structure of a tightly focused vortex beam in the presence of primary coma
TRANSCRIPT
Optics Communications 282 (2009) 1501–1510
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Optics Communications
journal homepage: www.elsevier .com/locate /optcom
Structure of a tightly focused vortex beam in the presence of primary coma
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 28 August 2008Received in revised form 22 November 2008Accepted 28 November 2008
Keywords:Optical vortexTopological chargePolarizationNumerical apertureIntensity distribution
0030-4018/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.optcom.2008.11.085
* Corresponding authors. Tel.: +91 11 26596580 (RE-mail addresses: [email protected] (R.K. S
c.in (K. Singh).
a b s t r a c t
Structure of a tightly focused circularly polarized vortex beam in the presence of primary coma has beenstudied by using Debye–Wolf diffraction integral. The role of topological charge and handedness of thepolarization on the focused structure has been discussed. Results have been presented for the total inten-sity and squares of the polarization components for left-, and right circularly polarized vortex beams.Impact of coma, on the dark core of an azimuthally polarized non-vortex beam is also investigated andcompared with the dark core of a circularly polarized vortex beam. The presence of comatic aberrationin the focusing system results in a positional shift of the high intensity lobes, and reduction of the inten-sity on one side of the center. Effect of coma on the focused structure has also been discussed in the con-text of STED microscopy.
� 2008 Elsevier B.V. All rights reserved.
1. Introduction
Structure of the focused optical beams by high-numerical-aper-ture (NA) systems has been an area of extensive studies for a longtime [1–23]. With increasing number of new applications, it be-comes important to investigate the focused structure of the beamby manipulating the complex amplitude profile, and polarizationdistribution of the incident beam. The polarization distribution ofthe beam can be ignored for the low NA systems in contrast tothe high NA systems, where polarization state of the beam is takeninto consideration in the diffraction integral [1]. Focus shaping byconsidering the polarization distribution of the input beam hasalso attracted considerable attention in recent times [3–5]. The un-ique structure of the vortex beams has also been used for the focusshaping [6]. A proper combination of the vortex charge, and hand-edness of the circular polarization decides the focal structure of thebeam [19–23]. The uniqueness of the beam is due to the presenceof singular point in the wave field, where the phase is undefined[24–26]. In the wave field, the phase singularity gives a helicalphase structure around the point of singularity, and the accumu-lated phase variation around this point is 2pm, where m is referredto as the topological charge [24,25]. Diffraction pattern of the beampossesses a dark core due to the destructive interference of second-ary waves which are out of phase.
The presence of dark core in the tightly focused vortex beam de-pends on the polarization distribution of the input beam [19–22]. A
ll rights reserved.
.K. Singh).ingh), [email protected]
tightly focused linearly polarized vortex beam gives non-zero focalpoint intensity due to longitudinal and transverse polarizationcomponents for beams with m = 1 and 2, respectively [6]. However,a tightly focused left circularly (LC) polarized singular beam withm = 1 shows a circularly symmetric dark core [19–21]. The polari-zation characteristics of the beam change in the high NA focusing,and a circularly polarized beam does not remain purely circularlypolarized in the focal plane [10]. The tight focusing of the circularlypolarized vortex beam can also be decomposed in the aberration-free case into three components, each with uniform polarizationin such a way that the total angular momentum due to polarizationand vortex is conserved in each component [10]. Impact of thespace-variant geometric phase on the high-numerical-aperturefocusing of light has also drawn attention in recent years [11].The geometric phase results in the formation of mode with helici-ties and phase singularities that differ from those of the originalbeam [11]. Effect of the tight focusing on the spin-to-orbital angu-lar momentum conversion has also been investigated [22].
