Xie et al.
Vol. 30, No. 10 / October 2013 / J. Opt. Soc. Am. A
1937
Effect of polarization purity of cylindrical vector
beam on tightly focused spot
Xiangsheng Xie,* Huayang Sun, Liangxin Yang, Sicong Wang, and Jianying Zhou
State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China
*Corresponding author: xxsh0711@gmail.com
Received June 27, 2013; revised August 20, 2013; accepted August 21, 2013;
posted August 21, 2013 (Doc. ID 193006); published September 6, 2013
The tightly focused spots of cylindrical vectors (CVs) are dependent on polarization composition. We experimentally demonstrate the effect of polarization purity (PP) of the CV beam on the tightly focused spot quantitatively,
which should be strictly controlled for the effective applications of the CV beam. The focal spots measured by a
knife-edge scanning method showed that the azimuthally polarized (AP) component increases the transverse field
and the size of the focal spots, while the radially polarized component results in a nonzero intensity distribution at
the center of the focus even in a high PP AP beam. © 2013 Optical Society of America
OCIS codes: (260.5430) Polarization; (260.1960) Diffraction theory; (110.1220) Apertures.
http://dx.doi.org/10.1364/JOSAA.30.001937
1. INTRODUCTION
The past decade has witnessed rapid progress on the research
of cylindrical vector (CV) beams [1], especially the radially polarized (RP) beams and the azimuthally polarized (AP) beams.
RP beams can be focused to generate fields with sizes
beyond the diffraction limit, which are desirable in optical imaging, optical data storage, particle acceleration, fluorescent
imaging, second-harmonic generation, and Raman spectroscopy. AP beams can generate a shaper doughnut field and
become promising tools for optical tweezers [2] and for the
stimulated emission of depletion (STED) microscopy [3].
The methods adapted to generate CV beams can be classified into two categories: the intracavity [4–6] and extracavity
methods [7–16]. The intracavity method forces the laser to oscillate with one CV mode of high quality by rigorous design
and arrangement. However, it is not convenient to switch
to other CV modes with the same cavity. The extracavity
methods generally generate CV beams by inserting polarization converters or combining two polarized beams via interferometer [8–11]. They draw a growing interest since CV
modes with different polarization distributions can be flexibly
generated [7–13]. The CV modes are switchable by simply
changing a number of optical devices or the polarization of
the incident beam. This flexibility, on the negative side, will
degrade the purity of the CV mode when the optical devices
were not precisely fabricated or the optical setup was not
perfectly arranged. In simulation and in theory, the structural
focal spots beyond the diffraction limit (i.e., sharper focused
spot, optical needle, optical cave, etc.) have been precisely
modulated with a pure CV incident beam [1]. Any deviation
of the modulating structural light field will distort the designed
focused profiles. The purity of CV mode, especially the purity
of the polarization (PP), generated by extracavity methods
should be carefully measured and improved to be qualified for
the further applications. Quabis et al. showed that the transmitted field of a polarization converter [14] (consisting of four
segments of half-wave plates) has an overlap of 75% with the
1084-7529/13/101937-04$15.00/0
ideal doughnut RP beam. They improved the PP to about
99% [15] (2% peak to valley) by a Fabry–Perot interferometer.
Other methods were reported to generate high PP of RP or AP
beams. Lai et al. [9] demonstrated an eight-segment spirally
varying retarder to generate a RP light with the PP over
96% at the far field. Ma et al. [16] designed a fiber-based beam
combiner to generate high purity CV beam with two orthogonally polarized LP11 modes. PPs as high as 95% for AP beam
and 97% for RP beam were obtained. However, a majority of
papers did not mention the PP of the generated CV beams.
And the effect of PP on the tightly focused spot of CV beams
has not been systematically discussed and presented to the
best of our knowledge.
