Optics Communications 281 (2008) 5939–5948

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Optics Communications

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Focusing of linearly-, and circularly polarized Gaussian background vortexbeams by a high numerical aperture system afflicted withthird-order astigmatism

Rakesh Kumar Singh *, P. Senthilkumaran, Kehar Singh*

Department of Physics, Indian Institute of Technology Delhi, New Delhi 110 016, India

a r t i c l e i n f o

Article history:Received 5 February 2008Received in revised form 3 September 2008Accepted 10 September 2008

Keywords:Optical vortexVectorial diffractionPolarization stateTopological chargeIntensity distribution

0030-4018/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.optcom.2008.09.036

* Corresponding authors. Tel.: +91 1126596580 (R.E-mail addresses: krakeshsingh@gmail.com (R.K.

ernet.in (K. Singh).

a b s t r a c t

Effects of third-order astigmatism on the focused structure of linearly and circularly polarized Laguerre–Gaussian beams have been investigated by using vectorial Debye–Wolf integral. The results have beenpresented for total intensity distribution and squares of the polarization components at the focal planeof a high numerical aperture system, for two values of the topological charge. Astigmatism results inthe stretching of the intensity pattern as well as of the squares of the polarization components. A splitis observed in the intensity pattern of a focused beam having double topological charge, and also inthe pattern of the longitudinal polarization component of circularly polarized beam even with unit topo-logical charge.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

An optical vortex beam possesses helical phase structure and apoint of undefined phase within the wave field [1–8]. Such a helicalphase structure has generated extensive interest because of itsclose relationship to the mechanical strength or orbital angularmomentum (OAM), and doughnut shaped intensity distributionof the beam. Strength of the OAM depends on the topologicalcharge m which refers to the number of complete cycles of phase(2 pm) change in a closed loop surrounding the vortex point. Anoptical vortex occurs as a particular solution to the wave equationin cylindrical coordinate systems; the doughnut mode of a laser isan example of an optical vortex beam. When an optical vortex ishosted within the Gaussian background, the resulting beam exhib-its an annular intensity profile with dark core, whilst maintaining ahelical phase structure [5,8]. A typical example of such a light fieldis a Laguerre–Gaussian (LG) beam. The LG modes that may emergefrom a cylindrical laser cavity belong to a family of solutions ofwave equation. This family of solutions is referred to as ‘La-guerre–Gaussian (LG) modes’ which are rotationally symmetricand exhibit an azimuthal angular dependence of the complex formexp(im/), where / is the azimuthal coordinate in the transverseplane [1–8].

ll rights reserved.

K. Singh).Singh), kehars@physics.iitd.

The structure of the focused beam depends on the polarizationdistribution of the input beam in a high numerical aperture (NA)system [9–26]. High NA focusing of a linearly polarized vortexbeam with m = 1 shows residual intensity at the center with twoside lobes [17,19,25], whereas the left circularly polarized vortexbeam with m = 1 shows an annular ring with a dark center[19,25,26]. Size and shape of the focal spot depend on the polariza-tion distribution of the input beam. In order to obtain a circular fo-cal spot, the polarization distribution of the input beam must berotationally symmetric [21]. Structure of the dark core in the dif-fraction pattern of an apertured LG beam has been an area ofattraction in recent years due to the increasing number of applica-tions e.g. in lithography [27,28], particle trapping and manipula-tion [29,30], and microscopy [19,26,31,32]. Most of the studieson the focusing of apertured vortex beams have been carried outeither for perfect or aberrated systems using scalar theory of dif-fraction [33–40]. The results of such investigations, however, arenot useful in high NA focusing. The structural modification of thefocused beam by the high NA systems is affected by imperfectionsor aberrations in the focusing system. Since possibilities of aberra-tions can not be ruled out even for well-corrected systems withhigh NA [11–13,20], the effect of aberrations in the high NA focus-ing has been carried out for non-vortex beam (m = 0) [12,13], andfor radially polarized vortex beams [20]. However, no detailedstudies were available on the effect of aberrations on the highNA focusing of vortex carrying beams having phase singularity, ex-cept investigations of Braat et al. [41].

