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Optical wavefront dislocations and their properties
15 September 1995
OPTlCS
COMMUNICATIONS ELSEVIER
Optics Communications 119 ( 1995) 6CH-612
Full length article
Optical wavefront dislocations and their properties I.V. Basistiy, MS. Soskin, M.V. Vasnetsov Instituteof Physics, Nutional Academy of Science of Ukraine, Praspect Nauki 46, 252650 Kiev-22, Ukraine Received 22 September 1994; revised version received6 April 1995
Abstract We analyze screw, edge and mixed screw-edge dislocations in monochromatic light waves. Methods for the experimental dete~nation of the topological charge value and helicity of screw dislocations are elaborated. By use of computer-synthesized binary gratings waves with edge and mixed screw-edge disl~ations were obtained and studied ~xpe~mentally. The transformation of a mixed screw-edge distocation into a single screw disl~ation was observed and explained.
1. Introduction
The imperfections of~gularity in optical wavefronts are known probably since 1950 [ 11. A detailed analysis and description of phase defects in wave trains and monochromatic waves was performed in 121. The prevalence of screw-type defects in a wavefront with complex structure (speckle field), so-caBed screw dislocations, or optical vortices, was demonstrated later [ 31, Regular optical waves possessing a single screw dislocation or an array of several dislocations were experimentally obtained by use of different techniques: generation of laser beams with helical wavefronts ]4,5], diffraction by specially syn~esized hologr~s [ 6-81 and transmission of light beams through a phase mask [ 9, IO]. Phase defects are easily observed in laser light scattered by a random phase diffuser ]3,11 J or transmitted through a multimode fiber [ 61. The appearance of optical vortices was also observed experimentally in the oscillation of ring cavities containing a photorefractive crystal as a gain medium [ 121, or in a linear cavity with a photorefractive phasereversal mirror [ 13] . For the existence of phase singularity it is necessary that the amplitude of a light wave E gets zero at this ~30~~1~/9S/$O9.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(95)00267-7
point, so that the real as well as the imaginary part of the complex amplitude equals zero and the phase becomes indete~inate. Another variant is the amplitude singul~ity, when the amplitude of a wave tends to infinity at the point of the phase defect [ 21. The totality of the results obtained ensures that we can reveal the main properties of optical waves with phase defects, namely structural stability of the unitycharged screw dislocation, conservation of the total topological charge in the processes of birth, annihilation and decay of the multicharged screw dislocations. It was shown theoretically and in experiments, that optical vortices behave like charged particles at the frame of a shared wavefront: equally charged dislocations repel each other, and dislocations of the opposite charges attracteach other and can annihilate in collision { 14,151. The aim of this paper is to give a general rule for the dete~ination of the major features of screw dislocations and to report the first analytical approach and experiments with holographic~ly synthesized monochromatic waves possessing edge and mixed screwedge dislocations.
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Communications
119 (1995) 604412
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2. Classification of phase defects According to the terminology, introduced by Nye and Berry [ 21, wavefront dislocations in a monochromatic wave are divided into three main types: (i) infinitely extended edge dislocation (nonlocalized interference fringe), (ii) screw dislocations, and (iii) limited edge dislocation (localized interference fringe). The infinitely extended edge dislocation produces a shift of a part of the wavefront surfaces by rr (half a wavelength) along the direction of wave propagation: a cut of the wavefront occurs on the dislocation line, as shown in Fig. 1. For simplicity, we shall use the term “edge dislocation” for this type of phase defect. As an example, we may refer to the diffraction on a circular aperture, where the far-field amplitude distribution is given by the well-known Airy formula, and where the change of the amplitude sign means a phase shift by rr, i.e. each of the dark rings where E = 0 is a closed edge-dislocation line. In laser beams which have the structure of transverse cavity modes, edge dislocations are black lines between neighboring (rrshifted in phase) mode spots. The simplest example is a TEM,, mode having two intensity maxima divided by a zero-intensity line with phase step r. Screw dislocation, unlike edge dislocation, is a point defect. It possesses a topological charge, equal to the integer m of a 27r phase change on any closed circuit around the dislocation center. A cophasal wavefront containing a screw dislocation has the form of a oneor multistart helical surface, i.e. one round-trip on the continuous phase surface around the dislocation axis will lead to the next (or preceding) coil with a pitch mh (A is a wavelength), Figs. 2a, b. We define the dislocation charge to be positive if a helical wavefront produces a right screw in space, and negative in the opposite case. The dislocation axis, on which the phase
Fig. I. Schematic view of the infinitely extended edge dislocation. Each of the wavefronts propagating along the z-axis is cut, and the upper isophase surfaces are shifted by h/2 with respect to the lower half space.
