influence of astigmatism and defocusing on the focusing of a singular beam

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Optics Communications 270 (2007) 128–138

Influence of astigmatism and defocusing on the focusingof a singular beam

Rakesh Kumar Singh *, P. Senthilkumaran, Kehar Singh

Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India

Received 28 April 2006; received in revised form 21 August 2006; accepted 7 September 2006

Abstract

Using the Fresnel–Kirchhoff diffraction integral, the intensity distribution and encircled energy of a singular beam at the focal plane ofa lens, has been numerically evaluated in the presence of astigmatism and defocusing. Study has been made for two values of topologicalcharge. The aberration results in flattening of dark core and the effect is more pronounced for beam with double topological charge.Twofold symmetry in intensity distribution is observed for selected values of defocusing. The results have been verified by the opticaltransfer function (OTF) approach.� 2006 Elsevier B.V. All rights reserved.

Keywords: Diffraction theory; Phase singularity; Topological charge; Astigmatism; Intensity; Encircled energy

1. Introduction

An optical beam possessing isolated points with indeter-minate phase and zero amplitude is called a singular beam,and such points are called singular points. The indetermi-nate phase value at the positions of phase singularitiesimplies that both, real as well as imaginary parts of thecomplex amplitude are equal to zero. In addition to havingzero amplitude, the accumulated phase change on a closedloop surrounding the singular point must be an integralmultiple ‘m’ (called topological charge) of 2p. Wave frontthat contains such points possesses helical shape and helic-ity is characterized [1,2] by m. A feature of singular beam isthat the singular points appear as isolated dark cores (zerointensity) in the diffraction patterns. An optical field withrandom space structure (speckle field) has on an averageone singular point per speckle, and there must exist somedegree of correlation between neighboring singular points

0030-4018/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2006.09.038

* Corresponding author. Tel.: +91 1126596580.E-mail addresses: [email protected], krakeshsingh@gmail.

com (R.K. Singh).

[3]. Phase singularities (vortices) can be categorized [3–5]into two types: canonical and non-canonical. The zerocrossing of the real and imaginary parts of the field makeright angle for a canonical vortex. Non-canonical vorticesare characterized by general angles at the zero crossing ofthe real and imaginary parts of the field.

An interesting aspect of helicity of the beam is thatsuch a beam possesses orbital angular momentum unlikethe spin angular momentum arising from polarization.This angular momentum is useful in several applicationsincluding optical manipulation of microparticles [6,7] andquantum teleportation [8]. Intensity distribution of the sin-gular beams possesses annular rings surrounding a darkcore that can trap [9] both low index particles (in dark core)and high index particles (in bright regions) simultaneously.In simultaneous trapping of high and low index particles,the astigmatism weakens the gradient forces [7]. The shapeand depth of the potential well in an optical trap is suscep-tible to the presence of geometrical aberrations in the lensand hence have drawn much attention of various workers[10,11]. It has been recently shown that optical vortexcan be employed for image processing and pattern recogni-tion [12–14], e.g. the filters that produce Bessel function

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R.K. Singh et al. / Optics Communications 270 (2007) 128–138 129

correlation output. Optical vortices are also helpful inexamining weak background signal hidden in a glare ofbright coherent star [15].

Recently, considerable interest has been shown in thestudies of a singular beam in the presence of astigma-tism. This includes optical vortex symmetry breakdown[16], transformation of orbital angular momentum [17],topological charge inversion [18], transformation ofphase profile [19], and propagation of LG beam withastigmatism [20]. Bekshaev et al. [21] have studied thetransformation of higher-order optical vortices by astig-matic lens using the concept of orbital angular momen-tum transformation and Hermite–Gaussian expansionmethod. However no study seems to have been madefor the diffraction of a singular beam by an aperturedsystem in the presence of astigmatism and defocusing.In view of the importance of the role of aberrations onfocusing of a singular beam, we have studied the influ-ence of astigmatism and defocusing for such a case,and have presented results for the intensity distributionand encircled energy, using Fresnel–Kirchhoff diffractionintegral. Results of intensity profile and encircled energyfor a non-singular beam in the presence of astigmatismhave been compared with known results [22–24]. In thesecond approach we have calculated optical transferfunction (OTF) of the apertured system in case of singu-lar and non-singular beams in the presence of astigma-tism. Results in case of non-singular beam, in thepresence of astigmatism, have been compared, and foundin agreement with those of Kapany and Burke [22],Mahajan [24] and De [25].

