Critical-point level-crossing geometry in random wave fields

Isaac Freund Vol. 14, No. 8 /August 1997 /J. Opt. Soc. Am. A 1911

Critical-point level-crossing geometry in randomwave fields

Isaac Freund

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

Received July 5, 1996; revised manuscript received November 18, 1996; accepted February 5, 1997

Computer simulations and analytical calculation are used to study the geometry of level crossings of the realand the imaginary parts of wave functions and their spatial derivatives at the critical points of these functions,the intensity, and the phase. Zero-crossing maps that locate these critical points are presented, and topologi-cal constraints that determine the ordering of the critical points along the zero crossings are reviewed. Avail-able theoretical calculations of the critical point number densities are also reviewed and are compared withmeasured data from the simulations. Quantitative results are given for the directions of the level crossings,their radii and centers of curvature, and the structure of the bifurcation lines of saddle points. Correlationsbetween these quantities are also discussed. © 1997 Optical Society of America[S0740-3232(97)02508-8]

1. INTRODUCTIONLevel crossing (contour) maps provide informative, widelyused representations of two-dimensional fields in manyareas. Of special interest are zero crossing maps, whichserve to locate the important critical points—maxima,minima, saddle points, and phase singularities orvortices1—that define the field structure. Since a level-crossing map is a field of lines, its topology is constrainedby requirements of continuity and single valuedness.2

These constraints were recently shown to lead to numer-ous, sometimes surprising correlations among the variouscritical points.3,4

Certain types of level crossing statistics in randomfields have been studied extensively following the pio-neering work of Rice on one-dimensional fields5 and themilestone studies of Longuet-Higgins and of Cartwrightand Longuet-Higgins on two-dimensional fields.6 A re-view and extension of these classic works from the per-spective of the mathematician has been provided byAdler,7 and recent, important advances in theory and ex-periment have been made by Barakat,8 Ebeling,9 Asakuraand colleagues,10 and others.11,12 The solution of two-dimensional problems is sometimes facilitated by a map-ping onto the corresponding one-dimensional case, andsuch problems are the ones most widely studied. Classicexamples are the probability-density functions (PDF’s) forthe up and down crossings along particular directions andPDF’s of the intervals between these crossings.5,8–11 Butthere are other important properties, such as criticalpoint number densities that represent difficult, fully two-dimensional problems. As critical points always occur atintersections of two different level crossings, these densi-ties are fundamental to the present study. They are re-viewed in Appendix B, where available theoretical resultsare compared with data from recent large-scale computersimulations.13

Except for number densities of their intersections, itappears that little else is currently known quantitatively

0740-3232/97/0801911-17$10.00 ©

about critical point level crossings in random fields.Here we study the geometry of the level crossings of thereal and the imaginary parts of the wave function at theircritical points, at the vortices, and at critical points of theintensity and phase. For the critical points of the realand the imaginary parts of the wave function and for thevortices we use both computer simulations and analyticalcalculations based on Gaussian statistics. As the ana-lytical calculations are relatively straightforward but te-dious, we relegate them to Appendixes A and C. Forlevel crossings at critical points of the intensity and phasewe obtain empirical results only, using computer simula-tions. These simulations are briefly reviewed in Section2. In Section 3 we present zero-crossing maps for thereal and the imaginary parts of the wave function and fortheir derivatives as well as for derivatives of the intensityand the phase. These maps also locate all critical points,and we review the topological constraints that determinethe ordering of critical points along zero crossings. InSection 4 we define the geometric parameters used tocharacterize the different level crossings. In Section 5we discuss level-crossing statistics for real and imaginaryparts of the wave function at the different critical points,and in Section 6 we do the same for derivatives of thesefunctions. We summarize our conclusions in Section 7.Throughout, we illustrate relevant features of the wavefield, using maps obtained from our simulations to pro-vide a more complete picture of the level crossing struc-ture of random fields than may have hitherto been avail-able.

2. COMPUTER SIMULATIONAs our computer simulation and its extensive verificationhave recently been described in detail,13 only those as-pects needed for the present study of critical points arebriefly reviewed here. Our simulation assumes a statis-tically uniform circular source composed of randomly situ-ated point sources, each of which radiates into the far

1997 Optical Society of America

1912 J. Opt. Soc. Am. A/Vol. 14, No. 8 /August 1997 Isaac Freund

zone (Fourier-transforming geometry) a wave with unitamplitude and random phase. The simulation provides acomplete wave function from which all properties can becalculated numerically. The data obtained in this wayare the equivalent of experimental data (albeit of superiorquality) and are analyzed as such.The coordinates of the critical points were located to

within ;1025 of the coherence length of the wave field byfinding the zeros of appropriately defined functions.With these coordinates, the 12 variables corresponding tothe real (R) and the imaginary (I) parts of the wave func-tion and all their first and second derivatives (Rx5 ]R/]x, Rxy 5 ]2R/]x]y, etc.) were calculated in singleprecision and stored for later use. Ten randomly differ-ent source realizations were generated, and the data pre-sented are an average over these ten realizations. Thevortices were found from the zeros of uRu 1 uIu, the sta-tionary points of R (I) were found from the zeros of uRxu1 uRyu (uIxu 1 uIyu), the stationary points of the intensityU were found from the zeros of uUxu 1 uUyu, and the sta-tionary points of the phase w were found from the zeros ofuwxu 1 uwyu. As the zero-amplitude minima of U also cor-respond to the vortices, the nearly perfect agreement be-tween the locations of these zeros and the directly deter-mined coordinates of the vortices provided anindependent check of the algorithms. Although no algo-rithm can be certain of finding every single one of thehundreds of often closely spaced critical points present ineach wave-field realization, by use of the checks describedpreviously13 it was possible to determine that a nearly100% efficiency was, in fact, achieved. This nearly per-fect efficiency is verified by the nearly perfect agreementbetween theory and measurement for the number densi-ties of the 13 different critical points discussed in Appen-dix B. It must be emphasized that every wave field prop-erty obtained from the simulation has always been foundto be in excellent quantitative agreement with Gaussiantheory in all those cases (approximately three dozen innumber) for which theory is available.3,13 This is as ex-pected, since the simulation is a numerical implementa-tion of the standard random phasor model of Gaussianfields universally used in development of the theory.14

Accordingly, I believe that the data presented here pro-vide a reliable, representative picture of isotropic Gauss-ian fields. These data are used not only to check the ana-lytical calculations but also to provide insight into thephysical meaning of calculated results. The data alsoprovide significant new information on a broad range ofwave-field properties that have not yet been calculated.

3. ZERO CROSSINGSHere we discuss zero-crossing maps of the real and theimaginary parts of the wave function [Fig. 1(a)] and theirderivatives [Fig. 1(b)] as well as maps of the zero cross-ings of the derivatives of the intensity [Fig. 1(c)] and de-rivatives of the phase [Fig. 1(d)]. These maps locate thevarious critical points and provide an overview of thezero-crossing structure of a generic random field.Vortices are located at intersections of the zero cross-

ings of the real and the imaginary parts of the wave func-tion [Fig. 1(a)].1 As can be seen, along any zero crossing

the vortex signs alternate because of a topological con-straint called the sign principle.3 This principle was re-cently expanded4 and has also been shown to give rise tomany remarkable near-neighbor vortex–vortexcorrelations.3

