Saddle point wave fields

15 May 1999

Optics Communications 163 Ž1999. 230–242

Full length article

Saddle point wave fields Isaac Freund Jack and Pearl Resnick AdÕanced Technology Institute and Department of Physics, Bar-Ilan UniÕersity, Ramat-Gan 52900, Israel Received 4 January 1999; accepted 11 March 1999

Abstract Analytic algorithms are presented for the construction of wave fields whose only critical points are saddle points at predetermined locations. A simple calculus is developed for the evolution of these fields embedded in Gaussian laser beams, and is used to study the propagation of generic examples. This calculus is also applied to the propagation of embedded vortex arrays whose complexity strains standard methods. Possible applications of saddle point optical fields are outlined. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction Two dimensional optical vortex wave fields and their increasingly important applications are of substantial current interest w1–17x. Although ‘‘natural’’ fields normally contain equal numbers of positive and negative vortices Žphase singularities. w18–21x, it is not difficult to create specific fields containing vortices of a single sign. Similarly, regular Žsingularity free. two-dimensional optical wave fields normally contain nearly equal numbers of extrema and saddle points. It is then natural to ask if it is possible to create specified fields that contain only extrema, or only saddle points? We show by explicit construction that the answer is yes for saddle points, and defer fields containing only extrema to a later study. But why would one want a saddle point field? One possible answer is that in addition to their intrinsic interest as a new type of wave field, saddle point fields may form the basis for a number of potentially useful applications. As a simple example, in Fig. 1 we show an optical circular hole saw obtained by embedding a combination of elementary saddle point fields in a Gaussian laser beam. As the unusual properties of these fields become better understood, other applications are likely to follow. The plan of this brief study is as follows. In the next section we show how to construct a saddle point field containing a predetermined number of saddle points at specified locations. In Section 3 we discuss the propaga-

tion dynamics of these fields embedded in Gaussian laser beams, developing an easily implemented calculus that may prove useful also for other applications. In Section 4 we describe the special properties of saddle point fields that satisfy Laplace’s equation, and in Section 5 we briefly summarize our most important findings. We also include two appendices that discuss a number of specific examples. In Appendix A we use our new calculus to study the propagation of vortex wave fields whose structures are more general than those considered in the past, and in Appendix B we study the evolution of our new saddle point fields. We emphasize that the present study is not exhaustive, that new methods for the creation of fields with a single type of critical point will undoubtedly emerge, and that many important properties of these fields still remain to be found.

2. Gradient mappings and saddle point wave functions In this section we develop a systematic method for placing an arbitrary number of saddle points at arbitrary positions in the xy-plane. We consider first ‘‘obvious’’, a priori plausible methods, show why these are in general unsatisfactory, and then proceed to describe a possibly non-obvious gradient mapping that solves our problem. We start by considering the structure of a single first order Žgeneric. saddle, of which the classic example is

0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 1 4 2 - X

I. Freundr Optics Communications 163 (1999) 230–242

231

Fig. 1. Optical circular hole saw. This potentially useful cutting tool is constructed from the coherent superposition of two elementary saddle X XX point fields ŽLaguerre modes. Fn s ReŽw x q iy x n . and Fn s ImŽw x q iy x n .. These fields are embedded in a Gaussian laser beam of width w X XX 2 2x 2. 2 Ž . Ž w as Fn s Fn q r Fn exp y x q y rw r 1 q r . The saw has 2 n teeth and is rotated by varying the phase and amplitude of the ratio r of the two component fields. This may be done electrooptically, so that very high rotation speeds are possible. The diameter of the saw is D ; '2 n w. For the example shown n s 3 and w s 1r '2 . Ža.,Žb. Three dimensional views of the saw intensity distribution for r s y10 and 0 respectively. Note the sharpness Žwidth ; p wr w 2'2 n x . and 100% depth of modulation of the teeth. These important properties are preserved for all values of n and w, so that for large n very sharp, very fine teeth may be obtained. Žc. – Žg. Contour maps of the intensity showing rotation of the saw for r s y10, y1, 0, 1, and 10. Since biological cells and other small particles may be trapped in or between strong optical maxima, this device or its linear variants could also serve as an optical chuck or clamp, as well as a micromotor drive source. This saw is one example of potentially useful optical tools that may be constructed from saddle point fields.

'

f 0 Ž x , y . s x 2 y y 2 s Ž x q y .Ž x y y . . The contour map of this function may be drawn by inspection. Passing through the saddle point, which is located at the origin and at which f 0 s 0, is the pair of lines y s "x. These special lines, that we call ‘‘bifurcation lines’’, are a self intersection of the contour f 0 Ž x , y . s 0. They divide Žbifurcate. the plane into four sectors within which the remaining contours form hyperbolas that asymptote to the bifurcation lines. This is the characteristic map of a first order saddle point, and it always appears locally around any self intersection of a contour. We can, if we wish, move the saddle point to say Ž x "1 , y . s Ž"1,0. simply by moving both bifurcation lines, writing in this case f "1 Ž x , y . s Ž x y x "1 q y .Ž x y x " 1 y y . . We now place three saddle points in the plane at say Žy1,0., Ž0,0., and Žq1,0.. This is easily accomplished by laying down three pairs of bifurcation lines, writing f 3 Ž x , y . s fy1 Ž x , y . f 0 Ž x , y . fq1 Ž x , y . . Examining the re-

sulting contour map in Fig. 2, we observe, however, rather more than was wanted. Instead of three saddle points we find nine! Even worse, we also find four extrema – two minima and two maxima. The unwanted saddles come about because there is no way of placing six lines in the plane to form only three intersections, while the extrema are a necessary consequence of the closed contours resulting from intersections of more than two nonparallel lines. In the special case that all saddles lie on a single line L, we can create exactly n saddles and no extrema by laying down n parallel lines that intersect L, but no solution using only straight lines is possible for the general case of arbitrary saddle point locations. We are thus forced to consider sets of curved lines that intersect only at specified locations and that form no closed contours anywhere in the xy-plane. For the general case this is a formidable algebraic problem for which a tractable solution appears unlikely.

