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Vortex derivatives
15 April 1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications I37 ( 1997) I 1% 126
Full length article
Vortex derivatives Isaac Freund Jack and Pearl Resnick Advanced Technology Institute and Department
ofPhysics,
Bar-llan
University, Ramat-Gan
52900. Israel
Received 12 September 1996;accepted 12 November 1996
Abstract
It has long been known that optical vortices are located at intersections of the zero crossings of the real and imaginary parts of the wave function. By examining singular (divergent) phase and other derivatives of all orders, we now show that there are infinitely many additional zero crossings passing through every vortex. Each zero crossing makes its own topological demands on the wave function, leading to infinfinifelymany topological constraints on the wave field structure.
1. Introduction
Phase dislocations [ l,2] are ubiquitous in optics, with screw dislocations, a.k.a. optical vortices, being the most common dislocation type [3]. These vortices appear in diverse optical fields ranging from speckle patterns [4,5] to laser beams 16-81 to vortex crystals [7,9]. In addition to their intrinsic interest, vortices are of practical importance in areas such as image processing [lO,l 11, and may have other, highly novel applications [ 12,131. But vortices are not confined to classical wave fields and are abundant also in condensed matter physics, being found in such diverse areas as quantum chaos [ 141, the Kosterlitz-Thouless transition in liquid helium [ 151, the quantum Hall effect [16], and the Abrikosov lattice in superconductivity [ 171. As a simple example of a screw dislocation that satisfies the wave equation, Nye and Berry [1] introduced a monochromatic, nondiffracting wave + that may be described in standard notation by ~,!~(x,y,z,t)= E(x,y)exp[i(kz-of)], where E:“(x,y)=(xfiy)j-A(j)exp(icp), the phase cp(* j) = +j@, and the polar angle 0 is measured counterclockwise from the x-axis when the wave is observed on a screen in reflection [ 181.Accordingly, for an +jth-order vortex the phase circulates through f2 jn along a counterclockwise closed path that encloses the vortex. Since cpat the origin depends upon the direction of approach (path), the phase at this point is singular (undefined). The product of the order j of the singularity and its
sign ( +) is a topological invariant conserved under continuous deformation of the wave function, and is usually called the “topological charge”. Extended wave fields often have equal numbers of positive and negative singularities and a net topological charge of zero [3-51. Higherorder singularities (j > 1) are generally unstable against small perturbation and will split into some appropriate number of (generic) first-order ( j = 1) singularities in a process in which topological charge as well as topological index [ 19,201 are conserved. As the wave function must be everywhere single valued, the amplitude A necessarily vanishes at the center of the singularity. Here for example, A(j) = rj, where r is the distance from the origin. Off the origin the phase is well defined, its circulation is positive (negative) for cp(+) [ cp(-_)I, and the phase front takes the form of a right- (left-) handed screw. One reason these screw dislocations merit the name (phase) vortices is that Vcp= (cp,’+ (p,2)‘/‘, the “velocity” of the rotary phase circulation, diverges at the vortex center. In less idealized, more common wave fields such as those found in scattering patterns and laser beams, the wave is diffracting and the wave function takes on an often complicated z-dependent form. Nonetheless, zeros generically still correspond to simple first-order vortices [3]. Writing the general wave function in the (x,y)-plane at some fixed value of z as E( x,y) = R( x,y> + U( x,y), where R is the real and I is the imaginary part of the wave function, the requirement that E(x,y) vanish at the vortex
0030-4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. P/I SOO30-4018(96)00739-O
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center implies that vortices are always found at intersections of the zero crossings of R and I (the set of curves in the (x,yJ-plane along which the function vanishes). It has recently been shown that vortex signs must alternate along these zero crossings and that this sign alternation induces numerous near-neighbor vortex-vortex correlations [2 l]. This sign alternation principle has also been extended to other zero crossings and other wave field features such as intensity maxima, minima, and saddle points, and has been shown to induce numerous topological correlations also between these structures 122,231. In addition to the zero crossings of R and I, there are other zero crossings that pass through a vortex. Since the wave field intensity U = R* + I* must be a (zero) minimum at a vortex, the zero crossings of U, and U, must also pass through the vortex. But the list does not end there. Although both cp, = acp/ax and ‘py- acp/ay generally diverge as one approaches the center of a vortex, vortices are nonetheless always located on intersections of the zero crossings of these derivatives [23]. This possibly surprising and seemingly counterintuitive result has profound implications for the phase field structure, since other critical points of phase such as maxima, minima, and saddle points, are necessarily also located on these same zero crossings [23]. More generally, we now show here that vortices are always located on the zero crossings of all higher-order phase derivatives, with m zero crossings of each mth-order derivative passing through every vortex. Thus, in addition to one zero crossing each of R, I, V,, U,,, cp,, and spy passing through every vortex (a running total of six), there are also present two zero crossings each of c.~, ‘dye, and po,, (running total of 12). rhree zero crossings each of vl,,,. ‘p,,, (pxyyand ‘pyy, (running total of 24), twenty zero crossings of the five fourth-order phase derivatives (running total of 441, etc. Accordingly, every vortex is located at the intersection of infinitely many zero crossings, and since for every one of these zero crossings there exists an extension of the sign alternation principle that constrains the phase field structure, there are infinitely many topological correlations between different points in the phase field. In the following sections we prove this remarkable result and extend it to higher-order vortices and to other functions such as the intensity. We also present a number of illustrative examples using data for a Gaussian random wave field.
2.1. Vortex derivatives
2. Low-order derivatives
cp =- J (u yyY2 - %X2). ry 2u2
In this section we start by first developing explicit forms for a number of low-order vortex derivatives for first-order vortices. Using mathematical induction, we then show in subsequent sections that the conclusions drawn from these explicit forms hold also for all derivatives of arbitrarily high order.
Similarly, for the four independent third-order derivatives we obtain
Writing
(Ia> we have RI., - IR ~ ‘p, =
u
> P,” =
RIY - IR, (I
(‘b)
’
where, as before, the intensity U = R2 + 12. Worth noting is that interchanging x and y everywhere yields q,,( x,y) = cp,(y,x), an important symmetry preserved in derivatives of all higher orders. Moving the origin to the vortex center and using the fact that the vortex is a simple first-order zero of the wave function, we expand R and I in a Taylor series as R(x,y)
= T,x+
r”y+
. . . . I(x,y)
= L,X+ ~.“y+ . . . . (2)
where TX= (aR/ax&, L, = (N/ax),, etc., and ( JO implies evaluating the enclosed derivatives at the origin (vortex center). Since the intensity has a zero minimum at the vortex center we obtain cp, = -(J/u)y,
(3a)
‘py= (J/u).5
(3b)
where the sign of the Jacobian J = TX”? - T,,L~ equals the sign (topological charge) of the vortex [24,25], and where u(x,y) = ~u,,x2 + iuyvy2 + $uXyxy, with u,, = (#~//&x2>,, etc. In writing Eqs. (3a) and (3b), we neglect all higher-order terms in x and y since analysis shows (surprisingly, perhaps) that to leading order in the small displacement from the vortex center r = (x2 + y*)‘/*, such terms make no contribution to derivatives of this and all higher orders. Using the fact that off the vortex center the phase is a regular (analytic) function of x and y so that, for example, (pXy= i)cp,/ay = 8(py/ax, after straightforward calculation we obtain for the three independent second-order derivatives PXX= -u-‘(U,,
X+UryY)%*
cpyy = --u-‘(u.ryx
cpXXX = u -2[$&xz +3u,*u,,xy]
+ UyvY)‘p, = cp,,(Y7X)?
