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Fraunhofer diffraction of a Laguerre–Gaussian laserbeam by fork-shaped gratingSuzana Topuzoski a & Ljiljana Janicijevic aa Faculty of Natural Sciences and Mathematics, Institute of Physics, Skopje 1000, Republicof MacedoniaPublished online: 13 Jan 2011.

To cite this article: Suzana Topuzoski & Ljiljana Janicijevic (2011) Fraunhofer diffraction of a Laguerre–Gaussian laser beamby fork-shaped grating, Journal of Modern Optics, 58:2, 138-145, DOI: 10.1080/09500340.2010.543292

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Journal of Modern OpticsVol. 58, No. 2, 20 January 2011, 138–145

Fraunhofer diffraction of a Laguerre–Gaussian laser beam by fork-shaped grating

Suzana Topuzoski* and Ljiljana Janicijevic

Faculty of Natural Sciences and Mathematics, Institute of Physics, Skopje 1000, Republic of Macedonia

(Received 6 September 2010; final version received 22 November 2010)

In this article we present a theoretical study for Fraunhofer diffraction of a Laguerre–Gaussian laser beam withzeroth radial mode number and azimuthal mode number l by a diffractive grating with embedded fork-shapeddislocations of integer order p. Analytical expressions describing the diffracted wave field amplitude and intensitydistributions in the Fourier plane are deduced and analyzed. They are also followed by the vortex radiiexpressions.

Keywords: Fraunhofer diffraction; Laguerre–Gaussian laser beam; fork-shaped gratings; vortex

1. Introduction

Diffractive optical elements (DOEs) that generate laserbeams with phase singularities in their wavefronts havebeen research objects throughout recent years. Thewavefront dislocations or topological defects representdiscontinuities of the wave phase. There are two puredislocations [1]: edge dislocation, located along a linein the transverse plane and traveling with the beam,and vortex or screw dislocation, which is characterizedwith a spiral (helicoidal) wave phasefront, rotating as ascrew around the line of the dislocation. Along thevortex dislocation line the wave intensity has zerothvalue and non-defined phase. Among the DOEs thatgenerate optical vortices are phase spiral plates [2,3],helical axicons [4–6] and spiral zone plates [7]. Theability of the fork-shaped holograms and diffractiongratings to produce vortex beams from an incidentGaussian laser beam was experimentally verified [8,9]and widely used for guiding cold atoms [10,11], opticalmanipulation of micrometer-sized objects [12,13], andin quantum information applications [14,15], sincethese vortex beams carry definite photon orbitalangular momentum [16].

In [17] the complete analytical solution for theproblem of Fraunhofer diffraction and Fresnel dif-fraction of a Gaussian beam incident out of waist by afork-shaped grating with integer topological charge p,is reported. Analytical expressions for the waveamplitude and intensity distributions are derived,describing their radial part in the mth diffractionorder by the product of the mpth-order Gauss-dough-nut function and a confluent hypergeometric(or Kummer) function, or by the product of the

first-order Gauss-doughnut function and the difference

of two modified Bessel functions whose orders do not

match the singularity number.A new family of paraxial laser beams that form an

orthogonal basis is named as hypergeometric (HyG)

modes by the authors in [18], since they have the

complex amplitude proportional to the confluent

hypergeometric function. Unlike those, the hypergeo-

metric-Gaussian modes carry a finite power and have

been generated in [19] with a liquid-crystal spatial light

modulator.Transformation of an incident beam carrying

topological charge, like a Laguerre–Gaussian (LG)

beam with non-zeroth azimuthal mode number, or

higher-order Bessel beam, by the DOE with embedded

phase singularity is also very interesting. In [20], the

problem of Fresnel diffraction of an incident, non-

diverging, vortex Bessel beam, having topological

charge n, by forked grating with integer topological

charge p, has been solved analytically: the diffracted

wave field amplitude in the higher (mth) diffraction

order is described as a sum of Gauss hypergeometric

functions, and was shown to carry topological charge

ðn�mpÞ, respectively, for the positive and negative

mth diffraction order.In [21] and [22], respectively, the transformation of

the LGðl Þn¼0 and LG

ðl Þn6¼0 beams, into diverging or

nondiverging Bessel beams, which can have increased,

decreased, or zeroth topological charge number, by

means of a helical axicon, has been shown.In [23] the Fresnel diffraction of a Laguerre-

Gaussian beam with zeroth radial mode number and

arbitrary azimuthal index by forked grating has

*Corresponding author. Email: suzana_topuzoski@yahoo.com; suzanat@pmf.ukim.mk

ISSN 0950–0340 print/ISSN 1362–3044 online

� 2011 Taylor & Francis

DOI: 10.1080/09500340.2010.543292

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been treated. The incident beam propagation axis isorthogonal to the diffraction grating plane and inter-sects it exactly in the center of the grating bifurcationpoint, while the beam waist is in the grating plane.

