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Y. Kozawa and S. Sato Vol. 24, No. 6 /June 2007/J. Opt. Soc. Am. A 1793

Sharper focal spot formed by higher-orderradially polarized laser beams

Yuichi Kozawa and Shunichi Sato

Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Katahira 2-1-1, Aoba-ku,Sendai 980-8577, Japan

Received October 17, 2006; accepted December 19, 2006;posted January 16, 2007 (Doc. ID 76186); published May 9, 2007

The intensity distributions near the focal point for radially polarized laser beams including higher-order trans-verse modes are calculated based on vector diffraction theory. For higher-order radially polarized mode beamsas well as a fundamental mode �R-TEM01

*� beam, the strong longitudinal component forms a sharper spot atthe focal point under a high-NA focusing condition. In particular, double-ring-shaped radially polarized mode�R-TEM11

*� beams can effectively reduce the focal spot size because of destructive interference between theinner and the outer rings with � phase shift. Compared with an R-TEM01

* beam focusing in a limit of NA=1, the full width at half-maximum values of the focal spot for an R-TEM11

* beam are decreased by 13.6% forthe longitudinal component and 25.8% for the total intensity. © 2007 Optical Society of America

OCIS codes: 050.1960, 140.3300, 260.5430, 260.3160.

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. INTRODUCTIONadially polarized laser beams are known to generate atrong longitudinal electric field at the focal point in thease of high-numerical-aperture (NA) focusing.1 Thistrong longitudinal component forms a tight focal spothat will advance high-resolution microscopy2,3 and theeneration of an evanescent Bessel beam by the excita-ion of a surface plasmon.4 In addition, advantages of ra-ially polarized beams were theoretically and experimen-ally found in many applications, such as opticalrapping,5 material processing,6 and particlecceleration.7 In calculations, however, the focusing beamas assumed to be a flat-top4,5 or single-ring-shapedeam, which is often referred to as a radially polarizedEM01

* �R-TEM01*� mode beam.1–3 On the other hand, a

ouble-ring-shaped beam was experimentally observed ashigher-order radially polarized mode �R-TEM11

*� di-ectly from a laser cavity.8 In addition, these higher-orderode beams may be generated by a particular laser cav-

ty designed to oscillate only with radial polarization.9,10

ecently, it was theoretically reported that a double-ring-haped radially polarized beam has the potential to formdark space surrounded by a intense light field around

he focal point under a particular focusing condition.11

his dark space is a so-called optical bottle beam12 or op-ical cage.11 In the case of an R-TEM11

* mode, a double-ing structure accompanies with � phase shift betweennner and outer rings, which brings the destructive inter-erence of the longitudinal component near the focal pointn high-NA focusing. Therefore, the intensity distributionear the focal point varies drastically with the degree ofhe truncation of the incident beam by a pupil.

The focusing of a higher-order radially polarized modeeam11,13 such as an R-TEM11

* is similar to the super-esolution techniques that require a complex annular pu-il or phase mask in imaging systems.2,14,15 A higher-

1084-7529/07/061793-6/$15.00 © 2

rder radially polarized mode beam intrinsicallyossesses a ring-shaped intensity distribution and phasehift in the beam cross section. Consequently, higher-rder radially polarized laser beams can be expected toorm a narrower focal spot compared with R-TEM01

*

eam focusing. In fact, calculation has shown that aightly focused R-TEM11

* beam generates a sharp focalpot of the longitudinal component.11

In this paper we investigate the focusing properties ofadially polarized beams including higher-order modesased on vector diffraction theory. In particular, the focalpot size for radially polarized beams is numerically stud-ed. For higher-order radially polarized modes, as well asor a single-ring-shaped R-TEM01

* mode, the strong lon-itudinal component forms a sharper spot at the focaloint under high-NA focusing conditions. A double-ring-haped R-TEM11

* mode beam can effectively reduce theocal spot size because of the destructive interference be-ween inner and outer rings. Compared with R-TEM01

*

eam focusing in a limit of NA=1, the full width at half-aximum (FWHM) values of the focal spot for an-TEM11

* beam are decreased by 13.6% for the longitudi-al component and 25.8% for the total intensity (the sumf the longitudinal and transverse components).

