Generation of an optical vortex with
a segmented deformable mirror
Robert K. Tyson, Marco Scipioni,* and Jaime Viegas
Department of Physics and Optical Science, University of North Carolina at Charlotte,
9201 University City Boulevard, Charlotte, North Carolina 28223, USA
*Corresponding author: email@example.com
Received 3 March 2008; revised 5 May 2008; accepted 28 October 2008;
posted 29 October 2008 (Doc. ID 93346); published 20 November 2008
We present a method for the creation of optical vortices by using a deformable mirror. Optical vortices of
integer and fractional charge were successfully generated at a wavelength of 633nm and observed in the
far field (2000mm). The obtained intensity patterns proved to be in agreement with the theoretical
predictions on integer and fractional charge optical vortices. Interference patterns between the created
optical vortex carrying beams and a reference plane wave were also produced to verify and confirm the
existence of the phase singularities.© 2008 Optical Society of America
050.4865, 260.0260, 030.7060, 070.7345, 350.4600.
An optical vortex (also known as a screw dislocation
or phase singularity) is a zero of an optical field, a
point of zero intensity . Light is twisted like a cork-
screw around its axis of propagation [2,3]. Because of
the twisting, the light waves at the axis itself cancel
each other out. An optical vortex looks like a ring of
light with a dark hole in the center. The vortex is gi-
ven a number, called the topological charge ℓ, related
to the orbital angular momentum of the field. The
wavefront of an optical vortex is a continuous surface
consisting of ℓ embedded helicoids, each with ℓλ pitch,
spaced from each other at one wavelength λ. As an
example, Fig. 1 represents the wavefront of a charge
ℓ ¼ 3 vortex propagating along the z axis, illustrating
the three intertwined helicoids.
The generalized functional form for a field hosting
an optical vortex is, in a plane transverse to propa-
gation direction, locally given by
fðr;θÞ ¼ Aðr;θÞeiℓθ;
continuous, and smooth complex amplitude wave
function in cylindrical polar coordinates. The phase
argument θ represents the distinctive, transverse
vortex phase profile, impressing a linear phase in-
crease in the azimuthal direction to the field. The
charge of a vortex can be an integer or fraction,
and also be positive or negative, depending on the
handedness of the twist. Figure 2 shows a map of
the phase profile of a vortex beam. The phase jumps
by a value ℓ2π at the discontinuity.
Vortex beams have been successfully employed in
optical tweezers applications [4–7] because they offer
the advantage of trapping and spinning low index
(with respect to the hosting medium) dielectric
particles in their zero-intensity region.
Vortex carrying beams also have interesting
potential for use in free-space optical communica-
tions [8–11]. Of particular interest is the ability of
vortex beams to conserve their charge through atmo-
spheric turbulence . Also, vortex beams “self-
heal” around obstacles  and experiments have
shown that vortices are conserved through fog .
These properties make it an ideal extension to con-
ventional coding schemes, such as on–off keying or
coherent modulation techniques.
can beany square integrable,
© 2008 Optical Society of America
6300 APPLIED OPTICS / Vol. 47, No. 33 / 20 November 2008
Optical vortices can be generated in a number of
ways. We briefly review the methods here. Details
of the methods and operation are found in the
Generation of an Optical Vortex
One commonly used device for the generation of an
optical vortex is the liquid-crystal spatial light mod-
ulator (LC SLM) . Commercial LC SLMs are
either optically or electrically addressed and can
modulate the amplitude, the phase, or both, for an
incident input field. Their main strength point is that
they are dynamically reprogrammable. Nematic
SLM, the most common, has a time response of
roughly 60Hz. When an SLM is used, any significant
incident beam power must be distributed in order to
avoid boiling the liquid-crystal element, so the
amount of incident power can be a limitation.
In the case of amplitude-only spatial light modula-
tors, an optical vortex of a given charge and wave-
length can be made from a computer-generated
hologram (CGH) . CGHs are the digitally calcu-
lated interferograms between a plane wave beam
and a beam carrying an optical vortex. The resulting
CGH resembles a diffraction grating with a charac-
teristic “fork” dislocation, with the number of prongs
Spatial Light Modulator
in the fork directly related to the topological
charge of the design vortex (number of prongs ¼
desired topological charge þ 1). The CGH is then
applied to the SLM.