(x, y)

Exit pupil plane φ

θ

yp

xp

yi

xi

Observationplane

yg

xg

Gaussianplane

z

z

f-z

P /

Fig. 1. The coordinate system employed in the diffraction integral.

2. Theory

The term ‘‘phase singularity’’ is also commonly referredto as wavefront dislocation [1]. One of the solutions of waveequation is the complex field (x + iy)exp(�ikz), wherek = 2p/k and k is the wavelength of light beam. The solutionrepresents a single optical dislocation located at the originof the coordinate system. Under coordinate transformationform Cartesian to polar, solution on transverse plane (forfixed value of z) can be written as (x + iy) = rexp (i Æ h)where r is the amplitude and h is the polar angle. The totalphase of the wave H = [arg (x + iy) � kz] with constantvalue traces helical surface and consequently beam isreferred to as helical beam. (x + iy)mexp(�ikz) is also oneof the solution of the wave equation, where m is integercalled topological charge. Accumulated phase changearound the singular point for one cycle is 2pm.

When the optical beam with singular characteristicspasses through a focusing system having aberrations, fol-lowing transformation is possible [18]

Canonical vortex ! Lens with circularly symmetric

aberrations ði:e: spherical lensÞ! No topological charge inversion

Canonical vortex ! Lens with circularly nonsymmetric

aberrations ði:e: cylindrical lensÞ! Topological charge inversion:

For our study, we consider the coordinate geometryshown in Fig. 1. The diffraction image centered at Gaussianimage point P 0 is aberration free, if converging sphericalwave emanating from exit pupil has center of curvatureat point P 0. In the presence of aberration, center of curva-ture of wave shifts from point P 0 because of the deviationof the actual wave front from the ideal wave front at theexit pupil. The actual wave front and the Gaussian refer-ence sphere [24], both pass through the center of exit pupil,and the origin of coordinate system lies at the center of theexit pupil. In the coordinate geometry, we have chosenthree mutually parallel planes orthogonal to z-axis, namelyexit pupil plane, observation plane, and Gaussian plane.The z-axis lies along the optical axis of the focusing system.

We have considered one optical vortex with topologicalcharge ‘m’ located at the origin of the coordinate system.Then the typical phase function can be represented as

H ¼ arg ½ðxþ iyÞm� � kz ð1ÞOptical beam carrying such a point of undefined phase withorder of helicity m at the exit pupil plane is represented bythe function U:

Uðq; hÞ ¼ U 0 exp ðimhÞ ð2Þwhere h is the angular coordinate in the polar coordinatesystem on the exit pupil plane of radius a. A point on theexit pupil is represented by position vector rp and normal-ized radial coordinate q is equal to rp/a. The amplitude var-iation U0 in Eq. (2) can be of any form with the conditionthat the amplitude has to be zero at the singular point. Thephase of the wavefront has a singularity at a point aroundwhich the line integralIrH � dl ¼ �2mp ð3Þ

for the field to be single valued. For simplicity, we have ta-ken [26] the amplitude distribution U0 as constant (unity)in Eq. (2). At the exit pupil plane (i.e. z = 0), Eq. (2) canbe written as

130 R.K. Singh et al. / Optics Communications 270 (2007) 128–138

Uðh; z ¼ 0Þ ¼ exp ðimhÞ ð4ÞDiffraction pattern at the observation plane of lens can bestudied by using Fresnel–Kirchhoff diffraction formula.The diffraction image is centered at the Gaussian imagepoint in the absence of any aberration [24]. The diffractionpattern of singular beam possesses dark core surroundedby alternate maxima and minima with reduced value ofbrightness. Size of dark core depends on the topologicalcharge of the singular beam. Complex amplitude aroundthe focal plane of a lens of radius a is given [24] by

Uðr;/; zÞ ¼ C1

Z 1

0

Z 2p

0

Uðh; z ¼ 0Þ exp ikW ðq; hÞf g

� exp �i2pakz

rq cosðh� /Þ� �

qdqdh ð5Þ

where C1 ¼ a2

ikz eikz exp ðipr2=kzÞ.Here W(q,h) is the aberration function representing the

wave aberration with polar coordinates (q,h). (r,/) are thepolar coordinates at the observation plane and m = (2a/kz)r, Eq. (5) can be written as