In Section 2 we describe the electric field distribution ofigh-NA focusing for radially polarized beams on the ba-is of vector diffraction theory. Then the mathematical ex-ression for higher-order R-TEMp1

* beams is presented.n Section 3 the numerical studies of the intensity distri-utions near the focal point for radially polarized lasereams with a higher-order transverse mode are pre-ented. In Section 4 the focusing properties of R-TEM11

*

ode beams are shown in detail.

. THEORYradially polarized beam is frequently referred to as a

ull centered doughnut mode beam, TEM *, which has

01

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1794 J. Opt. Soc. Am. A/Vol. 24, No. 6 /June 2007 Y. Kozawa and S. Sato

ompletely cylindrical symmetry in both intensity and po-arization distributions. The notation of R-TEM01

* corre-ponds to a single ring-shaped radially polarized beam asfundamental mode. Similarly, the higher-order modesith multiring beam patterns that are expressed as-TEMp1

* also have completely cylindrical symmetry and+1 rings. The intensity distributions of cylindricallyymmetric R-TEMp1

* modes are represented by usingaguerre–Gaussian distributions.16–18 Figure 1 shows thealculated intensity distributions and the instantaneousolarization states for several modes, R-TEM01

*,-TEM11

*, and R-TEM21*. Every mode has a ring-shaped

ntensity distribution as shown in the lower part of thegure. In the upper part, the phase relationship betweenhe rings is depicted. For example, the R-TEM11

* modeas two rings, and the phase difference between the rings

s �. This phase difference of � will cause destructive in-erference in the focal region.11

The intensity distributions of cylindrical vector beamshat have radial or azimuthal polarization in the focal re-ion were calculated by Youngworth and Brown,1 follow-ng the vector diffraction theory that deals with a polar-zation contribution to the focusing with a high-umerical-aperture lens.19 When a radially polarizedeam is tightly focused by an aplanatic lens, the electriceld in the focal region is divided into two components,amely, the radial (transverse) field Er and the axial (lon-itudinal) field Ez. No azimuthal (tangential) componentccurs through the focusing as long as an incident beam isompletely radially polarized. In the cylindrical coordi-ate, these components are written as1

Er�r,�,z� = A�0

�max

cos1/2���sin�2��l0���J1�kr sin ��

�exp�ikz cos ��d�, �1�

Ez�r,�,z� = 2iA�0

�max

cos1/2���sin2���l0���J0�kr sin ��

�exp�ikz cos ��d�, �2�

here �max=sin−1�NA/n�, which is the maximum angleelated to the numerical aperture (NA) of the objective,here n is the refractive index in image space, l0��� is the

ig. 1. Bottom, theoretical intensity distributions of radially po-arized TEM01

*, TEM11*, TEM21

* modes. Top, corresponding in-tantaneous polarization states.

elative amplitude of the electric field in front of a pupil, ks the wave number in image space, A is a constant, andm is the first kind of Bessel function of order m. The geo-etrical focal point is located at the origin �0,0,0�. We as-

ume that the electric field of the incident R-TEMp1*

eams is11

l0��� =�0

2 sin �

sin2 �maxexp�−

�02 sin2 �

sin2 �max�Lp

1�2�02 sin2 �

sin2 �max� , �3�

here Lp1 is the generalized Laguerre polynomial and �0

s the size parameter of the incident beam determined byhe ratio of the pupil radius to the incident beam waist inront of the objective. Therefore the incident R-TEMp1

*

eam that has a certain intensity distribution is more oress truncated by an objective according to �0. However,or example, in the case of an R-TEM11

* beam �0 shoulde greater than 1 because the outer ring of the R-TEM11

*

eam will be completely blocked by the pupil if �1�1. Inhis study, all length measurements are in units of wave-ength; therefore the wavelength of an incident beam is 1.he refractive index n is assumed to be 1 throughout thisaper. A is determined so that the truncated focusingeam has the same optical power in image space.