5940 R.K. Singh et al. / Optics Communications 281 (2008) 5939–5948

In view of the importance of the high NA focusing of vortexbeams, and their structure in the focal region, we have undertakena systematic study on the subject [42,43]. In this paper, we presentthe results of our investigations on the effect of third-order astig-matism in case of vortex beam with Gaussian background. Ourinvestigation makes use of the vectorial Debye–Wolf integral,and the results have been presented for total intensity distribution,and the squares of different polarization components. Results inthe case of a vortex non-vortex beam have been compared withthe known results and found to be in good agreement.

2. Theory

We consider the complex amplitude of the LG beam at the inputplane [1,5] of the optical geometry (Fig. 1) as

Eðq;/Þ ¼ A0ðffiffiffi2p

cqÞjmjLjmjp2q2

c2

� �expð�c2q2Þ expðim/Þ ð1Þ

where A0 is the characteristic amplitude, q is the radial distance of apoint from its center normalized by the radius a of the lens, and u isthe azimuthal coordinate on the input plane. Ljmjp ðxÞ is the Laguerrepolynomial with p and m as the radial and angular mode numbers,and c = (a/w) is a truncation parameter with w as a parameter forbeam size at the exit plane. The parameter w together with A0 scalesthe peak intensity of the beam. The case with p = 0 exhibits at themost, one-intensity minimum within the beam. The complexamplitude of the LG beam can be written as

E0ðq;/Þ ¼ A0ðffiffiffi2p

cqÞjmj exp½�c2q2� expðim/Þ ð2Þ

For optical geometry in Fig. 1, Eq. (2) can be written in terms ofthe angular coordinates as

E0ðh;/Þ ¼ A0

ffiffiffi2p

csin hsin a

� �jmjexp �c2 sin h

sin a

� �2" #

expðim/Þ ð3Þ

A1ðhÞ ¼ffiffiffi2p

csin hsin a

� �jmjexp �c2 sin h

sina

� �2" #

where q ¼ sin hsina, A1(h) is the amplitude distribution of the input

beam, a is the maximum angle of convergence (i.e. hmax = a), andq is the zonal radius which is equal to unity at the edge of the focus-ing system.

Following Richards and Wolf [9], the field distribution in the fo-cal volume of a high NA optical system is given by using diffractionintegral (also known now as Debye–Wolf integral) as

o

rPφP

φ

ρ

θP

z

xP

yP

x

y

θ

Fig. 1. Schematic representation of the optical geometry.

EðPÞ ¼ � ik2p

ZX

Zaðsx; syÞ

szexp ikfUðsx; syÞ þ s � rðPÞg

� �dsxdsy ð4Þ

where a is a strength factor, r(P) is the radius vector connecting thepoint P with the Gaussian focus which is also the origin of the coor-dinate system (Fig. 1), s = (sx,sy,sz) is the direction vector of a typicalray in the image space, U is the wave aberration function which de-notes the deviation of the actual wavefront from the ideal one, andkð¼ 2p

k Þ. The integral is taken over the entire surface of the wave-front leaving the exit pupil. In the presence of the astigmatism, ra-dial distance of any point on the wavefront depends on the angularcoordinates and written [13] as

rðh;/Þ ¼ f þ Aaq2 cos2 /

r!ðPÞ ¼ rðPÞðsin hP cos /Piþ sin hP sin /Pjþ cos hPkÞ

U ¼ Aaq2 cos2 /

ð5Þ

where q ¼ sin hsin a is zonal radius, (i,j,k) are unit vectors, f is the focal

length of the optical system, Aa is the astigmatic aberration coeffi-cient in the units of wavelength, and ðrP; hP;/PÞ are position coordi-nates of a point on the observation plane.

Using the concept of the two orthogonal tangent vectors in thepolar and azimuthal directions, the unit normal to the aberratedwavefront is given [13] as

sx ¼ 1r sin h cos /� 1

roroh cos h cos /þ 1

r sin horo/ sin /

sy ¼ 1

r sin h sin /� 1r

oroh cos h sin /� 1

r sin horo/ cos /

sz ¼ 1

r cos hþ 1r sin h or

oh

� � ð6Þ

where r is the normalization factor given [13] as

r ¼ 1þ 1r2

oroh

� �2

þ 1

sin2 h

oro/

� �2( )" #1=2

Using the binomial expansion and ignoring the higher orderderivative of the position vector due to small value of the aberra-tions, we write

