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doi:10.1016/j.op

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Optics and Lasers in Engineering 45 (2007) 773–782

www.elsevier.com/locate/optlaseng

Focusing of a vortex carrying beam with Gaussian background by a lensin the presence of spherical aberration and defocusing

Rakesh Kumar Singh�, P. Senthilkumaran, Kehar Singh

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

Received 9 January 2007; accepted 11 January 2007

Abstract

Diffraction pattern formed by a lens for a vortex containing truncated beam with Gaussian background, and in the presence of

spherical aberration and defocusing has been studied by Fresnel–Kirchhoff diffraction integral. For the study, two different values of

topological charge are selected. Compensation of aberration in the presence of appropriate value of defocusing is investigated. Presence

of spherical aberration results in an increase in the size of the dark core of the diffraction pattern. Results are also presented for the

encircled energy. Some results are also presented for the influence of truncation parameter of the beam, on the point spread function at

different observation planes.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Vortex; Laguerre–Gaussian beam; Spherical aberration; Truncation parameter; Intensity distribution; Encircled energy

1. Introduction

Vortex infected beams have generated considerableinterest in many researchers. These type of beams can beexperimentally generated by cooperative frequency lockingin a helium–neon laser [1], through conversion ofHermite–Gaussian beams by an astigmatic mode converter[2,3], by a spiral phase plate [4] or through a computergenerated hologram [5,6]. In recent years, propagation ofvortex containing beam in free space or through aperturedsystems has drawn attention of several groups [7–14]. Thepropagation dynamics of optical vortices is influenced byphase and intensity gradients of the background beam,giving rise to a radial motion of the vortex in the directionof the transverse energy flow [8]. The gradients of thebackground beam act like driving forces in the motion ofthe vortex. Cai and He [14] studied the propagation of aLaguerre–Gaussian (LG) beam through a slightly mis-aligned paraxial optical system. Kotlyar et al. [15–18]investigated the diffraction of various types of beams

e front matter r 2007 Elsevier Ltd. All rights reserved.

tlaseng.2007.01.005

ing author. Tel.: +911126591324.

esses: krakeshsingh@gmail.com (R. Kumar Singh),

s.iitd.ac.in (K. Singh).

through spiral phase plates, and have presented theoreticaland experimental results.When an optical vortex is hosted within a Gaussian

beam, the resulting beam exhibits an annular intensityprofile with a dark core, while maintaining a helical phasestructure. A typical example of such a light field is a LGbeam. The LG modes that may emerge from cylindricallaser cavity belong to a family of solutions of waveequations. This family of solutions is referred to as LGmodes which are rotationally symmetric and exhibit anazimuthal angular dependence of the complex formexp(im y), where y is the azimuthal coordinate in thetransverse plane [19].Helical structure of constant phase surface in the vortex

containing beam makes it unique and different from otherconventional beams. This unique feature is exploited inseveral applications ranging from optical trapping [20,21]to optical testing [22,23]. Annular intensity profile withdark core in the diffraction pattern of a beam isadvantageous in comparison to Gaussian beam from thepoint of view of trapping, and no refractive indexlimitation exists in this case [24]. In the annular intensitydistribution, both low and high index particles aresimultaneously trapped [21]. The shape and depth of the

ARTICLE IN PRESS

θ�

Exit pupil plane φ

yp

xp

yi

xi

Observation plane

yg

xg

Gaussian plane

z

z

f-zP /

O

r

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

R. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 773–782774

potential well in an optical trap is susceptible to thepresence of geometrical aberrations, and hence has drawnmuch attention of various workers [25,26]. Application ofvortex phase filter in edge enhancement [27,28], inastronomy [29,30], and in optical correlators [31] has alsobeen a subject of interest. Use of vortex phase filter in edgeenhancement leads to rotationally symmetric performance.Recently, Guo et al. [32] proposed LG spatial filter (LGSF)for edge enhancement. This LGSF presents smalleroscillations in the point spread function (PSF) in compar-ison to that produced by a spiral phase filter.