When the problem of diffraction of a specific laserbeam by a diffraction grating is considered, thesolution is usually considered as complete when bothFresnel diffraction and Fraunhofer diffraction aretreated. Thus, here we refer to the Fresnel diffractionof the LG

ðl Þ0 beam by the forked grating when incident

with its waist on the grating plane [23] and whenincident out of waist on the grating plane (the mainresults arising from [24] are given here). In this paperwe treat theoretically – as far as we know for the firsttime – the problem of the Fraunhofer diffraction of theLGðl Þ0 beam by the forked grating with arbitrary

integer topological charge p (including the specialcase when p¼ 0, i.e. the grating is an ordinaryrectilinear one, and the case when l¼ 0, p 6¼ 0).Moreover, analytical formulas for the vortex radii arederived, which has not been done before for thisproblem, and which is of importance in the experi-ments for optical manipulation of micrometer-sizedobjects and atom trapping and guiding. Namely, theholographic optical tweezers use a computer-designedDOE to split a single collimated laser beam into severalseparate beams, each of which is focused into anoptical tweezers by a strongly converging lens [25].

2. Fraunhofer diffraction of LGðl Þ0 beam by the

fork-shaped grating

The incident Laguerre-Gaussian beam has radial modenumber n¼ 0 and azimuthal mode number l (takenwith positive value), and has its waist in the forkedgrating plane Dðr,’Þ

Ulð Þ0 r, ’, � ¼ 0ð Þ ¼ Al,0

rffiffiffi2p

w0

!l

exp �r2

w20

� �exp il’ð Þ ð1Þ

where the amplitude coefficient is Al,0 ¼

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=ð1þ �0,lÞ�l!

p, k ¼ 2�=� is propagation constant,

and w0 is the beam waist radius.We treat the problem of Fraunhofer diffraction of

the beam (1) by a fork-shaped grating (FSG), havingtransmission function

T r, ’ð Þ ¼X1

m¼�1

tm exp �im2�

Dr cos ’� p’

� �� �, ð2Þ

where p is an integer topological charge order of thegrating, showing the number of its internal ‘teeth’.When p¼ 0 this transmission function describes arectilinear grating. D is the period of the rectilineargrating and plays the same role for the FSG far from

its pole. The specifications of the transmission coeffi-

cients tm depend on the type of the grating, and wedefine them as in [17], for the cases of amplitudehologram and amplitude binary grating, and theirphase versions. The incident beam enters normally tothe grating plane Dðr, ’Þ, passing with its axis throughthe pole of the grating. We calculate the diffractedwave field amplitude in the focal plane �ð�, �Þ of aconvergent lens with focal distance f, using thediffraction integral

U �, �, fð Þ

¼ C

ðð�

T r, ’ð ÞUðl Þ0 r, ’ð Þ exp

ik�

fr cos ’� �ð Þ

� �r dr d’

in which we involve the grating transmission function(2) and the incident beam expression (1). In the upperintegral C ¼ i=� f is a complex constant, while� denotes the grating area that contributes to thediffraction. The integration over the angular variable isperformed by means of introducing new variables

ð�m, �mÞ and ð��m, ��mÞ in the observation plane [17]

��m ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 þ

m�f

D

� �2�2m��f

Dcos �

s;

tan��m ¼� sin �

� cos � �m�f=D:

The performed integration over the azimuthal variable’ leads to the following expression

U �, �, fð Þ ¼ 2�CAl,0

ffiffiffi2p

w0

!l(t0 exp il � þ �=2ð Þ½ �Y0 �ð Þ

þX1m¼1

tþm exp isþm �þm þ �=2ð Þ½ �Yþm �þmð Þ

þX1m¼1

t�m exp is�m ��m þ �=2ð Þ½ �Y�m ��mð Þ

):

ð3Þ

In Equation (3) the integrals over the radial variableare denoted as

Y0 �ð Þ ¼

ð10

exp �r2

w20

� �Jl

k�r

f

� �rlþ1 dr;

