The electric and magnetic polarization singularities of paraxial waves

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1 INSTITUT OF PHYSICS PUBLISHING JOURNAL OF OPTICS A: PUR AND APPLID OPTICS J. Opt. A: Pure Appl. Opt. 6 ( PII: S ( The electric and magnetic polarization singularities of paraial waves MVBerry HHWills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK Received 3 July 23, accepted for publication 4 September 23 Published 14 April 24 Online at stacks.iop.org/jopta/6/475 DOI: 1.188/ /6/5/3 Abstract For superpositions of electromagnetic plane waves in space whose directions span a small angular range θ,thecircular polarization line singularities (of the full and transverse electric and magnetic fields form clusters, each of four closely-spaced lines. The separations of lines in each cluster are calculated analytically in terms of the transverse electric field and are of the order of λθ much smallerthanthe wavelength λ andsmaller still than the transverse scale λ/θ of the intensity variations of the field. The electric and magnetic surfaces of transverse linear polarization also lie close together and the separation is calculated analytically, as are the positions on these surfaces of the lines of linear polarization of the full fields. To the two lowest orders in θ, thelocal wavevector (geometric phase 1-form is the same for the electric and magnetic fields. The sub-wavelength singular structures are illustrated by numerical calculations. Keywords: polarization, singularities, electromagnetic, paraial 1. Introduction In a seminal paper, Nye and Hajnal [1] discovered that the polarization singularities for typical monochromatic electromagnetic fields in three dimensions are C lines, on which the polarization is purely circular, and L lines, on whichthe polarization is purely linear. In general the singular lines for the electric field (C and L, say and for the magnetic field (C B and L B,sayaredifferent. This picture was confirmed in detail by eperiments with microwaves [2]. For transverse fields, that is two-dimensional fields, or threedimensional fields where one component is neglected, the circular polarization lines (now called c and c B aredifferent from the C and C B lines [3] and the linear polarization singularities are surfaces (l and l B [4];these phenomena were also observed [5, 6]. My main purpose here is to eplore the interplay of these different situations, as it occurs in paraial optics (section 2. In this limiting regime, the plane waves into which freely propagating fields can be decomposed span a small range θ of directions, centred on the positive z direction e z. Then the longitudinal fields z, B z are small in comparison with the transverse fields t, B t. In the approimation of purely transverse fields, in which the z components are neglected, it follows from Mawell s equations that t B t =, and this orthogonality implies that the c and c B lines coincide, as do the l and l B surfaces. When the effects of the z components are included to the lowest order in θ, t and B t are nolonger perpendicular, and the c and c B lines separate and become distinct from the C and C B lines resulting from inclusion of the longitudinal fields (section 3. In addition, the l and l B surfaces separate and, when the longitudinal fields are included, condense onto the L and L B lines (section 4. In the paraial approimation, these separation and condensation phenomena can all be described analytically in terms of the transverse electric field t and its transverse derivatives(thealternative choice B t would give an equivalent representation. In section 5, the local wavevector, uniquely defined [1, 7, 8] to be nonsingular at the polarization singularities, is studied in the paraial approimation. The result is that, although in general the wavevectors are different for the electric and magnetic fields, in the lowest two orders in θ they are the same for the two fields, thereby defining a nontrivial natural position-dependent paraial propagation direction for the whole electromagnetic field. Section 6 illustrates the singularity phenomena with numerical calculations. Let the full electric and magnetic fields in free space, written in terms of the three-dimensional comple fields /4/5475+7$3. 24 IOP Publishing Ltd Printed in the UK 475

