Hyperbolic-symmetry vector fields

Xu-Zhen Gao; Yue Pan; Meng-Qiang Cai; Yongnan Li; Chenghou Tu; Hui-Tian Wang

doi:10.1364/OE.23.032238

As an important nature of light, the polarization demonstrates unique features and various novel applications. Manipulation of polarization has seen rapidly growing interest owing to the fact that vector fields with spatially variant polarization states [1–3 ] have resulted in many unexpected effects and applications, such as the far-field focusing beyond the diffraction limit [3–6 ], the light needle of a longitudinally polarized component [7], optical trapping and manipulation of particles [8, 9], and nonlinear optics [10].

Most previous reports focused mainly on the cylindrical vector optical fields with cylindrical symmetry [1–13 ]. However, the vector optical fields with other kinds of symmetry, such as the bipolar symmetry, the parabolic symmetry, and the elliptic symmetry, have also been attracted interest [14–16 ]. The local linear polarizations of these vector fields remind the same along the constant curves of these orthogonal coordinates. Here we present, design, and create a kind of hyperbolic-symmetry vector field with the local linear polarization, and extend to explore the modified hyperbolic-symmetry vector fields with the twofold and fourfold mirror-symmetry polarization states. This work suggests for us that constructing other kinds of new orthogonal coordinate systems makes us be able to more flexible assisting in designing and generating the vector fields. These novel vector optical fields can be used in many areas, such as optical micro-processing patterns, optical trapping, near-field optics, nonlinear optics and so on.

The experimental setup for generating the hyperbolic-symmetry vector fields is shown in Fig. 1, which is a common path interferometric configuration with the aid of a 4f system composed of a pair of identical lenses (L1 and L2), based on the wavefront reconstruction and the Poincaré sphere [2, 9, 12]. This scheme we presented is a universal method for generating various vector fields. An input linearly polarized light delivered from a laser (Verdi V5, Coherent Inc.) is split into two parts, which are achieved by the ±1st orders diffracted from the sine/cosine grating displayed on a phase-only spatial light modulator (SLM) placed in the input plane of the 4f system. The two parts carry the spatial variant phase, which is also achieved by the SLM simultaneously. The two parts pass through different optical paths, making the two parts have orthogonal polarization states by using a pair of λ/4 wave plates behind the spatial filter (SF) placed in the vicinity of the Fourier plane of the 4f system. The SF is used to filter other unnecessary orders of diffraction from the grating displayed on the SLM. The two orthogonally polarized parts are recombined by the Ronchi phase grating (G) placed in the output plane of the 4f system. The phase function of the holographic grating displayed on the SLM has the following form

t (x, y) = \frac{1}{2} + \frac{1}{2} \cos [2 π f_{0} x + δ (x, y)],

where f ₀ is the spatial frequency of the grating and δ (x,y) is the spatially variant phase carried by the grating. This additional phase δ (x,y) is a dominant fact determining the spatial distribution of polarization states of the created vector field. The field amplitude of light reflected by the holographic grating displayed on the phase-only SLM should be proportional to exp[jt(x,y)]. Clearly, this field will be diffracted into many orders, and only the ±1 orders are allowed to pass for generating the vector fields, which is the reason why the SF is used.

Fig. 1 Schematic of experimental setup for generating the desired hyperbolic-symmetry vector optical fields. The main configuration is a 4f system composed of a pair of identical lenses (L1 and L2). A spatial light modulator (SLM) is located at the input plane of the 4f system. Two λ/4 wave plates behind a spatial filter (SF) with two apertures are placed in the vicinity of the Fourier plane of the 4f system. A Ronchi phase grating (G) is placed in the output plane of the 4f system. A polarizer may be inserted in the field, then the intensity patterns can be observed by a PC through a CCD.

