Download Observing Angular Deviations in the Specular Reflection of a Light

PROPAGATION ’09
Observing Angular Deviations in the
Specular Reflection of a Light Beam
M. Merano, A. Aiello, M.P. van Exter and J.P. Woerdman
(a)
Air
in in

Glass
(b)
(c)
Polarization
modulator
< > s)
(p —
Glass
prism

Lens
Rotation
stage

Beam waist w0
Split detector
Lock-in
p-s
(versus )
100
<p >–<s > [rad]
T
he law of reflection states that when
a plane wave (or light ray) falls on
the boundary between two homogeneous
media, the angle of incidence is equal
to the angle of reflection. Plane waves
are infinitely extended and thus cannot
exist in nature. A beam is the closest approximation of a ray that we can use in a
laboratory. Is this law still valid for a light
beam? Starting in the 1970s, researchers
have predicted a small angular deviation
of the law of reflection for a beam.1,2,3
This is a diffractive consequence of the
fact that beams have finite extension.
The reflected beam maintains its
shape (if the angle of incidence is not
too close to the Brewster angle) but the
center of the reflected beam is angularly
displaced with respect to the predictions
of geometrical optics. The displacement
takes place in the plane of incidence;
it is proportional to the square of the
beam angular aperture and is polarization dependent: s or p polarized beams
suffer different angular shifts. Th is
phenomenon bears similarities with
the well-known Goos-Hänchen shift,4
a positional shift of the beam center
relative to its geometrical-optics position
that occurs in total internal reflection.
The angular deviation of the beam axis
instead occurs only in partial reflection.
We recently reported observing the
angular deviation of a beam upon reflection using a TEM00 Gaussian beam of
820 nm.5 The beam was reflected by an
air-glass (BK7) interface. Its position,
in the plane of incidence, was measured with a calibrated split detector.
We switched the incident polarization
between p and s and used synchronous
detection to deduce the polarizationdifferential angular shift of the beam,
<p >–<s >. This was done as a function of the angle of incidence . For
=20°-80°, agreement with the theoretical curve was very good.
75
50
25
0
–25
–50
–75
–100
20
30
40
50
60
70
80
 [degrees]
(a) A light beam hits an air-glass interface. The angular deviation of the axis of the reflected
beam, relative to the specular direction (), takes place in the plane of incidence and is
polarization dependent. (b) Reflection takes place at the surface of a right-angle glass prism
(n=1.51). Technical noise is suppressed by switching the polarization of the incident beam
between p and s, followed by synchronous detection of the signal produced by the split
detector. This yields the difference of the angular deviation for p versus s polarization. The
phenomenon is proportional to the square of the beam aperture angle (beam waist 59 μm).
(c ) Difference of the angular shift suffered by a p and s polarized beam as a function of the
angle of incidence . (Inset) Intensity profiles of the reflected beam, which remains Gaussian
within the experimental accuracy.
For simplicity, we emphasized an airglass interface. However, angular nonspecularity should occur for any mirror
with <100 percent reflectivity. We expect
this phenomenon to occur in optical
implementations of angular metrology.
Th is may happen in geodetic surveying, machine-tool operation, torsion
pendulum read-out and cantilever-based
surface microscopy. Another example is
the angular alignment of gravitational
wave detectors such as LIGO (a Michelson interferometer that implies oblique
incidence on a 50/50 beam splitter) or
LISA (a triangular interferometer).
An angular shift apparently violates
the conservation of light momentum that
is typically associated with the specular
reflection law; such conservation is a consequence of in-plane translational symmetry. However, one should also account for
the momentum carried by the transmitted
beam always present in partial reflection.
It follows then from Snell’s Law that
total momentum is conserved. 
M. Merano ([email protected]),
A. Aiello, M.P. van Exter and J.P. Woerdman are
with the Huygens Laboratory, Leiden University,
Leiden, the Netherlands.
References
1. J.W. Ra et al. SIAM J. Appl. Math. 24, 396-413 (1973).
2. Y.M. Antar and W.M. Boerner. Can. J. Phys. 52, 962-72
(1974).
3. C.C. Chan and C. Tamir. Opt. Lett. 10, 378-80 (1985).
4. F. Goos and H. Hänchen. Ann. Phys. (Leipzig) 1, 333-46
(1947).
5. M. Merano et al. Nature Photon. 3, 337-40 (2009).