Y�m ��mð Þ ¼

ð10

exp �r2

w20

� �J s�mj j

k��mr

f

� �rlþ1 dr

and the following signs are introduced

sþm ¼ lþmp; s�m ¼ l�mp; ðm ¼ 1, 2, . . .Þ;

and sm¼0 ¼ l: ð4Þ

The upper integrals are the well known integrals ofBessel functions [26], used in [17]. Their solutions lead

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to the final analytical form for the diffracted wave

field, which we write as a sum of a zeroth-diffraction-

order beam and higher, positive and negative diffrac-

tion-order beams

U �, �, fð Þ ¼ U0 �, �, fð Þ þX1m¼1

Uþm �þm, �þm, fð Þ

þX1m¼1

U�m ��m, ��m, fð Þ: ð5Þ

The diffracted beam in the zeroth diffraction order is

found as an LG beam with a phase singularity l along

its propagation axis (�¼ 0), where it has zeroth value

amplitude

U0 �, �, fð Þ ¼ Al,0iw0

wft0

ffiffiffi2p� �l

exp il � þ �=2ð Þ½ �

��

wf

� �l

exp ��2

w2f

!: ð6Þ

In Equation (6) the notation for wf ¼ �f=w0� has been

used.The beams in the higher diffraction orders, positive

and negative, deviate from the z-axis, on the right and

left sides, for angles ��m ¼ arctan m�=Dð Þ. Their wave

amplitudes are found in the form

U�m ��m, ��m, fð Þ

¼ Al,0iw0

wft�m

ffiffiffi2p� �l

exp is�m ��m � �=2ð Þ½ �

���mwf

� � s�mj j

exp ��2�mw2f

!� s�mj j þ lð Þ=2þ 1ð Þ

� s�mj j þ 1ð Þ

�Ms�mj j � l

2, s�mj j þ 1;

�2�mw2f

!: ð7Þ

From the expressions (7) it can be concluded that, in

the (�m)th diffraction order, a phase singularity of

s�mj jth order occurs. The topological charge is equal to

lþmp and l�mp, respectively, for the positive and

negative diffraction-order beams. The radial part of

the wave functions is represented as a product of a

Gauss-doughnut function of order s�mj j, ð��m=wf Þjs�mj

expð��2�m=w2f Þ, and a confluent hypergeometric

(Kummer) function of real argument, Mððjs�mj �

l Þ=2Þ, js�mj þ 1; �2�m=w2f Þ.

Further, we will analyze the change of the beam’s

topological charge for different correlation between

l and p:

l5 pðs�m 5 0Þ ðl4 0, p4 0Þ:

The zeroth-diffraction-order beam has phase singular-

ity order and helicity of its wavefront as those of the

incident beam. In the positive diffraction orders,

the topological charge is increasing as lþmp, and thewavefront’s helicity is in the same rotation direction asthe incident beam. However, in the negative diffractionorders, the wavefront’s helicity has an opposite rota-tion direction, compared with that of the incident beam(the topological charge l�mp has negative sign), while,the singularity order increases when going to the right,towards the higher negative diffraction orders (anexample is shown in Figure 1).

lj j � p ðl4 0, p4 0Þ:

The zeroth-diffraction-order beam has phase singular-ity order l and helicity of its wavefront the same as theincident beam.

The positive-diffraction-order beams have topolog-ical charges lþmp, and their wavefronts’ helicity is inthe same direction as for the incident beam. In a givennegative diffraction order m0, l�m0p ¼ 0 or m0 ¼ l=pmight be satisfied. In this diffraction order, then, thebeam is without phase singularity, having a non-zerothwave amplitude along its propagation axis. This place,now, will act as a ‘referent place’ – on its right side thewavefront’s helicity has an opposite direction fromthat of the incident beam, and the topological charge isequal to l�mp. On the left side from this ‘referentplace’, the wavefront’s helicity is in the same directionas the incident beam (see the example in Figure 2).

Diffraction order –2 –1 0 1 2

Topological charge –5 –2 1 4 7

Incident beamwith topological charge 1

Figure 1. Change of the incident LG beam topologicalcharge in the diffraction orders, for the case: l¼ 1, p¼ 3(radial number n¼ 1). The arrows show the direction of thephase wavefront helicity of the beams in the correspondingdiffraction orders.