1rffi 1�Hþ oðHÞ

where H ¼ 12r2

oroh

� �2 þ 1sin2 h

oro/

2� �

Using the approach of the Richards and Wolf [9] for the linearlypolarized beam and Visser and Wiersma [12] for the arbitrarypolarization, one can derive the contribution of the polarizationfactor in the evaluation of strength factor a

aPðh;/Þ ¼ ðs � kÞ1=2= s2

x þ s2y

A s2y þ s2

x sz

þ Bð�sxsy þ sxsyszÞ

Að�sxsy þ sxsyszÞ þ B s2x þ s2

y sz

Aðð�sxÞ þ Bð�syÞ� �

s2x þ s2

y

266664

377775ð7Þ

or aP ¼ A2ðhÞPðh;/Þwhere A2ðhÞ ¼ ðs � kÞ1=2 ¼ ðszÞ1=2, and

Pðh;/Þ ¼ 1

s2x þ s2

y

A s2

y þ s2x sz

þ Bð�sxsy þ sxsyszÞ

Að�sxsy þ sxsyszÞ þ B s2x þ s2

ysz

½Að�sxÞ þ Bð�syÞ� s2

x þ s2y

266664

377775

Here A(h,u), B(h,u) are respectively the strengths of the x-, andy-polarized input beams, A2(h) corresponds to the apodization fac-tor equal to cos1/2h for an aplanatic lens [9]. In the aberration-freecase, the position coordinates at the wavefront are independent ofthe polar and radial coordinates, and in this situation the unit nor-mal to the wavefront is transformed to the case of Richards and

R.K. Singh et al. / Optics Communications 281 (2008) 5939–5948 5941

Wolf [9]. The polarization distribution in the aberration-free case istransformed into polarization matrix of Helseth [15]. For aberra-tion-free case and x polarization (B = 0), the polarization distribu-tion at the exit pupil is transformed into results of the Richardsand Wolf [9] and Boivin and Wolf [10]. Expression for the unit nor-mal to the aberrated wavefront can be written as the sum of twoterms representing the vector along the unit normal to the idealwavefront and other representing the deviation such as

s ¼ ð1�HÞnþ 1r

Fðh;/Þ

where n ¼ sin h cos /iþ sin h sin /jþ cos hk

Fxðh;/Þ ¼ �1r

oroh

cos h cos /þ 1r sin h

oro/

sin /

� �

and Fyðh;/Þ ¼ �1r

oroh

cos h sin /� 1r sin h

oro/

cos /

� �

Fig. 2. Intensity distribution (|E|2) of an x-polarized GBVB at the focal plane of a lens with(e) 0.0, (f) 0.5, (g) 1.0, and (h) 1.5.

Fzðh;/Þ ¼1r

sin horoh

� �

Expressing the dsxdsy as a function of dhdu and using Eqs. (5)–(7), Eq. (4) can be written as

Eðu; vÞ ¼ � ikf2p

Z a

0

Z 2p

0A1ðhÞ expðim/ÞA2ðhÞPðh;/Þ

� expfik½U� Hn� 1r

F� �

� rðPÞ�g

� exp iu cos h

sin2 a

� �exp i

v sin hsina

� �cosð/� /PÞ

� �jJjd/dh

ð8Þwhere the optical coordinates (v,u) are defined as

v ¼ krP sin hP sinau ¼ krP cos hP sin2 a;

and jJj ¼ osxoh

osy

o/ �osxo/

osy

oh

.

a = 75�, with m = 1, c = 1 and Aa (a) 0.0, (b) 0.5, (c) 1.0, (d) 1.5; for m = 2, c = 1 and Aa,

5942 R.K. Singh et al. / Optics Communications 281 (2008) 5939–5948

Eqs. (5)–(8) are used to evaluate the field distribution in the fo-cal region of an optical system for any polarization distribution ofthe input beam. For x-polarized beam (B = 0 and A = 1), complexamplitude at the focal plane (u = 0) is given as

Exðvx;vyÞ

Eyðvx;vyÞ

Ezðvx;vyÞ

2664

3775 ¼ ð�if=kÞ

Z a

0

Z 2p

0A1ðhÞA2ðhÞexpðim/Þ 1

ðs2x þ s2

yÞ

�

s2y þ s2

x sz

ð�sxsy þ sxsyszÞ

�sx s2x þ s2

y

266664

377775exp ik½U� ðHn� 1

rFÞ � rðPÞ�

�

� exp½i vsina

sinh cosð/�/PÞ�jJjd/dh ð9Þ

Iðvx;vyÞ ¼ jExj2 þ jEyj2 þ jEzj2 ð10Þ

Fig. 3. Squares of the polarization components and contour lines of an x-polarized GBVB|Ez|2; and Aa = 1.0, (d) |Ex|2, (e) |Ey|2, and (f) |Ez|2.