Fig. 2. Helical structure of a wavefront around the screw-dislocation axis: (a) m = 1, the wavefront produces a one-start right screw, the pitch equals A; (b) m = - 3, the wavefront produces a three-start left screw, the pitch is 3A. In both cases the phase distance between neighboring equiphase surface is 2~.
of a wave is indeterminate, should also be the line of zero amplitude, producing a “dark beam” captured inside a light wave. The dislocation axis may not coincide with a direction of propagation of the host beam. The so-called “hybrid” laser TEM$ mode may serve as an example of an optical wave having a screw dislocation of charge f 1 [ 41. Screw dislocations of charge plus and minus one are stable with respect to small perturbations of the wavefront, in spite of the highly charged screw dislocations, which may decay into a group of nearly located unitycharged dislocations of the same sign with conservation of the total charge value [ 151. The reflection of a wave carrying screw dislocation leads to the reversal of a topological charge sign. Screw dislocation may be transformed to an edge dislocation, for instance by use of the cylindrical lens converter [ 16-l 81. Reciprocal transformation is also possible. We note that the topological charge of an edge dislocation is zero, as the phase jumps by 7~ and - rr compensate each other on a closed circuit around any point on the edge dislocation line. Apart from “pure” screw and edge dislocations, mixed-type dislocations (limited edge dislocation) may also exist in a monochromatic wave. Fig. 3 schematically shows one possible variant of the localized interference fringe, or a half-cut of a wavefront surface. In this case the line of a wavefront cut has a one-side
006
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beam, for instance a Gaussian beam. In this case a wave with an axial screw dislocation of charge m is expressed as follows [ 121:
Fig. 3. Schematic view of the mixed screw-edge dislocation. wavefronts are cut starting from a dislocation starting point.
The
limited extension on a wavefront surface. The phase has a difference of rr at the opposite sides of the halfcut and changes smoothly around the starting point of the half-cut. The problem of structural stability of these creations (edge and mixed screw-edge dislocations) is not evident, as the necessary condition of edge dislocation existence (coincidence of lines Re(E) = 0 and Im{ E} = 0) can be violated by the influence of perturbations.
3. Screw dislocation: determination value of charge
,
(1)
where p, 6, z are cylindrical coordinates, k is a wavenumber. Actually this expression cannot describe a real wave, as the amplitude grows infinitely with increasing p, but near the axis of dislocation it may serve as a good approximation, satisfying the scalar wave equation [ 21. The continuous cophasal surface of the wavefront shows itself as a helicoid, determined by the relation $-+-kz=const.,
(3)
where E. is a real amplitude, po is a waist parameter of a beam, @(p, #, z) is a phase having the form
t--arctang
@(p,+,z)=(]m]+l)
1
( ‘v*
-
2z -?-k2p;/2z
-m4-kz.
(34
An expression for the transverse phase dependence lows from (3a) :
fol-
of sign and
In this paragraph we consider one of the principal questions related to a screw dislocation, namely the determination of sign and value of its topological charge. We start with the simplest expression for a wave carrying unity-charged screw dislocation: R(p, 4, z)=pexp(+$-k)
Xev[i@z(p, $5z) 1 ,
(2)
while in the case of a plus in front of 4, according to the definition (the wavefront with a positively charged dislocation produces a right screw), the dislocation sign is negative, and vice versa. A real wave E(p) must have a finite amplitude and a border condition iim,,,E( p) = 0. According to this, it is necessary to take an appropriate wave as a host
where the radius of the wavefront curvature is R(z) = z + k2pi/4z. Let us examine the interference of the described wave with a coherent copropagating divergent spherical wave having at the space of the interference observation the wavefront curvature radius R,, which in general leads to the spiral interference pattern, exhibiting 1m 1 fringes starting from the center. The circulation of fringes is determined by the sign of m by the relation between the curvature radii of the wavefronts R(z) and Ro. The equation for the interference fringe maxima is
’ or -m$+kp2
R(z) -Ro 2R(z)Ro
=27rz,
(1=0,
+1, +2 ,... ).
(5a) The clockwise rotation of the fringes corresponds to the positive sign of the dislocation charge in the case of the plane reference wave (R, = co), and vice versa. In the case of equality, R(z) = Ro, the fringes are not bend, and the interference pattern will look like a
I.V. Basistiy et al. /Optics
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“star” with a number of rays equal to the modulus of the dislocation charge. The direction of the fringes rotation will change with the deviation of R,, from R(z) to one or another side. So, for a correct determination of the dislocation charge it is necessary to know the relation between the divergencies of the beams. The clockwise fringe rotation indicates a positive dislocation charge for the plane reference wave. To detect the presence of wavefront dislocations, offaxis interference can also be used. Let the wave with screw dislocation propagate along the z-axis, and the reference plane wave is tilted by an angle 8 to the zaxis with wavevector projections k sin 8 on the polar axis and k cos 0 on the z-axis. A nearly periodic intensity distribution will occur on a screen, in which the positions of the intensity maxima are determined by the equation (the relative phase difference is not taken into account) kp'
cos -M-
-([ml
2R(z)
+l)arctan~
kd
8 + kp cos 4 sin 0 - 2kz sin’ 2
+(]m]+l);
=l 1
Using formula (6)) we can draw on the p, 4 plane a two-dimensional grating, assuming z = const. and kp* < 2R(z). The equation for the intensity maxima is simplified to -m++kp
+l)?r/2,
607
rn=l
n1=-3
Fig. 4. Structure of the interference fringe maxima, calculated according to formula (6a). for m = 1 (a, b, c ) and m = - 3 ( d, e, f) , and for different G(z) values, as indicated.
or due to the change of direction of the reference wave (sign of 19). Thus, an essential conclusion from this consideration is the rule for the correct determination of the sign of a screw-dislocation charge from the interference pattern (the direction of the fringe circulation or the “forklets’ ’ orientation). Other methods of charge determination are also possible [ 8,181. A similar approach can be used also in the interference scheme proposed recently 1191. Experimental measurements of topological charge sign and value were performed for the beams created by laser beam diffraction on computer-synthesized gratings in order to verify the results obtained. The technique of binary grating synthesis was proposed and applied by us [ 6,7] on the basis of relation (6a). Analogous experiments and analyses were also reported in
[81.
