clear that we can apply an entirely similar
procedure for the transverse energy flux, thus
predicting (Fig. 3}a lateral shift of the reflected
beam with the expression
1
t
L
=
S'dz
S~
(12)
and explicit value
-j a
t
(r
*r
gi
E
'*E(')
7 t*v E')*E'
)
2ht
(1

a,
')'/'(E,
*'E'
+
E'*E')
' (13}
Additional remarks pertaining to the latter result
are (Fig. 1) the following:
(1) The integral
f
S',
"dz
is a sinusoidal function
of Z with zero mean value, so that
f
S',"dz
is
zero in the sense of Fourier integrals.
(2) As the discontinuity of S at z=0 is a simple
+6 0
step function,
f,
S,
dz -0 when e -0. Thus
f
Sdz
= f'"S,dz.
Among other cases, the transverse shift
L
is
zero when EI'I =0 or
Ei
~=0, which was evident from
a symmetry argument. Incidentally,
L,
is nonzero
in the case of oblique linear polarization of the in-
cident beamz (almost conserved in the quasilimit-
ing case); however, the transverse shift remains
very small in this case.
'
For given a, , tv( EiI!
and! Tt
E
J, II.
I
is maxi-
mum for ~ E,
'=+
jvIIE'II, that is, left or right cir-
cular polarization inside the evanescent wave. In
this case, formulas (2) and (7) yield +2&uS,
'=
S,
S',
in accord with Costa de Beauregard's" general
formula+2~,
=
B,S-B, S,. Thus, as far as the
magnitude of the effect is concerned, circular po-
larization of the incident beam is not the optimal
case. It is, however, the most convenient case,
and for that reason the one we have used in our
experimental studies. In the general case, the
formula
~ + i+ i
St
t J(~IIEII ~JEJ. TJ. EJ.. ~IIE ll)
tt
St
y
2 1It
EiWEi
1It'
EitItEg
8 x
II II Il II
(14)
follows from (2) and (7).
Using the general formulas (11) and (13) for the
longitudinal I.andtransverse
L,
shifts of the
reflected beam in the particular case of circular
polarization F.
'
=+
jEI'I, we find
i+T T
)
Lh
x
8v (a
2
a
2)1/2(]
'2)l/2 2& x x I
Lc
(c)
~X at(TJ Tll
+
Tgl
8tt a, (1

a
')'/'
(16)
8a, a,
'(1
a
)
2 2
+ 4(1 2)
O')
+ 0-')
(18)
where
v.
!I
and 7, denote the transmission coeffi-
cients of the two linear polarization states, a,
.
the
incidence angle's sine and
a,
=

## 1/n its limiting

value, X,
=
X/n the wavelength inside the medium;
the sign in (16) is negative for left and positive
for right circular polarization.
Using the well-known formulas
4 at'(1- ai')(1+ at'}
~
II+!I
+
~1.~g 4Ii 2x 2 2
&g
f1-
ei J+ ai

e)
CALCULATION AND EXPERIMENTAL PROOF. . .
AA BB
p
A
1
l
I 1
metallic
layer
FIG. 4. Principle of the Goos-Hanchen experiment.
we rewrite (15) and (16) as
~i'(1+ cii')(1 ~
)'"
27'
(CKi