Uðv;/; zÞ ¼ C1

Z 1

0

Z 2p

0

Uðh; z ¼ 0Þ expfi½W ðq; hÞ�g

� exp ½�ipmq cosðh� /Þ�qdqdh ð6Þ

The wave aberration function for astigmatism in presenceof defocusing is given by

W ¼ Aaq2 cos2 hþ Adq

2

where Aa and Ad are astigmatic and defocusing aberrationcoefficients in units of wavelength of light. Ad = 0 corre-sponds to the Gaussian image plane. Complex amplitudeat the observation plane in the presence of astigmatismand defocusing is written as

Uðv;/; zÞ ¼ C1

Z 1

0

Z 2p

0

exp ðimhÞ expfið2p=kÞ

� ½Aaq2 cos2 hþ Adq

2�g� exp ½�ipmq cosðh� /Þ�qdqdh ð7Þ

The phase factor of the multiplicative factor C1 does notplay any role in intensity and can be ignored. The Intensitydistribution at the observation plane is then given by

Iðv;/; zÞ ¼ jUðv;/; zÞj2 ð8Þ

Intensity distribution refers to intensity distribution in thefar-field image of a point object. In the intensity profilewe have normalized all the intensity results with respectto aberration free case. Hence, the intensity distributionis same as point spread function (PSF).

Another interesting parameter relating to property offocusing system is the encircled energy which is the fractionof total energy that falls in a circular domain of diffractionpattern of a specific radius and centered on the axis. There isa functional relationship between the encircled energy andthe optical transfer function (OTF) in view of the fact thatthe PSF and the OTF form a Fourier transform pair [27].

This functional relationship allows us to compute the PSFby two different routes, i.e. by evaluating the Fresnel–Kirchhoff integral to find the PSF, or computing the OTFby evaluating the autocorrelation of the pupil function.Intensity distribution by transfer function route [23] is givenby

Iðv;/Þ ¼ ð1=pÞZ 2

0

Z 2p

0

Cðs;uÞ exp ½imq cosðu� /Þ�sdsdu

ð9Þwhere C (s,u) is OTF in presence of astigmatism calculatedby normalized autocorrelation of the pupil function [28], s

is the normalized distance between two points in the aper-ture plane, u is the azimuthal angle in the pupil functionafter axis transformation [24] and / is the azimuthal angleon the observation plane. Encircled energy for specifiedreceiving plane and designated amounts of astigmatismcan be calculated [23] by using the following relation

Eðm0; zÞ ¼Z 2p

0

Z m0

0

Iðv;/Þv dvd/ ð10Þ

3. Results and discussion

An analytical solution of the diffraction integral (Eq.(7)) for arbitrary values of aberration coefficients is notpossible [24,29]. Hence, we have numerically evaluatedEqs. (7) to (10) for intensity distribution and encircledenergy for different values of Aa and Ad. Results of PSFat the focal plane are presented in Fig. 2(a)–(d) and (e)–(h) for m = 1 and m = 2, respectively. OTF was calculatedby evaluating the autocorrelation of the pupil function, andalso by Fourier transforming the PSF. Results of OTF fora system with non-singular beam in the presence of astig-matism have been compared with the results of Mahajan[24] and De [25] and found to be in agreement. Intensityprofiles and encircled energy in case of non-singular beamin the presence of astigmatism and defocusing have com-pared with those of Gupta et al. [23] and Mahajan [24].

3.1. Intensity profile and encircled energy at Gaussian plane

Intensity profiles of the singular beam with m = 1 at theGaussian plane are shown in Figs. 3(a)–(c) for / = 0, p/4and p/2. Continuous line graphs shown in these figures cor-respond to aberration-free case (Aa = 0), and result matcheswith the result reported by Swartzlander, [15]. Along / = 0(Fig. 3a), an increase in the value of Aa results in a decreaseof maximum intensity with stretching of dark core. For azi-muthal angle / = p/4, intensity maximum decreases forAa = 1.0 and 1.5, but increases for Aa = 0.5. The high inten-sity region is displaced for higher values of Aa. Howeveralong p/2 (Fig. 3c), continuous decrease in intensity isobserved with increase of Aa and intensity distribution issymmetric with respect to m = 0 with no positional shift ofmaxima. Intensity profiles at the Gaussian plane for thebeam with m = 2 are shown in Figs. 4(a)–(c) for azimuthal