where I(vx,vy) is the total intensity distribution in the focal plane, andvx;y ¼ k½xP; yP� sin a are the optical coordinates in the focal plane herexP ¼ rP sin hP cos /P and yP ¼ rP sin hP sin /P . From Eq. (10), the totalintensity is proportional to the sum of the squares of the x-, y-, andz-polarized components The linearly polarized vortex beam in anaberration-free case with m = 1 (or �1) produces |Ex|2 = 0, |Ey|2 = 0and |Ez|

2 – 0 at v = 0, and removes the dark core from the centre,whereas a beam with m = 2 (or �2) produces |Ex|2 – 0, |Ey|2 – 0,and |Ez|

2 = 0 at the focal point [25]. An intensity null exists at the focalpoint for a beam with m P 3, but circular symmetry of the pattern isnot maintained. In case of an x-polarized Gaussian beam (m = 0), thefield is purely x-polarized at the focal point (i.e. |Ex|2 – 0).

Let us consider the complex amplitude of the field at the exitpupil plane as

Eðh;/Þ ¼ eim/ð~E1 þ eiu~E2Þ ð11Þwhere ~E1 and ~E2 are respectively the electric fields due to x-, and y-polarized beams. An appropriate phase delay u between the x-, and

with m = 1, c = 1 at the focal plane a lens with a = 75�, Aa = 0.0 (a) |Ex|2, (b) |Ey|2, (c)

R.K. Singh et al. / Optics Communications 281 (2008) 5939–5948 5943

y-polarized input beams produces elliptic polarization. When|E1| = |E2| and u ¼ � p

2 ; the beam is circularly polarized; u ¼ þp=2and �p/2 correspond respectively to left circular (LC) and right cir-cular (RC) polarizations.

Let us assume that the amplitudes of both x- and y-polarizedbeams are equal, and with p/2 phase delay. The field distributionof circularly polarized vortex beam at the focal plane is obtainedby putting p/2 for the phase delay in Eq. (11). Using Eq. (8), we get

Exðvx; vyÞEyðvx; vyÞEzðvx; vyÞ

264

375 ¼ ð�if=kÞ

Z a

0

Z 2p

0A1ðhÞA2ðhÞ expðim/Þ 1

ðs2x þ s2

yÞ

�ðs2

y þ s2x szÞ � ð�sxsy þ sxsyszÞ

ð�sxsy þ sxsyszÞ � ðs2x þ s2

yszÞð�sxÞ � ð�syÞ� �

ðs2x þ s2

yÞ

2664

3775

Fig. 4. Squares of the polarization components and contour lines of an x-polarized GBV(c) |Ez|2; and Aa = 1.0 (d) |Ex|2, (e) |Ey|2, and (f) |Ez|2.

expfik½U� ðHn� 1r

FÞ � rðPÞ�g � exp½i vsin a

sin h cosð/� /PÞ�

� jJjd/dh ð12Þ

In the aberration-free case, the LC polarized beam with m = 1produces intensity null at the focal point. Zero complex field forLC polarization occurs at the focus, provided that the spin angu-lar momentum of the photons (related to the handedness ofcircular polarization) and the orbital angular momentum of thebeam (related to the topological charge) point in the same direc-tion. However the RC polarized beam with m = 1 produces anon-zero field at the focal point only due to longitudinalpolarization component. The LC and RC polarized helical beamswith m = 1 also act as basis for the radially polarized doughnutbeam.

B with m = 2, c = 1 at the focal plane a lens with a = 75� Aa = 0.0 (a) |Ex|2, (b) |Ey|2,

5944 R.K. Singh et al. / Optics Communications 281 (2008) 5939–5948

3. Results and discussion

Results are presented for a Gaussian background vortex beam(GBVB) in the focal plane of an air aplanatic lens with a = 75� inthe presence of third-order astigmatism. Results of the vortexbeam with uniform background in the aberration-free case havebeen compared with the results of Ganic et al. [17]. The resultsof LG beam in aberration-free case have been compared with thoseof Torok and Munro [19], Iketaki et al. [25], and Bokor et al. [26]. Itis seen that a good agreement is found with the respective results.Results of the non-vortex beam with uniform background are com-pared with the results of Kant [13] for astigmatic case and arefound to be in reasonably good agreement. When the NA of thefocusing system is low, the focal plane intensity distribution showssimilarity to the results obtained by scalar theory [17,38–40]. Inthis paper we have presented all the results for a lens of focallength equal to 5000k.

