Download Shifting the spherical focus of a 4Pi focusing system

Shifting the spherical focus of a 4Pi focusing
system
Shaohui Yan,1 Baoli Yao,1,* and Romano Rupp2
1
State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics,
Chinese Academy of Sciences, Xi’an 710119, China
2
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
* [email protected]
Abstract: In a 4Pi focusing system radially polarized laser beams can be
focused to a spherical focal spot. For many applications, e.g., for moving
trapped particles or for scanning a specimen, one would like to change the
position of focal spot along the optical axis without moving lenses or laser
beams. We demonstrate how this can be achieved by modulating the phase
of the input field at the pupil plane of the lens. The required phase
modulation function is determined by spherical wave expansion of the
plane wave factors in the Richards–Wolf integral.
©2011 Optical Society of America
OCIS codes: (110.2990) Image formation theory; (140.3300) Laser beam shaping; (260.5430)
Polarization.
References and links
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23),
233901 (2003).
K. S. Youngworth, and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt.
Express 7(2), 77–87 (2000).
Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A
73(4), 043402 (2006).
P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser
beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced
Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004).
Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382
(2004).
S. Yan, and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys.
Rev. A 76(5), 053836 (2007).
H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric
sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841
(2007).
T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and
azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008).
M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical
tweezers in air using a radially polarized beam,” Appl. Opt. 48(32), 6143–6151 (2009).
H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally
polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong
longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
N. Bokor, and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,”
Opt. Lett. 29(17), 1968–1970 (2004).
W. Chen, and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi
microscopy,” Opt. Lett. 34(16), 2444–2446 (2009).
B. Richards, and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an
aplanatic system,” Proc. Roy. Soc. A 253(1274), 358–379 (1959).
J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, New York, 1941).
1. Introduction
When radially polarized laser beams are focused by a lens of high numerical aperture (NA),
they exhibit unique focusing properties in comparison to linearly polarized ones: a smaller
#136303 - $15.00 USD
(C) 2011 OSA
Received 8 Oct 2010; revised 19 Nov 2010; accepted 7 Dec 2010; published 5 Jan 2011
17 January 2011 / Vol. 19, No. 2 / OPTICS EXPRESS 673
focus spot and a strong axial electric field component [1, 2]. This made them find applications
in many fields, e.g., electron acceleration [3, 4], spectroscopy [5], and particle trapping and
manipulation [6–10]. By imposing proper amplitude or/and phase modulations on radially
polarized input fields, unusual field distributions can be constructed at the focus. For
example, a uniform and non-diffracting axially polarized beam with sub-diffraction beam size
was obtained by focusing a radially polarized Bessel–Gaussian beam with a combination of a
binary-phase optical element and a lens of high NA [11, 12]. Recently, the possibility of
focusing a radially polarized beam to a sharp spherical focal spot was demonstrated
theoretically for a 4Pi focusing system by properly choosing the input field at the pupil plane
of the lens [13, 14]. Such spherical focal fields provide, e.g., equal axial and transverse
resolutions for confocal microscopy. In this paper, we generalize the approach of Refs [13,
14]. in order to achieve a dynamical spherical spot, i.e., a spherical spot that can be shifted
along the optical axis in real time.
2. Theory
Fig. 1. A 4Pi focusing system consists of two confocal high-NA objective lenses illuminated
by two counter-propagating radially polarized beams. Their fields at the pupil planes of the
lens are denoted by lleft (θ) with 0°θ90°and lright (θ) with 90°θ180°or, equivalently, l(θ)
with 0°θ180° for the total field.
Figure 1 sketches the geometry of a typical 4Pi focusing system. It consists of two objective
lenses of high numerical aperture (For the simplicity of mathematic derivation, we assume
that NA = 1). Two counter-propagating radially polarized light beams are focused by both
lenses such that the foci coincide. The input field at the pupil plane of each lens takes the
form of Einput = l(θ)eρ, where eρ (θ) is the unit vector in radial direction and l(θ) represents the
input field. We denote the input fields at the pupil planes by lleft(θ) and lright(θ) for the left and
right lens, respectively, where θ covers the range from 0° to 90° for lleft and from 180° to 90°
for lright, while the single input field l(θ) at the pupil plane of the lenses covers the full range
from 0° to 180°. When lleft and lright have a certain phase relation, the z-component of the
electric fields near the focus becomes remarkably strong thanks to constructive interference,
while the radial component experiences destructive interference and thus is very weak.
Mathematical description of this interference effect can be established by using the wellknown Richards–Wolf integral [2, 15]