cos 4 sin 8+ G(z)
=2?rz-(]m]
I19 (1995) 604412
(6a)
where G(z) = ( I m I + 1) arctan (kpi/2z) - 2kz sin’( o/2). The central parts of the gratings, calculated form = 1 and m = - 3, are depicted in Fig. 4, for different values of G(z). In the center of the pictures, I m 1 new fringes are born or vanished, thus producing a fringe splitting in the form of the “forklet”. A “forklet” in the interference pattern will be directed upwards or downwards, depending on the sign of 8: If in the interference arrangement the reference wave has a positive projection of a wavevector on the polar axis ( 8> 0), and m is positive, the “forklet” is directed upwards. The orientation will change to the opposite when m is negative,
A laser beam diffracted by a grating and carrying a screw dislocation of charge m interfered with a coherent reference slightly divergent beam, which curvature was controlled by a system of two spherical lenses. The interferograms of copropagated beam interference are shown in Figs. Sa,b,c for three possible cases: (i) the radius of curvature of the reference wave is larger than that of the wave with dislocation, (ii) the radii are equal, the (iii) the radius of curvature of the reference beam is less than that of the wave with dislocation. The rotation of the spirals changes direction with the change of curvature of the reference beam in perfect accordance with the analysis. For the off-axis interference scheme, two “forklet” orientations are shown for two variants of the reference beam disposition (Fig. 6). It means that the scheme of interference shown in Fig. 6b
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d Fig. 5. Interference pictures produced by a wave possessing a high-order screw dislocation m=4 and a reference wave: (a) the reference wavefront curvature is smaller than that of a screw-dislocation wavefront, the rotation of fringes is clockwise; (b) the wavefront curvatures are equalized, the fringes form a cross; (c) the reference wavefront curvature is larger than that of a screw dislocation wavefront, the direction of the fringes rotation has changed to the opposite. The corresponding schemes of interference are depicted in (d), (e), (f), where the screwdislocation wavefront is shown by the dashed line, and the reference wavefront by the solid line.
screen polar
v
--
b
C
Fig. 6. Off-axis interferograms (a. d) of a wavefront with a unity-charged screw dislocation (m = 1). and the corresponding interference schemes (b, c). A wave with screw dislocation propagates normally to the screen, the reference plane wave is tilted by an angle It 8. Interferogram (a) corresponds to the orientation (c), where the reference wave-vector projection on the polar axis is negative, and the “forklet” is directed downwards; interferogram (d) corresponds to the positive projection, and the “forklet” is directed upwards.
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permits in a most simple way to find the absolute helicity of a screw-dislocation wavefront.
4. Edge and mixed screw-edge
dislocations
As mentioned earlier, a trivial example of edge dislocation is a TEMel mode of a laser cavity, expressed in the form E(p,@)=
Esin4exp(-
$)exp(-
g)
(7)
((Y is the beam halfwidth, R is the wavefront radius of the curvature). The horizontal line dividing the beam into two parts corresponds to a zero value of the amplitude and phase hop on r. As it is known, a combination of the TEMe, and orthogonal TEMia mode with a necessary phase shift by + 7~12 produces a “doughnut” circular TEM$ mode:
609
location by diffraction of a plane wave on a synthesized grating. The off-axis interference will show a very simple picture of two periodic gratings shifted by half a period on a line of zero amplitude. SimpIified gratings were calculated without taking into account the wave curvature and the variation of fringes contrast over a grating (Fig. $a). For the mixed edge-screw dislocation, we need the existence of a zero-intensity line crossing a beam with starting point at the beam center. The complex amplitude distribution E( p, 4) may be expressed in this case as
which may be decomposed
into a sum of two waves,
2
E(p,&)
Xexp
=
i
(cos
&*i
sin 4) exp
( 1
IeM-W)-11
- $
Pa)
One of the equiphase lines, determined by (7), is shown in Fig. 7a. Let the phase @be zero on this line, thus we obtain a curve Im(E) = 0. We may draw a curve Re(E) = 0 which is an equiphase line corresponding to the relative phase @= ~12 in a similar way. On the horizontal axis, where the amplitude equals zero, both curves overlap. Following our technique, we can calculate an interference light distribution on a screen taking a plane reference wave and then restore a wave with edge dis-
The expression (8) due to the multiplier sin(4/2) possesses a line of zero amplitude corresponding to 4 = 0. The azimuthal dependence of the phase is given by a factor exp( - i4/2), which may be treated as a topological charge l/2 located at the point p = 0. Equiphase lines produce broken spirals, as shown in Fig. 7b. On the zero-amplitude line the conditions Re( E) = 0 and Im( E) = 0 are satisfied simultaneously. On the other hand, relation (8a) explains more clearly the structure of a mixed screw-edge dislocation.
+=0 Rc(E} = Im(E)=O
Fig. 7. Localization of the lines Re( F) = 0 (weak line) and Im( E) = 0 (solid line) for (a) a wavefront with edge dislocation a mixed screw-edge dislocation.
and
(b) one
with
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Fig. 8. Synthesized binary gratings for the creation of diffracted beams with a cut in the wavefront: (a) for pure edge dislocation, screw-edge dislocation. The corresponding near-field images of the diffracted beams are in photographs (b) and (d).
