Maybe the most typical quantity for characterizing three-dimensional (3D) optical vector fields is the spin (angular momentum) density vector (SDV), whose direction represents the orientation of the polarization ellipse, and the scale indicates the density of the spin angular momentum^[1,2]. Since its important role in 3D nano-optics, for instance, in controlling light–matter interaction^[3] and in the emission directivity of a dipole-like nano-particle^[4], the SDV has attracted much attention in recent years^[5–14]. It has been found that transverse spin density (SD) has a strong connection with the spin Hall effect of light^[5,6], and the purely transverse SDVs (tSDVs) lead to an interesting optical phenomenon—the ‘photonic wheel’^[7–9]; furthermore, the spiral behavior of the SDVs provides a new rotation freedom in optical tweezers^[10–12]. The new features of SDVs continue to be observed^[13,14].

The Gouy phase is an additional phase that the focused field acquires, compared to an ideal non-diffracted field^[15–17], and it plays a fundamental role in many applications, like in terahertz spectroscopy^[18] and optical calibration^[19,20]. Because of the polarization effect in strongly focused fields, there will be a Gouy phase difference between the field components with different polarizations^[20]. It has been found that this Gouy phase difference leads to the rotation of the SDVs during beam propagation^[10–12]. The vortex beams with different orbital angular momentum (OAM) modes (i.e., different topological charges) have been studied substantially in past decades^[21–27], and it has been demonstrated that the (accumulated) Gouy phase difference between the beams with different OAM modes can induce the rotation of the field distributions^[21,22]. Then, it will be of interest to examine the behaviors of the SDVs in a field with two distinguishing Gouy phase differences. Additionally, it is shown that in 3D optical fields, new types of optical singularities can exist: the SD singularities^[14], and these new singularities also have an effect on the behavior of the SDVs.

In this Letter, we compose a simple 3D field where two types of Gouy phase differences (i.e., the difference between the field polarization components and the difference between the field OAM modes) exist. It will be shown that the evolution of SDVs in such a 3D field is strongly dependent on these two Gouy phase differences. Also, as it will be seen, in the evolution of the SDVs, the SDV singularities can be observed, and their effect on the SDVs will be discussed.

Suppose that there are two vortex Gaussian beams with topological charges t1=+1 and t2=−2, and these two beams are superimposed collinearly. Then, the complex amplitude of the composite electric field at the beam waist w0 can be expressed as^[28]where (r,ϕ) are polar coordinates in the transverse plane (x=rcosϕ,y=rsinϕ), and A, B are measures of the amplitudes of the two beams. The intensity and the phase distribution of this composite field are shown in Fig. 1, where A=B. In Fig. 1(a), the intensity is normalized to its maximum at the waist plane. It is shown that the intensity of the composite field has three lobes, and, when the difference between A and B is increased, these lobes will be blurred. The white lines in Fig. 1(b) indicate the phase singularities, and the phases across the white line have a π phase jump.

Now, assume that this composite beam [Eq. (1)] is linearly polarized in the x direction and is also incident upon a strongly focused system with the entrance plane coincident with the waist plane of the composite beam. The strongly focused system is illustrated by Fig. 2, where the focal length is f, and a semi-aperture angle is denoted by α. By applying the Richards–Wolf vectorial diffraction theory^[29], we can calculate the electric field of the strongly focused composite field in the focal region at point (ρs,ϕs,zs) as [note that the following equations can be calculated by putting Eq. (1) as the incident field into Eq. (2) of Ref. [20] or referring to the derivation of Eq. (2.26) in Ref. [29]]andwhereand where k=2π/λ is wave number, and Jn(x) is the nth-order Bessel function of first kind. We can see that there are three field (polarization) components with different expressions in the focal region.

Before examining the behavior of the SDVs in this focused field, we will first review some related concepts.

The Gouy phase is a physical quantity to measure the phase difference between the actual field and the non-diffracted (ideal) field in the same conditions^[15], and, in a strongly focused field, a Gouy phase δi of the field component ei can be written aswhere Arg means the phase of the field component, and R=ρs2+zs2 is the distance of the point from the focus. For a laser mode, the Gouy phase is usually used to represent the accumulated phase difference of the whole beam, i.e., the ‘general’ Gouy phase, and, in that case, the Gouy phase (δg) of a Laguerre–Gaussian (LG) beam with topological charge t and radial mode p is given by^[21,22]where zR means the Rayleigh range. Generally speaking, the Gouy phase δg reflects the extra phase that the beam acquires after propagating through the focus, and, under the paraxial approximation, δg in Eq. (21) and δi in Eq. (20) are essentially the same. In order to avoid confusion, from here on, we call the Gouy phase of the whole beam as the ‘general Gouy phase.’ If the beam is singly ringed (p=0), and the general Gouy phase is cumulated from zs=−∞ to zs=+∞, the δg equals (|t|+1)π.

The SDV, sE of the electric field, according to its definition^[1,2] and Eq. (20), in the present 3D field can be expressed by the Gouy phase difference δij=δi−δj (i,j=x,y,z)^[2,9,10] aswhere ϵ0 denotes the permittivity of free space, and ω is the angular frequency. Im and * represent the imaginary part and the complex conjugate, respectively.

In the following, the evolution of SDVs in the focal region of this strongly focused composite field is discussed. Note that in this Letter only the case of A=B is considered.