E (  , z )   l ( ) X ( )sin 2 J1 (k  sin  )eih ( ) z d ,
(1)
0

Ez (  , z )  2i  l ( ) X ( )sin 2  J 0 (k  sin  )eih ( ) z d .
(2)
0
Here Eρ and Ez are the radial and axial components of the electric field at an observation point
R(ρ, z) near the focus, ρ and z are cylindrical coordinates, and X(θ) is the pupil apodization
function where X(θ) = (cosθ)1/2 for an aplanatic lens and X(θ) = 1 for a Herschel-type lens
(used in this paper), Jn(x) is the cylindrical Bessel function of first kind of order n, h(θ) =
kcosθ is the z-component of k, and k is the wave number.
Choosing the input field l(θ) to be l0(θ) = sinθ exp(2sin2θ) and substituting this into the
integrals (1) and (2), one can see that the intensity of the focal field, dominated by the zcomponent, exhibits an almost spherical distribution, i.e., a spherical focal spot, at z = 0. Our
#136303 - $15.00 USD
(C) 2011 OSA
Received 8 Oct 2010; revised 19 Nov 2010; accepted 7 Dec 2010; published 5 Jan 2011
17 January 2011 / Vol. 19, No. 2 / OPTICS EXPRESS 674
aim is to determine l(θ) such that it corresponds to a spherical (focal) spot translated to
another position z = z0 along the optical axis. As mentioned, the by far dominating
contribution to the spherical intensity distribution at the focus is coming from the zcomponent. Since |Eρ|2 is negligibly weak compared to |Ez|2, [for example, if l(θ) = l0(θ),
max(|Eρ|2)/max(|Ez|2) 0.1303 in the focal region] it is sufficient to consider only Eq. (2). As
we know, the z-component of the electric field obeys the scalar wave equation, whose
spherically symmetrical solution is the first kind of spherical Bessel function of zero order:
j0(kR)  sin(kR)/(kR), where R = |R|. For a fixed vector z = u3z0 on the optical axis (here u3 is
the unit vector in the z direction), the desired solution is j0(k|R z|) with an intensity |j0(k|R
z|)|2 that forms a spherical spot at the shifted position z0.
To derive the field l(θ) by which this can be achieved, we transform from cylindrical to
spherical coordinates by expressing the factor J0(kρsinθ)eih(θ)z in terms of the spherical wave
functions [16]:

J 0 (k  sin  )eih( ) z   i n (2n  1)Pn (cos  ) Pn (cos  ) jn (kR),
(3)
n 0
where Pn(x) denotes the Legendre polynomial of order n, α is the polar angle of the
observation point R, and jn(x) is the spherical Bessel function of the first kind with order n.
When Eq. (3) is inserted into Eq. (2), one obtains [Note that we use here X(θ) = 1]

Ez   An jn (kR) Pn (cos  ),
(4)
n 0
with

An  i n (2n  1)  l ( ) Pn (cos  )sin 2  d .
(5)
0
Now we have to find the coefficients An by which the right hand side of Eq. (4) transforms
into an expression of the form of j0(k|R z|) that represents a spherical spot centered at z0 on
the optical axis. Consider the addition theorem of the spherical Bessel functions [16]

j0 (k | R  u3 z0 |)   (2n  1) Pn (cos  ) jn (kz0 ) jn (kR).
(6)
n 0
Noting that jn(x) = (1)njn(x) and equating the right hand sides of Eqs. (4) and (6), we find that
An  (1)n (2n  1) jn (kz0 ),
(7)
Putting x = cosθ and g(x) = l(θ), Eq. (5) becomes
in
2n  1
An 
(1  x 2 )1/ 2 g ( x) Pn ( x)dx.
2
2 1
1
(8)
We recognize this as the coefficients of the Legendre series expansion of (1x2)1/2l(x). So, we
require the input field g(x) to be
in
An Pn ( x)
k 0 2

g ( x)  (1  x 2 )1/ 2 
(9)
Unfortunately, (1x2) 1/2 has a divergence at x = 1. To overcome this, we replace (1x2)1/2 by
g0(x). Here g0(x) = (1x2)1/2exp[2(1x2)] or l0(θ) = sinθ exp[2sin2θ], respectively, denotes
the fundamental radial polarization mode. Since Bokor and Davidson use the input field l0(θ)
#136303 - $15.00 USD
(C) 2011 OSA
Received 8 Oct 2010; revised 19 Nov 2010; accepted 7 Dec 2010; published 5 Jan 2011
17 January 2011 / Vol. 19, No. 2 / OPTICS EXPRESS 675
= sinθ exp(2sin2θ) to produce a spherical focal spot [13], we expect it to work in our case,
too. So Eq. (9) becomes
l ( )  A( )l0 ( )
(10a)
with
in
An Pn (cos  ),
n 0 2