The first part in (pa) is an ordinary wave with a unitycharge screw dislocation, and the other part is a wave without azimuthal phase dependence, but getting zero when p = 0:
E(p) = “xp(-
$)exp(--z),
2a
This is a super-Gaussian beam [ 201, and in a far-field zone it will transform into a wave having the form of a Hankel transform of a radial function (8b) : E(v)=
zJp2exp(-
$)exp(-
g)
0 xJo(2rrvp)
dp,
(8~)
where Y is the spatial frequency, Jo is the Bessel function of zero order. Relation (8~) describes a smooth amplitude distribution with a central maximum. The total far-field picture produced by the amplitude function (8a) will display an interference between the “doughnut” wave carrying screw dislocation, and the background coherent vortices-free wave. The presence
(c) for mixed
of the background causes a shift in the position of the vortex location from the center of the beam [ 151, and its rotation on a definite azimuth angle. To create a wave with a mixed screw-edge dislocation, we used the following description for the binary amplitude grating: T( p, 4) =
0,
cos( - 4/2 + Kp cos 4) < 0,
1,
cos( -4/2+Kpcos
(9)
+)>O,
where T( p, 4) is grating transmittance, K = 27r/A, A is the grating spacing. On the line I$= 0 the phase experiences hop by r (here the amplitude gets zero). This means that a wavefront of a wave diffracted by this grating has a cut from the center to the periphery along the line C#J = 0, and the point p = 0 is the starting point of the limited edge dislocation. Conventionally, a charge f l/2 may be attributed to the point p = 0. At the first order of diffraction, an object wave is restored which is a combination, as described above, of a doughnut wave and a coherent background wave. The phase is doubled at the second order of diffraction,
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and thus the azimuthal phase dependence is the same as for the pure doughnut wave. Typical gratings and near-field pictures of diffracted beams are exhibited in Figs. 8a-d. At the first order of diffraction, we observed for the grating in Fig. 8a two m-shifted bright spots divided by a black line (Fig. 8b) and a half-cut in a transverse section of a diffracted beam for the grating in Fig. 8c (Fig. 8d). In accordance with a theoretical prediction, at the second order of diffraction by a grating shown in Fig. 8a we observed an ordinary vortices-free wave and a unity-charged pure screw dislocation for a grating shown in Fig. 8c: this confirms the existence of charge 112 in the firstorder diffracted beam. The beam having a half-cut of a wavefront in a near-field zone (Fig. 9a) changes on its way, the phase cut tends to transform into several pairs of opposite-sign screw dislocations which disappear on the beam periphery, and finally in a far-field zone we observe the only presence of a single unity-charged screw dislocation (Figs. 9b, c) . It means that a mixed screw-edge dislocation does not propagate as a selfsimilar stable object, but transforms into a combination of a single screw dislocation and a background dislocation-free wave.
5. Conclusions
Fig. 9. Interferogram of a beam in the near-field picture (a) corresponding to Fig. Sd, and the far-field picture and interferogram of the same beam (b, c)
Summarizing, we have established important rules for the determination of a screw-dislocation charge on the basis of the introduced definition of the sign of screw-dislocation topological charge: the sign is positive if a helical wavefront surface produces a right screw in space. This definition does not depend on the choice of the coordinate systems and in this sense is absolute. In addition to the analysis and experiments with screw dislocations of different charges, we started (for the first time to our knowledge) to study edge dislocations (having zero charge) and mixed screw-edge dislocations, as structures having charge f l/2 at the end point of a cut of the wavefront. Our analysis and first experimental results have shown that the mixed edge-screw dislocation is not stable and decays in space producing in the near field several pairs of oppositely charged screw dislocations of unity charge and a single unity-charge screw dislocation in a far field.
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Acknowledgements We thank I. Marienko for assistance in the gratings preparation. This work was supported by the International Science Foundation, Grant U61000 and the Committee of Science and Technologies of Ukraine.
References [ 1 ] A. Sommerfeld, Optik, Vorlesung uber Theoretische Physik, band IV, Wiesbaden (Geest und Portig, Leipzig, 1950). 121 J.F. Nye and M.V. Berry, Proc. R. Sot. London A 336 ( 1974) 165. [ 31 N.B. Baranova, A.V. Mamaev, N.F. Pilipetskii, V.V. Shkunov and B.Ya. Zel’dovich, J. Opt. Sot. Am. 73 (1983) 525. [4] J.M. Vaughan and D.V. Willetts, J. Opt. Sot. Am. 73 (1983) 1018. [5] C. Tamm and CO. Weiss, J. Opt. Sot. Am. B 7 (1990) 1034. [ 61 V.Yu. Bazhenov, M.V. Vasnetsov and M.S. Soskin, JETPLett. 52 ( 1990) 429. [7] V.Yu. Bazhenov, M.S. Soskin and M.V. Vasnetsov, J. Mod. Optics 39 (1992) 985.
