Diffractive properties of obstructed vector Laguerre–Gaussian beam under tight focusing condition

$Page 1: Diffractive properties of obstructed vector Laguerre–Gaussian beam under tight focusing condition$
Diffractive properties of obstructed vectorLaguerre–Gaussian beam under

tight focusing condition

Sunil Vyas,* Masato Niwa, Yuichi Kozawa, and Shunichi Sato

Institute of Multidisciplinary Research for Advanced Materials, Tohoku University,Katahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan*Corresponding author: [email protected]

Received February 14, 2011; revised April 20, 2011; accepted May 6, 2011;posted May 11, 2011 (Doc. ID 142674); published June 10, 2011

Diffractive and focusing properties of vector Laguerre–Gaussian beams with obstacle are investigated under tightfocusing conditions. Using vector diffraction theory, intensity and polarization distributions near the focus atdifferent orthogonal planes are calculated and analyzed for vector Laguerre–Gaussian beams. It is observed thatthe beam is able to compensate the distortion produced by obstacles when the size of the obstacle is small. Thestructural changes in the polarization distribution are not the same in different orthogonal planes. The polarizationcharacteristics of the beam show a significant change when the size of the obstacle is large. A comparative study ofthe focusing and diffractive properties of vector Laguerre–Gaussian and vector Bessel–Gaussian beams has alsobeen performed. © 2011 Optical Society of America

OCIS codes: 050.1940, 260.5430, 260.1960.

1. INTRODUCTIONIn recent years, advances in nanotechnology and nanofabrica-tion have driven the need for a deeper understanding of thethree-dimensional structure of the beams in the focal region.Richards and Wolf [1] have provided vector diffraction theoryfor analyzing the focal structure of beams in high-numerical-aperture focusing conditions. Besides the intensity and phasedistributions, polarization of the incident beam plays a majorrole in formation of the shape and size of the focal spot. Vectorbeams are known to have a unique polarization distributionand corresponding focusing characteristics. Vector Laguerre–Gaussian beams are the paraxial solutions of the vectorHelmholtz equation and have cylindrical symmetries of ampli-tude as well as polarization [2]. The lowest-order radially po-larized Laguerre–Gaussian beams have a doughnut-shapedintensity distribution and an intensity null on the beam axisin the paraxial limit due to the polarization singularity. Variousmethods of generation of vector Laguerre–Gaussian beamshave been demonstrated [3–7]. A single-transverse higher-order vector Laguerre–Gaussian beam with multiple ringshas been generated in a Nd:YAG laser cavity by using a reflec-tivity-modified and polarization-selective photonic crystalmirror [8]. Very recently, vector Laguerre–Gaussian beamswere generated in a Nd:YAG laser by using a spot defectedmirror in the laser cavity [9]. The focusing properties of vectorbeams have been studied by various authors [10–14]. Tightlyfocused vector beams have many applications, such as confo-cal microscopy, optical tweezers, and optical data storage. Inall these applications, the spatial distribution of polarizationplays a crucial role.

The investigations of the nondiffracting properties ofoptical beams are of current interest. Nondiffracting beamscan propagate over a long distance without changing their in-tensity profile. These beams also have a so-called self-healing

effect, which shows the preservation of transverse intensityprofile during propagation even if some part of the beam isblocked. The best known example of a nondiffracting beamis a Bessel–Gaussian beam. Bessel–Gaussian beams havebeen used in many applications, for example in biological cellmanipulations [15], optical binding of microparticles [16], op-tical conveyor belt [17], optical levitation [18], nanopatterning[19], microscopy [20,21], and microlithography [22].

Apart from the Bessel beams, self-healing property is alsoobserved in other beams. Brokey et al. demonstrated boththeoretically and experimentally the resilience of Airy beamsagainst perturbation and explained the self-healing propertyduring propagation of the beam [23]. Bouchal showed theself-reconstruction ability of nondiffracting vortex beamsand verified that the self-healing effect appears even if thebeam is disturbed by an obstacle with a helical phase shiftwhen the interaction is accompanied by the exchange ofthe orbital angular momentum [24]. Self-reconstruction of asingular beam is experimentally demonstrated when the cen-tral part of the incident beam is cut off [25]. Using simulationand experiments, the self-reconstruction property of the frac-tional Bessel beam has also been described. It is shown thatthe length needed for self-reconstruction depends on thelocation, size of the obstacles, and the beam parameters [26].

