## Abstract

We present in principle and demonstrate experimentally a new kind of vector fields: elliptic-symmetry vector optical fields. This is a significant development in vector fields, as this breaks the cylindrical symmetry and enriches the family of vector fields. Due to the presence of an additional degrees of freedom, which is the interval between the foci in the elliptic coordinate system, the elliptic-symmetry vector fields are more flexible than the cylindrical vector fields for controlling the spatial structure of polarization and for engineering the focusing fields. The elliptic-symmetry vector fields can find many specific applications from optical trapping to optical machining and so on.

© 2014 Optical Society of America

Recently, with an increasing interest in vector fields, much research has focused on finding new kinds of vector fields to enrich the family of vector fields. As we know, the cylindrical vector fields have been well studied and a variety of properties have been proved in many areas such as optical trapping, laser machining and so on [1]. Inasmuch as the controllable parameters for cylindrical vector fields are limited, introducing new controllable degree of freedom to vector fields becomes an urgent need. The simplest one is to generate a kind of elliptic-symmetry vector fields.

Here we present, design, and generate a new kind of elliptic-symmetry vector field with local linear polarization. Such a kind of vector field breaks the cylindric symmetry of vector fields and enriches the family of vector fields. The geometric configurations of polarization provide additional degrees of freedom, assisting in controlling the field symmetry and in engineering the weak and tight focusing field distribution for different applications or requirements. A series of elliptic-symmetry vector fields in the elliptic coordinates system, we concerned here, is quite different from the vector fields with elliptical symmetry reported in [2].

To generate the elliptic-symmetry vector fields, a flexible approach in a common path interferometric configuration with the aid of a 4f system [3, 4] is still practicable. The schematic is shown in Fig. 1, which is very similar to those used in [3, 4]. This approach includes four key steps: (i) an input linearly polarized laser field is divided to two equi-amplitude parts, which are easily achieved by a sine/cosine grating displayed on a spatial light modulator (SLM) placed in the input plane of the 4f system; (ii) the two parts must carry the space-variant phase *δ*(*x*, *y*); (iii) the two parts must pass through different optical paths (not only is the separation in space), making the two parts have orthogonal states of polarization by using a pair of wave plates (such as 1/2 and 1/4 wave plates) behind the spatial filter (SF) placed in the Fourier plane of the 4f system; (iv) the two orthogonally polarized parts are combined by the Ronchi phase grating (G) placed in the output plane of the 4f system.

The amplitude transmission function of holographic grating displayed on SLM is [3]

If the two waveplates placed on the Fourier plane of the 4f system in our approach [3] are two 1/4 waveplates (i.e., the two bases for generating the vector field are the right- and left-circularly polarized fields), the output vector optical field can be represented as [4]*x*and

*y*components are always in phase. Arbitrary vector fields can be generated by changing the additional phase

*δ*, because

*δ*is allowed to possess arbitrary spatial distribution which can be flexibly realized by SLM.

The elliptic coordinate system is an essential orthogonal coordinate system [5]. We should first give an introduction to the elliptic coordinate system (*ε*, *η*), which has the following relation with a Cartesian coordinate system (*x*, *y*)

As shown in Fig. 2, the elliptic coordinate system (*ε*, *η*) has two foci, *F*_{1} and *F*_{2}, which are located at (−*f*, 0) and (*f*, 0) in the system (*x*, *y*), respectively. The curves of constant *ε* are a series of confocal ellipses (solid lines), while the curves of constant *η* are a series of confocal hyperbolas (dotted lines). These two kinds of curves can be written as

To generate elliptic-symmetry vector fields, it is of great importance to reference the generation of cylindrical vector fields. To generate the cylindrical vector fields, the key point is that the expression of the additional phase *δ* in the transmission function of the holographic grating displayed on SLM in Eq. (1) has the following form in the polar coordinate system [6]

*φ*and

*r*are the azimuthal and radial coordinates in the polar coordinate system,

*r*

_{0}is the radius of the circular vector field, and

*δ*

_{0}is the initial phase. Therefore, if the additional phase

*δ*has the following form as the vector field described by Eq. (2) should be the elliptic-symmetry vector field.

Furthermore, we should give the expressions of *ε* and *η* in terms of *x* and *y* as

*ε*is limited within a range of [0, +∞) and

*η*has a range of [0, 2

*π*), respectively.