Effect of aberrations on diffraction pattern in focalvolume of optical systems has been an area of interest for along time. It is well-known that the size and shape of thediffraction pattern depends on the properties of thefocusing system. Vast literature is available on the focusingproperty of optical systems afflicted by various types ofaberrations. In a number of papers, researchers havestudied the effects of aberrations in terms of opticaltransfer function (OTF) and/or PSF[33–38] etc. Mao et al.[39] and Mao and Zhao [40] investigated the propagationcharacteristics of the Kurtosis parameters of flat toppedand Hermite–Gaussian beams passing through fractionalFourier transformation system with a spherically aberratedlens. But the studies are mostly confined to the cases ofoptical systems in the absence of vortices.

In recent years, the effect of aberrations on thediffraction pattern of vortex containing beams hasattracted attention of a few groups [41–44]. Bekshaev etal. [41,42] studied the stability of higher-order opticalvortices upon focusing by an astigmatic lens. They haveused orbital angular momentum decomposition, andstudied the role of astigmatism. Wada et al. [43,44]investigated the role of astigmatism, and comatic aberra-tion on the propagation characteristics of a LG beam.However, detailed investigations on the intensity distribu-tion and encircled energy have so far not been carried outfor various apertured systems. There are several applica-tions where size and shape of the dark core in thediffraction pattern of vortex carrying beam play animportant role. Particle trapping, vortex mask as windowfor astronomical application, and optical processing aresome of the examples. Recently Singh et al. [45,46] studiedthe role of astigmatism and coma on the diffraction patternof a vortex containing beam with uniform amplitudedistribution. In the context of importance of the role ofaberrations, we have studied the diffraction of a beam witha vortex embedded in a Gaussian background by a lens inthe presence of spherical aberration and defocusing. Theresults of intensity distribution and encircled energy areevaluated by using Fresnel–Kirchhoff diffraction integral,and the results have been cross checked by OTF approach.

2. Theory

The term vortex is used to represent helical structure of aconstant phase surface with a point of undefined phase and

zero amplitude in the heart of the surface. This point isreferred to as a singular point, and zero amplitude at thesingular point is a consequence of the zero value of real andimaginary parts of the complex wave field which is locatedat the zero crossing of real and imaginary parts of the wavefield [47]. In addition, accumulated phase around a singularpoint is an integral multiple of 2p. This integral multiple isreferred to as the topological charge ‘m’. When an opticalvortex is hosted within a Gaussian beam, the resultingbeam exhibits an annular intensity profile with a dark core,while maintaining a helical phase structure. In our study,we have considered the optical geometry shown in Fig. 1.The position of a point in the exit pupil plane andobservation plane, is denoted by polar coordinates (r, y)and (r,f), respectively. r is the normalized positioncoordinate written as r ¼ rp/a where rp is the distance ofa point from the center on the exit pupil plane of radius a.The diffraction image centered at the Gaussian image pointP/ is aberration-free, if converging wave emanating fromthe exit pupil has center of curvature at point P/. In thepresence of aberration, center of curvature of wave shiftsfrom point P/ because of the deviation of the actual wavefront from the ideal wave front at the exit pupil. The actualwave front and the Gaussian reference sphere [36], bothpass through the center of the exit pupil, and the origin ofcoordinate system ‘O’ lies at the center of the exit pupil. Inthe coordinate geometry, we have chosen three mutuallyparallel planes orthogonal to z-axis, namely exit pupilplane, observation plane, and Gaussian plane (or focalplane). The z-axis coincides with the optical axis of thefocusing system.The LG modes are a family of exact and orthogonal

solutions of the paraxial wave equations in circularcoordinates. The transverse distribution of the fieldcontains Laguerre polynomial. In general, this solution ischaracterized by the separable phase term exp(im y) whichis referred to as the vortex term. The complex field for LGbeam at the z ¼ 0 plane is given [8,19] by