First, the SDVs can spin along the propagation axis (i.e., zs axis). From Eqs. (3)–(19), one can find that when ρs=0, the ex component and ey component only have a π/2 phase difference (δyx=−π/2), which means that the transverse component of the electric field is circularly polarized. In this case, the behavior of the SDVs is dependent on the Gouy phase difference δxz, and, in the monotonic interval of δxz, the special structure, ‘spiral SDVs’ (i.e., the SDVs rotate around the central axis), will be generated^[10,12]. The curves of δxz with different values of semi-aperture angle α are shown in Fig. 3. As one can see, the monotonic intervals of δxz are formed in each plot, and the range is increased with α, which is consistent with those observed in Ref. [10]. While the difference from that in Ref. [10] is that in Fig. 3(b), there exist two special points near zs=±2.9λ, and the δxz has a π phase jump at these two points. This phenomenon is caused by the null SD at these two points, i.e., the ‘SDV singularity’^[14]. Note that the SDV singularity describes a point at which the SD is zero, the SDV is thus undefined, and the topological structure around this singularity follows the same rule as the traditional vector singularity of electric fields does.

The corresponding SDVs on the zs axis are illustrated in Fig. 4, where the blue arrows denote the SDVs and the red curve is their envelope. We can see that in the monotonic intervals of δxz [for instance (−7λ,+7λ) in Fig. 4(b)], the SDVs rotate clockwise along the propagation axis, and, as α increases, the rotation range (corresponding to the monotonic interval in Fig. 3) expands [i.e., this range in Fig. 4(c) is bigger than that in Figs. 4(a) and 4(b)]. Especially, at the points with SDV singularity in Fig. 4(b) (denoted by ‘V’), the SDVs do not exist, while by passing through these points the SDVs reverse their directions. This is depicted more clearly in Fig. 4(d), where the SDVs at points zs=2.8λ,2.9λ have (nearly) opposite directions (note that the exact position of this singular point is between 2.8λ and 2.9λ).

Second, the SDVs exhibit special behaviors on the ys axis. When ϕs=±π/2, from Eqs. (3)–(19), we can obtain δzy=±π/2, δxz=0,±π, and δyx=±π/2, which means that here the Gouy phase differences are constants (their signs change at the points of phase singularity). This fact leads to sE(x)=±C0|ey||ez|, sE(y)=0, sE(z)=±C0|ex||ey| (here C0=ϵ0/2ω). Therefore, along this axis all of the SDVs are exactly perpendicular to the ys direction, and the SDV singularities can be observed easily. The Gouy phase differences δzy and δyx are shown in Fig. 5. One can see firstly that the values of δzy and δyx are very different on the +ys axis and the −ys axis, i.e., asymmetric with respect to the origin O. Second, δzy (or δyx) has a π jump at the phase singularities of the ey or ez component (or the ex or ey component). Also, δzy and δyx have their common ‘jump’ points, which can be seen along the dotted (vertical) lines in Fig. 5. At these common ‘jump’ points [denoted by ‘Vi’ (i=1,2,…,6)], the SDV singularities are formed, which also can be observed in Fig. 6.

In Fig. 6, the SDVs with the same parameter as Fig. 5 are displayed [note that here the plots (a) and (b) are the same figure but seen from different views]. One can see firstly that all the SDVs are perpendicular to the ys axis. Second, there are six ‘vector singularities’ from ys=−2.6λ to ys=+2.6λ, and the SDVs have opposite directions at the two sides of each ‘vector singularity.’ Furthermore, the SDVs have quite different distributions on the two half-axes. This asymmetry is mainly caused by the rotation of the spatial distribution of the SDVs, which will be discussed in the following part.

Third, the spatial distribution of the tSDVs rotates during beam propagation. The tSDVs, sE(t)=(sE(x),sE(y)), in the focal plane are shown in Fig. 7(a), where the arrows denote the tSDVs with their size and color indicating the density of the tSDVs (i.e., the transverse SD). The color-coded transverse SD of the same region is also shown in Fig. 7(b). We can see that there are six strong density spots, and they are distributed symmetrically with respect to the ys axis. The tSDVs on the transverse planes at different propagation distances are displayed in Fig. 8, from which one see that the spatial distribution of the tSDVs rotates clockwise as the beam propagates. To illustrate this rotation exactly, we selected two maxima density points denoted by a black circle and a black square, respectively (see Fig. 8), and their azimuthal angles at different transverse planes are plotted in Fig. 9.

From Fig. 9, we can find that the point denoted by the circle rotates from about 170° to 110°, and the other one circles from around 70° to 10°. Both total rotation angles are about 60°. This rotation can be explained by the difference of the general Gouy phases between two beams with different OAM modes. The incident field is composed by two beams with t1=+1 and t2=−2. As we discussed before, here, the general Gouy phases of these beams can be calculated from Eq. (21), and they are 2π and 3π. Due to this general Gouy phase difference, with the beam propagation, the composite field will acquire an angular change on the field distribution, and this change is π(|t2|−|t1|)t2−t1=−π/3^[22] (note that the value −π/3 is calculated for the beam propagating from −∞ to +∞). Therefore, the spatial distribution of the tSDVs gets an about 60° clockwise rotation.

In conclusion, the evolution of the SDVs of a strongly focused composite field is studied. It is found that in such a field, the SDVs of the points on the zs axis can rotate with the beam propagation, while the SDVs for the points on the ys axis are all perpendicular to the ys direction. These two phenomena can be explained by the Gouy phase difference between the field polarization components (i.e., the components with different polarization states). Particularly, the SDV singularities are also observed on these two axes, and the SDVs will reverse their directions by passing through these singularities. The spatial distribution of the tSDVs (the transverse component of SDVs) is examined, and it is found that the tSDVs’ distribution will have a 60° rotation during the beam propagation. It is quite interesting to see that this 60° rotation is caused by the general Gouy phase difference between the field OAM modes (i.e., the components with different OAM modes) rather than the field polarization components. The findings in this Letter may have applications in optical manipulations; on the other hand, this work also supplies a simple model for observing rich behaviors of SDVs and for distinguishing two types of Gouy phases.