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Vickers et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 823

Phase and interference properties of optical vortexbeams

John Vickers, Matt Burch, Reeta Vyas, and Surendra Singh*

Department of Physics, University of Arkansas, Arkansas 72701, USA*Corresponding author: ssingh@uark.edu

Received November 30, 2007; accepted January 8, 2008;posted January 24, 2008 (Doc. ID 90018); published February 27, 2008

Laguerre–Gauss vortex beams carrying different topological charges are generated from Hermite–Gauss laserbeams emitted by a gas laser, and their phase properties are explored by studying their interference with aplane wave. Interference of two Laguerre–Gauss vortex beams carrying equal but opposite topological chargeis also studied by using a modified Mach–Zehnder interferometer. Experimentally recorded intensity profilesare in good agreement with the theoretically expected profiles. © 2008 Optical Society of America

OCIS codes: 260.6042, 260.3160, 140.3300, 140.3295.

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. INTRODUCTIONaser beams are modeled in terms of stable solutions

modes) of the paraxial scalar wave equation [1–3]. Sev-ral sets of stable solutions are known, each correspond-ng to a particular coordinate system in which thearaxial wave equation can be solved by the method ofeparation of variables [4,5]. Each set of modes is com-lete in the sense that any paraxial laser beam can be ex-ressed as a linear superposition of the modes belongingo that set. From a practical point of view, Hermite–GaussHG) and Laguerre–Gauss (LG) beams are the most inter-sting. Most lasers naturally emit light in HG beams.hese beams do not carry orbital angular momentum

OAM). It should be noted that HG beams can carry spinngular momentum if the beam has circular or ellipticalolarization. On the other hand, certain LG beams, whoseransverse field distribution features an azimuthal angu-ar dependence of the form exp�i���, where � is an inte-er, carry OAM and are examples of the so-called opticalortices. Integer � is called the topological charge (alsohe azimuthal index), which determines the number of 2�hase shifts that occur in one revolution of the azimuthalngle �. Each photon in such a beam carries an OAM ��6]. The sign of � determines the handedness of angular

omentum.Apart from their fundamental importance, LG beams

ave found applications ranging from optical tweezersnd manipulation of Bose–Einstein condensates to opticalmaging [7–12]. They have been used in quantum infor-

ation [13] experiments to study entanglement, and theirse in quantum communication has been proposed [14].everal methods, including helical phase plates, spatialodulators, and astigmatic mode converters, have been

sed to generate LG beams [15–21]. The purpose of thisaper is to report on investigations to study their phasend interference properties using a modified Mach–ehnder interferometer [22]. Our interference setup usesdove prism in one arm, which allows us to study the in-

erference between LG beams with equal and opposite to-

1084-7529/08/030823-5/$15.00 © 2

ological charge, in addition to the interference betweenG beams and a plane wave.

. HERMITE–GAUSS ANDAGUERRE–GAUSS MODESor a monochromatic laser beam propagating in the z di-ection, the complex electric field in the paraxial and sca-ar approximation can be written as

E� �r�,t� = e��r��ei�kz−�t�, �1�

here e describes the state of polarization of the beam,hich will not concern us in this paper. The function ��r��escribing the transverse profile of the beam satisfies thearaxial wave equation

��2 ��r�� + 2ik

���r��

�z= 0, �2�

here the propagation constant k is related to wave-ength � by k=2� /� in free space. The transverse Laplac-an is given by

�2 = �

�2

�x2 +�2

�y2in Cartesian coordinates

1

�

�

���

�

��+

1

�2

�2

��2in cylindrical coordinates

. �3�

n Cartesian coordinates the stable solutions are HGodes with indices m ,n=0,1,2,3. . .

�mn�r�� =� 2

�m ! n ! 2m+n

1

we−�x2+y2�/w2

e−ik�x2+y2�/2Re−i�m+n+1��

Hm��2x/w�Hn��2y/w�. �4�

ere Hm�x� is the Hermite polynomial of order m, and weave suppressed the z dependence of laser beam spot size�z�, wavefront radius of curvature R�z�, and phase ��z�

or convenience of writing. These parameters are indepen-

008 Optical Society of America

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l

Fci

824 J. Opt. Soc. Am. A/Vol. 25, No. 3 /March 2008 Vickers et al.

ent of beam indices m and n and are given by

w�z� = w0�1 + �z/zR�2, �5a�

zR =1

2kw0

2 =�w0

2

�, �5b�

R�z� = z +zR

2

z= z�1 +

zR2

z2� , �5c�

��z� = tan−1� z

zR� . �5d�

e will denote a HG beam with indices m ,n by HGmn.Similarly, in cylindrical coordinates � and � (o���,

