a mathematical framework for wave dislocation lines

2s __ __ l!B ELSEVIER

IS March 1999

Physics Letters A 253 ( 1999) 57-65

PHYSICS LETTERS A

A mathematical framework for wave dislocation lines Yishi Duan, Hong Zhang *, Guang Jia

Instihrte of Theoretical Phvsics, Lmzhou University, Lmzhou 730000. China

Received 14 April 1998; revised manuscript received 4 December 1998; accepted for publication 29 December 1998

Communicated by P.R. Holland

Abstract

We construct a mathematical framework for wave dislocation lines (zero lines of a complex scalar wave) in light beams by making use of the @mapping topological current theory. The topological structure of wave dislocation lines is detailed in the neighborhoods of the branch points of the complex scalar wave. @ 1999 Published by Elsevier Science B.V.

PAW 42.6S.Sf; 42.50.Vk; 02.40.-k

Kqwordst Wave dislocation lines; Light beams

1. Introduction

Light beams possessing wave dislocations, i.e. phase singularities, have been studied intensively in many ways [ I-71. Wave dislocations are topological objects on wave-front surfaces and possess topological charges which can be attributed to the helicoidal spatial structure of the wave-front around a phase singularity. This structure is similar to a crystal lattice

defect, and therefore was first known as a wave dislocation [ 11. There are three types of dislocations: edge, screw and mixed edge-screw dislocations. An edge

dislocation is located along a line in the transverse plane and travels with the wave. A screw dislocation is called an optical vortex, the essential property of which is that the phase changes by 2n-m on a closed circuit around it. m is called the topological charge, which is positive for a right-screw helicoid (m > 0), and vice versa. At the locations of these three kinds

* Corresponding author; e-mail address: i@[email protected].

of dislocations the wave amplitude becomes zero and the phase is indeterminate [ 2,3].

The generation, annihilation, collision, split and merging of dislocations have been investigated from many aspects [ 1,2,6,8-l 31. The theory of wave dislocations in time-dependent complex scalar waves in (2 + 1)-dimensional space-time (x, z, t) was given

by Nye and Berry [ 11. They proved the existence of wave dislocations by exhibiting a number of special

solutions of the scalar wave equation that had the dislocation property. It is also interesting to discuss the annihilation, generation and intersection of dis-

location lines in three-dimensional space (x, y, z ) . Soskin et al. studied some explicit examples of exact solutions of the wave equation showing dislocation

lines intersecting, annihilating, generating, etc. in three-dimensional space [ 21. However, a general theory on the conditions for generating, annihilating, intersecting, splitting and merging of dislocation lines in three-dimensional space has not been given.

The &mapping topological current theory and the composed theory of gauge potential play an important

0375.9601/99/$ see front matter @ 1999 Published by Elsevier Science B.V. All rights reserved.

PIISO375-9601(99)00019-S

S8 Y Dunn rr d. /Plt~v.~ivsic~s Lettus A 253 (1999) 57-65

role in studying the topological invariant and struc-

ture of a physical system and have been used to study the topological current of a magnetic monopole, the

topological string theory, the topological characteris- tics of dislocations and disclinations continuum, the topological structure of the defects of space-time in

the early universe as well as its topological bifurcation, the topological structure of the Gauss-Bonnet-

Chern theorem, the topological structure of the Lon- don equation in a superconductor and point defect of a vector parameter [ 14- 171.

In this paper, we will show that the $-mapping topo-

logical current theory is a powerful tool to study the topological structure of wave dislocation lines. Among its advantages and without using any particular mod-

els or hypotheses, we give the topological charge density current of dislocation lines in three-dimensional space, therefore, one can get the most general properties of wave dislocation lines. According to the values

of the vector Jacobians of a complex scalar wave in three-dimensional space, the branch points are classi- fied into two types: limit points and bifurcation points.

Furthermore, it is found that the wave dislocation lines generate or annihilate at the limit points and encounter each other, split or merge at the bifurcation points of

the complex scalar wave. This paper is organized as follows: in Section 2 we

construct an elementary topological current in light beams. By means of the &mapping topological current theory, we derive the topological charge density current of the wave dislocation lines. In Section 3 from

the properties of the complex scalar wave, the conditions for generating, annihilating, intersecting, split-

ting and merging of wave dislocation lines are obtained and several crucial cases of branch process are discussed. The topological charge conservation near the branch points is also studied. We present our con- cluding remarks in Section 4.

