ARTICLE IN PRESS

Optics and Lasers in Engineering 48 (2010) 834–840

Contents lists available at ScienceDirect

Optics and Lasers in Engineering

0143-81

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

Vortices from wavefront tilts

Sunil Vyas n, P. Senthilkumaran

Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

a r t i c l e i n f o

Article history:

Received 6 January 2010

Received in revised form

23 April 2010

Accepted 23 April 2010Available online 23 May 2010

Keywords:

Singular optics

Phase

Spatial light modulator

Interference

66/$ - see front matter & 2010 Elsevier Ltd. A

016/j.optlaseng.2010.04.008

esponding author. Tel.: +91 11 2659 6580; fa

ail address: sunilk_vyas@yahoo.com (S. Vyas)

a b s t r a c t

We show evolution of optical vortices in the regions of a wavefront where circulating phase gradients

are present. Two different regions in a wavefront are given different tilts and it is shown that phase

singularities can evolve at discrete points along the line of phase discontinuity. The phase distribution

and the gradient in the neighborhood of these points are studied. Using a spatial light modulator (SLM)

we have experimentally demonstrated the vortex generation.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Optical vortices or phase singularities are three dimensionalthreads of darkness where amplitude of the light field goes to zeroand phase is undetermined. Vortices are naturally created and/orannihilated when a coherent beam of light passes throughturbulent media [1]. Vortex infested wavefronts have helicalshape. The growing range of applications of optical vortices callsfor reliable methods of their generation. Several methods such asthe use of spiral phase plate [2], helical mirror [3], deformablemirror [4] wedge plates [5–9], phase only diffractive opticalelement [10–12] and computer generated hologram [13] can beused for vortex generation.

In this paper we demonstrate a new method of vortexgeneration using only wavefront tilts. It is shown that if twodifferent regions in a wavefront are given different tilts, atdiscrete points on the line of phase discontinuity, phasesingularity can evolve upon propagation. Any two diametricallyopposite points in the immediate neighborhood of these discretepoints are out of phase and have opposite tilts.

The magnitudes of tilt required to generate vortices are verysmall. Hence a spatial light modulator (SLM) is used to provideopposite constant phase gradients (tilts), of small magnitude attwo different regions of the wavefront. It is interesting to see that2p helical phase variation can be realized by an element that canproduce phase retardation less than 2p radians. The resultspresented here are significant towards understanding the evolu-tion of vortices in diffracted optical fields. Vortex localization

ll rights reserved.

x: +91 11 2658 1114.

.

plays an important role in optical testing where vortices are usedas phase markers [14]. The occurrence of small phase gradients indifferent regions of the wavefront is common in wave propaga-tion. In a random phase distribution different regions of the samewavefront can have phase gradients different both in magnitudeand direction. We develop the subject in the following manner. InSections 2 and 3, we discuss about the nature of phase gradientsof optical vortices and of pure wavefront tilts. In Section 4 phasedistributions that can lead to vortex generation are discussed. InSection 5 the computational results of diffracted fields arepresented. In Section 6 vortex detection methods are dealtwith. Generation of vortices using SLM and their detection usingMach–Zehnder interferometer configuration are presented inSection 7. Results are discussed in Section 8.

2. Optical vortex

An isolated amplitude zero point on a wavefront around whichthe line integral satisfies the condition

Hrf � dl¼ 2pm charac-

terizes an optical vortex. Here rf is the phase gradient, f is thephase distribution and m is an integer representing topologicalcharge. In this paper we deal with vortices of charge 71. In theabsence of vortex the above line integral is zero. Since we areconcerned with singularity in the phase distribution, let usconsider an optical vortex whose complex amplitude is givenby [15]

Uðx,yÞ ¼ðx7 jyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2þy2p ¼ exp jfðx,yÞ

� �ð1Þ

ARTICLE IN PRESS

KTKI

y

E

S. Vyas, P. Senthilkumaran / Optics and Lasers in Engineering 48 (2010) 834–840 835

where j¼ffiffiffiffiffiffiffi�1p

and

f¼ Arg x7 jy½ �: ð2Þ

Only the transverse components of phase gradient in thevortex beam are considered. The transverse phase gradient r?fof an optical vortex of charge +1, is given by

r?f¼1

ry: ð3Þ

Note that there is no radial component.The Cartesian components of the phase gradient of vortex of

charge +1can be found by writing the phase as

fðx,yÞ ¼1

jln

xþ jyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þy2

p" #

ð4Þ

r?fðx,yÞ ¼xy�yx

x2þy2

� �: ð5Þ

The phase gradient in the x-direction (y-direction) is not aconstant for a vortex. From Eq. (3), one can observe that as r-0,i.e. near the vortex core, rf-N. The phase gradient at any pointdepends on how far the point is located from the vortex core. Atr¼r0, where r0 is a constant, the magnitude of the phase gradientis constant for all values of y and at y¼y0 and at y¼y0+p theseconstant phase gradients point in opposite directions. This isdepicted in Fig. 1.

z

x

Fig. 2. Introduction of tilt by optical element E on the incident plane wave.

