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Yan et al. Vol. 27, No. 9 /September 2010 /J. Opt. Soc. Am. A 2033

Generation of multiple spherical spots with aradially polarized beam in a 4� focusing system

Shaohui Yan,1 Baoli Yao,1,* Wei Zhao,1 and Ming Lei2

1State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics,Chinese Academy of Sciences, Xi’an 710119, China

2College of Life Science and Shaanxi Key Laboratory of Molecular Biology for Agriculture,Northwest A & F University, Yangling 712100, China

*Corresponding author: yaobl@opt.ac.cn

Received May 11, 2010; revised July 22, 2010; accepted July 22, 2010;posted July 26, 2010 (Doc. ID 128282); published August 18, 2010

We demonstrate the possibility of creating multiple spherical spots in a 4� focusing system with a radiallypolarized beam. Using spherical waves to expand the plane wave factor in the Richards–Wolf integral, it isfound that a proper spatial modulation in the amplitude of the input field with radial polarization can formmultiple spherical spots with a focusing system satisfying the Herschel condition. These spots are distributedsymmetrically about the focus on the optical axis with variable positions and intensities. Although we consideronly the case of three spherical spots in this paper, generalization to the multiple-spots case will present nodifficulty. © 2010 Optical Society of America

OCIS codes: 110.2990, 140.3300, 260.1960.

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. INTRODUCTIONadially polarized beams were shown to have importantpplications in many fields such as electron acceleration1,2], spectroscopy [3], and particle trapping and manipu-ation [4–7] due to their unique focusing properties ofmaller focus spot size and strong axial electric field com-onent compared to linearly polarized beams [8]. When aeld with radial polarization is properly modulated, it cane focused to construct unusual field distributions. For ex-mple, in a recent paper, Wang et al. [9] created a uniformnd non-diffracting axial light beam with a sub-iffraction beam size �0.43�� by focusing a radially polar-zed Bessel–Gaussian beam with a combination of ainary-phase optical element and a high-numerical-perture (NA) lens. Recently, radially polarized beamsave been theoretically studied with a hope for improvinghe axial resolution of confocal microscopy in the 4� con-guration [10,11], in which two counter-propagatingeams, coming from two opposing high-NA objectiveenses, coherently illuminate a fluorescent specimen. Theroper choice of the input field at the pupil plane can yieldsharp and spherical spot, resulting in equal axial and

ransverse resolutions. Bokor and Davidson [10] used aundamental radial polarization mode to create a nearlypherical central spot in the focal region, with a reducedpot volume and uniformly low sidelobe intensities. Chennd Zhan [11] generated a spherical spot by focusing a ra-ial polarization beam with spatial amplitude modulationbtained by solving the inverse problem of antenna radia-ion. With spatial phase modulation, they also realizedwo identical spherical spots on the optical axis. However,his can happen only when the distance between the twopots is an odd integer number times a half-wavelength.n this paper, we present a method of generating multiplen-axis spherical spots in a 4� focusing system with a ra-

1084-7529/10/092033-5/$15.00 © 2

ially polarized beam whose amplitude is properly modu-ated. These spherical spots are symmetrical about the fo-us in position and intensity to ensure that the inputelds at the pupils of two objective lenses maintain sym-etry. Our method may find applications in multiple par-

icle optical trapping and optical microscopy. Specifically,hen the position parameter varies continuously, we canake the single spherical focal spot move along the opti-

al axis.

. THEORETICAL MODELhe typical 4� focusing system is shown in Fig. 1, whichonsists of two high-NA �NA=1� objective lenses havingoincident foci. With a phase difference of � between thewo input fields as indicated in Fig. 1 (electric fields in si-ultaneously opposite directions at corresponding posi-

ions), the z-component of electric fields near the focusill be strengthened by constructive interference com-ared to focusing by a single objective lens, while the ra-ial component amplitude becomes negligibly weak due toestructive interference. The powerful tool for describinghe focusing of field by high-NA objective lenses is the soalled Richards–Wolf integral [12,13] involving the inputeld amplitude l��� at the pupils. Our task is to seek aroper l��� to lead the integral to yielding the desired re-ult. Before going further on this issue, we first consider aore general problem: Can the z-component of an electriceld yield the intensity of a multi-sphere shape on thexis? The answer is “yes.” As we know, the axial electriceld component obeys the scalar wave equation, whosepherically symmetrical solutions are the first kind of thepherical Bessel functions of zero order: j0�k�R−x��sin�k�R−x�� / �k�R−x��, where k is the wave number andand x stand for the position vectors with �y� denoting

