Distribution of angular momentum and vortex morphology in optical beams

Distribution of angular momentum and vortex morphology in optical beams

Optics Communications 242 (2004) 45–55 www.elsevier.com/locate/optcom Distribution of angular momentum and vortex morphology in optical beams Filippu...

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Optics Communications 242 (2004) 45–55 www.elsevier.com/locate/optcom

Distribution of angular momentum and vortex morphology in optical beams Filippus S. Roux

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School of Information Technology and Engineering, University of Ottawa, 800 King Edward Ave., Ottawa, Ont., Canada K1N 6N5 Received 25 May 2004; received in revised form 27 July 2004; accepted 4 August 2004

Abstract The amount of orbital angular momentum associated with an optical vortex depends on the helicity of the optical vortex, which forms part of the vortex morphology. It is shown that one can define a nontrivial morphology distribution for an optical beam, which, together with the distribution of the state of polarization, determines how the angular momentum is distributed over the cross-section of the beam. As an example, the morphology distribution of a Gaussian beam with an off-axis vortex is considered. Both the polarization distribution and the morphology distribution can be represented in terms of spinor fields. An expression is provided for the angular momentum distribution in terms of these spinor fields. It helps to reveal the relationship among the various spin representations. An interesting and potentially useful property of the morphology distribution is that one can in some cases associate a nontrivial monopole winding number with it. Ó 2004 Elsevier B.V. All rights reserved. PACS: 42.25.Bs; 42.25.Ja; 42.90.+m; 02.40.Re Keywords: Optical vortex morphology distribution; Orbital and spin angular momentum; Monopole winding number; Monopole index integral

1. Introduction Many of the challenges in the trend to make devices smaller have been resolved with optical tech-

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Tel.: +1 613 562 5800; fax: +1 613 562 5175. E-mail address: [email protected].

nology. The ability to trap and manipulate small particles with beams of light has led to breakthroughs in science and technology. Recently this optical capability has been expanded to the application of torque to small particles [1]. It now becomes feasible to use light to drive microscopic mechanical devices. To mature the capability of light to apply torque one needs to have a thorough understanding

0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.08.006

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F.S. Roux / Optics Communications 242 (2004) 45–55

of the way that light conveys angular momentum. Significant progress has already been made in this understanding. It is known that, while the polarization of light is associated with spin angular momentum, optical vortices [2] are associated with orbital angular momentum (OAM) – each photon in a rotationally symmetrical vortex bearing beam carries an OAM proportional to the topological charge of the optical vortex [3]. Bekshaev et al. [4] distinguished between ÔvortexÕ OAM and ÔasymmetricÕ OAM in specific types of asymmetrical beams, showing that OAM is not only associated with the vortices. OÕNeil et al. [5] distinguished between ÔintrinsicÕ and ÔextrinsicÕ OAM based on the axis of rotation, showing how OAM is distributed in Laguerre–Gaussian beams. Soskin et al. provided expressions for the distribution of the total angular momentum in specific combinations of vortexless and on-axis vortex bearing Gaussian beams [6], showing how the number of vortices depend on certain parameters of these beams. In this paper, we investigate how angular momentum is distributed over the cross-section of arbitrary beams, with specific attention given to the distribution of intrinsic OAM. The distribution of spin angular momentum follows directly from the distribution of the state of polarization over the cross-section of the beam. However, the distribution of OAM is more complicated because optical vortices are localized phenomena. Although a canonical (isotropic) optical vortex carries the above-mentioned fixed amount of OAM, a noncanonical (anisotropic) optical vortex carries a reduced amount of OAM. One can consider a noncanonical vortex as a linear combination of two canonical vortices with opposite topological charges, which partially cancel each other to leave a diminished amount of OAM for the combined noncanonical vortex. Hence, to understand how OAM is distributed through the beam one needs to consider vortex morphology. It turns out that one can define a morphology for every point in the beam irrespective of whether there is a vortex located at that point or not. In this paper, we investigate this morphology distribution and find that it determines the distribution of OAM. Knowledge of the morphology distribu-

Fig. 1. Torque with opposite directions applied to opposite ends of a small elongated object to cause a bending action.