In the case of phase-only SLMs, the phase profile is
the sum of the desired optical vortex phase and a
phase tilt needed to steer the reflected incident beam
away from the direction of incidence. The result is a
blazed phase grating that still has a fork feature in
its center. Because the blazed phase grating is not
perfect, multiple diffraction orders appear after the
beam is reflected off the SLM. The first diffraction
order contains the optical vortex with the desired to-
pological charge and is the most intense diffraction
order. The zero diffraction order is the specular
reflection off the SLM. The other diffraction orders
are vortex beams with topological charge equal to
the diffraction order number multiplied by the
topological charge of the desired vortex beam.
Hermite–Gaussian laser modes form an orthogonal
family of laser beams. An appropriately weighted
superposition of two Hermite–Gaussian beams, with
the right mode order, can result in a Laguerre–
Gaussian beam carrying an optical vortex of the
desired topological charge at its center. The super-
position is achieved through a system of cylindrical
lenses by making use of the Gouy effect . This set-
up presents alignment challenges and requires high-
order Hermite–Gaussian beams to obtain high-order
vortex beams, thus limiting the flexibility of the
A helical mirror was recently proposed  to create
optical phase singularities of various topological
charges. The mirror shape, controlled by a piezoelec-
tric actuator, provides a continuous phase variation
along the azimuthal direction, but also introduces
radial phase variations because of unavoidable
material stresses, thus lowering the quality of the
generated vortex beams.
By stacking dielectric wedges [19,20], it wasshown to
be possible to create a system capable of producing
optical vortices of topological charge higher than
one. The charge of the vortex beams corresponds
to the number of wedges used in the system.
A simple, adjustable spiral phase plate has also been
used to create vortex beams . The plate is con-
structed from a parallel-sided transparent plate with
polished surfaces in which a crack is induced starting
at one edge and terminating close to its center.
Static spiral phase plates (SPPs) are very common.
They are spiral-shaped pieces of crystal or plastic
that approximate the ideal spiral with a discrete
number of phase steps. SPPs are engineered specifi-
cally to the desired topological charge and incident
Spiral Phase Plates
Wavefrontof an optical vortexwith charge ℓ ¼ 3 (arbitrary
Fig. 2. Vortex phase in plane transverse to propagation.
20 November 2008 / Vol. 47, No. 33 / APPLIED OPTICS6301
wavelength [22–25]. They are efficient, yet expen-
sive, and show high topological charge purity only
for low topological charge ℓ .
A deformable mirror (DM) can be used to generate a
vortex. A conventional continuous faceplate DM is
not well suited for this action because the surface
must have a discontinuous line (not necessarily
straight) between the singularity and the edge. On
the other hand, a segmented DM, with discontinu-
ities already in place between the segments, can
be formed into a vortex shape, which is transferred
to the phase of a beam reflecting from the surface.
If, at the discontinuity, the surface jumps one-half
of the wavelength of the light, the reflected beam will
have a phase jump of one full wave and the beam will
have a vortex charge ℓ ¼ 1. By simply multiplying
the amplitudes of the segment pistons and tilts,
we can apply any charge to the beam, up to the
mechanical limits of the DM. This allows us a great
variability of charge and even fractional charges.
Because the mirror is simply a reflecting surface,
it can be used at multiple wavelengths.
We have performed a number of experiments with
our 37-segment deformable mirror and have shown
that we can generate a vortex. We can vary the
charge and have verified the charge in the pupil
plane and in the far field after propagation. The de-
vice used to demonstrate vortex generation is the Iris
AO S37-X segmented deformable mirror ; see
Fig. 3. The S37-X deformable mirror is fabricated
with micromachining technology, making it a com-
pact, low mass modulator. The 3:5mm aperture
DM consists of 37 hexagonal segments tiled into
The DM segment consists of an actuator platform
elevated above the substrate as a result of engi-
neered residual stresses in the bimorph flexures.
The actuator platform and underlying electrodes
form parallel plate capacitors. Placing a voltage
Results with a Segmented Deformable Mirror
across the capacitors generates Coulombic forces
that pull the segment toward the substrate. By vary-
ing the voltages on the lower electrodes, the actuator
can move in piston (pure vertical), tip, and tilt direc-
tions. The forces are solely attractive, so bidirectional
actuation is achieved by biasing the segment at the
Electrostatic actuation has a nonlinear response
between position and voltage. Furthermore, the seg-
ment piston/tip/tilt positions are coupled, making the
position versus voltage response more complicated.