Incident beamwith topological charge 2

Diffractionorder

Topologicalcharge

–4 –3 –m0=2 –1 0 1

–2 –1 0 1 2 3

Figure 2. Change of the incident LG beam topologicalcharge for the case l¼ 2, p¼ 1 (n¼ 1).

140 S. Topuzoski and L. Janicijevic

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It is interesting to notice the case when:

l�m0p ¼ 0. Then, from Equation (7), we get the

amplitude distribution in the negative m0th diffraction

order as follows

U�m0��m0,��m0

, f �¼ iAl,0t�m0

ffiffiffi2p� �l w0

wf� l=2þ 1ð Þ

� exp ��2�m0

w2f

!M �l=2, 1;

�2�m0

w2f

!: ð8Þ

Its intensity distribution along the propagation axis is

different from zero

I�m0��m0

¼ 0, ��m0, z

�/ w0=wf

�22l�2 l=2þ 1ð ÞM2 �l=2, 1; 0ð Þ 6¼ 0: ð9Þ

If it is satisfied that l�m0p ¼ 0 and l is an

even number at the same time, l¼ 2n0 (where n0 is

integer), then, from Equation (8), using the relation

between Laguerre polynomial and Kummer

function [26]

Mð�n0,þ 1;xÞ ¼ðn0Þ!

þ 1ð Þn0LðÞn0 xð Þ

we get the diffracted wave amplitude in this negative

diffraction order m0 in the form

U�m0��m0,��m0

, z �¼ iA2n0, 0t�m0

2n0

ðn0Þ!½ �2w0

wfexp �

�2�m0

w2f

!Lð0Þn0

�2�m0

w2f

!:

ð10Þ

One can see that the beam given by Equation (10) is a

pure LG mode, with radial mode number n0 and

azimuthal mode number equal to zero.The beams diffracted in the higher diffraction

orders, positive and negative, have phase singulari-

ties s�m ¼ l�mp of the variables ��m, in the

points Cþm m�f=D, 0ð Þ and C�m m�f=D,�ð Þ, where

��m ¼ 0. The topological charge in these diffraction

orders is changing in the way explained in Figures 1

and 2.The expressions (7) are similar to the far-field

approximations of the higher-diffraction-order ampli-

tudes, obtained in the process of Fresnel diffraction of

an out-of-waist incident LGðl Þn¼0 beam by a forked

grating, derived and discussed in detail in [24], and to

the results presented in [23]. In [24] the incident

LGðl Þ0 beam enters into the forked grating plane

normally, but, different from [23], its waist

is shifted a distance z¼ � from the grating.

The higher-diffraction-order beams in the far-field

approximation are found as [24]

U�m ��m, ��m, zð Þ

� Al,0t�mffiffiffi2p� �l w0

wðzÞw �ð Þw zð Þð Þ

s�mj j�l2

�i�

� z� �ð Þ

� � s�mj j�l2 � ðlþ s�mj jÞ=2þ 1ð Þ

� s�mj j þ 1ð Þ

� exp �i00g

� �exp is�m ��mþ �=2ð Þ½ �

��mwðzÞ

� � s�mj j

� exp ��2�mw2ðzÞ

� �M ð s�mj j � l Þ=2, s�mj j þ 1;

�2�mw2ðzÞ

� �ð11Þ

where

00g � kz�l� s�mj j

2

� �arctan

�

z0

� �

�lþ s�mj j

2þ 1

� �arctan

z

z0

� �

is the Guoy phase.In expressions (7), the variables w0 and wf are

present, instead of w(�) and w(z), respectively.

w zð Þ ¼ w0½1þ ðz=z0Þ2�1=2 is the beam transverse ampli-

tude profile radius for the fundamental (Gaussian)

mode, at distance z from its beam waist w0, z0 ¼ kw20=2

is the Rayleigh distance. These far-field approximate

expressions arise from the general solution of the

Fresnel diffraction of LGðl Þ0 beam incident with its

waist a distance z¼ � from the forked grating plane [24]

U�m ��m, ��m, zð Þ

¼ Al,0t�mis�mj j

ffiffiffi2p

wð�Þ

!lw0

wðzÞ

wð�Þ

wðzÞ

� � s�mj jþl2

�ik

2ðz� �Þ

� � s�mj j�l2 � ðlþ s�mj jÞ=2þ 1ð Þ

� s�mj j þ 1ð Þexp �ig

�� exp is�m ��mþ �=2ð Þ½ � exp �

�2�mw2ðzÞ

� �� s�mj j

�m

�M ð s�mj j � l Þ=2, s�mj j þ 1;�ikqð�Þ

2qðzÞðz� �Þ�2�m

� �:

ð12Þ

Now, the Guoy phase is:

g ¼ k zþ�2�m2

1

RðzÞ�

1

z� �

� �þ

�2

2ðz� �Þ

� �

�l� s�mj j

2

� �arctan

�

z0

� �

�lþ s�mj j

2þ 1

� �arctan

z

z0

� �,

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and, with qðzÞ ¼ zþ ikw20=2 the beam complex param-

eter is signed.