2 MVBerry and B,be Re[(r, z ep{i(kz ωt], Re[B(r, z ep{i(kz ωt], r ={, y. It is convenient to separate the transverse and longitudinal fields, that is = { t, z, B = {B t, B z, with t = { 1, y = 2 { L + R, i( L R B t = { B, B y = 1 2 {B L + B R, i(b L B R, where the left- and right-handed circular components of the fields have been introduced (with the convention that {, y ={1, i/ 2isright-handed. The C and c lines are loci where the full and transverse field vectors are nilpotent, that is [8 1] = (C lines, B B = (C B lines, t t = 2 L R = (c lines, B t B t = 2B L B R = (c B lines. These singularities are lines, because the vanishing of a comple quantity, e.g. implies two conditions. The L and l singularities are loci where the real and imaginary parts of the fieldvectors are parallel, that is Im = (L lines; Im B B = (L B lines Im t t = (l surfaces; Im B t B t = (l B surfaces. As can easily be seen, the L conditions are equivalent to requiring the parallelism of two real 3-vectors, e.g. Re and Im ;this corresponds to two conditions, so the L singularities are lines in space. The l singularities are equivalent to requiring the parallelism of two real 2-vectors, corresponding to one condition, so the l singularities are surfaces in space. In their original papers, Nye [3, 4] and Nye and Hajnal [1] use slightly different notation for the singularities: C T and L T (T for true for what are here called C and L lines, and C lines and S surfaces for what are here called c lines and l surfaces. 2. Paraial approimation This has been derived many times; it is summarized here in order to make the representation self-contained and to establish notation. We work with the comple fields defined in (1, with the z direction conveniently chosen, for eample as the direction of zero transverse momentum, that is, vanishing integral of the Poynting vector Re B over the r planewith z fied. Paraiality means that the z variation of the comple fields and B is slow in comparison with the wavelength λ = 2π/k = 2πc/ω,sothefactors involving k in (1 dominate the z derivatives (i.e.formally, k is a large parameter. (1 (2 (3 (4 Paraiality enables the longitudinal components of the fields to be approimated in terms of the transverse components, using the Mawell divergence equation: z i k t t, B z i k t B t, { where t =,. (5 y Manipulation of the two Mawell curl equations, incorporating the longitudinal fields (5 (or one of the curl equations, together with the vector paraial wave equation, now leads to ( B B t = B y ( {( 1 y k 2 2 y 2 y 2 2 y {( c k 2 2 y y. (6 y In the lowest paraial approimation, where the terms in 1/k 2 are neglected, the transverse fields t and B t are perpendicular, that is B =. Amorecompact form of (6 can be written in terms of the circular polarization components defined in (2 and comple coordinates ζ = +iy, ζ = iy. (7 Ashort calculation gives B L i ( L 2k 2 c 2 ζ 2 R B R i ( R 2k 2 c 2 ζ 2 L,. Finally, the longitudinal component B z in (5 can be epressed in terms of the transverse electric field t ;tolowest order in k B z i ( kc y y = i kc e z t t. (9 Allcomponents of both fields and B have now been epressed in terms of the transverse electric field t. 3. Circular polarization (c and C singularities Consider first the transverse fields. For given z, the c singularities are points in the r plane. The condition for a c point (third equation in (3 can be written L = (right-handed c point, (1 R = (left-handed c point. Similarly, a right-(left-handed c B point corresponds to the vanishing of B L (B R. The magnetic transverse components are given by (8. If the contributions in 1/k 2 are ignored, the conditions for c and c B are the same, so the electric and magnetic c points coincide. With the corrections included, the points separate, in a way now to be calculated. The c B points satisfy the fourth equation in (3, namely, to lowest order in k 2B L B R 2 [ c 2 L R 2 ( 2 k 2 L ζ 2 ] 2 L + R ζ 2 R =. (11 (8 476