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From the foregoing, it is clear that a purely algebraic approach is unlikely to yield a systematic means for creating fields whose stationary points consist only of saddle points, and a rather different approach is therefore required. This is provided by the gradient mapping reviewed below, which maps the stationary points of a regular function f Ž x , y . onto vortices w22x. Consider first the problem of placing N vortices with topological charges Žwinding numbers. qn s "1 at positions Ž x n , yn . . Using the fact that such vortices are firstorder zeros, the wave function Vn Ž x , y ;qn . of the nth vortex may be written Vn Ž x , y ;qn . s R n Ž x , y . q iIn Ž x , y . , R nŽ x , y . s w R x x n Ž x y x n . q R y In Ž x , y . s w I x x n Ž x y x n . q I y Fig. 2. Field produced by product of three saddle point functions of the form f nŽ x, y . s Ž x y x n q y .Ž x y x n y y ., with x n s 0," 1. In addition to the desired three saddles on the center line at Ž x, y . s Žy1,0., Ž0,0., and Ž1,0., the field contains six unwanted saddles at Ž0,y1., Žy1r2,y1r2., Ž1r2,y1r2., Žy1r2,1r2., Ž1r2,1r2., and Ž0,1., together with four extrema Žclosed contours., two maxima on the horizontal centerline and two minima on the vertical centerline.

n

n

Ž y y yn . ,

Ž y y yn . ,

Ž1.

where w x n implies evaluating the enclosed partial derivatives at the vortex center Ž x n , yn . . The vortex charge qn is then given by qn s sgn R x I y y R y I x n. Because these vortices are isolated point zeros, the total wave function V Ž x , y . of an array of N vortices may be written as the product N

VŽ x , y . s

Ł Vn Ž x , y . .

Ž2.

ns 1

Saddle points are also stationary points at which both f x s E frE x and f y s E frE y vanish. They therefore lie not only on self intersections of contours, but also on intersections of the zero crossings Žnodal lines. Z x of f x with the zero crossings Z y of f y . But any attempt to construct a saddle point field using these zero crossings not only suffers from the problems noted above for bifurcation lines, but also from the fact that at adjacent intersections of say Z y with a given Z x , saddles must always alternate with extrema w16,21x.

Product wave functions are, of course, not the only possible representations of arrays of vortices with specified charges and locations, but such products are widely used for simplicity and convenience, and for these reasons are extensively employed also here. We now map the stationary points of a regular function f onto vortices using the singularities of the gradient field = f Ž x , y . w22x. This mapping is reviewed in Table 1, where it may be seen that the topological charge Žwinding

Table 1 Mapping of stationary point of regular function f Ž x , y . onto vortex V Ž x , y . . Ž a . Mapping

Vortex

Stationary point

Function Field Components Zero crossings

V Ž x, y . s R Ž x, y . q iI Ž x, y . R, I RsIs0

= f Ž x, y . s f x Ž x, y . xˆ q f y Ž x, y . yˆ fx , fy fx sfy s0

Amplitude

A s R2 q I 2

L s f x2 q f y2

Phase angle

f s arctan Ž IrR . R s w R x x0 x q w R y x0 y

Q s arctanŽ f yrf x . f x s w f x x x0 x q w f x y x0 y

I s w Ix x0 x q w Iy x 0 y

f y s w fx y x0 x q w f y y x0 y

q s sgn w R x I y y R y I x x 0

Q s sgn f x x f y y y f x2y

Taylor series Topological charge Ž Winding number. Ža .

'

(

Partial derivatives enclosed by w x 0 are evaluated at the origin Žcritical point center..

0

I. Freundr Optics Communications 163 (1999) 230–242

number. associated with the nth singularity is Q n s sgn f x x f y y y f x2y n. But the Hessian H s f x x f y y y f x2y is positive definite for extrema and negative definite for saddle points, so that f-extrema map onto positive vortices and f-saddles map onto negative vortices. Based upon this mapping, we can create a saddle point field f with N saddles at locations Ž x n , yn . . We do this by first constructing a product wave function from negatiÕely charged vortex wave functions Vn Ž x y x n , y y yn ;y 1 . , and then setting N

ž žŁ

f x Ž x , y . s R Ž x , y . s Re

ns 1 N

f y Ž x , y . s I Ž x , y . s Im

/ /

Ł Vn Ž x y x n , y y yn ;y 1.

,

Vn Ž x y x n , y y yn ;y 1 . .

ns 1

Ž3. Now, in order that f x and f y be partial derivatives of a regular function f Ž x , y . , we must have f x y s f y x . This implies R y s Ix ,

Ž4.

which is a non-trivial restriction that does not necessarily hold for a given set of vortex wave functions  Vn 4 . Assuming, however, that do we succeed in building V Ž x , y . such that Eq. Ž4. is satisfied, the unique solution of Eq. Ž3. that satisfies all requirements may then be written for the finite polynomial wave functions of interest here,

233

expw iŽ kz y v t .x it becomes a Žnondiffracting. solution of Maxwell’s equations, since E satisfies both the vector wave equation = 2 E q k 2 E s 0 and the divergence requirement = P E s 0. Here the wave vector k s vrc, where v is the frequency and c is the velocity of light. As a first example illustrating the above considerations we construct a single saddle point at Ž x 1 , y 1 . using the vortex wave function V1 s aŽ x y x 1 . q bŽ y y y 1 . q iw cŽ x y x 1 . q dŽ y y y 1 .x. Then, the requirement R y s I x yields c s b and the requirement that the vortex charge be negative yields ad - 0. Solving for both solutions in Eq. Ž5. and inserting the above restrictions on parameters, we verify that Ž i . both solutions yield the same function f Ž x , y . s 12 Ž ax 2 q dy 2 . qbxy y Ž ax 1 qby1 . x y Ž bx 1 qdy1 . y, Ž ii . f has only one stationary point, which is indeed located at Ž x 1 , y 1 . , and Ž iii . f x x f y y y f x2y s ad y b 2 - 0, so this stationary point is in fact a saddle point. As a second example we construct a field with two saddle points at Ž "1,0 . , writing V s Vq1Vy1 , where V "1 s aŽ x y x " 1 . q by q iw cŽ x y x " 1 . q dy x with x " 1 s "1. Negative charges for both vortices now require only ad y bc - 0, while f x y s f y x requires either Ž i . d s ya and c s b or Ž ii . b s 2 a2 cr Ž a 2 y c 2 . and d s 2 ac 2r Ž a2 y c 2 . . The restrictions in Ž i . are acceptable and lead to the desired uniquely defined saddle point field shown in Fig. 3. Those in Ž ii ., however, are unacceptable because

f Ž x , y . s R Ž x , y . dx q I Ž 0, y . dy q f Ž 0,0 .

H

H

s I Ž x , y . dy q R Ž x ,0 . dx q f Ž 0,0 . .

