The concept of scalar fields with uniformly rotating intensity distributions and propagation-invariant radial scales is extended to the case of electromagnetic fields with rotating but otherwise propagation-invariant states of polarization. It is shown that the conditions for field rotation are different for scalar and electromagnetic fields and that the electromagentic analysis brings in new aspects such as the possibility that different components of a rotating electromagnetic field can rotate in opposite directions.
©2001 Optical Society of America
Certain non-square-integrable solutions of the Helmholtz equation in free space continue to attract considerable interest in optics. The primary classes of such fields are those of self-imaging fields [1, 2], propagation-invariant fields [3, 4], and rotating fields [5, 6, 7, 8, 9]. In case of propagation-invariant fields, which retain their intensity distributions upon propagation in free space, the results of scalar analysis have been extended to (vectorial) electromagnetic fields during the past decade [10, 11, 12, 13]. This analysis can be extended quite straightforwardly to self-imaging electromagnetic fields , which reproduce their transverse distributions of energy density periodically upon propagation in free space.
The main goal of this article is to extend the theory of rotating scalar optical fields into the electromagnetic domain, where the state of polarization of the field becomes an aspect of central concern. Instead of considering only the energy density (intensity), we pay attention also to the state of polarization of the field. In particular, we shall consider electromagnetic fields with similarly rotating radial and azimuthal components. Such considerations lead to conclusions that are rather different from those obtained within scalar theory.
We begin with a brief discussion of rotating scalar field, which is somewhat more general those given in the previous studies, partly to establish notation and partly to provide easy comparison with the new results for electromagnetic fields.
2 Rotating scalar fields
If we express both the position vector r and the wave vector k in circular cylindrical coordinates, i.e., r=(ρ, φ, z) and k=(α, ψ, β), the angular spectrum representation  of the complex wave function U(ρ, φ, z) takes the form
The condition that the field intensity is independent of the z coordinate, apart from a linear rotation upon propagation, is that
where Jm denotes a Bessel function of the first kind and order m and
Here we have used the Jacobi–Anger expansion 
Because Eq. (5) must hold for all ρ and ϕ the integrand must vanish identically. This yields
where q is an integer. However, because ξ (Δz) must be continuous and in addition ξ(0)=0, no other values of q than zero are possible. Thus, β may assume only discrete values
where β 0 is a constant. The angular spectrum of the field in now confident on a discrete set of concentric rings (known as Montgomery’s rings [1, 2]) with radii αm =(k 2- )1/2. Hence the field expression (1) takes the form
where M denotes the set of allowed values of m (those values for which 0≤βm ≤k) and am ≡am (αm ).
3 Relation of rotating fields with self-imaging and propagation-invariant fields
By definition, it is clear that a rotating field also self-images. The self-imaging distance, denoted by zT , is usually called the Talbot distance because Talbot  was the first to observe the phenomenon of self-imaging. In view of Eq. (4) it is clear that the field experiences a full rotation of 2π radians between planes separated by a distance
However, the intensity distribution may reproduce itself also at shorter distances
where Q is a positive integer. This situation, which is consistent with our original definition of self-imaging, takes place when the intensity distribution exhibits a Q-fold symmetry
where q=1,…, Q. Now, after propagating over the distance zT /Q, the field has rotated by 2π/Q radians.
Propagation-invariant fields are a special case of rotating fields in the limit that there is no rotation at all. In view of definition (4), this obviously occurs when η=0. Now, in view of (9) only one value of βm is allowed, i.e., the angular spectrum is confined on a single Montgomery’s ring. This is consistent with the results of Durnin [3, 4].
4 Rotating electromagnetic fields
The considerations presented above are valid also in the electromagnetic case provided that all scalar field components rotate in the same way and that only the intensity of the field is concerned. Let us next consider the case in which the field itself, rather than just its intensity distribution, rotates.