]8] N.R. Heckenberg, R. McDuff, C.P. Smith, H. RubinszteinDunlop and M.J. Wegener, Opt. Quantum Electron. 24 ( 1992) s951. ]9] S.N. Khonina, V.V. Kotlyar, M.V. Shinkaryev, V.A. Soifer and G.V. Uspleniev, J. Mod. Optics 39 ( 1992) 1147. I101 M.W. Beijersbergen, R.P.C. Coerwinkel, M. Kristiensen and J.P. Woerdman, Optics Comm. 112 ( 1994) 321. I 111 1.Found, N. Shvartsman and V. Freilikher, Optics Comm. 10 1 (1993) 247. I121 F.T. Arecchi, G. Giacomelli, P.L. Ramozza and S. Residoti, Phys. Rev. Lett. 67 (1991) 3749. ]13] S.R. Liu and G. Indebetouw, J. Opt. Sot. Am. B 9 (1992) 1507. L141 G. Indebetouw, J. Mod. Optics 40 ( 1993) 73. I151 I.V. Basistiy, V.Yu. Bazhenov, MS. Soskin and M.V. Vasnetsov, Optics Comm. 103 (1993) 422. [ 161 E. Abramochkin and V. Volostnikov, Optics Comm. 83 ( 1991) 123. [ 171 M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen and J.P. Woerdman, Optics Comm. 96 (1993) 123. [ 181 G. Slekys, C.O. Weiss, D.Y. Tang and M.F.H. Tarroja, J. Opt. Sot. Am. B 11 ( 1994) 2089. [19] M. Harris, C.A. Hill, P.R. Tapster and J.M. Vaughan, Phys. Rev.A49(1994)3119. [ 201 C. Palma and V. Bagini, Optics Comm. 111 ( 1994) 6.
OPTlCS
COMMUNICATIONS ELSEVIER
Optics Communications 119 ( 1995) 6CH-612
Full length article
Optical wavefront dislocations and their properties I.V. Basistiy, MS. Soskin, M.V. Vasnetsov Instituteof Physics, Nutional Academy of Science of Ukraine, Praspect Nauki 46, 252650 Kiev-22, Ukraine Received 22 September 1994; revised version received6 April 1995
Abstract We analyze screw, edge and mixed screw-edge dislocations in monochromatic light waves. Methods for the experimental dete~nation of the topological charge value and helicity of screw dislocations are elaborated. By use of computer-synthesized binary gratings waves with edge and mixed screw-edge disl~ations were obtained and studied ~xpe~mentally. The transformation of a mixed screw-edge distocation into a single screw disl~ation was observed and explained.
1. Introduction
The imperfections of~gularity in optical wavefronts are known probably since 1950 [ 11. A detailed analysis and description of phase defects in wave trains and monochromatic waves was performed in 121. The prevalence of screw-type defects in a wavefront with complex structure (speckle field), so-caBed screw dislocations, or optical vortices, was demonstrated later [ 31, Regular optical waves possessing a single screw dislocation or an array of several dislocations were experimentally obtained by use of different techniques: generation of laser beams with helical wavefronts ]4,5], diffraction by specially syn~esized hologr~s [ 6-81 and transmission of light beams through a phase mask [ 9, IO]. Phase defects are easily observed in laser light scattered by a random phase diffuser ]3,11 J or transmitted through a multimode fiber [ 61. The appearance of optical vortices was also observed experimentally in the oscillation of ring cavities containing a photorefractive crystal as a gain medium [ 121, or in a linear cavity with a photorefractive phasereversal mirror [ 13] . For the existence of phase singularity it is necessary that the amplitude of a light wave E gets zero at this ~30~~1~/9S/$O9.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(95)00267-7
point, so that the real as well as the imaginary part of the complex amplitude equals zero and the phase becomes indete~inate. Another variant is the amplitude singul~ity, when the amplitude of a wave tends to infinity at the point of the phase defect [ 21. The totality of the results obtained ensures that we can reveal the main properties of optical waves with phase defects, namely structural stability of the unitycharged screw dislocation, conservation of the total topological charge in the processes of birth, annihilation and decay of the multicharged screw dislocations. It was shown theoretically and in experiments, that optical vortices behave like charged particles at the frame of a shared wavefront: equally charged dislocations repel each other, and dislocations of the opposite charges attracteach other and can annihilate in collision { 14,151. The aim of this paper is to give a general rule for the dete~ination of the major features of screw dislocations and to report the first analytical approach and experiments with holographic~ly synthesized monochromatic waves possessing edge and mixed screwedge dislocations.
I.V. Basistiy et al. /Optics
Communications
119 (1995) 604412
605
2. Classification of phase defects According to the terminology, introduced by Nye and Berry [ 21, wavefront dislocations in a monochromatic wave are divided into three main types: (i) infinitely extended edge dislocation (nonlocalized interference fringe), (ii) screw dislocations, and (iii) limited edge dislocation (localized interference fringe). The infinitely extended edge dislocation produces a shift of a part of the wavefront surfaces by rr (half a wavelength) along the direction of wave propagation: a cut of the wavefront occurs on the dislocation line, as shown in Fig. 1. For simplicity, we shall use the term “edge dislocation” for this type of phase defect. As an example, we may refer to the diffraction on a circular aperture, where the far-field amplitude distribution is given by the well-known Airy formula, and where the change of the amplitude sign means a phase shift by rr, i.e. each of the dark rings where E = 0 is a closed edge-dislocation line. In laser beams which have the structure of transverse cavity modes, edge dislocations are black lines between neighboring (rrshifted in phase) mode spots. The simplest example is a TEM,, mode having two intensity maxima divided by a zero-intensity line with phase step r. Screw dislocation, unlike edge dislocation, is a point defect. It possesses a topological charge, equal to the integer m of a 27r phase change on any closed circuit around the dislocation center. A cophasal wavefront containing a screw dislocation has the form of a oneor multistart helical surface, i.e. one round-trip on the continuous phase surface around the dislocation axis will lead to the next (or preceding) coil with a pitch mh (A is a wavelength), Figs. 2a, b. We define the dislocation charge to be positive if a helical wavefront produces a right screw in space, and negative in the opposite case. The dislocation axis, on which the phase
Fig. I. Schematic view of the infinitely extended edge dislocation. Each of the wavefronts propagating along the z-axis is cut, and the upper isophase surfaces are shifted by h/2 with respect to the lower half space.