Vector Laguerre–Gaussian and vector Bessel–Gaussianbeams have many similar features. For instance, both beamsare the solutions of vector wave equation in the cylindricalcoordinates, and posses a polarization singularity on the beamaxis and a multiring structure with a phase shift of π betweenadjacent rings. In many applications of the cylindrical vectorbeams, for example, microscopy, particle trapping, andimaging, where high-numerical-aperture focusing is used,exact knowledge of the diffraction and polarization propertiesof the beam is important. Diffractive and focusing properties

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of vector Laguerre–Gaussian beams in the presence of anobstacle under tight focusing condition have not been studiedso far. Motivated by these facts, we have done the numericalstudies on diffractive properties of a vector Laguerre–Gaussian beam with an obstacle under tight focusing condi-tions and performed a comparative study with diffractiveand focusing properties of a vector Bessel–Gaussian beam.

The remaining part of the paper is organized as follows. InSection 2, we present a brief mathematical description of avector Laguerre–Gaussian beam. Diffractive properties of avector Laguerre–Gaussian beam in the presence of an obsta-cle are presented in Section 3. A comparative study of focus-ing and diffractive properties of vector Laguerre–Gaussianand vector Bessel–Gaussian beams is presented in Section 4.Discussion and conclusions are presented in Sections 5 and 6,respectively.

2. THEORYWe have considered the coordinate geometry shown in Fig. 1.The general expression of complex amplitude of a vectorLaguerre–Gaussian beam was derived by Tovar [2]. Solutionof the vector wave equation in the cylindrical coordinatessystem is given by

~EðvÞ ¼ ~Uðr;ϕ; zÞ expð−iωtÞ ð1Þ

~Uðr;ϕ; zÞ ¼ uradialir þ uazimuthal iϕ; ð2Þ

where ~Uðr;ϕ; zÞ is the field amplitude, expð−iωtÞ is theharmonic time dependence, uradial and uazimutal are the radialand azimuthal components, respectively, and ir and iϕ are theunit vectors along the radial and azimuthal directions,respectively. A simple expression for a vector Laguerre–Gaussian beam can be written as

~Uðr;ϕ; zÞ ¼ E00

ω0

ωðzÞ� ffiffiffi

2p

rωðzÞ

�n�1

× Ln�1p

�2r2

ω2ðzÞ�exp

�−

r2

ω2ðzÞ�

× exp�

ikr2

2RðzÞ þ ikz − i½2pþ ðn� 1Þ þ 1�ηðzÞ�

×

� cosðnϕÞiϕ∓ sinðnϕÞir� sinðnϕÞiϕ þ cosðnϕÞir

�; ð3Þ

where r is the radius, ϕ is the azimuthal angle, z is the distancefrom the beam waist, ω0 is the minimum beam radius at z ¼ 0,RðzÞ ¼ ðz2 þ z20Þ=z is the radius of curvature of the wavefront,z0 ¼ kω2

0=2, k is the wave number, ð2pþ ðn� 1Þ − 1ÞηðzÞ is theGouy phase shift with ηðzÞ ¼ arctanðz=z0Þ, and ωðzÞ ¼ω0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðz=z0Þ2

p. Ln�1

p is the Laguerre polynomial with radialindex p and azimuthal index n� 1. We define an azimuthalindex m ¼ n� 1. The lower term in Eq. (3) with m ¼ 1 andn ¼ 0 with the upper sign representing pure radially polarizedbeam.

We followed the vector diffraction theory of Richards andWolf [1] for calculation of intensity and polarization distribu-tion at the focal plane for high-numerical-aperture focusing.Electric field components of a tightly focused vectorLaguerre–Gaussian beam can be written as

