tight focusing of linearly and circularly polarized vortex beams; effect of third-order spherical...
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ARTICLE IN PRESS
Optics and Lasers in Engineering 47 (2009) 831–841
Contents lists available at ScienceDirect
Optics and Lasers in Engineering
0143-81
doi:10.1
� Corr
tory, Ou
E-m
d.ernet.
journal homepage: www.elsevier.com/locate/optlaseng
Tight focusing of linearly and circularly polarized vortex beams; effect ofthird-order spherical aberration
Rakesh Kumar Singh �, P. Senthilkumaran, Kehar Singh
Department of Physics, Indian Institute of Technology Delhi, New Delhi, 110016, India
a r t i c l e i n f o
Article history:
Received 7 November 2008
Received in revised form
28 January 2009
Accepted 28 January 2009Available online 17 March 2009
Keywords:
Optical vortex
Topological charge
Tight focusing
Polarization distribution
Debye–Wolf integral
Intensity distribution
66/$ - see front matter & 2009 Elsevier Ltd. A
016/j.optlaseng.2009.01.013
esponding author Present address: Universi
lu Sothern Institute, Vierimaantie 5, 84100, F
ail addresses: [email protected] (R.K
in (K. Singh).
a b s t r a c t
Tight focusing of linearly and circularly polarized vortex beams is studied in the presence of third-order
spherical aberration, using vectorial Debye–Wolf integral. Results for total intensity distribution are
presented for both polarizations. In addition, results for x-, y-, and z-polarization components are
presented for the circularly polarized beam. Generation of longitudinal optical vortex in the tightly
focused left circularly polarized beam has also been demonstrated by plotting its phase distribution.
Compensation for the effect of spherical aberration has been studied in the presence of defocusing.
Effect of aberration on the dark core of a tightly focused azimuthally polarized beam is also investigated.
& 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Smallest dark region in the diffraction pattern of a vortex beam isimportant and useful in many areas of optics, such as in opticaltrapping and manipulation [1,2], and in super-resolution fluores-cence microscopy [3–5]. A beam carrying vortex possesses a point ofundefined phase within the wave field, called ‘singular point’, anddoughnut structure in the diffraction pattern. Accumulated phasechange around the singular point must be an integral multiple of 2p[6,7], and this multiple is referred to as the ‘topological charge’ m.The helical structure of the beam provides it an orbital angularmomentum (OAM), which is different from the spin angularmomentum due to polarization of the beam [7]. Strong variationof the optical field around the singularities is highly sensitive tochanges in their neighborhoods, and this characteristic has beenused recently in the singular beam microscopy [8]. The diffractionpattern of the vortex beam possesses dark core due to thedestructive interference of secondary waves coming from the pointswhich are out of phase. For example, for m ¼ 1, they are located atdiametrically opposite points equidistant from the core.
In both the above-mentioned applications [1–5], size and shape ofthe dark core in the diffraction pattern of the beam play an importantrole. For example, the size of the focused annular ring influences thetransfer of mechanical strength to the trapped particles [1], and gives
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ty of Oulu, RFMedia Labora-
inland.
. Singh), [email protected]
the smallest fluorescent spot in stimulated emission depletion (STED)microscopy [3–5]. Tight focusing of a linearly polarized vortex beamyields intensity distributions markedly different from those obtainedby the scalar theory of diffraction [9] that is inadequate for studyingthe tight focusing or high numerical aperture (NA) focusing system[10,11]. A vortex beam shows formation of lobes perpendicular to theincident polarization direction, and the focal spot becomes elongatedin the polarization direction [3,9]. High NA focusing of light has alsodrawn much attention in recent years in the context of space-variantgeometric phase, since the geometric phase results in the formationof mode with helicities and phase singularities that differ from thoseof the original beam [12]. Recently, the effect of the tight focusing onthe spin-to-orbital angular momentum conversion has also drawnattention of the researchers [13]. Manipulation of focused structure ofa vortex beam using polarization and pupil function engineering hasdrawn attention in recent years [2,3,14]. High intensity lobes of atightly focused linearly polarized vortex beam are useful in thetrapping of small particles [2]. Role of intensity lobes in the trappingof small particles has recently been investigated by Jeffries et al. [2].Torok and Munro [3] have examined the applicability of vortex beamfor STED microscopy, and observed that due to the rotationalsymmetry and sharpness of the dark core, the left circularly polarizedvortex beam is more useful in comparison to a beam with linearpolarization. Also, the individual polarization components of thepump beam and the erase beam play an important role in theformation of the fluorescent spot [5].