Stationary points (maxima, minima, and saddle points)of R (I) are located at intersections of the zero crossingsof Rx and Ry (Ix and Iy), and a typical map of these zerocrossings is shown in Fig. 1(b). The zero crossings ofRx (Ry) can be seen to be elongated in the y (x) directionand tend to be equally spaced. This gives rise to a quasi-crystalline structure for the stationary points in whichthere is a distorted square unit cell with a maximum atthe cell center, saddle points along each side, and minimaat the corners.13 As can be seen from Fig. 1(b), along ev-ery zero crossing, extrema alternate with saddle points.This is a result of a topological constraint that can beviewed as an extension to stationary points4 of the signprinciple, which was originally formulated for vortices.There is a further, rather more subtle level of ordering

for the extrema, which is shown in Fig. 2, where positive(negative) regions of Rx [Fig. 2(a)] and Ry [Fig. 2(b)] arecolored white (black). The boundaries between black andwhite regions are the zero crossings of Fig. 1(b). As canbe seen, along any segment of a zero crossing that doesnot reverse its direction all extrema are of the same kind(either maxima or minima). This possibly unexpected or-der results from the fact that Rx increases (decreases) oncrossing a boundary from black to white (white to black),whereas at a maximum (minimum), ]Rx /]x 5 ]2R/]x2 ispositive (negative) definite [Fig. 2(a)]. An equivalent re-sult holds also for Ry [Fig. 2(b)].Stationary points of the intensity U are located at in-

tersections of the zero crossings of Ux and Uy and includethe vortices, which are minima of zero intensity. Typicalzero-crossing maps of Ux and Uy are shown in Fig. 1(c).Here too, topological constraints force extrema andsaddles to alternate along every zero crossing.4 An un-usual feature of the intensity is that small isolated clus-ters of maxima tend to be surrounded by chains of closelyspaced minima and saddles. This is evident from themap of Fig. 1(c), where there are long intertwined seg-ments of Ux and Uy that surround maxima clusters, withsaddles and minima alternating along these segments.Stationary points of the phase w are located at intersec-

tions of the zero crossings of wx and wy , and a typical mapof these zero crossings is shown in Fig. 1(d). Although wand all its derivatives are singular at a vortex, vorticesare also located at intersections of the zero crossings ofwx and wy .

4 This possibly unexpected behavior is evi-dent in Fig. 1(d). Phase extrema and saddles are notshown explicitly, but their positions on the map areuniquely determined by a topological constraint that re-quires saddles to alternate with either vortices orextrema.4 Accordingly, on every zero crossing of thismap vortices must be separated by an odd number of un-marked intersections. If a single intersection separates apair of vortices, then this intersection must contain asaddle, whereas if there are, say, three unmarked inter-sections in a row, the middle intersection must house anextremum. From Fig. 1(d) we observe that there are fewunmarked adjacent intersections, so extrema are evi-


Fig. 1. Zero-crossing maps of a generic random wave field. (a) Thick curves, real (R) and thin curves, imaginary (I) parts of the wavefunction; filled squares, positive vortices; open squares, negative vortices. (b) Thick curves, ]R/]x; thin curves, ]R/]y; filled squares,extrema (maxima or minima) of R; open squares, saddle points of R. (c) Thick curves, ]U/]x; thin curves, ]U/]y, where U is thewave-field intensity; filled squares, extrema (maxima or minima) of U; open squares, saddle points of U. (d) Thick curves, ]w/]x; thincurves, ]w/]y, where w is the phase; filled squares, vortices (positive and negative); open squares, saddle points of w. These maps are190 pixels wide (x axis) and 220 pixels high ( y axis), where the coherence length Lcoh of the wave field is 19.2 pixels (Appendix A). Allmaps cover the same region of the wave field and are discussed in Section 3.

dently rare. As was discussed recently,15 they are out-numbered by saddles by ;14:1. Finally, recalling that thetopological (Poincare) index2 of extrema and vortices (ofeither sign) is 11, whereas that of saddles is 21, and de-fining a generic derivative zero crossing as one that al-ways passes through the critical points, we can usefullysummarize the overriding topological rule4 that governsthe ordering of all the different types of critical pointsalong a derivative zero crossing, as follows:

Adjacent critical points on a generic derivative zerocrossing must have indices of opposite sign.

4. LEVEL-CROSSING GEOMETRYOf the various parameters that determine the structure(geometry) of the level crossing of a two-dimensional func-tion f(x, y) at point x0 , y0 , the most important is the di-rection (slope) of the curve:

tan u0 5 S dydx Dx0 ,y0

5 2fx~x0 , y0!

fy~x0 , y0!. (1)

We obtain Eq. (1) by noting that along a level-crossingcurve y(x) on which f assumes a constant value,


df 5 fxdx 1 fydy 5 0. The angle u, which is positiveas measured counterclockwise from the x axis, will be re-ferred to as the level-crossing angle. For simplicity, wecalculate u from the parent function f(x, y) through theequation u 5 2arctan( fx /fy) together with the restriction2p/2 < u < p/2, although it is possible to extend thedefinition consistently to include the range 2p to 1p.At the stationary points of f itself, where fx 5 fy 5 0,

the level crossings of the function have directions that aresingular [Eq. (1)]. At maxima and minima the curvesshrink to a point, whereas at saddles the level crossingsintersect themselves and their directions becomes for-mally undefined [Fig. 3(a)]. The component lines of theseself-intersections are referred to as bifurcation lines be-cause, as illustrated in Fig. 3(a), these lines divide the

Fig. 2. Maps of (a) the x derivative ]R/]x and (b) the y deriva-tive ]R/]y of the real (R) part of the wave function. The mapscorrespond to Fig. 1(b). Positive (negative) regions of the func-tions are colored white (black). Filled squares, maxima of R;open squares, minima of R. The boundaries between black andwhite regions are the zero crossings in Fig. 1(b). These mapsare discussed in Section 3.

field into separate regions. Of special interest is theangle between the lines, the bifurcation angle V. As dis-cussed previously4 and as illustrated in Figs. 3(b) and3(c), in a random wave field maxima and minima must beenclosed in a loop formed by the bifurcation lines of asaddle (bifurcation loop). If the bifurcation angle issmall, for example, the resulting loop will tend to be longand narrow and the enclosed extremum will thereforetend to be a highly elongated ellipse. Bifurcation anglestherefore yield useful information on the shapes ofsaddles and their associated extrema.In the principal axis system (X, Y) of a saddle [Fig.

3(a)] centered for simplicity on the origin, we can write, tofirst order in X and Y, f(X, Y) 5 f0 1 1/2( f1X

2

1 f2Y2), where the eigenvalues of the Hessian matrix

]( fx , fy)/](x, y) are f6 5 1/2@T 6 (T2 2 4J)1/2#, thetrace T 5 fxx 1 fyy , and the Jacobian J 5 fxx fyy 2 fxy

2 isnegative definite. At a saddle point one principal axis isa minimum and the other a maximum, and, as f1 . f2 , itfollows that f1 . 0 and f2 , 0. The equations of the bi-furcation lines are obtained as the roots Y6(x) off( X, Y) 2 f0 5 0, so we have for the bifurcation angleVX centered on the direction of steepest ascent (X axis)

VX 5 22 arctan~ f1 /f2!, 0 < VX < p, (2a)

and for its complement V(Y ) centered on the direction ofsteepest descent (Y axis)

VY 5 p 2 VX , 0 < VY < p. (2b)

Fig. 3. Saddle point (filled circles) and its bifurcation lines(dashed curves). (a) Contour map showing the principal axesX and Y (directions of steepest ascent and descent; thick lines)and associated bifurcation angles VX and VY . (b), (c) Bifurca-tion loops. (b) Figure eight with labeled interior (int.) and exte-rior (ext.) angles. The two loops of a figure eight contain thesame type of extremum (maximum or minimum). (c) Reentrantsaddle. The two loops of a reentrant saddle contain extrema ofopposite types. (d) The bifurcation lines of the phase saddlesare generically terminated by vortices with signs as shown.These graphs are discussed in Section 4.


The direction of steepest ascent (descent) leads to a maxi-mum (minimum), so loops containing VX must enclosemaxima and those containing VY must enclose minima.From Eq. (2b) it follows that the PDF’s P(VX) andP(VY) are mirror images, and so only P(VX) is displayed.As shown by the examples in Fig. 1, level crossing are

generally not straight lines but are curved. The radius ofcurvature r of the level crossing is by definition positiveand is given by16

r 5 uDuAfx2 1 fy2, D 5

fx2 1 fy

2

2 fx fyfxy 2 fx2fyy 2 fy

2fxx,

(3a)where the derivatives are evaluated at the critical point,x0 , y0 . The coordinates xc , yc of the associated center ofcurvature are given by16

xc 5 x0 1 fxD, yc 5 y0 1 fyD. (3b)

5. STATISTICS OF LEVEL CROSSINGS OFRANDOM FIELDSWe consider here the level-crossing statistics of R and Ifor the field as a whole, the stationary points of R andI, the stationary points of the intensity U and the phasew, and the vortices.