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2034 J. Opt. Soc. Am. A/Vol. 27, No. 9 /September 2010 Yan et al.

he length of vector y. For the solution j0�k�R−x�� with axed x (in this paper, x is assumed to be on the opticalxis to ensure cylindrical symmetry), its intensity

j0�k�R−x���2 takes on the spherical distribution with aenter at x. Therefore, if we linearly combine these solu-ions with different x’s, the resulting solution �Cxj0�k�Rx�� will be expected to have the desired intensity distri-ution: multiple spherical spots with different centers x’s,rovided that the distance between any two centers is suf-ciently large. We find that if l��� makes the Richards–olf integral for the z-component yield �Cxj0�k�R−x��,

he result of on-axis multi-spherical spot will be obtained.Consider a radially polarized beam whose amplitude

unction at the pupil of the objective lens is l���. When fo-used by a high-NA objective lens, the electric fields in theicinity of the focal spot can be calculated by theichards–Wolf vectorial diffraction integral method as

12,13]

E���,z� =�0

�max

l���X���sin 2� J1�k� sin ��eihzd�, �1�

Ez��,z� = i2�0

�max

l���X����sin ��2J0�k� sin ��eihzd�. �2�

ere E��� ,z� and Ez�� ,z� are the radial and axial electriceld components at the observation point R�� ,z� near the

ocus; �max=sin−1�NA� is the maximum converging angleetermined by the NA; X��� is the pupil apodization func-ion, and X���= �cos ��1/2 for an aplanatic lens and X���1 for a Herschel type lens (used in this paper); Jn�x� is

he Bessel function of the first kind with order n; and=k cos � is the z-component of k. For the 4� focusingystem �NA=1� shown in Fig. 1, the upper limit �max inntegrals (1) and (2) is replaced with �. Then the two in-egrals can be written as

E���,z� =�0

�

l���sin 2� J1�k� sin ��eihzd�, �3�

Ez��,z� = i2�0

�

l����sin ��2J0�k� sin ��eihzd�. �4�

ere we have used the condition X���=1.As mentioned above, in the 4� focusing system as

hown in Fig. 1, �E��2 is negligibly weak compared to �Ez�2.o, let us focus on Eq. (4). Now we want the right-handide of Eq. (4) to be of the form of �C j �k�R−x��, which is

ig. 1. (Color online) Geometry of the 4� focusing system con-isting of two confocal high-NA objective lenses, illuminated bywo counter-propagating radially polarized doughnut beams withrelative � phase shift.

x 0

he linear combination of spherical wave functions. It iseasonable to write the right-hand side of Eq. (4) in thepherical coordinate system. Note that only the factor0�k� sin ��eihz in the integrand is not expressed inpherical coordinates, which represents a cylindricalave. Expressing the factor J0�k� sin ��eihz in spherical

oordinates can be achieved by expanding the cylindricalave in terms of spherical wave functions [14],

J0�k� sin ��eihz = �n=0

�

in�2n + 1�Pn�cos ��Pn�cos ��jn�kR�,

�5�

here Pn�x� denotes the Legendre polynomial, � is the po-ar angle of the observation point R with R= �R�, and jn�x�s the spherical Bessel function of the first kind with order. The cylindrical symmetry of J0�k� sin ��eihz is ex-ressed by the independence of the spherical wave func-ion factor Pn�cos ��jn�kR� on the azimuthal angle. Substi-uting the spherical wave expansion (5) for0�k� sin ��eihz into Eq. (4) gives

Ez = i2�n=0

�

Anjn�kR�Pn�cos ��, �6�

ith An defined through

An = in�2n + 1��0

�

l���Pn�cos ���sin ��2d�. �7�

ow, the task is to specify the coefficients An’s to trans-orm the right-hand side of Eq. (6) into the linear combi-ation �Cxj0�k�R−x��, i.e., the field with intensity configu-ation of multiple spherical spots on the axis. First, weant some restrictions on the x’s and Cx’s. Since the two

nput fields shown in Fig. 1 are symmetric about the z0 plane, it is necessary that these spherical spots on thexis are symmetrical about the focus, both in position andn intensity, that is, Cx=C−x. Consider three sphericalpot distributions centered at z=0, ±z0; the correspondingeld may take the form