CRg ) [CRi (1

R. )+lX

(Ri ]
FIG. 6. Longitudinal Landlateral
I,
shifts of a
rectilinear object CAB by total reflection of a circular-
ly polarized light beam.
whence
yi \$(1
2)1/2
&r
(1-
&i )+%
L(c)
(~i ni ~i )
L(c}
k T
t
X p
with
(2o)
(21)
shift would reverse each time, and no multiplica-
tion would occur. Thus we must either modify the
Goos-HKnchen technique in order to reverse the
helicity sign between two successive alternating
reflections, or use a method where the successive
reflection angles will have the same sign. We have
used both of these techniques with equal success
and complete consi. steney of the experimental. re-
sults.
2 2
7j.~a+ 7ll~tl ki +kt
(22) T 3}C
~j.
~
II
+ 7
l1
T~ 2k) kt
k,
=
nor/c and fi,
=
&o/c denote the lengths of the
propagation vectors inside the medium and in
vapo.
It thus turns out that the transverse shift we are
investigating is considerably smaller than the
longitudinal Goos-HKnchen shift. For A,
=
6328 A,
n
=
1.8, and an incidence angle 5 min above the
limiting angle, which are the conditions in one of
our experiments, we find
I.
''
=8.36X=5.3 p.
,
I.' =~0.46z=~0.3 p.
Our detecting procedure will then have to be
more sensitive than Goos and HKnchen's. Fortu-
nately, the helicity dependence of I.
'~
will allow an
obvious differential procedure for doubling the
value of the measured displacement: We wiQ il-
luminate half of the field with left and the other
half with right circularly polarized light. As for
multiplying the effect by using N consecutive re-
flections, we cannot use alternating reflections on
a parallel face plate as Goos and HKnchen did
(Fig. 4), because then the sign of the transverse
III. EXPERIMENTAL METHOD
In order to test the transverse shift we are ex-
pecting, we must first mark the beam. To this
object
(a)
coated faces
~gF
&-------c
FIG. 5. The '%Volter object.
"
(b)
FIG. 7. The multiplying prism E: (a) cross-section
view; (b) perspective view.
792 CHRISTIAN IMBERT
end we use a rectilinear object which, in order to
reduce the diffraction pattern, we take to be a
W'olter
plate; that is, a paraUel face plate half of
which is covered by a layer of thickness such that
a w phase shift is introduced (Fig. 5). Figure 6
then shows how the image
C'A'8'
of the rectilinear
object CAB will be shifted, in a total reflection,
by the longitudinal amount
L,
and the lateral
amount L,
.
As previously said, we have used two different
multiplying prisms. The first one, F, has an index
n
=
1.52 and its section is an isosceles triangle
[Fig. 7(a)] with angles such that a beam undergoing
quasilimit total reflection on the basis \$T tra-
verses orthogonally the two sides SR and RT.
These sides are coated with a semitransparent
metallic layer reflecting part of the beam inside
the prism, as in a Parrot-Fabry interferometer.
I'EG. 8. The multiplying prism P {cross section).
Ni,
FIG. 9. The multiplying prism P, perspective view: {a) apparatus in working condition;
{b) slightly deranged apparatus showing light escaping.
FIG. 10. Over-all experi-
mental arrangement.
We know that circular polarization, including the
helicity sign, is preserved by the quasi-limit total
reflection on ST, whereas the helicity is obviously
reversed by each normal reflection on the sides
Sg and R 7, so that the successive transverse
shifts mill add. In order to separate the succes-
sive emerging beams we have given a slight ob-
liquity to the incidence planes with respect to the
cross-section plane of the prism [Fig. V{b)];
this
allowed us to verify that the circular polarization
of the incident beam was indeed preserved in each
of the 20 outgoing beams. The advantage of this
procedure is its intrinsic symmetry (or quasi-
symmetry), which facilitates both measurements
and discussions of the results. Its dramback is
that it absorbs much energy.
Our second multiplying prism, P, has an index
n
=
1.8 and its section is an equilateral triangle.
The light beam inside follows a helical polygon
(Figs. 9 and
9), the slope of which is controllable
in order to adjust the reflection angles very near,
and slightly above, the limiting value of total re-
flection. These angles have additive projections
on the section planes of the prism. In this case
we use 28 total reflections. Figure 9(b) shows a
slightly deranged experimental apparatus produc-
ing outgoing beams, while Fig. 9(a) shows the
prism in working conditions.
Figure I0 displays the whole experimental ar-
rangement. A laser beam, with horizontal linear
polarization, traverses a quarter-wave plate and
then a half-wave plate covering half of the beam;
the resulting beam, which illuminates the recti-
linear object CAB, is thus polarized circularly
with opposite helicities on its two halves. This
mill produce the differential doubling effect we
have alluded to. After illuminating the xectilinear
object CAB, the beam enters either of our two
multiplying prisms, and the image of CAB is fin-
ally observed. The high luminosity of the prism
