Observation of the Faraday effect via beam deflection in a longitudinal magnetic field

Abstract

We report the observation of the magnetic field induced circular differential deflection of light at the interface of a Faraday medium. The difference in the angles of refraction or reflection between the two circular polarization components is a function of the magnetic field strength and the Verdet constant. The reported phenomena permit the observation of the Faraday effect not via polarization rotation in transmission, but via changes in the propagation direction in refraction or in reflection. An unpolarized light beam is predicted to split into its two circular polarization components. The light deflection arises within a few wavelengths at the interface and is therefore independent of pathlength.

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arXiv:physics/0702063v1 [physics.optics] 7 Feb 2007
Observation of the Faraday effect via beam deflection in a longitudinal magnetic field
Ambarish Ghosh and Peer Fischer
The Rowland Institute at Harvard, Harvard University, Cambridge, Massachusetts 02142
We report the observation of the magnetic field induced circular differential deflection of light at
the interface of a Faraday medium. The difference in the angles of refraction or reflection between
the two circular polarization components is a function of the magnetic field strength and the Verdet
constant. The reported phenomena permit the observation of the Faraday effect not via polarization
rotation in transmission, but via changes in the propagation direction in refraction or in reflection.
An unpolarized light beam is predicted to split into its two circular polarization components. The
light deflection arises within a few wavelengths at the interface and is therefore independent of
pathlength.
PACS numbers: 33.55.Fi, 33.55.-b, 33.55.Ad, 78.20.Ek, 78.20.Fm, 78.20.Ls
Well established magneto-optical phenomena are the
Faraday, Cotton-Mouton-Voigt and the magneto-optical
Kerr (MOKE) effects, as well as magnetic circular dichro-
ism. These are described by changes in the azimuth (opti-
cal rotation) or the ellipticity of an electromagnetic wave.
Apart from changes in these Stokes parameters, a mag-
netic field may also influence the propagation direction
of light.
The deflection of a light beam in isotropic media sub-
ject to a homogenous [1] transverse magnetic field has
been reported by Rikken and Tiggelen who observed the
deflection in scattering [2] as well as in transmission [3],
and by Blasberg and Suter [4] who showed that angu-
lar momentum conservation causes a small lateral dis-
placement near resonance in an atomic vapor. Trans-
verse magnetic field induced refraction at the cesium va-
por/glass interface has been reported by Schlesser and
Weis [5]. However, their observed deflection appears to
be nonlinear in the strength of the magnetic field and
nonlinear in the light intensity [5]. A complete explana-
tion for this effect has not yet been provided [3, 5].
Here, we show that it is also possible to observe
magneto-optical deflection at an interface in the presence
of a longitudinal magnetic field. In particular, we show
that longitudinal magnetic field induced refraction and
reflection at an interface gives rise to a circular differen-
tial beam deflection, and that this is an alternate means
to determine Verdet constants.
A magnetic field renders any medium (isotropic or
oriented) optically active. In particular, any isotropic
medium becomes uniaxial in the presence of a magnetic
field, and its refractive indices for right- (+) and left-
(−) circularly polarized light are unequal, such that the
plane of polarization of a linearly polarized electromag-
netic wave rotates as the wave propagates along the di-
rection of the field. The Faraday rotation in radians de-
veloped by an electromagnetic wave at the wavelength λ
traversing a distance l is given by:
α = V B l =π l
λ
?
n(−)− n(+)?
(1)
where V is the frequency-dependent Verdet constant, and
B is the magnetic field strength. Magneto-optical ac-
δ
θ
(−)
θi
(+)
θ
B
θ
(−)
(+)
θ
B
δ'
(a)
(b)
+
n
( )
0 n
θi
FIG. 1: a) Double refraction at the boundary of a Faraday
medium (shaded) and a medium with negligible Verdet con-
stant characterized by the scalar refractive index n0. The
direction of the transmitted beam is different for left- and
right-circularly polarized waves. An unpolarized or a linearly
polarized beam will thus split into two. b) Reflection inside
a Faraday medium. The incident beam is subject to circu-
lar birefringence n(±)such that the left- and right- circular
components have different angles of reflection.
tivity should, however, not only manifest itself through
Faraday rotation in transmission, but should also be ob-
servable as a deflection of the light beam in reflection [6]
and in refraction at an interface. We have recently re-
ported the observation of related phenomena in the case
of natural optical activity (chirality) [7].
If one considers the refraction of an electromagnetic
wave at a boundary formed by a Faraday medium and
a medium with a negligible Verdet constant, as shown
in Fig. 1a, then the left- and the right-circularly po-
larized waves or wave-components must independently
obey Snell’s law. Circularly polarized waves that propa-
gate along the optical axis in the Faraday medium, will

