Non-iterative dartboard phase filter for achieving multifocal arrays by cylindrical vector beams

Jian Guan; Nan Liu; Chen Chen; Xiangdong Huang; Jiubin Tan; Jie Lin; Peng Jin

doi:10.1364/OE.26.024075

1. Introduction

Cylindrical vector beams have recently drawn a significant amount of interest due to their distinct superiorities for achieving focusing fields with novel properties [1–12]. Radially polarized (RP) beams can achieve smaller focal spots than homogeneously polarized beams when the marginal parts of the beams have more energy [1,13,14]. Complex amplitude modulated RP beams can also generate longitudinally polarized optical needles, which have attractive applications in particle acceleration and scanning microscopy [2,8,15–17]. Azimuthally polarized (AP) beams can be directly focused into doughnut-shaped spots [18], which can be potentially applied in the stimulated emission depletion (STED) microscopy [19] and the absorbance-modulation optical lithography (AMOL) [20]. Moreover, by the amplitude or phase modulation, transversely polarized optical tunnels [21] or needles [9,22,23] can also be generated by AP beams.

Multifocal arrays, which are a kind of focusing fields, can be implemented in parallel optical microscopy or parallel lithography to improve the measurement or processing speed [24–27]. Recently, the multifocal arrays with the controllable position and polarization were generated by cylindrical vector beams, which had promising applications in polarization-multiplexed data storage, polarization-sensitive nanophotonic devices and direct laser writing [27–30]. A phase filter composed of a two-dimensional (2D) grating function and a π-phase-step filter was first proposed to create a multifocal array with controllable polarization [27]. However, the amount of the focal spots that could be generated was limited, only 4. This limitation was caused by the 2D grating function. An iterative phase-retrieval method based on the Gerchberg-Saxton algorithm was proposed, which could achieve a three-dimensional (3D) multifocal array with the controllable polarization [28]. However, the design process requires a large number of iterations and the designed phase pattern is random. Afterwards, the fan-shaped phase filter [29] and the concentric annulus phase filter [30] were proposed to realize 2D multifocal arrays with the controllable polarization. Although these two filters were analytically designed, the division methods for the cell zones in these two filters were one-dimensional (1D) in polar coordinates, along either the azimuthal or radial direction. These 1D cell zone division methods involve only one design degree of freedom. The intensity uniformity of the generated multifocal arrays could only be improved by increasing cell zones in these two filters. When a multifocal array with more foci is desired, plenty of cell zones are needed in these two filters to retain the high intensity uniformity. This will drastically reduce the central angle of a fan-shaped subarea in the fan-shaped phase filter [29], and the width of an annulus in the concentric annulus phase filter [30], which was unconducive to the implement of these two filters.

Here, we proposed a non-iterative method using a dartboard phase filter (DPF) to achieve multifocal arrays by cylindrical vector beams. The polarization and spatial position of each focal spot in the generated multifocal arrays are controllable. The proposed DPF is two-dimensionally divided in polar coordinates, along the radial and azimuthal directions, which makes the filter more flexible to design and easier to implement. Meanwhile, a modulation factor was introduced into the DPF to improve the intensity uniformity of the generated multifocal arrays. The 2D cell zone division method of the DPF and the modulation factor make it easier to achieve multifocal arrays with higher uniformity. In Section 2, the calculating theory and design method for the DPF were introduced. In Section 3, the 1D, 2D and 3D subwavelength multifocal arrays generated by the DPF were presented. The effects of the design parameters of the DPF were discussed in Section 4. The conclusions were shown in Section 5.

2. Theory and method

2.1 Principle of moving the 3D position of the focal spot

The schematic of creating a multifocal array by a DPF is shown in Fig. 1. The DPF is placed in the pupil plane in Fig. 1(a). The generated focal electric field is calculated by [31, 32]:

\begin{array}{l} \vec{E} (x, y, z) = - i A \iint_{r \leq R_{\max}} \frac{P (θ, φ) {\vec{E}}_{T} (θ, φ) e^{i k z \cos θ}}{\cos θ} e^{- i 2 π (x ξ + y η)} d ξ d η \\ = - i A \cdot FT [\frac{P (θ, φ) {\vec{E}}_{T} (θ, φ) e^{i k z \cos θ}}{\cos θ}] \\ = - i A \cdot FT [G (ξ, η)], \end{array}

where A is a constant, λ is the wavelength in the medium and λ₀ = nλ is the wavelength in vacuum. n is the refractive index of the medium in the focal volume. ξ and η are