��2�) the stable solutions are the LG modes

�mn�r�� =� 2

�m ! n!

min�m,n�!

w

�− 1�min�m,n�e−�2/w2e−i�m+n+1��eik�2/2R

��2�

wei���m−n�

Lmin�m,n��m−n� �2�2

w2 � , �6�

here Lp��x� is the associated Laguerre polynomial and

�z� and R�z� are independent of the mode indices. Notehat normally we would denote the solution of Eq. (6) byp�, where p=min�m ,n� and �= �m−n�. In the present con-ext, it is more meaningful to use the notation of Eq. (6)nd denote the LG beam of indices m and n by LGmn.If we introduce N�m+n as the order of the beam, thenLG beam of order N can be expanded in terms of HG

eams of the same order as follows [6]:

LGmn�r�� = s=0

N

eis�/2b�m,n,s�HGN−s,s�r��, N = n + m,

�7�

ith the coefficients b�m ,n ,s� given by

ig. 1. Outline of the experimental setup. BS’s are beam splitylindrical lens pair for HGmn and LGmn mode conversions. The bnterference with a plane wave.

b�m,n,s� = � �N − s� ! s!

2Nn ! m! �1/2

1

s!

ds

dts ��1 − t�m�1 + t�n�t=0

.

�8�

nother useful relation is that a HG beam whose princi-al axes are inclined at an angle of 45° to the �x ,y� axesan be written as a linear combination of HG beams re-erred to the �x ,y� axes as [24]

Gmn��x + y�/�2,�x − y�/�2,z�

= s=0

N

b�m,n,s�HGN−s,s�x,y,z�, N = n + m, �9�

here b�m ,n ,s� is given by Eq. (8). A comparison of Eqs.7) and (9) shows that we can convert a HG mode into aG mode of the same order if we rephase the terms in de-omposition (9). This can be done by exploiting the Guoyhase shift that a beam undergoes in passing through aocus (waist) by suitably chosen astigmatic elements15,23,24]. Once this conversion is accomplished, we canhen explore the phase and interference properties of LGeams.

. EXPERIMENTe used a 30 cm long He:Ne gain tube with a high reflec-

ivity mirror of 60 cm radius of curvature hard sealed tone end of the tube and a Brewster window attached tohe other end. A standing wave laser cavity was formed bydding a plane output coupler of 1% transmission aroundhe gain tube. The mirror separation was 45 cm, and theaist was at the output mirror. Two fine cotton fibers, oneorizontal and the other vertical, were both mounted per-endicular to the laser axis on linear translators. Withouthe fibers in the cavity, the laser oscillates in many trans-erse HGmn. By moving the fibers in and out of the beam,he laser was forced to operate in a specific higher-orderode that has nodal lines at the location of the fibers.The HGmn mode from the laser was mode matched by a

ens L1 into a � /2 mode converter consisting of two cylin-

1 and L2 are the collimating lens pair, and C1 and C2 are thexpander setup (shown in gray) is used only to observe LG mode

ters, Leam e

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Vickers et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 825

rical lenses C1 and C2 with their axes parallel to one an-ther but at 45° to the principal axes of the input HGmnode. For HG-to-LG mode conversion, the cylindrical

enses must be separated by �2 fcyl, where fcyl is theirommon focal length (� /2 converter). The output of theode converter consists of a LGmn mode according to Eq.

7), which was sent to a modified Mach–Zehnder interfer-meter. One arm of the interferometer contains a Doverism, which changes the helicity of the angular momen-um associated with the input LGmn mode. The easiestay to see this is to recall that reflection at a mirror

hanges the sign of the ray vector component perpendicu-ar to the mirror. Thus if the plane of incidence is the y ,z

ig. 2. (Color online) Experimentally recorded intensity in a plaively, are the HGmn and LGmn mode patterns. Column 3 is the intave, and column 4 is the resulting pattern when LG modes of e

espond to zero radial index LG modes, whereas the bottom two

lane and the mirror lies on the x–z plane, the y compo-ent of the ray vector changes sign at reflection.The extra optical path inside the Dove prism in one

rm of the interferometer was compensated by a variableptical delay line in the other arm of the interferometer.hen the optical paths in the two arms are equal, theavefronts of the two beams traversing the two arms of

he interferometer match at the output. This means thathe two beams have the same wavefront radius of curva-ure and spot size at the output.