2. Topological charge density current of wave dislocation lines

Let us study a complex scalar wave Cc, (r, t) of a light beam directed along the z-axis at a fixed time, which is denoted as

v+(r) = 4’(r) + i&(r), (I)

where 4’ (r) and 4’(r) can be regarded as two com- ponents of a two-dimensional vector field

4 = ($V) (2)

on three-dimensional space II%-‘. We can construct a topological current of the complex scalar wave

J’ = ~~‘~“~,,~d,rl”~~n”, i, j, k = 1,2,3, (3)

which is a special case of the &mapping topological current theory [ 171. II” is the two-dimensional unit vector field of the complex scalar wave:

lzl’ = @l/11411, a = 1.2. (4)

satisfying

IPll“ = I.

From (4), it is easy to see that the zeroes of the wave

$ (r ) are just the singularities of n( T). Using (4) and

J’ changes into

By defining the vector Jacobians of ti (r) :

D’ f! 0 x

(5)

and making use of the Laplacian relation in 4 space:

we do obtain the S function like topological current

J’=6*(qb)Di f 0

(7)

Thus we have the important relation between the topological current J and the complex scalar wave g(r) in light beams:

J=@(+)D 9 0 x ’

(8)

E Dunn et al. /Phvsics Letters A 253 (1999) 57-65 .59

from which we see that the topological current J does not vanish only at the zero points of #J, i.e.

~‘(X,~,Z) =o, &(X,.v.Z) =O. (9)

The solutions of Eqs. (9) are generally expressed as

.r = .r; ( z ) 1 v = y;(z). i= 1,2 ,..., N, (10)

which represent N zero lines L; (i = I, 2,. . . , N) where 1+5(r) = 0 in three-dimensional space. The po- sitions of the wave dislocation lines are lines of intersection of surfaces representing the zeroes of the real

part and imaginary part of the complex scalar wave: Re(+) = r$’ = 0 and Im($) = c$~ = 0. So, the location and the direction of the ith wave dislocation line are determined by the ith zero line L; and the unit vec-

tor field m(r) = D( c$/x)/ID(c$/x) 1 on L; respectively.

In the theory of 6 function of (P(r) [ 181, one can prove that

where

02 0

= I ik ,cy &p” 7 E- ~mn --

ad ad . (12)

11 L

We stress that 2; is a planar element normal to L; at the point r,(Z) and u = (ui, ~2) are the intrinsic coordinates on 2;. The positive integer /3; is the Hopf index of $-mapping and

+ v1 = sgn D - 0 = fl,

u r; (13)

is the Brouwer degree of &mapping [ 171. The meaning of /3, is that when the point r covers the neighbor-

hood of r; on .Y; once, the vector field 4 covers the corresponding region p; times. Using (6), (9) and ( 12) as well as following the +-mapping topological current theory, we can obtain

(14)

then from ( I I ) and ( 14) we have

s?(g,o(;) =-&?;~;/dr,S’(r-r,), (15) /=I L

that is,

.J=&;v;/dt.;fi’(r-l;). i=l L,

(16)

It is obvious to see that the topological current ( 16) represents N isolated wave dislocation lines of which the ith wave dislocation is charged with the topological charge /3;~;. For the case of optical vortices (screw

dislocations), our result coincides with Refs. [ 1.3.41, and has a more straightforward and strict significance. The ith optical vortex in light beams shows itself as a

system of fi; helicoids, nested on the same axis, 7; is positive for a counter-clockwise helicoid and negative for a clockwise one.

Now, we discuss the physical meaning of the topological current 1. Let 2 be an arbitrary surface and

suppose M wave dislocation lines pass through it. Ac- cording to (8), one can prove that

which shows that the topological current J describes the density of dislocation lines in space. So, we call the topological current J the topological charge densit? current of dislocation lines.

The solution ( 10) of Eqs. (9) is based on the condition that the Jacobian

(18)

When the condition fails, the above results ( IO) will change in some way. It is interesting to discuss what will happen and what is the correspondence in physics when D’(c#J/x) II, = 0.

3. Topological bifurcation of wave dislocation lines

When D3( qb/x) lr, = 0 at some points along L,, it is shown that there exist several crucial cases of branch process. There are two kinds of branch points, namely limit points and bifurcation points. Each of them corresponds to different cases of branch process. We’ll discuss them in detail.

First, we study the case that the zeroes of the complex scalar wave ti,( r) include some limit points. The

limit points are determined by Eqs. (9))

= 0.

and

# 0, j= 1,2.

(19)

(20)

We denote one of the limit points along L; as r* = T;( I* ) = (x;” , yf , z: ) . Since the usual implicit

function theorem is of no use when the Jacobian

D3 ( c$/x) lrT = 0. For the purpose of using the implicit function theorem to study the branch properties of wave dislocation lines at the limit points, we use the Jacobian D' (4/x) instead of D3 (4/x) to search for the solutions of 9 (r) = 0. This means that we have

replaced x by z. For clarity we rewrite Eqs. (9) as

$“(z, y. x) = 0, u= 1,2. (21)