A( )  
(10b)
i.e., the input field l(θ) can be thought of as a modulation of the fundamental mode l0(θ) by
A(θ).
3. Numerical results
In the last section we have derived expression (10) for the input field l(θ). In this section, we
substitute l(θ) into Eqs. (1) and (2) to check whether this gives the desired result. For A(θ) =
1, the situation is the same as in [13], where a spherical focal spot is produced with the input
field l(θ) = l0(θ) at the origin, but what we want is a spherical spot at another position z0 on
the optical axis. For numerical demonstration we have chosen z0 = 1.5λ. We calculate the
coefficients An from Eq. (7), and the input field l(θ) from Eq. (10). In Fig. 2(a) the intensity
|l(θ)|2 of the input field is plotted. We find that lleft(θ) and lright(θ) have exactly the same
Fig. 2. Intensity (a) and phase (b) of the input field l(θ) for generating a spherical intensity spot
centered at a point z0 = 1.5λ. The range of θ is from 0° to 90° for l left (θ) and from 180° to 90°
for l right (θ). The insets show the gray scale values of the phase.
intensity distributions and each distribution exhibits the intensity property of a fundamental
radially polarized field: an annular intensity distribution with intensity minimum in the center,
i.e., both are still fields with radial polarization. In fact, it can be verified that the modulation
function has a constant modulus, suggesting that modulation does not change the intensity
distribution of the input field. In Fig. 2(b), we present the phase of the input field l(θ). From
the phase distribution, a two-belt phase structure is found for lleft(θ). The first belt covers the
range from 0 to about 70° and the second one goes from 70° to 90°. In the first belt, the phase
increases almost linearly from 0 to 2π. Then, after a transition to 0 at θ = 70°, it increases
linearly from 0 to π in the second belt. The phase of lright(θ) has the same two-belt structure as
lleft(θ), but with a difference of sign in the corresponding belts. Figures 2(a) and 2(b) suggest
that the input field l(θ) can be obtained from a fundamental radial field mode by a two-belt
phase modulation, which can be achieved with a spatial light modulator. Having determined
l(θ), we calculate from the integrals (1) and (2) the electric intensities in the focal region.
Figure 3 shows line scans of the total intensity I = |Ez|2 + |Eρ|2 in the axial direction (solid line)
and in the transversal direction (dashed line), respectively. The maximum of the intensity has
been normalized to unity. As can be seen, a nearly spherical intensity spot centered at z0 =
1.5λ can indeed be realized. Axial and transversal spot diameters are nearly identical and have
a width of approximately 0.5λ, in agreement with the results in [13] and [14] at z0 = 0.
#136303 - $15.00 USD
(C) 2011 OSA
Received 8 Oct 2010; revised 19 Nov 2010; accepted 7 Dec 2010; published 5 Jan 2011
17 January 2011 / Vol. 19, No. 2 / OPTICS EXPRESS 676
Fig. 3. Line scans of the normalized total intensity of the electric fields of a spherical spot
centered at z = 1.5λ. The solid line refers to the axial intensity distribution and the dashed line
corresponds to the transversal intensity, overlaid on the axial line scan.
By changing the parameter z0 continuously, it is evident that the spherical spot will move
along the optical axis, i.e., the goal of a real time shifting of a spherical spot thus is solved.
For illustration, we plot 2D (XZ plane) color graphs of the intensity in the vicinity of the focus
for different values of z0 in Fig. 4 and demonstrate by a movie (Media 1) the evolution of the
intensity in the vicinity of the focus when the parameter z0 is changed continuously from 2λ
to 2λ (The value of z0 increases by 0.05λ every 0.05s). Figures 4(a)-(e) are successive frames
extracted from the movie at t = 0s, 1s, 2s, 3s and 4s (or z0 = 2λ, 1λ, 0λ, 1λ and 2λ),
respectively. As desired, we obtain a series of spots centered at z0 that are excellent
approximations to the desired spherical spots of intensity and that all have a radius of
approximately 0.5λ. The corresponding movie (Media 1) proves that the intensity distribution
keeps its nearly spherical shape over the whole translation range. To quantify the extent of the
shape of the spot to spherical shape, we introduce the size mismatch parameter ∆ XZ, measured
by the difference between transverse and axial diameters. We find that the size mismatch ∆ XZ
(slightly elongated along the z axis) are all roughly 0.036λ for three z0 ( = 0λ, 1λ and 2λ),