Deformation in the size and shape of the dark core is possibledue to perturbation in the focused beam. This perturbation mayarise due to the refractive index mismatch between two media,defect or misalignment of the optical system. A refractive index
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mismatch introduces spherical aberration that increases withincreasing depth inside the medium. Consequently, the effect ofinterface between two media of different refractive indices hasalso been an area of investigation in recent years [3,15–17].Recently, Escobar et al. [17] have investigated the reduction of thespherical aberration effect in a high NA system. Investigations onthe phase singularities in the image plane have been carried outfor both low and high NA systems in aberration-free and aberratedcase [18–20]. Structural modifications in the focused structure ofthe vortex beam due primary aberrations in low NA system havebeen carried out in recent years [21–24]. However, investigationson the high NA systems are mostly limited to aberration-freesystems [25–30], or the aberrated system in the absence of phasevortices [31–34]. Filter performance parameters for investigatingthe pupil filters for aberration-free high NA focusing has drawnattention in recent years [35,36].
It is well-known that even well-corrected objectives suffer fromsmall amounts of aberration. An important investigation wasinitiated by Braat et al. [34], who used extended Nijboer–Zernikerepresentation of the vector field in the focal region of an aberratedhigh NA optical system. Structural modification of the focuseddoughnut beam due to aberration has been mentioned briefly byWillig et al. [4] in the context of STED microscopy. Biss and Brown[33] have investigated the effect of primary aberrations on thefocused structure of the radially polarized vortex beam. However,no detailed studies seem to have been made on the effect of primaryaberrations on the tight focusing of beams carrying a phase vortex.In view of the importance of the high NA focusing of vortex beamsand their structure in the focal region, we have undertaken asystematic study on the subject [37,38]. In this paper, we presentthe results of our investigations on the effect of spherical aberrationfor linearly and circularly polarized light in case of vortex beamswith uniform amplitude. Focused structure of an azimuthallypolarized non-vortex beam has also been investigated.
2. Intensity distribution in the focal region
For our study, we have considered the coordinate geometryshown in Fig. 1. The complex amplitude of the vortex beam can beexpressed [3] as
E0ðy;fÞ ¼ A1ðyÞ expðimfÞ (1)
where A1(y) represents the amplitude profile of the vortex beam.In our study, we have assumed a vortex beam with uniformamplitude. Following Richards and Wolf [25], the field distribu-tion in the focal volume of a high NA optical system is given by the
o
rP
θ
φP
φ θP
z
xP
yP
x
y
ρ
Fig. 1. Schematic representation of the focusing system.
diffraction integral (also known now as Debye–Wolf integral) as
~EðPÞ ¼ �ik
2p
ZZO
~aðsx; syÞ
szexp½ikfFðsx; syÞ �
~S �~rðPÞg�dsx dsy (2)
where a is a strength factor, r(P) is the radius vector connectingthe point P with the Gaussian focus which is also the origin of thecoordinate system (Fig. 1), s ¼ (sx, sy, sz) is the direction vector of atypical ray in the image space, F is the aberration function thatdenotes the deviation of the actual wavefront from the ideal one,and k( ¼ 2p/l). The integral is taken over the entire surface of thewavefront leaving the exit pupil. In the presence of the aberration,radial distance of any point on the wavefront depends on theangular coordinates, and for spherical aberration written [32] as
rðy;fÞ ¼ f þ Asr4 þ Adr2
~rðPÞ ¼ rðPÞfsin yP cos fP
_
iþ sin yP cos fP
_
jþ cos yP
_
kg
F ¼ Asr4 þ Adr2 (3)
where r ¼ sin y/sina is the zonal radius, f is the focal length of theoptical system, As and Ad are, respectively, the spherical aberrationand defocusing coefficient in units of wavelength, and (rP, yP, fP)are position coordinates 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 [32] as
sx ¼1
s sin y cos f�1
r
@r
@ycos y cos f
� �
sy ¼1
ssin y sin f�
1
r
@r
@ycos y sin f
� �
sz ¼1
s cos yþ1
rsin y
@r
@y
� �(4)
where s is the normalization factor given as
s ¼ 1þ1
r2
@r
@y
� �2( )" #1=2
Using the binomial expansion and ignoring the higher orderderivative of the position vector due to small value of theaberrations, [32] we write
1
sffi 1�Yþ oðYÞ
where
Y ¼1
2r2
@r
@y
� �2" #
Using the approach of Richards and Wolf [25] for the linearlypolarized beam and Visser and Wiersma [31] for the arbitrarypolarization, one can derive the contribution of the polarizationfactor in the evaluation of the strength factor a.