Size and shape of the dark core in the diffraction pattern of thevortex beam plays an important role in many applications; forexample the size of the focused annular ring influences the transferof mechanical strength to the trapped particles [26], and to givesmallest fluorescent spot in the case of stimulated emission anddepletion microscopy [19,21]. Interest has been shown to achievethe smallest dark core using the azimuthal polarization and filters[9]. Presence of aberration in the focusing system counterbalancesall the efforts made to achieve smallest dark core. Dark core ishighly sensitive to nonrotational symmetric aberrations relativeto focused structure of a normal beam and small distortion in thedoughnut structure is enough to degrade its utility [27]. Therefore,
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it is important to investigate the structural change in the focusedstructure of the dark core in the presence of comatic aberration.In addition, it is also important to investigate the susceptibilityof the dark core against polarization and topological charge inthe presence of aberrations in high NA focusing. Effects of primaryaberrations on the focused structure of a non-vortex beam, andradially polarized doughnut beam for high NA systems have beeninvestigated in the past [28–30]. Braat et al. [31] used extendedNijboer–Zernike representation of the vector field for evaluatingintensity in the focal region of an aberrated high-aperture opticalsystem. In view of the importance of the high NA focusing of vortexbeams, and their structure in the focal region, we have undertakena comprehensive study on the subject [32–34]. In this paper, wepresent the results of our investigations on the effect of primarycoma on the dark core of a vortex and an azimuthally polarizedbeam embedded into uniform background. Significance of thetopological charge and polarization distribution on the dark corehas been investigated in the presence of aberration. Finally impactof aberration on the focused structure has been discussed in thecontext of STED microscopy.
2. Theoretical background
We have considered the coordinate geometry shown in Fig. 1.The complex amplitude of the input beam is considered as:
E0ðq;/Þ ¼A
B
� �expðim/Þ; ð1Þ
where q is the normalized radial coordinate around the optical, u isthe polar coordinate in the plane perpendicular to the beam axis, m
is the topological charge, and AB
� �represents the polarization
strength of the beam. The field distribution in the focal plane ofthe focusing system is given [1] as:
~EðPÞ ¼ � ik2p
ZZX
~aðsx; syÞsz
exp½ikfUðsx; syÞ �~s �~rðPÞg�dsxdsy; ð2Þ
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 ¼ ðsx; sy; szÞ is the direction vector of a typ-ical ray in the image space, U is the path aberration function whichdenotes the deviation of the actual wavefront from the ideal one,and k ¼ 2p=k. The integral is taken over the entire surface of thewavefront leaving the exit pupil. In the presence of the aberration,
Fig. 1. Geometric configuration; h is conic angle, u and uP are the azimuthal anglesin the input and observation planes respectively, (rP, hP, uP) are position coordinateson the observation plane.
radial distance of any point on the wavefront depends on the angu-lar coordinates and for comatic aberration is written [29] as:
rðh;/Þ ¼ f þ Acq3 cos /;
~rðPÞ ¼ rðPÞðsin hP cos /P iþ sin hP sin /P jþ cos hPkÞ;U ¼ Acq3 cos /;
ð3Þ
where q ¼ sin hsin a is zonal radius, f is the focal length of the optical sys-
tem, a is the maximum angle of convergence (i.e. hmax = a), Ac is theaberration coefficient in units of wavelength, ði; j; kÞ are unit basevectors in the Cartesian coordinate systems, and ðrP ; hP ;/PÞ are po-sition coordinates of a point P on the observation plane. Using theconcept of two orthogonal tangent vectors in the polar and azi-muthal directions, the unit normal to the aberrated wavefront isgiven [29] as:
sx ¼1r
sin h cos /� 1r@r@h
cos h cos /þ 1r sin h
@r@/
sin /
� �;
sy ¼1r
sin h sin /� 1r@r@h
cos h sin /� 1r sin h
@r@/
cos /
� �;
sz ¼1r
cos hþ 1r
sin h@r@h
� �;
ð4Þ
where r is the normalization factor given as:
r ¼ 1þ 1r2
@r@h
� �2
þ 1
sin2 h
@r@/
� �2( )" #1=2
:
Using the binomial expansion and ignoring the higher orderderivative of the position vector due to small value of the aberra-tions [29], we write
1rffi 1�Hþ oðHÞ;
where H ¼ 12r2
@r@h
� �2 þ 1sin2 h
@r@/
� �2�
and o(H) represents other higherorder terms.