3 Paraial polarization singularities y Figure 1. nergy density + c 2 B B for the field (36 (38 with b a =.3, b b =.7, b c =.3andθ = 7.5 ( and y in wavelength units. Consider a right-handed c point at r c,thatis L (r c =, and let the corresponding c B point lie at r cb = r c + δ cb. (12 The shift δ cb can be found by perturbation theory, starting from (11, as the solution of δ cb t L (r c = 2 2 k 2 ζ R(r 2 c. (13 The observation that δ cb is a real vector leads to δ cb = 4Im[ ] e z t L 2 ζ 2 R k 2 Im[e z t L t L ], (14 where all quantities are evaluated at r c.foraleft-handed c B point, L and R are interchanged and ζ is replaced by ζ. It is instructive to estimate the magnitude of this splitting of the c and c B points. In a paraial field whose plane waves span asmall angular range θ, thescale of gross features of the field pattern (e.g. the maima of the field energy density + c 2 B B isλ/θ; seefigure 1, to be discussed later. Then, using / θ/λ,etc,itfollows that the longitudinal field components (5 and (9 are smaller than the transverse components by a factor of the order of θ, andforthe1/k 2 corrections in (6thefactor is of the order of θ 2,sotheparaial epressions (formally, large k indeed correspond to small θ. Incorporating this observation into the transverse derivatives in (14 gives δcb λθ. (15 This is smaller than the paraial transverse scale by a factor θ 2 and represents sub-wavelength geometrical structure in the field, as is familiar in singular optics (see, e.g., [11]. All subsequent shifts will have the same order of magnitude. Consider now the C singularities of the full threedimensional field, starting with the electric C points (again in the r plane with given z. From the first equation in (3, and using (5, these satisfy = t t + z 2 2 L R ( t t 2 =. (16 k 2 λ Supposing the C point to be shifted from the c point r c to r C = r c + δ C (17 (see (12, perturbation theory gives the shift as δ C = Im[e z t L ( t t 2 ] k 2 Im[e z t L t L ]. (18 This is for a right-handed C point; for a left-handed point, L is replaced by R. The magnetic C B points satisfy the second equation in (3, which with (8 and (9 can be written as B B 2 [ c 2 L R 2 ( 2 k 2 L ζ 2 2 L + R ζ 2 R (e z t t 2 ] =. (19 Assuming this to be shifted from the electric c point to r CB = r c + δ CB, (2 perturbation theory gives, for a right-handed C B point, δ CB = δ cb + Im[e z t L (e z t t 2 ] k 2 Im[e z t L ; (21 t L ] for a left-handed C B point, L is replaced by R. All four of the circular polarization points c,c B,C and C B form a paraially close cluster, in the sense that all their separations are of the order of λθ and all points in a cluster have the same handedness. 4. Linear polarization (l and L singularities Consider first the transverse fields. For given z, the l singularities are lines in the r plane. The condition for a l line (third equation in (4 can be written as Im e z t t = 2Im y = R 2 L 2 =. (22 The l B line, slightly displaced from this, satisfies the fourth equation in (4 and can be written R 2 L 2 4 ( k Re 2 2 R ζ 2 L L 2 ζ 2 R. (23 To calculate the displacement, we again use perturbation theory, starting from a point r l on the l line. Let δ lb be the perpendicular displacement of l B from l,inthedirection of t ( R 2 L 2.Then(23 gives δ lb = 4Re( R 2 ζ 2 L L 2 ζ 2 R. (24 k 2 t ( R 2 L 2 For the full three-dimensional fields, the L singularities, for given z, arepoints in the r plane. The L points lie on the line l (equation (22 determined by the z component of the first equation in (4; their location on this line is fied by the and y components of this vector equation. After using (5, this leads to Re( t t t =. (25 477