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In principle, the input laser incident on the SLM can have arbitrary profile in amplitude. We should emphasize that the input laser has the uniform distribution here, which can be easily achieved by expanding the output beam of the Verdi laser. To generate the local linearly polarized vector fields, the two parts carrying space-variant phase must be orthogonal right- and left-circularly polarized, which requires the two λ/4 wave plates to be used. Thus the generated vector fields in the output plane of the 4f system can be represented as [12]

E (x, y) = \frac{1}{\sqrt{2}} A_{0} [e^{- j δ (x, y)} {\hat{e}}_{+} + e^{j δ (x, y)} {\hat{e}}_{-}] = A_{0} [\cos δ (x, y) {\hat{e}}_{x} + \sin δ (x, y) {\hat{e}}_{y}] .

Here ê ₊ and ê ₋ are the unit vectors describing the right- and left-circular polarizations, and ê _x and ê _y are the unit vectors in the x and y directions, respectively. A ₀ indicates the space-invariant amplitude. This means that the output vector field is local linearly-polarized vector field, because the x and y components are always in phase. Arbitrary vector fields can be generated by changing the additional phase δ (x,y), because δ (x,y) is allowed to possess arbitrary spatial distribution which can be flexibly realized by the SLM.

To generate the hyperbolic-symmetry local linearly polarized vector optical fields, we should first construct a new orthogonal coordinate system—hyperbolic coordinate system (u,v). Before constructing the hyperbolic coordinate system, it should be very beneficial to give an introduction to the existing two dimensional orthogonal coordinate systems including the Cartesian coordinates (x,y), the Polar coordinate system (r,φ), the Parabolic coordinates (u,v), the Bipolar coordinates (u,v), and the Elliptic coordinates (u,v), as shown in Fig. 2. The curves of constant r and φ in Polar coordinate system are a series of circles and rays, as shown in Fig. 2(a). In the Parabolic coordinate system the constant u curves form a set of confocal parabolas that open upwards, whereas the constant v curves form another set of confocal parabolas that open downwards. The two sets of parabolas have a common focus located at the origin, as shown in Fig. 2(b). The bipolar coordinate system has two foci, F ₁ and F ₂, which are located at (− f,0) and (f,0) in the Cartesian system (x,y), respectively. The constant u and v circles are two groups of non-concentric circles and all the constant u circles pass through the two foci, as shown in Fig. 2(c). The elliptic coordinate system also has two foci (F ₁ and F ₂). The constant u curves are a series of confocal ellipses, while the constant v curves are a series of confocal hyperbolas, as shown in Fig. 2(d).

Fig. 2 Four kinds of two dimensional orthogonal coordinate systems. (a) The polar coordinate system. The constant r and φ curves are a series of circles and rays, respectively. (b) The parabolic coordinate system. The constant u and v curves are two sets of confocal parabolas. (c) The bipolar coordinate system. The constant u and v curves are two groups of non-concentric circles. (d) The elliptic coordinate system. The constant u and v curves are confocal ellipses and hyperbolas, respectively.

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Here we construct a new coordinate system that is referred to as the hyperbolic coordinate system (u,v), as shown in Fig. 3. This hyperbolic coordinates, u and v, are defined by the Cartesian coordinates x and y as follows

u = \sqrt{\frac{1}{2} sgn (x + y) sgn (y - x) (y^{2} - x^{2})},

v = \sqrt{sgn (x) sgn (y) x y},

where sgn(·) is the well-known sign function. Clearly, we easily confirm that the constant u and v curves are two sets of hyperbolas in the Cartesian coordinate systems (x,y). In particular, the newly constructed hyperbolic coordinate system is an orthogonal coordinate system, implying that it is a complete coordinate system.

Fig. 3 The new orthogonal hyperbolic coordinate system. Red and blue hyperbolas show the constant u and v hyperbolas, respectively.

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Similar to the generation of various vector fields [2, 3, 12–17 ], we can set the space-variant phase δ of the grating displayed on the SLM in our scheme to have the following form

δ = m u + n v + δ_{0} .