implying quantitatively the almost spherical shape of the spot during the whole translation.
Fig. 4. (Media 1) Evolution of the intensity of the spherical focal spot in the vicinity of the
focus as the parameter z0 changes continuously from 2λ to 2λ. (a)-(e) are successive frames
extracted from the movie at t = 0s, 1s, 2s, 3s and 4s (or z0 = 2λ, 1λ, 0λ, 1λ and 2λ),
respectively.
Before leaving this section, we want to make some remarks concerning the practical
applications of our design. First, the objective lens used in this paper is a Herschel-type lens,
while aplanatic lenses are much more widely used in practice. The Herschel-type lens has a
#136303 - $15.00 USD
(C) 2011 OSA
Received 8 Oct 2010; revised 19 Nov 2010; accepted 7 Dec 2010; published 5 Jan 2011
17 January 2011 / Vol. 19, No. 2 / OPTICS EXPRESS 677
special principal surface (see Fig. 1 in [13]), which can eliminate the first-order axial
aberration. Furthermore, for a Herschel-type focusing system the apodizer factor due to the
conservation of energy appearing in the Richards–Wolf integral (1) and (2) is simply X(θ) = 1,
for which a fundamental radial polarization mode is focused into a spherical spot centered at
the focus. For an aplanatic focusing system, the apodizer factor X(θ) = (cosθ)1/2, the resulting
focal spot is not of spherical shape but slightly transversely elongated (see Fig. 2 in [13]). To
obtain the spherical focal spot in the aplanatic focusing system, we can simply divide the field
amplitude l(θ) of the incoming beam by (cosθ)1/2, which means the introduction of an
amplitude modulation, as done by Chen and Zhan in [14]. Second, we have assumed the
maximum converging angle θmax to be 90° for the objective lens, which is absolutely
impossible for practical objective lens. Meanwhile, when the aplanatic focusing system is
used, the introduction of the amplitude modulation [l(θ)/(cosθ)1/2, as discussed above] also
requires θmax less than 90°. For two high numerical aperture objective lenses with θmax = 79.6°
[NA = 1.49 oil (1.515) immersion objective, Nikon], we find that the focal spot can still be
moved along the axis, but the spot is elongated along the transverse direction with the size
mismatch ∆XZ 0.1480λ (z0 = 2λ). However, we can reduce the size mismatch by properly
choosing the minimum converging angle θmin of the focusing system, which can be achieved
by blocking the central region of the incoming beam with an opaque disk. For the case
discussed here (θmax = 79.6°), if we put θmin = 25°, the size mismatch will be ∆XZ 0.082λ. The
final comment refers to the translation range of the spot, i.e, the maximum value of z0. In our
calculation, we find that for larger values of z0 our design still works. For example, when z0 =
10λ, we obtain a nearly spherical spot with ∆XZ -0.037λ. But the corresponding phase
distribution of the input field will become a ten-belt structure, while the phase structure for z0
= 1.5λ is of only two belt as shown in Fig. 2(b). With further increasing z0, the phase structure
becomes more complex. As a result, we conclude that our design can realize ± 10λ translation
of the spherical spot of intensity without complex phase modulation.
4. Conclusion
A method is proposed that allows shifting of an on-axis spherical spot in real time in a 4Pi
focusing system using a radially polarized beam. We have determined the proper input field
l(θ) that does the trick to produce a spherical intensity spot at any designated position z0. As
concerns the intensity, l(θ) has the same distribution as the fundamental radial polarization
mode, but its phase is modulated. In other words, l(θ) turns out to be simply a phase
modulated fundamental mode. We have shown that the position parameter z0 can be varied
continuously such that the spherical intensity distribution of the focus is maintained during
dynamical movement of the focal spot along the optical axis. In conclusion, we have pointed
out a way how to move a trapped particle or to scan a specimen without moving objective
lenses or laser beams.
Acknowledgments
This research is supported by the Natural Science Foundation of China (NSFC) (10874240,
61077005) and the Chinese Academy of Sciences (CAS)/State Administration of Foreign
Experts Affairs of China (SAFEA) International Partnership Program for Creative Research
Teams.
#136303 - $15.00 USD
(C) 2011 OSA
Received 8 Oct 2010; revised 19 Nov 2010; accepted 7 Dec 2010; published 5 Jan 2011
17 January 2011 / Vol. 19, No. 2 / OPTICS EXPRESS 678