H

H

Ž5.

Viewing f Ž x , y . as a surface above the xy-plane, the constant term f Ž 0,0 . in Eq. Ž5. controls the height of this surface above the plane but does not affect its shape, and in what follows we will usually find it convenient to set this constant equal to zero. One simple way to guarantee that Eq. Ž4. does hold is to write Vn Ž x , y ;y 1 . s Vn Ž x y iy . ,

Ž6.

in which case in addition to Eq. Ž4., the Cauchy–Riemann relations read R x s yI y . This choice for Vn also ensures that f Ž x , y . obeys the two-dimensional Laplace equation

E 2 frE x 2 q E 2 frE y 2 s 0,

Ž7.

thereby providing an iron clad guarantee that all stationary points are indeed saddles. Other far ranging implications of Eqs. Ž6. and Ž7. are discussed in Section 4. Eq. Ž6. has the added bonus that when f Ž x , y . is embedded in E Ž x , y ; v . s w f Ž x , y . xˆ q Ž ir k . f x Ž x , y . zˆ x

Fig. 3. Saddle point field constructed using Eq. Ž5. and a basis of two isotropic negative vortices at Ž x, y . s Ž"1,0.. The two resulting saddles are located at the same positions as the vortices and are indicated by the two dark crosses. Note the three directions of steepest ascent alternating with three directions of steepest descent around the boundaries of the figure that are characteristic of a single second-order saddle. This illustrates the fact that globally a saddle point field containing N saddle points is equivalent to a field containing a single higher-order saddle of order N.

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I. Freundr Optics Communications 163 (1999) 230–242

they lead to ad y bc s 0, corresponding to vortices Žand therefore stationary points. whose natures are undefined. We emphasize that the above two examples obey neither Eq. Ž6. nor Eq. Ž7., demonstrating that these equations are not intrinsic to the method. As the number of saddle points increases, however, so do the increasingly complex restrictions on parameters, and in order to preserve the simplicity and utility of the method one is ultimately driven to use Eq. Ž6.. As discussed in Section 4, use of this equation also leads to a drastic simplification of the whole method. Fields that include higher order saddles are also of interest. A first order saddle such as f Ž x , y . s Ž x q y . Ž x y y . contains two pairs of bifurcation lines Ž y s "x . separating directions of steepest ascent Ž y s 0. and descent Ž x s 0. that leave the saddle point. Possibly less familiar higher-order saddles contain additional numbers of bifurcations lines together with additional directions of steepest ascent and descent, and these higher order saddles may easily be incorporated into our saddle point fields using higher order vortices. The order of a saddle, which equals the absolute value of its topological index t w23,24x, is one less than the number of its bifurcation lines, and using Eq. Ž5. a saddle point of order m Žt s ym. may be constructed from a negative vortex of the same order Žtopological charge q s ym.. The interested reader may enjoy verifying that the second order Ž q s y2. vortex V2 2 s Ž x y iy . generates the second order Žt s y2. saddle f 2 Ž x , y . s x Ž x r ' 3 y y . Ž x r ' 3 q y . , th e 3 third order Ž q s y3. vortex V3 s Ž x y iy . generates the Ž . third order t s y3 saddle f 3 Ž x , y . s 14 Ž sq x y y . Ž sqx q y .Ž syx y y .Ž syx q y . , s "s Ž1 " '2 ., etc. As the topological index Žcharge. of a bounded area is the sum of the indices of the critical points contained within the area, the region surrounding say two first order saddles takes on the structure of a second order saddle. This may be seen in Fig. 3 which contains two first order saddles in its interior, but whose outer boundary contains three directions of steepest ascent alternating with three directions of steepest descent that are characteristic of a second order saddle. Reversing the above, it might appear that our problem could also be solved with the aid of catastrophe theory w25–29x by causing a higher order saddle to split Žunfold. into specified lower order saddles. But although this theory does classify unfoldings of higher order critical points under various perturbations, it does not provide a simple means for obtaining saddles of given orders at given locations, which remains an intractable algebraic problem. In contrast, as becomes apparent in Section 4, a gradient mapping based on Eq. Ž6. is simplicity itself. We note that although both the real and imaginary parts of say a second-order vortex are first-order saddle points, simply taking f Ž x , y . to be either the real or imaginary part of a vortex wave function containing N second-order vortices does not solve our problem. The reason is that the

topological charge of the vortex field is "2 N, so that globally its real and imaginary parts each correspond to a saddle of order 2 N y 1. Thus, these vortex generated saddle point fields must contain not only the desired N saddle points, but also N y 1 unwanted saddles – a conclusion verified by direct calculation.

3. Propagation dynamics of Gaussian beams A pure saddle point field constructed in the form of a finite two dimensional polynomial as in the preceding section is unbounded, and therefore not actually realizable in practice. Such saddle point fields can, however, be embedded in finite diameter beams, and the most likely optical realization of these fields involves embedding them within a Gaussian laser beam. This may be accomplished using recently developed holographic methods w1,3,8x. This embedding may always be carried out in such a way that the initial saddle point structure of the field is preserved at say the beam waist, where the composite field consists of the original saddle point field multiplied by a Gaussian envelope. This structure then changes as one moves away from the waist, and although the resulting field is still the product of a saddle point field multiplied by a now complex Gaussian envelope, the internal structure of the saddle point field is no longer that of the original. Propagation dynamics of these composite beams then becomes an important issue. Although embedding the saddle point field does affect its evolution, we find that this field and its Gaussian envelope evolve independently. Since the dynamics of the Gaussian envelope are well known, this independence permits us to always separate the saddle point field from its envelope function, and in later sections to study the evolution of the saddle point field alone. The classic method for handling propagation dynamics of highly structured laser beams is based on numerical evaluation of the paraxial approximation to Huygens integral using the Fast Fourier Transform ŽFFT.. But in addition to well-known problems such as aliasing, associated with the FFT is another, subtler problem that is especially important in the present application. Intrinsic to the FFT is the fact that it is a discrete Fourier transform whose output is necessarily periodic. This periodicity is normally ignored, and a single unit cell is taken as the output field. But it is easily seen that the net topological charge and topological index of the unit cell of a periodic array must be zero w30,31x. Thus, every unit cell must contain equal numbers of positive and negative vortices as well as equal numbers of extrema and saddle points. Under favorable circumstances, unwanted features introduced by this requirement are likely to be confined to a narrow region on the periphery of the cell, and depending on the width of this region may escape notice. But there is no simple way of guaranteeing or of even estimating when this occurs,