Fig. 5. Intensity distribution and contour lines of (|E|2) for an LC polarized GBVB at the focfor m = 2, c = 1 and Aa, (e) 0.0, (f) 0.5, (g) 1.0, and (h) 1.5.

3.1. Linearly polarized beam

The intensity distributions of an x-polarized GBVB with m = 1, 2,and c = 1, at the focal plane are shown in Fig. 2. Results for |E|2 dis-tribution of the GBVB with m = 1 at the focal plane are shown inFig. 2a–d for Aa = 0.0, 0.5, 1.0 and 1.5. For Aa = 0.0, the intensity dis-tribution possesses central non-zero intensity surrounded by highintensity region. The high intensity peaks are situated in the direc-tion orthogonal to the direction of polarization, with an elongationof the pattern along the direction of polarization. The presence ofastigmatism (Fig. 2b–d) results in a stretching of the intensity dis-tribution and this stretching increases with an increase in the valueof Aa. The residual intensity also decreases and is accompaniedwith a positional shift of the side lobes as a result of an increasein Aa. The side lobes in Fig. 2e are situated at larger distance incomparison to those in Fig. 2a. The presence of astigmatism in

al plane of a lens with a = 75�, with m = 1, c = 1 and Aa (a) 0.0, (b) 0.5, (c) 1.0, (d) 1.5;

R.K. Singh et al. / Optics Communications 281 (2008) 5939–5948 5945

the system stretches the intensity distribution with a split of thelow intensity region into two parts (Fig. 2f–h).

Distributions of |Ex|2, |Ey|2, and |Ez|2 are shown in Figs. 3 and 4

for beams with m = 1 and 2, respectively. Fig. 3 represents resultsof square of polariztion components for Aa = 0.0 and 1.0. The fieldat the focal point is purely z-polarized for a beam with m = 1,and the contribution of other polarization components increasesin the outer regions. Distribution of |Ex|2 possesses two side lobeswith its peak value higher than the peaks of |Ey|2 and |Ez|

2. Presenceof aberration in the system stretches the structure of x-, y-, and z-polarization components ( Fig. 3d–f) and the contribution of z-polarization component reduces at the focal point. In the case ofm = 2, the focal point intensity arises due to the transverse polari-zation components. Presence of astigmatism in the system causessplit in the dark core of |Ex|2, an effect also apparent from the total

Fig. 6. Squares of the polarization components and contour lines of an LC polarized GBVB(c) |Ez|2; Aa = 1.0, (d) |Ex|2, (e) |Ey|2, and (f) |Ez|2.

intensity distribution. An increase in the value of Aa results in therotation of intensity peaks with stretching in the patterns of |Ex|2,|Ey|2, and |Ez|

2.

3.2. Circularly polarized LG beam

Intensity distributions in case of the LC polarized GBVB withm = 1 and 2 are shown in Fig. 5 for Ac = 0.0, 0.5, 1.0 and 1.5.Fig. 5a and e shows circularly symmetric dark core with a highintensity ring, and the size of the dark core for beam with m = 2is greater than that of beam with m = 1. The presence of astigma-tism in the focusing system splits the circular ring into two highintensity lobes. An increase in Aa results in the stretching and rota-tion in the intensity lobes. Also, splitting of the dark core into twodark regions for a beam with m = 2 takes place in the presence of

with m = 1, c = 1 at the focal plane a lens with a = 75� and Aa = 0.0 (a) |Ex|2, (b) |Ey|2,

5946 R.K. Singh et al. / Optics Communications 281 (2008) 5939–5948

aberration. With an increase in the value of Aa, the dark core iselongated in a certain direction, with a further shift of the highintensity region.

Fig. 6a–c shows the distributions of |Ex|2, |Ey|2, and |Ez|2 of a

GBVB with m = 1 and c = 1 for an aberration-free case, whereasFig. 6d–f shows the results for Aa = 1.0. The distribution of |Ex|

2,and |Ey|2 possesses two high amplitude lobes with equal magni-tude, and their positions are interchanged in the orthogonal direc-tion in the aberration-free case. The distribution of |Ez|

2 (Fig. 6c)possesses circularly symmetric dark core, which is large in compar-ison to the dark core of the transverse field components. Highintensity lobes and dark core of the transverse field componentsrotate, and get stretched in the presence of aberration. Annularintensity ring in the distribution of |Ez|

2 (Figs. 6f) splits, and thedark core is separated into two parts. For Aa = 1.0, the high inten-sity region and the dark core are further stretched with an increasein the value of Aa.