3. The diffracted intensity distributions and vortex

radii in the Fourier plane

In the transverse profile, the central dark spot of the

zeroth-diffraction-order beam is surrounded by a

bright ring, whose radius is derived using the intensity

distribution of the beam (6)

I0 �, �, fð Þ ¼ Al,0t0 2 w0=wf

�22l �=wf

�2lexp �2�2=w2

f

� �,

by searching for its first extreme upon the radial

variable �, as

�0,max ¼ wf

ffiffiffiffiffiffil=2

p: ð13Þ

The analytical expressions for the vortex radii in the

higher-diffraction-order beams can be derived from

their intensity distributions

I�m ��m, ��m, fð Þ

¼ Al,0t�m 2 w0

wf

� �2

2l�2�mw2f

! s�mj j

exp �2�2�mw2f

!

��2 s�mj j þ lð Þ=2þ 1ð Þ

�2 s�mj j þ 1ð Þ

�M2 s�mj j � lð Þ=2, s�mj j þ 1;�2�mw2f

!ð14Þ

deriving them upon the radial variables ��m. The

transverse intensity profiles of the phase singularity

beams are rings with dark cores, whose radii can be

found from the equation

s�m � 2x�mh� gð Þ

h

� �M g, h;x�mð Þ

þ 2x2�mg h� gð Þ

h2ðhþ 1ÞM gþ 1, hþ 2; x�mð Þ ¼ 0 ð15Þ

where we have denoted: x�m ¼ �2�m=w

2f ; g ¼

s�m � lð Þ=2; h ¼ s�m þ 1 (the modulus of s�m is con-

sidered). The derivation of Equation (15) is explained

in more detail in Appendix 1 and 2. Its exact solution

is possible only for given values of m, p and l.

Considering x�m with a small value, thus neglecting

the term which contains x2�m in the previous Equation

(15), the analytical solution is evaluated as:

x�m ¼s�mh

2ðh� gÞ

or

��mð Þmax¼ wf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil�mp þ 1 �

l�mp

l�mp þ lþ 2

s: ð16Þ

The comparison between the radii values computed

from Equation (16) and those read from the graphics

of the radial intensity distributions, calculated using

Equation (14), shows that there is an excellent agree-

ment for not very high values of the topological chargemp. More precise results for the vortex radii can be

found by solving numerically the transcendental equa-

tion, obtained as a condition that the first derivative of

the intensity distribution upon the radial coordinate be

equal to zero (Equation (15)).In the graphs in Figures 3–5 the normalized radial

intensity distributions in the focal plane of a conver-

gent lens with focal length f¼ 30 cm, for given l, m and

I/I0

r1(mm)

Figure 4. Radial intensity profiles of the diffracted beams inthe first diffraction order (m¼ 1) for l¼ 1, and: p¼ 1(dot-dashed curve), p¼ 2 (solid curve), p¼ 3 (dashed curve).

r1 (mm)

I/I0

Figure 3. Radial intensity profiles of the diffracted beamsin the first diffraction order (m¼ 1) for p¼ 1, and: l¼ 1(dot-dashed curve), l¼ 2 (solid curve), l¼ 3 (dashed curve).

142 S. Topuzoski and L. Janicijevic

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p values are plotted based on the intensity distributions(14). The parameters used are: w0¼ 1mm and�¼ 800 nm.

In Figure 3 the radial intensity distributions in thefirst diffraction order, for a given value of p, anddifferent values of the topological charge s (which isvarying due to the change of the incident beamazimuthal mode number l) are plotted. It can be seenthat, for an incident beam with bigger topologicalcharge l, the radius of the vortex is increasing, but themaximum intensity value in the bright ring, surround-ing the core, is decreasing.