I. Freundr Optics Communications 163 (1999) 230–242

nor is there any simple criterion for deciding if a given feature is ‘‘real’’ – or an unwanted artifact of the periodicity. Clearly, an efficient alternative free of these uncertainties is required, and this is developed below. It has long been appreciated that many functions may be embedded in Gaussian laser beams using expansions in Hermite polynomials. Since the functions of interest here are themselves two-dimensional polynomials of low order in x and y, only a relatively small number of Gauss– Hermite functions of each variable are required in the expansion. But each Hermite polynomial itself involves a small number of different powers of x or y, so that instead of a formal expansion we may simply substitute for each power of x or y in the original function the appropriate low order polynomial in the same variable. Normally, this circular path would yield x n y m s x n y m and simply regenerate the original function. Here, however, under propagation each different Hermite function acquires a different phase shift, and the coefficients in the substitution polynomials become distance dependent. This dependence then generates a new function that includes the desired propagation dynamics of our original array of saddle points. We can thus reduce the propagation problem to one of elementary algebra, thereby producing a simple means for generating an exact solution of the paraxial wave equation. We note that this solution yields explicit analytical formulas well suited for both closed form analysis and efficient numerical evaluation. Indeed, we find that in actual practice this method not only substantially outperforms the FFT in speed of execution, but also yields output data of very much superior quality. Below, we develop the required substitution polynomials. The wave function of a Gaussian TEM p q laser beam with mode indices pq may written w32x TEM p q Ž x , y ;Z . s G Ž Z . G Ž x , y ;Z . Hp q Ž x , y ;Z . ,

Ž 8a.

where G Ž Z . s exp w i Ž ykZ q arctan Z . x ,

Ž 8b .

ž

235

1

kw 2

/Ž

x2qy2 .

Ž 8c .

Hp q Ž x , y ;Z . s b p Hp Ž x . = b q Hq Ž x . ,

Ž 8d.

G Ž x , y ;Z . s exp y

2

qi

4R

w s w Ž Z . s w 0'1 q Z 2 , R s R Ž Z . s Z0 Ž Z q 1rZ . ,

Ž 8e . bn s b n , b s b Ž Z,Zi . s exp w i Ž arctan Z y arctan Zi . x .

Ž 8f . Here and throughout, x s x Ž Z . and y s y Ž Z . are measured in units of the normalized beam radius w Ž Z . r '2 , Z is measured in units of the Rayleigh range Z0 s 12 kw 02 , the wave vector k s Ž vrc . z, ˆ and the frequency v , the velocity of light c, and the waist parameter w 0 are constant for all modes pq. As each of the three factors G Ž Z . , G Ž x , y ;Z . , and Hp q Ž x , y ;Z . in Ž8. propagates independently of the other two, in what follows we need only concern ourselves with Hp q Ž x , y ;Z . . Due to the Z-dependent Guoy phase shifts b p and b q , this term alone carries the propagation dynamics of interest here. Although one is usually interested in the dynamics of an initial field f i Ž x , y . embedded at the beam waist where Zi s 0 and b s 1, using Eq. Ž8f. an arbitrary starting position is easily accommodated. We embed the field of interest in a linear superposition of laser modes at the point of interest by expanding each power of x and y in f i in Hermite polynomials w33x, and then multiplying each polynomial Hn by the phase factor bn. For example, we write x s w x x = w y 0 x s 12 H1 Ž x . b 1 Ž Z . = w H0 Ž y . b 0 Ž Z . x , xy 2 s w x x = w y 2 x s 12 H1 Ž x . b 1 Ž Z . = 14 H2 Ž y . b 2 Ž Z . q 12 H0 Ž y . b 0 Ž Z . , etc. Carrying out the above program, expanding all Hermite polynomials in powers of their arguments and collecting terms, we obtain the substitution polynomials listed in Table 2. As simple examples of the

Table 2 Propagation substitution coefficients c m Ž N . for Gaussian beams.Ža . Nx

m™

0 1 2 3 4 5 6 7 8 9 10 Ža .

u

N

N r2. ´ ÝintŽ cm ms 0

Ž N .U

Ny2 m

0

1

2

3

4

5

1 1 1 1 1 1 1 1 1 1 1

1 3 6 10 15 21 28 36 45

3 15 45 105 210 378 630

15 105 420 1260 3150

105 945 4725

945

m

B , U s b u, B s

1 2

2

Ž1 y b . . b

is defined in Eq. Ž8f..

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I. Freundr Optics Communications 163 (1999) 230–242

use of these transformations, w x x = w y 0 x in f i Ž x, y . is replaced by wUŽ x .x = w1x s b x, w x x = w y 2 x is replaced by w U Ž x .x = w U Ž y . 2 q B x s b 3 xy 2 q 12 b Ž1 y b 2 . x, etc. We note that the transformations in Table 2 are unitary, so that for backwards propagation from Z to Zi one simply replaces b by b U . This useful feature permits easy assembly at the beam waist of the components required for a desired far field structure. As a first application of these results, we display in Figs. 4 and 5 the propagation dynamics of vortex arrays. Fig. 4 illustrates the rather complex dynamics associated even with a simple dipole consisting of one positive and

Fig. 4. Propagation dynamics of an anisotropic vortex dipole embedded in a Gaussian beam at its waist, Appendix A, Eq. ŽA.4. with g s1, asy0.25, bs 0.5, cs 0.75 and initial vortex separation 2 bs w 0 r'2 , where w 0 is the size of the beam waist. Similar results are obtained for other values of g . The vortices are located at intersections of the zero crossings of the real Žthick lines. and imaginary Žthin lines. parts of the wave function. Calculated vortex positions are shown in Ža. and Žl. for positive Žnegative. vortices by filled Žopen. squares. The propagation distances Z from the beam waist in units of the Rayleigh range are Ža. Zs 0 Žbeam waist., Žb. Zs 0.05, Žc. Zs 0.5, Žd. Zs 0.65, Že. Zs 0.8, Žf. Zs 0.9, Žg. Zs1, Žh. Zs1.1, Ži. Zs1.5, Ž j . Zs 5, Žk. Zs 20, Žl. Zs10 100 Ž ; infinity.. Vortex annihilation occurs in Žd. and pair production in Žj.. The sequence shown is typical for initial vortex separations of order w 0 . For much smaller initial separations Ž ; w 0 r5. this sequence is modified in that following vortex annihilation as in Žd. the real zero crossing continues to shrink and ultimately disappears, reappearing near Zs1 from which point on the sequence continues as from Žf. to Žl.. For much larger initial separations Ž ; 5w 0 . the real and imaginary zero crossings always overlap and vortex annihilation does not occur. The complex winding and unwinding of the zero crossings that drives the vortex dynamics, annihilations, and pair production are characteristic of vortex fields containing both positive and negative vortices.