. NUMERICAL APERTURE DEPENDENCEOR FOCAL SPOT SIZEigure 2 shows the intensity profiles at the focal point

z=0� for various radial mode numbers p. For the sake ofomparison with the result in Ref. 2, the NA is assumed toe 1 in our calculations. In addition, the value of �0 forach mode beam are determined so that 90% of the incom-ng intensity in front of the objective is focused. Each in-ensity profile is normalized to the maximum value of theotal intensity (solid curves) for the R-TEM01

* beam inig. 2(a). In any mode beam focusing, only the strong lon-itudinal component (dashed curves) emerges on theeam axis, whereas no radial component (dotted curves)s seen on the axis. The longitudinal peak intensity be-omes weaker for higher-order mode beam focusing, inontrast to the enhancement and broadening of the side-obes.

The longitudinal and total intensity distributions forach mode are plotted in Fig. 3. The peak intensities areormalized to 1. It is clearly seen that the center peaks ofhe longitudinal component for the higher-order modesre sharper than that of the R-TEM01

* mode as shown inig. 3(a), while the sidelobes become stronger. It is

hought that such size reduction of the focal spots is at-ributed to the � phase shifts of the multiring beam pat-ern for the higher-order modes, which causes destructiventerference of the longitudinal component near the focaloint. The values of the FWHM for the center focal spotre obtained to be 0.464�, 0.401�, 0.385�, and 0.370� forhe radial mode number p=0, 1, 2, 5, respectively. For theotal intensity distributions shown in Fig. 3(b), the corre-ponding FWHM are 0.582�, 0.432�, 0.403�, and 0.378�.ote that the focal spot size of the R-TEM11

* beam is re-uced by 13.6% in the longitudinal component and by5.8% in total intensity compared with the R-TEM01

*

eam focusing. The focal spot sizes calculated by Quabis

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Y. Kozawa and S. Sato Vol. 24, No. 6 /June 2007/J. Opt. Soc. Am. A 1795

t al.2 for radially polarized doughnut mode focusing in aigh-NA limit �NA=1� were 0.160�2 for the longitudinalomponent and 0.260�2 for the total intensity. These spotizes given in units of �2, which are determined by therea at half the maximum value, correspond to FWHM of.451� and 0.575� for the longitudinal component and theotal intensity, respectively. These values are very close tohose obtained for the R-TEM01

* beam in our calculation.hus the higher-order mode beam can create a smallerpot size than can the doughnut mode beam.

Figure 4 shows the intensity distributions of the longi-udinal component along the optical axis for each mode.he peak intensities are normalized to 1. From Fig. 4, wean see the elongation of the longitudinal component withncreasing radial mode number. This behavior is in strongontrast to the results obtained for the spot size, whichecreases with increasing radial mode number, as shownn Fig. 3. In other words, the spot size can become smallert the cost of elongation in the axial direction. Thus, onean manipulate the distribution of the longitudinal com-onent along the optical axis by choosing the proper modeumber of the radially polarized beam.In the previous calculations, the NA is assumed to be 1

s an extreme case. Table 1 gives the FWHM values of theocal spot size for each mode when the NA varies from 0.6o 1.0. Note that the FWHM values of the total intensity

ig. 2. (Color online) Calculated intensity profiles near the f-TEM21

*, (d) R-TEM51* modes, with a NA of 1. The total intens

adial component (dotted curves) are drawn in each figure. Evers in units of wavelength.

ocal point �z=0� for the focusing of (a) R-TEM01*, (b) R-TEM11

*, (c)ity (solid curves), the longitudinal component (dashed curves), and they profile is normalized to the peak intensity in (a). The horizontal axis

ig. 3. (Color online) Intensity profiles of (a) longitudinal com-onents and (b) total intensity in the focal plane for differentransverse modes. The peak intensity for each mode is normal-zed to 1.