A. Level-Crossing Directions (Whole Field)For the whole field the normalized PDF’s of the level-crossing directions of R and I can be shown to be indepen-dent of level height. This possibly surprising result is adirect consequence of Gaussian statistics and is discussedfurther in Appendix C. Since there are no preferred di-rections in an isotropic wave field, the level-crossingangles uR and uI for both the real and the imaginary partsof the wave function must be uniformly distributed whenthey are averaged over the whole field. This is confirmedfrom measurement in Fig. 4(a) and from Gaussian theoryin Eq. (C4) below. As R and I are also statistically inde-pendent when averaged over the whole field, the PDF ofthe difference angle DRI 5 uR 2 uI can be obtained by in-spection as the PDF of the separation uDRIu between twopoints uniformly and randomly distributed along a line oflength p, i.e., PF(DRI) 5 (p 2 uDRIu)/p2. Also, this isconfirmed from theory in Eq. (C5) below and from mea-surement in Fig. 4(b).

B. Bifurcation Angle VThe stationary points of the real and the imaginary partsof the wave function necessarily have the same statisticsas R and I and can be interchanged by a uniform phasetransformation that leaves the measurable statistics in-variant. Because at their respective stationary pointsR and I remain statistically independent, the level-crossing angle uI is uniformly distributed at the station-ary points of R, and the level crossing angle uR is uni-formly distributed at the stationary points of I.The PDF of the bifurcation angle VX at saddle points of

R (I) is shown in Fig. 5(a). Although VX is always cen-tered on the direction of steepest ascent (X axis), thisangle is not necessarily the interior angle of a bifurcationloop that encloses a maximum and can, instead, be the ex-

terior angle of a loop that encloses a minimum [Fig. 3(b)].P(VX) in and of itself is therefore insufficient to deter-mine the shape of bifurcation loops. As discussedpreviously,17 almost all maxima of R (I) are positive,whereas almost all minima are negative. We also findempirically that figure eights [Fig. 3(b)] are much morecommon than reentrant saddles [Fig. 3(c)], so a saddlepoint is almost always uniquely associated with either amaximum or a minimum but rarely with both. Accord-ingly, positive saddles (i.e., those for which f at the saddlepoint is positive) will tend to enclose maxima, whereasnegative saddles will tend to enclose minima. By exam-ining P(VX) separately for positive and negative saddleswe can therefore separate bifurcation loops containingmaxima from those containing minima. This is done inFig. 5(a), where the solid (dotted) curve is the PDF forpositive (negative) saddle heights. Because for maximaVX is the interior angle but for minima its complement

Fig. 4. Whole-field PDF’s PF of (a) the level-crossing angle [Eq.(1)] of the real part R of the wave function uR and (b) the differ-ence angle DRI 5 uR 2 uI , where uI is the level-crossing angle ofthe imaginary part I of the wave function. In this and succeed-ing figures histograms are data measured from our computersimulation discussed in Section 2; here and in Figs. 5, 9(a), and10–13 angles are measured in degrees with the PDF’s normal-ized to reflect this scale. The solid curve in (a) is the theoreticalcalculation of Appendix C [Eq. (C4)], and the curve in (b) is Eq.(C5). PF(uI) is the same as PF(uR) and is therefore not shown.These graphs are discussed in Subsection 5.A.


VY is the interior angle, we observe from Fig. 5(a) that atboth maxima and minima the interior angle has a distri-bution centered on ;60°. This implies that bifurcationloops and their enclosed extrema tend to be ellipses withmajor axes oriented approximately in the direction of thesaddle point [Fig. 3(b)]. This behavior is evident on themap of R shown in Fig. 6.

Fig. 5. PDF’s of the bifurcation angle VX in degrees [Fig. 3(a)]for saddle points of (a) the real part R of the wave function, (b)the intensity U, and (c) the phase w. In (a) the solid (dotted)curve is for positive (negative) saddle points whose heights aregreater than (less than) 0.1^R2&1/2 (20.1^R2&1/2), and in (b) thesolid (dotted) curve is for saddle points whose heights are greaterthan (less than) 1/2Û&, where ^ & are whole-field averages.These graphs are discussed in Subsection 5.B.

A bifurcation angle analysis is also of interest forsaddle points of intensity U. As shown in Fig. 5(b), herethe bifurcation angle distribution (the histogram) ishighly asymmetric, reflecting the intrinsic asymmetry be-tween intensity maxima and minima. The height distri-butions of intensity maxima and minima15 reveal thatmost maxima have a height substantially greater thanhalf of the average intensity 1/2Û&, whereas almost allminima have heights that are much less than 1/2Û&.We find empirically also for the intensity that figureeights [Fig. 3(b)] greatly outnumber reentrant saddles[Fig. 3(c)], so we can assume that saddle points whoseheights exceed 1/2Û& [solid curve in Fig. 5(b)] are associ-ated mostly with maxima and those with heights lessthan 1/2Û& [dotted curve in Fig. 5(b)] are associatedmainly with minima. This assumption is supported bythe fact that, of the 2181 saddle points in our intensitydata set, 767 (1414) have heights greater (less) than1/2Û&, which compares well with the 837 maxima and1341 minima that are present. From the dotted curve inFig. 5(b) we observe that bifurcation loops enclosingminima tend to have small bifurcation angles and aretherefore highly asymmetric. This is evident from Fig. 7,which shows a representative map of the intensity.Worth noting is that these elongated saddles and theirenclosed elliptical minima tend to form chains that sur-round isolated small clusters of oval shaped maxima.17

Figure 5(c) shows the distribution of bifurcation anglesfor phase saddles. Here, most bifurcation lines (.95%)do not form closed loops but terminate on vortices [Fig.3(d)].4 Figure 5(c) indicates that the resulting X-shapedstructures tend to be symmetric, with a most probable bi-furcation angle of 90°. Our simulation also verifies that

Fig. 6. Map of the real part of the wave function. Alternatelevels are colored black and white. Maxima and minima corre-spond to closed contours; saddle points are located at the centersof the distorted X-shaped black or white regions (bifurcationlines; Fig. 3). As can be seen, almost all saddles are figureeights [Fig. 3(b)]. This map is 100 pixels wide and 135 pixelshigh and is discussed in Subsection 5.B.


P(VX) is independent of saddle point height, as expectedfrom the fact that there can be no special values for thephase.

C. Level Crossings at Stationary Points of Intensity andPhaseAt the stationary points of both the intensity and thephase, isotropy of the wave field guarantees that uR anduI are uniformly distributed. However, except at vorti-ces, the difference angle DRI 5 uR 2 uI has the PDF

PU~DRI! 5 Pw~DRI! 5 d~DRI!. (4)

For the intensity this possibly startling result followsfrom U 5 R2 1 I2 together with Ux 5 Uy 5 0, U . 0,whereas for the phase it follows from w 5 arctan( I/R) to-gether with wx 5 wy 5 0, R2 1 I2 . 0. In both casesRx /Ry 5 Ix /Iy , so uR 5 uI . As Eq. (4) has the determin-istic derivation given here, a statistical derivation is notprovided in Appendix C. Note that the above argumentfails for the vortices that are zero minima of U whereR 5 I 5 0. These critical points are treated in Subsec-tion 5.D.Equation (4) implies that, at stationary points of the in-

tensity and the phase, level crossings of R and I are tan-gent to each other. We elucidate the structure of thesecontacts with the aid of the radii and centers of curvatureof the level crossings [Eqs. (3)]. In Fig. 8(a) we displaymeasured PDF’s for the radii of curvature r of R (I) at in-tensity maxima (M), saddle points (S), and nonzerominima (m). Here r is measured in units of the wave-field coherence length Lcoh 5 19.2 pixels for our simula-tion (Appendix A). As can be seen, the distribution formaxima is strongly peaked at the origin, which impliesthat the most probable contours are small closed curves.