The generated vector field should possess the hyperbolic symmetry of linear polarization. Analogous to the vector fields in the polar coordinate system, m and n can be referred to as the indices in the u and v dimensions, respectively. Notice that the initial phase δ ₀ can change the polarization direction of all points simultaneously. We can set δ ₀ = 0 here, without loss of generality, for exploring the fundamental properties of the vector fields with the hyperbolic symmetry of linear polarization.

By using the principle and the experimental scheme, as mentioned above, we can generate the hyperbolic-symmetry vector optical fields. Firstly, we explore the case when δ depends only on u (m ≠ 0 and n ≡ 0). As shown in Fig. 4, the total intensity pattern exhibits the near uniform distribution. This means that there has no polarization singularity in the generated hyperbolic-symmetry vector fields, which differs from the case of the cylindrical vector fields with one singularity. The simulated intensities of the x (y) component in the second (fourth) row are in good agreement with our experimental results in the third (fifth) row. The intensity of the x component at the centre is always nonzero, because the polarization of the generation vector field at the centre is always the x polarization. The polarization distributions depend only on u independent of v, implying that all positions in the constant u curves (hyperbolas) have the same polarization. So the intensity patterns of x- and y-components exhibit the hyperbolic shapes, and the two ±45° straight lines represent the asymptotic lines of the hyperbolas. These hyperbolas are in agreement with the rad solid curves in Fig. 3, which are the constant u curves. The number of the hyperbolas increases as m increases. There are four mirror-symmetric axes in the intensity patterns of x and y components: the x axis, the y axis and the y = ±x.

Fig. 4 Hyperbolic-symmetry vector fields for different m = 1,3,5,7,9 when n = 0 and δ ₀ = 0. The first row shows the experimentally measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm².

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Figure 5 shows the generated vector fields with the polarization distributions dependent only on v (m ≡ 0 and n ≠ 0). The simulated intensities of the x (y) component in the second (fourth) row are in good agreement with the measured ones in the third (fifth) row. The intensity of the x component at the centre is also nonzero, implying that the polarization of the generation vector fields at the centre is always the x polarization. In this case, the hyperbolas in the intensity patterns of the x and y components agree with the constant v curves (blue dashed curves) in Fig. 3. As stated above, the constant u curves are equivalent to that the constant v curves are rotated by an angle of 45°. So the intensity patterns of the x and y components in Fig. 5 are also rotated 45° with respect to the intensity patterns in Fig. 4. This accords well with the theory of the hyperbolic-symmetry vector fields. In particular, all the intensity patterns of the x (y) component in Figs. 4 and 5 exhibit the bright (dark) four-pointed star shape with four fold symmetry, respectively.

Fig. 5 Hyperbolic-symmetry vector fields for different n = 1,3,5,7,9 when m = 0 and δ ₀ = 0. The first row shows the experimentally measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm².

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We now discuss the case when the additional phase δ depends on both u and v, i.e. both m and n are nonzero simultaneously. Figures 6, 7, and 8 show the hyperbolic-symmetry vector fields, for different n = (1,3,5,7,9) when m = 5 and δ ₀ = 0, for different m = (1,3,5,7,9) when n = 7 and δ ₀ = 0, and for different combinations of m and n with m = n = (1,3,5,7,9) and δ ₀ = 0, respectively. We can see from the first row in the three figures that any total intensity pattern has no polarization singularity. The intensity patterns of the x and y components exhibit multi hyperbolas and the number of the hyperbolas increases as m and n increases. The y component intensity at the center is always zero. When m ≠ n the intensity patterns of the x and y components exhibit the fourfold symmetry, but when m = n the intensity patterns of the x and y components exhibit the nested eight-pointed star shape with the eightfold symmetry because of the equal contributions of u and v when m = n. We should emphasize that in the measured eight-pointed star patterns in the first columns in Figs. 6 and 7, four protrusions in the x and y axes are not clear.

Fig. 6 Hyperbolic-symmetry vector fields for different n = 1,3,5,7,9 when m = 5 and δ ₀ = 0. The first row shows the measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm².