We first simulate the local linear-polarization distributions of six typical elliptic-symmetry vector fields, as shown in Fig. 3. For cases when *m* ≠ 0 and *n* ≡ 0, implying that *δ* in Eq. (6) is a function of *η* independent of *ε* in the elliptic coordinate system (*ε*, *η*), the polarization distributions of the elliptic-symmetry vector fields are schematically shown in Fig. 3(a) for *m* = 0.5 and *δ*_{0} = 0, in Fig. 3(b) for *m* = 1 and *δ*_{0} = 0, and in Fig. 3(c) for *m* = 1 and *δ*_{0} = *π*/2, respectively. For the case of (*m*, *n*, *δ*_{0}) = (0.5, 0, 0) in Fig. 3(a), its polarization map has a singular ray along the +*x* direction, where the polarization has the uncertainty, therefore this kind of vector field will occur a zero-intensity dark ray in the +*x* direction. This is similar to the case when *m* = 0.5 reported in [3]. When *m* = 1 and *n* = 0, the two cases of *δ*_{0} = 0 and *δ*_{0} = *π*/2 in Eq. (6) are similar to the radially polarized (RP) and azimuthally polarized (AP) fields [3], respectively, therefore these two elliptic-symmetry vector fields can be referred to as the RP-like and AP-like vector fields, as shown in Figs. 3(b) and 3(c). In particular, the RP-like and AP-like vector fields are a pair of orthogonal vector fields, as the RP and AP vector fields. When *m* > 1, the polarization map occurs more spots of singularity. For the case when (*m*, *n*, *δ*_{0}) = (3, 0, 0), as shown in Fig. 3(d), there occur three points of singularity. For the above four cases, the polarization distributions are a function of *η* independent of *ε*. When *m* = 0, *n* = 1 and *δ*_{0} = 0, as shown in Fig. 3(e), the polarization map has no polarization singularity, and the polarization distribution is a function of *ε* independent of *η*. When *m* = *n* = 1 and *δ*_{0} = 0, as shown in Fig. 3(f), the polarization distribution depends on both *ε* and *η*.

We will now demonstrate experimentally the generation of elliptic-symmetry vector fields by using the method as mentioned above. Figure 4 shows six elliptic-symmetry vector fields for different *m* when *n* ≡ 0 and *δ*_{0} = 0. We can see from the first row that the total intensity pattern has some singular spots for the integer *m*, while that has also a singular ray besides some singular spots for the non-integer *m*, due to the polarization uncertainty. The simulated intensities of the *x* (*y*) components in the second (fourth) row are in good agreement with the measured those in the third (fifth) row. Indeed, the polarization distribution depends on *η* independent of *ε*, implying that all positions in the curves of constant *η* (hyperbolic curves) have the same polarization. In particular, we find from the second and third rows that the zero-intensity singularities for the non-integer *m* originate from the contribution of the *x* component only, while from the fourth and fifth rows that the zero-intensity singularities for the integer *m* are from the contribution of the *y* component only.

Figure 5 shows five elliptic-symmetry vector fields for different *n* when *m* ≡ 0 and *δ*_{0} = 0. We can see from the first row that any total intensity pattern exhibits a homogeneous distribution, while has no singularity for the integer *n*. In fact, the elliptic-symmetry vector field has always no singularity for any *n* (is not limited to the integer *n*) provided that *m* ≡ 0 (i.e. *δ* is independent of *η*). The simulated intensities of the *x* (*y*) components in the second (fourth) row are in good agreement with the measured ones in the third (fifth) row. Indeed, the polarization distribution depends on *ε* independent of *η*, implying that positions with the same polarization form the elliptic curves of constant *ε*. We can also see from the first and second rows that there is a bright line formed by the *x* component between two foci. For the *x* (or *y*) component, the number of the bright (or extinction) ellipses is equal to 2*n*.

