820 OPTICS LETTERS / Vol. 31, No. 6 / March 15, 2006

Focusing property of a double-ring-shaped radiallypolarized beam

Yuichi Kozawa and Shunichi SatoInstitute of Multidisciplinary Research for Advanced Materials, Tohoku University,

Katahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan

Received November 21, 2005; revised December 7, 2005; accepted December 12, 2005; posted December 13, 2005 (Doc. ID 66137)

The intensity distributions of a tightly focused radially polarized beam that has a double-ring-shaped trans-verse mode pattern were calculated based on vector diffraction theory. The distribution of the longitudinalcomponent near the focus varied drastically with the degree of truncation of the incident beam by a pupil.When the ratio of the pupil radius to the beam radius was �1.3, the longitudinal component disappeared atthe focal point, owing to destructive interference. This dark area surrounded by an intense light field was ofthe order of the wavelength, with excellent intensity symmetry. © 2006 Optical Society of America

OCIS codes: 140.3300, 110.2990, 260.5430, 260.3160.

Laser beams with radial polarization are character-ized by a strong longitudinal electric field1 and asmaller spot size2 at the focal point when the beamsare tightly focused. It is well known that these fea-tures are suitable for many applications such as op-tical trapping,3 material processing,4 particleacceleration,5 and high-resolution microscopy.1 Be-cause of the advantages of the radially polarizedbeam mentioned above, various methods for generat-ing the beam have been reported.6–13 A superpositionof two orthogonally polarized Gaussian modes,TEM10 and TEM01, outside or inside a laser cavitywas demonstrated to generate a doughnut-shaped(single-ring) radially polarized beam.7,8 However,only the lowest-order mode (R-TEM01

* mode) can begenerated at present because the generation ofhigher-order modes requires several higher-orderHermite–Gaussian or Laguerre–Gaussian modeswith appropriate polarization.

By contrast, direct generation of a radially polar-ized beam from a laser cavity has the potential ofreadily generating not only an R-TEM01

* mode butalso higher-order modes by use of a polarization-selective optical element or mechanism.10–13 This isso because the polarization-selective element ormechanism is insensitive to intensity distribution. Adouble-ring-shaped beam with radial polarization asa single transverse mode was demonstrated by use ofa polarization-selective grating mirror.12 Because thehigher-order mode can be expressed as a radially po-larized TEMp1

* (p is the radial mode number) modeby analogy to a linearly polarized Laguerre–Gaussian TEMp1

* mode,14 the double-ring-shapedmode demonstrated is consequently expressed as anR-TEM11

* mode.Figure 1 shows the calculated intensity distribu-

tion of an R-TEM11* mode and the instantaneous po-

larization state. The appearance of a node in the ra-dial direction implies a phase difference of � betweenthe inner and the outer rings, as shown in Fig. 1(b).Consequently, a peculiar behavior of the electric ormagnetic field is expected to occur because of the in-terference between the inner and the outer rings

*

when an R-TEM11 beam is tightly focused. Focusing

0146-9592/06/060820-3/$15.00 ©

properties of this beam, however, were not investi-gated.

In this Letter we describe a numerical study, basedon vector diffraction theory, of a property of a double-ring-shaped R-TEM11

* mode beam that is tightly fo-cused by a high-numerical-aperture objective. Thelongitudinal component of the light field disappearedat the focal point, producing a dark area under a spe-cific focusing condition. This dark area, surroundedby an intense light field, which we call an opticalcage, emerged probably as a result of the destructiveinterference between the inner and the outer rings ofthe R-TEM11

* mode beam. The optical cage is of theorder of a wavelength, with excellent intensity sym-metry.

The intensity distributions of radially and azi-muthally polarized Bessel–Gauss beams in the focalregion of an aplanatic lens were calculated by Young-worth and Brown,1 following the vector diffractiontheory of Richards and Wolf.15 We adopted this dif-fraction theory to analyze the behavior of a tightly fo-cused R-TEM11

* beam. The electric field in the focalregion can be written as

E�r,�,z� = Erer + Erez, �1�

where er and ez are the vectors in the radial andbeam propagation directions, respectively. Accordingto Youngworth and Brown, the electric fields of radialcomponent Er and longitudinal component Ez are ex-pressed as1

Fig. 1. (a) Theoretical intensity distribution of radially po-larized TEM11

* modes and (b) instantaneous polarization

state.