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1796 J. Opt. Soc. Am. A/Vol. 24, No. 6 /June 2007 Y. Kozawa and S. Sato

or R-TEM01*, R-TEM11

*, and R-TEM21* are not obtained

or the several lower NAs and are indicated by “−” be-ause the longitudinal component at the focal point be-omes smaller than the surrounding transverse (radial)omponent. As a result, these total intensity distributionsn the focal plane have a dent at the focal point. Thereforehe tight focal spots are no longer formed. Table 1 showslearly that the FWHM values of the longitudinal compo-ent and the total intensity decreases with increasing ra-ial mode number and NA.For comparison, the focal spot sizes for a linearly polar-

zed fundamental Gaussian �L-TEM00� beam calculatedlong the x and y axes are added to Table 1. The incident-TEM00 beam is assumed to be polarized along the xxis. The beam width of the beam is determined so that0% of the incoming intensity in front of the objective isocused. The FWHM values for L-TEM00 in Table 1 indi-ate that the focal spot stretches along its polarization di-ection because the FWHM along the x axis is larger thanhat along the y axis. This elongation of the focal spot inhe direction of the polarization is known to be the effectf the longitudinal component for linearly polarized beamocusing with a high-NA lens.19 It is also known that thislongation effect can be reduced by radially polarized il-

Table 1. FWHM „�−1… of the Focal Spot for

High-NA Focusinga

NA

Beam Mode 0.6 0.7 0.8 0.9 1.0

ongitudinal componentR-TEM01

* 0.809 0.691 0.602 0.530 0.464R-TEM11

* 0.675 0.579 0.506 0.450 0.401R-TEM21

* 0.642 0.551 0.483 0.430 0.385R-TEM51

* 0.614 0.527 0.461 0.411 0.370otal intensityR-TEM01

* — — 1.410 0.923 0.582R-TEM11

* — 1.119 0.749 0.551 0.432R-TEM21

* — 0.970 0.656 0.499 0.403R-TEM51

* 1.323 0.846 0.585 0.456 0.378L-TEM00,xaxis 1.020 0.899 0.818 0.767 0.756L-TEM00,yaxis 0.942 0.801 0.693 0.606 0.522

aA dash indicates that a focal spot is not obtained.

ig. 4. (Color online) Calculated intensity profiles of the longi-udinal component along the z axis for R-TEM01

*, R-TEM11*,

-TEM21*, R-TEM51

* mode focusing. The peak intensity for eachode is normalized to 1.

umination. In fact, the center focal spots in total inten-ity for the radially polarized beam are sharper thanhose for the linearly polarized beam, especially inigh-NA focusing as shown in Table 1. However, since theenter focal spot of the radially polarized beams consistsainly of the longitudinal component, the formation of a

harper spot size by using the radially polarized beams isimited to high-NA focusing.

As shown in Table 1, the reduction of the focal spot sizeith increasing NA is dominant for the R-TEM11

* beamompared with the R-TEM01

* beam. Although higher-rder modes than R-TEM21

* can also reduce the focal spotize in the high-NA condition, the size reduction ratio be-omes small. In addition, a gradual decrease of the peakntensity of the focal spot is shown in Fig. 2 besides therowth and the broadening of the sidelobes in the focallane in the higher-order radially polarized beam focus-ng. Accordingly, an R-TEM11

* mode beam may provideoth a sharper focal spot size and a comparatively strongeak intensity in high-NA focusing.

. BEAM WIDTH DEPENDENCE OF R-TEM11*

ODE BEAM FOCUSINGn the previous section, the incident beam width param-ter �0 for each mode was determined to focus 90% of thehole intensity in front of the objective. However, the in-

ensity distribution near the focal point will vary with �0hen an R-TEM11

* mode beam that has a � phase shift inbeam cross section is tightly focused.Figure 5(a) shows the peak intensity of the longitudinal

omponent, �Ez�2, at the focal point for the R-TEM01* mode

dashed curve) and R-TEM11* mode (solid curve) focusing

ith NA=1 when �0 varies in the range of 0��0�4.rom Fig. 5(a) it can be seen that the intensity of the lon-itudinal component for the R-TEM11

* beam changes

ig. 5. Peak intensity of the longitudinal component ��Ez�2� athe focal point plotted against the beam width parameter �0 for-TEM01

* (dashed curve) and R-TEM11* (solid curve) mode focus-

ng. (b) FWHM variation of the focal spot corresponding to (a).he FWHM value is in units of wavelength.