Fig. 7. Map of the intensity. This map covers the same wave-field region as Fig. 6 and is colored in the same way. Here al-most all maxima are overexposed and are white, whereas almostall minima are underexposed and are black. As can be seen,chains of saddles (Fig. 3) and highly elongated minima tend tosurround clusters of maxima. This map is discussed in Subsec-tion 5.B.

Since U 5 R2 1 I2, this distribution for maxima of Ucorresponds to being near either maxima or minima ofR and I. Supporting this conclusion is the systematicnarrowing of the distributions in Fig. 8(a) in going fromminima to saddles to maxima, which is the direction ofincreasing intensity15 and therefore also of increasingR and I.

Fig. 8. Level-crossing radii of curvature r [Eq. (3a)] of the real(R) and the imaginary (I) parts of the wave function at station-ary points of the intensity. r is measured in units of the wave-field coherence length Lcoh (Appendix A). (a) The PDF P(r) atintensity maxima (M), saddle points (S), and minima (m). (b)Correlation scatter plot of rI versus rR at intensity maxima,showing strong anticorrelation. (c) Exterior and (d) interior con-tacts of curves with different radii of curvature r1 and r2 . Asshown by the dashed and dotted curves, the number of possibili-ties (phase space) for exterior contacts exceeds that for interiorcontacts. These graphs are discussed in Subsection 5.C.


Figure 8(b) displays a correlation scatter plot of rI , theradius of curvature of I, versus rR , the radius of curva-ture of R, for intensity maxima. As can be seen, rI andrR are strongly anticorrelated, since large values of rI arepaired with small values of rR and vice versa. This im-plies that if, say, R is very close to one of its maxima, thenI is relatively much farther away from any of its maxima(or minima). This conclusion is directly verified by visualinspection of the appropriate contour plots (not dis-played), which show the relevant contours to be osculat-ing ellipses of rather different sizes. This finding is alsoconsistent with our recent discovery that at speckle spot(intensity) maxima R and I are themselves stronglyanticorrelated.17

Figures 8(c) and 8(d) display the two generic possibili-ties for the contacts between R and I at intensitymaxima. In obvious notation we label the contact in Fig.8(c) ‘‘exterior’’ and that in Fig. 8(d) ‘‘interior.’’ At anygiven critical point the type of contact can be identifiedfrom the locations of the centers of curvature [Eq. (3b)].Drawing vectors from the critical point (contact point) tothe centers of curvature of R and I, we label the angle be-tween these vectors cRI . cRI 5 0° corresponds to an in-terior contact [Fig. 8(d)], and cRI 5 180° corresponds toan exterior contact [Fig. 8(c)]. Denoting by r the averageratio of exterior/interior contacts, we obtain from oursimulations r 5 1.6 for intensity minima, r 5 2.2 for in-tensity saddles, and r 5 3.6 for intensity maxima(60.1). In all cases r . 1, presumably reflecting the factthat the phase space of exterior contacts always exceedsthat of interior contacts [Figs. 8(c) and 8(d)]. This is alsoconsistent with the trend in r in going from minima tomaxima since the relative advantage of exterior contactsincreases with decreasing radii of curvature. Simple geo-metric (phase space) arguments [Figs. 8(c) and 8(d)] sug-gest that r ; r* [ ^(rR 1 rI)/urR 2 rIu&, whereas ourcomputer simulation yields r ; (1/2)r* . We attributethis (not unreasonable) discrepancy to the anticorrelationbetween rR and rI shown in Fig. 8(b), which tends to re-duce the advantage of exterior over interior contacts.

D. Level Crossings at VorticesAs the vortices do not break wave-field isotropy, thePDF’s of the zero-crossing angles uR and uI are uniformlydistributed. The PDF of their difference, DRI 5 uR2 uI , is given in Eq. (C15) below as Pv(DRI) } (p2 uDRIu)sinuDRIu and is shown in Fig. 9(a). The first termin this expression reflects the uniform distribution andstatistical independence of the zero-crossing directionsand for this reason is equivalent to the whole-field PDFPF(DRI) [Subsection 5.A and Eq. (C5) below]. But a vor-tex is created only when the zero crossings of R and I in-tersect, so Pv(DRI) must also reflect the probability of in-tersection. Since parallel lines never meet, thisprobability vanishes when uR 5 uI and grows with in-creasing angular difference, reaching a maximum at DRI5 90°. This is the source of the sinuDRIu factor inPv(DRI). We note that, although it is derived in Appen-dix C from Gaussian statistics, Pv(DRI) is independent ofall wave-field parameters and so must also be obtainablefrom purely geometric considerations.

The measured PDF of the radii of curvature r of thezero crossings of R(I) at the vortices is shown in Fig. 9(b).Unlike the data in Fig. 8(a), which peak at the origin,Pv(r) starts at zero and rises to a maximum atr ; 0.5Lcoh , where as before Lcoh is the coherence lengthof the wave field. These data yield an average radius ofcurvature of ^r&v ; 7Lcoh . The whole-field PDF (not

Fig. 9. Geometry of zero crossings of R and I at vortices. (Seecaptions of Figs. 4 and 8 for explanations of symbols and units).(a) PDF of difference angle DRI 5 uR 2 uI . The solid curve isEqs. (C15). (b) PDF of radius of curvature r of the zero crossingof R. (c) PDF of cos cRI , where cRI is the angle between vectorsdrawn from the vortex center to the centers of curvature of R andI. The dashed line is Eqs. (5). These graphs are discussed inSubsection 5.D.


shown), which also vanishes at the origin, peaks atr ; 0.25Lcoh and has ^r&F ; 4Lcoh . These differencesindicate that at vortices the zero crossings of R and I tendto be significantly straighter than throughout the remain-der of the wave field, a conclusion that appears consistentwith Fig. 1(a). In retrospect this behavior is expected,since nonparallel straight lines always intersect whereascurved ones may veer away and never intersect.Figure 9(c) displays the PDF of cos(cRI), where cRI dis-

tinguishes exterior from interior contacts and was definedin Subsection 5.D. In contrast to the stationary points ofthe intensity and the phase, for the vortices Pv@cos(cRI)# isnot constrained to be nonzero only at cos(cRI) 5 61 but isfound empirically to be the same for all cos(cRI). Thesedata therefore imply that

Pv@cos~cRI!# 512 , Pv~cRI! 5

12 sin~cRI!. (5)

Although cRI and DRI are closely related, their PDF’s[Eqs. (5) and (C15)] can be seen to differ by a factor of(p 2 uDRIu). This difference gives relatively more weightto larger values of cRI and reflects the higher probabilityof exterior contacts (cRI . 90°). We add for complete-ness that for the whole field PF(cRI) 5 1/p, soPF@cos(cRI)# 5 @p (1 2 cos2 cRI)

1/2#21.

6. STATISTICS OF LEVEL-CROSSINGDIRECTIONS OF DERIVATIVES OF RANDOMFIELDSA. Derivative Level Crossings of the Whole FieldFor the whole field also the normalized PDF’s of deriva-tive level-crossing directions are independent of levelheight. This is again a direct consequence of Gaussianstatistics and is discussed further in Appendix C. Asshown in Fig. 1(b), differentiation breaks wave-field isot-ropy and the zero crossings of Rx tend to be elongated par-allel to the y axis whereas those of Ry are elongated par-allel to the x axis. This is reflected in the PDF’s of thelevel-crossing directions ux of Rx [Eq. (C6a) below] anduy of Ry [Eq. (C6b)], which are shown in Figs. 10(a) and10(b). Although the PDF of their difference, Dxy 5 ux2 uy [Eq. (C7)], diverges at Dxy 5 690°, it does so onlylogarithmically, so the tendency for perpendicular align-ment is weak.