P allows this image to be projected on a screen,
as shown in Fig. 10.
According to the previous explanation, it is ex-
pected that the final image, as processed by the
prism, assumes the form Z, or E, (Fig. 10) ac-
cording to the possible associations of the circu-
lar polarization states with the left and right
halves of the beam. It is of course easy to switch
fx om one association to the other by a 90 rotation
of the-quarter-wave plate in its plane, while a 180
rotation restores the initial state of affairs.
IV. EXPERIMENTAL DATA
The experimental results are shown on the two
photographs in Fig. 11, where the circular polari-
zation states of the two halves of the beam have
been indicated. It is seen that the expected effect
does exist, with the right sign.
%'e
have verified
that a
180'
rotation of the quarter-mave plate
restores in fact the original configuration, thus
excluding artifacts due to prism effects (inciden-
tally, it is hard to imagine w'hich prism effect
could cause the observed configurations, as the
quarter-wave plate is placed before the linear
object).
The measured magnitude of the effect has also
come out quite right, either with prism I or P.
As our most precise measurements were per-
formed with the P prism, we will presently dis-
cuss this case.
Due to the helical light path in the P prism, the
expected transverse shift per x eflection is I,'=
I,
~ cos8, where g denotes the angle between the inci-
dence plane and the cross-section plane of the
prism, to which the linear object CAB is parallel.
, CHRISTIAN IMBERT
FIG. 11. Image of a rectilinear object displaying the transverse shift of a circularly polarized light beam. The circular
polarization states of the two halves of the beam are indicated.
A straightforward calculation yields, in the limit-
ing case of total reflection where +=1/n, cos8
=[(n'-1)/3]' '; that is, with n= 1.8,
cos6= 0.885.
For confirmation we have also measured the angle
8'
between the light beam and the cross-section
plane, and found
8'=16';
as the relation cos8= ~n
x cos
g'
holds, we obtain in this way cose
=
0.87.
Now, as we have said, the wavelength of the
laser radiation was A, =0.6328 p.
,
whence, accord-
ing to formula (20),
L'=0.288 p, and
L'~'=0.
288
x0.865 p, . Taking into account the 28 successive
reflections and the measured magnification 300 of
our optical system, we finally obtain
Ly tgeor
=4.2 mm.
The measurements have yielded
y meas
=
4.I + 0.3 mm.
We thus conclude that the predicted effect indeed
exists with the right sign and the right magnitude.
V. ADDITIONAL VERIFICATIONS
AND REMARKS
Interposing a circular-light analyzer on the out-
going beam, we have verified, with both prisms
E and P, that the initial circular polarization is
conserved (as previously said, with the prism F
this verification was made for each individual re-
flection).
CALCULATION AND EXPERIMENTAL PROOF. . .
This preservation of the rotational symmetry of
the beam excludes artifacts that could come, with
prism I', from the longitudinal Goos-K6nchen
effect via the nonzero angle
P
between two consec-
utive lneldence planes.
An easy calculation yields p= 0.214m, so that
after 28 reflections the incidence plane is rotated
by 6m, thus excluding all possibility of testing the
Goos-Hhnchen displacement with this apparatus.
This we have nevertheless verified, by placing the
rectilinear object CAB perpendicular to the cross-
section plane of the prism, and illuminating each
of its halves with orthogonal states of linear po-
larization. As expected, we observed that under
these conditions the final image of our object re-
mained perfectly straight, thus excluding any con-
tribution from the Goos-Hanchen effect. (Inciden-
tally, it is easily seen that any multiple of four
reflections, as is 28, will exactly compensate the
Goos-HKnchen effect in its horizontal projection,
which is not the one we are using.
)
VI. CONCLUSIONS
One important conclusion of our work is to se-
lect, among the different formulas that have been
proposed for the transverse shift, the one that is
best supported by the experimental measurements.
It is certainly astonishing that, among the numer-
ous authors who have been concerned with the
transverse energy flux, only three have proposed,
before our experimentation, a formula for the
transverse shift, and that all three formulas are
different. Let us first recall them, for the ease of
e1rculRr polRr1ZRt1on 1nslde the evRQeseent wRve
Rnd the limiting value
i,
of the incidence angle":
x'
Costa de Beauregard (Ref. 11), 1964: I,'=+4
g ant'
f
Schilling (Ref. 13),1965:
xtani,
'
Imbert (Ref. 15), 1966, A.
'
I'=+
Rtcard (Ref. 19),
19't0
'
"
w sini, cosi,
Thus our value is 1/cos'i, times larger than
Schilling's and 4/cos'i, times larger than Costa
de Beauregard's. In the case of the prism E,
n=1.52, sini, =0.658, cos i,
=0.570, and icos i,
=
0.l42. In the case of the prism P,
n= 1.
8, sini,
=0.555, cos'i, =0.690, and

