1. Introduction

A 4Pi focusing system consists of two confocal high-NA objective lenses, by which two counter-propagating incident beams are focused to coherently interfere in the focal region. Compared to a single lens focusing system, the 4Pi focusing system has more robust ability to control focal fields as it contains the –z traveling plane waves (missing in a single lens). By properly modulating the input fields at the two pupil planes, various focal fields with different structures are formed. One example is the generation of the so-called spherical focal spot with a hope of resulting in equal axial and transverse resolutions in optical microscopy [1–3 ]. By focusing the input fields with radial polarization, the side lobe intensity occurring in linear polarization case can be reduced or moved away from the main lobe [4,5 ]. With polarization and amplitude modulation allowed, more complicated focal field configurations are obtained. A dark spherical spot can be realized by radially polarized beams with vortex phase [6]. By imposing a specially designed phase mask on the incident field, the focal fields like multiple spherical spots with tunable position [7–11 ], those with tunable polarization [12,13 ], and an long optical chain [14] are formed. These special focal field configurations have certain applications in various aspects. For example, in optical trapping, two counter-propagating beams in a 4Pi focusing system can reduce the scattering force [15–17 ] even eliminate it [18].

In this paper, we present a method of creating curved optical tubes in a 4Pi focusing system, that is, hollow focused beams with central-axis line described by a 3D curve. In [19–21 ], similar curve-like focal fields (with solid cross section) were created in a single lens focusing system. The methods there were essentially based on the translation of some basic focal spot. Our optical tubes presented in this paper may be looked on as the hollow version of those focal fields. The focal fields of optical tubes have interesting properties: the energy is concentered in the neighborhood of the 3D curve and the cross-sectional intensity distribution is of hollow shape. The spectral functions (the incident field as well) of these focused beams are obtained by linearly superposing the spectrum of transformed building block, an annular focal spot. The curve is rather arbitrary. Three examples of optical tubes are given, i.e., a torus, a hollow solenoid, and a hollow trefoil knot. The curved optical tubes may find applications in optical trapping for confining particles to a particular curve and transporting them along the curve.

2. Theoretical Model

In a 4Pi focusing system, the focused fields of an input field with linear polarization have a complicate vectorial structure. It is impossible to describe these fields by a scalar function. However, the use of a vector potential A(x, t) with uniform polarization may simplify greatly the problem considered. Suppose a vector potential as A(x) = u A(x), where u is a unit (uniform) polarization vector and A(x) the amplitude, and a harmonic time dependence exp(−iωt) is understood. Since the polarization vector u is uniform, we are led to treating the scalar function A(x) alone. In the focusing issue, the input field at the pupil plane actually corresponds to the spatial spectrum $\mathcal{A}$(θ, ϕ) of the focal field A(x). To establish this connection, we express A(x) in the Fourier integral as [22,23 ]

 

Fig. 1 Geometry of a 4Pi focusing system.

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In [19–21 ], various focusing fields like solid solenoid and knot were created by translating a fundamental focal spot (as a building block) δA ₀(x) to each point of a given curve. For the focusing of a plane wave, it is small spherical focal spot centered at the focus. At a given point x ₀ on the curve, the translated δA(x ₀) is δA ₀(x−x ₀). Denoting the spatial spectra (the input field as well) of δA ₀(x) and δA(x ₀) by δ ${\mathcal{A}}_0$ and δ $\mathcal{A}$(x ₀) respectively, it is clear that δ ${\mathcal{A}}_0$ = 1 and δ $\mathcal{A}$(x ₀) = exp(− i k∙x ₀) for the incident plane wave . Linearly supposing each contribution of δ $\mathcal{A}$ gives the final input field $\mathcal{A}$(θ, ϕ). Here, we want to construct some focusing fields of curved hollow tube structure in a similar manner. The natural choice of the building block is a annular focal spot generated by a vortex phase input field, i.e., δ${\mathcal{A}}_0$ = exp(imϕ). However, the method of translation used above falls here when we move the building block to each point on the curve bit by bit. The reason for this is that an annular focal spot has a symmetry axis on which the intensity vanishes, while the cross section of the tube also has a local symmetry axis, the tangent vector at each point of central-axis curve. Only when the translated annular focal spot has its symmetry axis coincide with the tangent vector, do we obtain a tube-like field. This means that, before translation, we need a rotation operation so that the resulted building block has its symmetry axis along the tangent vector to the central axis line at the target point.