Using the approach of the Richards and Wolf [1], one can derivethe contribution of the polarization factor in the evaluation ofstrength factor a
aPðh;/Þ ¼ ðs � kÞ1=2=ðs2
x þ s2yÞ
Aðs2y þ s2
x szÞ þ Bð�sxsy þ sxsyszÞAð�sxsy þ sxsyszÞ þ Bðs2
x þ s2y szÞ
Að�sxÞ þ Bð�syÞ �
ðs2x þ s2
yÞ
2664
3775ð5Þ
or aP ¼ A2ðhÞPðh;/Þwhere A2ðhÞ ¼ ðs � kÞ1=2 ¼ ðszÞ1=2, and note from Eq. (4), sx, sy arefunction of h and u
Pðh;/Þ ¼ 1ðs2
x þ s2yÞ
Aðs2y þ s2
x szÞ þ Bð�sxsy þ sxsyszÞAð�sxsy þ sxsyszÞ þ Bðs2
x þ s2yszÞ
Að�sxÞ þ Bð�syÞ �
ðs2x þ s2
yÞ
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3775:
Here A(h, u) and B(h, u) represent the strengths of the x-, andy-polarized input beams respectively, A2(h) corresponds to the apo-dization factor equal to cos1/2h in an aberration-free case [1] inwhich the position coordinates at the wavefront are independentof the polar and radial coordinates. In this situation, the unit nor-mal to the wavefront is transformed to the case of Richards andWolf [1]. The polarization distribution in the aberration-free caseis transformed into polarization matrix given by Helseth [9].Expression for the unit normal to the aberrated wavefront can bewritten [29] as the sum of two terms representing the vector alongthe unit normal to the ideal wavefront and other representing thedeviation such as:
~s ¼ ð1�HÞ~nþ 1r~Fðh;/Þ
R.K. Singh et al. / Optics Communications 282 (2009) 1501–1510 1503
where ~n ¼ sin h cos /iþ sin h sin /jþ cos hk
Fxðh;/Þ ¼ � 1r@r@h cos h cos /þ 1
r sin h@r@/ sin /
� �and Fyðh;/Þ ¼ � 1
r@r@h cos h sin /� 1
r sin h@r@/ cos /
� �Fzðh;/Þ ¼ 1
r sin h @r@h
� �
9>>>=>>>;:
Expressing dsxdsy as a function of dhdu and using Eqs. (3)–(5), Eq. (2)can be written as:
~Eðu;vÞ ¼ � ikf2p
Z a
0
Z 2p
0A2ðhÞ expðim/ÞPðh;/Þ
� exp ik Uþ ðH~n� 1r~FÞ �~rðPÞ
� � � exp �i
u cos h
sin2 a
� �
� exp �iv sin hsin a
� �cosð/� /PÞ
� jJjd/dh;
ð6Þ
where the optical coordinates (u, v) are defined as:
v ¼ krP sin hP sina;
u ¼ krP cos hP sin2 a
and Jacobian is given as: jJj ¼ @sx@h
@sy
@/ �@sx@/
@sy
@h
� �.
2.1. Linearly polarized beam
In case of an x- polarized vortex beam (A = 1, B = 0), the complexfield distribution in the focal plane (u = 0) can be written as:
Exðvx;vyÞEyðvx; vyÞEzðvx;vyÞ
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375 ¼ ð�if=kÞ
Z a
0
Z 2p
0A2ðhÞ
� expðim/Þ 1ðs2
x þ s2yÞ
ðs2y þ s2
x szÞð�sxsy þ sxsyszÞ�sxðs2
x þ s2yÞ
264
375
� exp ik½Uþ ðH~n� 1r~FÞ �~rðPÞ�
�
� exp½�iv
sinasin h cosð/� /PÞ�jJjd/dh:
ð7Þ
Also, Iðvx;vyÞ ¼ jExj2 þ jEyj2 þ jEzj2, where Iðvx;vyÞ is the totalintensity distribution in the focal plane, and vx;y ¼ k½xP; yP � sin aare the optical coordinates in the focal plane. xP and yP are theCartesian coordinates in the image plane (Fig. 1). Ex, Ey, and Ez
are respectively the optical field components in the x-, y-, and z-directions. From Eq. (7), the total intensity is proportional to thesum of the squares of the x, y, and z polarized components. The vor-tex beam with m = 1 (or �1) produces residual intensity at the fo-cal point due to longitudinal polarization component, whereas thebeam with m = 2 (or �2) produces such an intensity due to trans-verse polarization components. On the other hand, intensity at thecenter of the focal region remains zero for higher values of thetopological charge.