B. Derivative Level Crossings at Stationary Points of Rand IStationary points (maxima, minima, and saddle points) ofR and I are located at the intersections of the zero cross-ings of Rx and Ry or Ix and Iy [Fig. 1(b)]. Passing thewave through a suitable 180° phase shifter (uniform glassplate) turns maxima of R or I into minima and vice versa.But from the wave field alone it is not possible determinesuch a history, so the level crossing PDF’s of R, I, andtheir derivatives must be the same for both maxima andminima. The PDF of the zero-crossing direction ux ofRx is shown in Fig. 11(a) and can be seen to be sharplypeaked at uuxu 5 90° and to vanish for ux 5 0 [Eq. (C9a)].The PDF of the zero-crossing direction uy of Ry shown inFig. 11(b), on the other hand, is sharply peaked atuy 5 0° and vanishes for uuyu 5 90° [Eq. (C9b)]. ThePDF of their difference, Dxy 5 ux2uy , shown in Fig. 11(c)

vanishes when these zero crossings are parallel andreaches a maximum when they are perpendicular [Eq.(C10)]. These tendencies are evident in Fig. 1(b) and, asdiscussed previously, give rise to short-range quasi-crystalline order for the stationary points.13

In contrast to maxima and minima, at the saddle pointsof R (I) both ux and uy are uniformly distributed [Fig.12(a)]. As discussed in Appendix C, this possibly unex-

Fig. 10. Whole-field PDF’s PF for the directions of the levelcrossings of the x and y derivatives of R. (a) PDF of the direc-tion ux of ]R/]x. The curve is Eq. (C6a). (b) PDF of the direc-tion uy of ]R/]y. The curve is Eq. (C6b). (c) PDF of the differ-ence angle Dxy 5 ux 2 uy . The curve is Eqs. (C7). All anglesare in degrees. These graphs are discussed in Subsection 6.A.


pected behavior is a nonobvious mathematical conse-quence of wave-field isotropy [Eq. (C12)]. Physically, thisbehavior reflects the fact that, if these zero crossings wereas highly oriented parallel to the x and y axes at saddlepoints as they are at maxima and minima, then the wavefield as a whole would undoubtedly exhibit long-rangequasi-crystalline order. But this is impossible in a mac-roscopically isotropic field, and the necessary degree oflong-range randomness is evidently provided by the

Fig. 11. PDF’s for the directions of the zero crossings of the xand y derivatives of R at extrema (maxima and minima) of thisfunction. (a) PDF of the direction ux of ]R/]x. The curve is Eq.(C9a). (b) PDF of the direction uy of ]R/]y. The curve is Eq.(C9b). (c) PDF of the difference angle Dxy 5 ux 2 uy . Thecurve is Eqs. (C10). All angles are in degrees. These graphsare discussed in Subsection 6.B.

saddle points. However, short-range order is preserved.This is evident from the PDF of Dxy 5 ux2uy shown inFig. 12(b), which, just as for maxima and minima [Fig.11(c)], vanishes when the zero crossings are parallel andreaches a maximum when they are perpendicular [Eq.(C13)].

C. Derivative Level Crossings at VorticesAt vortices the level crossings of Rx and Ry (Ix and Iy) dis-played in Figs. 13(a) and 13(b) show only a weak tendencyfor alignment parallel to the y and x axes [the theory forthese PDF’s is given in the discussion preceding Eqs.(C16)]. However, the PDF of the difference angle Dxyshown in Fig. 13(c) does suggest a significant correlationbetween level-crossing directions that leads to a tendencytoward perpendicular orientation [Eqs. (C16)]. Althoughthese results have no obvious physical interpretation,they are included here for completeness.

7. SUMMARYCritical points of the real and the imaginary parts of thewave function, of the intensity, and of the phase are all

Fig. 12. PDF’s for the directions of the zero crossings of the xand y derivatives of R at saddle points of this function. (a) PDFof ux . The PDF of uy is equivalent and is therefore not shown.The curve is Eqs. (C12). (b) PDF of the difference angle Dxy .The curve is Eq. (C13). All angles are in degrees. Thesegraphs are discussed in Subsection 6.B.


located at the intersections of zero crossings. The topol-ogy and the geometry of these crossings determine thecritical point structure as well as correlations betweenneighboring critical points. Zero crossings of the x andy derivatives of the real and also of the imaginary parts ofthe wave function tend to form a distorted lattice that ex-hibits short-range order with a unit cell containing one

Fig. 13. PDF’s for the directions of the level crossings of the xand y derivatives of R at vortices. (a) PDF of ux . (b) PDF ofuy . The curves in (a) and (b) are, respectively, Eqs. (C6a) andEq. (C6b) with parameters listed in the discussion preceding Eqs.(C16). (c) PDF of the difference angle Dxy . The curve is Eqs.(C16). All angles are in degrees. These graphs are discussed inSubsection 6.C.

maximum, one minimum, and two saddle points. Zerocrossings of the x and y derivatives of the intensity tendto form intertwined segments containing chains of alter-nating saddles and minima that surround small clustersof oval speckle spots. Critical points of the phase are lo-cated at intersections of the zero crossings of the x andy derivatives of the phase. These critical points includethe vortices, at which all derivatives become singular.On every generic derivative zero crossing the topologi-

cal index of adjacent critical points must be of oppositesign. This important constraint generalizes a rule origi-nally formulated for vortex signs.Extrema are always enclosed in bifurcation loops

formed by saddles. The shapes of these loops determinethe shapes of the enclosed extrema. The interior angle ofthe bifurcation loops of the real and also of the imaginaryparts of the wave function has a distribution centered at;60°, and the associated extrema tend to be ellipses withmajor axes oriented in the direction of the saddle point.The interior angle of the bifurcation loops of the intensitythat contain minima has a distribution that peaks at 0°,and the enclosed minima tend to be highly elongated.Bifurcation lines of phase saddles are almost always

(.95%) terminated by vortices. Vortices that terminatea given bifurcation line must be of the same sign, whereasvortex signs alternate from one line to the other. Thedistribution of bifurcation angles for phase saddles peaksat 90°, resulting in a roughly symmetric X-shaped struc-ture for the saddle and its associated vortices.At the critical points of the intensity and phase that do

not correspond to vortices, level crossings of the real andthe imaginary parts of the wave function are always tan-gent. At intensity maxima these level crossings tend toform closed curves whose radii of curvature are anticorre-lated, so a small roughly circular crossing of the real partof the wave function will tend to be paired with a substan-tially larger elliptical crossing of the imaginary part andvice versa. Exterior contacts are ;4 times more prob-able that interior contacts, so the circle will most often lieoutside the ellipse rather than inside it.At extrema of the real or the imaginary parts of the

wave function the zero crossing of the x derivative neverparallels the x axis but is most likely to be perpendicularto this axis. The zero crossing of the y derivative be-haves similarly with respect to the y axis. At the saddlepoints of these functions all directions are equally prob-able for the zero crossings of both derivatives, but the di-rections of these zero crossings are highly anticorrelatedsuch that x- and y-derivative zero crossings are neverparallel but are most likely perpendicular. In a similarfashion, the directions of the zero crossings of the real andthe imaginary parts of the wave function are never paral-lel at vortices but are rather most likely to cross at;60°.Although the quantitative geometric distributions and

correlations given here are specific to an isotropic Gauss-ian field, many of the qualitative features described arelikely to be present also in other random wave fields.Knowledge of these structural features and their correla-tions may therefore also prove useful in the correctionand manipulation of a wide variety of abberated opticalfields.


APPENDIX A: GAUSSIAN STATISTICS OFISOTROPIC RANDOM FIELDSHere we review the Gaussian PDF of the set of 12 vari-ables that comprise R, I, and all their first and second de-rivatives. Denoting this PDF by PRI , and writing as be-fore Rx 5 ]R/dx, Rxx 5 ]2R/dx2, etc., we have

PRI~R, Rx , Ry , Rxx , Ryy , Rxy ; I, Ix , Iy , Ixx , Iyy , Ixy!

