Broadband nulling of a vortex phase mask
Grover A. Swartzlander, Jr.
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721
Received May 24, 2005; revised manuscript received June 15, 2005; accepted July 18, 2005
A pulse transmitted through a helical vortex phase mask undergoes a temporal Hilbert transform. The flu-
ence transmitted into the unfavorable plane wave mode is found to increase as the square of the bandwidth
and, to first order, is independent of the topological charge. © 2005 Optical Society of America
OCIS codes: 100.5090, 070.6610, 260.2030, 350.1260.
Optical vortices enjoy a wide range of applications for
quasi-monochromatic light, including optical spatial
filtering,1optical tweezers,2quantum cryptography3
and communication,4high-resolution spectroscopy,5
and semiconductor patterning.6This broad range of
uses is afforded by robust attributes: vortices exist as
natural modes in many systems, they exhibit a dis-
tinct point of destructive interference in coherent
light, and they may be readily produced in the labo-
ratory by various techniques. Additional applications
may be possible once the broadband properties of vor-
tices are fully explored. Colored vortices and other
optical singularities have been investigated.7Spec-
tral anomalies have been reported,8–10and temporal
coherence properties under broadband illumination
have been measured.11Here we explore the broad-
band transmission properties of a vortex phase mask,
giving special attention to the amount of light trans-
mitted into the nonvortex mode. This latter mode
limits the use of a vortex phase mask for spatial fil-
tering applications such as the search for extrasolar
planets.12The topological dispersion11of the vortex
phase mask provides a means to both null the peak of
a pulse via a temporal Hilbert transform and signifi-
cantly attenuate the on-axis fluence.
Optical vortices are modes that include a broad
family of separable cylindrical wave functions.13The
complex amplitude of the scalar field of such modes
may be expressed as a separable function in cylindri-
cal coordinates ?r,?,z?:
Fm?r,?,z? = Am?r,z?exp?im??,
where m is a mode number called the topological
charge. A lossless phase mask is frequently used to
transform an initial distribution, Fm?, into Fm
=t???Fm?where the azimuthally dependent transmis-
sion function is given by t???=exp?i?m−m????. Typi-
cally the mask is used to create a vortex of charge m
on a beam having m?=0. For example, the initial dis-
tribution may represent a uniform plane wave or a
Gaussian beam. We thus assume m?=0 and write
t??? = exp?im??.
A single mask that satisfies Eq. (2) across a broad
spectrum does not yet exist. On the other hand, it is
simple to design such a mask for a single wavelength
or over a small band of wavelengths. For example, a
vortex mask may be made by azimuthally varying
the thickness of a substrate by an amount
d??? = dbase+ ?d?1 − ?/2??,
where ?d=m0?0/?ns??0?−n0??0??, dbaseis the mini-
mum thickness, m0is the topological charge pro-
duced at the design wavelength ?0, and the refractive
indices of the substrate and the surrounding media
at the design wavelength are ns??0? and n0??0?, re-
spectively. The effective topological charge generated
at any frequency ? is given by
m??? = m0
ns??? − n0???
ns??0? − n0??0??,
where ?0=2?c/?0is the design frequency and ? is
the angular frequency associated with the light
source. We note that m??? is not necessarily an inte-
ger. For most materials the factor in brackets in Eq.
(4) differs much less than the ratio ?/?0. For math-
ematical convenience we therefore assume ns???
−n0????ns??0?−n0??0?, and we write
m??? = m0?/?0.
If m0is an integer, then m=m0p is also an integer
1,2,3,... . The frequencies satisfying ?=p?0will be
called principal frequencies.
Although the topological charge varies continu-
advantageous11,14to rewrite the field as a Fourier se-
ries of modes having an integer topological charge.
For this purpose we write
it is sometimes
Cl??? = ?2??−1?
where Cl??? shall be called the lth order vortex spec-
trum. Inserting Eq. (5) into Eq. (2), one may readily
Cl??? = sinc?m0??/?0− l??.
The vortex spectra, C0??? and C±1???, are plotted in
Fig. 1 for the case m0=1. As expected, at ?=?0only
C+1has a nonzero value. We also see that C0=0 at the
principal frequencies, varying linearly in the vicinity
OPTICS LETTERS / Vol. 30, No. 21 / November 1, 2005
0146-9592/05/212876-3/$15.00© 2005 Optical Society of America
of ?−l?0. The discussion below makes use of this lin-
ear variation to null the peak of the transmitted
pulse along the optical axis.
