optical vortices in gaussian random wave fields: statistical probability densities

1644 J. Opt. Soc. Am. A/Vol. 11, No. 5/May 1994

Optical vortices in Gaussian random wave fields:statistical probability densities

Isaac Freund

Jack and Pearl Resnick Institute of Advanced Technology, and Department of Physics, Bar-flan University, Ramat-Gan 52900, Israel

Received August 19, 1993; revised manuscript received October 18, 1993; accepted November 22, 1993

Simple, closed-form analytical expressions are given for the statistical probability densities of the six parame-ters that define an optical vortex (phase singularity) in a Gaussian random wave field. Good agreement isfound between calculation and a computer simulation that generates these vortices.

1. INTRODUCTIONIn a fundamental paper of far-reaching import, Nye andBerry1 showed that (even) in free space the electro-magnetic field could contain stable, propagating phasesingularities that they termed "dislocations."2 7 Thesephase singularities are closely related both to the spiralsolutions of the (nonlinear) complex Ginzburg-Landauequation,8 which have recently been the subject of ex-tensive investigations in nonlinear optics and in laserphysics, 9 -2 0 and to defects in material systems.2 1 2 2 Asthe wave propagates, the lines of constant phase sur-rounding a dislocation trace out a spiral in space or intime, so that these phase singularities are now usuallyreferred to as optical vortices, a term that is used inthis paper.

Although, as indicated, optical vortices have found theirmajor application in the study of nonlinear optical pro-cesses, they are, in fact, of the greatest intrinsic im-portance in the linear scattering of optical waves fromrandom media. Berry2 and, subsequently, Baranova andher co-workers 23' 24 have shown that in speckle patterns,for example, the density of optical vortices can be extraor-dinarily high, and in a fully developed (Gaussian) specklepattern there is, on average, one optical vortex accom-panying each speckle spot. Ramazza et al. ,1 on the otherhand, have studied spatial and temporal fluctuations inthe local number density of vortices in a (slowly) time-varying random wave field, finding substantial agreementwith a (random) Poissonian for the spatial distributionwith a temporal distribution modified by antibunchingeffects that are due to a finite refractory time. Surpris-ingly, these few studies, together with the recent work inthis area at Bar-Ilan University,2 5 appear to be the onlyprior applications of optical vortices to scattering fromrandom media.

In Ref. 25 Freund et al. showed, inter alia, that eachoptical vortex in a random wave field requires six pa-rameters for its complete description. In the currentpaper simple, closed-form expressions are obtained forthe statistical probability-density functions (PDF's) thatdescribe each of these parameters in the Gaussian limit.The results are compared with a computer simulation thatgenerates a Gaussian speckle pattern, and good agree-

ment is found between the calculated densities and theexperimental ones obtained from this simulation.

In Section 2 I briefly review the theory of Nye andBerry1 for a single optical vortex, in Section 3 I introducethe six vortex parameters, in Section 4 I describe thecomputer simulation, and in Section 5 I derive the vortexPDF's and compare these with the simulated PDF's.,

2. SINGLE VORTEXIn free space, a linearly polarized monochromatic opti-cal field at frequency , 'I(x,y,z)exp(iwt) satisfies theHelmholtz equation (V2 + k2)I = 0, where k = 2Tr/A andA is the wavelength. Writing the transverse componentof the field as E = uF(x,y)exp(-ikz), where u is a unitpolarization vector and F(x,y) satisfies Laplace's equa-tion a2 F/ax2 + 2 F/ay2 = 0, yields stable nondiffract-ing solutions that propagate without change along thez axis.26 Decomposing F into real and imaginary parts,F(x, y) = fre(X, y) + ifim(x, y), readily yields the importantclass of singular solutions, fre(X,Y) = are + brex + CreYand fim(x,y) = aim + bimx + cy. The nature of thesesingularities is clarified by consideration of the specialcase F = x + iy = A exp(iqo), where A is the amplitudeand p = arg(F) is the phase. Using cylindrical polarcoordinate r, , z, where x = r cos and y = r sin 0, wehave F = r exp(iO), so that the phase o equals the polarangle 0, and the amplitude A equals the distance fromthe origin r. Thus the phase map of this solution is acounterclockwise star with an amplitude that goes to zeroat the origin, where the phase becomes indeterminate.By convention, this phase singularity is positive. If wewrite F = x - iy, the phase circulates clockwise andthe singularity is negative. The total transverse opticalfield is E(x,y,z;t) = ur exp[i(( + t - kz)], so that thephase star rotates as the wave propagates in space orin time, leading to the currently used description of thephase singularity as an optical vortex. These singular-ities, which were originally called "screw dislocations"by Nye and Berry,1 are, as was first shown by Berry,2

the generic phase singularity in a random Gaussianwave field.