4 MVBerry The L B points lie on the l B line, which to leading order coincides with the l line. The further restriction, locating the points on this line, is the vanishing of the and y components of the second equality in (4, that is, using (9, Re( t e z t t =. (26 Because (25 and (26 do not involve k, paraiality does not imply that the L and L B points lie close to one another. It might seem that one of the two components of either of the transverse vector equations (25 and (26 is redundant, being implied by the other component, together with the z component (23, since these originate in the first two equations in (4, equivalent to the parallelism of two 3-vectors. But this isnot always so; sometimes, the simultaneous vanishing of all three components is required to fi the L points, as eplained in the appendi to [1]. 5. Local wavevectors With any wave in three dimensions that is not simply plane or spherical, the question arises of specifying a local propagation vector k(r, z. Some apparently simple choices [1, 7] are unsatisfactory because they break down on L or C lines. A choice that does not suffer from this defect is an adaptation[7, 12] of the Pancharatnam connection underlying the geometric phase 1-form [13 15]. For the electric field, the wavevector is k = Im[e ep( ikz (ep(ikze] = ke z +Im[e e], (27 where the scalar product connects e and e, and (r, z e (r, z = (28 (r, z (r, z is the locally normalized comple electric field in three dimensions. k is geometric in the sense that it depends only on this locally normalized field, and nonintegrable in the sense that k. The wavevector k B for the magnetic field is defined analogously. Of principal interest is the transverse component of k, namely k t = Im e t e. (29 To approimate this eact epression paraially, it is convenient to separate the transverse and longitudinal components and introduce the normalized transverse electric field: t e t =. (3 t t Ashort calculation, making use of (5 to eliminate z,gives, to order 1/k 2, where k t Im[(1 µ 2 e t t e t + µ t µ], (31 µ = t k t t = 1 k ( e t + 12 e t log t t. (32 Asimilar calculation for k B,using(6and(9, gives the same result if terms of the order of 1/k 2 (the terms involving µ areneglected. Thus k k B ke z +Ime t t e t. (33 The direction of this vector defines a natural local propagation direction for any paraial electromagnetic field. Like the eact epression (27, (33 is non-singular everywhere. An attractive feature of (33 is that the transverse component k t inherits the geometric and nonintegrable features of the full threedimensional vector k. In terms of the spread θ of the component plane waves, the transverse part k t is of the order of kθ, sothelocal direction of k deviates from the z direction by angles of the order of θ, asitshould. If terms of the order of 1/k 2 are included, k and k B differ and their directions differ by angles of the order of θ 3.Moreover, the geometric character is lost because of the appearance of the field intensity t t in (31 and (32 (and in the corresponding formulae for k Bt. 6. Numerical illustration We will eplore a slight generalization of a field studied eperimentally by Hajnal [2], consisting nominally of three same-handed elliptically polarized plane waves (a, b, c making the same angle θ with the z ais and symmetrically arrayed around it, that is with wavevectors k i = k{sin θ cos φ i, sin θ sin φ i, cos θ, i = {a, b, c φ a = 1 6 π, φ b = 5 6 π, φ c = 1 2 π. (34 The total electric field is (r, z = a + b + c, (35 where { ( a = ep ik z cos θ sin θ + 2 y sin θ {b a e 1a +ie 2a, { ( b = ep ik z cos θ sin θ + 2 y sin θ {b b e 1b +ie 2b, c = ep{ik(z cos θ y sin θ{e 1c +ib c e 2c. (36 The degree of polarization of the waves is specified by the ellipticities (1 bi 2,with ellipse aes determined by the orthogonal directions (perpendicular to k i e 1i = e cos θ e z sin θ cos φ i, cos2 θ +sin 2 θ cos 2 φ i e 2i = e sin 2 θ sin φ i cos φ i + e y ( cos 2 θ +sin 2 θ cos 2 φ i cos2 θ +sin 2 θ cos 2 φ i e z cos θ sin θ sin φ i. cos2 θ +sin 2 θ cos 2 φ i (37 For the fields a and b,thelong aes of the polarization ellipses, projected onto the r plane, are parallel to the ais; 478