tion is therefore of practical importance in the design of systems that are to transfer angular momentum to small objects, as effectively as possible. It is particularly important where different amounts (or even different directions) of rotation are required at different locations in a particular beam. This would for instance be the case when a bending action is to be applied to a small elongated object, as portrayed in Fig. 1. The morphology distribution also conveys other interesting information, such as the locations of degeneracy lines where all annihilations and revivals occur. Degeneracy lines represent points in the morphology distributions where vortices become edge dislocations. While the formalism for the state of polarization in terms of Jones vectors is widely known, the equivalent formalism for the morphology of optical vortices [7–9] is not. We shall therefore briefly review the Jones vector formalism in Section 2 and then in Section 3, by analogy, show that a similar formalism can be used for the morphology of optical vortices. The morphology distribution of an optical beam is then discussed in Section 4. For illustration, an analytical example of a Gaussian beam with a single off-axis vortex is considered. In Section 5, the angular momentum distribution is derived and expressed in terms of the polarization and morphology distributions. Some topological properties of morphology distributions are considered in Section 6 with the aid of a monopole index integral for spinor distributions, which is derived in Appendix A. A summary and conclusions are provided in Section 7. It will be assumed that all beams considered here are paraxial, coherent, fully polarized and, for definiteness, are propagating in the z-direction.

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2. Jones vectors Using the right-hand and left-hand circular polarization states, 1 1 ^ nR ¼ pffiffiffi ð^x  i^y Þ 2

and

1 ^ nL ¼ pffiffiffi ð^x þ i^y Þ; 2

ð1Þ

as polarization bases, one can define an arbitrary state of polarization as a linear combination, ^ nL , where |P|2 + |Q|2 = 1. The two n ¼ P^ nR þ Q^ coefficients, collected in a two component Jones vector, can be conveniently parameterized as     P cosðb=2Þ expðia=2Þ v¼ ¼ : ð2Þ Q sinðb=2Þ expðia=2Þ The two angles, a and b, are coordinates on the spherical surface, of the Poincare´ sphere – the configuration space of the Jones vectors. Each point on the Poincare´ sphere represents a unique state of polarization and all possible states of polarization are represented by points on the Poincare´ sphere. The ranges of these angles are 0 6 a < 2p and 0 6 b 6 p. The orientation of the state of polarization is represented by a (the inclination angle is a/2). Since b only extends to p one can represent it in terms of cos b, which ranges from 1 to 1 and represents the helicity of the state of polarization. A positive (negative) value of cos b implies a right-handed (left-handed) state of polarization. The ellipticity of the state of polarization is determined by the magnitude of the helicity, |cos b|. The connection between the state of polarization and the spin angular momentum is given by the fact that the Jones vectors are spinors (a spin-1/2 representation of the SU(2) group [10]). The connection will be made more precise in Section 5. The state of polarization can in general vary over the cross-section of an optical beam. One can therefore define a polarization distribution, v(x, y), given by a continuous mapping from any two-dimensional plane (xy-plane), perpendicular

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to the direction of propagation (z-direction), to the Poincare´ sphere. It is often convenient to express this distribution in terms of the scalar functions, cos b(x, y) and a(x, y). The latter is a phase function that can have phase singularities, which would always coincide with points where |cos b (x, y)| = 1, representing the circular polarization states.

3. Optical vortex morphology The morphology of an elementary optical vortex (one with unit topological charge) specifies its helicity and orientation in a way that is analogous to polarization states. In analogy to the polarization bases, one can use the two canonical (isotropic) elementary optical vortices, 2 1 1 mþ ¼ pffiffiffi ðx  iyÞ and m ¼ pffiffiffi ðx þ iyÞ 2 2

ð3Þ

as the morphology bases. An optical vortex with an arbitrary morphology can then be represented by a linear combination, V ¼ ^nmþ þ ^fm , where 2 2 j^nj þ j^fj ¼ 1. We collect these coefficients in a two component vector, which we call the morphology spinor and which is parameterized in a way similar to the Jones vectors [7,9] " #   ^n cosðw=2Þ expði/=2Þ ^g ¼ ¼ ; ð4Þ ^f sinðw=2Þ expði/=2Þ where the ˆ indicates that the spinor is normalized, ^gy ^g ¼ j^gj2 ¼ 1. The morphology spinors share many of the properties of the Jones vectors, including a spherical configuration space [8,9], which is here referred to as the morphology sphere, with coordinates / and w. Each point on the morphology sphere represents a unique vortex morphology and all possible elementary vortex morphologies are represented by points on the morphology sphere. Moreover, the morphology spinors form a spin-1/2 representation of the SU(2) group [10], which gives the connection be-

1

We use the definition that an observer looking in the propagation direction of a right-hand circularly polarized plane wave sees the electric field rotating clockwise as a function of time and we follow the convention that phase increases with time.