In addition, if the topological charge s has beenincreased, but now by increasing the grating’s topo-logical charge p (Figure 4), the radius of the dark spotand the radial distance of the maximum intensityvalue increase. The intensity decreases more abruptlyon the internal side of the vortex, compared to theexternal side.

In Figure 5 the radial intensity distribution in thefirst diffraction order for l¼ 1, p¼ –1 is shown, in orderto see that, when the output beam topological charge(s¼ l�mp) is equal to zero, the beam has a non-zerothintensity value (I0) along its propagation axis.

3.1. Specialization of the results when p^0

When the LGðl Þn¼0 beam is diffracted by a rectilinear

grating (p¼ 0), then, in the higher diffraction ordersthe wave amplitudes are

U�m ��m, ��m, zð Þ ¼ iAl,0t�m2l=2 w0

wf

��mwf

� � lj j

� exp il �=2þ ��mð Þ½ � exp ��2�mw2f

!

ð17Þ

while the zeroth-diffraction-order beam is equal to thereduced in amplitude (by t0) incident beam.

The beams diffracted in all diffraction orders haveequal topological charges, l, which are the same as thatof the incident beam. They also have equal vortexradii: �0 ¼ ��m ¼ wf

ffiffiffiffiffiffil=2p

. This specialization agreeswith the results reported in [27].

3.2. Specialization of the results when l^0

For the case of diffraction of a Gaussian beam by fork-shaped grating, from Equation (7), by substitutionl¼ 0 we get expressions where theKummer function canbe expressed through the difference of two modifiedBessel functions (see the identity (A6) in [17]). Thus,we arrive at the final result equal to Equation (42) in[17]. From Equation (6) for l¼ 0 we get the zeroth-diffraction-order beam as an ordinary Gaussian beam,as that one given by Equation (41) in [17].

For the vortex radii, from Equation (16), substitut-ing l¼ 0, the following expression is obtained

��mð Þmax¼ wf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimpþ 1ð Þmp

mpþ 2

s: ð18Þ

It differs from Equation (46) in [17] by a multiplicatorffiffiffi2p

, because there, the expression for the vortex radiiwas derived by introducing an approximate, paraxialformula for the modified Bessel function of a smallargument. In this article, the vortex radii expression isderived from the intensity distribution expressedthrough the Kummer function, without applying theBessel beam approximation. The expression (18) isconsidered as more exact. In both, the approximationof neglecting the quadratic terms ð�2�m=w

2f Þ

2 was done,so these formulas are more valid for not very highvalues of the topological charge mp. More preciseresults for the vortex radii can be found by solvingnumerically the equations obtained as a condition ofthe first derivative of the intensity distributions uponthe radial coordinate being equal to zero.

It is also interesting to point out that, in the far-field approximation, the derived expression for thevortex radii is same as that given by Equation (16); but,instead of the multiplicator wf there is wðzÞ (the beamtransverse amplitude profile radius for the fundamen-tal mode, a distance z from its beam waist).

4. Conclusions

Summarizing, we have obtained an analytical expres-sion for the paraxial solution of Fraunhofer diffractionof a Laguerre–Gaussian beam with zeroth radial modenumber and arbitrary azimuthal mode number l,

I/I0

r1 (mm)

Figure 5. Radial intensity profile of the diffracted beam inthe first diffraction order (m¼ 1) for p¼�1, l¼ 1.

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by a diffraction grating with integer forked disloca-tions p. It shows similarity with the far-field approx-imation solution for Fresnel diffraction of a LG

ðl Þn¼0

beam by a forked grating, which has been alsodiscussed. The change of the diffracted beam’s topo-logical charge for different correlation between l and phas been analyzed. In addition, we derived analyticalformulas for the vortex radii in the Fourier plane. Theresults obtained are being specialized for the caseswhen p¼ 0, and when the azimuthal mode number isequal to zero (l¼ 0). Each diffracted beam can betransformed by using the same hologram into a helicalmode with specific, prescribed topological charge bychanging the incident beam topological charge.

Two near-by optical vortices having oppositetopological charges exert torques with opposite direc-tions that together can be used to create a microfluidicpump. The resulting fluid flows can be reconfigureddynamically by changing the topological charges,intensities and positions of the optical vortices in anarray, which could be potentially useful for microfluidsand lab-on-a-chip applications [28].

References

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165–190.[2] Khonina, S.N.; Kotlyar, V.V.; Shinkaryev, M.V.; Soifer,

V.A.; Uspleniev, G.V. J. Mod. Opt. 1992, 39, 1147–1154.[3] Beijersbergen, M.W.; Coerwinkel, R.P.C.;

Kristensen, M.; Woerdman, J.P. Opt. Commun.