2.2. Circularly polarized beam
In order to study the circularly polarized beam, we consider aninput field in the form:
E0ð/Þ ¼E1
e�uE2
� �expðim/Þ ð8Þ
where E1 and E2 are the electric fields due to x-, and y- polarizedbeams respectively, and u the phase delay between them. Withan appropriate choice of the phase delay, Eq. (8) produces an ellip-tically polarized beam. In case of |E1| = |E2|, i.e A = B and u ¼ p=2,
the input field is termed left circularly (LC) polarized or right circu-larly (RC) polarized beam for positive and negative phase delayrespectively.
The field distribution of a circularly polarized vortex beam atthe focal plane is given as:
Exðvx;vyÞEyðvx;vyÞEzðvx;vyÞ
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375 ¼ ð�if=kÞ
Z a
0
Z 2p
0A2ðhÞ
� expðim/Þ 1ðs2
x þ s2yÞ
ðs2y þ s2
x szÞ � ið�sxsy þ sxsyszÞð�sxsy þ sxsyszÞ � iðs2
x þ s2y szÞ
ð�sxÞ � ið�syÞ �
ðs2x þ s2
yÞ
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� exp ik½Uþ ðH~n� 1r~FÞ �~rðPÞ�
�
� exp �iv
sinasin h cosð/� /PÞ
h ijJjd/dh: ð9Þ
Using the trigonometric relation in the aberration-free case, it isnoticed that the contribution of all the polarization components iszero at the focal point for an aberration-free LC polarized vortexbeam with m = 1, whereas the intensity distribution of the RCpolarized vortex beam with m = 1 possesses non-zero focal pointintensity due to contributions from the longitudinal polarizationcomponents. The longitudinal polarization components for the LCbeam with m = 1 give azimuthal phase factor with double topolog-ical charge, whereas the azimuthal phase factor of the RC polarizedbeam vanishes for m = 1. The focal point intensity arises for non-vortex LC and RC beams due to contributions from the x-, and y-polarized components. Intensity null at the focal plane occurs fora circularly polarized beam provided that the spin angular momen-tum of the photons related to the handedness of circular polariza-tion and the orbital angular momentum of the beam related to thetopological charge point in the same direction [20].
2.3. Azimuthally polarized beam
Field distribution in the focal plane (u = 0) for an azimuthallypolarized beam can be evaluated by substituting A = sinu andB = �cosu in the polarization distribution [9] and subsequentlyusing Eq. (6). Hence
Exðvx;vyÞEyðvx;vyÞEzðvx;vyÞ
264
375 ¼ ð�if=kÞ
Z a
0
Z 2p
0A2ðhÞ expðim/Þ 1
ðs2x þ s2
yÞ
�sin /ðs2
y þ s2x szÞ þ ð� cos /Þð�sxsy þ sxsyszÞ
sin /ð�sxsy þ sxsyszÞ þ ð� cos /Þðs2x þ s2
yszÞsin /ð�sxÞ þ ð� cos /Þð�syÞ �
ðs2x þ s2
yÞ
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3775
� exp ik½Uþ ðH~n� 1r~FÞ �~rðPÞ�
�
� exp �iv
sin asin h cosð/� /PÞ
h ijJjd/dh: ð10Þ
The field distribution at the focal plane for the azimuthallypolarized beam possesses only the transverse polarization compo-nents and contribution of the longitudinal polarization componentvanishes. Using the trigonometric identities for the integrationwith respect to azimuthal coordinates, the x-, y-, and z-polarizationcomponents in the aberration-free case are given as
Exðu; vÞ ¼ ½�i2pimþ1eiðmþ1Þ/IAmþ1 þ i2pim�1eiðm�1Þ/IA
m�1�;Eyðu; vÞ ¼ �½2pimþ1eiðmþ1Þ/IA
mþ1 þ 2pim�1eiðm�1Þ/IAm�1�;
Ezðu; vÞ ¼ 0;
ð11Þ
1504 R.K. Singh et al. / Optics Communications 282 (2009) 1501–1510
where
IAm�1ðu;vÞ ¼ ð�if=kÞ
Z a
0cos1=2 h sin hJm�1
vsina
sin h� �
� exp �iu
sin2 acos h
� dh
and Jmð�Þ is the Bessel function of the first kind, order m.An intensity null exist at the focal point in the absence of any
phase singularity in the wave field (m = 0). However, a beam withunit topological charge produces non-zero focal point intensity dueto contributions from the x-, and y- polarization components. Forother values of the topological charge, the intensity null at the focalpoint is maintained.