2

Here, our definition is that an observer looking in the propagation direction of a plane wave with a positive vortex sees the phase function rotating clockwise as a function of time.

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tween vortex morphology and the OAM in the optical beam, as is shown in Section 5. The ranges of the angles in Eq. (4) are 0 6 / < 2p and 0 6 w 6 p. The angle / represents the orientation of the vortex morphology (the orientation angle for an edge dislocation is given by //2) and cos w represents the helicity of the morphology. The sign of cos w represents the topological charge of the vortex and the magnitude, |cos b|, quantifies how (an)isotropic the vortex is. The components of the morphology spinor for a given optical vortex can be extracted from the complex (scalar) distribution of the beam by using the following differential operators [9]:   1 o o o ¼ pffiffiffi i ; ð5Þ oy 2 ox which have the property that o+m+ = om = 1 and o+m = om+ = 0. Operating on the complex scalar distribution over the beam cross-section, U(x, y), at the location of the vortex, these operators produce the spinor components multiplied by the background amplitude and phase (i.e., the ÔdressedÕ spinor components), oþ U ¼ n ¼ A0 expðih0 Þ^ n and o U ¼ f ¼ A0 expðih0 Þ^f;

ð6Þ

where A0 and h0 are, respectively, the background amplitude and phase. Note that although A0 is called an ÔamplitudeÕ it has the dimensions of a slope. The helicity and orientation of the morphology, as well as the background parameters, can then be obtained from the dressed spinor components by [9] cos w ¼

jnj2  jfj2

expði/Þ ¼

A0 ¼

2

2

jnj þ jfj

;

nf ; jnjjfj

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 jnj þ jfj ;

expði2h0 Þ ¼

nf ; jnjjfj

where * denotes the complex conjugate.

ð7Þ

ð8Þ

ð9Þ ð10Þ

4. Morphology distributions Although the morphology is associated with the optical vortices, which are localized in the beam, one can define a morphology distribution, similar to the polarization distribution defined in Section 2. This can be seen by noting that when a constant field is added to a vortex bearing beam the vortex shifts to a new location. One can in general produce a vortex at an arbitrary location in a beam by adding a constant field with an amplitude equal to minus the complex amplitude of the original beam at that location. The morphology of this new vortex can be obtained by applying the differential operators of Eq. (5) at that location. However, because these operators are purely differential operators, the morphology of the new vortex does not depend on the constant field that has been added. In fact one could determine the morphology of the would-be vortex at that location without having to add the constant field first. It thus follows that one can determine the morphology for all the would-be vortices at all points over the cross-section of the beam simply by applying the operators of Eq. (5) to the entire complex amplitude distribution of the beam. In general the morphology would vary from point to point. As a result, a general optical beam has a nontrivial morphology distribution, g(x,y), associated with every cross-section (xy-plane) along the propagation direction (z-direction). It can be defined as a continuous mapping from the two-dimensional cross-section to the morphology sphere. (One can combine all the two-dimensional distributions along the propagation direction into a three-dimensional distribution.) The two-dimensional morphology distribution can be conveniently represented by the scalar distributions of the helicity, cos w(x,y), and the orientation, /(x,y). As an example, we consider the morphology distribution in a Gaussian beam with a single vortex. At its waist a Gaussian beam with an off-axis canonical vortex located at (u,v) = (u0,0) can be expressed as    ð11Þ gðu; vÞ ¼ ðu  u0  ivÞ exp  u2 þ v2 ; where u(=x/x0) and v(=y/x0) represent normalized transverse coordinates, with x0 being the beam ra-