',
cos i,
=O.IV2. Quite
apart from tke fact that our reasoning is muck
more direct than both Costa de Beauregard's and
Schilling s, our experi Rental tPleasuretents ar8
definitely consistent cuith our formula and incom-
patible arith both Costa de Beauregard's and
Schilling's.
Another point of interest is the answer appropri-
ate to the question,
"Does our experimental result
entail or not that the velocity and momentum of
the spinning photon are noneollinear inside the
vacuum of Fresnel s evanescent waveF' We feel
that we have the right to express our feeling re-
garding this point, especially since the minority'
supporting the somewhat heretical view that they
are noncollinear has steadily increased through
the years and has recently received the strong
support of Hestenes's papers and Corben's24
paper and authoritative book.
When we rotate our quarter-wave plate by +90
in its plane, we observe a lateral shift of the re-
flected beam, that is, a shift from right to left
{or vice versa) of the transverse energy flux in-
side the evanescent wave. However, the expecta-
tion values of the photon momentum states 1.e.
of the Fourier components, are exactly the same
in the left or right circular polarization states of
the cylindrical incident beam. This is true also
for the inhomogeneous plane waves that stand in a
one-to-one correspondence with the incident plane
waves, each of which has its complex propagation
vector inside the same incidence plane as the cor-
responding incident plane wave. T/gsr8fore tks ro-
tation of our quarter-urave Plate cannot change the
probability distribution of the (complex) momentum
inside the evanescent zoave, zohereas it obviousLy
changes the energy fLux. Loosely speaking, it
does not change the momentum distribution, but
it does change the velocity distribution. A Fou-
rier-type analysis of the evanescent wave faith-
fully vindicates the foregoing views.
~
ACKNOWLEDGMENTS
The author expresses thanks to Dr. O. Costa de
Beauregard for having suggested the present theme
for theoretical and experimental study, and for his
many suggestions, criticism, and advice during
the work. His gratitude goes also to Professor A.
Kastler and Professor A. Marechal for their inter-
est in the problem and very active support. Also,
to Dr. J. Ricard, who has devoted much of his
thinking to these questions, for many stimulating
conversations and dlscusslons.
~8.
Boguslavrski, Physik. Z. 13, 393 (1912).
2A.
Wiegxefe, Ann. Physik 45, 465 (1914); 50, 2VV
(1916).
3H.
Rose and A. Wiegrefe, Ann. Physik 50, 281 (1916}.
4J. Pioht, Ann. Physik 3, 433 (1929); Physik. Z. \$0,
905 (1929).
SF.
Goos and H. HKnchen, Ann. Physik 1, 333 (1947);
5, 251 (1949).
796
CHRISTIAN IMBERT
~P.
Acloque and C. Guillemet, Compt. Rend. 250, 4328
(1960).
~H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54,
1078 (1964}.
F. I. Fedorov, Dokl. Akad. Nauk SSSR 105, 465 (1955).
BJ.
Frenkel, Z. Physik 37, 243 (1926); M. Mathisson,
ibid. 67, 270 (1931); Math. Ann. 107, 400 (1933); A. Pro-
ca, Ann. Phys. (Paris) 20, 347 (1933); O. Costa, de Beau-
regard, Compt. Rend. , 214, 904 (1942); J. Math. 22, 85
(1943); J. Weyssenhof and A. Raabe, Acta Phys. Polon.
9, 7 (1947); 9, 9 (1947); J. Weyssenhof, ibid. 9, 26
(1947); 9, \$4 (1947}; 9, 46 (1947).
O. Costa de Beauregard, Phys. Rev. 134, 8471 (1964);