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[4] Vasara, A.; Turunen, J.; Friberg, A.T. J. Opt. Soc. Am.

A 1989, 6, 1748–1754.[5] Davis, J.A.; Carcole, E.; Cottrell, D.M. Appl. Opt. 1996,

35, 593–598.[6] Kotlyar, V.V.; Kovalev, A.A.; Skidanov, R.V.; Moiseev,

O.Yu.; Soifer, V.A. J. Opt. Soc. Am. A 2007, 24,

1955–1964.[7] Heckenberg, N.R.; McDuff, R.; Smith, C.P.; White,

A.G. Opt. Lett. 1992, 17, 221–223.[8] Bazhenov, V.Yu.; Vasnetsov, M.V.; Soskin, M.S. Pis’ma

Zh. Eksp. Teor. Fiz. 1990, 52, 1037–1039.

[9] Heckenberg, N.R.; McDuff, R.; Smith, C.P.;

Rubinsztein-Dunlop, H.; Wegener, M.J. Opt. Quant.

Electron. 1992, 24, S951–S962.[10] Franke-Arnold, S.; Leach, J.; Padgett, M.J.; Lembessis,

V.E.; Ellinas, D.; Wright, A.J.; Girkin, J.M.; Ohberg, P.;

Arnold, A.S. Opt. Express. 2007, 15, 8619–8625.

[11] Clifford, M.A.; Arlt, J.; Courtial, J.; Dholakia, K.

Opt. Commun. 1998, 156, 300–306.[12] He, H.; Heckenberg, N.R.; Rubinsztein-Dunlop, H.

J. Mod. Opt. 1995, 42, 217–223.[13] Cojoc, D.; Garbin, V.; Ferrari, E.; Businaro, L.;

Romanato, F.; Di Fabrizio, E. Microelectron. Eng.

2005, 78–79, 125–131.

[14] Liu, Y; Gao, C.; Gao, M.; Qi, X.; Weber, H.

Opt. Commun. 2008, 281, 3636–3639.[15] Langford, N.K.; Dalton, R.B.; Harvey, M.D.; O’Brien,

J.L.; Pryde, G.J.; Gilchrist, A.; Bartlett, S.D.; White,

A.G. Phys. Rev. Lett. 2004, 93, 053601.[16] Allen, L.; Barnett, S.M.; Padgett, M.J. Optical Angular

Momentum; Institute of Physics Publishing: Bristol,

2003.

[17] Janicijevic, L.; Topuzoski, S. J. Opt. Soc. Am. A 2008,

25, 2659–2669.[18] Kotlyar, V.V.; Skidanov, R.V.; Khonina, S.N.; Soifer,

V.A. Opt. Lett. 2007, 32, 742–744.[19] Karimi, E.; Zito, G.; Piccirillo, B.; Marrucci, L.;

Santamato, E. Opt. Lett. 2007, 21, 3053–3055.

[20] Janicijevic, L.; Topuzoski, S., Eds. Proceedings of the

Sixth International Conference of the Balkan Physical

Union, Melville, NY, 2007; American Institute of

Physics: AIP Conference Proceedings Volume 899.[21] Topuzoski, S.; Janicijevic, L. Opt. Commun. 2009, 282,

3426–3432.[22] Topuzoski, S.; Janicijevic, L. Acta Phys. Pol. A 2009,

116, 557–559.[23] Bekshaev, A.Ya.; Orlinska, O.V. Opt. Commun. 2010,

283, 1244–1250.

[24] Topuzoski, S. Diffractive Optical Elements which

Generate Beams with Phase Singularities. Ph.D.

Dissertation, University Ss Cyril and Methodius, 2009.[25] Curtis, J.E.; Koss, B.A.; Grier, D.G. Opt. Commun.

2002, 207, 169–175.[26] Abramowitz, M.; Stegun, I.A. Handbook of

Mathematical Functions; Dover: New York, 1964.

[27] Janicijevic, L.; Jonoska, M.; Mitreska, Z. Optik. 1998,

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1144–1149.