Fig. 5. Propagation dynamics of a Big Dipper composed of seven vortices embedded initially at the waist of a Gaussian beam ŽAppendix A.. Positive Žnegative. vortices are shown by filled Žopen. squares, and real Žimaginary. zero crossings by thick Žthin. lines. Ža. Initial vortex configuration at beam waist Ž Zs 0.. The initial coordinates Ž x, y . n of vortex n are: Ž0.7,y0.5.1 , Ž0.2,y 0.8 . 2 , Žy0.2,y 0.6 . 3 , Ž0,0 . 4 , Žy0.1,0.2. 5 , Žy0.4,0.7 . 6 , Žy0.9,0.8. 7 in units of w 0 r'2 , where w 0 is the size of the beam waist. Žb. Real and imaginary zero crossings corresponding to Ža.. Žc. Zs 0.2. By this point vortices 4 and 5 have collided and annihilated one another as have vortices 3 and 7, leaving behind only vortices 1, 2 and 6. These three vortices survive into the far field. Žd. Far field configuration. Note that all seven vortices are present with missing vortices having been regenerated by pair production. The winding and unwinding of the zero crossings, together with vortex annihilations and ultimate regeneration due to pair production that lead to conservation of initial vortex number in the far field appear to be characteristic of vortex arrays with no special symmetries Ždegeneracies.. The field of view in Žc. is enlarged to three times, and in Žd. to five times that of Ža. and Žb..

one negative vortex, while Fig. 5 describes the dynamics of seven vortices arranged initially in the form of the Big Dipper in Ursa Major. Characteristic of both figures is a complex sequence of vortex collisions and annihilations followed by pair production that ultimately restores the initial number of vortices in the far field. This far field conservation of vortex number appears to be characteristic of generic arrays Žthose with no special symmetry.. Further details are given in Appendix A.

4. Laplacian wave fields As mentioned in Section 2, when isotropic negative vortices are used as a basis wEq. Ž6.x, the initial saddle

I. Freundr Optics Communications 163 (1999) 230–242

237

m

tions u m ´ Ž b u . for all m. All this then leads to the following very simple algorithm for constructing the final

Fig. 6. Propagation dynamics of a Big Dipper composed of seven Laplacian saddles embedded initially at the waist of a Gaussian beam ŽAppendix B.. The saddle point locations in Ža. are the same as the vortex positions in Fig. 5a. Zero crossings of the x-derivatives Žthick lines. and y-derivatives Žthin lines. are shown for the real part of the wave function in Ža., Žc., and Že., and for the imaginary part in Žb., Žd., and Žf.. Ža., Žb. Beam waist. Žc. – Žf. Far field. During propagation from the beam waist to the far field the zero crossings wind and unwind while saddle points recede to infinity along one direction and return along another. These dynamics are characteristic of saddle point fields. Note that in Žc. and Žd. the saddle points form centrosymmetric arrays, which is a characteristic of the far field of Laplacian vortices. The fields of view in Že. and Žf. are 100 times that of Ža. – Žd., and again illustrate ŽFig. 3. that globally a saddle point field containing N saddle points is equivalent to a field containing a single saddle of order N.

point field f Ž x, y . in Eq. Ž5. satisfies the two-dimensional Laplace equation wEq. Ž7.x. Accordingly, f may be written as say the real part of some function of the complex variable x y iy. Now, we find by direct calculation that under these circumstances substituting the transformations of Table 2 into f Ž x , y . to obtain the final wave function F, yields F Ž x , y . s f Ž b x , b y . . This in turn implies that Ž i . F also satisfies Laplace’s equation, and Ž ii . that the transformations in Table 2 can be replaced by the substitu-

Fig. 7. Propagation of a centrosymmetric Laplacian saddle point field containing seven saddles initially arranged in the form of a figure eight at the waist of a Gaussian beam ŽAppendix B.. Zero crossings of the x-derivatives Žthick lines. and y-derivatives Žthin lines. are shown for the real part R of the wave function in Ža., Žc., Že., and Žg., and for the imaginary part I in Žb., Žd., Žf., and Žh.. In Ža. – Žd. the saddle point at the origin is of first order and the net index of the array is odd. In Že. – Žh. this saddle is of second order and the index of the array is even. Ža., Žb., Že., Žf. Beam waist. Žc., Žd., Žg., Žh. Far field. Note that all figures maintain a center of symmetry, that for the odd index fields Ža. – Žd. the far-fields of R and I are 908 rotations of their respective beam waist configurations, while for the even index fields Že. – Žh. a crossover occurs and the far-field of I is a 908 rotation of the near field configuration of R and vice versa.

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238

saddle point field F Ž x , y . . Writing the initial vortex product as N

˜ .s V Ž bz

˜ y zn . , Ł Ž bz

Ž9.

ns 1

where z s x y iy, zn s x n y iyn , and x n , yn are the coordinates of the vortex centers, the final Laplacian saddle point field is F Ž x , y . s Re

žH Ž ˜ . Ž ˜ . / V bz d bz

˜

.

Ž 10.

bs b

Here, the real constant b˜ serves as a surrogate for the complex transfer function b . As b is not a ‘‘player’’ in our application of the theory of complex variables, it is only substituted into Eq. Ž10. after the real part of the integral is taken. Using computer assisted analysis, we have verified Eq. Ž10. by Žautomatic. term-by-term comparison with wave functions generated using Table 2 for final wave functions F containing up to ; 64 000 terms Ža practical upper limit set by execution times..