- \sin θ \cos φ / λ

and

- \sin θ \sin φ / λ,

respectively. FT(·) denotes the Fourier transform. (r, φ) are the polar coordinates in the pupil plane. The maximal radius of the pupil plane is R_max. θ is the converge angle and its maximum is α = arcsin(NA/n). k = 2π/λ is the wave number in the medium. For an aplanatic lens,

P (θ, φ) = \cos^{1 / 2} θ \cdot L (θ) T (θ, φ),

where L(θ) is the amplitude of the incident beam. In this article, the Bessel-Gaussian beam is employed as following:

L (θ) = \exp [- β^{2} {(\frac{\sin θ}{\sin α})}^{2}] J_{1} (2 β \frac{\sin θ}{\sin α}),

where β is the ratio between the pupil radius and the beam waist [18]. J₁(x) is the first kind of Bessel function of first-order. T(θ,φ) is the transmittance function of the proposed DPF.

{\vec{E}}_{T} (θ, φ)

is the intermediate vector denoting the polarization, which is calculated by [3]:

{\vec{E}}_{T} (θ, φ) = [\begin{matrix} \sin^{2} φ + \cos θ \cos^{2} φ & (\cos θ - 1) \cos φ \sin φ & \sin θ \cos φ \\ (\cos θ - 1) \sin φ \cos φ & \cos^{2} φ + \cos θ \sin^{2} φ & \sin θ \sin φ \\ - \cos φ \sin θ & - \sin φ \sin θ & \cos θ \end{matrix}] [\begin{array}{l} p_{x} {\vec{e}}_{x} \\ p_{y} {\vec{e}}_{y} \\ p_{z} {\vec{e}}_{z} \end{array}],

where

{\vec{e}}_{x}

,

{\vec{e}}_{y}

and

{\vec{e}}_{z}

are the unit vectors along the x-, y- and z-axes, respectively.

[p_{x} {\vec{e}}_{x}, p_{y} {\vec{e}}_{y}, p_{z} {\vec{e}}_{z}]^{'}

is the polarization vector of the incident beam.

[p_{x}, p_{y}, p_{z}]^{'}

is

[\cos φ, \sin φ, 0]^{'}

and

[- \sin φ, \cos φ, 0]^{'}

for a RP and an AP incident beam, respectively.

Fig. 1 Schematic of achieving a multifocal array by a DPF. (a) Schematic for the calculation of the focal field. (b) Configuration of the proposed DPF and two examples of the cell zone distribution in a DPF.

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According to the translation property of the Fourier transform, one can obtain the equation that $\vec{E} (x - Δ x, y - Δ y) = - i A \cdot FT [G (ξ, η) \cdot e^{i 2 π (Δ x ξ + Δ y η)}]$ . Moreover, the variable z in Eq. (1) only appears in the term of $e^{i k z \cos θ}$ . Therefore, the following equation can be obtained:

\vec{E} (x - Δ x, y - Δ y, z - Δ z) = - i A \cdot FT [G (ξ, η) \cdot e^{i 2 π (Δ x ξ + Δ y η)} e^{- i k Δ z \cos θ}],

which is the result after translating

\vec{E} (x, y, z)

along the x-, y- and z-axes by Δx, Δy and Δz, respectively. Therefore, the position of the focal spot of the focusing lens can be translated when a phase filter with the following phase function is placed in the pupil plane:

Φ (Δ x, Δ y, Δ z) = - k (Δ x \sin θ \cos φ + Δ y \sin θ \sin φ + Δ z \cos θ) .

2.2 Dartboard phase filter

To achieve multifocal arrays, a phase filter based on the aforementioned principle (as shown in Sec. 2.1) of moving the 3D position of the focal spot was proposed. The configuration of the proposed phase filter is like a dartboard, as shown in Fig. 1(b). For creating a multifocal array with M focal spots, the proposed DPF is divided into N concentric annuli of the equal area where N >M. Each annulus is then divided in to M sectors. Each sector is a cell zone of the DPF. The location of a cell zone in the DPF can be represented by an integer pair of (p, q). p and q respectively are the ordinal numbers of the annulus and sector where the cell zone is located, as shown in Fig. 1(b). Each cell zone of the DPF corresponds to one focal spot in the multifocal array, and the cell zone implements the phase transmittance for this focal spot according to Eq. (5). The ordinal number of a single focal spot in the multifocal array is defined as S. The relationship between S and the cell zone (p, q) in the DPF is described by the function S(p, q), which can be calculated by:

S (p, q) = Γ (q - p + 1, M), p = 1, 2, ..., N; q = 1, 2, ..., M

where

Γ (x, M)

is a function defined as:

Γ (x, M) = {\begin{cases} \mod (x, M) for \mod (x, M) \neq 0, \\ M otherwise . \end{cases}

In Eq. (7), mod(·) is the modulo operation and x represents an independent variable. The formation of a DPF can be explained by a more clear method. In an annulus of the filter, M sectors are successively formed anticlockwise. The M sectors respectively correspond to the focal spots numbered 1 to M in the multifocal array. For the p-th annulus of the filter, the start boundary of the first formed sector in this annulus is the polar axis

φ = \mod [2 π (p - 1) / M, 2 π]

, where p = 1, 2, …, N. By this method, the whole DPF is formed.