We now focus on the cases when *δ* is a function of both *ε* and *η*, i.e. *m* and *n* must be nonzero simultaneously. Figure 6 shows the elliptic-symmetry vector fields for different *n* = (0.5, 1, 1.5, 2, 2.5, 3) when *m* ≡ 1 and *δ*_{0} = 0. We can see from the first row that any total intensity pattern exists a zero-intensity singular segment between the two foci. The intensity patterns of the *x* component exhibit a non-common-origin dual-spiral structure, and the origins of the two spiral curves are located at the two foci. In contrast, the intensity patterns of the *y* component has also a dual-spiral structure, and the origins of the two spiral curves are connected by a bright segment between the two foci, which is different from the *x* component. In particular, the zero-intensity singular segment in any total intensity pattern is from the contribution of the *y* component only. Figure 7 shows the elliptic-symmetry vector fields for different *m* = (0.5, 1, 1.5, 2, 2.5, 3) when *n* ≡ 2 and *δ*_{0} = 0. Figure 8 shows the elliptic-symmetry vector fields for different combinations of *m* and *n* with *m* = *n* when *δ*_{0} = 0. As shown in the first rows in Figs. 7 and 8, in the total intensity patterns there have the zero-intensity singular spots due to the polarization uncertainty caused by the nonzero *m* in the cases of the integer *m*. However, in the total intensity patterns there have also the zero-intensity singular ray besides the zero-intensity singular spots in the cases of the half-integer *m*. The intensity pattern of the *x* or *y* component exhibits a single- or multi-spiral structure depending on *m*. The singular ray originates from the contributions of both the *x* and *y* components for the half-integer *m*. From Figs. 6 and 8, it is clear that the arms of the spiral curves in the intensity pattern of the *x* or *y* component is equal to 2|*m*| independent of *n*, while the separation between turns of the spiral curves decreases as |*n*| increases.

Besides the parameters *m* and *n*, the interval 2 *f* between the two foci as an important degree of freedom can also be used to manipulate the elliptic-symmetry vector fields. Figure 9 shows the elliptic-symmetry vector fields for five different *f* = (0.1, 0.2, 0.3, 0.4, 0.5) in the two cases of (*m*, *n*, *δ*_{0}) = (2, 0, 0) and (*m*, *n*, *δ*_{0}) = (0, 1.5, 0). The total intensity patterns have two zero-intensity singular spots located at two foci for the case of (*m*, *n*, *δ*_{0}) = (2, 0, 0). The intensity patterns of the *x* and *y* components exhibit the confocal hyperbolic shapes. As *f* decreases to zero, the elliptic coordinate system degenerates into the traditional polar coordinate system (two foci coincide to an origin). Correspondingly, the hyperbolic curves describing the trajectory with the same polarization become to the radial rays in the polar coordinate system.

The total intensity patterns have no zero-intensity singular spot for the case of (*m*, *n*, *δ*_{0}) = (0, 1.5, 0). The intensity patterns of the *x* and *y* components exhibit the confocal elliptic shapes because the trajectory drawn by the positions with the same polarization. As *f* decreases to zero, the ellipticity of the elliptically-shaped intensity patterns of the *x* and *y* components becomes small until to zero, implying that the confocal ellipses degenerate into the concentric circles. In fact, when *f* = 0, all the cases in the elliptic coordinate system degenerate into those in the polar coordinate system reported in [3].

In discussion, the elliptic-symmetry vector field seeks to be very similar to the vector field reported in [2], but both are quite different. The vector field reported in [2] is considered as deformed cylindrical vector field, which is in fact a flattened cylindrical vector field and has at most one polarization singularity. However, the elliptic-symmetry vector fields can have two polarization singularities located at the two foci. The Ince-Gaussian, Mathieu, and Mathieu-Gaussian fields are also associated with the elliptic coordinate system, which are the same as the elliptic-symmetry vector fields we generated here. The Ince-Gaussian and Mathieu fields are both nondiffracting or invariant fields. The Ince-Gaussian fields constitute the third complete family of exact and orthogonal solutions of the paraxial wave equation, and their transverse distributions are described by the Ince polynomials and have an inherent elliptic symmetry [7]. The Mathieu fields are solutions of the paraxial wave equation in elliptic coordinates, which are described by the radial and angular Mathieu functions [8]. The Mathieu-Gaussian fields are not the nondiffracting solutions of the paraxial wave equation and can be considered as a modulation of Mathieu beams [9]. The Ince-Gaussian, Mathieu, and Mathieu-Gaussian fields have space-invariant distribution of polarization. However, the elliptic-symmetry vector fields we generated here have the space-variant distribution of polarization, which exhibit the elliptic symmetry. The elliptic-symmetry vector fields are not the eigen solutions of the paraxial wave equation, that is to say, which are not the nondiffraction eigenmodes while can be decomposed into a series of eigenmodes.