F.S. Roux / Optics Communications 242 (2004) 45–55

dius in the waist. The dressed spinor components of the morphology distribution are obtained by applying Eqs. (5), (7) and (11). The resulting spinor components are then used in Eqs. (7) and (8) to compute the helicity and orientation as a function of the transverse coordinates. The distribution of the helicity, cos w(x,y), is shown in Fig. 2 for u0 = 2. White (black) represents a helicity of +1 (1). The Ô0Õ denotes the location of the beam axis at (u,v) = (0,0) and the ÔVÕ denotes the location of the vortex at (u,v) = (u0,0). One of the first things to notice is that the distribution has a mirror symmetry with respect to the line through u = u0/2. Both points marked by Ô0Õ and ÔVÕ have helicities of +1. The dark areas next to these two points contain points with helicities of 1. There are therefore four special points where the morphology is isotropic. These four isotropic points lie on the u-axis (v = 0). The distribution of the orientation, /(x,y), for u0 = 2 is shown in Fig. 3 as a phase function where white (black) represents 0 (2p). There are four sin-

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Fig. 3. Distribution of the morphological orientation for a Gaussian beam with one off-axis canonical vortex (u0 = 2). White (black) represents 0 (2p). The four singularities, marked by ÔAÕ, ÔBÕ, ÔCÕ and ÔDÕ, coincide with the four isotropic points, in Fig. 2.

gularities on the u-axis, marked by ÔAÕ, ÔBÕ, ÔCÕ and ÔDÕ, which coincide with the four isotropic points in Fig. 2 (point-Ô0Õ and point-ÔVÕ in Fig. 2, respectively, coincide with point-ÔBÕ and point-ÔCÕ in Fig. 3). Because the orientation is undefined when the morphology is isotropic, such isotropic points will always be represented by singularities in the distribution of the orientation. For u, v ! 1, we find that cos w ! 0 – the vortices become degenerate (edge dislocations) – and the orientation of the edge dislocations are perpendicular to the direction from the beam axis. The asymptotic behavior of the morphology distribution at u,v ! 1 that we see here is generic for Gaussian beams with polynomial prefactors.

5. Angular momentum distribution Fig. 2. Distribution of the morphological helicity for a Gaussian beam with one off-axis canonical vortex (u0 = 2). The helicity ranges from 1 (black) to +1 (white). The Ô0Õ denotes the location of the beam axis (the centre of the Gaussian beam) and the ÔVÕ denotes the location of the vortex.

Here, we consider the angular momentum distribution in an optical beam as a result of nontrivial distributions of polarization and morphology. The aim is to determine how much angular

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momentum can be transferred to a small particle that absorbs a localized portion of light in the beam. We will assume that the amount of angular momentum that is transferred to such a small particle, located at x0 ¼ x0^x þ y 0 ^y , is equal to the angular momentum in the beam around x0, integrated over a small circular disk, centered at x0, that represents the cross-section of the particle. The z-component of the (complex) angular momentum at x0 is therefore given by Z ij J z ðx0 Þ ¼ ð½x  x0   ½E  ðr  EÞÞ  ^z d2 x; Ax A ð12Þ where j is a constant that depends on the medium; A represents the area of integration; x is the angular frequency; x ¼ x^x þ y^y ; E is the electric field strength; and * denotes the complex conjugate. For paraxial beams, the polarization vector, denoted by ^ nE , is perpendicular to the z-direction ð^nE  ^z ¼ 0Þ. If the area of integration is small enough one can assume that the polarization vector is constant over the area. Furthermore, the (scalar) electric field in a small enough area around the point x0 can be expressed in terms of a Taylor series expansion, where only the constant term and terms linear in the coordinates need to be retained. One can then rearrange the linear terms to express them as a linear combination of the morphology bases given in Eq. (3). Finally, without loss of generality we will set x0 = 0. So the electric field in such a small area is given by   1 1 E ¼ U^ nE ¼ U 0 þ n0 pffiffiffi ðx  iyÞ þ f0 pffiffiffi ðx þ iyÞ ^ nE ; 2 2 ð13Þ where n0 ¼ ðoþ U Þx¼0

and

f0 ¼ ðo U Þx¼0

ð14Þ

represent the dressed spinor components of the morphology distribution, and 1 i ^ nE ¼ pffiffiffi ðP þ QÞ^x  pffiffiffi ðP  QÞ^y : 2 2