As in the scalar case, the rotating electromagnetic field also self-images and hence all of its cartesian components satisfy Eq. (10). However, because the polarization state of the field rotates the condition for rotating must differ from Eq. (4). The most natural way to handle the rotation of an electromagnetic field is to use the circular cylindrical field components defined by
(the radial field component) and
(the azimuthal field component), with the z-component of the field being unchanged. The conditions for rotation are now
i.e., the ratio of the radial and the azimuthal components as well as their amplitudes are required to remain exactly the same.
where am (α) and bm (α) are the functions defined in Eq. (6) for the x and the y field components, respectively. By inserting condition (16) to Eqs. (17) and (18) we obtain, similarly as in the scalar case,
In order to satisfy (19) we must choose
As in the scalar case, this result discretizes the values of kz to
where β 0n is a constant and n is a sign function, which can assume a values n=-1 or n=+1 for each m. Thus two different Bessel field modes, with m=p+1 and m=p-1, are allowed on each Montgomery’s ring. This is contrary to the prediction of scalar theory, within which only one value of m is allowed for each ring.
The field expressions (17) and (18) now take the forms
where amn ≡am (αmn ) and αmn are the radii of the Montgomery’s rings. One should notice that although α m+1,-1=α m-1,+1 the coefficients a m+1,-1 and a m-1,+1 are independent.
By using the Maxwell’s divergence equation for the electric field in cylindrical coordinates,
we obtain an expression for the z-component of the electric field:
We immediately see that also this component fulfills the rotation condition (24). This is, of course, clear because the other two cylindrical components remain shape- and scale-invariant, except for rotation.
By examining Eqs. (25), (26), and (28), we find that the field consist of either left-handed or right-handed circularly polarized components. However, if n is the same for all m, we obtain a purely circularly polarized field, which of course belongs to the class of rotating fields. In that case Eqs. (9) and (24) coincide and thus both cartesian and cylindrical field components rotate in the same way.
5 Relations between scalar and electromagnetic rotating fields
The main difference between scalar and electromagnetic theories of rotating fields is that each Montgomery’s ring can contain two Bessel field modes in the electromagnetic case but only one in the scalar case. Since the nature of x and y components of a rotating electromagnetic field is determined by scalar theory, these components can not rotate if any one of the rings contains two modes, while the radial and azimuthal components can still rotate. An example of this situation is shown in Fig. 1, where the squared absolute values of Eρ , Ez , and Ex are illustrated as a function of distance (the parameters assumed in the calculations are given in Table 1). The other two components, Eϕ and Ey , behave essentially as Eρ and Ex , respectively, and are thus not shown. On the other hand, if the requirement (22) is not met, the cylindrical components can not rotate even though the two cartesian components may still do so.
Let us next consider the case in which one Montgomery’s ring contains only one Bessel mode and bm (α)=±iam (α). Because of the difference between the cartesian and cylindrical coordinates, the relation between γ=2π/Pz T and η=2π/Qz T, where P and Q are integers, is not clear. Let us denote the z components of the wave vectors associated with two different Montgomery’s rings by βj and βm . By using Eqs. (9) and (24) we obtain
Thus, P can take only values Q-2, Q, or Q+2. An example of this is represented in Fig. 2 in which pentagonally symmetric cylindrical components are associated with triangularly symmetric cartesian components. These field components are calculated with the parameters given in the Table 1. The connection between P and Q implies also that if Q=1 it is possible to have P=-1 which means that the cartesian x and y components of the field rotate in the opposite direction of the cylindrical components. This is illustrated in Fig. 3, with the parameters given in Table 2.
By examining Figs. 2 and 2 we find that the rotation of the z component is identical with the radial component, rather than the x component, of the field. This phenomenon arises from the cylindrical geometry, in which the x and y components of the field have no special meaning. An another example of this is represented in Fig. 4 in which the x component rotates but Eρ and Ez do not.
We have provided the basic theory of rotating but scale-invariant electromagnetic fields and presented examples, which show that interesting new phenomena appear when the vectorial nature of light fields is considered. Therefore it can be concluded that scalar theory does not provide a complete picture of the nature of rotating optical fields.
The work Jari Turunen was supported by the Academy of Finland. Jani Tervo thanks the Finnish Academy of Science and Letters for generous grant.
References and links
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