Ann. Inst. Henri Poincare 2, A131 (1965}.
O. Costa, de Beauregard, Cahiers Phys. 18, 471
(1964); Phys. Rev. 139, 81446 (1965).
~20.
Costa. de Beauregard, Compt. Rend. 272, 81116
(1971).
~3H.
Schilling, Ann. Physik 16, 122 (1965).
~4F.
Noether, Ann. Physik 11, 141 (1931}.
~~C.
Imbert, Compt. Rend. 267, 81401 (1968).
~6N.
Kristoffel, Scient. Acta, Tartu State Univ.
,
1956
(private communication}.
~R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964).
C. Imbert, Compt. Bend. 269, 81227 (1969); 270,
8529 (1970); Phys. Letters 31A, 337 (1970).
~SJ.
Ricard, Nouv. Revue d'Optique 1, 275 (1970).
H. Wolter, Handbuch der Physik, edited by S. Flugge
(Springer, Berlin, 1956), Vol. 24, p. 582.
2~J.
Pavageau, Compt. Rend. 263, 8276 {1966).
In fact our formula is stationary near the limiting an-
gle.
23D.
Hestenes and R. GurQer, Am. J. Phys. 39, 1028
(1971); D. Hestenes, ibid. (to be published).
H. Corben, Nuovo Cimento 20, 529 (1961}; Classical
and Quantum Theories of Spinning Particles (Holden-
Da,
y,
San Francisco, 1968).
PHYSI CA L REVI EW D VOLUME 5, NUMBER 4 15 FEBRUARY 1972
Frequency Dependence of the Speed of Light in Space
Z. Bay and J. A. White*
National Bureau of Standards, S'ashington, D. C. 20234
(Received 16 August 1971)
To characterize the possible dispersion of the velocity of light in space (vacuum) a Cauchy-
type formula, n =1+A/v +8v, is used. It is shown that relativity only allows a nonzero
A term, independent of the nature of the waves or a quantization thereof. Recent experimen-
tal data provide upper bounds for A and B, limiting thereby the dispersion in the microwave,
infrared, visible, and ultraviolet regions of the spectrum to less than one part in
1020.
Recent observations of radio-wave,
'
visible,
'
and x-ray' emissions from pulsars have been in-
terpreted to provide experimental bounds on the
dispersion of light in interstellar space.
' '
The
dispersion or lack thereof has been discussed us-
ing the expression"'
modified Cauchy expression of the form'
n =1+

+Bv
A
V
or
(2)
to relate measurements in different regions of the
spectrum giving different limits of dispersion ~c
in the velocity. It was pointed out by
Brown' that
P
clearly cannot be a good constant to characterize
the variation of velocity with energy, because it
would be infinite for A.
,
=A.
2.
Even though that dif-
ficulty could be avoided by the introduction of
(X,

A, )/X,
in place of
X,/A.
,
in the definition of
p,
there remains the more serious objection against
the concept of
p,
in our opinion, that it suggests a
linear dependence of c on X.
It appears preferable to the present authors to
represent the dispersion (if there is any) via a
where n is defined by c~b,
=
co/n and
c, is the ve-
locity of light in the absence of dispersion. The
corresponding group velocities, to be used in the
analysis of the experimental data, are readily
calculated from the above expressions.
In anticipation of their use in regions of the
spectrum remote from resonances, only the lead-
ing terms in
v'
and (l/v)', or
X'
and (l/A)', are
retained in Eqs. (2) and (3). The absence of odd
powers of v or A. is assured by the presumed sym-
metry with respect to reversal of the direction of
time.
'
Expressions (2) and (3) describe the frequency
dependence of the speed of light in any dispersive
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