aPðy;fÞ ¼ ð~s �~kÞ1=2=ðs2x þ s2
y Þ
Aðs2y þ s2
x szÞ þ Bð�sxsy þ sxsyszÞ
Að�sxsy þ sxsyszÞ þ Bðs2x þ s2
yszÞ
Aðð�sxÞ þ Bð�syÞ� �
ðs2x þ s2
y Þ
2664
3775 (5)
or
aP ¼ A2ðyÞPðy; fÞ
where
A2ðyÞ ¼ ð~s �~kÞ1=2¼ ðszÞ
1=2
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R.K. Singh et al. / Optics and Lasers in Engineering 47 (2009) 831–841 833
and
Pðy;fÞ ¼1
ðs2x þ s2
y Þ
Aðs2y þ s2
x szÞ þ Bð�sxsy þ sxsyszÞ
Að�sxsy þ sxsyszÞ þ Bðs2x þ s2
yszÞ
Að�sxÞ þ Bð�syÞ� �
ðs2x þ s2
y Þ
2664
3775
Exðvx;vyÞ
Eyðvx;vyÞ
Ezðvx;vyÞ
2664
3775 ¼ ð�if=lÞ
Z a
0
Z 2p
0A2ðyÞ expðimfÞ
1
ðs2x þ s2
y Þ
ðs2y þ s2
x szÞ � ið�sxsy þ sxsyszÞ
ð�sxsy þ sxsyszÞ � iðs2x þ s2
yszÞ
ð�sxÞ � ið�syÞ� �
ðs2x þ s2
y Þ
26664
37775
exp ik Fþ Y~n�1
s~F
� ��~rðPÞ
� �� � exp �i
v
sinasin y cosðf� fPÞ
h ijJjdf dy (7)
Here A(y, f), B(y, f) are, respectively, the strengths of the x-, andy-polarized input beams , A2(y) corresponds to the apodizationfactor equal to cos1/2y for an aplanatic lens [25]. In the aberration-free case, the position coordinates at the wavefront are indepen-dent of the polar and radial coordinates, and in this situation theunit normal to the wavefront is transformed into the case ofRichards and Wolf [25]. The polarization distribution in theaberration-free case is transformed into polarization matrix ofHelseth [28]. For aberration-free case and x-polarization (B ¼ 0),the polarization distribution at the exit pupil is transformedinto results of Richards and Wolf [25]. Expression for the unitnormal to the aberrated wavefront can be written [32] as thesum of two terms representing the vector along the unit normal tothe ideal wavefront, and the other representing the deviationsuch as
~S ¼ ð1�YÞ~nþ1
s~Fðy; fÞ
where
~n ¼ sin y cos f_
iþ sin y sin f_
jþ cos y_
k
and
Fxðy; fÞ ¼ �1
r
@r
@ycos y cos fþ
1
r sin y@r
@fsin f
� �
Fyðy; fÞ ¼ �1
r
@r
@ycos y sin f�
1
r sin y@r
@fcos f
� �
Fzðy; fÞ ¼1
rsin y
@r
@y
� �
Expressing the dsxdsy as a function of dydf and using Eqs.(3)–(5), Eq. (2) can be written as
Eðu; vÞ ¼ �ikf
2p
Z a
0
Z 2p
0A2ðyÞ expðimfÞPðy; fÞ
� exp ik Fþ Y~n�1
s~F
� ��~rðPÞ
� ��
� exp �iu cos ysin2 a
� �exp �i
v sin ysin a
� �cosðf� fPÞ
� �jJjdfdy
(6)
where the optical coordinates (v, u) are defined as
v ¼ krP sin yP sin a
u ¼ krP cos yP sin2 a
and Jacobian is given as |J| ¼ ((qsx/qy)(qsy/qf)�(qsx/qf)(qsy/qy)).