Ur;radialðr0;ϕ0; z0Þ

¼ −iAπ

Z α

0

Z2π

0

ffiffiffiffiffiffiffiffiffiffiffiffiffifficosðθÞ

psinðθÞ cosðθÞ cosðϕ − ϕ0Þ

× uradial expfik½z0 cosðθÞ þ r0 sinðθÞ cosðϕ − ϕ0Þ�gdϕdθ ð4Þ

Uϕ;radialðr0;ϕ0; z0Þ

¼ −iAπ

Z α

0

Z2π

0


psinðθÞ cosðθÞ sinðϕ − ϕ0Þ

× uradial expfik½z0 cosðθÞ þ r0 sinðθÞ cosðϕ − ϕ0Þ�gdϕdθ ð5Þ

Uz;radialðr0;ϕ0; z0Þ

¼ iAπ

Z α

0

Z2π

0


psin2ðθÞ × uradial expfik½z0 cosðθÞ

þ r0 sinðθÞ cosðϕ − ϕ0Þ�gdϕdθ ð6Þ

Ur;azimuthalðr0;ϕ0; z0Þ ¼iAπ

Z α

0

Z2π

0


psinðθÞ sinðϕ − ϕ0Þ

× uazimuthal expfik½z0 cosðθÞþ r0 sinðθÞ cosðϕ − ϕ0Þ�gdϕdθ ð7Þ

Uϕ;azimuthalðr0;ϕ0;z0Þ ¼−iAπ

Z α

0

Z2π

0


psinðθÞcosðϕ−ϕ0Þ

×uazimuthal expfik½z0 cosðθÞþ r0 sinðθÞcosðϕ−ϕ0Þ�gdϕdθ ð8Þ

Uz;azimuthalðr0;ϕ0; z0Þ ¼ 0; ð9Þ

where A is a constant, α is the semi-aperture angle on the im-age side and related to the numerical aperture NA of the lensas α ¼ sin−1ðNAÞ, θ is the polar angle, and ϕ is the azimuthalangle on the transverse plane perpendicular to the z axis,and ðr0;ϕ0; z0Þ are the cylindrical coordinates in the focalregion. For all the numerical calculations, λ ¼ 632:8 nm, therefractive index n ¼ 1, the numerical aperture of an objectiveNA ¼ 0:95, ω0 ¼ 3mm, focal length f ¼ 3mm, and z ¼ 0.

3. DIFFRACTIVE PROPERTIES OF VECTORLAGUERRE–GAUSSIAN BEAMFirst we have considered the diffraction of a tightly focusedradially polarized Laguerre–Gaussian beam. The most inter-esting feature of a radially polarized Laguerre–Gaussian beamis the formation of a sharper, stronger, and nonpropagatingFig. 1. (Color online) Schematic representation of optical geometry.

1388 J. Opt. Soc. Am. A / Vol. 28, No. 7 / July 2011 Vyas et al.

longitudinal electric field component around the focal point.We have done numerical calculations for the total intensityand polarization distribution for the single-ring (m ¼ 1, p ¼ 0)radially polarized Laguerre–Gaussian beam obstructed by asector-shaped obstacle. It is assumed that the sector-shapedobstacle is placed at the pupil plane and it partially hinders theinput beam. The size of the obstacle is represented by acenter angle of sector g. Following the vector diffractiontheory, we have used Eqs. (4)–(9) for our calculations.Figures 2(a)–2(c) show normalized intensity and transversepolarization components at the pupil plane (x–y plane). Forthe single-ring Laguerre–Gaussian beam, β0 is chosen suchthat 90% of the incident beam passes through an objectivelens. The total intensity distribution at the focal plane is thesum of the longitudinal and transverse components.

Results of total intensity distribution and polarization com-ponents at the focal plane are shown in Figs. 2(d)–2(g). Thetop row of Fig. 2 depicts the field distribution for an unob-structed beam. In this case, the value of g is equal to zero.The lower rows of Fig. 2 show the results for the total intensityand polarization components at the focal plane for differentsizes of obstacles. From the first two rows, it can be seen thatthe intensity and polarization distributions at the focal planeare similar in their structure for obstructed and unobstructedcases, except that there is slight elongation in the total inten-sity distribution and the radial component of the electric field.In other words, the missing field component due to the obsta-cle is recovered at the focal plane and there is a small healingeffect in the beam. By increasing the size of the obstacle,however, the distortion in the intensity and polarization dis-tributions increases. When half of the beam is blocked thenthe structure of the field significantly changes from a circularshape to a rectangular shape. In addition, there is elongationin the total intensity and the polarization distribution in the

direction perpendicular to the direction of the obstacle. Theradial component of the field could not maintain its dark cen-ter for the larger size obstacle. With increasing the size of theobstacle, a weak azimuthal component appears and graduallyincreases. Appearance of the azimuthal component at thefocal plane indicates that the beam is no longer purely radiallypolarized and the polarization property of the beam is altered.It means that the beam could not sustain its initial polarizationcharacteristics when the distortion is very large. From theseresults, we can state that a vector Laguerre–Gaussian beamhas a self-healing effect only for small obstruction.