In this article, we experimentally demonstrate the tightly
focused spots generated by CV Gaussian beams with high
NA annular aperture. The CV beam consists of RP and AP
beams, and the proportion of the AP beam can be adjusted
from 5% to 97%. Our results show that the AP component will
increase the transverse field and the size of the focus. When
the PP of the RP beam is larger than 95%, this increment can
be neglected, while the AP beam with little impurity (PP as
high as 97%) will result in a nonzero intensity distribution
at the center of the focus and was not qualified for effective
application in the STED microscopy. We further simulate the
effect of PP on the optical needle formed by a RP beam with
phase modulation. Our results can be expanded to higher
order CV beams and CV beams with phase modulation.
2. THEORY
Richards and Wolf [17,18] developed a vectorial diffraction
method to study the focusing of a paraxial optical field by
an aplanatic optical lens. Youngworth and Brown [19] developed the Richards and Wolf theory by taking into account the
incident field distribution in the cases of RP and AP beams. In
practice, the RP beam and AP beam always exist and compete
in the same system, i.e., the fiber [16], the cavity [4–6,20], the
interferometer [8–11], and the segment polarization converter
© 2013 Optical Society of America
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J. Opt. Soc. Am. A / Vol. 30, No. 10 / October 2013
Xie et al.
[14,15]. The electric field near the focus illuminated by a
CV beam consisting of RP and AP beams has the following
form [21]:
Z
E ρ ρ; z cos φ0
α2
0
P 0 θcos1∕2 θ sin 2θJ 1 kρ sin θ
× exp−2ikz sin θdθ
Zα
2
E φ ρ; z 2 sin φ0
P 0 θcos1∕2 θ sin θJ 1 kρ sin θ
0
× exp−2ikz sin θdθ
Zα
2
E z ρ; z 2i cos φ0
P 0 θcos1∕2 θ sin2 θJ 0 kρ sin θ
0
× exp−2ikz sin θdθ;
(1)
where ρ and z are the cylindrical coordinates and α2 arcsinNA∕n is the maximum divergence angle of the objective. P 0 θ is the apodization function for a Gaussian beam
with its waist in the annular aperture. φ0 is the angle between
polarization and radial directions. Therefore, PP of RP beam
can be calculated by PP cos φ0 2 , which denotes the
ratio of power that is RP [16], while the PP of AP beam
can be calculated by PP sin φ0 2 .
3. EXPERIMENT
The schematic of the experimental setup is shown in Fig. 1
with all optical devices placed on an active vibration isolation
system (TS-150, Table Stable Ltd., Germany). A linear polarized beam from a continuous-wave Nd:YVO4 laser at 532 nm is
spatially filtered, expanded, and collimated to produce a clean
Gaussian beam. The collimated beam then passed through
a liquid-crystal (LC) polarization converter (ARCoptix,
Switzerland) and an annular aperture before being focused
by an objective lens (Olympus, X100, NA 0.9). The beam
waist and the inner and outer radii of the annular aperture
are 1.8, 1.8, and 2.2 mm, respectively. The LC polarization converter consists of a 90° twisted cell, a θ-cell, and a phase
shifter. The 90° twisted cell is capable of controlling the proportion of vertical and horizontal polarized beam by the bias
voltage. The θ-cell [8] can convert the entrance vertical or
horizontal polarized beam into AP or RP beam respectively
with a π phase step, which can be compensated by the phase
shifter.
By fine adjusting the applied bias voltage of the 90° twisted
cell, CV beams formed by different proportion of RP and AP
beams can be obtained. An analyzer and a laser beam analysis
system (LBA-USB-SP620, Ophir-Spiricon Corp.) are inserted
Fig. 1. Experimental setup of the focused light beam measurement.
to measure the polarization property of the incident beam.