Exðvx;vyÞ
Eyðvx;vyÞ
Ezðvx;vyÞ
264
375 ¼ ð�if=lÞ
R a0
R 2p0 A2ðyÞ expðimf
exp ik Fþ Y~n� 1s~F
��~rðPÞ
h in
Eqs. (1)–(6) are used to evaluate the field distribution in thefocal region of an optical system for any polarization distributionof the input beam. For an x-polarized beam (B ¼ 0 and A ¼ 1 ), thecomplex amplitude at the focal plane (u ¼ 0) is given as
Iðvx; vyÞ ¼ jExj2 þ jEyj
2 þ jEzj2 (8)
where I(vx, vy) is the total intensity distribution in the focal plane,and vx,y ¼ k[xP, yP]sina are the optical coordinates in the focal plane.Ex, Ey, and Ez are, respectively, the optical field components in x-, y-,and z- directions. From Eq. (8), the total intensity is proportional tothe sum of the squares of the x-, y-, and z polarized components.
The vortex beam with m ¼ 1 (or �1) produces |Ex|2¼ 0, |Ey|2 ¼ 0,
and |Ez|2a0 at v ¼ 0, and this leads to residual intensity at the focal
point, whereas the beam with m ¼ 2 (or�2) produces |Ex|2a0, |Ey|2a0, and |Ez|
2¼ 0 at v ¼ 0, which produces non-zero central intensity at
the focal plane. On the other hand, intensity at the center of the focalregions remains zero for higher value of the topological charge.
In order to deal with the circularly polarized beam, weconsider the complex amplitude of the input beam as
E0ðy; fÞ ¼ eimfð~E1 þ eij~E2Þ (9)
where ~E1 and ~E2 are, respectively, the electric fields due to x- andy-polarized helical beams, and j is the phase delay between thex- and y-polarizations. With an appropriate choice of the phasedelay j in Eq. (9), the sum of ~E1 and ~E2 produces a circularlypolarized vortex beam provided |E1| ¼ |E2|. Cases j ¼ +p/2 andj ¼ �p/2 correspond to the left circular (LC) and right circular(RC) polarizations, respectively. The field distribution of circularlypolarized vortex beam at the focal plane is obtained bysubstituting Eq. (9) with p/2 phase delay into Eq. (6)
Exðvx;vyÞ
Eyðvx;vyÞ
Ezðvx;vyÞ
264
375 ¼ ð�if=lÞ
R a0
R 2p0 A2ðyÞ expðimfÞ 1
ðs2xþs2
y Þ
ðs2y þ s2
x szÞ
ð�sxsy þ sxsyszÞ
�sxðs2x þ s2
y Þ
2664
3775
exp ik Fþ Y~n� 1s~F
��~rðPÞ
h in o� exp �i v
sin a siny cosðf�fPÞ� �
jJjdf dy
(10)
The LC-, and RC-polarized beams correspond to the positive andnegative sign column matrix, respectively, in Eq. (10). In theaberration-free case, zero central intensity is generated for the LCvortex beam for all the values of the NA. However, the RC-polarized vortex beam creates strong axial components [4] forunit topological charge, and non-zero transverse electric fieldcomponents for the double topological charge.
Field distribution in the focal region for an azimuthallypolarized beam can be evaluated by substituting A ¼ sinf andB ¼ �cosf in the polarization distribution [28] and subsequentlyusing Eq (6). Hence,
Þ 1ðs2
xþs2y Þ
sin fðs2y þ s2
x szÞ þ ð� cos fÞð�sxsy þ sxsyszÞ
sin fð�sxsy þ sxsyszÞ þ ð� cos fÞðs2x þ s2
yszÞ
sin fð�sxÞ þ ð� cos fÞð�syÞ� �
ðs2x þ s2
y Þ
2664
3775
o� exp �i v
sin a sin y cosðf� fPÞ� �
jJjdf dy
(11)
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The field distribution at the focal plane for the azimuthallypolarized beam possesses only the transverse polarizationcomponents and contribution of the longitudinal polarizationcomponent vanishes.
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m ¼ 2 and As (d) 0.0, (e) 0.5, and (f) 1.0.
