Self-healing of tightly focused scalar and vectorBessel–Gauss beams at the focal plane
Sunil Vyas,* 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 December 22, 2010; revised February 25, 2011; accepted February 26, 2011;posted March 2, 2011 (Doc. ID 140120); published April 21, 2011
1. INTRODUCTIONRecently, increasing attention has been paid to nondiffractingbeams because they offer many potential applications. A de-scription of the nondiffracting property of optical beams is im-portant from the point of view of comprehension of diffractionphenomena and the nature of the electromagnetic field. Dur-nin [1] introduced Bessel beams, which are quite attractivebecause of their fascinating properties of nondiffraction aswell as self-reconstruction. Ideal Bessel beams are not physi-cally realizable, as they carry infinite power through any crosssection normal to their propagation direction. The intensityprofile of a zero-order Bessel beam has a high-intensity centralcore surrounded by a series of concentric rings, whereas ahigher-order Bessel beam has a dark central core due to phasesingularity. Various applications have been reported for theBessel beams. For example, it has been used in optical micro-scopy [2], interboard optical data distribution [3], and opticaltrapping [4]. Vector Bessel beams are the solutions of the vec-tor Helmholtz wave equation and the superposition of the vec-tor component of the angular spectrum [5]. A scalar Besselbeam is the specific case of more general vector Bessel beam.To overcome the difficulties of the physical realization of Bes-sel beams, Gori and Guattari [6] introduced the Bessel–Gauss(BG) beam, which has finite energy. BG beams are character-ized by a Bessel function with a Gaussian envelope. Thesebeams can be experimentally realized and have the abilityto propagate without significant divergence.
Many studies have been performed on the focusing proper-ties of nondiffracting beams using scalar diffraction theory.Using Huygens–Fresnel diffraction, Lu et al.calculated thethree-dimensional intensity distribution of focused BG beams[7]. They concluded that, by suitable choice of the systemparameters, the intensity distribution of linearly polarizedBG beam can form a spot or annulus. Bagini et al. [8] de-scribed a superposition model for the BG beam, modifiedBG beam, and generalized BG beam. They analyzed the effect
of the lens on the propagation of the generalized Bessel–Gaussian beam. It was observed that, by considering the effectof the lens on the field, these different sets of BG beams cantransform into one another [9]. In the case of the focusing of alight beam by a high numerical aperture (NA) objective, scalardiffraction theory is not adequate. Richards and Wolf [10] pro-vided a basic formulation for analyzing the focal field of tightlyfocused polarized beams.
In recent years, interest in vector beams has been growingbecause of their unique features of cylindrically symmetricpolarization distribution. The focusing properties of vectorbeams have been studied by various authors [11–17]. Tightlyfocused vector beams find many applications, such as confo-cal microscopy, optical tweezers, and optical data storage. Inall these applications, the spatial distribution of polarization ofthe incident field plays a crucial role. Vector BG beams are thesolution of vector wave equations [18]. Various methods forthe generation of cylindrical vector beams based on activeand passive methods have been reported [19,20]. Ito et al.[21] experimentally demonstrated the generation of higher-order modes of vector BG beams by using a spot defect mirrorin an Nd:YAG laser cavity. They pointed out that there is apoint in common between the scalar and vector beams withrespect to the requirement of cylindrical symmetry for thelaser oscillation. Scalar and vector beams are known to havea singularity on the beam axis of phase and polarization, re-spectively, which results in the formation of intensity null onthe beam axis. Self-healing properties of the scalar beamshave been studied by many authors [22–27]. In all these stu-dies, the self-healing property of the beam is demonstrated byobserving the intensity distribution at different planes perpen-dicular to the propagation direction. However, to the best ofour knowledge, the self-healing property of vector BG beamsin the tight focusing condition have not been studied so far.Particularly, the healing of polarization at the focal region isexpected to be a new aspect of vector beams in addition tothat of intensity distribution observed for scalar beams. Such
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studies are useful for better understanding of scalar and vec-tor beams at the focal region. In this paper, we investigate theself-healing property at the focal plane for both scalar and vec-tor BG beams in a tight focusing condition. It is shown that notonly intensity profile but also the polarization components ofan input beam, which are affected partially by an obstacleplaced at the pupil plane, are substantially recovered at thefocal plane. The recovery was observed even though halfof the beam was obstructed.