ð15Þ

As a result the angular momentum density becomes

ðx  ½E  ðr  EÞÞ  ^z   ¼ U  ðx  ^nE Þ ½rU  ð^z  ^nE Þ:

ð16Þ

Now we integrate the angular momentum density in Eq. (16) over a small circular area with radius R, centered at x0( = 0). Because of the symmetry of the area of integration, unsymmetrical contributions to the angular momentum drop out. (This removes the extrinsic OAM component [5] that would give rise to a linear momentum for the small particle.) The result is Z R Z 2p ij Jz ¼ 2 ðx  ½E  ðr  EÞÞ  ^zq dh dq pR x 0 0  jR2 h 2 jn0 j  jf0 j2 jP j2 þ jQj2 ¼ 8x   þ n0 f0  f0 n0 ðPQ þ QP  Þ    jn0 j2 þ jf0 j2 jP j2  jQj2    ð17Þ  n0 f0 þ f0 n0 ðPQ  QP  Þ ; Jz ¼

jR2 A20 ½cosðwÞ  cosðbÞ 8x þ i sinðwÞ sinðbÞ sinð/  aÞ;

ð18Þ

where the last step follows from Eqs. (2) and (4). The real part of the angular momentum distribution is proportional to the algebraic sum of the morphological helicity distribution and the (negative) polarization helicity distribution, which, respectively, represent the orbital and spin angular momentum distributions. (The imaginary part has a more complicated dependence, which includes a dependence on the relative orientation angle of the morphology and the polarization.) Apart from the sign of the polarization helicity (which we discuss below), the result in Eq. (18) agrees with those that were obtained previously for Laguerre–Gaussian beams [3] and for rotationally symmetrical beams [11]. Here, it is found to be locally true in an arbitrary beam. Notice that the angular momentum distribution is still proportional to R2 in spite of the fact that it has been divided by the area of the disk. The reason is that the contribution of the angular momentum density at a distance R from the axis is proportional to R4.

F.S. Roux / Optics Communications 242 (2004) 45–55

The variation of the OAM over the cross-section of the beam is given by A20 cos w – both A20 and cos w are functions of the transverse coordinates. For a Gaussian beam, A20 is proportional to the Gaussian profile. An interesting consequence of this is that in the example of a Gaussian beam with an off-axis vortex, shown in Figs. 2 and 3, the OAM distribution has its largest values near the beam axis and not at the location of the vortex. To address the negative sign of the polarization helicity in Eq. (18), we digress to consider a circularly polarized rotationally symmetrical beam without a vortex, E ¼ f ðr; zÞð^x þ iT ^y Þ:

ð19Þ

Here T(=±1) denotes whether the polarization is right-handed orpleft-handed and the beam only deffiffiffiffiffiffiffiffiffiffiffiffiffiffi pends on r ¼ x2 þ y 2 and z. To determine the sign of the spin angular momentum it suffices to consider I z ¼ RefiE  ðr  EÞg  ^z:

ð20Þ

The result is 2

Iz ¼ T

r ojf ðr; zÞj ; 2 or

ð21Þ

which is proportional to T as expected. However, it is also proportional to the radial derivative of the magnitude squared of the beam profile. As is known, the angular momentum for a circularly polarized constant beam (plane wave) vanishes. The derivative in Eq. (21) would in most cases be negative, however, if there is an optical vortex in the centre of the beam the slope would be positive around the centre. From this, we conclude that it is not possible to make a direct association between the sign of the polarization helicity and the sign of the spin angular momentum. The definition of the helicity of the state of polarization is therefore a matter of convention, which is the reason why we are so pedantically explicit in the definitions of the polarization and morphology bases. We now derive an expression for the angular momentum distribution in terms of the spinor distributions. For this purpose we use the Pauli spin matrices. Together with the two-by-two identity matrix, these spin matrices are denoted by rl for l = 0, 1, 2, 3, where

 r0 ¼  r1 ¼  r2 ¼ 



1

0

0

1

0

1

1

0

0

i

i

0

1 r3 ¼ 0

51

;

ð22Þ

;