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Fig. 7 Hyperbolic-symmetry vector fields for different m = 1,3,5,7,9 when n = 7 and δ ₀ = 0. The first row shows the measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm².

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Fig. 8 Hyperbolic-symmetry vector fields for different combinations of m and n with m = n when δ ₀ = 0. The first row shows the measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm².

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The intensity patterns of the x component have 4 protrusions when (m ≡ 0, n ≠ 0) or (m ≠ 0,n ≡ 0) and 8 protrusions when m ≠ 0 and n ≠ 0. These protrusions are along the x = 0, the y = 0 and y = ±x lines. Such a phenomenon can be understood as follows. We can rewrite the hyperbolic coordinates (u,v) by the polar coordinate system (r,φ) as

u = \frac{1}{\sqrt{2}} r \sqrt{| \cos (2 φ) |} and v = \frac{1}{\sqrt{2}} r \sqrt{| \sin (2 φ) |} .

For the x component, the radial coordinate r should have the form

r = \frac{\sqrt{2} l π}{m \sqrt{| \cos (2 φ) |} + n \sqrt{| \sin (2 φ) |}},

where the integer l is defined as mu + nv = lπ (l = 0,±1,±2,…).

To discuss the variation trend of the intensity of the x component with the azimuthal angle φ, we need to determine the derivative of r in the above formula with respect to φ, as follows

\frac{\partial r}{\partial φ} = \frac{2 l π [m sgn [\cos (2 φ)] {| \sin (2 φ) |}^{3 / 2} - n sgn [\sin (2 φ)] {| \cos (2 φ) |}^{3 / 2}]}{{[m \sqrt{| \cos (2 φ) |} + n \sqrt{| \sin (2 φ) |}]}^{2} \sqrt{| \sin (4 φ) |}} .

We can find from Eq. (7) that r should have the minimum value when ∂r/∂φ = 0, i.e.

{| \tan (2 φ) |}^{3 / 2} = \frac{n}{m} .

Based on Eq. (8), we can understand the intensity patterns of the x component. (i) In the case when n ≡ 0 and m ≠ 0 as shown in Fig. 4, there occur minimum values of r at φ = 0, π/2, π and 3π/2. There exists one protrusion between two minimum values of r, because the monotonicity of r along φ is opposite between the adjacent two minimum values of r. As a result, the intensity pattern of x component has 4 protrusions which are along y = ±x lines (i.e. at azimuthal positions of φ = π/4, 3π/4, 5π/4 and 7π/4, as shown in Fig. 4). (ii) In the case when m ≡ 0 and n ≠ 0 as shown in Fig. 5, there occur minimum values of r at φ = π/4, 3π/4, 5π/4 and 7π/4. In the same way, we can conclude that the intensity pattern of x component has 4 protrusions which are along x = 0 and y = 0 lines (i.e., at azimuthal positions of φ = 0, π/2, π and 3π/2, as shown in Fig. 5). (iii) In the case of m = n (as shown in third row of Fig. 6, fourth row of Fig. 7, and Fig. 8), there occur minimum values of r at azimuthal positions of φ = π/8, 3π/8, 5π/8, 7π/8, 9π/8, 11π/8, 13π/8, and 15π/8. As a result, the intensity pattern of the x component has 8 protrusions which are along x = 0, y = 0 and y = ±x lines (i.e., at azimuthal positions at φ = 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, and 7π/4), as shown in third row of Fig. 6, fourth row of Fig. 7, and Fig. 8. (iv) In the case of m ≠ n ≠ 0 (as shown in Figs. 6 and 7), there occur minimum values of r at φ = (1/2)arctan(n/m)² ^/ ³ + kπ/4 (where k is an integer). The 8 protrusions of the x component intensity are still along x = 0, y = 0, and y = ±x lines (i.e., at azimuthal positions of φ = 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, and 7π/4), and the positions of the minimum values of r in the intensity patterns of the x component are mirror-symmetric about the x = 0, y = 0, and y = ±x lines, as shown in Figs. 6 and 7. The discussion of the y-component is completely similar to that of x-component, because the intensity patterns of the y-component are complementary to those of the x-component.