The remaining parts of the paper are organized as follows.In Section 2, a theoretical background of the high NA focusingof scalar and vector BG beams is presented. Numerical studiesof the focusing property of vector BG beams is presented inSection 3. Section 4 deals with the self-healing property in thefocal planes of vector BG beams. In Section 5, the effect of thesize of the aperture on the self-healing property of vector BGbeams is described. Section 6 deals with the self-healing prop-erty in the focal plane of a linearly polarized BG beam for com-parison with that of the vector beam. Sections 7 and 8 presentthe discussion and conclusion, respectively.
2. THEORYA BG beam may be considered a superposition of Gaussianbeams whose wave vectors lie on the surface of a cone. It ischaracterized by two real parameters β and ω0 whose valuesdetermine the propagation feature of the beam [4]. β is thelength component, orthogonal to the direction of propagation,of anywave vector belonging to oneof thewaves producing thebeam [6]. For a particular value of β and Gaussian beam widthω0, both the Gaussian beam and the diffraction-free Besselbeam can be obtained. The value of the β controls the widthof the Bessel component of the BG beam. The choice of inde-pendent parameters β and ω0 can produce a wide range ofbehaviors in the BG beams’ transverse intensity profiles [28].
A. Scalar BG BeamsA scalar BG beam was first demonstrated as a solution of theparaxial wave equation [6], which belongs to the class of non-diffracting beams and can be experimentally realized easily.Here, we consider a linearly polarized BG beam. General ex-pression of a BG beam of order n with linear polarization inthe x direction is given by [9]
~EðsÞ ¼ Uðr;ϕ; zÞ expð−iωtÞix; ð1Þ
Uðr;ϕ; zÞ ¼ E0ω0
ωðzÞ in exp½ikz − iψðzÞ�
× exp
�−r2
�1
ω2ðzÞ −ik
2RðzÞ��
QðzÞ
× JnðuÞ expðinϕÞ; ð2Þ
where E0 is a constant and ix is the unit vector along the xdirection, ω0 is the beam radius at z ¼ 0, u ¼ βr=ð1þ iz=z0Þ, RðzÞ ¼ ðz2 þ z20Þ=z is the radius of curvature ofthe wavefront, k is the wavenumber, ψðzÞ ¼ arctanðz=z0Þ,ωðzÞ is the width of the Gaussian beam defined as ωðzÞ ¼ω0
0=2, and QðzÞ ¼ expf−iβ2z=½2kð1þ iz=z0Þ�g with a constant β. The zero-order Besselbeam has an on-axis-maximum, whereas the higher-orderBessel beams have a intensity null at the center due to phasesingularity.
B. Vector BG BeamsThe general expression of the vector BG beam was derived byHall [29]. In the cylindrical coordinates system, a solution of avector wave equation is given by
~EðvÞ ¼ ~Uðr;ϕ; zÞ expð−iωtÞ; ð3Þ
~Uðr;ϕ; zÞ ¼ uradialir þ uazimutaliϕ; ð4Þ
where ~Uðr;ϕ; zÞ is the field amplitude, uradial and uazimutal arethe radial and azimuthal components, respectively, and ir andiϕ are the unit vectors along the radial and azimuthal direc-tions, respectively.