ð23Þ

 

 0 : 1

ð24Þ

ð25Þ

The different combinations of the polarization spinor components in Eq. (17) can then be represented by  2 2 jP j þ jQj ¼ vy r0 v ¼ 1; ð26Þ ðPQ þ QP  Þ ¼ vy r1 v;

ð27Þ

iðPQ  QP  Þ ¼ vy r2 v;  2 2 jP j  jQj ¼ vy r3 v;

ð28Þ ð29Þ

where   represents the Hermitian adjoint. In a similar way, one can also represent the different combinations of the morphology spinor components in Eq. (17) in terms of the spin matrices,    jn0 j2 þ jf0 j2 ¼ A20 ^gy r0 ^g ¼ A20 ; ð30Þ      n0 f0 þ f0 n0 ¼ A20 ^gy r1 ^g ;

ð31Þ

    i n0 f0  f0 n0 ¼ A20 ^gy r2 ^g ;

ð32Þ

   jn0 j2  jf0 j2 ¼ A20 ^gy r3 ^g :

ð33Þ

The right-hand sides of Eqs. (30)–(33) are all proportional to the square of the background amplitude, A20 , while the background phase cancels out. Substituting the matrix expressions of Eqs. (26)–(28), (30)–(33) into Eq. (17), we obtain R2 A20  y  y   y  y  ^g r3 ^g v r0 v  i ^g r2 ^g v r1 v 8x  y  y   y  y   ^g r0 ^g v r3 v þ i ^g r1 ^g v r2 v   y  R2 A20  y ^g rl ^g Clm ð34Þ ¼ z v rm v ; 8x

Jz ¼

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where we assume summation over repeated indices l and m in the last line, and where 2

0 0 0 6 60 0 i 6 Clm z ¼ 6 6 0 i 0 4

1

1

0

0

0

3

7 0 7 7 7: 0 7 5

ð35Þ

The expression for the z-component of the (complex) angular momentum distribution, provided in Eq. (34), represents an exercise in Lie group representation theory [10]. The morphology spinor, ^g, and the Jones vectors, v, represent spin-1/2 fields. The tensor product of two spin-1/2 objects can be expressed as the sum of a spin-0 and a spin-1 object. The Pauli spin matrices are the Clebsch– Gordon coefficients [10] that extract the spin-1 field from the tensor products of these spin-1/2 fields. In this way the morphology spinors combine to form the OAM distribution, which is a spin-1 field, given by ^ gy rl ^ g (up to a multiplicative constant). In similar fashion the spinors of the state of polarization (Jones vectors) combine to form the spin angular momentum distribution, also a spin-1 field, given by v rmv (up to a multiplicative constant). The tensor product of two spin-1 fields can be expressed as the sum of a spin-0, a spin-1 and a spin-2 field. In this case the matrix in Eq. (35) represents the Clebsch–Gordon coefficient that extracts the z-component of the spin-1 field. Thus the z-component of the angular momentum, in Eq. (34), is formed from the combination of two spin-1 fields, namely the spin angular momentum distribution and the OAM distribution. We need to clarify one more point. The angular momentum distributions, ^ gy r l ^ g and v  rmv, are four-dimensional vectors and are therefore representations of the Lorentz group, SO(3,1). However, angular momentum is associated with the rotation group SO(3), which is a subgroup of SO(3,1). The different spin representations that are being referred to here (spin-1/2, spin-1, etc.) are therefore those of the SU(2) group, which is the cover group of SO(3).

6. Morphology monopole index An interesting property of the morphology distribution is the fact that it can have a nontrivial ÔmonopoleÕ winding number, which is different from the winding number (or topological charge) associated with vortices. While the vortex winding number comes from the mapping of closed curves on the xy-plane onto the circular configuration space of phase values, the monopole winding number comes from the mapping of areas on the xyplane onto the spherical configuration space of morphological states. 3 If the mapping of such an area onto the morphology sphere completely covers the surface of the sphere an integer number of times, then there is a winding number equal to that integer associated with that area of the morphology distribution. The surface of the morphology sphere can be covered in either a positive or a negative orientation. If a specific mapping contains different portions that are mapped onto the surface in opposite orientations, their contributions to the winding number would (partially) cancel each other. To make these statements more precise one needs to formulate an index integral for the monopole winding number. The index integral for the vortex winding number is given by I n2p ¼ rhðx; yÞ  d^l; ð36Þ C

where n is an integer representing the vortex winding number. The tangential component of the gradient of the phase function h(x,y) is integrated along a closed contour C which encloses the vortex. The index integral for the monopole winding number in a normalized spinor distribution, ^gðx; yÞ, is given by (see Appendix A) Z  T    ^g r^g  ^gy r^g  ^z dx dy; n4p ¼ ð37Þ A