In the above, we focus on the generation of various hyperbolic-symmetry vector fields. It is very interesting to explore the tight focusing behaviors of such a kind of vector fields. As mentioned above, the polarization states of the hyperbolic-symmetry vector fields are neither vertically nor horizontally symmetric, but the intensity patterns, including the intensity patterns of the x and y components, exhibit the vertical and/or horizontal symmetry, because the polarization at symmetric positions besides the x and y axes are the same. The hyperbolic-symmetry vector fields have no mirror symmetry in the polarization states, this fact results in that the tight focusing patterns are also a lack of the mirror symmetry. For many applications including optical machining, we may need tightly focused fields with specific mirror symmetric intensity distributions, so we now will explore the tight focusing behaviors of the modified hyperbolic-symmetry vector fields with the mirror symmetric polarization states. Here we focus on two kinds of modified vector fields which have the twofold and fourfold symmetries of polarization states. The space-variant phase δ has two modified forms as follows

δ^{'} = (m u + n v - π / 2) sgn (x) sgn (y) - (π / 2) sgn (y) + π,

\begin{matrix} δ^{″} = 2 π - (π / 2) sgn (y) + (m u / 2 + π / 4) sgn (y) sgn (x - y) \\ + (m u / 2 - 3 π / 4) sgn (y) sgn (x + y) + (m u / 2 - π / 2) sgn (x) sgn (x - y) \\ - (m u / 2 + π / 2) sgn (x) sgn (x + y) . \end{matrix}

When selecting δ′, the polarization states of the vector fields in the first quadrant in the Cartesian coordinate system (x,y) are retained, while the polarization states in other three quadrants are redesigned according to the polarization states in the first quadrant to achieve the twofold symmetry about the x and y symmetric axes. For the case of δ″, the polarization states of the vector optical fields in the section (0 < φ < π/4) are retained, while the other sections are redesigned based on the principle of symmetry. In this case, the polarization states of the vector optical fields have four mirror symmetric axes (the x and y axes and the y = ±x lines).

For comparison, we first simulate the local linear polarization distribution of the original and modified vector fields, as shown in Fig. 9. The polarization states of the original vector fields are schematically shown in Fig. 9(a) for (m,n) = (5,0), in Fig. 9(b) for (m,n) = (0,3), and in Fig. 9(c) for (m,n) = (5,3), respectively. We can see from Figs. 9(a)–9(c) that the polarization states of the original vector fields have no mirror symmetry. The polarization states are the same in the constant u and v curves in theory, but it is difficult to find this phenomenon in Figs. 9(a)–9(c) due to the limited number of points in simulation. The polarization states of the modified vector fields with twofold and fourfold mirror symmetries are shown in Figs. 9(d)–9(f) and Figs. 9(g)–9(i), respectively. In detail, the symmetric axes in Figs. 9(d)–(f) are the x and y axes, and those in Figs. 9(g)–9(i) are the x and y axes and the y = ±x lines.

Fig. 9 Stimulated polarization states of the hyperbolic-symmetry vector fields for different parameters. The first row shows the polarization states of the original vector fields. The second and third rows correspond to the cases of the twofold and fourfold mirror symmetries of polarization states, respectively. The first, second, and third columns correspond to the cases of (m,n,δ ₀) = (5,0,0),(0,3,0),(5,3,0), respectively.