A simple expression for the vector BG beam can be writtenas
~Uðr;ϕ; zÞ ¼ E00
ω0
ωðzÞ exp½ikz − iψðzÞ�
× exp
�−r2
�1
ω2ðzÞ −ik
2RðzÞ��
QðzÞTðr;ϕ; zÞ; ð5Þ
where E00 is a constant. Tðr;ϕ; zÞ has two families, TE field
solution Te and TM field solution Tm [21], expressed by
Teðr;ϕ; zÞ ¼ ½Jm−1ðuÞ − Jmþ1ðuÞ��− sinðmϕÞcosðmϕÞ
�iϕ
þ ½Jm−1ðuÞ þ Jmþ1ðuÞ��cosðmϕÞsinðmϕÞ
�ir ; ð6Þ
Tmðr;ϕ; zÞ ¼ −½Jm−1ðuÞ þ Jmþ1ðuÞ��cosðmϕÞsinðmϕÞ
�iϕ
þ ½Jm−1ðuÞ − Jmþ1ðuÞ��− sinðmϕÞcosðmϕÞ
�ir ; ð7Þ
where JmðuÞ is the Bessel function of the first kind of ordermand u ¼ βr=ð1þ iz=z0Þ. For m ¼ 0, the lower part of the firstsquare bracket of Te represents the case of pure azimuthalpolarization and the lower part of the last square bracketof Tm represents pure radial polarization of the electric field.Form ≥ 1, the polarization variation along the azimuthal direc-tion is similar to that of the vector LG beam [21].
Here, we follow the formalism derived by Youngworth andBrown [30], who applied the vector diffraction theory ofRichards and Wolf to cylindrically polarized beams. Cylindri-cal components in the focal region of radially or azimuthallypolarized beams are given by
where focal length f ¼ 3mm, A is a constant, α is the semia-perture angle on the image side and related to the NA of thelens as α ¼ sin−1ðNAÞ, θ is the polar angle, ϕ is the azimuthalangle on the transverse plane perpendicular to the z axis, andðr0;ϕ0; z0Þ is the cylindrical coordinate in the focal region.From Eqs. (4)–(13), the field distribution near the focus is cal-culated for the vector BG beam represented by Eq. (5). For allthe calculations, we have used λ ¼ 632:8 nm, the refractive in-dex n ¼ 1, the NA of an objective NA ¼ 0:95, ω0 ¼ 3mm,f ¼ 3mm, and z ¼ 0. We assumed a circular pupil with a diam-eter of 5:7mm. In Figs. 1–6, the dimensions at the pupil planeand at the focal plane are 5:7mm × 5:7mm and 6 μm × 6 μm,respectively.
3. FOCUSING PROPERTIES OF VECTOR BGBEAMSFigures 1(a)–1(c) show the total intensity distribution andtransverse components jEr j2 and jEϕj2 at the pupil planefor pure radially polarized BG beams. For calculation of
the intensity and polarization distribution of pure radially po-larized beams, we used Eq. (5) with m ¼ 0 and the lower partof the last square bracket of Eq. (7). For pure azimuthally po-larized beams, we took m ¼ 0 in Eq. (5) and the lower part ofthe first bracket in Eq. (6). Figures 2(a)–2(c) show the totalintensity distribution and transverse components jEr j2 andjEϕj2 at the pupil plane for pure azimuthally polarized BGbeams. For the same values of ω0, the number of rings ofthe BG beam inside an input pupil is determined by the βparameter. For smaller values of β, only the central ring ofthe BG beam appears and, as the β value increases, the num-ber of rings inside the pupil increases as shown in Figs. 1(a)and 2(a). In our calculations, the value of β is varied in therange 0:00005k ≤ β ≤ 0:0005k. For β ¼ 0:00005k only a singlebright ring appears inside the pupil. By contrast, many ringsappear in the intensity pattern for β ¼ 0:0005k. Results of thetotal intensity distribution and polarization components at thefocal plane for pure radially and azimuthally polarized BGbeams for the various values of β are shown in Figs. 1(d)–1(g) and 2(d)–2(g). The total intensity distribution at the focalplane is the sum of transverse and longitudinal components.In the case of the radially polarized BG beam, for smaller β,the intensity distribution has a small bright spot as discussedby Youngworth and Brown [30]. When β is increased, the totalintensity at the center decreases and the bright rings sur-rounding the center become dominant, as shown in Fig. 1(d).Similar behavior is observed for the longitudinal componentof the electric field at the focal plane, as shown in Fig. 1(g). Asthe beam is radially polarized, the azimuthal component iscompletely zero. In the case of a pure azimuthally polarizedBG beam, there are no radial and longitudinal componentsbut only the azimuthal component of the electric field pre-sents, as shown in Fig. 2. The azimuthal component accompa-nies a dominantly bright ring whose size increases withincreasing β. Note that the pure radially polarized BG beamwith a multiring structure at the pupil plane forms a large ringpattern at the focal plane as shown Fig. 1 for large β. This pat-tern shows a strong contrast to that obtained for a radiallypolarized Laguerre-Gauss beam with a multiring structure,which forms a sharper spot mainly due to the longitudinalcomponent [11]. Although both beams have pure radial
Fig. 1. (Color online) Calculated intensity distribution and polarization components for a radially polarized BG beam for three values of β:0:00005k, 0:000275k, and 0:0005k. 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.