2006 Optical Society of America

March 15, 2006 / Vol. 31, No. 6 / OPTICS LETTERS 821

Er�r,�,z� = A�0

�max

cos1/2���sin�2��l0���

�J1�kr sin ��exp�ikz cos ��d�, �2�

Ez�r,�,z� = 2iA�0

�max

cos1/2���sin2���l0���

�J0�kr sin ��exp�ikz cos ��d�, �3�

where �max=sin−1�NA/n�, which is the maximumangle and depends on the numerical aperture (NA) ofthe objective, where n is the refractive index in imagespace, l0��� is the relative amplitude of the electricfield of the incident beam at a pupil, k is the wavenumber in image space, and A is a constant. J0 andJ1 are Bessel functions of the first kind of orders 0and 1, respectively. Note that an azimuthal compo-nent is zero everywhere in image space because ofthe cylindrical symmetry of the radially polarizedbeam. To calculate Eqs. (2) and (3) requires using theform of relative amplitude l0��� that corresponds toan R-TEM11

* mode. In the case of a single doughnut-shaped radially polarized beam, a Bessel–Gaussianbeam obtained as a paraxial solution of Maxwell’svector wave equation was employed.1 However, itwas theoretically16,17 and experimentally12 reportedthat the higher-order radially polarized R-TEMp1

*

beams that have cylindrical intensity symmetry andp+1 rings are represented by use of the Laguerre–Gaussian distribution. Therefore we chose the formof l0��� for the R-TEMp1

* as

l0��� = �02

sin �

sin2 �maxexp�− ��0

sin �

sin �max�2�

�Lp1�2��0

sin �

sin �max�2� . �4�

1

Fig. 2. Calculated intensity profiles at the focal point �z=0� for �0= �a� 2.5, (b) 2.0, (c) 1.5, (d) 1.3, with a NA of 1.2and n=1.33. The total intensity (solid curve), the longitudi-nal component (dashed curve), and the radial component(dotted curve) are represented in each figure. Every inten-sity profile is normalized to the maximum of the total in-tensity in (b). The horizontal axis is in units of wavelength.

Here Lp is the generalized Laguerre polynomial and

Ll1�t�=2− t for the R-TEM11

* mode. �0 is the ratio ofthe pupil radius to the incident beam radius in frontof the focusing lens. Note that �0 should be greaterthan 1 because the outer ring of the R-TEM11

* beamwill be completely truncated by the pupil if �0�1. Inthis study the NA is assumed to be 1.2, with n=1.33.A is determined such that each truncated beam hasthe same optical power.

Figures 2(a)–2(d) show the intensity �E2� profilesat the focal point �z=0� for �0=2.5,2.0,1.5,1.3, re-spectively. Every intensity profile is normalized tothe maximum value of the total intensity (solidcurves) for �0=2.0. It is shown that the longitudinalcomponent (dashed curves) on the beam axis variesdrastically with �0, whereas the radial component(dotted curves) maintains the null on the axis andthe annular intensity distribution. Moreover, bothcomponents almost disappear at the focal point (r=0, z=0) when �0=1.3, as shown in Fig. 2(d). The in-tensity variation for �0, that is, for the incident beamradius, was caused mainly by destructive interfer-ence between the longitudinal components of the in-ner and outer rings of the R-TEM11

* mode beam ow-ing to the intrinsic � phase shift between the tworings. Because when the beam radius is large (small�0) the intensity distribution of the outer ring ap-proaches that of the inner ring near the focal region,the destructive interference between the two ringsbecomes dominant.

Figure 3 shows the intensity distributions of eachcomponent in the z–r plane for �0=1.3. As shown inFig. 3(a), the radial component on the beam axis isnull at any z position. This characteristic is similar tothe focusing property of a tightly focused R-TEM01

*

beam, which is calculated in the same manner. How-ever, it should be noted that the longitudinal compo-nent forms two symmetrical bright spots on the beamaxis sandwiching a dark region in the focal area, asshown in Fig. 3(b). The appearance of this peculiardistribution of the longitudinal component is notshown in the case of R-TEM01

* beam focusing.Figure 4 shows the total intensity distribution ob-

tained by a summation of the distributions of Figs.3(a) and 3(b). We found that the dark space sur-rounded by the intense light field emerges in the fo-cal region. The inset depicts the intensity profilesalong (a) the beam propagation direction, (b) the ra-

Fig. 3. Calculated intensity distributions of (a) the radialcomponent and (b) the longitudinal component in the focalarea for �0=1.3 with a NA of 1.2 and n=1.33. Both axes are

in units of wavelength.