3. Numerical results and discussion
The complex amplitude of a circularly polarized vortex beam inthe focal plane of a lens system is obtained by Eq. (9), which is
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Fig. 2. Intensity distribution of an LC polarized vortex beam (m = 1) in the focal plane of a0.5 (g) 1.0 (h) 1.5.
numerically evaluated by applying point by point integration usingMATLAB. For implementation of the numerical technique, we havedivided the integration domain into radial and azimuthal zones.The radial zone corresponds to the sampling of the conic angle.The complex function in the sampling interval can be expandedusing linearization of the function. Due to small sampling interval,the first and higher order gradients of the amplitude and phasefunction can be ignored and only mid point value of the functionis taken into consideration. We have taken 125 samples in eachzone. Near the vortex core, 125 sample points are sufficient to takecare of the high phase gradient of the azimuthal phase term. Theradial dependence is taken care of by the number of zones usedin the computation. The results of intensity distribution are pre-sented for a lens with a ¼ 75; and for Ac = 0.0, 0.5, 1.0, and 1.5. Re-sults are normalized by maximum value of the total intensity inthe aberration-free case. The results of vortex beam in the aberra-tion-free case are compared with the results of Iketaki et al. [12],Török and Munro [19], and Bokor et al. [20,21] for the correspond-
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R.K. Singh et al. / Optics Communications 282 (2009) 1501–1510 1505
ing special case of non-uniform amplitude, and good agreement isfound between them. For non-singular beam (m = 0), the resultsare compared with the results of Bomzon [10] and have been foundto be in good agreement. In this paper we have presented all theresults for a lens of focal length equal to 5000k.
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Fig. 3. Distribution of squares of the polarization components of the LC polarized vortex b|Ez|2; and Ac = 1.5 (d) |Ex|2 (e) |Ey|2 (f) |Ez|2.
3.1. LC polarized vortex beam
Fig. 2 shows results of the total intensity distribution for an LCpolarized vortex beam. The intensity distribution possesses anintensity null at the focal point with a circularly symmetric high
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1506 R.K. Singh et al. / Optics Communications 282 (2009) 1501–1510
intensity ring for the aberration-free case. Presence of the comaticaberration results in a positional shift of the dark core, and reduc-tion of intensity of the ring on one side. Also the intensity ringssplit into side lobes. With an increase in the value of Ac, the highintensity lobes come closer, and the dark core is stretched alongwith stretching of the high intensity regions. With increasing valueof Ac, the dark core is transformed into a triangular shape. Intensitydistribution of the double topological charge beam possesses acentral dark core, which is larger in size, in comparison to the darkcore for the beam with m = 1. For double topological charge, thedark core splits into two parts with an increase in the value ofAc. The positional shift of the dark core also depends on the topo-logical charge and it increases with an increase in the value of m.
The distributions of the squares of the polarization components|Ex|2, |Ey|2, and |Ez|2 are shown in Figs. 3 and 4 for beam with m = 1and 2, respectively. The intensity null at the focal point for the LCpolarized beam arises due to the zero contribution of all the polar-ization components. Due to high amplitudes, the transverse polar-ization components play an important role in shaping the totalintensity distribution. The distributions of |Ex|2, and |Ey|2possesstwo high amplitude peaks with equal amplitude, but positions ofthe peaks are interchanged in the transverse direction for x-, andy- polarization. However, distribution of and |Ez|2 possesses a cir-cularly symmetric ring with a dark core in the center. Size of thedark core for the longitudinal polarization component is largerthan the size of dark region for the transverse polarizationcomponents.