(4a) (4b)
+ (2uf, - $4,,uyy )Y2 CPXV
(5a)
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(5b) %,vrvy(X~Y) = cp,.r,(Y,X),
(5c)
= cp,,.dYJ)~
(5d)
'4vyv(X*Y)
2.2. Vortex zero crossings
We now use Eqs. (3)-(5) to deduce the structure of the vortex zero crossings. Returning to Eq. (3a) and recalling that the intensity (u) is positive definite, we note that off the vortex center, both for positive and negative x, ‘pXis positive (negative) whenever the product Jy is less than (greater than) zero. For positive vortices J > 0, so excluding the origin, cp, is positive for negative y and is negative for positive y, while for negative vortices this situation is reversed. Now, since ‘p, is regular everywhere between vortex centers, its transition between positive and negative values is continuous and must pass through zero. We may therefore conclude that in the immediate vicinity of a vortex the boundary between positive and negative values, the x-axis, is tangent to a zero crossing of ‘p,, and that by a limiting process this zero crossing passes through the vortex center. Similarly, we may conclude from Eq. (3b) that: in the immediate vicinity of a vortex the y-axis is tangent to a zero crossing of ‘p, that also passes through the vortex center, and that for positive vortices ‘p, is positive (negative) for positive (negative) x, while for negative vortices this situation is reversed. All these conclusions are fully borne out by the data in Fig. 1, which shows results obtained from recent large-scale computer simulations of a Gaussian random wave field [25]. The above results may also be obtained by inspection using simple geometric arguments. Inserting Eq. (2) into Eq. (lb) we have cp=arctan[(l +AtanB)/(B+Ctane)], where tanO=y/x, A = L,,/L~, B = ‘T’/L_~, and C = Ty/L,. To first order cp is independent of the radial distance r, so to this order contours of constant phase are straight lines that radiate outward from the vortex center. ‘p, (‘p,) therefore vanishes along the x-axis ( y-axis) which is a line of constant phase. The signs of ‘pX and ‘p, throughout the (x,y)-plane are also easily obtained geometrically by noting that increasing x causes 0 to decrease for positive y and increase for negative y, while increasing y causes 8 to increase (decrease) for positive (negative) x. For the second-order vortex derivatives Eq. (4a) shows that one zero crossing of ‘pXXthat passes through the vortex center coincides with the zero crossing of cpX,i.e. the x-axis. Writing x = rcos0 and y = rsin 0, we find a second vortex zero crossing tangent to the line 0= - arctan[u,,/u,,]. We note that this line is also tangent to the zero crossing of U, at this point. From Eq. (4b) we find one zero crossing of (py,, passing through the vortex
(a)
03
(cl Fig. 1. (a) Zero crossing maps of ‘pX= flcp/ax (thick line) and ‘p, = ?lcp/ay (thin line), where cp is the phase, showing n positive and 0 negative vortices at intersections of the zero crossings. As discussed in Ref. [23], along any given zero crossing vortices must always be separated by phase saddles (unmarked intersections). (b) Map of cp,. (c) Map of qY. In (b> and (c) positive (negative) regions of the function am colored white (black). All maps shown here and in Figs. 2 and 3 cover the same region of the wave field and are 199 pixels wide (x-axis) and 110 pixels high (y-axis), where the transverse coherence length of the field is 19.2 pixels.
center coinciding with the zero crossing of ‘pY, i.e. the y-axis, together with a second zero crossing tangent to the line 0- -arctan[u,,/u,,] (also the tangent to the zero crossing of U,,). Finally, from Eq. (4c) we find two zero crossings of ‘p_rYpassing through the vortex center. These are tangent to the lines 8,s +arctan[(u,,/u,,)“2], where both u,, and uYYare positive definite since ZJ is a minimum at the vortex center. In Fig. 2 we present data obtained from our computer simulations that confirm as well as illustrate these results. In similar fashion we find from Eqs. (5) for the thirdorder derivatives three zero crossings of ‘p,,, passing through the vortex center. One coincides with the zero crossing of cp,, 01,XxX= 0, n, and the other two are
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ing each of ‘pI, and qDr, two zero crossings each of ‘p, ‘i, ‘pyy. and V+ and three zero crossings each of P_~*~, cpyvv. (P.~.~~,and ‘P,,,. Although this suggests a trend, it is clearly not feasible to carry out an unbounded number of (increasingly complex) explicit calculations in order to establish that this trend holds for all higher-order derivatives. Accordingly, we now turn to the use of mathematical induction to demonstrate that m zero crossings of every mth-order derivative do, in fact, pass through every vortex. With this established, we then extend these findings to higher-order vortices, to the wave field amplitude, and to regular functions such as the intensity.
3. Higher-order
(4
(c) Fig. 2. Zero crossing maps of (a) qAA, (b) ‘p,,. and (c) ‘pAy showing hvo zero crossings of each derivative passing through both n positive and 0 negative vortices.
to the lines 02,rxr = -arctan(3~,,[(3H)‘/~ + - 3u,,]~‘]. 3n.J’1 and %x.rr = arctan{3u,,[(3H)‘/Z Here, the Hessian H = u,,uyv - uf, is positive definite since U is a minimum at the vortex center. For ‘p,,, we also find three zero crossings passing through the vortex center. One coincides with the zero crossing of ‘p,, O,,lyy = n/2, 31~/2 and the other two are tangent to the lines and 03.yyy = e LYYY = - arctan(3 uy ,[(3 H) ‘1’ + 3u,,]-‘} For (pxl,, we find from arctan{3u ,,[(3 H) ‘/2 - 3u,,]-‘). Eq. (SC) three zero crossings that pass through the vortex center. Writing t = tan0, these may be obtained from the roots of t3 + a,t2 + a3t = 0, where a, = 3~,,/(2u,,) and Since the discriminant A = a3 = --u~,/(~u,~u,~). u4xy ) is negative definite, all three roots are - 4@,/(& real and distinct, and when required may be written down explicitly using standard methods. Finally, the three zero crossings of (erVy may be obtained with the aid of these same roots simply by replacing tan0 with cot0. This replacement follows from ‘px,,( x, y) = ce,,,( y, x) [Eq. tangent
(5c)l. We have therefore shown by explicit calculation that passing through the vortex center there is one zero cross-
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derivatives
Returning to Eqs. (3a) and (3b), we introduce two successive linear transformations. The first is a principal axis transformation, i.e. rotation through an angle cy that diagonalizes the intensity. In the resulting x’,y’ coordinate system, u becomes u(x’,y’) = fA+x” + +A_$‘, where A, are the eigenvalues of the Hessian matrix H (H,, = u.u, H22 = uyy’ H,, = H,, = u.~J. The second transformation is a dilation of the y’-axis by D = ( A+/A )‘I’ that produces our final coordinates X, Y. As a result of these transformations, the intensity becomes isotropic and may be written u(X,Y)= +A+$, where p=(X’+ Y 2)‘/2. This isotropy makes it convenient to work in polar coordinates, and we define a polar angle (T measured counterclockwise from the X-axis, and write X = pcosc~, Y = psina. Due to the combination of rotation and dilation, our original x- and y-axes are no longer orthogonal and are oriented at peculiar angles relative to the (X,Y )-coordinate system. We therefore calculate vortex derivatives along some arbitrary direction s = dY/dX using df/ds [(af/dX>, + s(af/aY ),]/(I + s2)‘i2. Then by choosing s so that the direction of differentiation corresponds to the normal to the y- or x-axes (s, or s,,, respectively), (af/kx), = K,(df/ds,) and (af/ay), = K,(df/ds,), where K, and K,” are constants. In this way the mth-order directional phase derivatives (f = cp> multiplied by K(~)(x,. . ,y _ . ), the appropriate produCt of the K., and K~, correspond to vortex derivatives such as ‘P.~,~,-
[ d4’( X.X-X.X> = K_:], (p+xyyy[K(‘)( Xryvv) = +;I,
etc.
As the above linear transformation does not change the order of the wave function zeros, Eqs. (l)-(3) are applicable with X replacing x and Y replacing y. Using dp
- -(coso+ z-j&2
--
ssina),
-sina+
scosu
(f-9
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we have
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Eq. (8) implies that every m&r-order vortex derivative has m zero crossings passing through the vortex center. These are tangent to the lines o-= --m-‘arctan[ [(“)/J/(“)]. Noting from Eq. (9g) that these m zero crossings in X,Y map uniquely into m equivalent zero crossings in x, y completes the demonstration.
where J, = 7’, ~~ - Ty~x is the transformed Jacobian in X,Y. We now invoke mathematical induction. If 4. Extension to higher-order
d’b
cp?.! = ds,...ds, cos(ncr)
+
rl? [
I
@aI
’
P”
where $ and 5 are constants, the superscript (n) denotes the order of the derivative, and the subscript . . . . represents differentiation along an arbitrary sequence of directions s,, sz,. . . ,s,, then d”+ ‘(p P!(:Its’)= ds
, . ..ds.ds
I
cos[(n + l)a]
+ p+ 1) . . ..I [
P
I?+1 I?
@b)
5.!“.:‘)=
n[3?.1!-P.?!l &---&
(8~)
.
As this is of the same form as Eq. (8a), it leads to a recursion loop that closes on itself if the initial condition is met. Since the starting point, Eq. (7a) for p(l), is, in fact, precisely of the form required by Eq. (8a), Eq. (8b) describes vortex derivatives of arbitrary order. These derivatives may, if required, be expressed in terms of r,8 or x,y in the original x,y-coordinate system with the aid of the following formula summary: X=xcosa+ysincr, tan(2cr) =
Y=D(-xsina+ycosa),
(9a)
2u,, (9b)
U .1I -
UYY
D=dE, A,=
f(T+
(9c)
tw-G>,
T=u,,+L+
s,= K,
-Dtancr, =
(94
H=u,,u~,-u:~, s,=Dcot(~,
D’sin’a! +
COS2a,
K~ =
(9e) D2COS2a
i-
Sin2a
,
WI
6=o+arctan[D-‘tana],
(9g)
p-r[co~~(~-(~)+D~sin~(~-a)]“~.