In order to probe further the field structure at the focalplane, we have performed numerical calculations for the in-tensity and polarization distributions in the x–z and y–zplanes. Figures 3 and 4 show the calculated results in thex–z plane and y–z plane, respectively. Diffraction by the ob-stacle has different effects on the intensity and polarizationcomponents in the x–z and y–z planes, as seen in Figs. 3and 4. In the x–z plane, the radial and longitudinal compo-nents maintain their shape and they are not affected muchby the variation of the size of the obstacle. For the larger ob-stacle, a weak azimuthal component emerges at the focalplane, as shown in Fig. 3(f). Total intensity and the polariza-tion components seems to be less distorted when the beam ishalf blocked as compared to that for g ¼ 0:6π. In the y–zplane, as shown in Figs. 4(a)–4(g), the total intensity andthe polarization components could not maintain their shapewhen the obstacle size is large. The radial component forthe half blocked beam changes its shape from a two-lobepattern to a single-lobe pattern and total intensity distributionalso gets affected. From the bottom two rows of Fig. 4, weobserve a tilted and elongated pattern of the total intensityand longitudinal component for the larger obstacles. Avector Laguerre–Gaussian beam has cylindrical symmetry of

Fig. 2. (Color online) Calculated intensity distribution and polarization components for radially polarized Laguerre–Gaussian beam for m ¼ 1,p ¼ 0, β0 ¼ 1:39 in x–y plane. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Fielddistribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. Thedimensions at the pupil plane and at the focal plane are 5:7 × 5:7mm2 and 1 × 1 μm2, respectively.


phase, intensity, and polarization. A higher-order vectorLaguerre–Gaussian beam has polarization distribution similarto petals of flowers. To make the analysis more general wehave calculated intensity and polarization distribution for ahigher-order mode (m ¼ 2with n ¼ 3) obstructed by differentsizes of obstacles. Results for total intensity and polarizationcomponents are shown in Figs. 5–7 for different orthogonalplanes (x–y, x–z, and y–z planes, respectively). A higher-order

mode contains both the azimuthal and radial polarizationcomponents, with each intensity lobe polarized in the direc-tion opposite to those on either side. It can be seen that thereis no significant difference in the diffractive property of avector Laguerre–Gaussian beam for lower (m ¼ 1) and higher(m ¼ 2) order modes. In both cases, the beam is able torecover the missing field component only for the smaller sizeobstacle and a certain level of healing effect can be observed.

Fig. 3. (Color online) Calculated intensity distribution and polarization components for radially polarized Laguerre–Gaussian beam for m ¼ 1,p ¼ 0, β0 ¼ 1:39 in x–z plane. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Fielddistribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. Thedimensions at the pupil plane and at the focal plane are 5:7 × 5:7mm2 and 1 × 1 μm2, respectively.

Fig. 4. (Color online) Calculated intensity distribution and polarization components for radially polarized Laguerre–Gaussian beam for m ¼ 1,p ¼ 0, β0 ¼ 1:39 in y–z plane. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Fielddistribution at the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. Thedimensions at the pupil plane and at the focal plane are 5:7 × 5:7mm2 and 1 × 1 μm2, respectively.


4. DIFFRACTIVE PROPERTIES OFRADIALLY POLARIZED BESSEL–GAUSSIANBEAMTo study the diffractive property of a vector Laguerre–Gaussian beam in detail, we have done a comparative studyof the diffractive property of a multiring radially polarizedLaguerre–Gaussian and a radially polarized Bessel–Gaussianbeam. As mentioned previously, both the beams have similar

distributions of intensity and polarization and belong to theclass of cylindrical vector beams. One of the important attri-butes of a vector Bessel–Gaussian beam is the nondiffractingnature, which makes this beam robust against distortions.Comparison of the diffractive property of a vector Laguerre–Gaussian with a vector Bessel–Gaussian beam, which has aninherent property of self-healing, is more appropriate.