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thus refract in the medium characterized by the polar-
ization independent refractive index n0 with angles of
refraction θ(−)and θ(+), depending on whether they
are, respectively, left- or right-circularly polarized. Sim-
ilarly, if an unpolarized or linearly polarized wave is in-
cident from the Faraday medium with angle of incidence
θi, then it will split into two beams, one left- and the
other right-circularly polarized. The angular divergence
δ = θ(+)−θ(−)between the two refracted circular polar-
ization components in Fig. 1a is
δ ≈
?n(+)− n(−)?
n0
sinθi
cosθ,
(2)
where θ is the average of the two angles of refraction.
It follows that B-field induced deflection of light at an
interface can be used to determine circular birefringences,
and hence Verdet constants, or in the case of a known
Verdet constant the strength of an applied magnetic field:
δ ≈ −λ sinθi
π n0cosθV B .
(3)
It is interesting to note that unlike Faraday rotation (Eq.
(1)), which is a function of the light path through the
medium, magneto-optical double refraction arises within
a few wavelengths at the interface. This could for in-
stance be of use in the study of ultrathin transparent
samples [7].
One may also consider the components of an electro-
magnetic wave that reflect inside the Faraday medium.
Because a circularly polarized wave reverses its circular-
ity upon reflection, the incident and the reflected waves
are necessarily associated with different refractive in-
dices. Hence, in an optically active medium the angle
of reflection of a circularly polarized wave will in gen-
eral not equal the angle of incidence [6, 7].
polarized or linearly polarized wave can therefore split
into its two circularly polarized components upon reflec-
tion [7]. The theoretical description of magnetic double
reflection is complicated by the fact that the reflected
wave no longer propagates along the optic axis of the
system. The reflected beam is thus potentially subject
to circular birefringence (Faraday effect) as well as the
birefringence due to a transverse magnetic field [3]. To
simplify the discussion we will consider a reflected beam
that propagates in a direction perpendicular to the mag-
netic field as shown in Fig 1b, such that it experiences
no birefringence due to the longitudinal component of
the magnetic field. Furthermore, we neglect any trans-
verse B-field induced birefringence and assume that the
reflected waves are only subject to an average refractive
index n = (n(−)+ n(+))/2. The angular divergence δ′is
then given by
An un-
δ′≈ −λ tanθi
π n
V B .(4)
Depending on the Fresnel reflection coefficients for a
given interface, the circular components may not fully
(10-6 radian)
δ
Magnetic field (Tesla)
light source

polarizer
PEM
sample
cell
PSD
reflection
refraction
B
0.00.51.0
0
1
2
0.00.51.0
0
1
2
3
refraction
reflection
PSD
a)
b)
c)
FIG. 2: a) Schematic of the experimental arrangement. Light
passes through a photoelastic modulator (PEM) where its po-
larization is switched between left- and right-circular at ∼50
kHz before refracting in a glass prism (solid line), or reflect-
ing inside a liquid cuvette (dotted line). The prism and the
prismatic cuvette are respectively mounted inside an electro-
magnet (not shown). The position of the beam is recorded
with a position sensitive detector (PSD). b) Refraction data
for SF11 glass at 532 nm. c) Reflection data for pure CS2 at
∼473 nm.
reverse their circularity upon reflection and become ellip-
tically polarized. This can be accounted for by including
the appropriate Fresnel coefficients.
We have observed the magnetic field induced double
refraction and reflection phenomena in an experimental
arrangement schematically depicted in Fig 2a. In the re-
fraction measurements light from a 532 nm diode laser
was modulated between left- and right-circular polarized
with a Hinds photoelastic modulator (PEM) at ∼ 50
kHz and then passed through a transparent glass (SF11)
prism mounted between the pole pieces of an electromag-
net (Walker HV7). The position of the laser beam was
recorded with a position sensitive diode and a lockin am-
plifer. The difference in the angles of refraction for the
two circular polarization components is shown in Fig. 2b
as a function of the applied magnetic field strength. From
a linear fit to the data and using Eqn. (3) a Verdet con-
stant of 29.37 ± 0.96 T rad−1m−1is obtained. This is in
good agreement with the tabulated values of the Verdet
constant for SF11 Schott glass [8], from which we ex-
trapolate a Verdet constant of 30.44 T rad−1m−1. Each
data point in Fig. 2b is measured with an uncertainty
that is approximately the size of the symbol. However,
the photoelastic modulator itself gives rise to an angular
deviation that fluctuates on the time scale of the mea-
surement and causes the data points to deviate from a