The elliptic-symmetry vector fields may have some more useful properties with respect to the cylindric-symmetry vector fields in the polar coordinate system, because it has an additional controllable degree of freedom (the focal length *f* of the elliptic coordinate system). Here we focus on their tight focusing property by an objective with high numerical-aperture (NA). The tightly focused vector fields have been already studied carefully [10–13]. The tight focusing field can be expressed as [1, 11]

*ρ*and

*ϕ*are the radial and azimuthal coordinates, and

*E*and

_{ρ}*E*are the radial and azimuthal components of the incident vector fields in the polar coordinate system (

_{ϕ}*ρ*,

*ϕ*) attached on the input plane, respectively.

*r*,

*φ*and

*z*are the radial, azimuthal and longitudinal coordinates in the cylindric coordinate system (

*r*,

*φ*,

*z*) attached on the focal plane.

*k*= 2

*π/λ*is the wavenumber and

*λ*is the wavelength in free space, and

*F*is the focal length of the objective.

*P*(

*θ*) is the pupil plane apodization function, which can be chosen to be $P(\theta )=\sqrt{\text{cos}\theta}$ with sin

*θ*=

*ρ/F*.

*θ*is the maximum ray angle passing through the objective, defined as sin

_{m}*θ*=

_{m}*NA*. The incident field is a round field has a radius of

*ρ*=

_{m}*F*sin

*θ*=

_{m}*F*·

*NA*. Applying Eq. (8) to the elliptic-symmetry vector fields we presented above, we can yield the tight focusing fields.

Figure 10 shows the intensity distributions of cylindric- and elliptic-symmetry vector fields in the geometric focal plane. In the case of *f* = 0, the elliptic coordinate system becomes the traditional polar coordinate system, correspondingly, the elliptic-symmetry vector fields degenerates into the cylindric vector fields. We consider only the case of *n* = 0 here. As an example, we give a comparison between the cylindric- and elliptic-symmetry vector fields in two cases *δ*_{0} = 0 and *δ*_{0} = *π*/2 when *m* = 3. The first column in Fig. 10 shows the intensity distributions of the tight focusing fields for the cylindric vector fields, which exhibit clearly the fourfold rotation symmetry. As *f* increases, the cylindric symmetry is broken. This results show the redistribution of the intensity of the tight focusing field in the focal plane. The fourfold rotation symmetry of the tight focusing field intensity pattern becomes to the mirror symmetry (twofold rotation symmetry), as shown from the first column to the sixth column. For instance, for the total intensity in the case of *δ*_{0} = 0, the zero intensity at the centre of the pattern shown in the first column and in the top row becomes the nonzero intensity at the centre shown in the sixth column and in the top row. For the total intensity in the case of *δ*_{0} = *π*/2, as *f* increases, the tight focusing field is evolved to a pattern composed of a pair of “ears” when *f* = 0.9*F*. The transverse and longitudinal components of the tight focusing fields are also shown in Fig. 10. For the transverse components, four strong spots become to two strong spots as *f* increases for both cases of *δ*_{0} = 0 and *δ*_{0} = *π*/2. We can also find that as *f* increases, for the longitudinal components, four relatively strong spots become to one strong spot when *δ*_{0} = 0, while only the intensity weakens when *δ*_{0} = *π*/2 (the intensity pattern is changed slightly). Clearly, due to the flexibly engineerable focal field, the elliptic-symmetry vector fields can be useful in many areas such as optical trapping, optical tweezers, laser machining and so on [1, 12, 14–19].

In summary, we have presented and generated a new kind of elliptic-symmetry vector fields. The elliptic-symmetry vector fields are generated in elliptic-symmetry coordinate system. The properties of this kind of vector fields are similar to the cylindrical vector fields except that one additional controllable parameter *f* is introduced. As the researches of cylindrical vector fields have been very mature, it is necessary to expand the family of vector fields. We present the tight focusing of the elliptic-symmetry vector fields in order to illustrate that the applications of cylindrical vector fields can be expanded to the elliptic-symmetry vector fields, and the new vector fields can also have new properties and be more flexible than the cylindrical vector fields. We hope more conclusions can be gotten by using elliptic-symmetry vector fields.

## Acknowledgments

This work is supported by the National Basic Research Program (973 Program) of China under Grant No. 2012CB921900 and the National scientific instrument and equipment development project 2012YQ17004, the National Natural Science Foundation of China under Grant Nos. 11274183 and 11374166, 111 Project Grant No. B07013, Tianjin research program of application foundation and advanced technology 13JCZDJC33800 and 12JCYBJC10700.

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