3

2

1

0

-1

-2

-3

3. Wavefront tilts

Let unit amplitude on-axis plane wave be incident on anoptical element E as shown in Fig. 2. The wavefront tilt introducedby the optical element can be given by writing the transmittance

expðj2pða xþb yþg zÞÞ ¼ejðK!

TU r!Þ

ejðK!

IU r!Þ

ð6Þ

where K!

I and K!

T are the wave vectors of the incident and

transmitted waves, respectively, and k!

2pl

� �z.

al¼ KTUx

bl¼ KTUy

gl¼ KTUz

ð7Þ

where kT ¼ k= k and a, b, and g are spatial frequencies along x, y,

and z directions, respectively. l is the wavelength of light. Such atilted wavefront can be generated using a mirror, or a wedge or aSLM. The phase gradient for a tilted wavefront is

rf¼ 2pða xþb yþg zÞ: ð8Þ

r0

�0

Fig. 1. Phase gradients at (r0,y0) and (r0,y0+p) in a vortex. In the beam cross-

section the dark spot indicates the vortex core.

As before we consider only transverse components of phasegradient

r?f¼ 2pða xþb yÞ: ð9Þ

Unlike an optical vortex, here the phase gradient is constantand does not depend on the location. Our aim is to generatevortex by bringing wavefronts with two different constant phasegradients near to each other and allow them to propagate.

4. Wavefront tilt configurations

We start our analysis by studying two wavefront tilt config-urations shown in Fig. 3. These two configurations, which will bediscussed later, are shown to generate vortices. In configuration Icorresponding to Fig. 3(a) the complex amplitude in the

3

2

1

0

-1

-2

-3

Fig. 3. (a) Configuration I, two linear phase variations in opposite directions in the

upper and the lower region of the wavefront. (b) Configuration II, a linear phase

variation in the upper region and a constant phase in the lower regions of the

wavefront.

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Q

P

(x0, 0)

θ

∇⊥�

∇⊥�

r0

Line of phase discontinuity

Fig. 4. r0 neighborhood around (x0,0).

S. Vyas, P. Senthilkumaran / Optics and Lasers in Engineering 48 (2010) 834–840836

transverse plane is

cI ¼expðj2paxÞ when y40

exp j2pað1�xÞð Þ when yo0

(ð10Þ

x varies from 0 to 1 with 1 representing full length across the beamcross-section. Spatial frequency a gives rate at which the linear phasevariation occurs across the beam and is related to the tilt. It indicatesthe number of 2p phase variations in unit length across the transverseplane. In the upper portion of the wavefront (y40) and in the lowerportion of the wavefront (yo0) the tilts are opposite. These tworegions are separated by a line of phase discontinuity.

In configuration II, corresponding to Fig. 3(b), the complexamplitude is

cII ¼expðj2paxÞ when y40

expðjf0Þ when yo0

(ð11Þ

where f0 is a constant. The upper portion has a tilt in positivex-direction, while the lower portion has no tilt and has a constantphase. In Eq. (10) and (11) the term exp(j2pax) indicates linearphase variation in the transverse plane (xy plane). It correspondsto a tilted plane wave whose propagation vector k

!is in the xz

plane. Hence k!

U r!¼ kxxþkzz. In the transverse plane (xy plane;

z¼const), the phase variation is kxx¼2pax. The constant phasefactor kzz can be dropped. Configuration II is similar to the onepresented in reference [8]. This is presented along with config-uration I, to emphasize the fact that even though the presence ofcirculating phase gradients are not as strong as that of config-uration I, vortex can evolve in the diffracted field.

4.1. Points of interest

In both the configurations along the line of phase discontinuityy¼0, we find Df(x), which is the phase difference between thetwo points—one point in the upper region with coordinate (x,y0)and one point in the lower region with coordinate (x,�y0) where9y09 is small.