For arbitrary nonzero values of the six parameters are,bre, cre, aim, bm, and cij, we always obtain an optical vor-

0740-3232/94/051644-09$06.00 © 1994 Optical Society of America

Isaac Freund

Vol. 11, No. 5/May 1994/J. Opt. Soc. Am. A 1645

tex somewhere in the x, y plane; but important propertiesof this vortex, such as sign or even location, are not readilyapparent from the values of these parameters. Accord-ingly, in the following section I define a new, operationallyuseful and physically meaningful set of parameters withwhich to describe the morphology of a vortex.

3. MORPHOLOGICAL VORTEXPARAMETERS

Although the results of the previous section provide asimple, soluble model of an optical vortex, this model is toorestrictive, and more generally the optical field E(x, y, z)will not be a nondiffracting solution of the wave equa-tion, nor will the transverse component F satisfy Laplace'sequation. In the far field, however, which is the regionof interest here, the internal wave-field structure simplyexpands uniformly (on the surface of a sphere) with in-creasing distance Z from the sample, and the wave-fieldstatistics remain unchanged during propagation. Underthese circumstances the internal structure of the wavefield is a function of the momentum transfers Ak = k8Oxand Aky = k80y, where the (assumed) small scatteringangles are 30, = x/Z and 80y = y/Z. Accordingly, in thefar field the distance Z from the sample serves only as acoordinate scaling parameter, and we may once again de-scribe the wave function by F(x, y) = fre(X, Y) + fim(X, y)-

In general, the functions fre(X, y) and fm(x, y) may berepresented by two surfaces that we call the real and theimaginary surfaces. In a random wave field these sur-faces can be expected to have both positive and negativeregions and to exhibit a complicated structure containingmany different hills, valleys, ridges, etc. The character-istic length scale over which these variations occur is thecoherence length of the speckle field. As a given surfacegoes from positive to negative values, it crosses zero in thex-y plane, and the curves defined by f (x, y) = 0 (wheref = fre, fm) are referred to as the zero crossings of thecorresponding function. At the center of a vortex wherethe phase becomes indeterminate, F = fre + ifim mustbe identically zero, since the total optical field must beeverywhere single valued. Accordingly, vortex centersare always located at the intersections in the x-y planeof the zero crossings of fre and fim. In the immediatevicinity of such an intersection, the real and the imagi-nary surfaces can generally be approximated by their tan-gent planes, so that the zeros of F are of first order.Higher-order zeros, although not absolutely forbidden, areexpected to be rare, since they correspond to extrema orsaddle points of the real and the imaginary surfaces co-inciding (accidentally) with a zero crossing. In the com-puter simulation described in Section 4 below, a searchthrough several hundred vortices yielded no example ofa higher-order zero. In our previous study of vortices inexperimental speckle patterns,2 5 an apparent example ofa second-order zero (doubly degenerate) vortex was found.On the basis of the results of the present computer simu-lation, it now appears likely that what was observedexperimentally were two closely spaced, unresolved vor-tices of the same sign, which can mimic a second-orderzero. Indeed, the distinction between two closely spacedfirst-order zeros and one second-order zero will almost al-ways be simply a question of resolution. Accordingly, in

what follows it is assumed that all vortices correspond tofirst-order zeros and that in the immediate vicinity of avortex both fre and fim are well approximated by theirtangent planes.

If we assume that the vortex center is at x0, yo andreplace fre and farm by their tangent planes, we may write

fre(Xy) = rxo(x - Xo) + r(y - yo),

fm(xy) = i(x - X) + i(y - yo),

(la)

(lb)

where we adopt the notation r. = afre/ax, ix = afn/max,ry = afre/ay, and i = afi/may, and where the super-scripts 0 imply evaluation of the derivatives at x = x0, y =yo. Thus six fundamental parameters are needed to de-scribe a (linear) vortex: the two coordinates of the vortexcenter and the four partial derivatives of the real and theimaginary surfaces.