3. Results and discussion
The complex amplitudes for linearly and circularly polarizedvortex beams has been obtained by Eqs. (7) and (10), respectively.
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ith a ¼ 751with m ¼ 1 and spherical aberration As (a) 0.0, (b) 0.5, and (c) 1.0; with
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The results of aberration-free case for linearly polarized beam arecompared with the results of Ganic et al. [9] and found to be ingood agreement. Results calculated by using Eq. (7) for a non-vortex beam (i.e., m ¼ 0) and zero aberration agree with theresults of Richards and Wolf [25] and Chon et al. [27]. Goodagreement is also noticed between our results and those of Visserand Wiersma [31] and Kant [32] for a non-vortex beam in thepresence of spherical aberration. In the low NA case, results tendtowards the results obtained by scalar diffraction theory [21,22].In this paper, we have presented all the results for a lens of focallength equal to 5000 l.
3.1. Linearly polarized vortex beam
Fig. 2 represents results of the total intensity distribution andcontour plots in the focal plane of a lens with a ¼ 751 for an x-
polarized vortex beam with two values of the topological charge.The results for the beams with m ¼ 1 and 2 are shown in Fig.2(a–f) respectively for three values of As. Results are normalizedwith respect to the maximum intensity for the aberration-freecase. Tight focusing of a linearly polarized vortex beam producestwo side lobes, and their positions with respect to the centerdepend on the NA. The intensity peaks in the side lobes areorthogonal to the direction of the incident beam polarization. Sidelobes are more compressed towards the focal point and theresidual intensity increases with an increase in the angle ofconvergence. For a beam with m ¼ 1 and NA ¼ 1, the focal pointintensity is approximately 48.8% of the maximum intensity [9],while for NA ¼ 0.965 (Ea ¼ 751) the corresponding value isapproximately 43.6%.
The intensity distribution in the focal plane undergoessignificant changes in the presence of spherical aberration. Thepeak intensity region with side lobes undergoes a rotation aboutthe center, and the side lobes are more dispersed in comparison to
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Fig. 3. Intensity distribution of a LC-polarized vortex beam with m ¼ 1 in th
those for the aberration-free case. The residual intensity at thefocal point also decreases with an increase in As. Figs. 2(d–f) showthe intensity distribution and contour lines for a beam with m ¼ 2.Intensity peaks and side lobes for double topological charge aresituated at a distance larger than those for m ¼ 1 case. Positions ofthe side lobes are changed, and the intensity distribution isdispersed in the presence of spherical aberration (Fig. 2e and f).
3.2. Circularly polarized vortex beam
j ¼7p/2 in Eq. (9) corresponds to the circular polarizationcase, in which the central intensity in the focal plane touches itslowest value for the LC case, and has a non-zero value for the RCcase. The LC-polarized beam with m ¼ 1 produces an intensitynull at the focal point because of the zero contribution from all thepolarization components. Results of the total intensity distribu-tion and for an LC-polarized beam with m ¼ 1 are shown in Fig. 3for As ¼ 0.0, 0.5, and 1.0. A rotationally symmetric dark core isproduced by focusing of the LC-polarized vortex beam, and thesize of the dark core depends on the topological charge and theNA. Intensity null at the focal point is maintained with a reductionof the peripheral intensity in the presence of aberration.Sharpness of the dark core decreases with an increase in thevalue of As. Aberration leads to a reduction of the total intensity atthe periphery of the dark core with a reduction in the intensitygradient.
Fig. 4(a–c) shows the distribution of |Ex|2, |Ey|2, and |Ez|2 in the
aberration-free case, whereas Fig. 4(d–f) shows the results forAs ¼ 0.5. Results for |Ex|2 and |Ey|2 possess maximum of intensitylobes along x-, and y-directions, respectively. On the other hand,the longitudinal component |Ez|
2 possesses rotationally sym-metric distribution with small amplitude in comparison to that forthe case of transverse polarization components. The effect ofaberration is more visible on the transverse polarization compo-
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Fig. 4. Distribution of the polarization components of a LC-polarized vortex beam in the focal region of a lens with a ¼ 751; and As ¼ 0.0 (a) |Ex|2, (b) |Ey|2, and (c) |Ez|2;
As ¼ 0.5 (d) |Ex|2, (e) |Ey|2, and (f) |Ez|2.