3 Formally, vortex and monopole winding numbers can be distinguished on the basis of the homotopy groups [12] (pn) that give rise to them. For vortices p1 = Z (the set of all integers) and p2 = 0, while for monopoles p1 = 0 and p2 = Z [13].

F.S. Roux / Optics Communications 242 (2004) 45–55

where A is the area over which the integral is evaluated. Although n in Eq. (37) represents the monopole winding number it is not necessarily an integer, because A is not a closed surface. A more complete analogy between the monopole winding number and the vortex winding number would exist if one evaluates the monopole index integral on a closed surface in a three-dimensional function. In such a case its result will always be an integer (the winding number) times 4p. If the integer is nonzero, the closed surface encloses one or more monopoles. Although the integral in Eq. (37), which is evaluated over an arbitrary area, does not in general give an integer times 4p, one can find some areas on the two-dimensional xyplane, for certain morphology distributions, where the monopole winding number is a nonzero integer. As an example, consider again the Gaussian beam with an off-axis canonical vortex for which the morphology distribution was considered in Section 4. We want to determine whether there is an area over which the monopole index for this distribution would be an integer. Viewing the morphology sphere from the north pole (positive isotropic morphology) one observes the orientation angle increasing in a counter-clockwise direction, while viewing it from the south pole (negative isotropic morphology) one observes the orientation angle increasing in a clockwise direction. Hence, if the mapping from the xy-plane to the morphology sphere has a positive orientation the morphological orientation angle must increase counter-clockwise around positive isotropic points and clockwise around negative isotropic points. Applying these observations to Fig. 3, one can determine that the regions in the vicinity of points B, C and D are mapped with positive orientations but the region in the vicinity of A is mapped with a negative orientation. For an area to give an integer winding number the mapping must cover the entire morphology sphere an integer number of times, and therefore must contain equal numbers of positive and negative isotropic points all with the same mapping orientation. From this we deduce that there exists an area with a winding number of n = 1, which includes points C and D in Fig. 3, but excludes points A and B – the monopole in-

53

dex integral over this area would be 4p. It is however not easy to determine the boundary of this region. Fortunately, it turns out that the positive and negative mapping orientations of the remaining part of the xy-plane exactly cancel, with the result that the monopole index integral evaluated over the entire xy-plane gives 4p. Apart from being interesting in its own right, this result may help to elucidate the behavior of the morphology distribution and, by implication, also the behavior of the OAM distribution in Gaussian beams.

7. Summary and conclusions The distribution of angular momentum (and in particular OAM) over the cross-section of an optical beam is investigated in terms of the distribution of the state of polarization and the morphology distribution. The spinor formalisms for the state of polarization in terms of Jones vectors and for the morphology of optical vortices are briefly reviewed. It is shown that the morphology can be defined for every point on the cross-section of a beam, giving rise to a morphology distribution. Considering the portion of angular momentum that can be transferred to a particle at any point on the cross-section of an arbitrary beam under paraxial approximation, one finds that the angular momentum distribution is proportional to the sum of the helicities of the polarization state distribution and the morphology distribution. The angular momentum distribution can also be expressed in terms of the two spinor distributions, which shows how the different spinor fields combine to give the spin-1 field of the angular momentum distribution. To get a better understanding of the physical significance of the morphology distribution, we provide preliminary information about a monopole winding number that can be associated with it. One can show that in some cases a nontrivial morphology monopole winding number can be associated with (portions of) the morphology distribution. As an example it is shown that the morphology distribution over the entire xyplane in the waist of a Gaussian beam with a single off-axis canonical vortex has a unit

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monopole charge (winding number = 1) associated with it.