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polarization and a multiring structure with a phase shift of πbetween adjacent rings, the intensity patterns at the focalplane are different. While it is interesting to investigate theorigin of this difference, we return to the main subject of thispaper, which is the self-healing effect of BG beams at thefocal plane.
4. SELF-HEALING OF VECTOR BG BEAMSIN THE FOCAL PLANENumerical calculations of the intensity distribution and polar-ization components of the electric field at the focal plane areperformed. It is assumed that a sector-shaped obstacle is keptat the pupil plane, which partially hinders the incident beam.The size of the obstacle is represented by a center angle of asector g. First, we calculated the intensity distribution and po-larization components of the electric field at the focal planefor tightly focused radially and azimuthally polarized BGbeams. An obstacle with g ¼ 0:2π is assumed to be placedat the pupil plane of an objective lens with an NA of 0.95.Figure 3 shows the calculated intensity distribution and polar-ization components. In the top and bottom rows, the resultsfor radially and azimuthally polarized BG beams are shown,respectively. The dark parts corresponding to the obstacle
are seen in Figs. 3(a)–3(c). The intensity distribution and po-larization components at the focal plane for β ¼ 0:0005k andm ¼ 0 are shown in Figs. 3(d)–3(g). For β ¼ 0:0005k, there is abalance in the Bessel and Gaussian character of the beam. It isclearly seen that the intensity patterns are almost similar tothose of nonobstructed beams, shown in Figs. 1 and 2, exceptfor slight distortion at symmetric positions with respect to thebeam axis. This symmetric feature in the intensity pattern canbe qualitatively explained as follows. If a beam is decomposedinto rays or plane waves, all the waves direct toward the geo-metrical focal point and deliver energy symmetrically with re-spect to the focal point. Thus, the obstructed part is mainlycompensated by the opposite part of the input beam resultingin symmetric distortion at the focal plane. The similarity in thepatterns observed at the focal plane indicates that vector BGbeams have a healing effect from the point of view of recoveryof field distribution because a distorted pattern was well re-covered. It is interesting that the polarization componentsare also recovered in the same way as shown in Figs. 3(e),3(d), 3(f), and 3(g). This indicates that the vector BG beamshave a self-healing effect of not only intensity but also polar-ization at the focal plane. We have also analyzed the self-healing behavior of vector BG beams for different values of
Fig. 2. (Color online) Calculated intensity distribution and polarization components for azimuthally polarized BG beam for three values of β:0:00005k, 0:000275k, and 0:0005k. 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.
Fig. 3. (Color online) Calculated intensity distribution and polarization components for radially and azimuthally polarized BG beams with β ¼0:0005k andm ¼ 0. Top and bottom rows correspond to radially and azimuthally polarized beams, respectively. 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) radialcomponent, (f) azimuthal component, and (g) longitudinal component.