F = i2C0j0�kR� + i2C�j0�k�R − ezz0�� + j0�k�R + ezz0���,

�8�

here ez is the unit vector in the z direction, C0 and C areonstants, and the proportionality factor i2 is introducedor convenience of comparison with Eq. (6) later. Clearly,his field corresponds to a spherical focal spot centered athe focus and two identical spherical spots symmetricallyligned on the two sides of the focus. The peak intensityatio of the central focal spot to that of the side spots isetermined by C0 /C. The last two spherical fields j0�k�Rezz0�� and j0�k�R+ezz0�� in expression (8) are expressedith respect to two different centers ±z0, while the right-and side of Eq. (6) is expressed with respect to the singleenter z=0. For the convenience of comparison, we mustransform j0�k�R−ezz0�� and j0�k�R+ezz0�� into the sameoordinate system as for Eq. (6). This is done by the addi-ion theorem of the spherical Bessel functions [14]:

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Yan et al. Vol. 27, No. 9 /September 2010 /J. Opt. Soc. Am. A 2035

j0�k�R ± ezz0�� = �n=0

�

�2n + 1�Pn�cos ��jn�±kz0�jn�kR�.

�9�

hen this expression is inserted into expression (8), webtain a sum over n:

F = i2�n=0

�

�2n + 1���0nC0 + 2Cjn�kz0��jn�kR�Pn�cos ��.

�10�

n going from Eq. (9) to Eq. (10), we have used the rela-ion jn�x�= �−1�njn�x�. So the odd terms in the sum vanish.quating the right-hand sides of Eqs. (10) and (6), we find

hat if An’s satisfy

An = �0nC0 + 2C�2n + 1�jn�kz0�, n = even,

An = 0, n = odd, �11�

e may obtain the expected field. Having obtained the co-fficients An’s in the field expression (6) which now repre-ents the superposition of three spherical fields, the finaloal of determining the input field l��� can be reached byesolving Eq. (7). With the change of variable x=cos �, theoefficient expression (7) becomes

i−n

2An =

2n + 1

2 �−1

1

�1 − x2�1/2l�x�Pn�x�dx. �12�

his is nothing but the coefficients of the Legendre seriesxpansion of �1−x2�1/2l�x�. So, we require the amplitudeunction l�x� to be

l�x� = �1 − x2�−1/2�k=0

�

�− 1�kA2k

2P2k�x�. �13�

owever, the divergence of �1−x2�−1/2 at x=1 occurs. Tovercome this, we replace �1−x2�−1/2 with l0�x�, where0�x�= �1−x2�1/2exp�−2�1−x2�� or �sin ��exp�−2�sin ��2� ishe fundamental radial polarization mode. Since Bokornd Davidson used l0���= �sin ��exp�−2�sin ��2� to producespherical focal spot [10], we expect it to work in our pro-

ram. So Eq. (13) becomes

l�x� = l0�x��k=0

�

�− 1�kA2k

2P2k�x� = l0�x�A�x�. �14�

he amplitude function l�x� given by Eq. (14) can behought of as a modulation of the fundamental mode l0�x�y A�x�=1/2��−1�kA2kP2k�x�. In fact, the modulation fac-or A�x� acts to translate the single spherical field (cen-ered at the focus) along the optical axis by z0 and −z0.

. NUMERICAL RESULTSow, we present the simulation results. First, we want to

ealize three identical spherical spots: one being the focalpot at the focus and the other two being centered at z0

±10�. Take C0=C=1 to ensure equal intensity. Withhese parameters, the focused field will take the form EE +E +E , where E is the central spherical field and

0 L R 0

L and ER are two side spherical fields, which are ob-ained by translating E0 by z0 and −z0, respectively, asentioned above. From Eq. (11), we obtain the coeffi-

ients An’s and then the input field amplitude l��� at theupil by Eq. (14), whose graph is plotted in Fig. 2 [the dot-ed line l3���]. Using integrals (3) and (4), we calculate theelds, and thus the total intensity, at the focal region. Fig-re 3(a) shows the total intensity I= �Ez�2+ �E��2 of the elec-ric field in the neighborhood of the focus. The linescans ofhe axial and transversal intensity distributions arehown in Fig. 3(b). All intensities are normalized to theaximum of the total intensity, and all spatial coordi-ates are measured in wavelength. From the calculatione see that �E��2 always contributes very little to the total

ntensity I, especially at the locations of the three brightpots, which implies that �Ez�2 is dominant, as we pointedut previously. As seen from Figs. 3(a) and 3(b), three ap-roximately identical spherical spots (the central one cor-esponds to the focal spot) have been obtained. Each ofhem has nearly equal axial and transversal spot sizes, asell as the radius, approximately equal to 0.5�. We note