Eðr; y; z ¼ 0Þ ¼ E0ðffiffiffi2p

g rÞ mj jL mj jp 2g2r2� �

expð�g2r2Þ

� expðim yÞ, ð1Þ

ARTICLE IN PRESSR. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 773–782 775

where E0 is the characteristic amplitude, r is the radialdistance of a point from its center normalized by its radiusa, y is the azimuthal coordinate on the z ¼ 0 plane, L mj j

p ðxÞ

is the Laguerre polynomial with p and m as radial andangular mode numbers, and

g ¼ ða=wÞ (2)

is a truncation parameter with w as a parameter for beamsize at the z ¼ 0 plane [36]. The parameter w together with

0.45

0.4

0.3

0.2

0.1

0.35

0.25

0.15

0.05

0-1.

0.45

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0-1.5

0.45

0.4

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0.25

0.15

0.05

0

-1.5

a

b

c

Fig. 2. Amplitude distribution at the exit pupil plane of m ¼ 1

E0 scales the peak intensity of the beam. The case withp ¼ 0 exhibits at the most, one intensity minimum withinthe beam. The complex amplitude at the transverse planez ¼ 0 in this particular case is written as

Eðr; y; z ¼ 0Þ ¼ E0ðffiffiffi2p

g rÞ mj j expð�g2r2Þ expðim yÞ. (3)

The vortex hosted in the beam is referred to as r vortexbecause of the variation of dark core size as a function ofthe radial coordinate. A pupil is referred to as the LG pupil

5 -0.5 0.5 1.510-1

Profile of (a)

-0.5 0.5 1.510-1

Profile of (b)

-0.5 0.5 1.510-1

Profile of (c)

for (a) g ¼ 0.7 (b) g ¼ 1.0 (c) g ¼ 3.0 with radial profiles.

ARTICLE IN PRESSR. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 773–782776

because the transmittance variation across the pupil is inthe form of LG mode given by Eq. (1). In our study, wehave considered a case with single vortex at the origin ofthe transverse plane embedded in the Gaussian back-ground, and complex amplitude is given by Eq. (3). In this

0.05 0.05

0.1

0.15

0.2

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0.020.01

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0.04

0.05

0.06

0.07

0.08

0.09

0.04

0.06

0.08

0.1

0.12

a b

e f

Fig. 3. Intensity distribution of LG beam at the plane corresponding to Ad ¼ �

(d) 1.5 for m ¼ 1; and As (e) 0.0 (f) 0.5 (g) 1.0 (h) 1.5 for m ¼ 2.

= As

I

v

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 1 2 3 4 5

0.0

0.5

1.0

1.5

m = 1

Fig. 4. Intensity profile for LG beam with m ¼ 1 and 2 in the presence of sp

Ad ¼ �As/2.

case, the amplitude profile is governed by power factor andexponential term, and maximum amplitude is located atr ¼ (1/g)O (m/2). This results in a shift of maximumtowards lower values of radial coordinate with an increasein the value of g. The variation of amplitude with radial

0.01

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0.005

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0.025

0.005

0.01

0.015

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0.025

0.03

0.035

0.04

0.045

c d

g h

As/2 with values of spherical aberration coefficient As (a) 0.0 (b) 0.5 (c) 1.0

= As

I

1

0.9

0.8

0.7

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0.5

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0.3

0.2

0.1

0

v0 1 2 3 4 5

0.0

0.5

1.0

1.5

m = 2

herical aberration (As ¼ 0.0, 0.5, 1.0 and 1.5), at plane corresponding to

ARTICLE IN PRESS

As =As =

v’ v’

E0

E0

1

0.9

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0

1

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00 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

m = 1 m = 2

0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

Fig. 5. Encircled energy for LG beam with m ¼ 1 and 2 in the presence of spherical aberration (As ¼ 0, 0.5, 1.0 and 1.5) at plane corresponding to

Ad ¼ �As/2.