Although the four partial derivatives in Eqs. (1) containa complete description of the vortex structure, they encodethis information in a form that is not readily deciphered.Consider, for example, the sign (+/-) of a vortex, whichwe denote sgn(v). As was indicated in Section 2, sgn(v)is positive if the phase so circulates counterclockwise andis negative if the phase circulates clockwise; i.e.,

sgu(v) = sgn(dqp/d0)o=o, (2)

where 0 is the polar angle measured counterclockwisefrom the x axis and where we may conveniently evaluatethe derivative at 0 = 0, since its sign is independent of0. Writing

tan (x, y) = fim(xy) = iZ0 + i tan 0,tan p~x~) =fre (X, Y) -rX

0 + r tan 0 (3)

noting that sgn(dqp/d0) = sgn[d(tan q')/do], and insertingEq. (3) into Eq. (2), we obtain

sgn(v) = sgn( i | (4)

Clearly, it would be desirable to have a description of avortex from which the sign (and other important parame-ters) could be read off by inspection, rather than havingto evaluate expressions such as in Eq. (4).

In light of the above, a description of a vortex is nowintroduced in terms of the geometry of the real and theimaginary tangent planes.25 As shown in Fig. 1(a), therelevant geometric parameters for these planes are (1)Pre and pm,, the angles made by the normals to theirzero crossings relative to the x axis, and (2) tan Yre andtan Arm, the slopes of the planes. The phase map of thevortex, however, depends on the ratio of fim to fre, so thatrather than using absolute angles and absolute slopes, wefind relative values to be more informative.

I first define a new coordinate system, x', y', as an in-ternal vortex coordinate system in the x-y plane suchthat x' is perpendicular to the zero crossing of the realtangent plane, y' is perpendicular to the zero crossingof the imaginary tangent plane, and the origin of thiscoordinate system is at the vortex center x0, yo [Fig. 1(b)].

Isaac Freund


O a oo,

-0 - a - o,

-r p • r,

-7r/2 c a c •/2.

V

(a)

y

0~~~~~~~~~~~1

X

(b)Fig. 1. (a) Geometry of the real/imaginary tangent plane; anglep measures the orientation of the plane relative to the laboratoryx axis, and tan V/ is the slope of the plane. (b) Vortex internalcoordinates x',y'; p measures the rotation of the vortex coordi-nate system relative to the laboratory x,y frame, E; measures theangle between y' and x', and skewness parameter o- measuresthe deviation from orthogonality of the internal vortex frame.

In this coordinate system we have the simple descrip-tion F = (tan Vfre)X' + i(tan qiim)y' = a(x' + iay'), wherea = tan IAr and a = tan q'n/tan re. As becomes ap-parent, sgn(v) = sgn(a). The anisotropy a is one of twofundamental parameters that determine the internalstructure of the vortex phase map. The amplitude fac-tor a, on the other hand, is of no interest for the phasemap but does determine how rapidly the overall opticalintensity increases away from the vortex center.

I now define p = Pre measured counterclockwise from xand an angle X as the angle between y' and x' measuredcounterclockwise from x'. p determines the rotation ofthe internal vortex x', y' coordinate system relative to thelaboratory x,,y frame, and E measures the skewness of thenonorthogonal x', y' coordinate system. However, sinceit is the deviation from orthogonality that is important, Idefine a new angle o = - /2 with which to describethe vortex skewness [Fig. 1(b)]. As becomes apparent,the skewness - is the second fundamental parameter(together with a) that determines the internal structureof the vortex phase map. We thus have for the wavefunction of a single optical vortex centered at xo,yo,

F(x,y) = a{(x - xo)cos p + (y - yo)sin p

+ ia[-(x - x)sin(p + a) + (y - yo)cos(p + )]}.