Fig. 7. Squares of the polarization components and contour lines of an LC polarized GBV|Ey|2, (c) |Ez|2; and Aa = 1.0 (d) |Ex|2, (e) |Ey|2, and (f) |Ez|2.

The distribution of squares of the polarization components forGBVB with m = 2 and c = 1 are shown in Fig. 7a–c for Aa = 0.0.Fig. 7d–f shows the results for Aa = 1.0. Fig. 7a and b shows theintensity lobes at large distance from the focal point in comparisonto the case of m = 1. The dark core in the distribution of transversepolarization components is separated into two parts for m = 2, andis accompanied with a stretching of the high intensity lobes. How-ever, the effect of astigmatism on the distribution of |Ez|

2 is com-pletely different from the distribution of the transversepolarization components. The dark core in the distribution of|Ez|

2 is separated into three parts, and the stretching of the dark re-gions increases with Aa (Fig. 7f). The results for the longitudinalpolarization components also verify results of the generation ofthe longitudinal vortex in the tight focusing of circularly polarizedbeam [44], and split in the structure takes place for higher topolog-ical charge. Impact of the longitudinal polarization component isnot visible on the total intensity distribution due to its low inten-

B with m = 2, c = 1 at the focal plane of a lens with a = 75� and Aa = 0.0 (a) |Ex|2, (b)

R.K. Singh et al. / Optics Communications 281 (2008) 5939–5948 5947

sity in comparison to that of the transverse polarization compo-nents, and due to superposition of the dark core of |Ez|

2 on the highintensity regions of |Ex|2, and |Ey|2. The peak amplitude of the lon-gitudinal polarization component can be modified by centrallyobstructing the focusing system.

Intensity distribution and square of the polarized componentsat the focal plane for the right circularly polarized GBVB beamare shown in Fig. 8a–d for aberration-free case, and in Fig. 8e–hfor Aa = 1.0. The total intensity distribution in the aberration-freecase is circularly symmetric with non-zero value at the center.The intensity distribution around the focal point is not completelyflat but possesses a small dip at the center which can be modulatedby an appropriate pupil function. This non-zero intensity distribu-tion at the focal point arises due to contribution of the longitudinalpolarization component, whereas the transverse polarization com-ponents have two lobes around the center. Presence of circularlysymmetric ring around the center for the longitudinal polarizationcomponent is related to the conservation of total angular momen-

Fig. 8. Intensity distribution and squares of the polarization components of RC polarized|Ex|2, (c) |Ey|2, (d) |Ez|2; and Aa = 1.0, (e) |E|2, (f) |Ex|2, (g) |Ey|2, and (h) |Ez|2.

tum which depends on the nature of both helical phase structureand polarization. Stretching in the intensity distribution and inthe square of the polarization components takes place, and as a re-sult the circular symmetry of the distribution of total intensity andlongitudinal polarization component is lost.

4. Conclusion

The presence of primary astigmatism in the focusing systemintroduces stretching of the intensity pattern, and high intensitylobes are shifted from the center with an increase in the value ofastigmatism. Splitting in the intensity distribution of double topo-logical charged beam is also found along with stretching. However,squares of the longitudinal polarization components for the circu-larly polarized Gaussian background vortex beam split into twoparts even for unit topological charge, and in three parts in caseof a beam with double topological charge. This verifies the gener-ation of the longitudinal vortex in the left circularly polarized

GBVB with m = 1,c = 1 at the focal plane of a lens with a = 75� and Aa = 0.0 (a) |E|2, (b)

5948 R.K. Singh et al. / Optics Communications 281 (2008) 5939–5948

beam, and results for both left and right circular polarizations con-firm the conservation of total angular momentum.

Acknowledgment

Rakesh Kumar Singh is thankful to the Council of Scientific andIndustrial Research India (CSIR) for awarding Senior Research Fel-lowship. CSIR research grant 03/(1034)/05/EMR II is thankfullyacknowledged. The authors wish to thank Dr. K. N. Chopra for help-ful discussion.

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