DfðxÞ ¼f x,y0ð Þ�f x,�y0ð Þ: ð12Þ

This scanning along y¼0 line is done to identify the locations(xi,0), at which Df¼7(2n+1)p (where n is an integer). Thesepoints are of interest to us. For 0.5oao1, with n¼0 we have twopossible points of interest. The minimum and maximum phasevalues in the phase distributions (corresponding to Eqs. (10) and(11)) are 0 and 2pa, respectively. By applying Df¼7p in Eq. (10)the x-coordinates of these two points are

x0 ¼1

4a ð2aþ1Þ ð13Þ

and

x1 ¼1

4að2a�1Þ: ð14Þ

And for configuration II, Df¼7(2n+1)p condition is satisfied atonly one location for n¼0:

x0 ¼1

2a 1þf0

p

� �: ð15Þ

Both upper and lower regions of the wavefront in configura-tion I, have phase gradients where as only the upper region inconfiguration II has a phase gradient. That is why for the givenvalue of 0.5oao1, we have two possible points of interest inconfiguration I and only one in configuration II. For a41, we canhave multiple points of interest where the conditionDf¼7(2n+1)p is satisfied. As we have seen for n¼0 and0.5oao1, the phase variation across the wavefront in bothupper and lower regions are limited to 2p radians. Such a small

phase variation can be easily realized using SLM, where as usingconventional optics is very much demanding.

4.2. Phase distribution around these points of interest

We will now consider the phase distribution in the neighbor-hood of these points in the two configurations. For the points(Fig. 4) on the circle centered at (x0,0) of radius r0, for every pointP with coordinate x0þr0 cosy,r0 sinyð Þ, it can be seen that thecorresponding diametrically opposite point Q with coordinatex0þr0 cosðyþpÞ,r0 sinðyþpÞð Þ is out of phase in configuration I. If

the point x0þr0 cosy,r0 sinyð Þ is in the region y40, thediametrically opposite point is in the region yo0 and viceversa. This phase difference is

2pa x0þr0 cosyð Þ�2pa 1� x0þr0 cosðyþpÞð Þð Þ ¼ p ð16Þ

where x0 is given by Eq. (13).One can also see that this is true for any r0 in configuration I.

Similarly for the second point (x1,0) where x1 is given by Eq. (14),we have f(P)�f(Q)¼�p.

In configuration II, the out of phase condition, Eq. (16), for twodiametrically opposite points about (x0,0) requires that

x0 ¼1

2a1þ

f0

p

� ��r0 cosy: ð17Þ

For a point (x0,y0) just above and a point (x0,�y0) just belowthe point (x0,0), the value of y is p=2 and 3p=2, respectively, andDf¼p, provided the value of x0 given by Eq. (15). For other valueof y, Eq. (15) is valid if the term r0 cosy in Eq. (17) is negligible.This means that r0 should be very small. Hence in configuration II,the out of phase condition for diametrically opposite points to(x0,0) is valid for points only in the immediate neighborhood. Inboth the configurations, we exclude diametrically opposite pointsthat lie on the line of discontinuity.

4.3. Phase gradients around the points of interest

The phase gradient for the complex field of Eq. (10) can begiven in polar form as

rf? ¼2pacosy r�2pasinyy for 0oyop�2pacosy rþ2pasinyy for poyo2p:

(ð18Þ

The y component of the phase gradient for the complex fieldgiven in Eq. (10) is given by

rf?� �

Iy¼ �2pasinyy for 0oyop

2pasinyy for poyo2p:

(ð19Þ

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ˆ r

(x0, 0)

θ

∇⊥�

∇⊥�

∇⊥�

Line of phase discontinuity

ˆ r

(x0, 0)

θ

∇⊥� = 0

Line of phase discontinuity

�

�

Fig. 5. (a) y component of the phase gradient at a distance r0 from (x0,0) for the

phase distribution shown in Fig. 3a. (b) y component of the phase gradient at a

distance r0 from (x0,0) for the phase distribution shown in Fig. 3b.

ˆ r

θ

�

Fig. 6. Plot of phase gradient at a distance r0 from vortex core corresponding to an

optical vortex.

S. Vyas, P. Senthilkumaran / Optics and Lasers in Engineering 48 (2010) 834–840 837

The plot of y component of the phase gradient along a circle ofradius r0 around point (x0,0) is shown in Fig. 5a.

Similarly the y component of the phase gradient for thecomplex field given in Eq. (11) is

rf?� �

IIy¼ �2pasinyy for 0oyop

0 for poyo2p:

(ð20Þ

The plot of y component of phase gradient along a circle ofradius r0 around point (x0,0), corresponding to configuration II isgiven in Fig. 5b.