Fig. 2. Intensity distribution of singular beam with m = 1 at the Gaussian plane with values of astigmatic aberration coefficient Aa. (a) 0.0, (b) 0.5, (c) 1.0,(d) 1.5. Intensity distribution of singular beam with m = 2 at the Gaussian plane with values of astigmatic aberration coefficient Aa. (e) 0.0, (f) 0.5, (g) 1.0,(h) 1.5.


angles / = 0, p/4 and p/2. Case Aa = 0 corresponds to aber-ration-free case and the result is in agreement with that ofSwartzlander, [15]. In the presence of astigmatism, along/ = p/4 (Fig. 4b) dark core at the focal plane is separatedinto two dark regions whereas along / = 0 and p/2, tworegions of lower intensity are obtained. The extent of sepa-ration increases with increase in the value of Aa, with theappearance of bright region between the two.

Encircled energy at the Gaussian focal plane is com-puted by using Eq. (10) and the results for the beam withm = 1, for four values of Aa are shown in Fig. 5a. Note thatin the encircled energy plots, term v 0 = pv. Two continuouslines in Fig. 5a correspond to the beam with m = 0 and 1 inthe aberration free case. Initial variation of encircledenergy of the beam with m = 1 is smaller than the beamwith m = 0 due to the presence of dark core near theGaussian focal point. Axial stretching of PSF of singularbeam in the presence of astigmatism reduces the amountof energy confined within the circle of the specific radius.This explains the low energy variation near the Gaussianfocal point with increase in Aa. Encircled energy for beamwith m = 2 at the Gaussian focal plane is shown in Fig. 5b.The encircled energy curves in this figure are consistentwith the corresponding intensity distributions. Increase ofencircled energy, for aberrated case, near the center ofcoordinate system is due to emergence of high intensityregion in between dark cores. However, the increment ofbrightness of this region reduces for high values of Aa

which leads to the reduction of the encircled energy.

3.2. Intensity profile and encircled energy at defocused planes

Results of intensity distribution for the beam with m = 1and 2 at defocused plane (corresponding to Ad = �Aa/2)

are shown in Figs. 6(a)–(d) and (e)–(h). Due to the non-rotational symmetry in the PSF, intensity profiles are plot-ted for three azimuthal angles / = 0, p/4 and p/2. Results ofintensity profiles for beam with m = 1 are shown in Figs.7(a)–(c). For / = 0 (Fig. 7a), stretching of dark core isnoticed with reduction in intensity maxima. For azimuthalangle / = p/4 (Fig. 7b), dark core is sharpened withincrease in intensity of first maximum for Aa = 0.5. But thistendency is reversed with increase in the value of aberrationcoefficient. However, for the azimuthal angle p/2 (Fig. 7c),continuous decrease in intensity with maintenance of sym-metry around m = 0 is observed. Results of intensity profilesfor the beam with m = 2 at defocused planes for three valuesof / are shown in Figs. 8(a)–(c). From the intensity profiles,it is clear that the intensity values around the Gaussian focalpoint increase, leading to removal of dark core from thecenter. The effect is more prominent for Aa = 1.0 case.

Encircled energy for beam with m = 1 at the defocusedplanes is shown in Fig. 9a for different values of Aa. Slowvariation of encircled energy, in comparison to aberrationfree case, near the center of the observation plane is dueto stretching of dark core and twofold symmetry in thePSF. Encircled energy for beam with m = 2 and for variousvalues of Aa at the defocused planes is shown in Fig. 9b.Due to splitting of dark core into two dark regions andemergence of high intensity in between them, there is anincrease in the encircled energy near the center of the obser-vation plane. Initial high encircled energy for Aa = 1.0 caseis due to presence of high intensity points near the center ofthe observation plane. This effect is nullified for Aa = 1.5case due to spreading and reduction of high intensity valuearound the center.

As a representative practical case, we have presented(Fig. 10) results of intensity distribution along / = 0 for

Fig. 3. (a) Intensity profile for singular beam with m = 1 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), at plane corresponding to Ad = 0 forazimuthal angle / = 0. (b) Intensity profile for singular beam with m = 1 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), at plane correspondingto Ad = 0 for azimuthal angle / = p/4. (c) Intensity profile for singular beam with m = 1 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), at planecorresponding to Ad = 0 for azimuthal angle / = p/2.