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straight line. The goodness of the straight-line fit is thus
a more appropriate measure of the experimental error.
The circular differential reflection in a longitudinal mag-
netic field was observed in a right-angle prismatic cuvette
filled with carbon disulfide (CS2). A mirror was mounted
parallel to the hypotenuse on the inside of the liquid cu-
vette. The polarization modulated beam from a ∼473
nm diode pumped solid state laser travelled along the
direction of the magnetic field in the liquid and upon re-
flection exited the liquid perpendicular to the magnetic
field and normal to the window of the cuvette. From
the detected angular divergence shown in Fig. 2c we de-
duce a Verdet constant of 23.7 ± 0.8 T rad−1m−1. We
suspect that the small difference with the reported con-
stant of 0.0694 min G−1cm−1(20.2 T rad−1m−1) at
476.5 nm [9] is due to ellipticity in the reflected beam,
which has not been accounted for. We stress that even
though the experimental geometry is chosen such that
the medium is (approximately) uniaxial, the effects de-
scribed here will in general be exhibited by any wave that
refracts or reflects at the interface of a Faraday medium.
Similar effects are expected to arise in diffraction [10].
In summary, we have shown that the Faraday effect
can be observed via double refraction or reflection at an
interface in the presence of a longitudinal magnetic field.
We have demonstrated that the difference in the propa-
gation directions of the two refracted (or reflected) cir-
cular polarization components is an alternative means to
determine Verdet constants or magnetic field strengths.
The effects reported here distinguish themselves in a
number of ways from magneto-optical measurements re-
ported hitherto. The magnetic double refraction and re-
flection phenomena may be observed both with polarized
or with unpolarized electromagnetic waves. Further, the
effects arise within a few wavelengths at the boundary,
and may thus find application in the study of ultrathin
samples. Finally, the phenomena do not suffer from the
n-π ambiguity, which can plague Faraday rotation mea-
surements in large magnetic fields [11], or in space as-
tronomy. This prompts us to ask whether the deflection
phenomena of this paper also manifest themselves in as-
trophysical observations.
[1] A light beam may also deflect due to a variation in the
index of refraction as a result of a magnetic field gradient:
R. Holzner et al., Phys. Rev. Lett. 78 3451 (1997).
[2] G. L. J. A. Rikken and B. A. van Tiggelen, Nature 381,
54 (1996).
[3] G. L. J. A. Rikken and B. A. van Tiggelen, Phys. Rev.
Lett. 78, 847 (1997).
[4] T. Blasberg and D. Suter, Phys. Rev. Lett. 69, 2507
(1992).
[5] R. Schlesser and A. Weis, Opt. Lett. 17, 1015 (1992).
[6] M. P. Silverman and R. B. Sohn, Am. J. Phys. 54, 69
(1986).
[7] A. Ghosh and P. Fischer, Phys. Rev. Lett. 97, 173002
(2006).
[8] Optical Glass Catalog, (Schott-Glass, Mainz, Germany,
1985). Verdet constant of SF11 glass: 18.3, 23.6, 28.5,
and 49.5 T rad−1m−1at respectively 656.3, 589.3, 546.1,
and 439.5 nm.
[9] A. B. Villaverde and D. A. Donatti, J. Chem. Phys. 71,
4021 (1979).
[10] F. Fazal, A. Ghosh, and P. Fischer, manuscript in prepa-
ration (2007).
[11] A different scheme to overcome the n-π ambiguity is pre-
sented here: M. Tatarakis, I. Watts, F. N. Beg, E. L.
Clark, A. E. Dangor, A. Gopal, M. G. Haines, P. A. Nor-
reys, U. Wagner, et al., Nature 415, 280 (2002).