Mathematically, the above procedure is expressed as follows. Let x ₀(s) be the central axis curve of the tube parameterized by s and δ $\mathcal{A}$(x ₀) the input field of the building block after being rotated followed by the translation to the point x ₀. Let R and T denote the rotation and translation operators and δl the line element length of the curve at x ₀. Then the weighted δ $\mathcal{A}$(x ₀) = TRδ${\mathcal{A}}_0$δl, where the transform R rotates the coordinates axes in such a way as to align the symmetry axis of building block, usually the z axis, with the unit tangent vector T(x ₀) of the curve at x ₀. To specify R, we need to know its action on the x and y axes. As we know, the plane transversal to the tangent vector T(x ₀) is spanned by the principal normal n(x ₀) and the binormal vector B(x ₀) ≡ T(x ₀) × n(x ₀). Assume that the x and y axes, after being acted on by the rotation R, are to be along the directions of n(x ₀) and B(x ₀), respectively. Then the rotation R is determined completely. For a given curve x ₀(s), the tangent vector T(x ₀) is calculated to be

After being rotated by R, the input field δ${\mathcal{A}}_0$ becomes a new field δ$\mathcal{A}$(x ₀) related to δ${\mathcal{A}}_0$ by δ$\mathcal{A}$(x ₀) = δ${\mathcal{A}}_0$(R⁻¹(θ, ϕ)). Explicitly, the new angular variables (θ', ϕ') = R⁻¹(θ, ϕ) are related to the original variables (θ, ϕ) by

Here, the primed variable ϕ' is expressed as a function of (θ, ϕ) through the relation (5). To guarantee the propagating property of the field, we have add a dynamic phase factor exp[ißl(s)] with l(s) the length of curve from 0 to s and β a propagating constant . Having obtained the spectrum $\mathcal{A}$(θ, ϕ), the amplitude A(x) comes out directly from the integral in Eq. (1). Then the vector potential A(x) is formed by combining A(x) with a uniform unit vector u. Here we put u = e _y, a unit vector in the y direction. Then A(x) = e _yA(x). With the Lorenz gauge, the electric field E(x) is calculated to be E(x) = iω[A(x) + ∇∇∙A(x)/k ²] where ω is the angular frequency of the field. It is clear that the electric field E(x) satisfies exactly the Maxwell equations.

3. Numerical Results

As illustration of curved optical tube, we first present an optical torus, the central-axis line of the tube being a circle. Intuitively, it may be thought of being obtained by cutting off a segment of a higher-order Gauss-Laguerre beam and pasting the two ends together in a way such that its central-axis line forms a circle. Assume that the central-axis line is described by x ₀(s) = (Rcos(s), 0, Rsin(s)) with 0≤s≤2π, a circle contained in the xz plane with center at the focus and radius R. To completely determine the potential $\mathcal{A}$(θ, ϕ) via the integral in Eq. (6), we need the values of R, m and β. Here, we put R = 50λ/π, m = 3 and β = 0.55k. For the purposes of discussion, all the lengths are measured in units of wavelength λ throughout the theoretical analysis.