Laplacian vortex fields are also described by Eq. Ž9. with b˜ s b . This is true not only for arrays that are all negative, but also for arrays that are all positive, in which case z s x q iy and zn s x n q iyn. Higher-order vortices k k Ž z y z k . w Ž z y z k . x located at Ž x k , yk . with charge yk w k x may also be included in Eq. Ž9. simply by setting k Žsuccessive. zn s z k . We note that these results imply that in a Laplacian vortex field containing arbitrary numbers of vortices of different orders, each vortex n propagates independently of all others, moving on a circle of radius rn s x n2 q yn2 through an angle Dun s .Žarctan Z y arctan Zi ., where the negative Žpositive. sign is taken for positive Žnegative. vortex arrays. This striking rule, however, is not original, having first been obtained by Indebetouw using a very different, rather more difficult route than the one used here w4x. Since Indebetouw’s rule exhausts the propagation dynamics of all possible Laplacian vortex fields, in Appendix A we concentrate on the propagation of arrays of anisotropic vortices. Laplacian saddle point fields have unusual propagation dynamics. These are illustrated in Figs. 6 and 7. Fig. 6,

(

Fig. 8. Coherent addition of two saddle point fields provides an optical implementation of the arithmetical operation of subtraction. An addendum field with saddle of order n is added to a base field with saddle of order m s 7. Zero crossings of the x-derivatives Ž y-derivatives. are shown by thick lines Žthin lines.. The resulting field contains m y n first-order saddles on a circle of unit radius Ždotted line. at angles u j s "jprŽ m y n., j s 1,3,5, . . . , m y n Žopen squares.. It also contains a saddle of order n at the origin, so that the net topological index of the base field is conserved. The arithmetic operation of addition cannot be carried out using these fields alone. Ža. n s 0. Žb. n s 2. Žc. n s 3. Žd. n s 6.

I. Freundr Optics Communications 163 (1999) 230–242

which displays evolution of the real and imaginary parts of the wave function for a saddle point array initially in the form of the Big Dipper in Ursa Major, illustrates an important general property, namely that the far field of an arbitrary initial saddle point configuration is always centrosymmetric. If the initial configuration is itself centrosymmetric, than additional symmetries develop under propagation. These are illustrated in Fig. 7 and discussed in detail in Appendix B. Finally, Fig. 8 illustrates implementation of the arithmetic operation of subtraction using saddle point fields, suggesting possible application of these fields also for information processing.

5. Summary We have presented two algorithms for creating optical fields whose only critical points are saddle points at specified locations. Both algorithms are based upon mapping the singularities of the gradient of a regular function onto negative vortices ŽTable 1.. The more general algorithm, Eq. Ž5., uses a set of general anisotropic vortices as its basis, but suffers from the need to adjust the parameters of these vortices so as to satisfy often complicated algebraic constraints. The simpler algorithm, Eq. Ž10., creates a Laplacian saddle point field using isotropic vortices for its basis. An easily implemented calculus for the propagation dynamics of finite order polynomial fields embedded in Gaussian beams has been presented ŽTable 2.. This calculus has been applied to the evolution of arrays of anisotropic vortices ŽAppendix A., as well as to arrays of Laplacian saddle points ŽAppendix B.. An optical circular hole saw constructed by embedding elementary saddle points fields in a Gaussian laser beam has been presented as one of a number of possible applications for these fields ŽFig. 1.. Other uses for fields that contain only a single type of critical point will undoubtedly emerge as the unique properties of these fields become better understood.

Acknowledgements I am pleased to acknowledge useful conversations with Prof. Michael Wilkinson during the early stages of the work. This study was supported in part by the Israel Science Foundation of the Israel Academy of Arts and Sciences.

Appendix A. Vortex propagation In this appendix we consider propagation of a number of model vortex fields embedded in the waist of a Gaussian laser beam. These dynamics are not only of interest in their own right, but also provide a basis for comparison with the analogous saddle point fields that are studied in the following appendix.

239

For a single unit charge vortex initially at Ž x , y . s Ž a,b . , we write V1Ž 0 .s x y a q ig Ž y y b . , where the magnitude of g measures the vortex anisotropy and the sign of g is the vortex charge Ž"1.. Although this is not the most general form for the wave function of an anisotropic vortex, it already goes significantly beyond the previously studied isotropic case Žg s 1., and immediately yields unexpected results. Inserting the appropriate substitutions from Table 2 and writing b s Ž 1 q iZ . r '1 q Z 2 , we have RŽ Z . s

x y g Zy y a'1 q Z 2

, '1 q Z 2 Zx q g y y g b'1 q Z 2 IŽ Z . s , '1 q Z 2

Ž A.1 .

from which follow the vortex coordinates x ÕŽ Z . s

a q g bZ

'1 q Z 2

,

yÕ Ž Z . s

aZ y g b

'1 q Z 2

.

Ž A.2 .

For the case g s 1, as Z increases a positive Žnegative. vortex simply slides clockwise Žcounterclockwise. along a circle of constant Žrelative to the beam diameter. radius r s 'a2 q b 2 , moving by u s 908 as the far field is reached. This is in accord with a well-known rule first obtained by Indebetouw w4x, who showed that for Ža product wave function. an arbitrary array of isotropic vortices all with the same sign preserves its shape under propagation and simply rotates by 908 on passing to the far field. In the general anisotropic case, however, this result no longer holds, and not only is r s x Õ2 q yÕ2 a rather complicated function of Z, but the far-field angle through which the vortex moves is u s arctan yar Ž g 2 b . y arctan w bra x , which can differ substantially from 908. For two identical vortices located at Ž a," b . with arbitrary g we write the initial product wave function V2 Ž 0 . s w x y a q ig Ž y y b . x w x y a q ig Ž y q b . x . Although we have obtained analytical formulas for arbitrary Z, these are too cumbersome to present here, and we discuss only the important case of the far field. For g s 1 we find the far-field vortex coordinates Ž x Õ , yÕ . to be Ž qb, yÕ . and Ž yb, yÕ . , where yÕ s .a for g s "1. This corresponds to a simple clockwise Žcounterclockwise. rotation by 908 with no change in separation of the positive Žnegative. pair, and is in full accord with Indebetouw’s rule w4x. But for g / 1 we find a completely different far field result. Here several cases need be distinguished.

(

Ž i . b - 1, g s 1r'1 y b 2 , x Õ s 0, yÕ s yarg , Ž A.3a . Ž ii . b - 1, g ) 1r'1 y b 2 ,

(

x Õ s 0, yÕ s ya " g 2 Ž 1 y b 2 . y 1 rg ,

Ž A.3b .

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240

Ž iii . b - 1, g - 1r'1 y b 2 ,

(

x Õ s " 1 y g 2 Ž1 y b 2 . ,

yÕ s yarg ,

Ž A.3c .