For a multiring radially polarized Laguerre–Gaussian beam,we took m ¼ 1, p ¼ 3. Figures 8(a)–8(c) and Figs. 8(d)–8(g)

Fig. 5. (Color online) Calculated intensity distribution and polarization components for higher-order mode m ¼ 2, p ¼ 0, β0 ¼ 1:59 in x–y plane.Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane(d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupil planeand at the focal plane are 5:7 × 5:7mm2 and 1 × 1 μm2, respectively.

Fig. 6. (Color online) Calculated intensity distribution and polarization components for higher-order mode m ¼ 2, p ¼ 0, β0 ¼ 1:59 in x–z plane.Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane(d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupil planeand at the focal plane are 5:7 × 5:7mm2 and 1 × 1 μm2, respectively.


show the calculated intensity and polarization profile at thepupil plane and at the focal plane, respectively, in the x–yplane. In spite of the expectation that the multiring structureof the beam would help to overcome the distortions as is pre-sumed from a Bessel–Gaussian beam, there is no such effectfor a vector Laguerre–Gaussian beam. As shown in Fig. 6, thedistortion remains the same, especially for larger obstacles.

Here we will follow our earlier work done on the self-healing properties of a vector Bessel–Gaussian beam [27].

Figure 9 shows the total intensity distribution and polarizationcomponents at the focal plane for a radially polarized Bessel–Gaussian beam when sector-shaped obstacles with differentsizes are placed at the pupil plane. It is important to note thatdespite the change in the size of the obstacle, the intensityand polarization distributions are able to retain their shape.It is surprising that the intensity distribution as well as thepolarization distribution is well recovered at the focal planeeven when half of the beam is obstructed at the pupil plane.

Fig. 7. (Color online) Calculated intensity distribution and polarization components for higher-order mode m ¼ 2, p ¼ 0, β0 ¼ 1:59 in y–z plane.Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focalplane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions at the pupilplane and at the focal plane are 5:7 × 5:7mm2 and 1 × 1 μm2, respectively.

Fig. 8. (Color online) Calculated intensity distribution and polarization components for multiring radially polarized Laguerre–Gaussian beam form ¼ 1, p ¼ 3, β0 ¼ 5. Field distribution at the pupil plane (a) total intensity, (b) radial component, and (c) azimuthal component. Field distributionat the focal plane (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component. The dimensions atthe pupil plane and at the focal plane are 5:7 × 5:7mm2 and 6 × 6 μm2, respectively.


The nondiffracting property of the beam plays an importantrole in the self-healing effect of the beam, which can be under-stood by comparing the intensity and polarization distributionshown in Figs. 8 and 9 for vector Laguerre–Gaussian andBessel–Gaussian beams, respectively. Comparison of the fielddistribution at the focal plane for these two beams clearly de-monstrates that self-healing occurs for the vector Laguerre–Gaussian beam but the degree is small as compared to theBessel–Gaussian beam. Deviation of the intensity and polar-ization distributions for larger obstacles shows that the heal-ing effect is not a prominent feature in the case of the vectorLaguerre–Gaussian beam. By contrast, the axial symmetryof the polarization and intensity distributions at the focalplane is maintained for the vector Bessel–Gaussian beamdue to the inherent property of the nondiffraction and self-reconstruction. Even though both vector Laguerre–Gaussianand Bessel–Gaussian beams have the cylindrical symmetriesof the intensity and polarization, the Laguerre–Gaussian beamloses the symmetries when obstructed by a large obstacle,whereas the Bessel–Gaussian beam maintains them.

5. DISCUSSIONTo understand the influence of the size of obstacles on thepolarization properties of a Laguerre–Gaussian beam, wehave done systematic numerical study of the focal field distri-bution of the single-ring and multiring beams with varioussizes of obstacles placed in the pupil plane. One can see fromthe calculated results that the focal field properties of aLaguerre–Gaussian beam are degraded as the size of the ob-stacle is increased and the cylindrical symmetric polarizationdistribution of the beam is lost. A significant difference in thediffraction property of vector Laguerre–Gaussian beam andvector Bessel–Gaussian beams appears when the beam is halfblocked. Because of the typical property of self-reconstruc-tion of a vector Bessel–Gaussian beam intensity and polariza-

tion components are recovered, but for the vector Laguerre–Gaussian beam, the intensity and polarization distributionsare affected drastically.