R.K. Singh et al. / Optics and Lasers in Engineering 47 (2009) 831–841836
nents in terms of the rotation of the maximum value of the |Ex|2
and |Ey|2. On the other hand, |Ez|2 maintains its rotational
symmetry even in the presence of aberration with a spread inthe distribution. The presence of circularly symmetric dark core in
the longitudinal polarization component arises due to thepresence of helical modes with double topological charge. Thisresult is in accordance with the results of conservation of angularmomentum [13].
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Fig. 5. Phase distribution of the longitudinal polarization component at the focal plane of a lens with a ¼ 751 for beam with m ¼ 0 and As (a) 0.0 and (b) 0.5; for m ¼ 1 and
As (c) 0.0 and (d) 0.5.
Fig. 6. Variation of dip ratio with defocusing of a linearly polarized vortex beam
with m ¼ 1 focused by a lens with NA ¼ 1.0 and As ¼ (a) 0.0, (b) 0.5, and (c) 1.0;
NA ¼ 0.965 and As (d) 0.0, (e) 0.5, and (f) 1.0; linearly polarized beam with m ¼ 2
focused by lens with NA ¼ 0.965 (a ¼ 751) and As (a0) 0.0, (b0) 0.5, and (c0) 1.0.
Fig. 7. Radial intensity profile of a LC-polarized vortex beam with m ¼ 1 focused
by a lens with a ¼ 751 and As ¼ 0.0 (a) Ad ¼ 0.0; As ¼ 0.5 (b) Ad ¼ �0.55 (c)
Ad ¼ 0.0; and As ¼ 1.0 (d) Ad ¼ 0.0.
R.K. Singh et al. / Optics and Lasers in Engineering 47 (2009) 831–841 837
Fig. 5 shows the phase distribution of the longitudinalpolarization component of the optical beam with m ¼ 0 and 1 atthe focal plane of a lens with a ¼ 751. The multiplicative factorwith negative sign outside of the integral has been includedin the phase evaluation. The phase profile (Fig. 5a) of the Gaussianbeam represents a helical structure and this represents thegeneration of the longitudinal optical vortex [13]. The presenceof aberration leads to the appearance of curvature in the phaseprofile (Fig. 5b). The phase structure of the longitudinalcomponent of the focused vortex beam is shown in Fig. 5c and dfor As ¼ 0.0 and 0.5, respectively. An increase in the multipleof 2p phase variation takes place and the longitudinal optical
vortex is transformed to the topological charge (m+1) in thelongitudinal polarization component. Presence of the additionaltopological charge arises due to the conversion of spin angularmomentum into the orbital angular momentum. A smallmismatch between the amplitudes of the x- and y-polarizationsof the input beam (i.e., elliptical polarization) results in a split ofthe dark core.
3.3. Aberration compensation
The intensity profiles of the linearly and circularly polarizedvortex beams are shown, respectively, in Figs. 6 and 7 in thepresence of spherical aberration and defocusing. The results arenormalized with respect to the maximum intensity for the
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R.K. Singh et al. / Optics and Lasers in Engineering 47 (2009) 831–841838
aberration-free case, and the normalized intensity at the center isdefined as ‘dip ratio’ [25] in Fig. 6. The residual intensity at thegeometrical focal point with respect to the maximum of the totalintensity for NA ¼ 1.0 in the aberration-free case is the same asreported by Ganic et al. [9]. The curves (a–c) represent thevariation of dip ratio ‘D’ with defocusing for a lens with NA ¼ 1.0for beam with m ¼ 1, whereas (d–f) and (a0–c0) show the resultsfor a lens with a ¼ 751 for beams with m ¼ 1 and 2, respectively.In the presence of aberration, the position of the maximum ‘dipratio’ shifts from the geometrical focal point, and the shiftincreases with an increase in As. Symmetry in the dip ratiodistribution also vanishes in the presence of aberration. The effectof aberration is reduced for a proper combination of thedefocusing and aberration. Maximum improvement in the ‘dipratio’ takes place at Ad ¼ �0.6 for As ¼ 0.5, and at Ad ¼ 1.2 forAs ¼ 1.0 for a vortex beam with m ¼ 1 focused by a lens withNA ¼ 1.0. A decrease in the NA results in the compensation of the
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Fig. 8. Distribution of the polarization components of a RC-polarized vortex beam in th
|Ez|2; As ¼ 0.5 (e) |E|2, (f) |Ex|2, (g) |Ey|2, and (h) |Ez|
2.
effect of spherical aberration towards the lower value of Ad.Maximum compensation takes place for Ad ¼ �As in case of a lowNA system, as in scalar theory. The value of the topological chargealso affects the aberration compensation condition. For example,the best-balanced condition is changes for a beam with m ¼ 2focused by a lens with a ¼ 751.