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β. Results of total intensity and polarization distribution at thefocal plane for radially polarized BG beams for three values ofβ are shown in Figs. 4(a)–4(c). It is clearly observed by takingaccount of the results in Fig. 1 that the self-healing behaviorremains the same for different values of β since the beamsrecover their intensity and polarization distributions forsingle as well as multiring-shaped intensity distributions. InSection 5, the intensity and polarization patterns at the focalplane will be shown for different sizes of an obstacle for ahigher-order beam.
5. DEPENDENCE OF THE SIZE OFOBSTACLETo investigate the effect of the size of an obstacle, we calcu-lated the total intensity and polarization components at thefocal plane for a TM BG beam with m ¼ 3 and β ¼ 0:0005k.
Higher-order vector BG beams have a peculiar intensityand polarization distribution similar to a petal, as shown in thetop row in Fig. 5. In this case, the value of g is equal to zero.Results for total intensity distribution and polarization compo-nents at the focal plane for different-sized obstacles at the pu-pil plane are presented in the remaining rows in Fig. 5. Despiteincreasing the size of the obstacle, the distortion of both in-tensity and polarization distributions are quite small as shownin Figs. 5(d)–5(g). It is surprising that the intensity distributionas well as the polarization distribution are well recovered atthe focal plane even when the half of the beam is obstructedas shown in the bottom row in Fig. 5. In addition, an axial sym-metry with respect to the optical axis is also recognized for allcases. Furthermore, a sixfold symmetry of the intensity distri-bution at the focal plane observed for g ¼ 0 is well preservedeven for g ¼ 1:0π, indicating that the phase distribution is also
Fig. 4. (Color online) Calculated intensity distribution and polarization components for a radially polarized BG beam for m ¼ 0 and for threedifferent values of β. Field distribution at the pupil plane: (a) total intensity distribution, (b) radial component, and (c) azimuthal component. Fielddistribution at the focal plane: (d) total intensity, (e) radial component, (f) azimuthal component, and (g) longitudinal component.
Fig. 5. (Color online) Calculated intensity distribution and polarization components for a vector BG beam with β ¼ 0:0005k and m ¼ 3 fordifferent-sized 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.
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preserved at the focal plane. This is because the sixfold sym-metry is the result of π phase shifts of the input beam at thepupil plane. It is noted that self-healing is observed for pureradially and azimuthally polarized beams as shown in the topand bottom rows of Figs. 3(e)–3(g), respectively, while the ef-fect is slightly inferior compared to those in Fig. 5.
6. SELF-HEALING OF SCALAR BG BEAMSIN THE FOCAL PLANETo compare the self-healing behavior of a vector BG beamwith that of a scalar beam, we carried out a similar numericalsimulation for the focal field distribution of a tightly focused,linearly polarized BG beam of β ¼ 0:0005k and n ¼ 1. We haveused the equations derived by Richards and Wolf [10] to cal-culate the intensity distribution near the focus. The calculatedresult using Eq. (1) is shown in Fig. 6. The beam is assumed tobe polarized in the x direction, which is horizontal at the pupilplane. The intensity distribution and polarization componentsof the electric field at the pupil plane without an obstacle(g ¼ 0) are shown in the top row of Fig. 6. The total intensitydistribution is very close to that of the x-polarization compo-nent at the focal plane, revealing a single dominant ring withsome weak concentric rings. In Fig. 6, we have normalized allthe polarization components with respect to the maximumtotal intensity. The y component of the electric field is veryweak but not zero. A weak longitudinal component (z-polarization component) is also observed. This feature is simi-lar to the case of the tight focusing of a linearly polarizedGauss beam, which shows a two-lobe pattern along withthe polarization direction (x direction in this case). The effectof obstacles placed at the pupil plane is investigated by chan-ging the size of a sector-shaped obstacle. Results are shown inthe lower rows of Fig. 6. Increasing a center angle of the sec-
tor from 0:2π to 1:0π, the intensity distribution and polariza-tion components are gradually distorted. As a whole, theintensity and polarization patterns are well recovered, indicat-ing that the self-healing effect is observed again. However, thedistortion in both intensity and polarization seems to be stron-ger than that in Fig. 5. In addition, an axial symmetry is notobserved in contrast to those in Fig. 5. This behavior maybe attributed in part to the symmetry of an input beam. Whilevector BG beams have a cylindrical symmetry of intensity,phase, and polarization, a linearly polarized beam has a cylind-rical symmetry of intensity only. In particular, a spiral phaseshift of the wavefront may play an important role in the sym-metry breaking at the focal plane. It should be emphasizedthat the self-healing of a linearly polarized beam at the focalplane is also observed even when the half of the beam isobstructed at the pupil plane.