5 W~R, Rx , Ry , Rxx , Ryy , Rxy!

3 W~I, Ix , Iy , Ixx , Iyy , Ixy!, (A1a)

W~ f, fx , fy , fxx , fyy , fxy!

51

~2p!3AKexpH 2

12 FAf 2 1 ~1/b !~fx

2 1 fy2!

1 B~fxx2 1 fyy

2! 11dfxy

2

1 2Cf~fxx 1 fyy! 1 2Dfxx fyyG J , (A1b)

K 5 Mb2d, M 5 ~c 2 d !@a~c 1 d ! 2 2b2#, (A1c)

A 5c2 2 d2

M, B 5

ac 2 b2

M,

C 5b~c 2 d !

M, D 5

b2 2 adM

, (A1d)

a 5 ^R2&, b 5 ^Rx2&,

c 5 ^Rxx2&, d 5 ^Rxy

2&. (A1e)

PRI , although obtained by well-known standardmethods,18 does not appear to be readily available else-where in so explicit a form. In Eqs. (A1) use has beenmade of the fact that for isotropic fields

^R2& 5 Î2&, ^Rx2& 5 ^Ry

2& 5 Îx2& 5 Îy

2&,

^Rxx2& 5 ^Ryy

2& 5 Îxx2& 5 Îyy

2&,

^RRxx& 5 ^RRyy& 5 ÎIxx& 5 ÎIyy& 5 2^Rx2&,

^RxxRyy& 5 ÎxxIyy& 5 ^Rxy2&.

The correlators a, b, c, and d can be obtained fromthe autocorrelation function mRR(Dx, Dy)5 ^R(x, y)R(x1 Dx, y 1 Dy)& 5 mII(Dx, Dy) 5 Î(x, y)I(x 1 Dx, y1 Dy)& as19

a 5 @mRR#0,0 , b 5 2F ]2mRR

]~Dx !2G0,0

,

c 5 F ]4mRR

]~Dx !4G0,0

, d 5 F ]4mRR

]~Dx !2]~Dy !2G0,0

, (A2a)

where @ #0,0 implies evaluating the indicated derivativesat the origin, Dx 5 0 and Dy 5 0. For the uniform cir-cular source of diameter Ds and far-field (Fourier-transforming) geometry used in our computer simulation,the Van Cittert–Zernike theorem14 gives mRR(Dx, Dy)5 2^R2&J1(u)/u, where u 5

12kDs@(Dx)

2 1 (Dy)2#1/2,J1 is the Bessel function of order 1,20 and k 5 2p/(lZ),

with l the wavelength of light and Z the separation be-tween the source and the (distant) screen on which thewave field is observed. For this autocorrelation functionthe coherence length of the far-field speckle pattern isconventionally taken14 as Lcoh 5 j1,1 /(1/2kDs), where j1,15 3.83... is the first zero of J1 .

20 From Eq. (A2a) we thenhave

a 5 ^R2&, b 5^R2&k2Ds

2

16,

c 5^R2&k4Ds

4

128, d 5

c3. (A2b)

As becomes apparent in Appendix C, the fact thatd 5 c/3 is of special importance. This relationship is adirect consequence of the rotational symmetry of the wavefield, as is easily demonstrated by expanding the generalrotationally symmetric autocorrelation function in a Tay-lor series about the origin.The source in our computer simulation (Section 2) con-

sists of 104 randomly phased point sources, each radiat-ing a wave with unit amplitude, so the calculated averageintensity in the speckle pattern is Û& 5 ^R2& 1 Î2&5 10,000, and since ^R2& 5 Î2&, the calculated value of^R2& is 5,000. With convenient units in which k 5 1(which gives Lcoh53.83/0.2519.2 pixels), calculated (mea-sured) parameters in Eq. (A2b) are a 5 5,000 (4,7206 150), b 5 50 (46 6 2), c 5 1 (0.94 6 0.03), andd 5 1/3 (0.30 6 0.02). Here the measured values are anaverage over both the 66,000 data points that composethe wave field of a given source realization and the ten in-dependent realizations studied, and the indicated uncer-tainties are standard deviations of the mean.

APPENDIX B: CRITICAL-POINT NUMBERDENSITIESThe number density h of each of the 13 critical pointsstudied here are reviewed in this appendix. We beginwith the vortices, whose number density was first calcu-lated by Berry.21 Using the Van Cittert–Zernike theo-rem, we can write Berry’s results for h (vortices) in termsof the Gaussian curvature of mRR at the origin as

h~vortices! 51

2pmRR~0 ! H F ]2mRR

]~Dx !2G0,0

F ]2mRR

]~Dy !2G0,0

2 F ]2mRR

]~Dx !]~Dy !G0,0

2J 1/2

. (B1)

For the isotropic wave field studied here, with use of Eq.(A2a), Eq. (B1) reduces to h (vortices) 5 ^Rx

2&/@2p^R2&#5 b/(2pa) 5 1.59 3 1023 (in units where k 5 1), whichis in excellent (;1%) agreement with the measured value1.61 3 1023.The number densities of maxima, minima, and saddle

points of R, which are the same as the number densitiesof the corresponding stationary points of I, havebeen calculated by Longuet-Higgins and Cartwrightand Longuet-Higgins.6 For maxima the theoreticalvalue is h (Rmax) 5 ^Rxx

2&/@6p(3)1/2^Rx2&# 5 c/@6p(3)1/2b#


5 6.13 3 1024, in excellent agreement with the mea-sured (average of R and I) value 6.14 3 1024. Forminima theory yields h (Rmin)/h (Rmax) 5 1, in excellentagreement with the measured ratio 1.01, and for saddlepoints theory predicts h (Rsad)/h (Rmax) 5 2, in perfectagreement with the measured ratio 2.00. Worth notingis that h (Rsad) 5 h (Rmax) 1 h (Rmin), in accordancewith an important sum rule for regular functions dis-cussed by Longuet-Higgins and Cartwright andLonguet-Higgins.6

Theoretical calculations of the number densities ofmaxima, minima, and saddle points of the intensity Uhave been discussed by Weinberg22 and Weinberg andHalperin.23 As shown by these authors,23 the differentnumber densities all depend on a single dimensionless pa-rameter that can be written as lWH 5 ^Rx

2&2/@^R2&3 ^Rxy

2&# 5 b2/(ad) 5 1.5 for our source geometry.Our simulation yields for the number densities of saddlepoints h(Usad) 5 3.37 3 1023. From the data givenin the caption of Fig. 2 of Ref. 23 we have fromtheory h(Usad) 5 0.337^Rx

2&/^R2& 5 0.337b/a 5 3.373 1023, in perfect agreement with the measured value.Weinberg and Halperin pointed out that also for theintensity h(Usad) 5 h(Umax) 1 h(Umin). The minima ofU are conveniently subdivided into two classes, onethat corresponds to vortices for which U 5 0 and one forwhich U is finite. We label the nonzero minimaof U as Unz2min . With this subdivision thesum rule reads as h (USad) 5 h (UMax) 1 h (vortices)1 h (Unz2min). Our measurements yield h(UMax)5 1.30 3 1023 and h (U)nz2min 5 0.46 3 1023,so h(UMax) 1 h (vortices)1 h (Unz2min) 5 (1.30 1 1.611 0.46) 3 1023 5 3.363 1023, in excellent agree-ment with the theoretical value 3.37 3 1023 based on theabove sum rule. Unfortunately, calculation of h (UMax)and h (Unz2min) individually requires the eval-uation of still-unsolved multidimensional integrals,so at present there are no theoretical expressions avail-able for these quantities that could be compared with ourdata.There are also no direct theoretical calculations of

the number densities of stationary points of phase.Recently, however, I showed that for the phase (an irregu-lar function) there is a sum rule that reads as h (wsad)5h (vortices) 1 h (wMax) 1 h (wmin) and that h (wMax)5 h (wmin).