Taking into account (20) and using the implicit func-

tion theorem, we have a unique solution of Eqs. (21) in the neighborhood of the limit point rt:

L = z(x), .v = y(x), (22)

with z; = z (xf ). In order to show the topological structure of wave dislocation lines, we will investigate the Taylor expansion of (22) in the neighborhood of

rr In the present case, from ( 19) and (20)) we get

i.e.

dz

Z r,. = 0. (23)

Thus, the Taylor expansion of z = z(x) at the limit

point r: is

;=z(x,*)+g (X-x:)+;$ (x-q2 r : r:

= z;* , d2z

+ 2----Q (x - XT)?. r*

Therefore,

z- _* , d2z *I = Tdx’ (x - XT)2, (24)

‘:

which is a parabola in the x-z plane. From (24) we can obtain two solutions x’ ( z ) and x2 ( z ) , which give the branch solutions of wave dislocation lines at the

limit point. If (d*z/dx*) Jr; > 0, we have the branch solutions for z 3 z,* (see Fig. 1 a), otherwise, we have the branch solutions for z < z,* (see Fig. 1 b). The former is related to the origin of the wave dislocation lines, and the latter is related to the annihilation of the wave dislocation lines along the z-axis in three-

dimensional space. Since the topological current is identically con-

served, the topological charges of these two generated or annihilated wave dislocation lines must be opposite at the limit point, i.e.

P1771 + P27l2 = a

which shows the generation and the annihilation of a pair of dislocation lines. For the case of optical vortices, it corresponds to the generation and the annihilation of a vortex-antivortex pair.

Then, let us turn to the other case, in which the

restrictions are

= 0.

r : r : (25)

These two restrictive conditions will lead to an important fact that the functional relationship between z and x is not unique in the neighborhood of rt. In our topological current theory, this fact is easily seen from

dx D’(+) d;= @(4/x) ,;’

(26)

which under (25) directly shows that the direction of the integral curve of (26) is indefinite at r:. Therefore the very point r: is called a bifurcation point of the complex scalar waves. With the aim of finding the different directions of all branch curves of Eqs. (9) at the bifurcation point, we suppose that

K Dunn et al. /Physics Letrem A 253 (1999) 57-65 61

Z‘

Z Coordinate

Z’ Z Coordinate

Fig. I. The branch solutions for (24). (left) A pair of wave dislocation lines with opposite charges generate at the limit point, i.e. the

origin of wave dislocation lines. (right) Two wave dislocation lines annihilate in collision at the limit point.

From 4’ (x, y, z ) = 0, the implicit function theorem In order to explore the behavior of wave dislocation

says that there exists one and only one functional re- lines at the bifurcation points, let us investigate the

lationship Taylor expansion of

y=y(x.z).

Substituting (28) into @‘, we have

@(x,_v(x.i),i) = 0,

which gives

(28) M(x,z) =4*&Y(x,z),z) (30)

in the neighborhood of r:, which according to ( 19)

must vanish at the bifurcation point, i.e.

M(r:) = 0. (31)

From (30), the first order partial derivatives of M( X, z ) with respect to x and z can be expressed by

where the partial derivatives are

From these expressions it is easy for one to calculate the values of f;:, fl, fix, f& and fl’ at rr.

dM a42 dM d4' d4' rlx=ax

-+gj-;, -jy= z+$. (32)

By making use of (29), (32) and Cramer’s rule, it is easy to prove that the two restrictive conditions (25) can be rewritten as

which give

C?M - =o, g dx r;

= 0,

r : (33)

62 Y Duczn et al. /Physics Lettm A 253 (I 999) 57-65

by considering (27). The second order partial derivatives of the function M are

which at r: are denoted by

A - a2M ax’ y

B-d’M d2M

axa; ,:’ C=-

az’ FT. (34)

Then, from (3 1)) (33) and (34), we obtain the Taylor expansion of M(.r, ; 1,

M(x. z) = ;A(x - .x,~)‘+ B(x - xf)(: - z;)

+ gx - c,* I’.

which by (30) is the behavior of 42 in the neighbor-

hood of r,*. Because of the second equation of (9), we get

which leads to

A$ ( > 2

. +ZB~+C=O,

,’

and

2

+2B$+A=O.

(35)

(36)

The solutions of Eq. (35) or (36) give different directions of the branch curves (zero lines of the complex scalar wave, i.e. wave dislocation lines) at the bifurcation point. There are four important cases:

Case I (A # 0): For A = 4(B* - AC) > 0, from Eq. (35) we get two different directions of wave dislocation lines in three-dimensional space R”.

ds _Biv/_

K ,.2= A ’ (37)

which is shown in Fig. 2, where two zero lines inter-

Fig. 2. Bifurcation solution for (37): two wave dislocation lines

intersect with different directions at the bifurcation point.

sect with different directions at r:. This shows that two wave dislocation lines intersect at the bifurcation point.