Acknowledgement This research was supported in part by the Natural Sciences and Engineering Research Council of Canada.

Appendix A. Monopole index integral in terms of spinor fields Consider a vector field

V a ¼ gy ra g:

ðA:7Þ

When these spinor expressions are substituted into the vector components in Eq. (A.6) one can derive an expression for the monopole index integral for spinor fields, given by 4 ! I i2½ðgT rgÞ  ðgy rg Þ n4p ¼  d^s; ðA:8Þ 2 ðgy gÞ S

VðxÞ ¼ V x ðxÞ^x þ V y ðxÞ^y þ V z ðxÞ^z;

ðA:1Þ

where x ¼ x^x þ y^y þ z^z. To determine the presence of a monopole in this vector field one needs an auxiliary vector field M{V(x)} such that, I M  d^s ¼ n; ðA:2Þ S

where n is an integer, which represents the net number of enclosed monopoles, and S represents the closed surface of integration. An obvious choice for such an auxiliary vector field is 1 ðr cos h  r/Þ; 4p



Eq. (A.1) is a spin-1 field that can, with the aid of the Pauli spin matrices, ra (a = x,y,z) [given in Eqs. (23)–(25) for (x,y,z) = (1,2,3)], be expressed in terms of spin-1/2 fields (spinors). The components of the vector field are related to the spinor fields by,

ðA:3Þ

where the superscripts T, * and  , respectively, represent the transpose, the complex conjugate and the Hermitian adjoint; and where   0 1 ¼ : ðA:9Þ 1 0 Note that the denominator of Eq. (A.8) only depends on the magnitude of the spinor. Therefore, if the spinor is normalized, the index integral becomes I     n4p ¼ i2 ^gT r^g  ^gy r^g  d^s: ðA:10Þ S

where References

Vx cos h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2x þ V 2y þ V 2z and / ¼ arctan

ðA:4Þ

Vy : Vx

ðA:5Þ

From Eqs. (A.2)–(A.5), it then follows that the monopole index integral for three-dimensional vector fields is given by [14] n4p ¼

[1] See for example: M.J. Padgett, J. Courtial, L. Allen, Phys. Today (May) (2004) 35, and references therein. [2] J.F. Nye, M.V. Berry, Proc. R. Soc. Lond. A 336 (1974) 165. [3] L. Allen, M.W. Beijerbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 8185. [4] A.Y. Bekshaev, M.S. Soskin, M.V. Vasnetsov, J. Opt. Soc. Am. A 20 (2003) 1635. [5] A.T. OÕNeil, I. MacVicar, L. Allen, M.J. Padgett, Phys. Rev. Lett. 88 (2002) 053601.

# I " V x rV y  rV z þ V y rV z  rV x þ V z rV x  rV y

 d^s:

ðV 2x þ V 2y þ V 2z Þ

3=2

ðA:6Þ

To find the equivalent expression for a spinor field one can make use of the fact that the vector field of

4 There is a resemblance between the monopole index integral for spinor fields given here and a similar expression derived from the perspective of quantum mechanics given in [15].

F.S. Roux / Optics Communications 242 (2004) 45–55 [6] M.S. Soskin, V.N. Gorshkov, M.V. Vasnetsov, J.T. Malos, N.R. Heckenberg, Phys. Rev. A 56 (1997) 4064. [7] Y.Y. Schechner, J. Shamir, J. Opt. Soc. Am. A 13 (1996) 967. [8] M.J. Padgett, J. Courtial, Opt. Lett. 21 (1999) 430. [9] F.S. Roux, J. Opt. Soc. Am. B 21 (2004) 664. [10] Wu-Ki Tung, Group Theory in Physics, World Scientific, Philadelphia, 1985.

55

[11] M.V. Berry, Proc. SPIE 3487 (1998) 6. [12] J.R. Munkres, Topology, A First Course, Prentice-Hall, London, 1975. [13] S. Weinberg, The Quantum Theory of Fields, vol. II, Cambridge University Press, Cambridge, 1996. [14] J. Baez, J.P. Muniain, Gauge Fields, Knots and Gravity, World Scientific, Singapore, 1994. [15] M.V. Berry, Proc. R. Soc. Lond. A 392 (1984) 40.