(9h)
vortices
Returning to the simplest vortex model of Nye and Berry [I] reviewed in Section 1, we write again cp(rtj) = +j6. This describes an isotropic vortex. Using Eq. (6) with 0 replacing o and r replacing p, it may be seen that for this vortex the results of the previous section hold also for arbitrary j. Since for these isotropic vortices the linear transformation in Eq. (9a) is the identity transformation ((Y= 0, D = I), it follows from Eq (8b) that for all values of j and m, the mth-order derivative zero crossings form a symmetric star whose m arms have equal angular separation A r3crn)= n/m. As vortices are topologically stable features of the wave field, asymmetric vortices may be obtained from isotropic ones by continuous deformation. For first-order vortices (j - 1) it is apparent from Section 2 that these deformations simply cause the zero crossings to rotate to arbitrarily different angles, thereby destroying the symmetric arrangement obtained in the isotropic case. But we cannot be certain that this is all that occurs for higher-order vortices. In particular, we need be concerned that for the j&order vortex the zero crossings for the isotropic case might be say j-fold degenerate, and that this degeneracy would be split in going to the asymmetric case. A simple argument suggesting such a possibility is that if an isotropic jth-order vortex were to split into j closely spaced firstorder vortices, each of these latter would carry m zero crossings, so that when viewed from a distance as a single object the number of zero crossings radiating outward from this vortex grouping could total j X m. There are of course other possibilities, and we explore the question of degeneracy by examining in detail a simple example of a second-order vortex (j = 2). We begin by noting that for the isotropic second-order vortex (x + iy)‘, for which R = x2 - y2 and I = Zxy, both R and I correspond to saddle points. This remains true also for the general asymmetric case. The reason is that the phase circulates through 47~ for a second-order vortex, while contours of constant phase cp= 0, P, etc. correspond to zero crossings of I and contours of cp= ~/2, 3rr/2, etc. correspond to zero crossings of R. Accordingly, there must be MO zero crossings each of R and of I intersecting at the vortex. A self-intersection such as this corresponds to an ordinary (first order) saddle point. As the sign of dp/d0 corresponds to the vortex sign and must be independent of 8, the zero crossings must be interleaved
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I. In addition, for a saddle point the Hessian Rf ,(I,, I,, - l.zy) must be negative definite. LR,, These conditions can be met by writing R = ( y - ux)( y a), I = ( y - bx)( y - d.r), with a < b < c < d. Choosing as a simple example that has a sufficient degree of generalitya=l,b=2,~=3,andd=4,wefindfromEq.(lb) for cp+ that the zero crossings of this vortex derivative are given by the roots of 13-_2-9t+ 12=0, where as before I = tanf3. Since the discriminant A = ( - 28/9)3 + ( - 241/54)* is negative, there are three real distinct roots and therefore three nonoverlapping zero crossings of cp,. Using qJ( x, y) = M,( y, x), we also find that there are three nonoverlapping zero crossings of ‘p, that pass through the vortex center (these are the roots of tK3 - tA2 - 9r-’ + 12 = 0). The individual zero crossings of the isotropic case are therefore now seen to be three-fold degenerate, rather than two-fold degenerate as in our initial guess. Although the physical basis for this higher degree of degeneracy remains unclear, these results can be generalized to produce an upper limit of m(2j - 1) for the number of zero crossings of the &h-order derivative passing through an asymmetric vortex of order j. From Eq. (lb) it is apparent that independent of vortex order, the &h-order phase derivative can be written (pcrn) * Pm/Urn, where P, is some polynomial in x and y whose order we now determine. Since the phase circulates through 2 jrr for a vortex of order j, for such a vortex R=(y-a,xXy-a,x)...(y-ajx)-yi. Then, the intensity II - R* - yzi, while by dimensional arguments (rn) N l/y”, so we conclude that P,,,- y"'(*j-‘). As u is cp positive definite, the zero crossings of (pcrn) are given by the roots of P,,,(x, y) = 0. An upper limit on the number of real distinct roots is m(2j - I), which is an upper limit on the number of zero crossings. We therefore conclude that the number of zero crossings N(m,j) of each mth-order phase derivative passing through a vortex of order j is bounded by m zz N(mj) I m(2 j - 1), where the lower limit is the value of N(mj) for isotropic vortices, and as a conjecture based upon the specific example discussed above, the upper limit is its value for fully asymmetric vortices. R,
I, R,
5. Extension to the amplitude Although the intensity U and all its higher derivatives are regular everywhere, this is not the case for the amplitude A = U ‘/*. Since A, = (RR, + IIX>/lJ’12 and A, = all vortex derivatives of A are singu(RR.”+ uJ/u’/2, lar. Explicit calculations for a few low-order derivatives reveal the presence of degeneracies even for first-order For example, A,, = HY~/(~u~/~), A,, = vortices. so that both Hx*/(2 u3/*), and A.,, = -Hxy/(2u3/*), A .Y.t and A,, each have one doubly degenerate zero crossing while A.,, has two nondegenerate zero crossings
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passing through the vortex center. Similarly, A.,,( x, y) = -3H(u,,x + uXyy)y2/(4u5/*), and A,,,(x,y) = A ,,,( y, x) both have one doubly degenerate and one nondegenerate zero crossing passing through the vortex, while the three vortex zero crossings of A,,,(x, y) = - Hy(2u,,y2 + uXYxy - u,,x2)/(2u5/*) and of Alvy( x, y> = Al+$ y, x) are all nondegenerate. Although the methods of Section 3 permit us to write A = (A+/2)‘/$, application of Eq. (6) does not lead to a closed system, so that currently no general expression for Acm)is available. For higher-order vortices we may repeat the arguments of the previous section by writing the m&t-order derivative as Atm)- Q
6. Extension to other functions We now extend these results to the intensity U( x, y), to to any regular (i.e. singularity free) function F( x, y). Defining the vector field (gradient) VF(x, y) = F,x + FYy, where x,y are unit vectors along x, y, we note that since F is regular everywhere, so is VF. For such a vector field we may define a complex order parameter whose amplitude is the length of the vectors, A(F; x,y> = (VFI = (F,’ + F.z)‘/*, and whose phase is the angle (direction) the vectors make relative to the x-axis, @(F; x, y) = arctan( FJF,). Since the gradient always points in the direction of maximum increase of the function, these vectors radiate outward from a minimum and inward towards a maximum, so their directions at maxima and minima become undefined. The same is true for a saddle point which is a maximum along one principal axis and a minimum along the other. O(F) is therefore regular everywhere except at the stationary points (maxima, minima, and saddle points) of F(x,y) where it is singular, since here F, = FY = 0 (regularity of VF is preserved at these points since A = 0). The singularities of O(F) are easily shown to be the same as the singularities of the phase field cp, i.e. first-order vortices. Moving the origin to a stationary point and noting that Ffx.y) is regular everywhere, we have (in obvious notation) I;, =frxx +frYy + . .., F,”=frXx +fruy + . . . , which is completely analogous to Eq. (2) with F, replacing R and FY replacing I. All the other equations for the singularities of cp also become applicable to the singularities of 0 upon replacing 7” by f,,, ‘1 by f,,, L, by We note that this replacement changes f yxrand ‘Y by fyy. the symmetry condition ‘py(x, y) = cp,( y, x), which now becomes O,(F;x, y) = - OJF; y,x) [Eqs. (3a) and (3b)l.
R( x, y), to I( x, y). and in general
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eter 0’ = arctan(OY/@~r). At stationary points of 0. 0, = 0.” = 0, so at these points 0’ is singular. As before, these singularities correspond to first-order vortices whose sign is given by the sign of H’ = O,Yx@Y,- @_v,@,,r, so that extrema of 0 correspond to positive first-order @‘-vortices and saddle points of 0 correspond to negative first-order @‘-vortices. But to what do @-vortices (i.e. the stationary points of F) correspond? Using Eqs. (3a) and (3b) in the form 0, = - Hy/( F,’ + F,‘) and 0, = Hx/( c: + F,‘) we obtain for positive (negative) @-vortices, 0’ = ~9+ n/2 (31~/2), so that all stationary points of F correspond to isotropic positive first-order @-vortices 1231. Explicit forms for 0, and 0, are easily found to be
q X,Y> =
Vx, - F,F,x F.; + F,’
’
O,( x,y) = - @,(Y,X).
(10) (W
w Fig. 3. (a) Zero crossing maps of the X, y-derivatives of the phase O(R) of the complex order parameter of VR (Section 6). Here R is the real part of the wave function. (Thick line) O,(R), (thin line) @&R). (b) Map of O,(R). (c) Map of B,(R). These maps also show R-stationary points, n extrema and Cl saddles. In (b) and (c) positive (negative) regions of the function are colored white (black). As discussed in Section 7, along any given zero crossing R-stationary points must always be separated by O(R)saddles (unmarked intersections).
Worth noting is that from this it follows that at stationary points of 0 (OX= OY=O, IF,I>O, IF,I>O) the Hessian H = F,, F,,, - F& = 0. The corollary of this, that say 0, = H = 0 implies 0, = 0 and hence a stationary point of 0 is not necessarily true, however, since zero crossings of both 0, and H but not necessarily of 0: always pass through zero crossing intersections of F.,, and TX.,,. In summary, stationary points of F are located at the simultaneous intersections of zero crossings of F,, F,, @.,, and 0,. The zero crossing intersections of F, and FY define the singularities of 0, and the zero crossing intersections of 0, and 0, define the singularities of 0’. F-extrema correspond to positive, and F-saddles to negative O-vortices, while both types of F-stationary points correspond to isotropic, positive @‘-vortices. Stationary points of 0 are located at intersections of the zero crossings of (9x and OY where H = 0, while @-extrema conespond to positive and @-saddles correspond to negative @‘-vortices. In the next section we explore some of the topological correlations implied by these results.