Appendix 1

The intensity in the higher-diffraction-order beams depends

upon the radial coordinate, according to Equation (14), as

I / exp �2x�mð Þxs�m�mM2 g, h;x�mð Þ ð19Þ

where the following notations are used

x�m ¼�2�mw2f

; g ¼s�m � l

2; h ¼ s�m þ 1: ð20Þ

We are interested in the first derivative of the intensity (19)

upon the radial coordinate

dI

d��m¼

dI

dx�m

dx�md��m

¼2��mw2f

dI

dx�m, ð21Þ

taking into consideration that

dI

dx�m¼ exp �2x�mð Þxðs�m�1Þ�m M g, h;x�mð Þ

hs�mM g, h;x�mð Þ

þ 2x�mdM g, h; x�mð Þ

dx�m� 2x�mM g, h;x�mð Þ

i: ð22Þ

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After replacing Equation (22) into Equation (21) we search

for the root of Equation (21). Except in the vortex centre

��m ¼ 0, this derivative is also annulated when being

satisfied with the following identity

s�mM g,h;x�mð Þþ 2x�mdM g,h;x�mð Þ

dx�m� 2x�mM g,h;x�mð Þ ¼ 0:

ð23Þ

Further, in Equation (23) the following relation has been

applied [26]

dM g, h;x�mð Þ

dx�m¼M g, h; x�mð Þ �

h� gð Þ

hM g, hþ 1; x�mð Þ

ð24Þ

after which we obtain

sM g, h; x�mð Þ � 2x�mh� gð Þ

hM g, hþ 1; x�mð Þ ¼ 0: ð25Þ

Following the definition of the Kummer function we show

that it is valid (see Appendix 2)

M g, hþ 1;x�mð Þ

¼M g, h;x�mð Þ �x�mh

g

ðhþ 1ÞMð gþ 1, hþ 2;x�mÞ:

ð26Þ

This, after being involved in Equation (25), leads to the final

equation

s�m � 2x�mh� gð Þ

h

� �M g, h;x�mð Þ

þ 2x2�mg h� gð Þ

h2ðhþ 1ÞM gþ 1, hþ 2;x�mð Þ ¼ 0: ð27Þ

Appendix 2 (derivation of Equation (26))

We start from the definitions of the confluent hypergeometric

or Kummer functions and the Pochhammer symbols

M g, h;xð Þ ¼X1�¼0

gð Þ�hð Þ�

x�

�!; gð Þ�¼

� gþ �ð Þ

� gð Þ:

Further, the Kummer function M g, hþ 1;xð Þ is transformedin the following way

M g, hþ 1;xð Þ ¼X1�¼0

gð Þ�hþ 1ð Þ�

x�

�!¼X1�¼0

ðhþ �� �Þ

ðhþ �Þ

gð Þ�hð Þ�

x�

�!

¼X1�¼0

gð Þ�hð Þ�

x�

�!�X1�¼0

�

ðhþ �Þ

gð Þ�hð Þ�

x�

�!

¼Mð g, h; xÞ �X1�¼0

�

ðhþ �Þ

gð Þ�hð Þ�

x�

�!

¼Mð g, h; xÞ �X1�¼1

1

ðhþ �Þ

gð Þ�hð Þ�

x�

ð�� 1Þ!

ð28Þ

where, we have applied the equation

hþ 1ð Þ�¼� hþ 1þ �ð Þ

� hþ 1ð Þ¼

hþ �ð Þ

h

� hþ �ð Þ

� hð Þ¼

hþ �ð Þ

hhð Þ�:

ð29Þ

From the upper equation the coefficient hð Þ� is expressed and

replaced in Equation (28), leading to

M g, hþ 1;xð Þ ¼M g, h;xð Þ �x

h

X1�¼1

gð Þ�hþ 1ð Þ�

x��1

ð�� 1Þ!

¼M g, h;xð Þ �x

h

X1�0¼0

gð Þ�0þ1hþ 1ð Þ�0þ1

x�0

�0!: ð30Þ

Since it is valid:

gð Þ�0þ1¼g

g

� gþ �0 þ 1ð Þ

� gð Þ¼

g� gþ �0 þ 1ð Þ

� gþ 1ð Þ¼ g gþ 1ð Þ�0 ,

the previous equation can be rewritten as

M g, hþ 1;xð Þ ¼M g, h;xð Þ �x

h

g

ðhþ 1Þ

X1�0¼0

gþ 1ð Þ�0

hþ 2ð Þ�0

x�0

�0!

¼M g, h;xð Þ �x

h

g

ðhþ 1ÞMð gþ 1, hþ 2;xÞ:

ð31Þ

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