0.05

0.1

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0.1

0.02

0.04

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0.08

0.1

0.12

a b c

d e f

Fig. 6. Intensity distribution of LG beam at the plane corresponding to Ad ¼ �As with values of spherical aberration coefficient As (a) 0.5 (b) 1.0 (c) 1.5

for m ¼ 1; and As (d) 0.5 (e) 1.0 (f) 1.5 for m ¼ 2.

R. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 773–782 777

ARTICLE IN PRESSR. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 773–782778

coordinate at the exit pupil plane has been shown in Fig. 2for three different values of the truncation parameter. Outof these, we have selected the case g ¼ 1 with m ¼ 1 and 2for our computations and E0 ( ¼ 1) is taken as constant.

The complex amplitude at the observation plane can beevaluated by using the Fresnel–Kirchhoff diffractionintegral, and around the focal plane of a lens of radius a

is given [36] by

Eðr;f; zÞ ¼ C1

Z 1

0

Z 2p

0

Eðr; y; z ¼ 0Þ expfikW ðr; yÞg

� exp �i2pa

lzrr cos ðy� fÞ

� �r dr d y, ð4Þ

where C1 ¼ a2=ilzeikz expðipr2=lzÞ:Here W (r, y) is the aberration function representing the

wave aberration with polar coordinates (r, y). The waveaberration function for spherical aberration in the presenceof defocusing is given by

W ¼ Adr2 þ Asr4,

where Ad and As are the defocusing and sphericalaberration coefficients in units of wavelength of light.Ad ¼ 0 corresponds to the Gaussian image plane.

Complex amplitude at the observation plane in thepresence of spherical aberration and defocusing is therefore

= As

I

v

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 1 2 3 4 5

m = 1

0.0

0.5

1.0

1.5

Fig. 7. Intensity profile for LG beam with m ¼ 1 and 2 in the presence of sph

Ad ¼ �As.

written as

Uðv;f; zÞ ¼ C1

Z 1

0

Z 2p

0

ðffiffiffi2p

g rÞ mj j expð�g2r2Þ expðim yÞ

� expfið2p=lÞ½Adr2 þ Asr4�g

� exp½�ipn r cosðy� fÞ�rdrdy, ð5Þ

where n ¼ (2a/lz)r. The phase factor in the multiplicativefactor C1 does not play any role in intensity and can beignored. The amplitude factor a2/lz in C1 is only a scalingfactor and does not affect the functional form of theintensity distribution. Intensity distribution PSF at theobservation plane is then given by

Iðv;f; zÞ ¼ Uðv;f; zÞ�� ��2. (6)

Intensity distribution here refers to the intensity distribu-tion in the far-field image of a point object. Hence, theintensity distribution is the same as the PSF. In theintensity profile, we have normalized all the intensityresults with respect to the aberration-free case.The performance of an optical system can also be

evaluated by using encircled energy as an image assessmentparameter. The encircled energy describes the averageintegrated nature of the PSF, and is the fraction of totalenergy that falls within a circle of specific radius in the

= As

I

v

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 1 2 3 4 5

m = 2

0.00.5

1.0

1.5

erical aberration (As ¼ 0, 0.5, 1.0 and 1.5), at the plane corresponding to

ARTICLE IN PRESSR. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 773–782 779

diffraction pattern. There is a functional relationshipbetween the encircled energy and the OTF in view of thefact that the PSF and the OTF form a Fourier transformpair [35,38]. This functional relationship allows us tocompute the PSF by two different routes, i.e. by evaluatingthe Fresnel–Kirchhoff integral to find the PSF, orcomputing the OTF by evaluating the normalized auto-correlation of the pupil function. Intensity distribution bytransfer function route [37] is given by

Iðv;fÞ ¼ ð1=pÞZ 2

0

Z 2p

0

Cðs;jÞ exp½inr cos ðj� fÞ�sdsdj.