Fig. 4. (a) Intensity profile for singular beam with m = 2 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), at the Gaussian plane for azimuthalangle / = 0. (b) Intensity profile for singular beam with m = 2 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), at the Gaussian plane forazimuthal angle / = p/4. (c) Intensity profile for singular beam with m = 2 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), at the Gaussian planefor azimuthal angle / = p/2.


Fig. 5. (a) Encircled energy at the Gaussian plane for beam with m = 1 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5). (b) Encircled energy at theGaussian plane for beam with m = 2 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5).


a beam with m = 1 for an optical system with numericalaperture 0.5 and wavelength k = 0.6328 lm. Parameterschosen (shown by arrow) are Aa = 1.0 with Ad = �Aa/2,

Fig. 6. Intensity distribution of singular beam with m = 1 beam at the defocuseAa (a) 0, (b) 0.5, (c) 1.0, (d) 1.5. Intensity distribution of singular beam withastigmatism aberration coefficient Aa (e) 0.0, (f) 0.5, (g) 1.0, (h) 1.5.

and Aa = 1.0 with Ad = 0.0. For purpose of comparison,intensity profile of non-singular beam in aberration-freecase is also shown. It is observed that for parameter

d plane corresponding to Ad = �Aa/2 with astigmatic aberration coefficientm = 2 beam at the defocused plane corresponding to Ad = �Aa/2 with

Fig. 7. (a) Intensity profile for singular beam with m = 1 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), at the defocusing plane corresponding toAd = �Aa/2 for azimuthal angle / = 0. (b) Intensity distribution for singular beam with m = 1 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), atthe defocusing plane corresponding to Ad = �Aa/2 for azimuthal angle / = p/4. (c) Intensity distribution for singular beam with m = 1 in the presence ofastigmatism (Aa = 0, 0.5, 1.0 and 1.5), at the defocusing plane corresponding to Ad = �Aa/2 for azimuthal angle / = p/2.


Fig. 8. (a) Intensity profile for singular beam with m = 2 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), at the defocused plane corresponding toAd = �Aa/2 for azimuthal angle / = 0. (b) Intensity profile for singular beam of m = 2 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and 1.5), at thedefocused plane corresponding to Ad = �Aa/2 for azimuthal angle / = p/4. (c) Intensity profile for singular beam with m = 2 in the presence ofastigmatism (Aa = 0, 0.5, 1.0 and 1.5) at the defocused planes corresponding to Ad = �Aa/2 for azimuthal angle / = p/2.


Fig. 9. (a) Encircled energy at the defocused plane corresponding to Ad = �Aa/2 for beam with m = 1 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and1.5). (b) Encircled energy at the defocused plane corresponding to Ad = �Aa/2 for beam with m = 2 in the presence of astigmatism (Aa = 0, 0.5, 1.0 and1.5).

Fig. 10. Intensity profile at the observation plane for beam with m = 1 along / = 0 with parameters Aa = 0.0, Ad = 0.0; Aa = 1.0, Ad = 0.0; and Aa = 1.0,Ad = �0.5. Airy pattern for the system is shown by broken line curve.



Aa = 1.0 with Ad = 0, size of the dark core (at half inten-sity) increases from 0.47 lm (aberration-free case) to1.47 lm. For balanced aberration case with parameterAa = 1.0 with Ad = �0.5, dark core size is 0.52 lm.

4. Conclusion

Effects of astigmatism at different planes are studied byusing Fresnel–Kirchhoff diffraction formula and results arecrosschecked by OTF approach. Dark core of singularbeam shows axial elongation with rotation in the presenceof astigmatism at Gaussian plane. For double topologicalcharge, dark core is separated into two parts due to theemergence of high intensity values between dark regions.This effect leads to more encircled energy near the Gauss-ian point. On the other hand, twofold symmetry is foundin PSF at defocused plane in presence of astigmatism.The splitting in case of double topological charge intotwo separated dark regions is also observed at the defo-cused plane, leading to more encircled energy near the cen-ter of observation plane. Effect of astigmatism becomesmore pronounced with increase in the topological charge.

Acknowledgements

R.K.S. is thankful to the Council of Scientific andIndustrial Research (CSIR) India for the award of a Re-search Fellowship. Helpful discussions with Dr. K.N. Cho-pra are also thankfully acknowledged.

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influence of astigmatism and defocusing on the focusing of a singular beam

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