December 2009 | 27
PROPAGATION
Curved Plasma Channel Generation in Air
Using Ultra-Intense Self-Bending Airy Beams
U
ltrafast laser filamentation is a rich,
interdisciplinary branch of physics
that addresses propagation of intense
laser pulses in transparent media.1,2 Applications range from terahertz generation to lightning control.
When an ultra-intense, ultra-short
laser pulse propagates in air, the defocusing effect of the plasma generated via
multi-photon ionization dynamically
balances the self-focusing of the beam
and prevents it from collapsing into a
singularity. The hot core of the beam,
composed of the high-intensity laser
field and generated plasma, is referred to
as the filament. Filaments are typically
about 100 µm in diameter and exhibit
self-guided, sub-diffractive propagation
over long distances.
High optical intensities inside filaments facilitate efficient nonlinear wavelength conversion, leading to forward
emission of broadband radiation. Analysis of the angularly resolved spectra of
this radiation yields insights into the
pulse propagation dynamics.3
In early studies of femtosecond laser
filamentation, axially symmetric beam
profiles were used, such as Gaussian,
flat-top and Bessel beams. Accordingly,
filaments were generated along straight
lines. The broadband forward emissions
generated at different points along a
straight filament tend to overlap in the
far-field, leading to spectra that are difficult to interpret.
We recently reported generating
optical Airy beams.4 These non-axially
symmetric beams are approximately
diffraction free, and their main intensity
features freely self-bend (or accelerate)
on propagation in the absence of any
refractive-index gradients.
We also conducted experiments on
the filamentation of ultra-intense Airy
beams in air.5 We observed an unusual
filamentation regime in which the linear
28 | OPN Optics & Photonics News
(a)
(b)
3
2
1
0
–40 –20
0
20
40
Distance from Fourier plane [cm]
(c)
Beam displacement [mm]
Pavel Polynkin, Miroslav Kolesik, Jerome Moloney, Georgios Siviloglou
and Demetrios Christodoulides
Beam displacement [mm]
’09
3
2
1
0
–40 –20
0
20
40
Distance from Fourier plane [cm]
(d)
(Top row) Numerical simulations for plasma density generated along the beam path at low
pulse energy (5 mJ, panel a) and high pulse energy (10 mJ, panel b). (Bottom row) Burn
pattern produced by the beam on aluminum foil (c); numerical simulation for the transverse
intensity profile of intense Airy beam (d).
self-bending property of the beam pattern competed against nonlinear selfchanneling effects. The plasma channel
generated by the dominant intensity
feature of the beam followed the curved
beam trajectory. In this regime, broadband forward emission by the curved
filament is angularly resolved in the
far-field, thus enabling detailed study
of this emission along the optical path.
Extended curved filaments generated by
self-bending Airy beams may find applications in remote sensing.
We started with 35-fs pulses with a
Gaussian beam profile and transformed
the beam into a 2-D Airy beam using a
combination of a cubic phase mask and
a focusing lens. The pulse energy was
varied between 5 and 15 mJ. Analysis of
forward emission by the curved filament
suggests that the plasma channel was
continuous at low pulse-energy levels but
developed split-off channels at several locations along the propagation direction
when pulse energy was increased.
At high values of pulse energy, the
transverse beam profi le exhibited a
nonlinear reshaping and developed a
whisker-like ghost beam and lagging
satellite lobe. These features, clearly
visible in the burn patterns produced
on aluminum foil, were reproduced in
numerical simulations. 
This work was supported by AFOSR under contracts FA9550-07-1-0010 and FA9550-07-10256. The contribution of G.S. and D.C. was
partially supported by Lockheed Martin Corp.
Pavel Polynkin ([email protected]),
Miroslav Kolesik and Jerome Moloney are with the
College of Optical Sciences, University of Arizona,
Tucson, Ariz., U.S.A. Georgios Siviloglou and
Demetrios Christodoulides are with the College of
Optics and Photonics, University of Central Florida
in Orlando, Fla., U.S.A.
References
1. A. Couairon and A. Mysyrowicz. Physics Reports 441,
47-190 (2007).
2. L. Berge et al. 70, 1633-713 (2007).
3. M. Kolesik and J. Moloney. Opt. Express 16, 2971-88
(2008).
4. G. Siviloglou et al. Phys. Rev. Lett. 99, 213901 (2007).
5. P. Polynkin et al. Science 324, 229-32 (2009).
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