Iris AO has developed a controller that linearizes
this response. The user simply enters desired pis-
ton/tip/tilt positions and the controller, using a cali-
brated model, determines the required voltages and
sets them on the drive electronics. The controller has
demonstrated open-loop positioning of 30nmrms
residual surface figure errors. Thus, a vortex can
be created with the DM in open-loop operation.
Figure 4 shows the spiral ramp generated by the
DM compared to the ideal, infinitely smooth spiral
Beams carrying topological charge jℓj > 1 are
highly unstable to small symmetry-breaking azi-
muthal perturbations and decompose, upon propaga-
tion, into elementary charge ℓ ¼ 1 vortices of the
same sign, symmetrically distributed around the
Fig. 3.Iris AO S37-X segmented deformable mirror.
Fig. 4.(a) Ideal linear spiral ramp and (b) ramp approximated by Iris AO deformable mirror.
6302APPLIED OPTICS / Vol. 47, No. 33 / 20 November 2008
center of the beam, thus conserving the initial net
topological charge . Astigmatism in the beam
or small defects in the diffracting/reflecting optical
device are the probable causes of this fragmentation.
Figure 5 illustrates the unfolding. A dipole in Fig. 5
(b), a tripole in Fig. 5(c), and a quadrupole in Fig. 5(d)
are shown from the decay of charge two, three, and
four optical vortex beams created with the Iris AO
We also verified the optical charge from interfer-
ence patterns in the pupil plane. The interference
patterns between a reference plane wave and a beam
carrying an optical vortex were generated by using a
The resulting interference patterns shown in Fig. 6
reveal the typical fork pattern, which is an indicator
of the presence of the phase singularities in the beam
reflecting off the deformable mirror. In Fig. 6(b) the
interference fringes represent the deformable mirror
commanded to a flat profile. In Fig. 6(d) the two-
pronged fork pattern for a charge 1 vortex is shown
and in agreement with the simulation. For Fig. 6(f),
we placed amplitudes on the deformable mirror that
would generate a charge 5 optical vortex. The simu-
lated and experimental patterns are different be-
cause, upon the short propagation length within
the interferometer (a few centimeters), the charge
5 vortex apparently unfolded into five elementary
charge 1 vortices, as expected. We interpret the inter-
ference pattern tobe the presence of five two-pronged
forks within the pattern, indicating the presence of a
charge 5 vortex.
charge exhibit, in the near field, a radial line of
low intensity attributed to the presence of a chain
of charge 1 vortices of alternating sign along the
radial phase discontinuity [28,29].
In the far field, however, only a finite number of
same-sign vortices appear near the beam axis
[30,31]. The number results from rounding the
fractional charge ℓ of the vortex to the nearest higher
integer. For example, if ℓ ¼ 0:5, the far field will show
one charge 1 vortex, and, similarly, three charge 1
vortices, if 2:5 ≤ ℓ < 3:5.
In Fig. 7, each dark region indicates the presence of
an optical vortex in the field. By increasing the
height of the phase discontinuity in discrete incre-
ments, it is interesting to follow the evolution of
the intensity of the beam as the discontinuity
changes from one integer value of the wavelength
to the next higher one. It is apparent from Fig. 7,
as predicted from theory [31,32], that as the discon-
tinuity passes a half-integer value of the wavelength
λ, a new vortex fully appears in the beam, migrating
(d) charge ¼ 4 (close-up view).
Decay of multiple-charge optical vortices into charge 1 vortices: (a) charge ¼ 1, (b) charge ¼ 2, (c) charge ¼ 3 (close-up view),
20 November 2008 / Vol. 47, No. 33 / APPLIED OPTICS 6303
from the periphery along the radial phase disconti-
nuity to the central area of the beam.
We made use of the discontinuous surface of a seg-
mented DM to create an optical vortex that, by defi-
nition, requires a phase discontinuity. The reflective
surface allows for generation of vortices of any wave-
length and the simple open-loop nature of the
controller allows for integer and fractional vortex
charge at any wavelength.
The authors wish to thank Michael A. Helmbrecht
and Iris AO, Inc., for their support and for providing
the deformable mirror.
deformable mirror illustrating fork patterns due to the phase singularities present in the beam. (a) Simulated interference flat mirror,
(b) experimental interference flat mirror, (c) simulated interference charge 1 vortex, (d) experimental interference charge 1 vortex,
(e) simulated interference charge 5 vortex, (f) experimental interference charge 5 vortex.
Interference pattern [(a), (c), (e), simulated and (b), (d), (f), experimental] between reference plane wave and wave reflected off
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