5 Paraial polarization singularities -.36 c C -1 y λ Figure 2. Zero contours of real and imaginary parts of Q = ( t t (B t B t ( (B B, forthefieldoffigure 1, over a magnified range. Intersections mark the zeros of Q,whichare the circular polarization singularities of the field; 16 lines enter the apparent singularities, indicating that each is an unresolved cluster of four elementary singularities. for c,thelong ais is parallel to the y ais. In Hajnal s eperiment [2], the nominal values of the parameters were θ = 15, b a = b b = b c =.8 (i.e. eccentricity.6. With the convention in(2 (opposite to Hajnal s, all polarizations are right-handed for b i >. The corresponding magnetic field is B = B a + B b + B c, where B i = k i i. (38 ck Since for all field components the z dependence consists of the same phase factor, all polarization singularities are independent of z (lines c, C and L, and surfaces l; therefore it suffices to study these in the r plane (points c, C and L, and lines l. In numerical computations, we will work with the eact fields, even forsmallθ. Toevaluatethe theoretical epressions for the separations of the singularities, derived in sections 3 and 4, it suffices to use the small-θ limiting form for the transverse electric field, ignoring the irrelevant z dependence, namely { ( 1 3 t,paraial (r = ep ikθ 2 y + 2 {b a, i { ( 1 3 +ep ikθ 2 y 2 {b b, i +ep{ ikθ y{i, b c. (39 Figure 1 shows the power distribution in an electromagnetic field of the form (36 (38, with three different polarization ellipses. The variations are roughly on the epected paraial scale λ/θ 7.6andthereisnohintofthefine-scale singularity phenomena being eplored here. Figure 2 shows a three-fold linear magnification, plotting instead the zeros of the real and imaginary parts of the product Q of all four comple quantities in (3. At each of the separate c and C singularities, the Re Q and Im Q contours cross, so four lines should be attached to each singularity. In figure 2, however, 16 lines appear attached to the singularities, suggesting that each is an unresolved cluster of four. y -.38 λ/1 c B Figure 3. Magnification of the top left cluster of figure 3, revealing the four c and C singularities..5 y λ C B Figure 4. Linear polarization singularities for the field of figure 1 but with θ = 15,showing l lines (full curves, l B lines (broken curves, L points (triangles and L B points (squares; full symbols are numerical calculations and open symbols are the theoretical singularities, calculated from the theory of section 4. Table 1. Displacements of circular polarization singularities in figure 3 from the c point, for the fields (29 (31 with b a =.3, b b =.7, b c =.3andθ = 7.5,compared with the theory given in section 3. Numerical/ Numerical Theory theory δ cb {.499,.222 {.5,.221 {.998, 1.2 δ C {.132,.3 73 {.134,.3 66 {.986, 1.19 δ CB {.547,.38 {.539,.299 {1.15, 1.35 The further 1-fold magnification in figure 3 shows the resolution of one of these clusters into the four types of circular polarization singularity described in section 3, identified individually by plotting the four quantities t t, etc, separately. The separations of the singularities are smaller than would be epected on the basis of the rough estimate λθ.13, but comparable with λθ/2π.2. Table 1 shows a comparison of the separations measured from figure 3 with the theoretical formulae (14, (18 and (21, confirming the accuracy of the paraial theory. The corresponding linear polarization singularities are shown in figure 4. Note that the l and l B lines are close through most of their length, as predicted by (24, but have different topologies. This reflects the presence of a saddle in the quantity R 2 L 2 whose gradient appears in the 479

6 MVBerry denominator of (24, indicating a region where the simple theory fails and the separation of l and l B is larger than O(1/k 2. As epected, the L points lie on the l lines and the L B points lie on the l B lines. vidently, the positions of these points are determined accurately by the theoretical equations (22, (25 and (26. To study the approach to paraiality, we now consider the case where b a = b b = b c = b in (36, so all three polarization ellipses have the same eccentricity. The circular components of the paraial field (39 then take the simple form (1 b L = ep( ikyθ 2 [ ( 3 1 2cos 2 kθ ( ] 3 ep 2 ikyθ R = (1+b ep( ikyθ 2 [ ( ( ] cos 2 kθ ep 2 ikyθ. (4 Apart from a phase, these components are periodic, so the singularities form lattices. The circular polarization singularities c,c B,C and C B form clusters close to the zeros of L and R,forwhichwe choose as eamples { = { = 2π kθ3 3, y = 4π kθ3 3, y = ( L =, i.e. right-handed ( R =, i.e. left-handed. (41 The paraial displacements of the c B,C and C B points from the c point can now be calculated analytically from the formulae in section 3, leading to 2θ(1+b 3k(1 b e (right-handed δ cb = 2θ(1 b 3k(1+b e (left-handed θ(1+b 12k(1 b e (right-handed δ C = (42 θ(1 b 12k(1+b e (left-handed δ cb θ(1+b 12k(1 b e (right-handed δ CB = θ(1 b δ cb + 12k(1+b e (left-handed. Several conclusions follow from these results. First, the four singularities in each cluster lie on a straight line (parallel to e,; this feature, specific to the case where all b i are the same, appears in some of the singularities observed eperimentally, as can be seen by superposing figures 2 and 4 of [2]. Second, the separations of singularities in left-handed clusters are smaller than those in right-handed clusters by a factor (1 b 2 /(1 +b 2,whichforb =.8 is.12. This seems to be reflect the non-rigorous distinction made by Hajnal between strong and weak c points: strong c points are those δ cb δ C (a.4 (b (c δ cb δ CB Figure 5. Separations of circular polarization singularities, for b a = b b = b c = b =.8, calculated from the eact fields (36 (38 (dots and from the paraial theory (42 (full lines, (a separation of c B and c,(bseparation of C and c,(cseparation of c B and C B. The upper limit of θ is 21. with the opposite hand to the three waves comprising the field and are less vulnerable to paraial perturbation than the weak c points, which have the same hand as the waves in the field; the distinction is not a general one but reflects the fact that all waves in the beam have the same hand. Third, because of the numerical factors in (42, the separations between the two singularities (c and C, and between the two B singularities (c B and C B, are much smaller (up to 1 times than the separations between the two c singularities and the two C singularities. Figure 5 shows the convergence of the separations between the right-handed ( weak circular polarization singularities to those predicted by the paraial theory (42 as θ becomes smaller. The range over which the paraial theory is accurate is much larger for the two c singularities (figure 5(a than for the other separations. Acknowledgments Ithank Dr M R Dennis and Professors J H Hannay and J F Nye for helpful conversations. My research is supported by the Royal Society. θ θ θ 48

7 References [1] Nye J F and Hajnal J V 1987 The wave structure of monochromatic electromagnetic radiation Proc. R. Soc. A [2] Hajnal J V 199 Observation of singularities in the electric and magnetic fields of freely propagating microwaves Proc. R. Soc. A [3] Nye J F 1983 Lines of circular polarization in electromagnetic wave fields Proc. R. Soc. A [4] Nye J F 1983 Polarization effects in the diffraction of electromagnetic waves: the role of disclinations Proc. R. Soc. A [5] Hajnal J V 1987 Singularities of the transverse fields of electromagnetic waves. I. Theory Proc. R. Soc. A [6] Hajnal J V 1987 Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field Proc. R. Soc. A [7] Nye J F 1991 Phase gradient and crystal-like geometry in electromagnetic and elastic wavefields Sir Charles Frank, O.B..; anightieth Birthday Tribute ed R G Chambers, Paraial polarization singularities Jnderby, A Keller, A R Lang and J W Steeds (Bristol: Adam-Hilger pp [8] Berry M V and Dennis M R 21 Polarization singularities in isotropic random waves Proc. R. Soc. A [9] Berry M V 21 Geometry of phase and polarization singularities Singular Optics 2 vol 443, ed M Soskin (Alushta, Crimea: SPI pp 1 12 [1] Dennis M R 22 Polarization singularities in paraial vector fields: morphology and statistics Opt. Commun [11] Berry M V 1998 Wave dislocations in nonparaial Gaussian beams J. Mod. Opt [12] Nye J F 1999 Natural Focusingand Fine Structureof Light: Caustics and Wave Dislocations (Bristol: Institute of Physics Publishing [13] Berry M V 1984 Quantal phase factors accompanying adiabatic changes Proc. R. Soc. A [14] Berry M V 1987 The adiabatic phase and Pancharatnam s phase for polarized light J. Mod. Opt [15] Series G W 1975 The Collected Works of S Pancharatnam (Oford: Oford University Press 481