To describe the spatiotemporal effects of the vortex
mask, we write the scalar input field
Ein?r,?,z;t? = G?t?F?r,z?exp?i?t?exp?− ikzz?,
where G?t? describes the temporal profile of the
beam, ? is the carrier frequency, and kzis the projec-
tion of the carrier wave vector along the optical axis.
Applying the paraxial approximation, we set kz
=n?/c, where n is the refractive index. The temporal
pulse and its spectrum are related by the Fourier
If the maximum thickness of the mask, dbase+?d, is
much less than the characteristic diffraction length,
the field at the output plane of the mask may be writ-
Eout?r,?,z = 0;t? =?
El?r,?,z = 0;t?,
where we have assigned the plane z=0 to the output
plane of the mask. The transmitted vortex modes are
El?r,?,z = 0;t? = Gl?t?F?r,z = 0?exp?il??exp?i?t?,
Gl?t? = ?2??−1exp?− i?t??
gl??? = Cl???g???.
Material dispersion in the thin layer has been ig-
nored in Eq. (12). When the source spectrum is band
limited and Cl??? is an odd function over the spectral
band, Eq. (12a) may be interpreted as a temporal Hil-
bert transform. In this case the peak of the pulse be-
comes zero valued.
The zeroth-order vortex spectrum, C0???, is of spe-
cial interest because it is the only mode whose inten-
sity does not vanish along the optical axis. From a
spatial filtering point of view, this limits the nulling
ability of a vortex mask. Thus we see that the Fourier
series decomposition in Eq. (6) provides a means of
quantifying the nulling efficiency of a vortex mask
without the need to determine the net field. The net
field is complicated and may contain vortices in dif-
ferent locations.14,15Below we determine the relative
fluence, ?, transmitted into this mode. For conve-
nience we assume that the input field is planar,
?El=0?r,?,z = 0;t??2dt
?Ein?r,?,z = 0;t??2dt
Parseval’s theorem has been invoked to obtain the
right-hand side of Eq. (13).
The following analytical and numerical calcula-
tions establish important characteristics of Eq. (13).
Analytical results are made possible by assuming a
uniform band-limited spectrum. Let g???=?2???−1/2
for ?−??????+?? and g???=0 otherwise, such
ably simplified by assuming that ?=?0, integrating
over the variable ??=?/?−1, and expanding the in-
tegrand to O???3?, assuming that ???1 and m0is an
??g????2d?=1. An evaluation of ? is consider-
of the plane-wave mode through a vortex mask (design pa-
rameters, m0=1 and ?0), normalized to the net fluence of
the input pulse having a uniform band-limited frequency
distribution of half-width ?? and center ?.
Transmitted fluence (joules per square centimeter)
Vortex spectra for the orders l=0,+1,−1 for the
November 1, 2005 / Vol. 30, No. 21 / OPTICS LETTERS
??? + 1?2
Numerical integration of Eq. (13), whose values are
plotted in Fig. 2, reveals that the right-hand side of
relation (14) is an excellent approximation. Further-
more this result is seen in Fig. 2 to be valid in the
broad bandwidth regime when ???0. On the other
hand, when the bandwidth is small, direct integra-
tion shows that ???/?→0=C0
An ideal vortex-nulling filter requires that ??1
over a broad bandwidth. In the small-bandwidth re-
gime design parameter errors (?0or m0) significantly
affect the nulling efficiency. An extinction of more
than 8 orders of magnitude is shown in Fig. 2 when
m0=1, ??=0.1, and ?0is selected to coincide with the
center frequency of the source, ?. Errors in ?0/? as
small as 0.1% limit the extinction to, at best, 6 orders
of magnitude. The limited performance of the mask is
attributed to the linear frequency dependence of the
effective topological charge shown in Eq. (4). In prin-
ciple an ideal frequency-independent vortex mask
may be produced by using a high-dispersion material.
The author is grateful to Greg Foo (University of
Arizona) and David Palacios (Jet Propulsion Labora-
tory) for their comments. This work was supported by
the U.S. Army Research Office and the State of Ari-
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