The phase distribution of the LG modes generated athe output of the mode converter was explored by super-mposing them on a copropagating coherent plane wave.

sverse to the direction of propagation. Columns 1 and 2, respec-pattern when the LGmn mode of column 2 interferes with a planend opposite topological charge interfere. The first four rows cor-orrespond to nonzero radial index modes.

ne tranensityqual arows c

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826 J. Opt. Soc. Am. A/Vol. 25, No. 3 /March 2008 Vickers et al.

he resultant interference fringes are phase contours ofhe LG mode. To observe this interference, a small frac-ion of the HGmn beam derived from the beam splitter BS,laced just outside the laser, is expanded greatly andligned (grayed setup in Fig. 1) so that it is superimposedn the LGmn beam exiting the mode converter such thathe LG mode falls within one of the greatly expandedobes of the HG mode. The expanded HG mode acts as alane wave coherent with the LG mode. The intensityatterns of the superimposed field are shown in Fig. 2,olumn 3 for different LG modes with topological charge �anging from 1 to 4. The interference pattern consists of �piral-shaped fringes fanning out from the center.

For comparison, theoretically expected intensity pat-erns corresponding to Fig. 2 are shown in Fig. 3. Whenwo beams of intensity I1 and I2 are superposed, the in-ensity of the superposed beam is given by

I = I1 + I2 + 2�I1I2 cos , �10�

here is the phase difference between the beams. Thentensity pattern when a LGmn mode �p=min�m ,n� ,� = �m−n � � interferes with a plane wave is given by

ig. 3. Theoretically expected intensity patterns in a plane tran, the first four rows correspond to zero radial index LG modes, w

Imn = Ip + ILG�2�2

w2 ��

e−2�2/w2Lp��2�2

w2 �2

+ 2�IpILGe−�2/w2Lp

��2�2

w2 �cos��� + k�2/2R + �N + 1��0�.

�11�

ere Ip is the intensity of the plane wave and ILG2Pp ! / ���2�p+ l� ! �, where P is the power of the LG beam.

ntensity patterns for LG mode interference with a planeave, shown in Fig. 3, column 3, agree with the experi-entally observed patterns for modes with � ranging

rom 1–4 in Fig. 2.We also studied the interference of LG modes of the

ame radial index but equal and opposite topologicalharge. Experimentally measured interference patternshen two LG modes with the same radial index �p� butqual and opposite topological charges ±� are shown inig. 2, column 4 for � ranging from 1–4. Apart from theadial variation of the intensity, the interference patternonsists of 2� lobes spaced equally on a circle concentricith the beam axis.

to the direction of propagation corresponding to Fig. 2. As in Fig.the bottom two rows correspond to nonzero radial index modes.

sverse

t−

TcTpp

c2e=ffmii

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4WbmpasTvoctattqn

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

Vickers et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 827

Theoretically expected intensity profile when equal in-ensity mode-matched LGmn beams [Eq. (6) with �= ± �mn� and p=min�m ,n�] interfere is given by

Imn = 4ILGe−2�2/w2�2�2

w2 ��Lp����2�2

w2 �2

cos2 ��. �12�

his intensity distribution for LG beams with topologicalharge � ranging from 1–4 is shown in Fig. 3, column 4.he corresponding intensity profiles recorded in the ex-eriment for zero radial index LG modes agree with theserofiles.LG modes with nonzero radial index �p�0� have more

omplex interference patterns. The last two rows of Figs.and 3 show experimentally recorded and theoretically

xpected interference patterns for LG modes with p1, � =1 and p=1, � =2, respectively. For LG mode inter-

erence with a plane wave, we see one and two spiralringes, as expected, but in addition, we also see radialodulation of spiral fringes. Experimentally measured

ntensity profiles are in agreement with the correspond-ng theoretical intensity profiles in Fig. 3.

The interference pattern that results when two LGeams with p=1 and �= ±1, interfere is shown in columnof the fifth row in Fig. 2, and the interference pattern

hat results when two LG beams with p=1 and �= ±2 in-erfere is shown in column 4 of the sixth row in Fig. 2.hese interference patterns are in agreement with theheoretically expected interference patterns shown in Fig., column 4. These patterns can be understood as radialodulations of the corresponding �p=0, � =1,2� patterns

n Fig. 2.

. SUMMARYe have described the generation of high-quality LG

eams of nonzero topological charge using an astigmaticode converter and have studied their interference and

hase properties. These beams, which have an azimuthalngular dependence of ei��, carry OAM in addition to thepin angular momentum (associated with polarization).heir phase properties were explored in experiments in-olving their interference with a plane wave. Interferencef two beams carrying equal and opposite topologicalharge was also studied. Phase and propagation charac-eristics of LG and other paraxial electromagnetic beamsre of great current research interest, driven by not onlyhe fundamental curiosity but also the potential for use ofhese properties in a myriad of applications ranging fromuantum communications to optical trapping and ma-ipulation of microscopic particles.

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