A positional shift in the high amplitude lobes of the transversepolarization components is observed in the presence of coma. Oneof the amplitude lobes of |Ex|2 diminishes, whereas the high ampli-tude lobes of the y- polarization components come closer with anincrease in Ac. For Ac = 1.5 (Fig. 3d and e), the dark core in the trans-verse polarization components is transformed into triangularshape. Positional shift and reduction in the intensity of the ringalso result for the longitudinal polarization component. ForAc = 1.5, the dark core of the longitudinal polarization componentsappears to be separated into two regions even for beam with m = 1(Fig. 3f). The squares of the polarization components for the beamwith m = 2 are shown in Fig. 4 for Ac = 0.0, 0.5, and 1.5. Fig. 4a–cshow the results for an aberration-free case, whereas Fig. 4d–fshow results for the aberrated case. Presence of the aberration inthe system produces a positional shift with a reduction of theamplitude in one direction (Fig. 4d–f). For Ac = 1.5, the dark corein the distribution of the squares of the transverse polarizationcomponents (Fig. 4d and e) is separated into two regions, and intothree regions for |Ez|2.
Fig. 5 shows the phase distribution of the longitudinal polariza-tion component for the LC polarized beam at the focal plane. Phasedistributions are shown for the vortex beams with m = 1 and 2 forAc = 0.0 and 1.5. In the aberration free case, the longitudinal polar-
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Fig. 5. Phase distribution of the longitudinal polarization component at the focal plane of(d) 1.5.
ization component possesses azimuthal phase structure in terms ofmultiple of 2p. The longitudinal field component of the tightly fo-cused LC polarized vortex beam possesses a longitudinal vortexwith (m + 1) topological charge [22]. Existence of the helical phasestructure or vortex term in the longitudinal component of thetightly focused circularly polarized beam can be seen by trans-forming Eq. (9) into a single integral using the trigonometric iden-tities [19,22,23,32]. Presence of comatic or non-rotationalsymmetric aberration results in the splitting of helical phase struc-ture of the longitudinal polarization component according to thetopological charge of the incident beam as shown in Fig. 5. Splittingin the helical structure of higher topological charge takes place be-cause of the strong sensitivity of the phase against perturbation[35]. For example, an incident beam with m topological chargeshows split in the (m + 1) number of charges in the longitudinalpolarization component in the presence of aberration (Fig. 5).
3.2. RC polarized vortex beam
The total intensity distribution at the focal plane for the RCpolarized vortex beam is shown in Fig. 6. Fig. 6a–d and e–h showthe results for m = 1 and 2, respectively for Ac = 0.0, 0.5, 1.0 and1.5. For Ac = 0.0, the intensity distribution shows a rotational sym-metry and flattened intensity distribution around the focal point.The non-zero contribution at the focal point arises due to contribu-tion from the longitudinal polarization component. For a ¼ 75 thedip ratio (ratio of the focal point intensity with respect to the max.intensity) is approx. 96.7% and this decreases to approx. 75.2% fora ¼ 60. The dip ratio at the origin reduces to approx. 16.3% withrespect to the aberration-free case in a focusing system witha ¼ 75 for Ac = 0.5. Position of the peak intensity is shifted withan increase in the value of Ac. Positional displacement of the peakintensity also depends on the topological charge. Modification inthe focal point intensity is also possible by central obstruction ofthe focusing system. For example the intensity distribution aroundthe origin becomes flat (dip ratio unity) for 50% central obstructionbut its intensity value is reduced to 63.8% in comparison to theunobstructed case. The intensity distribution of beam with m = 2(Fig. 6e) shows non-zero focal point intensity due to x-, andy-polarized components. The dip ratio for the RC beam withm = 2 reduces to approx. 20.9% for a ¼ 75. The presence of aberra-tion results in the removal of rotational symmetry of the intensitydistribution, with a positional shift. The reduction of intensity onone side of the ring also occurs with an increase in Ac, and theintensity distributions are transformed from the rotational sym-metry into axial symmetry around the x axis.
The squares of the x-, y-, and z-polarization components for abeam with m = 1 are shown in Fig. 7 for Ac = 0.0 and 1.5. Fig. 7a–c show the results for Ac = 0.0, and Fig. 7d–f for Ac = 1.5. Fig. 7aand b possess amplitude lobes similar to those of Fig. 4a and b.
