a mathematical framework for wave dislocation lines
Post on 04-Jul-2016
214 Views
Preview:
TRANSCRIPT
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 topologi- cal 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 dislo- cation [ 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@K@lzu.edu.cn.
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 dis- locations 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 the- ory 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 bifurca- tion, 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 $-mapping topo-
logical current theory is a powerful tool to study the topological structure of wave dislocation lines. Among its advantages and without using any particular mod-
els or hypotheses, we give the topological charge den- sity current of dislocation lines in three-dimensional space, therefore, one can get the most general proper- ties of wave dislocation lines. According to the values
of the vector Jacobians of a complex scalar wave in three-dimensional space, the branch points are classi- fied into two types: limit points and bifurcation points.
Furthermore, it is found that the wave dislocation lines generate or annihilate at the limit points and encounter each other, split or merge at the bifurcation points of
the complex scalar wave. This paper is organized as follows: in Section 2 we
construct an elementary topological current in light beams. By means of the &mapping topological cur- rent theory, we derive the topological charge density current of the wave dislocation lines. In Section 3 from
the properties of the complex scalar wave, the condi- tions for generating, annihilating, intersecting, split-
ting and merging of wave dislocation lines are ob- tained and several crucial cases of branch process are discussed. The topological charge conservation near the branch points is also studied. We present our con- cluding remarks in Section 4.
2. Topological charge density current of wave dislocation lines
Let us study a complex scalar wave Cc, (r, t) of a light beam directed along the z-axis at a fixed time, which is denoted as
v+(r) = 4’(r) + i&(r), (I)
where 4’ (r) and 4’(r) can be regarded as two com- ponents of a two-dimensional vector field
4 = ($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 topo- logical 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 inter- section of surfaces representing the zeroes of the real
part and imaginary part of the complex scalar wave: Re(+) = r$’ = 0 and Im($) = c$~ = 0. So, the loca- tion and the direction of the ith wave dislocation line are determined by the ith zero line L; and the unit vec-
tor field m(r) = D( c$/x)/ID(c$/x) 1 on L; respec- tively.
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 mean- ing 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 topo- logical 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 con- dition 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 com- plex 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 vor- tices, it corresponds to the generation and the annihi- lation 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 im- portant 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+$. (32)
By making use of (29), (32) and Cramer’s rule, it is easy to prove that the two restrictive conditions (25) can be rewritten as
which give
C?M - =o, g dx r;
= 0,
r : (33)
62 Y Duczn et al. /Physics Lettm A 253 (I 999) 57-65
by considering (27). The second order partial deriva- tives of the function M are
which at r: are denoted by
A - a2M ax’ y
B-d’M d2M
axa; ,:’ C=-
az’ FT. (34)
Then, from (3 1)) (33) and (34), we obtain the Taylor expansion of M(.r, ; 1,
M(x. z) = ;A(x - .x,~)‘+ B(x - xf)(: - z;)
+ gx - c,* I’.
which by (30) is the behavior of 42 in the neighbor-
hood of r,*. Because of the second equation of (9), we get
which leads to
A$ ( > 2
. +ZB~+C=O,
,’
and
2
+2B$+A=O.
(35)
(36)
The solutions of Eq. (35) or (36) give different di- rections of the branch curves (zero lines of the com- plex 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 con- tact, i.e. two wave dislocation lines tangentially en- counter 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 differ- ent 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 dislo- cation 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 disloca- tion 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 the- ory, 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 im- plicit 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 dislo- cation 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 gen- erating, annihilating, intersecting, splitting and merg- ing 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.
References
I I] J.F. Nye, M.V. Berry, Proc. Roy. Sot. Lond. A 336 ( 1974)
165.
121 MS. Soskin. V.N. Gorshkov, M.V. Vasnetsov, J.T. Males,
N.R. Heckenberg, Phys. Rev. A 56 ( 1997) 4064.
131 I.V. Basistiy. V.Yu. Bazhenov, M.S. Soskin, M.V. Vasnetsov,
Opt. Commun. 103 (1993) 422.
141 N.R. Heckenberg, M. Vaupet, J.T. Males, C.O. Weiss, Phys.
Rev. A 54 (1996) 2369.
5 J P. Coullet. L. Gil. F. Rocca, Opt. Commun. 73 ( 1989) 403.
61 N.B. Baranova. A.V. Mamaev, N.F. Pilipetskii. V.V. Shkunov.
B.Ya. Zel’dovich, J. Opt. Sot. Am. 73 (1983) S2S.
71 1. Freund, N. Shvattsman, V. Freilikher, Opt. Commun. 101
(1993) 247.
8 1 J.F. Nye, Proc. Roy. Sot. Lond. A 378 ( 198 I ) 2 19.
91 J.F. Nye, J. Opt. Sot. Amer. A IS ( 1998) I 132.
I101 M.V. Berry, Much ado about nothing: Optical dislocation
lines, in: Singular Optics, Vol. 3487, ed. MS. Soskin (SPIE.
Frunzenskoe Crimea, 1998).
I I I I A.V. Ilyenkov. A.I. Khizniak, L.V. Kreminskayn, MS.
Soskin, M.V. Vasnetsov. Appl. Phys. B 62 (1996) 46.5.
1121 G. Indebetouw. J. Mod. Opt. 40 (1993) 73.
I I31 H. He, N.R. Heckenberg, H. Bubinsztein-Dunlop, J. Mod.
Opt. 42 (1995) 217.
I I41 Y.S. Duan, S. Li. G.H. Yang, Nucl. Phys. B 514 (1998) 705.
I151 Y.S. Duan, H. Zhang. S. Li, Phys. Rev. B 58 (1998) 125.
1161 Y.S. Duan, H. Zhang, L.B. Fu, Phys. Rev. E 59 (1999) 528.
I I7 I Y.S. Duan. X.H. Meng, J. Math. Phys. 34 ( 1993) 1149.
I I81 G.H. Yang, Ph.D. thesis. Lanzhou University, China ( 1997).
top related