Fig. 2. Helical structure of a wavefront around the screw-dislocation axis: (a) m = 1, the wavefront produces a one-start right screw, the pitch equals A; (b) m = - 3, the wavefront produces a three-start left screw, the pitch is 3A. In both cases the phase distance between neighboring equiphase surface is 2~.
of a wave is indeterminate, should also be the line of zero amplitude, producing a “dark beam” captured inside a light wave. The dislocation axis may not coincide with a direction of propagation of the host beam. The so-called “hybrid” laser TEM$ mode may serve as an example of an optical wave having a screw dislocation of charge f 1 [ 41. Screw dislocations of charge plus and minus one are stable with respect to small perturbations of the wavefront, in spite of the highly charged screw dislocations, which may decay into a group of nearly located unitycharged dislocations of the same sign with conservation of the total charge value [ 151. The reflection of a wave carrying screw dislocation leads to the reversal of a topological charge sign. Screw dislocation may be transformed to an edge dislocation, for instance by use of the cylindrical lens converter [ 16-l 81. Reciprocal transformation is also possible. We note that the topological charge of an edge dislocation is zero, as the phase jumps by 7~ and - rr compensate each other on a closed circuit around any point on the edge dislocation line. Apart from “pure” screw and edge dislocations, mixed-type dislocations (limited edge dislocation) may also exist in a monochromatic wave. Fig. 3 schematically shows one possible variant of the localized interference fringe, or a half-cut of a wavefront surface. In this case the line of a wavefront cut has a one-side
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I I9 (1995) 604-612
beam, for instance a Gaussian beam. In this case a wave with an axial screw dislocation of charge m is expressed as follows [ 121:
Fig. 3. Schematic view of the mixed screw-edge dislocation. wavefronts are cut starting from a dislocation starting point.
The
limited extension on a wavefront surface. The phase has a difference of rr at the opposite sides of the halfcut and changes smoothly around the starting point of the half-cut. The problem of structural stability of these creations (edge and mixed screw-edge dislocations) is not evident, as the necessary condition of edge dislocation existence (coincidence of lines Re(E) = 0 and Im{ E} = 0) can be violated by the influence of perturbations.
3. Screw dislocation: determination value of charge
,
(1)
where p, 6, z are cylindrical coordinates, k is a wavenumber. Actually this expression cannot describe a real wave, as the amplitude grows infinitely with increasing p, but near the axis of dislocation it may serve as a good approximation, satisfying the scalar wave equation [ 21. The continuous cophasal surface of the wavefront shows itself as a helicoid, determined by the relation $-+-kz=const.,
(3)
where E. is a real amplitude, po is a waist parameter of a beam, @(p, #, z) is a phase having the form
t--arctang
@(p,+,z)=(]m]+l)
1
( ‘v*
-
2z -?-k2p;/2z
-m4-kz.
(34
An expression for the transverse phase dependence lows from (3a) :
fol-
of sign and
In this paragraph we consider one of the principal questions related to a screw dislocation, namely the determination of sign and value of its topological charge. We start with the simplest expression for a wave carrying unity-charged screw dislocation: R(p, 4, z)=pexp(+$-k)
Xev[i@z(p, $5z) 1 ,
(2)
while in the case of a plus in front of 4, according to the definition (the wavefront with a positively charged dislocation produces a right screw), the dislocation sign is negative, and vice versa. A real wave E(p) must have a finite amplitude and a border condition iim,,,E( p) = 0. According to this, it is necessary to take an appropriate wave as a host
where the radius of the wavefront curvature is R(z) = z + k2pi/4z. Let us examine the interference of the described wave with a coherent copropagating divergent spherical wave having at the space of the interference observation the wavefront curvature radius R,, which in general leads to the spiral interference pattern, exhibiting 1m 1 fringes starting from the center. The circulation of fringes is determined by the sign of m by the relation between the curvature radii of the wavefronts R(z) and Ro. The equation for the interference fringe maxima is
’ or -m$+kp2
R(z) -Ro 2R(z)Ro
=27rz,
(1=0,
+1, +2 ,... ).