(7)

Here C(s,f) is the OTF in presence of aberration, s is thenormalized distance between two points in the apertureplane, j is the azimuthal angle in the pupil function afteraxis transformation [36] and f is the azimuthal angle onthe observation plane. Encircled energy for specifiedreceiving plane and designated amounts of sphericalaberration has been calculated [34] by using the followingrelation:

Eðn0; zÞ ¼Z 2p

0

Z n0

0

Iðv;fÞ v dv df. (8)

As =

E0

v '

1

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0.8

0.7

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0.5

0.4

0.3

0.2

0.1

00 1 2 3 4 5 7 8 9 106

m = 1

0.0

0.5

1.0

1.5

Fig. 8. Encircled energy for LG beam with m ¼ 1 and 2 in the presence of

Ad ¼ �As

3. Results and discussion

Eqs. (5–8) are used to compute the intensity distributionand encircled energy on the observation plane by twodifferent approaches, which are related to each otherby Fourier transform. Analytical solution of integral (5)is not possible for any arbitrary value of the aberrationcoefficient. Hence, we have used numerical methodsfor evaluating this integral. In the aberration-freecase, Eq. (5) is transformed to mth order Hankel trans-form, where m stands for the topological charge. Intensitydistribution and encircled energy for a non-singularbeam are obtained by setting m ¼ 0, and the effect ofGaussian background is minimized by making the trun-cation parameter tend towards zero. Results in case ofnon-singular beam are compared with the results ofKapany and Burke [33], Stamnes [34], Mahajan [36],Sanyal and Ghosh [38], and are found to be in goodagreement. In the other approach, we have evaluated OTFby using normalized autocorrelation of the pupil functionand also by evaluating the Fourier transform of the PSF.Results of the OTF for a system with non-singular beam byboth routes are compared and are found to be in goodagreement with the results of Williams and Becklund [35],Mahajan [36].

As =

E0

1

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0.2

0.1

0

v '

0 1 2 3 4 5 7 8 9 106

m = 2

0.0

0.5

1.0

1.5

spherical aberration (As ¼ 0, 0.5, 1.0 and 1.5) at plane corresponding to

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3.1. Intensity distribution and encircled energy at the

defocused planes

The intensity distribution of the LG beam with m ¼ 1and 2 at the defocused plane corresponding to Ad ¼ �As /2are shown in Figs. 3(a)–(h) for As ¼ 0.0, 0.5, 1.0 and 1.5.Radial profiles of the intensity distribution are shown inFig. 4. Since the results have symmetry about the y-axis,only half the figure (for positive values of v) has beenplotted. Fig. 5 shows the results for encircled energy in caseof m ¼ 1 and 2. Continuous line graph in these figurescorrespond to the aberration-free case. Result in thelimiting case of zero dark core size and uniform back-ground at the exit pupil plane tends towards the resultreported by Kotlyar et al. [17] and Swartzlander [29]. Theresults of aberration-free case with constant amplitudeprofile, and azimuthal phase factor in the Gaussianbackground are also compared with the results of Kotlyar

0.05

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0.06

0.08

0.1

0.12

0.05

0.1

0.15

0.2

0.25

0.3a b

d e

g h

Fig. 9. Intensity distribution for LG beam with truncation parameter g ¼ 0.7,

(d–f) at the defocused plane (Ad ¼ �As/2) with As ¼ 1.0. (g–i) at the defocuse

[18] and are found to be in good agreement. In addition,results in case of amplitude PSF with parameters w ¼ 0.74a

and 0.64a were also calculated and compared with those ofGuo et al. [32]. Good agreement has been found betweenthe two results.Impact of spherical aberration on the PSF is visualized

in terms of spreading of size of the dark core with areduction of intensity maximum. This effect is moreprominent with an increase in the value of aberrationcoefficients and appearance of outer rings in the PSF.Quantitative impact of the spherical aberration is investi-gated by plotting radial intensity profiles and encircledenergy for two different values of the topological charge.Encircled energy at the defocused plane corresponding toAd ¼ �As /2 is shown in Fig. 5 for m ¼ 1 and 2. Note thatin the encircled energy plots, parameter v0 ¼ pv and theencircled energy is normalized with respect to the aberra-tion-free case. Spreading of the dark core with a reduction

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14

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18

x 10-3

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0.012

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0.022c

f

i

1.0, and 3.0 with m ¼ 1. (a–c) at the Gaussian plane without aberration.

d plane (Ad ¼ �As ) with As ¼ 1.0.