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Fig. 6. Intensity distribution of an RC polarized vortex beam (m = 1) in the focal plane of a lens with a ¼ 75 and Ac = (a) 0.0 (b) 0.5 (c) 1.0 (d) 1.5; with m = 2, and Ac = (e) 0.0(f) 0.5 (g) 1.0 (h) 1.5.
R.K. Singh et al. / Optics Communications 282 (2009) 1501–1510 1507
However, the impact of aberration on the transverse polarizationcomponents (Fig. 7d and e) is different in comparison to those ofthe transverse polarization components of the LC polarized beamwith m = 1 (Fig. 4d and e). The contribution of the longitudinalpolarization component dominates in the total intensity distribu-tion, and it possesses rotationally symmetric structure. In the pres-ence of aberration, a positional shift in the distribution of |Ez|
2
takes place with an appearance of the side lobes. The squares ofthe polarization components for the beam with m = 2 are shownin Fig. 8 for Ac = 0.0, and 1.5. The contribution of both x-, and y-polarization components is the same at the focal point, and separa-tion in the dark region of the transverse polarization componentstake place due to the non-zero focal point value. The distributionof longitudinal polarization component |Ez|2 shows a rotationallysymmetric dark core surrounded by a bright ring, because of the
longitudinal vortex with unit topological charge. Presence of aber-ration results in a positional shift, and the dark core of the trans-verse polarization components are stretched with an increase inAc. The rotational symmetry of the longitudinal vortex also disap-pears with stretching of the core and positional shift. The presenceof aberration results in a positional shift of the peak intensity, thepositional shift being larger for a vortex beam in comparison to anon-vortex beam.
3.3. Azimuthally polarized beam
The total intensity distribution of the azimuthally polarizedbeam has been calculated using Eq. (10), and only the transversepolarization components contribute to shaping the focused struc-ture of the beam. Intensity distribution of the azimuthally polar-
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Fig. 7. Distribution of squares of the polarization components of RC polarized vortex beam with m = 1 in the focal plane of a lens with a ¼ 75; and Ac = 0.0 (a) |Ex|2 (b) |Ey|2 (c)|Ez|2; and Ac = 1.5 (d) |Ex|2 (e) |Ey|2 (f) |Ez|2.
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ized beam with m = 0 possesses a dark core surrounded by sharphigh intensity ring (Fig. 9a) in comparison to the high intensity ringof an LC polarized vortex beam with m = 1. In the presence of coma,the intensity ring in suppressed on one side of the dark core, and apositional displacement also takes place (Fig. 9b–d). Position dis-placement increases with an increase in the value of aberrations,and the ring is transformed into high intensity lobes.
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The circularly symmetric dark core with smallest size is usefulin the stimulated emission depletion microscopy for the depletionof fluorescent spot [19,21,36]. Size of the fluorescent spot in a STEDmicroscope depends on the overlapping area of the depletion beam(erase beam) and the Gaussian beam (pump beam). Maximumintensity of the fluorescent spot is allowed at the center due tothe presence of intensity null for the depletion beam. Reduction
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Fig. 9. Intensity distribution of azimuthally polarized beam at the focal plane of a lens with a ¼ 75 and Ac (a) 0.0 (b) 0.5 (c) 1.0 (d) 1.5.
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Fig. 10. Intensity profiles for fluorescence spot in the focal plane for azimuthal-linear polarization setup. Along x axis (a) Ac = 0.0, (b) Ac = 0.25; along y axis for (c)Ac = 0.0, (d) Ac = 0. 25.
R.K. Singh et al. / Optics Communications 282 (2009) 1501–1510 1509
in the circular symmetry of the dark core with suppression ofperipheral intensity on one side reduces the effectiveness of thevortex beam in the depletion of fluorescent spot. In addition a dif-ferent positional shift of the Gaussian and vortex beams also re-duces the area of the overlap.