4 Although not strictly exact, for all practicalpurposes this sum rule is expected to describe extendedrandom phase fields correctly.4 The measured values areh (wsad) 5 1.73 3 1023, h (wMax) 5 (0.051 6 0.013)3 1023, and h (wmin) 5 (0.072 1 0.014) 3 1023, soh (wmax) 5 h (wmin) to within their combined uncertainties.Using the previously determined value for h (vortices), wehave h (vortices) 1 h (wmax) 1 h (wmin) 5 (1.61 1 0.0511 0.072) 3 1023 5 1.74 3 1023, in nearly perfectagreement with h (wsad), as required by the above sumrule.We conclude by noting that the various number densi-

ties can easily be converted to other units such asAcoh

21, where Acoh is some conveniently defined coherencearea, by computing Acoh from, say, mRR with k 5 1 andthen multiplying the number densities listed here by thisarea.

APPENDIX C: LEVEL-CROSSINGPROBABILITY-DENSITY FUNCTIONSWe calculate here a number of level-crossing PDF’sfor the whole field (F), for R (I), and for the vortices.PDF’s for U and w are problems of special difficulty thatare not attempted. Denoting the set of 12 wave-fieldvariables by $R, I% 5 $R, Rx , Ry , Rxx , Ryy , Rxy ,I, Ix , Iy , Ixx , Iyy , Ixy%, we have at the critical point g forthe joint probability density of two arbitrary variables aand b assumed expressible in terms of $R, I%:

Pg~a, b! 5 Ng E ...E2`

`

d$R, I%PRI~$R, I%!

3 Dgd @a 2 fa~$R, I%!#d @b 2 fb~$R, I%!#,

(C1a)

where PRI($R, I%) is given in Eqs. (A1), the functionfa ( fb) defines the dependence of a (b) on $R, I%, and

Ng 5 F E ...E2`

`

d$R, I%PRI~$R, I%!DgG21

. (C1b)

When PRI($R, I%) is itself normalized as in Eqs. (A1),Ng 5 @h (g)#21, where h (g), as before, is the number den-sity of the critical point g (Appendix B). In Eqs. (C1),Dg , given explicitly below for each type of critical point,serves to select only those isolated points in the wave fieldthat correspond to the specified type of critical point g.6,21

Of the various geometrical level-crossing parameters[Eqs. (1)–(3)], only the angle u [Eq. (1)] can be calculatedin practice. We obtain the PDF for this and relatedquantities for R and I, for example, by first writinga 5 tan uR and b 5 tan uI , where, as before, uR (uI) isthe angle that the level crossing of R (I) makes with thex axis. Inserting these forms for a and b into Eqs. (C1),using Eq. (1) to write tan u in terms of $R, I%, transform-ing the delta functions in Eqs. (C1) as required by stan-dard rules for such transformations,14 and carrying outthe indicated integrations yield as a first step P(a, b)5 P@tan uR , tan uI#. This is then transformed by stan-dard methods14 into the PDF P(uR , uI), which in turn isconverted into the PDF of the difference angle DRI 5 uR2 uI by use of

P~DRI! 5 EE duRduIP~uR , uI!d~DRI 2 @uR 2 uI# !

3 Q~p/2 2 uuRu!Q~p/2 2 uuIu!,

where the Heaviside step function Q(v) 5 1 for v > 0and is zero otherwise. The marginal PDF’s of the indi-vidual angles P(uR) and P(uI) are obtained by standardtransformations from the PDF’s P(a) 5 * dbP(a, b) andP(b) 5 * daP(a, b). The corresponding PDF’s for ux ,the level crossing direction of Rx , and uy , the level direc-tion of Ry , and for their difference Dxy 5 ux 2 uy , aresimilarly calculated, except that now we write in the firststep a 5 tan ux and b 5 cot uy .


Although special care is sometimes required to ensurethat only valid regions of phase space are included in theintegrations, the calculations are relatively straightfor-ward, with preservation of normalization providing a use-ful check against errors. Since the intermediate PDFP(a, b) is easily obtained but not directly used for com-parison with the data, in what follows only P(u) andP(D) are listed. These PDF’s are conveniently expressedin terms of two integrals:

Jg~t; g, h ! 5 Eg

h

dwtw 2 1

~t2 1 w2 1 k1,g tw 1 k2,g!5/2,

(C2)

Cg~u; m; g8, h8!

5 Eg8

h8dw

usin w 2 sin uum

~Q1,g sin2 w 1 Q2,g sin w 1 Q3,g!m11/2 ,

(C3)

where the constants ki,g (i 5 1, 2) and the functionsQj,g(u) ( j 5 1–3) are different for different level cross-ings and different critical points g. The integral J can beexpressed in a rather cumbersome closed form that canalso be obtained automatically by using any one of severalcomputer math packages. But the kernel in Eq. (C2) ismore easily programmed than this closed form, whereaslimiting values of the function and the required graphsare easily generated from the integral representation, sothe closed-form expression for J is not presented. Theintegral C should be expressable in closed form in termsof what would appear to be rather opaque combinations ofthe elliptic integrals E and F,20 although this reduction isbeyond the capabilities of current math packages andwould need to be done by hand. But, once again, the ker-nel in Eq. (C3) is (much) more easily programmed thansuch a closed form, whereas limiting values of the inte-gral and the required graphs of the PDF’s are all easilygenerated from the integral representation, so we are notmotivated to attempt here a closed form expression for C.

1. Whole FieldNote from Eqs. (A1) that R and its first-order derivativesRx and Ry are statistically independent, so the normal-ized PDF of the direction [Eq. (1)] of any given level cross-ing, tan uR 5 2Rx /Ry , is independent of the height of thegiven level. Accordingly, for the whole field (F), in Eqs.(C1) DF 5 1 (i.e., all points in the wave field are includedin the averages), and for the level-crossing angles of Rand I we find that

PF~u! 51p

QS p

22 uuu D , (C4)

where u 5 uR or u 5 uI and where, as before, the Heavi-side step function Q(v) 5 1 for v > 0 and is zero other-wise. In Fig. 4(a) Eq. (C4) is compared with the mea-sured data.For the difference angle DRI 5 uR 2 uI we have

PF~DRI! 51

p2 ~p 2 uDRIu!Q~p 2 uDRIu!, (C5)

which is compared with the data in Fig. 4(b) and dis-cussed in Subsection 5.A.Since from Eqs. (A1), Rx is statistically independent of

Rxx and Rxy , also the normalized PDF of the directiontan ux 5 2Rxx /Rxy of a given level crossing of Rx [Eq. (1)]is independent of level height. The same is true for levelcrossings of Ry for which tan uy 5 2Rxy /Ryy . Accord-ingly, for the whole-field PDF’s of the derivative level-crossing angles ux and uy we have

PF~ux ; kg! 5kg /p

sin2 ux 1 kg2 cos2 ux

Q~p/2 2 uuxu!,

(C6a)

PF~uy ; kg! 5kg /p

kg2 sin2 uy 1 cos2 uy

Q~p/2 2 uuyu!,

(C6b)

where kg 5 kF 5 (c/d)1/2 5 (3)1/2. Worth noting is that,except for trivial differences in Q functions, PF (uy)5 PF (p/2 2 ux). This equivalence is expected, as it re-flects the underlying isotropy of the wave field. Equa-tions (C6) are compared with the data in Figs. 10(a) and10(b) and discussed in Subsection 6.A.For the difference in level-crossing angles Dxy 5 ux

2 uy we have the PDF

PF~Dxy! 5 KFCF~ uDxyu; 1; 2p 1 uDxyu, p 2 uDxyu!

3 Q~p 2 uDxyu!, (C7a)

KF 5 F c2 2 d2

8pcAcdG 5A39p

,

Q1,F 5~c 2 3d !~c 1 d !

4cd5 0,

Q2,F 5 2c2 2 d2

2cdsinuDxyu 5 2~4/3!sinuDxyu,

Q3,F 5 1 1~c 2 d !2

4cdsin2 Dxy 5 1 1 1/3 sin2 Dxy .