The magnitude and phase of $\mathcal{A}$(θ, ϕ) with these settings are shown in Figs. 2(a)-2(d) , where the coordinates are transverse wavenumber k_x = ksinθcosϕ and k_y = ksinθsinϕ. Furthermore, the k_z( = kcosθ) > 0 part denotes the field $\mathcal{A}$(θ, ϕ) incident on the left lens while k_z< 0 part corresponds to that incident on the right lens, as mentioned above. We note that the magnitudes as well as phases of the spectrum on the two lenses exhibit symmetry. This is expected since the optical torus constructed is symmetric about the z axis. In Figs. 2(e) and 2(f), the distributions of the total electric field intensity |E|² in the xy and xz planes are plotted, respectively. From the xy plane distribution in Fig. 2(e), two hollow intensity spots (marked by dashed circles) symmetrically located on the x axis are observed. The distance of the center of each spot to the focus is about 16.1λ, being equal roughly to the value of R( = 50λ/π). Ideally, we may expect these two hollow spots to be of annular shape, as it should be for a torus. However, the nature of coherence of the fields makes it difficult to obtain a completely desired annular shape when constructing such a complicated focal field as a torus. Fortunately, two hollow spot of approximately annular shape are obtained. In Fig. 2(f), the intensity distribution in the xz plane is shown characterized by two main concentric circles, as expected for a torus. Besides, a series of side circles also arise. Although not desired, these side circles are an inevitable result of the coherence of fields. We note that in Fig. 2(e) there are some hyperbola-like patterns which are symmetric about the y axis. Since the torus can be obtained by revolving Fig. 2(e) around the y axis, these patterns are to form some hyperboloids. Figure 2(g) give a 3D view of the electric field intensity inside a focal region of volume [−20,20]λ × [−20,20]λ × [−20,20]λ. As predicted, we see clearly a torus accompanied by some patterns resembling hyperboloid of one sheet. In the integral in Eq. (6), we note that there is a phase gradient along the curve, suggesting that the energy flow of the fields may also form a torus. It is very much like the energy current of a Gauss-Laguerre beam flowing along the propagation direction.

 

Fig. 2 Generation of an optical torus and the corresponding incident field $\mathcal{A}$(θ, ϕ) in the 4Pi focusing system. (a),(b): Magnitudes of $\mathcal{A}$(θ, ϕ) on the left (k _z > 0) and right (k_z < 0) lenses. (c),(d): Phases of $\mathcal{A}$(θ, ϕ) on the left (k _z > 0) and right (k_z < 0) lenses. (e),(f): Distributions of the total intensity |(E)|² in the xy and xz planes. (g): 3D view of the total intensity.

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A second illustration of curved optical tubes is a hollow optical solenoid. In [20], the solid version of solenoid was realized by means of translating a fundamental spot. With the modified method presented in the integral in Eq. (6), we can construct a hollow version. The central-axis curve of the solenoid is parameterized as x ₀(s) = R(cos(s), sin(s), 2(s−π)/π) with R = 10λ and 0≤s≤2π, which describes a spiraling along the z axis of radius R and pitch 2R. As shown in Fig. 2, we put m = 3 and β = 0.55k. Figures 3(a)-3(d) show the magnitude and phase of the spectrum $\mathcal{A}$(θ, ϕ), in which the notation k_z > 0 (< 0) refers to the incident field on the left (right) lens. Unlike the spectrum of the potential of the optical torus in Fig. 2, $\mathcal{A}$(θ, ϕ) here exhibits no symmetry either in magnitude [Figs. 3(a) and 3(b)] or in phase [Figs. 3(c) and 3(d)]. The incident field on the left lens has most of the energy, as shown in Fig. 3(a), where a quite clearly bright annular spot is noted. This is reasonable, since the solenoid considered here is set to travel along the z axis. A wave packet traveling along the z axis certainly contains most plane wave components with k_z > 0. The in-plane intensity distributions in the xy and yz planes are illustrated in Figs. 3(e) and 3(f), respectively. In Fig. 3(e), a hollow (approximately) elliptical spot (marked by dashed circles) is seen located roughly at (−R, 0), agreeing with the analytical values of the coordinates of intersection point made by the central-axis line of solenoid and the xy plane. The shape of ellipse arises from the fact that the tangent vector to the curve at the intersection point is not parallel to the normal to the xy plane. From the yz plane distribution of intensity in Fig. 3(f), we see two hollow spots located roughly at (R, R) and (−R, −R), being equal to the analytic values of coordinates of intersection points. Figures 3(e) and 3(f) verify that the in-plane intensity distributions show a good agreement with those expected. Figure 3(g) shows the 3D distribution of intensity in a focal region of volume [−20,20]λ × [−20,20]λ × [−20,20]λ. As desired, a solenoid of one loop is observed. Also we find some side patterns of intensity near the solenoid.