The intensity distributions of horizontal and vertical polarized
light are shown in the left and the center columns of Fig. 2,
respectively, with the direction of the analyzer indicated by
the white arrows. The local polarization at any point of the
incident beam can be obtained according to [16]. The calculation results are depicted in the right column of Fig. 2, where
the red arrows denote the directions of the main axis of the
local polarization. The highest PPs (ratio of power that is
demanded) of AP beam [Fig. 2(a)] and of RP beam [Fig. 2(b)]
are 97% and 95%, respectively.
The profile measurement of tightly focused laser beam is a
crucial process of the experiment. Dorn et al. [15,22] introduced the scanning knife-edge technique (SKET) in different
scanning directions to retrieve a two-dimensional (2D) profile
of a nonaxially symmetric focused beam. We realized a
convenient 2D profile measurement based on a 90° doubleknife-edge device [23,24] by taking two derivatives of the
transmitting image with respect to the scanning directions.
For rotationally symmetric incident light beam, the focal spot
should be rotational symmetry and one representative measurement with SKET would in principle suffice [15]. Figure 3(a)
highlights the knife-edge measurements clipped from the full
measurements (shown as the inset) of the focal spots of different CV incident beams. The measurement data must be monotonic because the knife-edge blocks more light while
scanning in one direction and less in the opposite direction,
i.e., the ideal knife-edge measurement of a Gaussian beam is
an error function or a complementary error function. The
slope of the intensity ramp profile can indicate the shape
Fig. 2. Polarization converter can produce CV beams with different
proportion of RP and AP beam, from (a) high PP AP beam (97%) to
(b) high PP RP beam (95%) and arbitrary combination of AP and RP
beam, i.e., (c) 55% AP and 45% RP beams. The intensity distributions
of horizontal and vertical polarized light are shown in the left and
the center columns. The local polarizations at the main axis of the
incident beam are depicted in the right column.
Xie et al.
Vol. 30, No. 10 / October 2013 / J. Opt. Soc. Am. A
1939
Fig. 4. Simulation cross sections at the focal plane for the case of
(a) 97% AP, (b) 55% AP, (c) 10% AP, and (d) 5% AP.
Fig. 3. (a) Highlights of the knife-edge measurements clipped from
the full measurements (shown in the inset) and (b) the corresponding
cross sections of the reconstructed profiles. The inset in (b) shows the
2D reconstruction profiles of the focus of the 97% AP beam.
and size of the focal spot. The RP beam with high
PP (5% AP) has a steep ramp profile corresponding to a
sharper focal spot, while the AP beam (97% AP) has a tworamps profile that indicates a small intensity distribution at
the center of the focus. Moreover, the derivative of the knifeedge measurement data is the one-dimensional projection
[15,22] of the focused beam. The reconstruction of the 2D
intensity distribution can be obtained by extending the
projection into other directions and applying the Radon backtransformation. Figure 3(b) shows the cross sections of the
reconstructed profiles of different CV beams. The intensities
of 5% AP and 10% AP are multiplied by 2 for a better view. It is
obvious that the RP beam, with low ratio of AP (5%) has a
sharpest focus with 329 nm full-width at half-maximum
(FWHM). When the PP of RP beam decrease to 90% (ratio
of AP is 10%), the FWHM of the focal spot increases into
393 nm. In the case of AP beam, a total zero intensity distribution at the center of the focus cannot be obtained even with
the highest PP of AP beam (97%). Moreover, when the PP of
AP beam decreases from 97% to 94% or 92%, the intensity at
the center of the focus increases substantially. The inset in
Fig. 3(b) shows the 2D reconstruction profiles of the focus
of the 97% AP beam.
Figure 4 shows the numerical simulation of the cross
sections of the intensity at the focal plane based on Eq. (1)
and the experiment settings. It is well known that the intensity
at the focus of a RP beam consists of the longitudinal and
transverse components. Sharper focus can be obtained by
enhancing the contribution of the longitudinal component
by increasing the annular factor of the annular aperture
[24]. In our experiment, the annular factor is set to 0.8; hence
the shape of the focal spot generated by the RP beam remains
the same (black line). The intensity at the focus of an AP beam
only contains the azimuthal component and a doughnut shape
structure is formed at the focal plane. The cross section of
sum intensity (Isum) depends on the ratio of AP and RP
beams. The numerical simulations are generally in good agreement with the experimental results. However, there are a few
deviations between the theoretical and experimental results.