Case 2 (A # 0): For A = 4(B’ - AC) = 0, from Eq. (35) we get only one direction of wave dislocation lines in three-dimensional space,

which includes three important cases according to Fig. 3. Firstly, two zero lines tangentially make contact, i.e. two wave dislocation lines tangentially encounter each other at the bifurcation point (Fig. 3a). Secondly, two zero lines merge into one zero line, i.e. two wave dislocation lines merge into one wave dislocation line at the bifurcation point (Fig. 3b). Finally, one zero line resolves into two, i.e. one wave dislocation line splits into two at the bifurcation point

(Fig. 3~). Case 3 (A = 0, C # 0): For A = 4(B2 - AC) = 0,

from Eq. (36) we have

dz -Bid=

z ,*= C (39)

As shown in Fig. 4, there are two important cases: (a) One zero line resolves into three, i.e. one wave dislocation line splits into three at the bifurcation point (Fig. 4a). (b) Three zero lines merge into one zero

Y Dunn et ul. /Physics Leilers A 253 (I 999) 57-65

2’

7 L7.7dlnate 2’

2 Coordinate

Z'

63

Z ,Ckc-dinate

Fig. 3. Bifurcation solutions for (38): wave dislocation lines have the same direction when they encounter each other. (top. left) Two

wave dislocation lines tangentially make contact at the bifurcation point. (top, right) Two wave dislocation lines merge into one wave

dislocation line at the bifurcation point. (bottom) One wave dislocation line splits into two wave dislocation lines at the bifurcation point.

line, i.e. three wave dislocation lines merge into one at the bifurcation point (Fig. 4b).

Case 4 (A = C = 0): Eqs. (35) and (36) give respectively

dx 0, dz 0

dz= dx=. (40)

This case is obvious as in Fig. 5, which is similar to Case 3.

The remainder component dy/dz can be given by

where partial derivative coefficients fi and fi have been calculated in (29). Now we get all the different directions of the branch curves. These mean the behavior of wave dislocation lines at the bifurcation

points is detailed. The above solutions reveal the topological structure

of wave dislocation lines in three-dimensional space IR3. Besides the intersection of wave dislocation lines, i.e. two wave dislocation lines intersect at the bifurca-

64 E Duun et ul./Physics Letten A 253 (1999) 57-65

2’

t Coordinate

L’

Z Coordinab

Fig. 4. Two important cases of (39). (left) One wave dislocation line splits into three wave dislocation lines at the bifurcation point

(right) Three wave dislocation lines merge into one wave dislocation line at the bifurcation point

Z Coordinate

Z’

Z Coordinate

Fig. 5. Two important cases of (40): two wave dislocation lines intersect normally at the bifurcation point. (left) Three wave dislocation

lines merge into one at the bifurcation point. (right) One wave dislocation line splits into three wave dislocation lines at the bifurcation point.

tion point (see Figs. 2 and 3a), splitting and merging

of wave dislocation lines are also included. When a multicharged wave dislocation line passes through the bifurcation point, it may split into several wave dislocation lines along different branch curves (see Figs. 3c, 4a and 5b), moreover, several wave dislocation lines can merge into one wave dislocation line at the bifurcation point (see Figs. 3c, 4b and 5a). The iden- tical conservation of the topological charge shows the sum of the topological charges of final wave dislocation line(s) must be equal to that of the initial wave

dislocation line(s) at the bifurcation point.

4. Conclusions

Firstly, with the &mapping topological current theory, we obtain the topological charge density current of dislocation lines in three-dimensional space. The topological charge density current of dislocation lines takes the form of generalized function 6* ( C/J) where 4’ and c$’ are the real and imaginary parts of a com-

E Dmm et al. /Phy.ws Letters A 253 (1999) 57-65 6.5

plex scalar wave +(r) of light beams. Secondly, when the Jacobian D3 (qb/x) = 0, it is shown that there exist the crucial cases of branch process. Based on the implicit function theorem and the Taylor expansion, the branch solutions at the limit points and the different

directions of all branch curves at the bifurcation points

are calculated. In the limit points case, two wave dislocation lines generate or annihilate along the z-axis. It can he looked upon as the origin and the annihilation of a pair of wave dislocation lines. At the bifurcation points, wave dislocation lines are found to intersect,

split or merge. So, we obtained the conditions for generating, annihilating, intersecting, splitting and merging of dislocation lines in three-dimensional space. Fi- nally. we would like to point out that all the results in this paper are obtained only from the viewpoint of topology without using any concrete or special mod-

els or hypotheses.

Acknowledgement

This work was supported by the National Natural Science Foundation of China.

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a mathematical framework for wave dislocation lines

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