The reason for this is that J(x,y) = -J(y,x) is replaced by H(x,Y) =f,,f,, -fryJ;.v = H(yd where fyx =.frf,,. With these replacements, extrema (maxima and minima) of F, for which H > 0, correspond to positive @-vortices, and saddle points of F, for which H < 0, correspond to nega-
7. Topological correlations
tive O-vortices. As the results of Sections 2 and 3 are applicable also to @-vortices, we may conclude that m zero crossings of each of the mth-order derivatives 0’“‘) pass through every maximum, minimum, and saddle point of F(x,y). Shown in Fig. 3 is an example for m = 1 that involves zero crossings of O,(R) and O,(R) passing through stationary points of R. This figure may be seen to be equivalent in every respect to Fig. 1, reflecting the equivalence of 0 and cp. In the next section we will need also to analyse the vector field VO and the phase of its complex order param-
Application of the sign alternation principle [21,23] to O-vortices immediately leads to the (possibly unexpected) result that F-extrema (positive @-vortices) and F-saddles (negative @vortices) must alternate along zero crossings of F, and F,. This has been shown to lead to a quasicrystalline arrangement for these stationary points in a Gaussian random wave field [26]. Similarly, application of the sign principle to @‘-vortices immediately leads to the analogous result that F-stationary points and @-extrema (both positive @‘-vortices) must alternate with @-saddles (negative @‘-vortices) along zero crossings of @%and 0,.
I. Freund/Optics
Cotnmunicutions 137 (1997) 118-126
The alternation of F-extrema and F-saddles along zero crossings of F,r and FY is also the mathematical statement that the structure of the (wave) function must be such that along certain lines defined by jirst-order derivatives (F, = FY = 01, a particular combination of second-order derivatives (H) must alternate in sign. This connection between nominally independent derivatives of different order is an unexpected, indeed striking topological constraint. Similarly, alternation of F-stationary points and O-extrema with O-saddles along zero crossings of 0, and 0, is equivalent to the mathematical statement that the structure of the wave function must also be such that along certain lines formed by combinations of first- and secondorder derivatives (@_,= 0, 0.” = 01, a particular combination of first-, second- and third-order derivatives (H’) must alternate in sign. By defining 00’ and the phase of its complex order parameter 0”. we can generate a third statement that constrains fourth-order derivatives along lines defined by lower-order derivatives, and since we can extend this procedure indefinitely, an indefinite number of constraints of this sort can be generated for derivatives of arbitrarily high order. We emphasize that these constraints are necessary in the sense that simply laying down an arbitrary network of zero crossings of say 0, and 0, suffices to force H’ to alternate along the network. The above constraints give rise to numerous near neighbor correlations that are similar to those found for near neighbor vortices [2 I I, while many additional topological constraints and their associated correlations may be deduced by taking VO,, VO,,v, etc., instead of VO as the starting point for the development in Section 5. As all these correlations differ from those already described in degree rather than in kind, we do not develop them further here.
8. Summary We have shown that M zero crossings each of every mth-order phase derivative pass through every first-order vortex, and have argued that for jth-order vortices the number of zero crossings is increased by up to a factor 2 j - I. These findings have been extended formally to the amplitude, and also to regular functions by relating the stationary points of such functions to singularities of their gradients. Unusual topological correlations connecting higher-order derivatives of the function to lower-order derivatives have been found, together with new relationships governing the spatial arrangements of zero crossings. As phase derivatives are important in determining ray directions and other wave field properties, the results described here may prove useful in areas such as image processing and adaptive optics [27], as well as in recently proposed, innovative vortex-specific applications. They
125
may also be of interest in other areas where vortices are of prime importance, such as two-dimensional solid state physics.
Acknowledgements This work was supported by the Israel Science Foundation of the Israel National Academy of Sciences and Humanities.
References [I] J.F. Nye and M.V. Berry, Proc. R. Sot. A 336 (1974) 165. [2] M. Berry, Singularities in waves and rays, in: Physics of Defects, eds. R. Balian, M. Kleman and J.-P. Pokier (NorthHolland, Amsterdam, 1981) pp. 453-543. [3] M.V. Berry, J. Phys. A I I (1978) 27. [4] N.B. Baranova, B.Ya. Zel’dovich, A.V. Mamaev, N.F. Pilipetskii and V.V. Shukov, JETP Lett. (1981) 195; N.B. Baranova and B.Ya. Zel’dovich, Sov. Phys. JETP 53 (1981) 925; N.B. Baranova, A.V. Mamaev, N.F. Pilipetskii. V.V. Shukov and B.Ya. Zel’dovich, J. Opt. Sot. Am. 73 (1983) 525. [5] 1. Freund, N. Shvartsman and V. Freilikher, Optics Comm. 101 (1993) 247; N. Shvartsman and I. Freund, Optics Comm. I 17 (1995) 228 [Erratum: 120 (1995) 3691; I. Freund and D.A. Kessler, Optics Comm. 126 (I 995) 321. [6] F.T. Arecchi, G. Giacomelli, P.L. Ramazza and S. Residori. Phys. Rev. Lett. 65 (1990) 2531. [7] M. Brambilla, F. Battipede, L.A. Lugiato, V. Penna, F. Prati, C. Tamm and CO. Weiss, Phys. Rev. A 43 (1991) 5090. 5114. [8] G. Indebetouw, J. Mod. Optics 40 (1993) 73; S.R. Liu and G. Indebetouw, Optics. Comm. 10 I (I 993) 442. [9] I. Dana and I. Freund, Vortex lattice wave fields, Optics Comm. 136 (1997) 93. [IO] H.V. Deighton, M.S. Scivier and M.A. Fiddy, Optics Lett. IO (1985) 250; P.-T. Chen and M.A. Fiddy, J. Opt. Sot. Am. I I (1994) 2210. [I I] C.C. Wackerman and A. Yagle, J. Opt. Sot. Am. I I (1994) 2016; P.J. Bonnes, C.R. Parker, B.I. Satherly and R.W. Watson, J. Opt. Sot. Am. 12 (1995) 1842. [I21 A. Ashkin, Biophys. J. 61 (1992) 569; K.T. Gahagan and G.A. Swartzlander, Jr., Optics Lett. 21 (1996) 827. 131 E. Abramochkin and V. Volostnikov. Optics Comm. 125 ( 1996) 302. 141 P. Leboef, J. Kurchan, M. Feingold and D.P. Arovas, Phys. Rev. Lett. 65 (1990) 3076; 1. Dana, Phys. Rev. E 52 f 1995) 466. 151 J.M. Kosterlitz and D.J. Thouless, Two dimensional physics, in: Progress in Low Temperature Physics, Vol. VII B, ed. D.F. Brewer (North-Holland, Amsterdam, 1978) pp. 371433. 161 D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. den Nijs, Phys. Rev. Lett. 49 (1982) 405; R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. 171 A.A. Abrikosov, Sov. Phys. JETP 5 (1957) 1174; D.E. Farrell, Transformations of the vortex solid, in: Physical
126
[ 181
[ 191
[20] [21]
I. Freud/
Optics Communications
Properties of High Temperature Superconductors, Vol. IV, ed. D.M. Ginsberg (World Scientific, Singapore, 1994) pp. 7-59. An important class of experimentally realizable (nearly) nondiffracting beams are based on Bessel functions: J. Dumin, J.J. Miceli Jr. and J.H. Eberly, Phys. Rev. Lett. 58 (1987) 1499; J. Dumin, J. Opt. Sot. Am. A 4 (1987) 651; F. Gori, Cl. Guattari and C. Padovani, Optics Comm. 64 (1987) 491. V.I. Arnold, Ordinary Differential Equations (MIT Press, Cambridge, Mass., 1973); S.H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA, 1994). J.F. Nye, J.V. Hajnal and J.H. Hannay, Proc. R. Sot. A 417 (1988) 7. N. Shvartsman and 1. Freund, Phys. Rev. Lett. 72 (1994) 1008; 1. Found and N. Shvartsman, Phys. Rev. A 50 (1994)
[22] [23] [24]
[25] [26] [27]
137 (1997) 118-126 5164; N. Shvartsman and I. Freund, J. Opt. Sot. Am. 11 (1994) 2710. I. Freund, Phys. Lett. A 198 (1995) 139. I. Freund, Phys. Rev. E 52 (1995) 2348. B.I. Halperin, Statistical mechanics of topological defects, in: Physics of Defects, eds. R. Balian, M. Kleman and J.-P. Poirier (North-Holland, Amsterdam, 1981) pp. 814-857. 1. Freund, J. Opt. Sot. Am. A 11 (1994) 1644. I. Fteund and N. Shvartsman, Phys. Rev E 51 (1995) 3770. The Schack-Hartmann lenslet array, which measures firstorder phase derivatives, is widely used as a wave front sensor in adaptive optical systems: L.A. Thompson, Physics Today 47 [ 12]( 1994) 24; See also: Adaptive Optics, Optical Society of America 1996 Technical Digest Series, Vol. 13.