(

Ž iv . b P 1, x Õ s " g 2 Ž b 2 y 1 . q 1 , yÕ s yarg . Ž A.3d . For the case of two vortices initially at Ž "b,a . instead of at Ž a," b . , the above results hold with x Õ and yÕ interchanged and g replaced by y1rg . We note that, surprisingly perhaps, case Ži. corresponds to collision and fusion of both vortices to form a new vortex. By conservation of topological charge this new vortex must have charge q2 Žy2. for positive Žnegative. g , a result directly verified from the detailed form of the wave function. This case is an example of automatic self assembly during propagation of a given far field structure from appropriate starting components, a possibly useful phenomenon briefly discussed at the end of the Section 3. For the rather idealized case g ' 1, we again recover wfrom Žiii. and Živ.x Indebetouw’s rule of a 908 rigid body rotation for all b, but in general, inevitable perturbations of the internal vortex structure must lead to g / 1, which requires use of the full set of Eq. ŽA.3.. For two vortices with opposite signs, propagation typically involves a complex sequence of vortex collisions and annihilations followed ultimately by pair production, with the far-field normally containing a positivernegative vortex dipole regardless of the initial vortex separation ŽFig. 4.. Placing the positive, gqs qg , wnegative, gys yg x vortex at Ž a,b . w Ž c,y b . x and taking g ) 0, we obtain the far-field wave function V` s 1 q g 2 y x 2 y g 2 y 2 q 2g bx q Ž c y a . g y q ac y g 2 b 2 q i Ž a q c .Ž g b y x . ,

Ž A.4 .

from which follow the far-field vortex positions x Õ s g b, yÕŽ " . s

1 2

(

2

c y a " Ž a q c . q 4Ž 1 q g 2 . .

Ž A.5 . Here the vortex at yÕ Ž" . has sign .sgn Ž a q c . , while interchanging the initial vortex signs changes the sign of x Õ. When a q c ' 0, the imaginary part of V` vanishes and Eq. ŽA.5. no longer hold. This occurs because these equations correspond to taking the lim Z™ `wŽ c q a. Z x equal to infinity instead of setting this limit identically equal to zero. Putting b s 0 for simplicity and recalculating, we obtain the general propagation wave function of a dipole initially located on the x-axis and centered on the origin, V Ž Z . s b 2 Ž x 2 q g 2 y 2 . q 12 Ž 1 q g 2 . Ž 1 y b 2 . y a 2 q 2 aigb y.

Ž A.6 .

Using computer assisted analysis to perform the required tedious manipulations, we find the vortex positions for arbitrary Z to be x ÕŽ" . Ž Z .

(Ž1 q Z . Ž 1 q g 2

s"

yÕ Ž Z . s

2

. Ž 4 a2 y g 2 y 1 . Z 2 q 4 a4

2 a Ž1 q Z 2 .

Ž 1 q g 2 y 2 a2 . Z

(

2 a Ž1 q Z 2 .

.

Ž A.7 .

From Eq. ŽA.7. follows that vortex collision w x Õ Ž Z . s 0x and annihilation occurs at a propagation distance ZU s

2 a2

(Ž 1 q g

2

. Ž 1 q g 2 y 4 a2 .

,

Ž A.8 .

(

and can therefore take place only when a F 12 1 q g 2 . Conversely, when this latter condition is met the far-field wave function is always vortex free, in contrast to the general far-field case in Eq. ŽA.4. for arbitrary a, b, and c that always contains a vortex pair. We note that in the limit g ' 1, Eq. ŽA.7. are in accord with the symmetric vortex dipole result of Indebetouw w4x, who studied only this limit. In practical experiments, however c ' ya and g ' 1 is unlikely to be met in practice, so that starting with a positive-negative vortex dipole at the beam waist one may generally expect to recover a dipole in the far field. Propagation dynamics of larger arrays are also of interest, and in Fig. 5 we display results for seven isotropic vortices Ž g s 1. arranged initially in the shape of the Big Dipper in Ursa Major. Since the number of terms in a product wave function grows exponentially with the number of vortices n as C 14 n2 4 n , for n G 7 it is impractical Ževen with computer assisted analysis. to keep vortex positions, signs, and anisotropies as algebraic constants, and numerical values for these parameters need be built into the wave function from the beginning. When this is done, the study of very large arrays should prove entirely feasible. As may be seen from Fig. 5, a complex sequence of vortex collisions, annihilations, and rearrangements occurs during propagation, but once again, due to pair production all seven vortices are present in the far field. Whether-or-not this far-field conservation of vortex number is a general result for generic arrays Ži.e. arrays with no special symmetries. remains an open question. Arrays containing up to 10 vortices are easily handled using Table 2, while still larger arrays may be studied by straightforward extension of this table. We note in this regard that although it is clear that there must exist a Žsimple. closed form expression for the entries in this table, we have not yet succeeded in obtaining this. In spite of current interest in the propagation of vortex arrays we do not pursue these matters further here, but turn instead in the following appendix to the propagation dynamics of Laplacian saddle point arrays.

,

I. Freundr Optics Communications 163 (1999) 230–242

241

Appendix B. Propagation of Laplacian saddle point fields

For regime A of R Ž Z F ZRU ., we have saddle point coordinates

Under propagation the initial real field F0 Ž x , y . develops an imaginary component, and we study here the dynamics of both components for simple arrays that exhibit generic behavior. Now, normally the division of a complex field into a real R Ž x , y ;Z . and an imaginary I Ž x , y ;Z . part is somewhat ambiguous, since R and I are mixed together under a uniform phase shift Dw that can usually be made arbitrary. But in our case since b s Ž1 q iZ .r '1 q Z 2 f 1 q iZ y Z 2r2 q ..., we have R Ž x , y ;0 . s F0 , I Ž x , y ;0 . s 0, which suffices to resolve all ambiguities. We start by considering a single saddle point located at the beam waist at Ž x 1 , y 1 . with x 1, y 1 G 0. Using Eq. Ž10., we find for the saddle point of R the Z-dependent coordinates x R Ž Z . s x 1 '1 q Z 2 r Ž 1 y Z 2 . , yR Ž Z . s y 1'1 q Z 2 r Ž 1 y Z 2 . . These coordinates imply that the saddle moves along a straight line L1 that passes through the origin with slope tan u s y 1rx 1, and that on this line its distance from the origin is L Ž Z . s L0'1 q Z 2 r 1 y Z 2 , where L0 s x 12 q y 12 . For L0 / 0, as Z ™ 1 the saddle point zooms off to q`. It then comes back from y` along the opposite side of L1 and settles down at the origin for large Z. The imaginary part of the wave function also contains a single saddle point, but with coordinates x I Ž Z . s 12 x 1'1 q Z 2 , yI Ž Z . s 12 y 1'1 q Z 2 . For L0 / 0, this saddle also moves along the line L1, now simply receding to q` with increasing Z. When L0 ' 0, however, both the real and imaginary saddle points remain at the origin over the whole range 0 F Z F `. The dynamics of a single saddle are thus seen to differ completely from that of a single vortex, which, as discussed in Appendix A follows Indebetouw’s rule w4x and simply moves through 90 0 along a circle of constant radius relative to the beam diameter. The general case of two saddle points at arbitrary initial locations is rather complicated, so as was the case for two vortices ŽAppendix A., we discuss only a partially symmetric configuration in which the saddles are initially located at the beam waist with coordinates Ž a," b . . For R we find two different regimes depending on whether Z is less than Žregime A. or greater than Žregime B . a characteristic distance