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v '

E0

1

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0.5

0.4

0.3

0.2

0.1

00 1 2 3 4 5 6 7 8 9 10

a

a'

b

b'

c

c'

Fig. 10. Encircled energy for LG beam with truncation parameter g ¼ 0.7,

1.0, and 3.0 with m ¼ 1. (a–c) at the Gaussian plane for aberration-free

case. (a0–c0) in presence of spherical aberration (As ¼ 1.0 ) at the defocused

plane (Ad ¼ �As).

R. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 773–782 781

in the intensity maximum in PSF is also supported by thesecurves. The slow variation of encircled energy near theorigin of the observation plane is due to the presence ofdark core in the diffraction pattern. Reduction of intensitymaxima with spreading of dark core in the presence ofspherical aberration leads to a slow variation of encircledenergy in comparison to the aberration-free case. Thiseffect is more pronounced for higher values of the sphericalaberration coefficient.

The PSFs at the plane corresponding to Ad ¼ �As arepresented in Fig. 6 for As ¼ 0.5, 1.0 and 1.5 for m ¼ 1 and2. Radial intensity profiles and encircled energy at thisplane are shown, respectively, in Figs. 7 and 8. The positionof first maximum after dark core is maintained in thepresence of spherical aberration at the defocused planecorresponding to Ad ¼ �As. Nature of intensity profile isdifferent from intensity profile shown earlier for differentobservation planes. The size of the dark core has beennearly maintained in the presence of spherical aberration atthe observation plane corresponding to the defocusingcoefficients Ad ¼ �As. On this plane, the impact ofspherical aberration is found to be compensated to a largeextent in comparison to anyother selected plane.

Impact of the truncation parameter g on the PSF at theGaussian plane is shown in Figs. 9(a–c) for non-aberratedcase for g ¼ 0.7, 1.0 and 3.0. Figs. 9(d–f) shows the resultsat the defocused plane (Ad ¼ �As/2) while Figs. 9(g–i)

show the results for Ad ¼ �As and As ¼ 1.0. Amplitudedistribution in the LG beam in the exit pupil plane has zeroamplitude at the singular point. Maximum amplitudeoccurs at r ¼ (1/g) O (m/2) and then the amplitude beginsto decrease with r. Contribution of outer rays of the exitpupil at the focal plane can be changed by changing thetruncation parameter. There is an increase in the suboscillations in the PSF and a reduction in intensity with adecrease in the truncation parameter. There is a reductionof intensity of the PSF for higher values of the truncationparameter. Encircled energy for truncation paratmeterg ¼ 0.7, 1.0 and 3.0 is shown in Fig. 10 for aberration freeand spherically aberrated (As ¼ 1.0) case at the observationplane corresponding to Ad ¼ �As. A large reduction in theencircled energy for g ¼ 3.0 is due to confinement of high-intensity regions within a very small region of the pupil.

4. Conclusion

Fresnel–Kirchhoff diffraction integral is used for evalu-ating the diffraction pattern of an LG beam with m ¼ 1and 2, and results are verified by the OTF approach.Impact of spherical aberration leads to an increase in thesize of the dark core with a reduction in intensitymaximum. The impact is minimized by selecting theappropriate values of the defocusing coefficient. The effectof truncation parameter of the LG beam on the PSFhas been investigated. The impact of aberration on the PSFhas also been investigated in the presence of sphericalaberration.

Acknowledgment

Rakesh Kumar Singh is thankful to the Council ofScientific and Industrial Research (CSIR) India for theaward of a Research Fellowship.

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