(5)

The six parameters appearing in this form of the wavefunction will be referred to as the morphological vortexparameters. So that different sets of parameters yielduniquely different wave functions and all possible com-binations of real and imaginary tangent planes are in-cluded, the morphological vortex parameters a, a, p, andC. are bounded as follows:

(6a)

(6b)

(6c)

(6d)

Figure 2 illustrates the effects of varying a and o.As shown, for o- = 0 and lal > 1 the lines of constantphase are initially pushed to the x axis. For Ia I < 1(not shown) the lines of constant phase initially crowdaround the y axis. In both instances, however, the con-tours o = 0 and s = ir/2 remain orthogonal. Sincethe so = 0 (f m = 0) and s = r/2 (fre = 0) contours arelocked to the internal vortex x',y' coordinate system,the angle between these contours changes only with oa.Changing p rotates the phase star but does not changeits shape, and changing the sign of a reverses the cir-culation of o. For any combination of parameters thecontours of constant so remain straight lines given byS = arctan[a sin(0 - p - o)/cos(0 - p)] and d/dO =a cos o-/[cos2(0 - p) + a 2 sin2(0 - p - a-)]. These re-sults explicitly display the rotational invariance of thephase star and the dependence of the vortex sign on thesign of a,

sgn(v) = sgn(a), (7)

since cos a- 2 0 as a is bounded by ± r/2 [inequality (6d)].The interferograms of Fig. 2 could, in principle, permit

the experimental measurement of the morphological pa-rameters for well-isolated vortices. In practice, however,the density of vortices is so high that there is substan-tial overlap between the interferograms of neighboringvortices, which would make it difficult if not impossibleto obtain accurate experimental parameter values. Thisproblem is aggravated by the fact that the scattered inten-sity fades to zero at each vortex. Accordingly, a computersimulation such as is described in Section 4 appears toprovide the only feasible experiment for obtaining the pa-rameters of the several thousand vortices that are neededfor comparison with the theory of Section 5.

4. COMPUTER SIMULATION

A random wave field F(x,y) was synthesized in a com-puter simulation as

NF(x,y) = a exp[i(xu. + yV. + 40n)],

n=1(8)

where x,y are appropriately scaled coordinates of momen-tum transfer in the plane of observation that was assumedto be in the far field, Un, Vn are the coordinates of the nthsource point in a (small) planar source oriented parallelto the plane of observation, and an and 40n are the cor-responding amplitude and phase. Equation (8) is thusthe usual description of a far-field wave function as theFourier transform of the source distribution. We synthe-sized a pure phase plate by setting all an = 1, since thissimplified the calculations and for a given N maximizedthe effective number of randomly phased sources. Ex-perimentation showed that (as expected) randomly vary-ing the an had no important effect on the final results.

Isaac Freund

fY

1�1

11

.1.1

I

1

1


you ~ ~ -4

180X4O 111

(a) (am)

90

-451804 0

V90 45 -

180 (C (l

90 450

80(d) (d')

Fig. 2. Phase maps of a single positive vortex illustrating the effects of vortex anisotropy a and skewness a. The linear equiphasesof the phase star are plotted at increments of 22.50. (a) a = 1,a = 00, isotropic vortex; (b) a = 5, a = O, anisotropic vortex;(c) a = 1, a- = 600, skewed vortex; (d) a = 5, a- = 600, general distorted vortex. (a')-(d') Calculated two-beam interferograms 2 5

corresponding to (a)-(d).

The un, v0 were randomly distributed uniformly within a uniformly between ±32 r (deep phase plate). For all thesquare of side s, and the 400 were randomly distributed data presented here, N = 104, which approximates a

Isaac Freund


typical experimental situation where the incident laserbeam diameter is, say, 0.5 mm and the correlation lengthof the phase plate is, say, 5 tem.

The resultant wave field was found to have the expectedcircular Gaussian statistics. We found that the distribu-tion of the measured phase was substantially uniform be-tween 0 and 2r, as predicted for a Gaussian wave field.Within a statistical uncertainty of - 3 -15% (depending onthe order of the moments involved), we found for momentswith 0 n + m • 4 that (1) (fre2n+lfim

2m+1) << N+m+l,

implying that all odd moments vanish, as is required forcircular statistics; (2) (fre2

fim2m)/(fre 2

) (f 2m)- 1, in-

dicating that fre and f m are statistically independent, asexpected; and (3) (f2n)/(f2)n _ (2n - 1)!! for both and fm, in accordance with Gaussian PDF's for the fieldcomponents.