x RŽA . Ž Z . s a

(

ZRU Ž r . s

(

1 2

(Ž3 r q 1. q 4r y Ž3 r q 1. 2

,

Ž B.1 .

'1 q Z 2 Ž1 y Z 2 . 1 y 3Z 2

(Ž1 q Z . 1 y 3Z y Z Ž1 q Z .rr 2

yRŽ A. Ž Z . s b

2

2

2

2

1 y 3Z 2

.

Ž B.2 . ŽA .

In this regime x R Ž Z . increases monotonically with increasing Z. yR ŽA . Ž Z . , on the other hand, rapidly falls to zero as Z approaches ZRU , causing the saddles to collide and fuse, thereby forming a single second-order saddle whose creation marks the end of regime A. In regime B of R Ž Z G Z U ., the saddle points, which are confined to the x-axis, have coordinates Ž x R Ž B" . ,0., where x RŽ B " . Ž Z . sa

½

'1q Z 2 Ž Z 2 y1."'Ž1q Z 2 . w Z 4 qŽ1q3 r . Z 2 y r x 3Z 2 y1

5

.

Ž B.3 . As Z increases past ZRU , the second-order saddle created at the end of regime A fissions into two first-order saddles. We label these SR Ž" . on the basis of their coordinates x R Ž B" . . SR Žq . initially starts to move towards the origin, but it soon turns around Žat Z s 1 for r s 1. and moves off to q`, its distance from the origin growing linearly with Z for large Z. In contrast, SR Žy . rapidly recedes from the origin, reaching q` at Z s 1r '3 . It then returns from y`, and asymptotically approaches the origin for large Z. Finally, for R in the limit a s 0 where the initial coordinates of the saddles are Ž 0," b . , fusion never occurs. Instead, in the region 0 F Z F 1r '3 the saddles separate along the y-axis to "`, and as Z increases further they return from "` along the x-axis, ultimately settling down at x s "br '3 for large Z. The imaginary part of the wave function also contains two saddles whose propagation dynamics roughly mimic those of the real part of the wave function. For r ) 1r3 the imaginary saddles have initial coordinates x I Ž Z s 0 . s 2 ar3, yI Ž Z s 0 . s " Ž br3 . 3 y 1rr . As Z increases both saddles move symmetrically towards the x-axis, colliding and fusing to form a second-order saddle at Z s ZIU s Ž 3 r y 1 . r Ž r q 1 . . As Z increases past ZIU this second-order saddle fissions, and with increasing Z the resulting first-order saddles SI Ž" . move only on the x-axis. SI Žq . initially approaches the origin, but then turns, moves away, and asymptotically settles down at x I Žq . Ž ` . s qa 1 q r . SI Žy . recedes to q` at Z s '3 , returns from y`, and settles down at x I Žy . Ž ` . s ya 1 q r . For r ' 1r3 the imaginary saddles start at Z s 0 as a single second-order saddle located on the x-axis at x I Ž 0 . s 2 ar3. This saddle immediately fissions, and

'

'

'

where r s Ž bra . . Note that ZRU is a monotonically increasing function of r , with ZRU Ž 0 . s 0 and ZRU Ž ` . s 1r '3 .

,

'

I. Freundr Optics Communications 163 (1999) 230–242

242

the two resulting first-order saddles, which remain on the x-axis, behave substantially as described above. Similarly, for r - 1r3 the imaginary saddles initially appear at x I Ž" . Ž 0 . s Ž ar3 . Ž 2 . 1 y 3 r . , yI Ž" . Ž 0 . s 0. They again remain on the x-axis, and move substantially as described above. From the foregoing it is evident that in contrast to the rigid body rotations of Laplacian Õortex fields, there is no universal prescription for the propagation dynamics of Laplacian saddle point fields. These fields, however, do exhibit an important far-field symmetry. From Eqs. Ž9. and Ž10. follows that in the far-field where b s i, the real R Žimaginary I . part of the wave function is a symmetric Žantisymmetric. function of coordinates x, y. This implies that also the first-order derivatives R x and R y Ž I x and I y . are antisymmetric Žsymmetric. functions. Now, since q0 s y0, the zero Žlevel. crossings of both symmetric and antisymmetric functions are themselves always symmetric, and since the critical points of the field are located at intersections of these zero crossings, for both symmetric and antisymmetric fields all critical points necessarily form centrosymmetric arrays. This symmetry is displayed by the examples discussed above, and is strikingly illustrated in Fig. 6 which shows the far-field of a Big Dipper formed initially from seven saddle points at the beam waist. Laplacian saddle point arrays that are themselves initially centrosymmetric have additional special symmetries: Ži. Regardless of the propagation dynamics – fusion, fission, expulsion to infinity, etc. – the saddle point arrays of both R and I maintain centers of symmetry from the beam waist to the far field. Žii. For arrays whose net initial topological index is odd Žwhich requires an odd-order saddle at the origin., the far fields of both R and I are simple 908 rigid body rotations of their initial beam waist configurations, but with zero crossings Z x and Z y interchanged. Žiii. Arrays with even values for the index, on the other hand, exhibit a crossover in which the far field of the imaginar y part of the wave function is a 908 rotation of the initial saddle point configuration of the real part of the wave function, and Õice Õersa. In this case no interchange of Z x and Z y occurs. These special symmetries are illustrated in Fig. 7 using an array of saddle points in the form of a figure eight. We close by noting that combinations of saddle point fields may often exhibit unexpected, rather unusual properties. This is illustrated in Fig. 8 which demonstrates the arithmetical operation of subtraction using these fields.

'

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