3.4. Effect of coma on the fluorescent spot size in STED microscope
Working principle of STED microscopy is based on the fact ofdepletion of the fluorescence process by using the stimulatedemission [19,21,36]. The pump beam of the Gaussian profile isused to excite the sample and transition of the molecule fromthe higher state can be either through spontaneous (fluorescence)or stimulated emission. This stimulated emission is promoted withthe use of erase beam which possesses doughnut shaped profile.When a pump beam together with an erase beam is focused ontoa sample, the fluorescence is allowed in the low intensity regionsand depleted at the periphery. As a result, the fluorescence spotis shrunken and overcomes the diffraction limit. However, modifi-cation in the structure of the focused structure due to aberrationresults in a change in the structure of fluorescent spot. To demon-strated the effect of coma on the fluorescent spot, we have consid-ered circular–circular polarization, and the azimuthal-linearpolarization setups. In the circular–circular polarization setup,both erase and pump beam possess circular polarization, whereasin the azimuthal-linear polarization setup, the erase beam pos-sesses azimuthal polarization distribution and pump beam has lin-ear polarization. Using the same parameters as used by Bokor et al.[21], we have assumed that the maximum erase beam photon fluxis limited to Fe,max = 1026 photons/cm2 s. The erase (ke = 599 nm)and pump (kp = 532 nm) beams are both focused with an oil-immersion lens of NA = 1.4, and n = 1.515. An annular aperturewith an inner radius of 60% of the entrance pupil was assumed.The depletion ratio is given [21] as
Dvect ¼ 1=½1þ C 0 � ð~nd �~EeÞ2�;
where Ee is the amplitude of the erase beam, C0 is molecular con-stant (=9.75 � 10�25 cm2 s), and ~nd is a unit vector pointing along
the molecular transition dipole axis. The intensity distribution If
of the super-resolved fluorescent spot is given [21] as:
If ðxP ; yPÞ / DðxP; yPÞ � IPðxP ; yPÞ;
where IP (xP,yP) is the intensity distribution of the pump beam.In the case of azimuthal-linear polarization setup, the fluores-
cent spot is not circularly symmetric as shown in Fig. 10. Curve a
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Fig. 11. Intensity profiles of fluorescence spot in the focal plane for circularpolarization setup: (a) Ac = 0.0; along x axis (b) Ac = 0.25; along y axis (c) Ac = 0.25.
1510 R.K. Singh et al. / Optics Communications 282 (2009) 1501–1510
represents the fluorescent spot profile along the x axis, whereascurve c represent the fluorescent spot along the y axis. Presenceof small value of comatic aberration has significant effect on thesize of the fluorescent spot because of change in the overlappingarea of the pump and erase beam, and also due to loss of the rota-tional symmetry. Curves b and d show the fluorescent spot alongthe x and y axes, respectively for Ac = 0.25. Size of the fluorescentspot is increased along the x axis due to an increase in the lowintensity regions, whereas the effect of positional shift is clear fromthe fluorescent spot size along the y axis. Modification in the size ofthe fluorescent spot for the circularly–circularly polarization set upis shown in Fig. 11. In the aberration-free case, the fluorescent spotis symmetric as shown by Bokor et al. [21]. However, the presenceof aberration (Ac = 0.25) destroys rotational symmetry. An increasein the size of the fluorescent spot in the x direction and shifting inthe y direction occurs in the presence of aberration as shown bycurves b and c, respectively.
4. Conclusion
Impact of coma on the focused structure of the circularly polar-ized vortex beam and azimuthally polarized non-vortex beam hasbeen studied using the Debye–Wolf diffraction integral. Positional
shift and reduction of the intensity on one side occurs in the fo-cused structure in the presence of aberration. With an increase inthe aberration coefficient, the dark core in the intensity distribu-tion transforms into a triangular shape with its stretching. Thesplitting in the dark core also results with an increase in the aber-ration. The longitudinal component of the circularly polarized vor-tex beam shows the presence of azimuthal phase factor, and thesplit in the structure is governed by the topological charge of thebeam. Effect of coma on the focused structure has been discussedin the context of STED microscopy.
Acknowledgments
Rakesh Kumar Singh is thankful to the Council of Scientific andIndustrial Research (CSIR) India for the award of a Research Fellow-ship. Financial support from Department of Science and Technol-ogy (DST), India under the Grant SR/S2/LOP-10/2005 is alsoacknowledged.
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