 

Fig. 3 Generation of an optical hollow solenoid and the corresponding incident field $\mathcal{A}$(θ, ϕ) in the 4Pi focusing system. (a),(b): Magnitudes of $\mathcal{A}$(θ, ϕ) on the left (k _z > 0) and right (k_z < 0) lenses. (c),(d): Phases of $\mathcal{A}$(θ, ϕ) on the left (k _z > 0) and right (k_z < 0) lenses. (e),(f): Distributions of the total intensity |(E)|² in the xy and yz planes. (g): 3D view of the total intensity.

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A third example of curved optical tubes is an optical hollow trefoil knot, i.e, a tube with its central-axis curve represented by a trefoil knot. In [21], an optical trefoil knot with a solid cross-section was realized. A typical trefoil knot curve can be parameterized as x ₀(s) = R(sin(s) + 2sin(2s), cos(s)−2cos(2s), −sin(3s)) with 0≤s≤2π. In calculation, we find that if the value of R is small, the knot is to be contained in a small region of space. This leads to a structure breaking in interlace region because of strong coherence existing in such a complex field. Also a large value of m gives rise to the breaking in the cross-sectional shape. To minimize these two disadvantages, we set R = 15λ and m = 2. For the value of β, we set it equal to 0.5k.

Figures 4(a)-4(d) show the magnitudes and phases of the spectrum $\mathcal{A}$(θ, ϕ) on the left lens (k_z > 0) and the right lens (k_z < 0), respectively. Since the knot is more complicated than either the torus or the solenoid presented above, the magnitudes of the spectrum $\mathcal{A}$(θ, ϕ) are seen to be more intricate on either left or right lens. The in-plane intensity distributions in the xy and xz planes are illustrated in Figs. 4(e) and 4(f), respectively. From Fig. 4(e), we see six hollow spots (marked by dashed circles) scattered across the region, each having a shape different from the others. Analytically, the knot curve is indeed to pass through six times the xy plane. Due to the fact that the tangent vectors at these points are not the same, the cross-sectional shapes need not be identical, as seen in Fig. 4(e). In Fig. 4(f), four hollow spots are observed, the number of which is identical to that of the knot curve crossing through the xz plane. From the 3D view of the intensity distribution in a focal region of volume [−40,40]λ × [−40,40]λ × [−40,40]λ shown in Fig. 4(g), we see that the field energy is indeed confined to the neighborhood of a trefoil knot. Back to the integral in Eq. (6), we note there a phase gradient along the knot curve. More specifically, the phase increases with increasing of s. Then, we may expect that the energy flows along the direction of increasing s, forming an energy flow knot. The properties of intensity and energy flow may help such an optical knot find its application in optical trapping: confining particles to a knot and transporting them along the curve.

 

Fig. 4 Generation of an optical hollow trefoil knot and the corresponding incident field $\mathcal{A}$(θ, ϕ) in the 4pi focusing system.. (a),(b): Magnitudes of $\mathcal{A}$(θ, ϕ) on the left (k _z > 0) and right (k_z < 0) lenses. (c),(d): Phases of $\mathcal{A}$(θ, ϕ) on the left (k _z > 0) and right (k_z < 0) lenses. (e),(f): Distributions of the total intensity |(E)|² in the xy and xz planes. (g): 3D view of the total intensity.

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4. Conclusions

In conclusion, we have presented the method of creating curved optical tubes in a 4Pi focusing system. To form such optical fields, we need the incident fields both on the left and on the right lens. The method of obtaining the incidental fields is a generalization of previous ones for creating curve-like focal fields. Our generalization consists essentially of rotation of the fundamental focal spot of a vortex beam with particular value of topological charge m followed by a translation to an appropriate point on the prescribed curve. The incident field is then obtained by linearly superposing the spectra of these transformed fundamental focal spots. The translation transform assures of a curve-like optical field while the rotation transform guarantees that this curve-like field has a hollow cross section. As illustration of curved optical tubes, a torus, a solenoid and a trefoil knot are presented numerically. As expected, the resulting fields show tube-like intensity patterns. Furthermore, by the phase gradient along the curve, the energy is expected to flow along the curve. These properties may help such focal fields find their applications in optical trapping: confining particles to a particular curve and transporting them along the curve. Although only three examples of optical tubes are given, creation of other optical tubes presents no difficulty since the method imposes no limit on the central-axis curve of tube.