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications I37 ( 1997) I 1% 126
Full length article
Vortex derivatives Isaac Freund Jack and Pearl Resnick Advanced Technology Institute and Department
ofPhysics,
Bar-llan
University, Ramat-Gan
52900. Israel
Received 12 September 1996;accepted 12 November 1996
Abstract
It has long been known that optical vortices are located at intersections of the zero crossings of the real and imaginary parts of the wave function. By examining singular (divergent) phase and other derivatives of all orders, we now show that there are infinitely many additional zero crossings passing through every vortex. Each zero crossing makes its own topological demands on the wave function, leading to infinfinifelymany topological constraints on the wave field structure.
1. Introduction
Phase dislocations [ l,2] are ubiquitous in optics, with screw dislocations, a.k.a. optical vortices, being the most common dislocation type [3]. These vortices appear in diverse optical fields ranging from speckle patterns [4,5] to laser beams 16-81 to vortex crystals [7,9]. In addition to their intrinsic interest, vortices are of practical importance in areas such as image processing [lO,l 11, and may have other, highly novel applications [ 12,131. But vortices are not confined to classical wave fields and are abundant also in condensed matter physics, being found in such diverse areas as quantum chaos [ 141, the Kosterlitz-Thouless transition in liquid helium [ 151, the quantum Hall effect [16], and the Abrikosov lattice in superconductivity [ 171. As a simple example of a screw dislocation that satisfies the wave equation, Nye and Berry [1] introduced a monochromatic, nondiffracting wave + that may be described in standard notation by ~,!~(x,y,z,t)= E(x,y)exp[i(kz-of)], where E:“(x,y)=(xfiy)j-A(j)exp(icp), the phase cp(* j) = +j@, and the polar angle 0 is measured counterclockwise from the x-axis when the wave is observed on a screen in reflection [ 181.Accordingly, for an +jth-order vortex the phase circulates through f2 jn along a counterclockwise closed path that encloses the vortex. Since cpat the origin depends upon the direction of approach (path), the phase at this point is singular (undefined). The product of the order j of the singularity and its
sign ( +) is a topological invariant conserved under continuous deformation of the wave function, and is usually called the “topological charge”. Extended wave fields often have equal numbers of positive and negative singularities and a net topological charge of zero [3-51. Higherorder singularities (j > 1) are generally unstable against small perturbation and will split into some appropriate number of (generic) first-order ( j = 1) singularities in a process in which topological charge as well as topological index [ 19,201 are conserved. As the wave function must be everywhere single valued, the amplitude A necessarily vanishes at the center of the singularity. Here for example, A(j) = rj, where r is the distance from the origin. Off the origin the phase is well defined, its circulation is positive (negative) for cp(+) [ cp(-_)I, and the phase front takes the form of a right- (left-) handed screw. One reason these screw dislocations merit the name (phase) vortices is that Vcp= (cp,’+ (p,2)‘/‘, the “velocity” of the rotary phase circulation, diverges at the vortex center. In less idealized, more common wave fields such as those found in scattering patterns and laser beams, the wave is diffracting and the wave function takes on an often complicated z-dependent form. Nonetheless, zeros generically still correspond to simple first-order vortices [3]. Writing the general wave function in the (x,y)-plane at some fixed value of z as E( x,y) = R( x,y> + U( x,y), where R is the real and I is the imaginary part of the wave function, the requirement that E(x,y) vanish at the vortex
0030-4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. P/I SOO30-4018(96)00739-O
I. Freund/Optic.s Communications 137 (1997) 118-126
119
center implies that vortices are always found at intersections of the zero crossings of R and I (the set of curves in the (x,yJ-plane along which the function vanishes). It has recently been shown that vortex signs must alternate along these zero crossings and that this sign alternation induces numerous near-neighbor vortex-vortex correlations [2 l]. This sign alternation principle has also been extended to other zero crossings and other wave field features such as intensity maxima, minima, and saddle points, and has been shown to induce numerous topological correlations also between these structures 122,231. In addition to the zero crossings of R and I, there are other zero crossings that pass through a vortex. Since the wave field intensity U = R* + I* must be a (zero) minimum at a vortex, the zero crossings of U, and U, must also pass through the vortex. But the list does not end there. Although both cp, = acp/ax and ‘py- acp/ay generally diverge as one approaches the center of a vortex, vortices are nonetheless always located on intersections of the zero crossings of these derivatives [23]. This possibly surprising and seemingly counterintuitive result has profound implications for the phase field structure, since other critical points of phase such as maxima, minima, and saddle points, are necessarily also located on these same zero crossings [23]. More generally, we now show here that vortices are always located on the zero crossings of all higher-order phase derivatives, with m zero crossings of each mth-order derivative passing through every vortex. Thus, in addition to one zero crossing each of R, I, V,, U,,, cp,, and spy passing through every vortex (a running total of six), there are also present two zero crossings each of c.~, ‘dye, and po,, (running total of 12). rhree zero crossings each of vl,,,. ‘p,,, (pxyyand ‘pyy, (running total of 24), twenty zero crossings of the five fourth-order phase derivatives (running total of 441, etc. Accordingly, every vortex is located at the intersection of infinitely many zero crossings, and since for every one of these zero crossings there exists an extension of the sign alternation principle that constrains the phase field structure, there are infinitely many topological correlations between different points in the phase field. In the following sections we prove this remarkable result and extend it to higher-order vortices and to other functions such as the intensity. We also present a number of illustrative examples using data for a Gaussian random wave field.
2.1. Vortex derivatives
2. Low-order derivatives
cp =- J (u yyY2 - %X2). ry 2u2
In this section we start by first developing explicit forms for a number of low-order vortex derivatives for first-order vortices. Using mathematical induction, we then show in subsequent sections that the conclusions drawn from these explicit forms hold also for all derivatives of arbitrarily high order.
Similarly, for the four independent third-order derivatives we obtain
Writing
(Ia> we have RI., - IR ~ ‘p, =
u
> P,” =
RIY - IR, (I
(‘b)
’
where, as before, the intensity U = R2 + 12. Worth noting is that interchanging x and y everywhere yields q,,( x,y) = cp,(y,x), an important symmetry preserved in derivatives of all higher orders. Moving the origin to the vortex center and using the fact that the vortex is a simple first-order zero of the wave function, we expand R and I in a Taylor series as R(x,y)
= T,x+
r”y+
. . . . I(x,y)
= L,X+ ~.“y+ . . . . (2)
where TX= (aR/ax&, L, = (N/ax),, etc., and ( JO implies evaluating the enclosed derivatives at the origin (vortex center). Since the intensity has a zero minimum at the vortex center we obtain cp, = -(J/u)y,
(3a)
‘py= (J/u).5
(3b)
where the sign of the Jacobian J = TX”? - T,,L~ equals the sign (topological charge) of the vortex [24,25], and where u(x,y) = ~u,,x2 + iuyvy2 + $uXyxy, with u,, = (#~//&x2>,, etc. In writing Eqs. (3a) and (3b), we neglect all higher-order terms in x and y since analysis shows (surprisingly, perhaps) that to leading order in the small displacement from the vortex center r = (x2 + y*)‘/*, such terms make no contribution to derivatives of this and all higher orders. Using the fact that off the vortex center the phase is a regular (analytic) function of x and y so that, for example, (pXy= i)cp,/ay = 8(py/ax, after straightforward calculation we obtain for the three independent second-order derivatives PXX= -u-‘(U,,
X+UryY)%*
cpyy = --u-‘(u.ryx
cpXXX = u -2[$&xz +3u,*u,,xy]
+ UyvY)‘p, = cp,,(Y7X)?