Aberration correction also takes place for a circularly polarizedbeam. Results of intensity profiles of an LC-polarized vortex beamare shown in Fig. 7. The intensity distribution possesses arotationally symmetric dark core surrounded by a high-intensityring. The full-width half-maximum (FWHM) for the aberration-free case (curve a) is approximately 2.5 ( ¼ Dv) and this increasesto approximately 2.8 for As ¼ 0.5 (curve c) and 5.6 (curve d) forAs ¼ 1.0. Effect of aberration on the intensity distribution isreduced with proper value of defocusing, and an improvement inthe peripheral intensity also takes place. For example, forAd ¼ �0.55 and As ¼ 0.5, the FWHM is approximately the same
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e focal region of a lens with a ¼ 751 and As ¼ 0.0 (a) |E|2, (b) |Ex|2, (c) |Ey|2, and (d)
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Fig. 9. Intensity distribution of azimuthally polarized beam at the focal plane of a lens with a ¼ 751 and As (a) 0.0, (b) 0.5, and (c) 1.0.
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u
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Fig. 10. Intensity distribution in the x–z plane of a lens with a ¼ 751 for a LC-polarized vortex beam with m ¼ 1; (a) As ¼ 0.0, (b) As ¼ 0.5, and (c) As ¼ 1.0; for azimuthally
polarized beam with m ¼ 0; (d) As ¼ 0.0, (e) 0.5, and (f) As ¼ 1.0.
R.K. Singh et al. / Optics and Lasers in Engineering 47 (2009) 831–841 839
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R.K. Singh et al. / Optics and Lasers in Engineering 47 (2009) 831–841840
as for the aberration-free case (curve b). The peripheral intensityalso increases to approximately 0.95.
Distributions of intensity and square of the polarized compo-nents at the focal plane, of an RC-polarized vortex beam withm ¼ 1, are shown in Fig. 8(a–d) for As ¼ 0.0 and Fig. 8(e–h) forAs ¼ 0.5. The total intensity distribution in the aberration-freecase is circularly symmetric with non-zero intensity dip at thefocal point. The focal point intensity arises due to the contributionfrom only the longitudinal polarization component, and non-zerovalue of the z-polarization components has a close relation withthe conservation of total angular momentum [13]. The transversepolarization components possess lobes around the center with nocontribution at the focal point. Aberration in the focusing systemreduces the intensity dip with a reduction in the intensity peaksand spreading of the pattern. This results due to decrease in thecontribution of longitudinal polarization component. Effect ofaberration on the transverse polarization components appears inthe form of rotation of side lobes from the orthogonal directionand its value also decreases.
0 2 4 60
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0 2 4 60
0.1
V0 V0
Fig. 11. Encircled energy at the focal plane of a lens with a ¼ 751; for an
azimuthally polarized beam with m ¼ 0 for As ¼ (a) 0.0 and (a0) 0.5; for LC-
polarized vortex beam with m ¼ 1 and As ¼ (b) 0.0 and (b0) 0.5; for RC-polarized
vortex beam with m ¼ 1 and As ¼ (c) 0.0 and (c0) 0.5; encircled energy at the focus
plane of a lens with a ¼ 751 for As ¼ (d) 0.0 and (d0) 0.5.