We obtained vortex parameters by locating all thezeros of fro and fm within a rectangle of size XY in somedozen different runs in which the random-number gen-erator was reseeded randomly for each run. The size ofthe rectangle scanned varied from run to run, but typi-cally a few hundred vortices were harvested per run,with a total of 4,102 vortices being measured. Since wewere unable to locate an existing useful algorithm, weobtained the vortex parameters by the following straight-forward, brute-force procedure. We found the zeros Offreand fm by scanning in raster fashion a coarse x-y grid forthe function g(x,y) = Ifr(x,y)I + Ifim (x,y)I in order to lo-cate all local minima for which g(x,y) < N 2/10. Whensuch a minimum was found, the corresponding grid valueof x,y was assumed to be within the region of validityof the linear approximation for the functions; and, usingEq. (8) to obtain a local value for the field derivatives, weused Eqs. (1) to locate the vortex center in first approxi-mation. The values of fre and fm were tested at this loca-tion, and if they exceeded 10-'N 1 2 (this limit being set byrounding of errors of the single-precision arithmetic usedin the calculation and summation of the 104 sine and co-sine functions), we again used Eqs. (1) and (8) to obtain asecond estimate of the vortex center. Usually only two tothree iterations were required, but if this process did notconverge within six iterations (a limit determined fromexperiment), it was considered that the initial minimumdid not represent a true zero, and the search proceeded tothe next minimum. For each true zero the four partialderivatives of the real and the imaginary surfaces werecalculated from Eq. (8), and these, together with the coor-dinates of the vortex center, were stored for later analy-sis. Since this method often resulted in more than onehit being made on a particular vortex, only vortices whosecenters differed in each coordinate by more than 10-5the coherence length of the speckle field (twice the ac-curacy to which the centers were determined) were con-sidered distinct.

The major problem in any simulation is being cer-tain that the algorithms locate every one of the hun-dreds of vortices contained within the desired region.This was accomplished with the aid of numerous prelimi-nary experiments in which a color-coded phase map of theregion being searched was displayed, together with a mapof the zero crossings of the real and the imaginary sur-faces. Each hit was recorded on these maps as it oc-curred, and in this way the algorithms were refined (and

debugged) until the success rate became 100% as deter-mined by the visual tracking of several hundred hits inover a dozen different test runs.

5. VORTEX STATISTICALPROBABILITY DENSITIESWe obtained the morphological vortex parametersfrom the results of the computer simulation by com-paring Eqs. (1) and (5). This yielded the equationsrx = a cos p, ry = a sin p, i° = -aa sin(p + a-), andiy -a a cos(p + a-), which were then numerically solvedhierarchically as

p = arctan(ryo/r=°),

a- = -arctan(ix0 /iy) -pI

a = rX0 /cos p = ryo/sin p,

(9a)

(9b)

(9c)

a = -ix°/[a sin(p + a-)] = i°/[a cos(p + a)]. (9d)

Ambiguities of ± 7r in Eq. (9a) were resolved by recourseto the relevant quadrants, and - was folded back intothe region ± r/2 as required. The fact that Eqs. (9c)and (9d) overdetermine both a and a provides a usefulcheck on the procedures employed, and the near-perfectagreement between the two possible values of a obtainedfrom Eq. (9c), together with the near-perfect agreementamong the four possible values of a (including sign) ob-tained from Eq. (9d), serve to verify that the parameterlimits in inequalities (6) encompass all possible vortexmorphologies.

Histograms of the parameter distributions were pre-pared for each run, and the bin contents were summedfor all runs to yield the final experimental histograms.These histograms were then converted into probabilitydensities with use of the fact that the number of eventsN(qi) in bin q with bin width Aq is N(qi) = NqPq(qi)Aq,where Pq is the PDF for variable q and Nq is the totalnumber of events recorded.

With these preliminaries completed, I turn now to ana-lytical calculations of the probability densities and a com-parison of these with the results of the simulation.

A. Vortex DensityBecause there are no special positions within a fully devel-oped Gaussian speckle pattern, the distribution of vortexcenters is purely random, and the (constant) probabilitydensities for the vortex-center coordinates x0 and yo areof little interest. Related to these PDF's, however, is thenumber density of vortices, 7vortex, a quantity of great in-terest, indeed.2" 4' 1 5 2 3 25 In our previous research25 weargued as follows: In a Gaussian random wave field theonly length scale is the coherence length 1coh, and the onlyscale of area is the coherence area Aoh. Accordingly, alldensities must be referred to Aoh. A speckle pattern,however, consists of bright coherence areas (speckle spots)and dark coherence areas, but the vortices are necessarilyconstrained to lie only in the dark areas, so we concludeda priori that25

71vortex 1/(2Ac0 h)- (10)

We also showed that this result was in substantial, al-though not exact, agreement with prior explicit calcula-

Isaac Freund


tions of 7vortex 2 '23-

25 We attribute the small differencesbetween calculation and Eq. (10) to the various approxi-mations used in the calculations as well as to differencesin the somewhat arbitrary definition of Acoh.