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Five columns in Fig. 10 from the left to the right show the vector fields with five kinds of hyperbolic-symmetry vector fields, with different n = 1, 3, 5, 7, and 9 and the same m = 5 and δ ₀ = 0, respectively. The first and second rows in Fig. 10 show the patterns of the total intensity and the x component intensity of the original hyperbolic-symmetry vector fields, respectively. In this case, although the polarization states have no mirror symmetry, the total intensity is spatially homogenous and the x component intensity patterns have four mirror-symmetric axes (the x and y axes, and the y = ±x lines) and exhibit the fourfold rotation symmetry. In particular, for the case of m = n as shown in the third column, the x component intensity patterns have eight mirror-symmetric axes [the x and y axes, the y = ±x lines, the y = ±tan(π/8)x lines, and the y = ±tan(3π/8)x] as well as exhibit the eightfold rotation symmetry. The third and fourth rows in Fig. 10 show the first kind of modified hyperbolic-symmetry vector fields corresponding to δ′ in Eq. (9a), with their polarization states having the mirror symmetry about the x and y axes. The total intensity patterns have the zero-intensity singular short lines along the x and y axes due to the polarization uncertainty and the x component intensity patterns exhibit the mirror symmetry about the x and y axes. As n increases, the number (length) of the singular short lines has the increasing (decreasing) tendency. In the intensity patterns of the x components, there are also two symmetric axes when m ≠ n and four symmetric axes when m = n.

Fig. 10 The modified hyperbolic-symmetry vector fields for different n = 1,3,5,7,9 when m = 5 and δ ₀ = 0. The first and second rows show the original hyperbolic-symmetry vector fields, for comparison. The third and fourth rows show the modified hyperbolic-symmetry vector fields with twofold mirror-symmetric polarization states. The fifth and sixth rows show the modified hyperbolic-symmetry vector fields with fourfold mirror-symmetric polarization states. Any picture has the same dimension of 4 × 4 mm².

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The fifth and sixth rows in Fig. 10 show the second kind of modified hyperbolic-symmetry vector fields corresponding to δ″ in Eq. (9b). In this case, their polarization states have four mirror-symmetric axes (the x and y axes, and the y = ±x lines). The total intensity patterns have the zero-intensity singular short lines along the above four mirror-symmetric axes due to the polarization uncertainty, and exhibits the mirror symmetry about the above four mirror-symmetric axes and fourfold rotation symmetry. However, the x component intensity patterns exhibit the mirror symmetry about the x and y axes only. The intensity patterns of the x components have two mirror-symmetric axes (the x and y axes). As n increases, the number (length) of the singular short lines has the increasing (decreasing) tendency.

We now explore the tight focusing property of vector optical fields, due to its great interest [3–10 , 14–20 ]. Based on the theory proposed by Richards-Wolf [4, 19], we simulate the tightly focused fields of the modified hyperbolic-symmetry vector fields by an objective with NA = 0.9, as shown in Fig. 11. We select the modified hyperbolic-symmetry vector fields instead of the hyperbolic-symmetry vector fields, owing to the following reasons. As the polarization states of the hyperbolic-symmetry vector fields are a lack of mirror symmetry, the tightly focused field have also no mirror symmetry. For many applications, it is needed that the tightly focused fields should have mirror-symmetric intensity distributions. In addition, this also gives a compare with the vector fields with cylindrical symmetric polarization states (radically and azimuthally polarized vector optical fields). The top row of Fig. 11 shows the tightly focused fields of the vector fields with their polarization states having twofold mirror symmetry about the x and y axes. When (m,n,δ ₀) = (0.5,0.5,0), the focal field is a sharp line along the x axis with the figure of merit being 0.7, which is superior to those in [15, 16, 20]. When (m,n,δ ₀) = (2,3,0), the focal field becomes two strong spots along the y axis. When(m,n,δ ₀) = (0.5,5.5,π/2), the focal field exhibits four strong spots. When (m,n,δ ₀) = (3,3,π/2), the focal field exhibits a pattern similar to a pair of ears. When (m,n,δ ₀) = (7.5,5.5,π/2), the focal field is similar to a ladder attaching a pair of sub-strong ears.

Fig. 11 Simulated tightly focused field for the modified hyperbolic-symmetry vector fields with different m, n and δ ₀ by using an objective with NA = 0.9. The first and second rows correspond to the vector fields with the twofold and fourfold mirror-symmetric axes for polarization states, respectively. Any picture has a dimension of 4λ × 4λ.