(4a) (4b)
+ (2uf, - $4,,uyy )Y2 CPXV
(5a)
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I. Freund/Optic.s Communications 137 (1997) 118-126
(5b) %,vrvy(X~Y) = cp,.r,(Y,X),
(5c)
= cp,,.dYJ)~
(5d)
'4vyv(X*Y)
2.2. Vortex zero crossings
We now use Eqs. (3)-(5) to deduce the structure of the vortex zero crossings. Returning to Eq. (3a) and recalling that the intensity (u) is positive definite, we note that off the vortex center, both for positive and negative x, ‘pXis positive (negative) whenever the product Jy is less than (greater than) zero. For positive vortices J > 0, so excluding the origin, cp, is positive for negative y and is negative for positive y, while for negative vortices this situation is reversed. Now, since ‘p, is regular everywhere between vortex centers, its transition between positive and negative values is continuous and must pass through zero. We may therefore conclude that in the immediate vicinity of a vortex the boundary between positive and negative values, the x-axis, is tangent to a zero crossing of ‘p,, and that by a limiting process this zero crossing passes through the vortex center. Similarly, we may conclude from Eq. (3b) that: in the immediate vicinity of a vortex the y-axis is tangent to a zero crossing of ‘p, that also passes through the vortex center, and that for positive vortices ‘p, is positive (negative) for positive (negative) x, while for negative vortices this situation is reversed. All these conclusions are fully borne out by the data in Fig. 1, which shows results obtained from recent large-scale computer simulations of a Gaussian random wave field [25]. The above results may also be obtained by inspection using simple geometric arguments. Inserting Eq. (2) into Eq. (lb) we have cp=arctan[(l +AtanB)/(B+Ctane)], where tanO=y/x, A = L,,/L~, B = ‘T’/L_~, and C = Ty/L,. To first order cp is independent of the radial distance r, so to this order contours of constant phase are straight lines that radiate outward from the vortex center. ‘p, (‘p,) therefore vanishes along the x-axis ( y-axis) which is a line of constant phase. The signs of ‘pX and ‘p, throughout the (x,y)-plane are also easily obtained geometrically by noting that increasing x causes 0 to decrease for positive y and increase for negative y, while increasing y causes 8 to increase (decrease) for positive (negative) x. For the second-order vortex derivatives Eq. (4a) shows that one zero crossing of ‘pXXthat passes through the vortex center coincides with the zero crossing of cpX,i.e. the x-axis. Writing x = rcos0 and y = rsin 0, we find a second vortex zero crossing tangent to the line 0= - arctan[u,,/u,,]. We note that this line is also tangent to the zero crossing of U, at this point. From Eq. (4b) we find one zero crossing of (py,, passing through the vortex
(a)
03
(cl Fig. 1. (a) Zero crossing maps of ‘pX= flcp/ax (thick line) and ‘p, = ?lcp/ay (thin line), where cp is the phase, showing n positive and 0 negative vortices at intersections of the zero crossings. As discussed in Ref. [23], along any given zero crossing vortices must always be separated by phase saddles (unmarked intersections). (b) Map of cp,. (c) Map of qY. In (b> and (c) positive (negative) regions of the function am colored white (black). All maps shown here and in Figs. 2 and 3 cover the same region of the wave field and are 199 pixels wide (x-axis) and 110 pixels high (y-axis), where the transverse coherence length of the field is 19.2 pixels.
center coinciding with the zero crossing of ‘pY, i.e. the y-axis, together with a second zero crossing tangent to the line 0- -arctan[u,,/u,,] (also the tangent to the zero crossing of U,,). Finally, from Eq. (4c) we find two zero crossings of ‘p_rYpassing through the vortex center. These are tangent to the lines 8,s +arctan[(u,,/u,,)“2], where both u,, and uYYare positive definite since ZJ is a minimum at the vortex center. In Fig. 2 we present data obtained from our computer simulations that confirm as well as illustrate these results. In similar fashion we find from Eqs. (5) for the thirdorder derivatives three zero crossings of ‘p,,, passing through the vortex center. One coincides with the zero crossing of cp,, 01,XxX= 0, n, and the other two are
I. Frrund/
Optics Communicuriwzs
(4
ing each of ‘pI, and qDr, two zero crossings each of ‘p, ‘i, ‘pyy. and V+ and three zero crossings each of P_~*~, cpyvv. (P.~.~~,and ‘P,,,. Although this suggests a trend, it is clearly not feasible to carry out an unbounded number of (increasingly complex) explicit calculations in order to establish that this trend holds for all higher-order derivatives. Accordingly, we now turn to the use of mathematical induction to demonstrate that m zero crossings of every mth-order derivative do, in fact, pass through every vortex. With this established, we then extend these findings to higher-order vortices, to the wave field amplitude, and to regular functions such as the intensity.
3. Higher-order
(4
(c) Fig. 2. Zero crossing maps of (a) qAA, (b) ‘p,,. and (c) ‘pAy showing hvo zero crossings of each derivative passing through both n positive and 0 negative vortices.
to the lines 02,rxr = -arctan(3~,,[(3H)‘/~ + - 3u,,]~‘]. 3n.J’1 and %x.rr = arctan{3u,,[(3H)‘/Z Here, the Hessian H = u,,uyv - uf, is positive definite since U is a minimum at the vortex center. For ‘p,,, we also find three zero crossings passing through the vortex center. One coincides with the zero crossing of ‘p,, O,,lyy = n/2, 31~/2 and the other two are tangent to the lines and 03.yyy = e LYYY = - arctan(3 uy ,[(3 H) ‘1’ + 3u,,]-‘} For (pxl,, we find from arctan{3u ,,[(3 H) ‘/2 - 3u,,]-‘). Eq. (SC) three zero crossings that pass through the vortex center. Writing t = tan0, these may be obtained from the roots of t3 + a,t2 + a3t = 0, where a, = 3~,,/(2u,,) and Since the discriminant A = a3 = --u~,/(~u,~u,~). u4xy ) is negative definite, all three roots are - 4@,/(& real and distinct, and when required may be written down explicitly using standard methods. Finally, the three zero crossings of (erVy may be obtained with the aid of these same roots simply by replacing tan0 with cot0. This replacement follows from ‘px,,( x, y) = ce,,,( y, x) [Eq. tangent
(5c)l. We have therefore shown by explicit calculation that passing through the vortex center there is one zero cross-
121
137 f 1997) I IS-126
derivatives
Returning to Eqs. (3a) and (3b), we introduce two successive linear transformations. The first is a principal axis transformation, i.e. rotation through an angle cy that diagonalizes the intensity. In the resulting x’,y’ coordinate system, u becomes u(x’,y’) = fA+x” + +A_$‘, where A, are the eigenvalues of the Hessian matrix H (H,, = u.u, H22 = uyy’ H,, = H,, = u.~J. The second transformation is a dilation of the y’-axis by D = ( A+/A )‘I’ that produces our final coordinates X, Y. As a result of these transformations, the intensity becomes isotropic and may be written u(X,Y)= +A+$, where p=(X’+ Y 2)‘/2. This isotropy makes it convenient to work in polar coordinates, and we define a polar angle (T measured counterclockwise from the X-axis, and write X = pcosc~, Y = psina. Due to the combination of rotation and dilation, our original x- and y-axes are no longer orthogonal and are oriented at peculiar angles relative to the (X,Y )-coordinate system. We therefore calculate vortex derivatives along some arbitrary direction s = dY/dX using df/ds [(af/dX>, + s(af/aY ),]/(I + s2)‘i2. Then by choosing s so that the direction of differentiation corresponds to the normal to the y- or x-axes (s, or s,,, respectively), (af/kx), = K,(df/ds,) and (af/ay), = K,(df/ds,), where K, and K,” are constants. In this way the mth-order directional phase derivatives (f = cp> multiplied by K(~)(x,. . ,y _ . ), the appropriate produCt of the K., and K~, correspond to vortex derivatives such as ‘P.~,~,-
[ d4’( X.X-X.X> = K_:], (p+xyyy[K(‘)( Xryvv) = +;I,
etc.
As the above linear transformation does not change the order of the wave function zeros, Eqs. (l)-(3) are applicable with X replacing x and Y replacing y. Using dp
- -(coso+ z-j&2
--
ssina),
-sina+
scosu
(f-9
I. Freund/
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Optics Communications
we have
137 (1997) 118-126
Eq. (8) implies that every m&r-order vortex derivative has m zero crossings passing through the vortex center. These are tangent to the lines o-= --m-‘arctan[ [(“)/J/(“)]. Noting from Eq. (9g) that these m zero crossings in X,Y map uniquely into m equivalent zero crossings in x, y completes the demonstration.
where J, = 7’, ~~ - Ty~x is the transformed Jacobian in X,Y. We now invoke mathematical induction. If 4. Extension to higher-order
d’b
cp?.! = ds,...ds, cos(ncr)
+
rl? [
I
@aI
’
P”
where $ and 5 are constants, the superscript (n) denotes the order of the derivative, and the subscript . . . . represents differentiation along an arbitrary sequence of directions s,, sz,. . . ,s,, then d”+ ‘(p P!(:Its’)= ds
, . ..ds.ds
I
cos[(n + l)a]
+ p+ 1) . . ..I [
P
I?+1 I?
@b)
5.!“.:‘)=
n[3?.1!-P.?!l &---&
(8~)
.
As this is of the same form as Eq. (8a), it leads to a recursion loop that closes on itself if the initial condition is met. Since the starting point, Eq. (7a) for p(l), is, in fact, precisely of the form required by Eq. (8a), Eq. (8b) describes vortex derivatives of arbitrary order. These derivatives may, if required, be expressed in terms of r,8 or x,y in the original x,y-coordinate system with the aid of the following formula summary: X=xcosa+ysincr, tan(2cr) =
Y=D(-xsina+ycosa),
(9a)
2u,, (9b)
U .1I -
UYY
D=dE, A,=
f(T+
(9c)
tw-G>,
T=u,,+L+
s,= K,
-Dtancr, =
(94
H=u,,u~,-u:~, s,=Dcot(~,
D’sin’a! +
COS2a,
K~ =
(9e) D2COS2a
i-
Sin2a
,
WI
6=o+arctan[D-‘tana],
(9g)
p-r[co~~(~-(~)+D~sin~(~-a)]“~.