The center intensity of the focal spot with 97% AP incident
beam (3.8% comparing to the maximum of the cross section)
is smaller than the simulation result (4.9%), and the difference
of FWHM of the focal spot between the 5% AP beam (0.628λ)
and 10% AP beam (0.739λ) are larger than the simulation results (0.684 and 0.730λ for 5% AP and 10% AP respectively).
The reason is that when we optimize the PP of AP or RP beam,
the residuary light supposed to be RP or AP beam in theory is
probably out of phase and contributes little to the focal spot.
Otherwise, we can continue to improve the PP of the light
beam. When we degrade the PP of incident AP beam, i.e., from
97% to 94%, it is not simply adding 3% RP beam into the residuary light. The nonideal light (out of phase) derived from the
fabrication error will distribute into the beam. Hence the
difference of the focus between the highest PP beam and
other beams increases.
4. DISCUSSION AND CONCLUSION
For applications, phase plates have been widely used in modulating the CV beam for producing beams with special phase
distributions and focal profiles. Different focal spots beyond
the diffraction limit, i.e., sharper focused spot, tight dark spot,
optical needle, optical cave, etc., have been generated by CV
beam combining with the phase plates (i.e., circular π-phase
plates, annular multiphase plates, helical phase plates, phase
Fresnel zone plates, and binary-phase optical elements (BOE).
Most of these methods are based on the phase modulations of
the RP or AP beams, which would be affected by the PP of the
incident beam. As an example, we demonstrate the effect of
PP on the “pure” longitudinal light beam [1] with a subdiffraction beam size (0.43λ) without divergence at a distance of
about 4λ. The total energy density distributions, as shown
in Fig. 5, are focused by a RP Bessel–Gaussian beam passing
through a BOE with all simulation parameters setting as in [1],
where r1–r5 are 0.091, 0.391, 0.592, 0.768, and 1, respectively.
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J. Opt. Soc. Am. A / Vol. 30, No. 10 / October 2013
Fig. 5. Contour plots for the total energy density distributions in the
y-z plane focused by a RP beam passing through a BOE with different
PP. (a) 100% RP, (b) 90% RP, 10% AP, (c) 80% RP, 20% AP, (d) 45% RP,
55% AP, (e) 3% RP, 97% AP, and (f) the BOE structure.
The FWHMs at the focus (plane z 0) of the beams in
Figs. 5(a)–5(e) are 0.43λ, 0.45λ, 0.48λ, 0.79λ, and 0.93λ, respectively. It is obvious that the focal intensities formed by the RP
beam and the AP beam are independent of each other. The
aforementioned effect of PP of the CV beam on the focused
spot can be expanded to more general cases, i.e., higher order
CV beams and CV beam with phase modulation.
We have demonstrated the effect of PP of CV beam on the
tightly focused spot experimentally and numerically. The PP is
controlled by adjusting the proportion of RP beam and AP
beam from 5% AP beam to 97% AP beam. The focal spot is
measured by a knife-edge scanning method. Our results show
that the AP component will increase the transverse field and
the size of the focus. When the PP of the RP beam is larger
than 95%, this increment can be neglected, while the RP component in a high PP AP beam results in a nonzero intensity
distribution at the center of the focus and cannot be qualified
for effective application in the STED microscopy. This result
can be expanded to higher order CV beams and CV beam with
phase modulation.
ACKNOWLEDGMENTS
This work is supported by the National Basic Research
Program of China (grant number 2012CB921904) and by the
Chinese National Natural Science Foundation (grant numbers
10934011 & 61205018).
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