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The bottom row of Fig. 11 shows the tightly focused fields of the vector fields with their polarization states exhibiting fourfold mirror symmetry about the x and y axes and the y = ±x lines. We can see that all the five focal patterns exhibit the fourfold rotation symmetry and have four mirror-symmetric axes (the x and y axes and the y = ±x lines). The focal field for (m,n,δ ₀) = (0,3.5,0) has a cross-shaped intensity pattern. For (m,n,δ ₀) = (0,5,π/2), the focal field has a pattern composed of four strong rounded rectangles. For (m,n,δ ₀) = (5.5,0.5,π/2), the focal field exhibits a pattern composed of four strong rounded trapezoids. For (m,n,δ ₀) = (1.5,3,π/2), the focal field is composed of four strong rounded triangles. For (m,n,δ ₀) = (9,4,π/2), the focal field is a nested structure with four inner strong stripes and four outer strong strips.

To confirm the simulated focusing behaviors of the hyperbolic-symmetry vector optical fields, we use them to ablate the microstructures on the K9 glass surface by using an objective with NA = 0.75. In our experiment, the used laser source was a femtosecond laser with a pulse duration of 35 fs operating at a wavelength of 800 nm and a repetition rate of 1 kHz. The experiment confirmed that when the power is beyond 0.30 mW, the microstructures can appear on the K9 glass. As examples, we select three modified hyperbolic-symmetry vector fields for fabricating the microstructures. For the first vector field with (m,n,δ ₀) = (0.5,0,π/2), its polarization states have the twofold mirror symmetry (two mirror-symmetry axes, the x and y axes). The second vector field with (m,n,δ ₀) = (0.5,6.5,0) and the third one with (m,n,δ ₀) = (10,5.5,π/2) have the fourfold mirror symmetry (four mirror-symmetry axes, the x and axes as well as the y = ±x lines). The simulated tightly focused fields of the three vector fields are shown in the top row of Fig. 12. For the first vector field, its focal field has two strong spots. The focal field of the second vector field has four strong spots. The focal field of the third vector field is similar to the bottom-right pattern in Fig. 11. We can see that under the condition when the laser power is 0.39 mW, the microstructures fabricated by ablating the K9 glass surface are in good agreement with the focal field patterns of the three vector fields, as shown in the bottom row of Fig. 12.

Fig. 12 The intensity patterns of the focal fields of three hyperbolic-symmetry vector fields by an objective with NA = 0.75 and the micro-structures on the K9 glass surface fabricated by them. The top row shows the intensity patterns of the tightly focused fields of the three hyperbolic-symmetry vector fields with (m,n,δ ₀) = (0.5,0,π/2),(0.5,6.5,0),(10,5.5,π/2). The bottom row shows the micro-structures on the K9 glass surface fabricated by the corresponding focal fields in the top row, which are the optical microscopic CCD image under the illumination of white light. Any picture has the same dimension of 10 × 10 μm².

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In summary, we present and construct a new orthogonal coordinate system, called as the hyperbolic coordinate system. Based on this newly constructed coordinate system, we successfully generate a new kind of hyperbolic-symmetry vector fields and modified hyperbolic-symmetry vector fields and then explore their novel tight focusing properties. We also fabricate the microstructures on the K9 glass surfaces, by using the tight focal fields of the hyperbolic-symmetry vector fields and modified hyperbolic-symmetry vector fields. The results demonstrate that the fabricated microstructures are in good agreement with the simulated tight focal fields of the hyperbolic-symmetry vector fields. This work provides for us a suggestion that constructing other kinds of new orthogonal coordinate systems makes us be able to more flexible for designing and generating the vector fields.

Acknowledgments

This work is supported by the National Basic Research Program (973 Program) of China under Grant No. 2012CB921900 and the National Natural Science Foundation of China under Grant No. 11534006.

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Hyperbolic-symmetry vector fields

Abstract

Acknowledgments

References and links

Cited By

Figures (12)

Equations (11)

Optics Express