(9h)
vortices
Returning to the simplest vortex model of Nye and Berry [I] reviewed in Section 1, we write again cp(rtj) = +j6. This describes an isotropic vortex. Using Eq. (6) with 0 replacing o and r replacing p, it may be seen that for this vortex the results of the previous section hold also for arbitrary j. Since for these isotropic vortices the linear transformation in Eq. (9a) is the identity transformation ((Y= 0, D = I), it follows from Eq (8b) that for all values of j and m, the mth-order derivative zero crossings form a symmetric star whose m arms have equal angular separation A r3crn)= n/m. As vortices are topologically stable features of the wave field, asymmetric vortices may be obtained from isotropic ones by continuous deformation. For first-order vortices (j - 1) it is apparent from Section 2 that these deformations simply cause the zero crossings to rotate to arbitrarily different angles, thereby destroying the symmetric arrangement obtained in the isotropic case. But we cannot be certain that this is all that occurs for higher-order vortices. In particular, we need be concerned that for the j&order vortex the zero crossings for the isotropic case might be say j-fold degenerate, and that this degeneracy would be split in going to the asymmetric case. A simple argument suggesting such a possibility is that if an isotropic jth-order vortex were to split into j closely spaced firstorder vortices, each of these latter would carry m zero crossings, so that when viewed from a distance as a single object the number of zero crossings radiating outward from this vortex grouping could total j X m. There are of course other possibilities, and we explore the question of degeneracy by examining in detail a simple example of a second-order vortex (j = 2). We begin by noting that for the isotropic second-order vortex (x + iy)‘, for which R = x2 - y2 and I = Zxy, both R and I correspond to saddle points. This remains true also for the general asymmetric case. The reason is that the phase circulates through 47~ for a second-order vortex, while contours of constant phase cp= 0, P, etc. correspond to zero crossings of I and contours of cp= ~/2, 3rr/2, etc. correspond to zero crossings of R. Accordingly, there must be MO zero crossings each of R and of I intersecting at the vortex. A self-intersection such as this corresponds to an ordinary (first order) saddle point. As the sign of dp/d0 corresponds to the vortex sign and must be independent of 8, the zero crossings must be interleaved
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I. In addition, for a saddle point the Hessian Rf ,(I,, I,, - l.zy) must be negative definite. LR,, These conditions can be met by writing R = ( y - ux)( y a), I = ( y - bx)( y - d.r), with a < b < c < d. Choosing as a simple example that has a sufficient degree of generalitya=l,b=2,~=3,andd=4,wefindfromEq.(lb) for cp+ that the zero crossings of this vortex derivative are given by the roots of 13-_2-9t+ 12=0, where as before I = tanf3. Since the discriminant A = ( - 28/9)3 + ( - 241/54)* is negative, there are three real distinct roots and therefore three nonoverlapping zero crossings of cp,. Using qJ( x, y) = M,( y, x), we also find that there are three nonoverlapping zero crossings of ‘p, that pass through the vortex center (these are the roots of tK3 - tA2 - 9r-’ + 12 = 0). The individual zero crossings of the isotropic case are therefore now seen to be three-fold degenerate, rather than two-fold degenerate as in our initial guess. Although the physical basis for this higher degree of degeneracy remains unclear, these results can be generalized to produce an upper limit of m(2j - 1) for the number of zero crossings of the &h-order derivative passing through an asymmetric vortex of order j. From Eq. (lb) it is apparent that independent of vortex order, the &h-order phase derivative can be written (pcrn) * Pm/Urn, where P, is some polynomial in x and y whose order we now determine. Since the phase circulates through 2 jrr for a vortex of order j, for such a vortex R=(y-a,xXy-a,x)...(y-ajx)-yi. Then, the intensity II - R* - yzi, while by dimensional arguments (rn) N l/y”, so we conclude that P,,,- y"'(*j-‘). As u is cp positive definite, the zero crossings of (pcrn) are given by the roots of P,,,(x, y) = 0. An upper limit on the number of real distinct roots is m(2j - I), which is an upper limit on the number of zero crossings. We therefore conclude that the number of zero crossings N(m,j) of each mth-order phase derivative passing through a vortex of order j is bounded by m zz N(mj) I m(2 j - 1), where the lower limit is the value of N(mj) for isotropic vortices, and as a conjecture based upon the specific example discussed above, the upper limit is its value for fully asymmetric vortices. R,
I, R,
5. Extension to the amplitude Although the intensity U and all its higher derivatives are regular everywhere, this is not the case for the amplitude A = U ‘/*. Since A, = (RR, + IIX>/lJ’12 and A, = all vortex derivatives of A are singu(RR.”+ uJ/u’/2, lar. Explicit calculations for a few low-order derivatives reveal the presence of degeneracies even for first-order For example, A,, = HY~/(~u~/~), A,, = vortices. so that both Hx*/(2 u3/*), and A.,, = -Hxy/(2u3/*), A .Y.t and A,, each have one doubly degenerate zero crossing while A.,, has two nondegenerate zero crossings
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passing through the vortex center. Similarly, A.,,( x, y) = -3H(u,,x + uXyy)y2/(4u5/*), and A,,,(x,y) = A ,,,( y, x) both have one doubly degenerate and one nondegenerate zero crossing passing through the vortex, while the three vortex zero crossings of A,,,(x, y) = - Hy(2u,,y2 + uXYxy - u,,x2)/(2u5/*) and of Alvy( x, y> = Al+$ y, x) are all nondegenerate. Although the methods of Section 3 permit us to write A = (A+/2)‘/$, application of Eq. (6) does not lead to a closed system, so that currently no general expression for Acm)is available. For higher-order vortices we may repeat the arguments of the previous section by writing the m&t-order derivative as Atm)- Q
6. Extension to other functions We now extend these results to the intensity U( x, y), to to any regular (i.e. singularity free) function F( x, y). Defining the vector field (gradient) VF(x, y) = F,x + FYy, where x,y are unit vectors along x, y, we note that since F is regular everywhere, so is VF. For such a vector field we may define a complex order parameter whose amplitude is the length of the vectors, A(F; x,y> = (VFI = (F,’ + F.z)‘/*, and whose phase is the angle (direction) the vectors make relative to the x-axis, @(F; x, y) = arctan( FJF,). Since the gradient always points in the direction of maximum increase of the function, these vectors radiate outward from a minimum and inward towards a maximum, so their directions at maxima and minima become undefined. The same is true for a saddle point which is a maximum along one principal axis and a minimum along the other. O(F) is therefore regular everywhere except at the stationary points (maxima, minima, and saddle points) of F(x,y) where it is singular, since here F, = FY = 0 (regularity of VF is preserved at these points since A = 0). The singularities of O(F) are easily shown to be the same as the singularities of the phase field cp, i.e. first-order vortices. Moving the origin to a stationary point and noting that Ffx.y) is regular everywhere, we have (in obvious notation) I;, =frxx +frYy + . .., F,”=frXx +fruy + . . . , which is completely analogous to Eq. (2) with F, replacing R and FY replacing I. All the other equations for the singularities of cp also become applicable to the singularities of 0 upon replacing 7” by f,,, ‘1 by f,,, L, by We note that this replacement changes f yxrand ‘Y by fyy. the symmetry condition ‘py(x, y) = cp,( y, x), which now becomes O,(F;x, y) = - OJF; y,x) [Eqs. (3a) and (3b)l.
R( x, y), to I( x, y). and in general
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(a)
137 (1997) 118-126
eter 0’ = arctan(OY/@~r). At stationary points of 0. 0, = 0.” = 0, so at these points 0’ is singular. As before, these singularities correspond to first-order vortices whose sign is given by the sign of H’ = O,Yx@Y,- @_v,@,,r, so that extrema of 0 correspond to positive first-order @‘-vortices and saddle points of 0 correspond to negative first-order @‘-vortices. But to what do @-vortices (i.e. the stationary points of F) correspond? Using Eqs. (3a) and (3b) in the form 0, = - Hy/( F,’ + F,‘) and 0, = Hx/( c: + F,‘) we obtain for positive (negative) @-vortices, 0’ = ~9+ n/2 (31~/2), so that all stationary points of F correspond to isotropic positive first-order @-vortices 1231. Explicit forms for 0, and 0, are easily found to be
q X,Y> =
Vx, - F,F,x F.; + F,’
’
O,( x,y) = - @,(Y,X).
(10) (W
w Fig. 3. (a) Zero crossing maps of the X, y-derivatives of the phase O(R) of the complex order parameter of VR (Section 6). Here R is the real part of the wave function. (Thick line) O,(R), (thin line) @&R). (b) Map of O,(R). (c) Map of B,(R). These maps also show R-stationary points, n extrema and Cl saddles. In (b) and (c) positive (negative) regions of the function are colored white (black). As discussed in Section 7, along any given zero crossing R-stationary points must always be separated by O(R)saddles (unmarked intersections).
Worth noting is that from this it follows that at stationary points of 0 (OX= OY=O, IF,I>O, IF,I>O) the Hessian H = F,, F,,, - F& = 0. The corollary of this, that say 0, = H = 0 implies 0, = 0 and hence a stationary point of 0 is not necessarily true, however, since zero crossings of both 0, and H but not necessarily of 0: always pass through zero crossing intersections of F.,, and TX.,,. In summary, stationary points of F are located at the simultaneous intersections of zero crossings of F,, F,, @.,, and 0,. The zero crossing intersections of F, and FY define the singularities of 0, and the zero crossing intersections of 0, and 0, define the singularities of 0’. F-extrema correspond to positive, and F-saddles to negative O-vortices, while both types of F-stationary points correspond to isotropic, positive @‘-vortices. Stationary points of 0 are located at intersections of the zero crossings of (9x and OY where H = 0, while @-extrema conespond to positive and @-saddles correspond to negative @‘-vortices. In the next section we explore some of the topological correlations implied by these results.