4.4. Vortices from wavefront tilts

We felt that the points (x0,0) and (x1,0) where the x-coordinatesgiven by Eqs. (13) and (14), have neighborhood phase distributionsuitable for the evolution of vortices for the following tworeasons.

Reason 1: If the complex field of Eq. (10) is allowed topropagate, the Huygen’s secondary wavelets generated from anytwo diametrically opposite neighborhood points of (x0,0) or (x1,0)will destructively interfere to form intensity null at these points.From Eq. (16) we have seen that the phase difference is p for thediametrically opposite points. A similar situation can be wit-nessed in the propagation of a vortex of unit topological charge. Ina vortex phase distribution (Eq. (4)) any two diametricallyopposite points situated around the vortex core are out of phaseand hence the secondary wavelets generated from the primarywavefront leads to destructive interference at the vortex core.

Reason 2: The transverse phase gradient of a vortex given byEq. (3) has been plotted along a circle of radius r0 about the vortexcore and is shown in Fig. 6. In a vortex, diametrically oppositepoints about the vortex core have opposite phase gradients. Inconfiguration I all the diametrically opposite points have y

component of phase gradient that are opposite and equal inmagnitude. From Figs. 5a and 6 one can see circulating phasegradients are present in both the cases. The y component of thephase gradient corresponding to configuration II is shown inFig. 5b. Comparison of Fig. 5b with that of vortex phase gradient(Fig. 6), shows that the circulating phase gradient in Fig. 5b is notstrong enough. Later in our experiment we observe that thissituation also leads to vortex generation.

5. Study of propagated fields

The propagation of the complex field from the phase mask iscomputed by the Fresnel Kirchhoff diffraction integration, whichis given by [16]

~U ðx,ZÞ ¼ ejkz

jlz

Z1�1

Z1�1

c exp jk

2zðx�xÞ2þðy�ZÞ2h i �

dxdy ð21Þ

where (x,y) and (x,Z) correspond to object and the diffracted fieldplanes, respectively. k

¼ 2p=l is the magnitude of propagationvector, and z is the distance between object and the diffractedplane.

Since line phase discontinuity is present in the complex fieldsof in Eqs. (10) and (11), closed analytical solution of Eq. (21) maynot be possible for arbitrary values of a and f0. Hence weattempted numerical solution to view the propagated (diffracted)field. Intensity at the observation plane is

Iðx,ZÞ ¼ ~U x,Z� � 2: ð22Þ

We have taken cI and cII and evaluated the diffracted fields inboth cases using Eq. (21). Fig. 7a and b show intensitydistributions of diffracted beams for the two configuration,respectively. The dark cores are indicated by the circles in thesefigures.

6. Vortex detection methods

Detection of phase singularity is one of the importantrequirements in the field of singular optics. Diffractive [17] and/or interferometric techniques [18–21], can be used for thedetection of phase singularity in an optical field. We are usinghere an interferometric technique, which is the most commontechnique for vortex detection. The interferometric techniqueuses two beam interference in which a vortex beam is made tointerfere with a plane or spherical beam resulting in the formation

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Fig. 7. (a) Simulated intensity distribution corresponding to the propagated field

of configuration I. (b) Simulated intensity distribution corresponding to the

propagated field of configuration II.

Fig. 8. (a) Simulated interference pattern between propagated field of cI and a

tilted plane wave. (b) Simulated interference pattern between propagated field of

cII and a tilted plane wave.

SF

BS1 M1

SLML

Nd:YAG Laser

S. Vyas, P. Senthilkumaran / Optics and Lasers in Engineering 48 (2010) 834–840838

of fork or spiral fringes, respectively. Birth of a new fringe at thevortex core is the unique feature in the interference pattern.

We have added a field corresponding to a tilted plane wave tothe computed complex field ~U ðx,ZÞ to produce interferencepattern. They are presented in Fig. 8. Formation of forks at thevortex points confirms our prediction about the creation ofvortices. The locations of the vortex cores are highlighted bydrawing circles around them.

BS2M2

CCDNDF

Fig. 9. Mach–Zehnder interferometer SF—spatial filter, BS1, BS2—beam splitters,

M1 and M2—mirrors NDF—neutral density filter, CCD–camera and SLM—spatial

light modulator assembly.