3.4. Azimuthally polarized beam
The total intensity distribution of the azimuthally polarizedbeam has been evaluated using Eq. (11). Only the transversepolarization components contribute to shaping of the intensitydistribution. The intensity for m ¼ 0, at the focal plane is shown inFig. 9a for an aberration-free case and in Fig. 9b and c for As ¼ 0.5and 1.0. The focal structure of the beam possesses doughnutshape, and size of the dark core is controlled by the focusingsystem. Dark core in the azimuthally polarized beam is sharperthan that of LC-polarized beam with m ¼ 1. Fig. 10 shows theintensity distributions on the (x, z) plane of a lens with a ¼ 751 foran aberration-free and aberrated cases, respectively, for a focusedLC-polarized singular beam with m ¼ 1, and azimuthally polarizednon-singular beam. A two-fold symmetric distribution with adark region in the center exists for an aberration-free condition(Fig. 10a). Position of the side intensity lobes depends on the valueof As. With an increase in the aberration, the intensity distributionis stretched with a positional shift of the peak, and also areduction in the peak intensity (Fig. 10b and c). A two-foldsymmetry in the intensity distribution on the x–z plane also existsfor an azimuthally polarized non-singular beam (Fig. 10d).However, the intensity lobes are orthogonal to the direction oflobes in case of the LC-polarized singular beam. In presence ofaberration, intensity lobes shift from the center along withspreading (Fig. 10e and f).
To get more information about the effect of aberration andpolarization distribution, we have calculated ‘encircled energy’which is defined as the total energy lying within a circle ofspecified radius at the aberration plane. Encircled energy isdefined as
Eðn0Þ ¼
Z 2p
0
Z n0
0Iðv; fÞv dv df (12)
Results of the encircled energy at the focal plane of high and lowNA systems are presented in Fig. 11 for various polarizationdistributions and topological charge. Curves a and a0 represent theencircled energy of the azimuthally polarized non-singular beam(m ¼ 0) at the focal plane for As ¼ 0.0 and 0.5. Encircled energy fora unit charged LC-polarized singular beam is shown by curves band b0 for As ¼ 0.0 and 0.5. Sharp variation of the encircled energyfor curve a and comparison to that of b verifies that the doughnutstructure of an azimuthally polarized beam is more confined incomparison to that of an LC-polarized singular beam. Decrease invariation of the encircled energy in aberrated case results due to
an increase in the size of dark core. Encircled energy for anRC-polarized singular beam with unit topological charge is shownin curves c and c0 for As ¼ 0.0 and 0.5, respectively. Encircledenergy variation is also rapid for RC-polarized singular beam withunit topological charge due to non-zero intensity distributionaround the center. However, this decrease is due to a spread in thestructure in the aberrated case (curve c0). With reduction of conicangle, results of encircled energy for aberration-free and aberratedcases (curve d and d0) match with our results of encircled energyobtained by scalar diffraction theory [21]. Smallest dark core withhigh-intensity gradient in the diffraction pattern of the azimuth-ally polarized beam is well suited for application in the STEDmicroscopy as an erase beam. Presence of spherical aberration inthe system reduces intensity gradient, with a spread in itsdistribution. However, the dark core in the LC-polarized beamwith m ¼ 1 is sharper in comparison to that of azimuthallypolarized beam with As ¼ 1.0. Reduction in the intensity gradientdue to aberration has impact on the fluorescent spot of STEDmicroscope. Impact of aberration on the dark core of the beam canbe analyzed in the context of its application in the STEDmicroscopy using either the LC-polarized or an azimuthallypolarized beam as the erase beam [5].
4. Conclusion
We conclude that the presence of spherical aberration resultsin a reduction of the residual intensity at the focal point and aspread of the peak intensity region in the side lobes in case of alinearly polarized beam. The position of side lobes also changes inthe presence of aberration. Intensity distribution of circularlypolarized beam maintains its rotational symmetry along with a
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R.K. Singh et al. / Optics and Lasers in Engineering 47 (2009) 831–841 841
reduction in the sharpness of the dark core. The impact ofspherical aberration on the circularly polarized beam is morevisible for the transverse component of the field; whereas thelongitudinal field component maintains its shape with appear-ance of secondary rings. Tight focusing of the circularly polarizedbeam also results in the generation of the longitudinal opticalvortex, but this ceases to exist for the elliptically polarized inputbeam. Effect of spherical aberration on the intensity distributioncan be compensated by choosing a proper value of defocusing, andthis compensation changes with the numerical aperture and thetopological charge of the beam.
Acknowledgments
Rakesh Kumar Singh is thankful to the Council of Scientific andIndustrial Research (CSIR) India for the award of a ResearchFellowship. Financial support from the Department of Science andTechnology (DST), India under the Grant SR/S2/LOP-10/2005 isalso acknowledged.
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