We can qualitatively understand the phenomena of self-healing by method of angular spectrum of the plane wave.The angular spectrum of the focal field distribution is com-posed of the plane waves. These planes waves are superposedat the focal plane and maintain their phase and polarizationproperties. Thus, the intensity and the polarization compo-nents can significantly recover their shape, even if someof the wave components are missing at the pupil plane.From the viewpoint of the angular spectrum, the differencebetween vector Bessel–Gaussian and vector Laguerre–Gaussian beams arises from the fact that the Bessel–Gaussianbeam is composed of a large number of Gaussian beams,whose axis is lying on the surface of the cone [28], resultingin the property of nondiffraction, whereas this property isabsent in the case of the Laguerre–Gaussian beam. In mostof the previous studies, the self-healing property of the beamis demonstrated by observing the intensity distribution at dif-ferent planes perpendicular in the direction of propagation[23–26]. With the present understanding of the subject thereare no theoretical treatments that can explain the self-healingbehavior of vector beams in the focal region. Numerical cal-culation is a useful and effective tool that enables us to drawsome useful conclusion concerning the focal field character-istics of vector beams. The fundamental property that is mak-ing a difference in the diffractive behavior of these beams isnot clear, and further investigations are required.

Vector Laguerre–Gaussian and vector Bessel–Gaussianbeams have cylindrical symmetry of intensity, phase, andpolarization distribution. Insertion of obstacles in the pupilplane changes the symmetric properties of the beam at thefocal plane. The vector Laguerre–Gaussian beam loses itscylindrical symmetry of intensity and polarization distribution

Fig. 9. (Color online) Calculated intensity distribution and polarization components for radially polarized Bessel–Gaussian beam with β ¼0:0005k and m ¼ 0 for different sizes of obstacles. Field distribution at the pupil plane (a) total intensity distribution, (b) radial component,and (c) azimuthal component. Field distribution at the focal plane (d) total intensity, (e) radial component, (f) azimuthal component, and (g)longitudinal component. The dimensions at the pupil plane and at the focal plane are 5:7 × 5:7mm2 and 3 × 3 μm2, respectively.


for lower- as well as for higher-order modes. Even for the pureradially polarized beam due to this symmetry breaking, a weakbut not zero azimuthal component is obtained at the focalplane, which should be otherwise zero. The degree of asym-metry increases with an increase in the size of the obstacle.The effects of the obstacle on transverse and longitudinalcomponents in x–y, x–z, and y–z planes are different. Butfor all the cases an axial symmetry with respect to the opticalaxis is recognized. The field redistributes itself at the focalplane by breaking its cylindrical symmetry, which also resultsin the azimuthal symmetry breakings. Breaking of inherentcylindrical symmetry of vector beams due to the presenceof the obstacle in the pupil plane and their correlation hasnot been studied so far.

The diffractive properties of vector Laguerre–Gaussianbeams and in particular the self-healing of the polarizationcomponents is prospective for many fields, where a focusedvector beam is used, for example, laser scanning microscopy,optical trapping, and optical data storage. In analogy with theBessel beams [29] and Airy beams [30], the property of self-healing of vector Laguerre–Gaussian beams may be used forthe multiple-plane particle manipulation, where the opticalfield distorted by the trapped particles in the one plane is ableto reconstruct its shape in the follow-up plane.

6. CONCLUSIONSIn summary, we investigated the diffractive properties of avector Laguerre–Gaussian beam in tight focusing conditions.It is observed that the vector Laguerre–Gaussian beam canheal the intensity distribution and the polarization compo-nents when the obstacle size is small. Comparative studyof vector Laguerre–Gaussian beams with vector Bessel–Gaussian beams was performed. It is found that the self-healing behavior is the salient feature of the Bessel–Gaussianbeam, and it is manifested in recovery of the intensity as wellas the polarization distribution at the focal plane. We believesuch studies have important implications in the understandingof the complex nature of the polarization of cylindrical vectorbeams in the focal region.

ACKNOWLEDGMENTSThis work was supported in part by the Japan Science andTechnology Corporation (JST), Core Research for EvolutionalScience and Technology (CREST).

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Diffractive properties of obstructed vector Laguerre–Gaussian beam under tight focusing condition