(5a) The clockwise rotation of the fringes corresponds to the positive sign of the dislocation charge in the case of the plane reference wave (R, = co), and vice versa. In the case of equality, R(z) = Ro, the fringes are not bend, and the interference pattern will look like a
I.V. Basistiy et al. /Optics
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“star” with a number of rays equal to the modulus of the dislocation charge. The direction of the fringes rotation will change with the deviation of R,, from R(z) to one or another side. So, for a correct determination of the dislocation charge it is necessary to know the relation between the divergencies of the beams. The clockwise fringe rotation indicates a positive dislocation charge for the plane reference wave. To detect the presence of wavefront dislocations, offaxis interference can also be used. Let the wave with screw dislocation propagate along the z-axis, and the reference plane wave is tilted by an angle 8 to the zaxis with wavevector projections k sin 8 on the polar axis and k cos 0 on the z-axis. A nearly periodic intensity distribution will occur on a screen, in which the positions of the intensity maxima are determined by the equation (the relative phase difference is not taken into account) kp'
cos -M-
-([ml
2R(z)
+l)arctan~
kd
8 + kp cos 4 sin 0 - 2kz sin’ 2
+(]m]+l);
=l 1
Using formula (6)) we can draw on the p, 4 plane a two-dimensional grating, assuming z = const. and kp* < 2R(z). The equation for the intensity maxima is simplified to -m++kp
+l)?r/2,
607
rn=l
n1=-3
Fig. 4. Structure of the interference fringe maxima, calculated according to formula (6a). for m = 1 (a, b, c ) and m = - 3 ( d, e, f) , and for different G(z) values, as indicated.
or due to the change of direction of the reference wave (sign of 19). Thus, an essential conclusion from this consideration is the rule for the correct determination of the sign of a screw-dislocation charge from the interference pattern (the direction of the fringe circulation or the “forklets’ ’ orientation). Other methods of charge determination are also possible [ 8,181. A similar approach can be used also in the interference scheme proposed recently 1191. Experimental measurements of topological charge sign and value were performed for the beams created by laser beam diffraction on computer-synthesized gratings in order to verify the results obtained. The technique of binary grating synthesis was proposed and applied by us [ 6,7] on the basis of relation (6a). Analogous experiments and analyses were also reported in
[81.
cos 4 sin 8+ G(z)
=2?rz-(]m]
I19 (1995) 604412
(6a)
where G(z) = ( I m I + 1) arctan (kpi/2z) - 2kz sin’( o/2). The central parts of the gratings, calculated form = 1 and m = - 3, are depicted in Fig. 4, for different values of G(z). In the center of the pictures, I m 1 new fringes are born or vanished, thus producing a fringe splitting in the form of the “forklet”. A “forklet” in the interference pattern will be directed upwards or downwards, depending on the sign of 8: If in the interference arrangement the reference wave has a positive projection of a wavevector on the polar axis ( 8> 0), and m is positive, the “forklet” is directed upwards. The orientation will change to the opposite when m is negative,
A laser beam diffracted by a grating and carrying a screw dislocation of charge m interfered with a coherent reference slightly divergent beam, which curvature was controlled by a system of two spherical lenses. The interferograms of copropagated beam interference are shown in Figs. Sa,b,c for three possible cases: (i) the radius of curvature of the reference wave is larger than that of the wave with dislocation, (ii) the radii are equal, the (iii) the radius of curvature of the reference beam is less than that of the wave with dislocation. The rotation of the spirals changes direction with the change of curvature of the reference beam in perfect accordance with the analysis. For the off-axis interference scheme, two “forklet” orientations are shown for two variants of the reference beam disposition (Fig. 6). It means that the scheme of interference shown in Fig. 6b
hOX
1.V. Basistiv et al. / Outics Communications 119 (1995) 604612
d Fig. 5. Interference pictures produced by a wave possessing a high-order screw dislocation m=4 and a reference wave: (a) the reference wavefront curvature is smaller than that of a screw-dislocation wavefront, the rotation of fringes is clockwise; (b) the wavefront curvatures are equalized, the fringes form a cross; (c) the reference wavefront curvature is larger than that of a screw dislocation wavefront, the direction of the fringes rotation has changed to the opposite. The corresponding schemes of interference are depicted in (d), (e), (f), where the screwdislocation wavefront is shown by the dashed line, and the reference wavefront by the solid line.
screen polar
v
--
b
C
Fig. 6. Off-axis interferograms (a. d) of a wavefront with a unity-charged screw dislocation (m = 1). and the corresponding interference schemes (b, c). A wave with screw dislocation propagates normally to the screen, the reference plane wave is tilted by an angle It 8. Interferogram (a) corresponds to the orientation (c), where the reference wave-vector projection on the polar axis is negative, and the “forklet” is directed downwards; interferogram (d) corresponds to the positive projection, and the “forklet” is directed upwards.
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permits in a most simple way to find the absolute helicity of a screw-dislocation wavefront.
4. Edge and mixed screw-edge
dislocations
As mentioned earlier, a trivial example of edge dislocation is a TEMel mode of a laser cavity, expressed in the form E(p,@)=
Esin4exp(-
$)exp(-
g)
(7)
((Y is the beam halfwidth, R is the wavefront radius of the curvature). The horizontal line dividing the beam into two parts corresponds to a zero value of the amplitude and phase hop on r. As it is known, a combination of the TEMe, and orthogonal TEMia mode with a necessary phase shift by + 7~12 produces a “doughnut” circular TEM$ mode:
609
location by diffraction of a plane wave on a synthesized grating. The off-axis interference will show a very simple picture of two periodic gratings shifted by half a period on a line of zero amplitude. SimpIified gratings were calculated without taking into account the wave curvature and the variation of fringes contrast over a grating (Fig. $a). For the mixed edge-screw dislocation, we need the existence of a zero-intensity line crossing a beam with starting point at the beam center. The complex amplitude distribution E( p, 4) may be expressed in this case as
which may be decomposed
into a sum of two waves,
2
E(p,&)
Xexp
=
i
(cos
&*i
sin 4) exp
( 1
IeM-W)-11
- $
Pa)
One of the equiphase lines, determined by (7), is shown in Fig. 7a. Let the phase @be zero on this line, thus we obtain a curve Im(E) = 0. We may draw a curve Re(E) = 0 which is an equiphase line corresponding to the relative phase @= ~12 in a similar way. On the horizontal axis, where the amplitude equals zero, both curves overlap. Following our technique, we can calculate an interference light distribution on a screen taking a plane reference wave and then restore a wave with edge dis-
The expression (8) due to the multiplier sin(4/2) possesses a line of zero amplitude corresponding to 4 = 0. The azimuthal dependence of the phase is given by a factor exp( - i4/2), which may be treated as a topological charge l/2 located at the point p = 0. Equiphase lines produce broken spirals, as shown in Fig. 7b. On the zero-amplitude line the conditions Re( E) = 0 and Im( E) = 0 are satisfied simultaneously. On the other hand, relation (8a) explains more clearly the structure of a mixed screw-edge dislocation.