7. DISCUSSIONIn the description of the recovery of field distribution of BGbeams at the focal plane, we have used the term self-healingrather than self-reconstruction because the intensity distribu-tion and polarization components are not exactly the same asthat of the nonobstructed case. We believe that this is the firstobservation of the self-healing of the polarization componentsat the focal plane in addition to the intensity pattern. In mostof work on the self-reconstruction of nondiffracting beams,scalar beams were mainly studied with a focus on their inten-sity distribution. Vector BG beams have a cylindrically sym-metry of polarization and phase in addition to intensity,making these beams more robust against the distortion com-pared to a scalar BG beam. By contrast, the propagation of ahalf-obstructed scalar Laguerre-Gauss beam in the paraxiallimit has been reported, showing that the intensity pattern
Fig. 6. (Color online) Calculated intensity distribution and polarization components for a linearly polarized (x-polarized) BG beam with β ¼0:0005k and n ¼ 1 for different-sized obstacles. Field distribution at the pupil plane: (a) total intensity. Field distribution at the focal plane: (b) totalintensity, (c) x-polarized component, (d) y-polarized component, and (e) longitudinal component.
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is rotated with the propagation, and the direction of the rota-tion depends on the sign of the topological charge [31]. Thisphenomenon can be attributed to the spiral phase shift of thebeam. In fact, the calculations for vector and scalar BG beamsshowed slight differences with regard to the healing effect.The vector BG beam is less deformed in comparison with alinearly polarized BG beam probably due to a higher symme-try of intensity, phase, and polarization of the vector beam. Tomake the analysis more general, we have varied the mode or-der m of the beam, and it was found that the property of self-healing is same for lower- and higher-order vector BG beams.This result is in accordance with the notion that self-healing isthe inherent property of nondiffracting beams and it is samefor different mode orders. We have also varied the focal lengthf and NA of the focusing lens, and it was observed that there isno significant difference in the self-healing property of thebeam by changing these parameters. However, for obstacleswith g larger than π, self-healing is diminished and the beamloses its axial symmetry of both intensity and polarizationdistribution.
In order to recover the missing field by an obstacle, all theparameters, such as intensity (amplitude), phase, and polari-zation must be compensated somehow. In this paper, the self-healing effect was discussed at the focal plane for convergingbeams. As mentioned earlier, all the waves in a beam directthe focal point with imperfect but highly symmetric intensity,phase, and polarization distribution with regard to the focalpoint. The self-healing effect before and after the focal planeis attractive and will be the focus of a future work. The self-healing of polarization at the focal plane observed even whenthe half of the beam is obstructed is a new observation. Thisfeature may play an important role in many fields wherefocused vector beams are used, such as laser scanning micro-scopy, optical trapping, and optical data storage.
8. CONCLUSIONSRecovery of the spatial distribution of polarization at the focalplane of obstructed BG beams is discussed based on the vec-tor diffraction theory. We showed the property of self-healingof both scalar and vector BG beams. It is found that the vectorBG beam is more resistant to the obstruction in comparisonwith the scalar BG beam. In addition, the intensity distributionand polarization components exhibit an axial symmetry evenif half of the input beam at the pupil plane is obstructed.
ACKNOWLEDGMENTSThis work was supported in part by the Japan Science andTechnology Agency (JST) Core Research for EvolutionalScience and Technology (CREST).
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