For a uniformly illuminated square of side s, the co-herence area on a distant screen (far field) at z = R isconventionally27 defined as A~oh = (AR/s)2 . In Eq. (8), x,which has the units of an inverse length, is, as alreadymentioned, a scattering wave vector related to the smallangle of scattering 60 by x k80. The solid angle corre-sponding to Acoh is dfcoh = ACoh/R 2 = (80coh)

2 . Accord-ingly, in the units employed in Eq. (8), the coherence areain the x-y plane is (xy)coh = (2r/s)2. If there are Nvortexvortices in a region of area XY in the x-y plane, then7vortex(meas) = Nvortex/XY, which is to be compared with

the calculated value 7vortex(calc) = 1/[2(xy)coh]. In mak-ing this comparison we write 7/vortex = 77/[2(XY)coh] andextract 77 from the data.

Obtaining 77vortex(meas) for each run from the measuredNvortex for that run and the associated value of XY andaveraging all runs yields 77(meas) = 1.065 ± 0.036, wherethe quoted uncertainty is the standard deviation of theruns. For a square source, we calculated previously2 5

77 = 7r/3 = 1.047, which within statistical uncertaintyis in satisfactory agreement with our current measure-ments. We note that within the framework of these cal-culations, -q depends on the source shape, so that fora uniform circular source, for example, we obtained2 5

77 = ii, 2/16 = 0.918, where ji,i is the first zero of theBessel function J1 .2 8

Although the coherence area of a random wave field isa fundamental property of the field, its current definitionin terms of the width of the autocorrelation function ofthe field is unsatisfactory. If the sample is a uniformlyilluminated square, for example, the width of the autocor-relation function is normally taken as the distance to itsfirst zero along a symmetry axis, whereas if the sample isilluminated by a Gaussian intensity distribution the l/epoint is normally chosen for the width.27 It is this arbi-trariness in the definition of ACh that gives rise to slightlydifferent values for . Clearly a definition of Acoh thatis free of such arbitrary choices is highly desirable. Ac-cordingly, I propose that Eq. (10) be used to provide anunambiguous definition of the coherence area in termsof 7

7vortex, a well-defined, directly calculable, and directlymeasurable quantity that is free of arbitrary assumptions.Such a definition also expresses well the notion that theloss of information implied by a loss of coherence in astatic system occurs only when a singularity is encoun-tered, and the definition is in full accord with previousstudies for the dynamic nonlinear case14"15 that show thatin these systems the mean nearest neighbor separationof vortices is of the order of the coherence length ofthe field.

B. Vortex Partial DerivativesI now calculate the joint probability density for the fourpartial derivatives rx°, r, 0 , ij°, and i, 0 for an assumedGaussian random wave field. The sixfold joint probabil-ity density p(fre,fim, r., r, i, i,) is by hypothesis Gauss-ian. Since we assume a uniformly illuminated squaresource, we may, following Ochoa and Goodman27 [theirEqs. (37), (38), and (53)], write

P(frefim, r., ry i, i0)

- 1 exp_ fre 2+ f, _2rx2 + r + i + i A

-8 r3a2b2 x 2 2 2b ) I(11)

where parameters a and b depend on the scaling of xand y, the sample dimension s, and the source intensity.Recalling that vortices occur only at the intersections ofthe zero crossings of fre and fim, we have

P (r. , ry 0, io i °)

[ dfre JL-X oo

dfim8( )

X a fA- )P(frefim r, r i Yi)] rx-tx 0 ryry0

=_ 1 exp (r 0 )2 + (ryO) 2 + (i. 0 )2 + (iyO)2]

4v 2 b 2 exL-2bj. (2

In Subsections 5.C.1-5.C.5 below, this result is usedto calculate the probability densities of the morphologi-cal vortex parameters a, a, p, and -. In perform-ing these calculations I make explicit use of therotational symmetry that appears in Eq. (12), whichcontains only the circular combinations (r, 0)2 + (rY0)2and (iX0)2 + (iY0) 2 . All quantities that depend onlyon these partial derivatives, such as angles p and -[Eqs. (9)], must have rotationally symmetric PDF's, sincefor these angles there can be no special directions in thespeckle pattern. These PDF's are to be contrasted withthe autocorrelation function, for example, which has onlythe symmetry of the (square) source.