822 OPTICS LETTERS / Vol. 31, No. 6 / March 15, 2006

dial direction, and (c) the direction of minimal inten-sity. The distances from the focal point (zero point) tothe half-point of each maximum intensity along the(a) and (b) directions are approximately 0.47 and0.34 �, respectively. Note that the curves along direc-tions (a) and (b) have an excellent similarity of nearlyup to position 0.5 �. Even for the curve along (c), theintensity variation is quite similar. Accordingly, thesize of the dark region is of the order of a wavelength,and the intensity symmetry is quite good in all direc-tions. Moreover, it should be noted that this distribu-tion has cylindrical symmetry owing to the radial po-larization, in contrast to an asymmetric distributionby linearly polarized beams.18 A similar dark area ap-peared in the focal region.19–21 The optical bottlebeam reported by Arlt and Padgett19 arose fromdestructive interference between two linearly polar-ized Laguerre–Gaussian modes, e.g., modes LG00 andLG20 with different Gouy phases. In Fig. 4 the opticalbarrier even in the direction of minimal intensity[curve (c)] is approximately one half of the maximumintensity in the beam propagation direction. The low-est optical barrier was one quarter of the maximumintensity in the case of a bottle beam.19 It should beemphasized that focusing the R-TEM11

* mode beamproduces a higher degree of intensity symmetry,yielding a nearly spherical optical barrier in additionto the cylindrical symmetry in the polarization. Inthis sense, we call this optical barrier achieved by thefocusing of the R-TEM11

* mode beam an optical cage.An optically dark spot has obvious applications for

trapping particles with refractive indices lower thanthat of the surrounding medium and the single-beamblue-detuned trapping of atoms in a low-intensityregion.19 In the optical cage proposed in this study, aparticle in the center of the cage will receive a force

Fig. 4. Total intensity distribution in the focal area ob-tained by summation of the results of Fig. 3. Inset, inten-sity profiles in (a) the beam propagation direction, (b) theradial direction, and (c) the direction of minimal intensity.Both axes are in units of wavelength.

with almost spherical symmetry. We expect that thepolarization symmetry as well as the intensity sym-metry will provide isotropic features for the trappingapplications. Note that one can generate the opticalcage by simply focusing a single R-TEM11

* beam withappropriate truncation without any complicated in-terference instruments.

In conclusion, the intensity distributions of atightly focused, radially polarized beam �R-TEM11

*�with a double-ring-shaped pattern were calculatedbased on the Richards–Wolf vector diffraction theory.When parameter �0, the ratio of the pupil radius tothe beam radius, was changed, the longitudinal com-ponent at the focal point varied drastically. A smalldark area of the order of the wavelength surroundedby an intense light field appeared for �0=1.3, withn=1.33 and a NA of 1.2 owing to destructive interfer-ence between the outer and the inner rings with a �phase shift. In this case, excellent intensity symme-try was predicted, yielding an almost spherical opti-cal barrier, the so-called optical cage.

Y. Kozawa’s e-mail address is yk-zw@mail.tagen.tohoku.ac.jp.References

1. K. S. Youngworth and T. G. Brown, Opt. Express 7, 77(2000).

2. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G.Leuchs, Opt. Commun. 179, 1 (2000).

3. Q. Zhan, Opt. Express 12, 3377 (2004).4. V. G. Niziev and A. V. Nesterov, J. Phys. D 32, 1455

(1999).5. S. C. Tidwell, G. H. Kim, and W. D. Kimura, Appl. Opt.

32, 5222 (1993).6. G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda,

and K. Ohbayashi, Appl. Phys. Lett. 84, 3855 (2004).7. L. Novotny, M. Beversluis, K. Youngworth, and T.

Brown, Phys. Rev. Lett. 86, 5251 (2001).8. S. C. Tidwell, D. H. Ford, and W. D. Kimura, Appl. Opt.

29, 2234 (1990).9. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z.

Bomzon, and E. Hasman, Appl. Phys. Lett. 77, 3322(2000).

10. D. Pohl, Appl. Phys. Lett. 20, 266 (1972).11. Y. Mushiake, K. Matsumura, and N. Nakajima, Proc.

IEEE 60, 1107 (1972).12. T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux,

M. A. Ahmed, and T. Graf, Appl. Phys. B 80, 707(2005).

13. Y. Kozawa and S. Sato, Opt. Lett. 30, 3063 (2005).14. A. V. Nesterov and V. G. Niziev, Appl. Phys. B 33, 1817

(1999).15. B. Richards and E. Wolf, Proc. R. Soc. London Ser. A

253, 358 (1959).16. A. A. Tovar and G. H. Clark, J. Opt. Soc. Am. A 14,

3333 (1997).17. A. A. Tovar, J. Opt. Soc. Am. A 15, 2705 (1998).18. A. Boivin and E. Wolf, Phys. Rev. 138, 1561 (1965).19. J. Arlt and M. J. Padgett, Opt. Lett. 25, 191 (2000).20. T. Freegarde and K. Dholakia, Phys. Rev. A 66, 013413

(2002).21. Y. Zhao, Q. Zhan, Y. Zhang, and Y. Li, Opt. Lett. 30,

848 (2005).