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Y. Kozawa and S. Sato Vol. 24, No. 6 /June 2007/J. Opt. Soc. Am. A 1797

rastically with the �0 values, in contrast to the-TEM01

* beam. The maximum and minimum-peak in-ensities in the R-TEM11

* beam focusing in the range of01 are obtained at �0=1.926 and �0=1.285, respec-ively. Note that the minimum intensity at �0=1.285 cor-esponds to the optical cage that forms a dark space sur-ounded by an intense light field at the focal point.11 Theorresponding FWHM values of the longitudinal spot athe focal point are shown in Fig. 5(b). For �01.285, theWHM of the R-TEM11

* is always smaller than that ofhe R-TEM01

*. Thus, Fig. 5 suggests that both a strongongitudinal component and a sharper focal spot are sat-sfactorily formed at �0=1.926 for an R-TEM11

* modeeam.Figures 6(a)–6(f) show the intensity profiles at the focal

oint of the R-TEM11* beam with �0=4.0, 3.0, 1.926, 1.5,

.285, 1.2, respectively. The NA is assumed to be 1 in thisalculation. These profiles in Figs. 6 are normalized to theeak intensity at the focal point in Fig. 6(c). From Fig. 6 its clearly seen that the intensity distribution in the focallane drastically varies with �0 as well as the intensityeak at the focal point as shown in Fig. 5(a). In the case of0=1.926 in Fig. 6(c), we can see that the center spot forhe total intensity (solid curve) at the focal point is alsoharper. In our calculation, the FWHM of the center spot

ig. 6. (Color online) Calculated intensity profiles near the focaloint �z=0� of R-TEM11

* mode focusing for �0= (a) 4.0, (b) 3.0, (c).926, (d) 1.5, (e) 1.258, (f) 1.2 with a NA=1. The total intensitysolid curves), the longitudinal component (dashed curves), andhe radial component (dotted curves) are drawn in each figure.very profile is normalized to the peak intensity in (c). The hori-ontal axis is in units of wavelength.

or the longitudinal and total intensity in Fig. 6(c) are.399 and 0.428, respectively.As can be seen in Fig. 6, a double-ring-shaped

-TEM11* mode beam has two different focusing proper-

ies due to destructive interference near the focal point.ne is the dark spot formation at the focal point, the so-

alled optical cage property,11 and the other is the sharperocal spot of the longitudinal component being discussedere. The situation can be conveniently selected by choos-

ng proper incident beam width, i.e., the value of �0, inront of an objective. In other words, one can generate theptical cage near the focal point by simply focusing aingle R-TEM11

* beam with appropriate truncation ��01.285�. On the other hand, the sharper spot of the lon-itudinal component with proper peak intensity near theocal point can be obtained when the incident beam widths adjusted so as to generate the longitudinal componentfficiently ��0=1.926�.

. CONCLUSIONhe intensity distributions near the focal point for radi-lly polarized laser beams including higher-order trans-erse modes are calculated based on vector diffractionheory. For higher-order radially polarized beams, thetrong longitudinal component forms a sharper spot athe focal point under a high-NA focusing condition. Inarticular, double-ring-shaped radially polarized modeR-TEM11

*� beams, for which we have already reportedhe optical cage property,11 can effectively reduce the focalpot size because of destructive interference between thenner and the outer rings with a � phase shift. Comparedo R-TEM01

* beam focusing in a limit of NA=1, the fullidth at half-maximum values of the focal spot for an-TEM11

* beam are decreased by 13.6% for the longitudi-al component and 25.8% for the total intensity.Y. Kozawa’s e-mail address is yk-zw

mail.tagen.tohoku.oc.jp.

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