C. Vortex Morphological Parameters

1. pSince there is no preferred direction in the speckle pat-tern, vortex rotation angle p must be uniformly dis-tributed between ± vr, so

PP(P) = 1 KW -1PI), (13)

where the Heaviside step function 0(u) is unity for u 2 0and vanishes otherwise. In Fig. 3, Eq. (13) is comparedwith the measured values of Pp (p). As there are no ad-justable parameters, the good agreement between the-

C

C

102p C

.3 _71

C

-270 -180 -90 0p (deg)

90 180 270

Fig. 3. Probability density Pp of vortex rotation angle p mea-sured in degrees. The solid line is the theory of Eq. (13), withv = 180°.

Isaac Freund


1.

0.

O.

12 , I, I X a a i X '

,1 1i A0.

a.:

zceu -u -0 -30 0 30 60 90 120cr (deg)

ing to polar coordinates and using the circular symmetryof the PDF's of the partial derivatives [Eq. (12)]. Con-verting from a2 to a and then to a/(a) in the usual way29

yields

(a = 2a a ] (18)

where (a) = (b/2)V 2 . In Fig. 5, Eq. (18) is comparedwith the data from the computer simulation. The goodagreement is self-evident, and since there are no ad-justable parameters this agreement confirms the appli-cability of Eq. (18).

Fig. 4. Probability density P, of the vortex skewness parametero- measured in degrees. The solid line is the theory of Eq. (15),with r = 1800.

ory and computer experiment verifies the applicability ofEq. (13).

2. o-In Section 3, X was defined as the angle between the nor-mals to the zero crossings of fim and fre; i.e., Z = Pun - Pre,and a-, the angle of interest here, was defined in terms of

= vr/2 + a-. Since there is no preferred direction in thespeckle pattern, the PDF for the vortex skewness o- mustbe symmetric, i.e, must depend only on low-, whereas bothPre and pin must be uniformly distributed. Accordingly,we may write

P,.(-) = K f dpi, f dpreo(pim - Pre - 7r/2 - -1),

(14)

where the normalization constant K, which must alsoreflect the fact that a- is folded back into the interval+7r/2, is conveniently fixed at the end of the calculation.We thus obtain

( ) r 2( 21- )(7r/2 -lo-).(15)

In Fig. 4, Eq. (15) is compared with the measured proba-bility density as obtained from the computer simulation,and satisfactory agreement is found between theory andexperiment.

3. aIn contrast to the somewhat informal derivations ofEqs. (13) and (15), I calculate the PDF of the vortexamplitude a by using Eq. (12). Since a is positive defi-nite [inequality (6a)], it is simpler first to obtain the PDFof a2 . Recalling from Section 3 that a is defined by theslope of the real tangent plane, we have

a2 = (r. 0)2 + (r 0 )2 .

4. aWe can observe from Eq. (4) that interchanging two rowsor two columns of the determinant changes the sign ofa. Since the PDF of the partial derivatives [Eq. (12)] isinvariant under all permutations of the derivatives, thePDF of vortex anisotropy a must be symmetric, i.e., de-pend only on a . Accordingly, we may first convenientlycalculate the probability density of a2 and from this ob-tain the density function for a itself. Recalling that inSection 3, a is defined as the ratio of the slope of theimaginary tangent plane to the slope of the real tangentplane, we have

a2 = (i= 0)2 + (iy0)2

(r.0)

2+ (ryO)2

P,2(a 2 ) = f dix f diY0 7 drxo f dryoF 2_ i ) i 1

(19)

X 8 a (ix0 )2 + (iy) 2 p(rx0 ,ryoix ,iy0 ).