The reason for this is that J(x,y) = -J(y,x) is replaced by H(x,Y) =f,,f,, -fryJ;.v = H(yd where fyx =.frf,,. With these replacements, extrema (maxima and minima) of F, for which H > 0, correspond to positive @-vortices, and saddle points of F, for which H < 0, correspond to nega-
7. Topological correlations
tive O-vortices. As the results of Sections 2 and 3 are applicable also to @-vortices, we may conclude that m zero crossings of each of the mth-order derivatives 0’“‘) pass through every maximum, minimum, and saddle point of F(x,y). Shown in Fig. 3 is an example for m = 1 that involves zero crossings of O,(R) and O,(R) passing through stationary points of R. This figure may be seen to be equivalent in every respect to Fig. 1, reflecting the equivalence of 0 and cp. In the next section we will need also to analyse the vector field VO and the phase of its complex order param-
Application of the sign alternation principle [21,23] to O-vortices immediately leads to the (possibly unexpected) result that F-extrema (positive @-vortices) and F-saddles (negative @vortices) must alternate along zero crossings of F, and F,. This has been shown to lead to a quasicrystalline arrangement for these stationary points in a Gaussian random wave field [26]. Similarly, application of the sign principle to @‘-vortices immediately leads to the analogous result that F-stationary points and @-extrema (both positive @‘-vortices) must alternate with @-saddles (negative @‘-vortices) along zero crossings of @%and 0,.
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The alternation of F-extrema and F-saddles along zero crossings of F,r and FY is also the mathematical statement that the structure of the (wave) function must be such that along certain lines defined by jirst-order derivatives (F, = FY = 01, a particular combination of second-order derivatives (H) must alternate in sign. This connection between nominally independent derivatives of different order is an unexpected, indeed striking topological constraint. Similarly, alternation of F-stationary points and O-extrema with O-saddles along zero crossings of 0, and 0, is equivalent to the mathematical statement that the structure of the wave function must also be such that along certain lines formed by combinations of first- and secondorder derivatives (@_,= 0, 0.” = 01, a particular combination of first-, second- and third-order derivatives (H’) must alternate in sign. By defining 00’ and the phase of its complex order parameter 0”. we can generate a third statement that constrains fourth-order derivatives along lines defined by lower-order derivatives, and since we can extend this procedure indefinitely, an indefinite number of constraints of this sort can be generated for derivatives of arbitrarily high order. We emphasize that these constraints are necessary in the sense that simply laying down an arbitrary network of zero crossings of say 0, and 0, suffices to force H’ to alternate along the network. The above constraints give rise to numerous near neighbor correlations that are similar to those found for near neighbor vortices [2 I I, while many additional topological constraints and their associated correlations may be deduced by taking VO,, VO,,v, etc., instead of VO as the starting point for the development in Section 5. As all these correlations differ from those already described in degree rather than in kind, we do not develop them further here.
8. Summary We have shown that M zero crossings each of every mth-order phase derivative pass through every first-order vortex, and have argued that for jth-order vortices the number of zero crossings is increased by up to a factor 2 j - I. These findings have been extended formally to the amplitude, and also to regular functions by relating the stationary points of such functions to singularities of their gradients. Unusual topological correlations connecting higher-order derivatives of the function to lower-order derivatives have been found, together with new relationships governing the spatial arrangements of zero crossings. As phase derivatives are important in determining ray directions and other wave field properties, the results described here may prove useful in areas such as image processing and adaptive optics [27], as well as in recently proposed, innovative vortex-specific applications. They
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may also be of interest in other areas where vortices are of prime importance, such as two-dimensional solid state physics.
Acknowledgements This work was supported by the Israel Science Foundation of the Israel National Academy of Sciences and Humanities.
References [I] J.F. Nye and M.V. Berry, Proc. R. Sot. A 336 (1974) 165. [2] M. Berry, Singularities in waves and rays, in: Physics of Defects, eds. R. Balian, M. Kleman and J.-P. Pokier (NorthHolland, Amsterdam, 1981) pp. 453-543. [3] M.V. Berry, J. Phys. A I I (1978) 27. [4] N.B. Baranova, B.Ya. Zel’dovich, A.V. Mamaev, N.F. Pilipetskii and V.V. Shukov, JETP Lett. (1981) 195; N.B. Baranova and B.Ya. Zel’dovich, Sov. Phys. JETP 53 (1981) 925; N.B. Baranova, A.V. Mamaev, N.F. Pilipetskii. V.V. Shukov and B.Ya. Zel’dovich, J. Opt. Sot. Am. 73 (1983) 525. [5] 1. Freund, N. Shvartsman and V. Freilikher, Optics Comm. 101 (1993) 247; N. Shvartsman and I. Freund, Optics Comm. I 17 (1995) 228 [Erratum: 120 (1995) 3691; I. Freund and D.A. Kessler, Optics Comm. 126 (I 995) 321. [6] F.T. Arecchi, G. Giacomelli, P.L. Ramazza and S. Residori. Phys. Rev. Lett. 65 (1990) 2531. [7] M. Brambilla, F. Battipede, L.A. Lugiato, V. Penna, F. Prati, C. Tamm and CO. Weiss, Phys. Rev. A 43 (1991) 5090. 5114. [8] G. Indebetouw, J. Mod. Optics 40 (1993) 73; S.R. Liu and G. Indebetouw, Optics. Comm. 10 I (I 993) 442. [9] I. Dana and I. Freund, Vortex lattice wave fields, Optics Comm. 136 (1997) 93. [IO] H.V. Deighton, M.S. Scivier and M.A. Fiddy, Optics Lett. IO (1985) 250; P.-T. Chen and M.A. Fiddy, J. Opt. Sot. Am. I I (1994) 2210. [I I] C.C. Wackerman and A. Yagle, J. Opt. Sot. Am. I I (1994) 2016; P.J. Bonnes, C.R. Parker, B.I. Satherly and R.W. Watson, J. Opt. Sot. Am. 12 (1995) 1842. [I21 A. Ashkin, Biophys. J. 61 (1992) 569; K.T. Gahagan and G.A. Swartzlander, Jr., Optics Lett. 21 (1996) 827. 131 E. Abramochkin and V. Volostnikov. Optics Comm. 125 ( 1996) 302. 141 P. Leboef, J. Kurchan, M. Feingold and D.P. Arovas, Phys. Rev. Lett. 65 (1990) 3076; 1. Dana, Phys. Rev. E 52 f 1995) 466. 151 J.M. Kosterlitz and D.J. Thouless, Two dimensional physics, in: Progress in Low Temperature Physics, Vol. VII B, ed. D.F. Brewer (North-Holland, Amsterdam, 1978) pp. 371433. 161 D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. den Nijs, Phys. Rev. Lett. 49 (1982) 405; R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. 171 A.A. Abrikosov, Sov. Phys. JETP 5 (1957) 1174; D.E. Farrell, Transformations of the vortex solid, in: Physical
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[ 181
[ 191
[20] [21]
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Properties of High Temperature Superconductors, Vol. IV, ed. D.M. Ginsberg (World Scientific, Singapore, 1994) pp. 7-59. An important class of experimentally realizable (nearly) nondiffracting beams are based on Bessel functions: J. Dumin, J.J. Miceli Jr. and J.H. Eberly, Phys. Rev. Lett. 58 (1987) 1499; J. Dumin, J. Opt. Sot. Am. A 4 (1987) 651; F. Gori, Cl. Guattari and C. Padovani, Optics Comm. 64 (1987) 491. V.I. Arnold, Ordinary Differential Equations (MIT Press, Cambridge, Mass., 1973); S.H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA, 1994). J.F. Nye, J.V. Hajnal and J.H. Hannay, Proc. R. Sot. A 417 (1988) 7. N. Shvartsman and 1. Freund, Phys. Rev. Lett. 72 (1994) 1008; 1. Found and N. Shvartsman, Phys. Rev. A 50 (1994)
[22] [23] [24]
[25] [26] [27]
137 (1997) 118-126 5164; N. Shvartsman and I. Freund, J. Opt. Sot. Am. 11 (1994) 2710. I. Freund, Phys. Lett. A 198 (1995) 139. I. Freund, Phys. Rev. E 52 (1995) 2348. B.I. Halperin, Statistical mechanics of topological defects, in: Physics of Defects, eds. R. Balian, M. Kleman and J.-P. Poirier (North-Holland, Amsterdam, 1981) pp. 814-857. 1. Freund, J. Opt. Sot. Am. A 11 (1994) 1644. I. Fteund and N. Shvartsman, Phys. Rev E 51 (1995) 3770. The Schack-Hartmann lenslet array, which measures firstorder phase derivatives, is widely used as a wave front sensor in adaptive optical systems: L.A. Thompson, Physics Today 47 [ 12]( 1994) 24; See also: Adaptive Optics, Optical Society of America 1996 Technical Digest Series, Vol. 13.