7. Experimental

Diffractive optics sculpts the propagating beam of light togenerate complex intensity and phase structure. It is achieved byimposing a particular phase or intensity pattern on the incidentbeam. An SLM in phase modulation mode can imprint a phasecomponent directly on the incident beam. We use a computer-generated phase mask that is designed to represent the phasedistribution of linear phase variation. The complex transmittancefunction of the phase mask with linear phase variation used in ourexperiments can be written as

tðx,yÞ ¼exp if1ðx,yÞ

� �for y40

exp if2ðx,yÞ� �

for yo0:

(ð23Þ

In configuration I the transmittance function t(x,y) of the phasemasked displayed on the SLM produces field distributioncorresponding to Eq. (10) and in configuration II the transmit-tance function t(x,y) of phase mask produces field distributiongiven by Eq. (11). Consider a Mach–Zehnder interferometer asshown in Fig. 9. A linearly polarized light from a diode pumped

solid state (DPSS) Nd:YAG laser emitting light at 532 nm isspatially filtered, expanded and collimated. The collimated beamof light is incident on to a beam splitter BS1 that divides the beaminto two parts. In one arm of the Mach–Zehnder interferometer,an SLM (Holoeye LC-2002 with 832�624 number of pixel andpixel pitch 32 mm) in phase modulation mode is used. The phasemasks corresponding to linear phase variations that are discussedin Section 4 are drawn as gray-level images on a personalcomputer and directed via the VGA output to the SLM. Thediffracted light from the SLM is made to interfere with the beamfrom the other arm of the Mach–Zehnder interferometer using

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Fig. 10. Experimentally observed (a) Intensity distribution of propagated field of cI and (b) its interference with a plane wave from the other arm of the Mach–Zehnder

interferometer.

Fig. 11. Experimentally observed (a) intensity distribution of propagated field of cII, (b) its interference with tilted plane wave and (c) its interference with spherical wave.

S. Vyas, P. Senthilkumaran / Optics and Lasers in Engineering 48 (2010) 834–840 839

BS2. The distance between the SLM and CCD in the setup is 60 cm.To enhance the quality of the interference fringes a neutraldensity filter is used which makes the amplitude of the referencewave equal to that of the vortex beam.

8. Results

We obtained phase modulation in the range of 0 to 1.5p atl¼532 nm using our SLM. The presence of vortices in the opticalfield is detected by interference with plane/spherical referencewave in the interferometer. Formation of fork and spiral fringeswithin the cross-section of the beam gives the experimentalevidence for evolution of vortex as predicted. Fig. 10(a) shows theintensity distribution observed at CCD when the fieldcorresponding to configuration I is displayed on SLM. Fig. 10(b)shows the presence of two fork fringes corresponding to twovortices. The two vortices are of same sign and their locations areindicated by circles in the interference pattern. Since themaximum phase shift realized by SLM is 1.5p, generation of twovortices is possible. When ao0.5 the phase variation in Eq. (10) isbetween 0 and a value that is less than p. Hence along line ofphase discontinuity, Df¼p condition is not satisfied anywhereinside the beam cross-section. This means that no vortexgeneration is possible for ao0.5 Using the SLM that canintroduce a maximum phase shift of 1.5p, the complex field ofEq. (11) is generated. In this field cII, the condition Df¼p issatisfied at only one location in the beam. In this case the phasevariation in the upper regions is between 0 and 1.5p. The intensitydistribution of the vortex carrying beam observed at CCD plane isshown in Fig. 11(a). Interference pattern between the generated

vortex beam and plane wave and interference pattern betweenvortex beam and spherical wave are shown in Fig. 11(b) and (c),respectively. Interference of vortex beam with plane wave resultsin fork fringes and interference of vortex beam with sphericalwave results in spiral fringes. Both these observations indicate theformation of a single vortex in the beam. The sign of vorticesproduced depends on the direction of increase of linear phasevariation. With the present method, two vortices of oppositecharge cannot evolve from the line of phase discontinuity.However, if we reverse the direction of tilts locally at variousregions, which is possible by using SLM, oppositely chargedvortices in linear arrays can be produced [22].

9. Conclusions

We have demonstrated for the first time that vortices can begenerated from pure wavefront tilts alone. Discrete pointscharacterized by Df(x)¼7(2n+1)p along the line of phasediscontinuity that separates two regions of wavefronts havingopposite phase gradients can develop phase singularities uponpropagation. We performed experiments by displaying phasemask on the SLM and have shown the generation of single vortexas well as vortices.

Acknowledgement

Research grant from Department of Science and Technology(No. SR/S2/LOP-10/2005) is thankfully acknowledged.

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