hat this value agrees with the results in [10,11]. Thishould come as no surprise. This value R=0.5� happenso be the distance such that kR is the first zero of thepherical Bessel function j0�kR�. It can also be verifiedhat �j0�kR��2 and the total electrical intensity distributionf the spherical focal spot behave very similarly in theange 0�R�0.5�. So the job of creating a spherical focalpot is equivalent to realizing a field distribution of theorm j0�kR� in the range 0�R�0.5�. From Fig. 3(b), wend that the intensity peak of the central spherical spot islightly below those of two side spherical spots. This cane explained as follows. As mentioned above, the focaleld is equal to E=E0+EL+ER, where E0, EL, and ER de-ne three almost identical spherical fields with three dif-erent centers: z=0 (the focus), z=z0 and z=−z0, respec-ively. The intensity will be �E�2= �E0�2+ �EL�2+ �ER�2+C .T.,here C .T. is the crossing term which represents the in-

erference of three spherical fields. The graph of E0 (notlotted in this paper) shows that on the optical axis thealues of E0 at z0 and 2z0 are E0�z0��0 and E0�2z0�=,here is a small positive amount. Since EL and ER are

ig. 2. (Color online) Input field amplitude as a function of �:0��� corresponds to the fundamental radial polarization modesolid line), l3��� corresponds to generation of three identicalpherical spots (dotted line), and l2��� corresponds to generationf two identical spherical spots (dashed line).

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2036 J. Opt. Soc. Am. A/Vol. 27, No. 9 /September 2010 Yan et al.

btained by translating E0 by z0 and −z0, we have EL�0�0 and EL�z0�=, and ER�0��0 and ER�−z0�=. This in-

icates that the crossing term acts to enhance the inten-ity at z0 and −z0, while keeping the intensity at the focusnchanged. As a result, the intensity peak of the centralpherical spot is smaller than those of the two side spheri-al spots as plotted in Fig. 3(b). Since the field is symmet-ic about the z=0 plane, the intensity peaks of two sidepherical spots must be equal. Furthermore, we note inig. 3(b) that the centers of the two side spherical spotsre at z= ±3.162�, which are in good agreement with thexpected locations ±10�.

In the above numerical simulations, if we put C=0, weill obtain from Eqs. (11) and (14) the input field l�x�l0�x�, which is identical to that used by Bokor andavidson [10]. So, a single spherical focal spot will be pro-uced. While putting C0=0, we are dealing with whathen and Zhan have done in [11], creating two identicalpherical spots. However, their model limits the distance

ig. 3. (Color online) Normalized intensity distributions of elec-ric fields of (a), (b) three and (c) two spherical spots with all in-ensities normalized to the maximum of the total intensity. (a)he total intensity distribution in XZ plane. (b) Linescans of cor-esponding transversal and axial intensity distributions of (a);0�x�, I−�x�, and I+�x� are transversal linescans for the threepherical spots overlaid on the axial linescan I�z�. (c) Intensityinescans of electric fields of two identical spherical spots with−�x� and I+�x� denoting the transversal intensity distribution.

etween the two spots to an odd integer number times aalf-wavelength. This does not occur in our model. In thebove example this distance is 210�. If we want the dis-ance between the two spherical spots to be 6�, we simplyet z0= ±3� and C0=0 and C=1. The corresponding inputeld amplitude function versus � is given in Fig. 2 [theashed line l2���], and the linescans of the total intensityre shown in Fig. 3(c). As desired, two identical sphericalpots with respective centers ±3� are obtained. From Eq.8), generalization to the case of more spherical spots wille possible by just introducing more C’s and z0’s.

. CONCLUSIONSn conclusion, we have presented the method of creatingultiple on-axis spherical spots in a 4� focusing systemith a radially polarized beam. Using a spherical wavexpansion technique, we obtain the proper input fields athe pupils, which is a fundamental radial polarizationode times an amplitude modulation function. These

pherical spots are distributed symmetrically about theocus on the optical axis with variable positions and inten-ities. For example, the case of three identical sphericalpots can be transformed into single spherical spot andwo identical spherical spot distributions with proper pa-ameters. Although we consider only the case of threepherical spots, the generalization to the multiple-spotsase will present no difficulty. Our method may find ap-lications in multiple particle optical trapping and opticalicroscopy. Specifically, when only a spherical focal spot

s created and the position parameter z0 varies continu-usly, we may expect a dynamical move of the focal spotlong the optical axis. This will provide an alternative tooving the trapped particle and scanning the specimenithout moving the objective lens or laser beams.

CKNOWLEDGMENTShis research is supported by the National Natural Sci-nce Foundation of China (NSFC) (10874240, 60678023)nd the Shaanxi Province 13115 Science and Technologynnovative Project (2008ZDKG-68).

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