(C7b)

Note that the important simplification Q1,F 5 0 is a di-rect consequence of the fact that d 5 c/3 and is thus ageneral property of isotropic wave fields. From the inte-gral representation of C [Eq. (C3)] it is easily seen thatPF (u) diverges logarithmically at uuu 5 p/2 and thatPF (0) 5 4KF . Equations (C7) are compared with thedata in Fig. 10(c) and discussed in Subsection 6.A.

2. Stationary Points of the Real and the Imaginary Partsof the Wave FunctionAs discussed in Section 4, at stationary points (s) of Rand I where Rx 5 Ry 5 0, the level crossings of R and Iare undefined, whereas as is evident from Eqs. (2) thePDF of the bifurcation angles of saddle points is not easilycalculated. Accordingly, we consider here only the zerocrossings of Rx , Ry (Ix , Iy).For maxima (M) and minima (m) we have for Dg in

Eqs. (C1) (Ref. 6)


DM 5 uJsud~Rx!d~Ry!Q~2Rxx!Q~2Ryy!Q~Js!, (C8a)

Dm 5 uJsud~Rx!d~Ry!Q~Rxx!Q~Ryy!Q~Js!, (C8b)

Js 5 RxxRyy 2 Rxy2. (C8c)

In spite of apparent differences between DM and Dm , thestatistical properties of the minima and maxima of R arenecessarily identical, since, as discussed in Subsection6.B, these can be interchanged by a uniform 180° phaseshift. This equivalence is reflected in the form of PRI inEqs. (A1), which is invariant under the set of transforma-tions Rxx ← 2Rxx , Ryy ← 2Ryy , and R ← 2R.For the zero-crossing angles ux and uy we have

PM/m~ux! 5 Ks

Js~ utan uxu; u1/tan uxu, `!

cos2 uxQS p

22 uuxu D ,

(C9a)

PM/m~uy! 5 Ks

Js~ ucot uyu; u1/cot uyu, `!

sin2 uyQS p

22 uuyu D ,

(C9b)

Ks 59A3~c2 2 d2!2

4pc3Acd5

16

3p,

k1,s 522dc

5 223; k2,s 5

c2 2 d2

cd5

83.

(C9c)

Again, as expected from wave-field isotropy, except fortrivial differences in Q functions, PM/m(uy) 5 PM/m(p/22 ux). From the integral representation of J [Eq. (C2)]we have PM/m(uuxu 5 p/2) 5 PM/m(uy 5 0) 5 4Ks /@3(21 k1,s)

2#, whereas PM/m(ux 5 0) 5 PM/m(uuyu 5 p/2)5 0. Equations (C9) are compared with the data in Figs.11(a) and 11(b) and discussed in Subsection 6.B.For the difference angle Dxy 5 ux 2 uy we have the

PDF

PM/m~Dxy! 5Ks

4sinuDxyuCs~ uDxyu; 2; uDxyu, p/2!

3 QS p

22 uDxyu D , (C10a)

Q1,s 5k1,s 1 k2,s

42

12

5 0,

Q2,s 5 2k2,s2

sinuDxyu 5 243sinuDxyu,

Q3,s 5 1 1 S k2,s 2 k1,s4

212 D sin2 Dxy

5 1 1 ~1/3!sin2 Dxy . (C10b)

As before, the simplification Q1,s 5 0 follows from d5 c/3 (wave-field isotropy). From the integral represen-tation of C we have PM/m(uDxyu 5 p/2) 5 (9Ks/64)@1 1 (3/2)1/2 ln(21/2 1 31/2)#. Equations (C10) arecompared with the data in Fig. 11(c) and discussed inSubsection 6.B.For the saddle points (S) of R we have

DS 5 uJsud~Rx!d~Ry!Q~2Js!. (C11)

We then obtain for the zero-crossing angles ux and uythe PDF’s

PS~ux! 5 Ks

Ss~ utan uxu!

cos2 uxQS p

22 uuxu D

51

pQS p

22 uuxu D , (C12a)

PS~uy! 5 Ks

Ss~ ucot uyu!

sin2 uyQS p

22 uuyu D

51

pQS p

22 uuyu D , (C12b)

Ss~t ! 5 Js~t; 2`, 0! 2 Js~t; 0, 1/t !. (C12c)

The second (nonobvious) equality in Eqs. (C12) holdswhen k1,s 1 k2,s 5 2, which is always the case for an iso-tropic wave field. Equations (C12) display the now famil-iar symmetry between ux and uy and are compared withthe data in Fig. 12(a) and discussed in Subsection 6.B.For the difference angle Dxy we have the PDF

PS~Dxy! 5Ks

8sinuDxyu H Cs~ uDxyu; 2; 2p 1 uDxyu, uDxyu!

3 QS p

22 uDxyu D 1 Cs~ uDxyu; 2; 2p

1 uDxyu, p 2 uDxyu!QS uDxyu 2p

2 D3 Q~p 2 uDxyu!J . (C13)

From the integral representation of C [Eq. (C3)] it followsthat PS(uDxyu 5 p/2 diverges logarithmically. Equations(C13) are compared with the data in Fig. 12(b) and dis-cussed in Subsection 6.B.

3. VorticesFor the vortices we have21

Dv 5 d~R !d~I !uJvu, (C14a)

Jv 5 RxIy 2 RyIx . (C14b)

Because of the Jacobian in Eq. (C14b), Rx,y and Ix,y are nolonger statistically independent at a vortex. Nonethe-less, because Eqs. (1), (A1), and (C14a) are all invariantunder the transformation R ↔ I, the level crossings ofR and I have the same statistics. The sign (6) of a vor-tex is given by sgn(Jv),

13,24 so changing the sign of a vor-tex is equivalent to interchanging R and I. As the levelcrossing statistics are invariant in such an interchange,positive and negative vortices also have the same statis-tics.For the level-crossing angles uR and uI we have Pv(u)

5 PF(u), where PF(u) is given in Eq. (C4) and, as be-fore, u 5 uR , uI . The PDF of the difference angle DRI5 uR2 uI , however, differs importantly from Eq. (C5),and for this we obtain


Pv~DRI! 512p

~p 2 uDRIu!sinuDRIuQ~p 2 uDRIu!.

(C15)

Equation (C15) is compared with the data in Fig. 9(a) anddiscussed in Subsection 5D.For the derivative level-crossing angles ux and uy we

have Pv(ux) 5 PF (ux ; kv) and Pv (uy) 5 PF (uy ; kv),where PF (u; kg) is given in Eqs. (C6) and where kg

5 kv 5 @(ac 2 b2)/(ad)#1/2 5 (3/2)1/2. These PDF’s arecompared with the data in Figs. 13(a) and 13(b) and dis-cussed in Subsection 6.C.For the difference angle Dxy we have the PDF

Pv~Dxy! 5 KvCv~ uDxyu; 1; 2p 1 uDxyu, p 2 uDxyu!

3 Q~p 2 uDxyu!, (C16a)

Kv 51

8pA a

dB3M5

1

6pA2

3,

Q1,v 514 S 2DB 1

1Bd D 2

12

5 0,

Q2,v 5 21

2BdsinuDxyu 5 2

23sinuDxyu,

Q3,v 5 1 1 S 14Bd

2D2B

212 D sin2 Dxy

5 1 213sin2 Dxy . (C16b)

As before, Q1,v 5 0 can be shown to be a direct conse-quence of the fact that for isotropic fields d 5 c/3. Fromthe integral representation of C [Eq. (C3)] we havePv(Dxy 5 0) 5 4Kv , whereas Pv(uDxyu 5 p/2) again di-verges logarithmically. Equations (C16) are comparedwith the data in Fig. 13(c) and discussed in Subsection6.C.Finally, we note that since level-crossing statistics are

independent of source intensity and diameter [^R2& andDs in Eqs. (A2b)], the various numerical factors listedhere are applicable to any uniform circular source.

ACKNOWLEDGMENTThis study was supported by the Fund for Basic Researchof the Israel Academy of Arts and Sciences.

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Critical-point level-crossing geometry in random wave fields