(20)

The integrals are easily carried out in polar coordinatesby use of the circular symmetry of Eq. (12), yielding

Pa(2) 1l2(1 + a2)2 (21)

In Fig. 6, Eq. (21) is compared with the measured his-tograms, and satisfactory agreement is found betweentheory and experiment. Note that Eq. (21) implies thatthe most probable value for [l is (3)-l/2 = 0.577, so thatmost vortices have a substantial degree of anisotropy.

(16) Pa/<a>

Accordingly,

Pa2 (a2) = f di° 7 diyo f dr2° f dry°

X 8[a 2- (r 0)2 - (rY0)2]p(r 0, ry °, i 0 , iy°). (17)

The integrals over i and iy° are standard, and one caneasily carry out the integrals over rxo and ry° by switch-

4a/<a>

Fig. 5. Probability density Paj(a) of normalized vortex amplitudea/(a). The solid curve is the theory of Eq. (18).

1o2

p-

Isaac Freund

l


Q4

0.3 NI I

0.2 nl~

alt I L-4 -I U

aFig. 6. Probability density Pa of vortexsolid curve is the theory of Eq. (21).

A

V

both must be independent of a and a. This is confirmedby the appropriate scatterplots. Accordingly, we have forthe joint probability density of all four vortex parameters,

Pa,a,p,cr(a, a,p, a) = Paa(a, a)Pp(p)P,(u). (23)

6. DISCUSSION

Because of the circular symmetry of Eq. (12), the PDF's ofthe morphological vortex parameters-anisotropy a, ro-tation p, and skewness a-which determine the shape

2 4 6 of the vortex phase map, are independent of the parame-ters of the source. Since all sources with threefold orhigher symmetry are described by Eq. (12), all symmet-ric sources yield exactly the same density functions as inEqs. (13), (15), and (21) for the morphological vortex pa-rameters. This is in stark contrast to the PDF's of thepartial derivatives, which, although all Gaussian, havewidths that depend explicitly not only on the source sizebut even on the intensity. For asymmetric sources theform of the probability densities of the morphological vor-tex parameters remains uncertain. Also completely un-known at present is how the density functions evolveas the random wave field propagates from the near tothe far field. Still, the morphological vortex parameterswith their universal probability densities for symmetric

*; 'e > sources clearly provide a fundamentally more useful setof parameters for the description of optical vortices thando the partial derivatives. The results given here shouldbe applicable to any random scalar wave field obeyingcircular Gaussian statistics, including electron waves andneutron waves. More generally, we can anticipate that

vortices in such waves, which have scarcely been stud-ied at all, will undoubtedly be found to have the samefundamental importance as they do in random opticalwave fields.

01 I 1-4 -2

Fig. 7. Joint probabilityanisotropy a: (a) scattercoded map of Eq. (23).

0 2 4density of vortex amplitude a andplot of a/(a) versus a, (b) gray-scale

ACKNOWLEDGMENTS

I acknowledge useful discussions with I. Kantor,V. Freilikher, and D. Eliyahu. I also thank the re-viewer for suggesting the term morphological vortexparameters. This work was supported by the IsraelScience Foundation of the Israel Academy of Sciencesand Humanities.

5. a, a, p, a: oint Probability DensitiesVortex amplitude a and anisotropy a are highly corre-lated. Figure 7(a) displays a scatterplot that illustratesthe strong correlation that exists between these two pa-rameters. Converting Eq. (12) to polar coordinates andusing Eqs. (16) and (19), one can easily calculate the jointPDF's of a and a in the usual way,2 9 yielding

a~Ial - a2(1 + a 2)]

Pa, a (a,at) = 2b2 exp 2b .(22)

A gray-scale coded map of this function is shown inFig. 7(b) and may be seen to be in accord with the dataof Fig. 7(a).

Because there are no special directions for angles Preand pi, P and a- must be independent of each other, and

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Pa

. .

Isaac Freund

-b


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26. Since we assume a linearly polarized wave, formally, alongitudinal component is required in order for V * = 0also to be satisfied (see Ref. 7, first entry). As a practicalmatter this longitudinal component is unmeasurable, andnormally it may be neglected. Taking u = x, for our casethe full formal solution of Maxwell's equations is I(x, y, z) =[iF(x,y) - z(i/k)0F(x,y)/dx]exp(-ikz), where () is a unitvector oriented along the x axis (z axis).

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Isaac Freund

optical vortices in gaussian random wave fields: statistical probability densities

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