+=0 Rc(E} = Im(E)=O
Fig. 7. Localization of the lines Re( F) = 0 (weak line) and Im( E) = 0 (solid line) for (a) a wavefront with edge dislocation a mixed screw-edge dislocation.
and
(b) one
with
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Fig. 8. Synthesized binary gratings for the creation of diffracted beams with a cut in the wavefront: (a) for pure edge dislocation, screw-edge dislocation. The corresponding near-field images of the diffracted beams are in photographs (b) and (d).
The first part in (pa) is an ordinary wave with a unitycharge screw dislocation, and the other part is a wave without azimuthal phase dependence, but getting zero when p = 0:
E(p) = “xp(-
$)exp(--z),
2a
This is a super-Gaussian beam [ 201, and in a far-field zone it will transform into a wave having the form of a Hankel transform of a radial function (8b) : E(v)=
zJp2exp(-
$)exp(-
g)
0 xJo(2rrvp)
dp,
(8~)
where Y is the spatial frequency, Jo is the Bessel function of zero order. Relation (8~) describes a smooth amplitude distribution with a central maximum. The total far-field picture produced by the amplitude function (8a) will display an interference between the “doughnut” wave carrying screw dislocation, and the background coherent vortices-free wave. The presence
(c) for mixed
of the background causes a shift in the position of the vortex location from the center of the beam [ 151, and its rotation on a definite azimuth angle. To create a wave with a mixed screw-edge dislocation, we used the following description for the binary amplitude grating: T( p, 4) =
0,
cos( - 4/2 + Kp cos 4) < 0,
1,
cos( -4/2+Kpcos
(9)
+)>O,
where T( p, 4) is grating transmittance, K = 27r/A, A is the grating spacing. On the line I$= 0 the phase experiences hop by r (here the amplitude gets zero). This means that a wavefront of a wave diffracted by this grating has a cut from the center to the periphery along the line C#J = 0, and the point p = 0 is the starting point of the limited edge dislocation. Conventionally, a charge f l/2 may be attributed to the point p = 0. At the first order of diffraction, an object wave is restored which is a combination, as described above, of a doughnut wave and a coherent background wave. The phase is doubled at the second order of diffraction,
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119 (1995) 60442
611
and thus the azimuthal phase dependence is the same as for the pure doughnut wave. Typical gratings and near-field pictures of diffracted beams are exhibited in Figs. 8a-d. At the first order of diffraction, we observed for the grating in Fig. 8a two m-shifted bright spots divided by a black line (Fig. 8b) and a half-cut in a transverse section of a diffracted beam for the grating in Fig. 8c (Fig. 8d). In accordance with a theoretical prediction, at the second order of diffraction by a grating shown in Fig. 8a we observed an ordinary vortices-free wave and a unity-charged pure screw dislocation for a grating shown in Fig. 8c: this confirms the existence of charge 112 in the firstorder diffracted beam. The beam having a half-cut of a wavefront in a near-field zone (Fig. 9a) changes on its way, the phase cut tends to transform into several pairs of opposite-sign screw dislocations which disappear on the beam periphery, and finally in a far-field zone we observe the only presence of a single unity-charged screw dislocation (Figs. 9b, c) . It means that a mixed screw-edge dislocation does not propagate as a selfsimilar stable object, but transforms into a combination of a single screw dislocation and a background dislocation-free wave.
5. Conclusions
Fig. 9. Interferogram of a beam in the near-field picture (a) corresponding to Fig. Sd, and the far-field picture and interferogram of the same beam (b, c)
Summarizing, we have established important rules for the determination of a screw-dislocation charge on the basis of the introduced definition of the sign of screw-dislocation topological charge: the sign is positive if a helical wavefront surface produces a right screw in space. This definition does not depend on the choice of the coordinate systems and in this sense is absolute. In addition to the analysis and experiments with screw dislocations of different charges, we started (for the first time to our knowledge) to study edge dislocations (having zero charge) and mixed screw-edge dislocations, as structures having charge f l/2 at the end point of a cut of the wavefront. Our analysis and first experimental results have shown that the mixed edge-screw dislocation is not stable and decays in space producing in the near field several pairs of oppositely charged screw dislocations of unity charge and a single unity-charge screw dislocation in a far field.
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Acknowledgements We thank I. Marienko for assistance in the gratings preparation. This work was supported by the International Science Foundation, Grant U61000 and the Committee of Science and Technologies of Ukraine.
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