Two examples for the distribution of S(p, q) in the proposed DPF with N = 6, M = 5 are shown in Fig. 1(b). In the first example, each sector has the same central angle $2 π / M .$ However, the central angles of the sectors in the second example with F₁ = 0.75 are not equal, because a modulation factor F_j is introduced into the filter. The modulation factor F_j is a number less than 1 and acts on all the cell zones for the j-th focal spot in the desired multifocal array. In a DPF with F_j, the central angle of the cell zones for the j-th focal spot in the multifocal array becomes 2πF_j/M. And the remaining cell zones have the same central angle. If the modulation factor F_j is not introduced, the total area of the cell zones for each focal spot is equal in the DPF, and the j-th focal spot has the highest intensity. However, F_j is introduced into the DPF, which reduces the total area of the cell zones for the j-th focal spot in the DPF. Therefore, the intensity of the j-th focal spot can be reduced by F_j and the uniformity of the multifocal array can be further improved. The detailed effects and usage rule of the modulation factor F_j are discussed in Sec. 4.2.

3. Results

To verify the feasibility of the proposed DPF to achieve multifocal arrays, DPFs with different phase distributions were designed. Various generated multifocal arrays with subwavelength transverse sizes distributed along the z-axis, in the focal plane and in the 3D space are presented. In the following calculations, the parameters NA = 1.4, n = 1.518, λ₀ = 532 nm and β = 1 were employed. Moreover, due to the periodicity of the phase, the final phases of the designed DPFs are the remainders after dividing by 2π.

3.1 Axial 1D multifocal array

A DPF with N = 44 shown in Fig. 2(a) was designed to generate a multifocal array with M = 3 along the z-axis. The designed 3D position shifts for the three focal spots were Δx = 0, Δy = 0 and Δz = [5λ₀, 0, −5λ₀]. The initial phase distribution of the DPF was designed according to Eq. (5). The final phase was the remainder of the initial phase divided by 2π. The modulation factor F_j was not introduced into this DPF, because the uniformity of the multifocal array generated by a RP beam was as high as 99.6%. The uniformity for a generated multifocal array is defined as Q = 1− (I_max−I_min) / (I_max + I_min), where I_max and I_min are the maximal and minimal intensities in the foci, respectively [29]. For the three generated focal spots, the lateral normalized intensity distributions and profiles are shown in Fig. 2(b). For comparison, the lateral normalized intensity profile of the focal spot generated by the same RP beam without the DPF is also plotted in Fig. 2(b), which is denoted by the curve marked with theory. The intensity profiles for the three focal spots generated by the DPF and the single focal spot generated without the DPF almost completely coincide. The full widths at half maximum (FWHMs) of the normalized intensity profiles along the x- and y-directions are [0.50λ₀, 0.49λ₀], [0.50λ₀, 0.50λ₀] and [0.49λ₀, 0.50λ₀] for the three generated spots in the planes of z = −5λ₀, 0 and 5λ₀, respectively. These results also coincide with the FWHM of 0.50λ₀ for the focal spot generated without the DPF. The proposed DPF does not dramatically change the spot shape when it achieves the multifocal array. To make the generated multifocal array more intuitive, the 2D intensity distribution of the generated focal field in the yoz plane is shown in Fig. 2(c). The axial intensity profile, which is shown in Fig. 2(d), indicates that the axial intensity peaks appear at the designed positions. These results demonstrate that the proposed DPF is able to achieve an axial 1D subwavelength multifocal array by a cylindrical vector beam.

Fig. 2 Axial multifocal array generated by a RP beam. (a) Phase distribution of the DPF with M = 3 and N = 44. (b) Lateral normalized intensity for each focal spot of the multifocal array generated by the DPF in (a). (c) and (d) Normalized intensity distribution and profile of the generated multifocal array in the yoz plane and on the z-axis, respectively.

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From Fig. 2(c), it can be found that the two focal spots in the planes with z = −5λ₀ and 5λ₀ are slightly deformed, which indicates these two generated focal spots are not strictly symmetrical about the z-axis. Moreover, for these two spots, there are tiny differences 0.01λ₀ between their each own FWHMs along the x- and y-directions. The asymmetry about the z-axis of these two spots is caused by the asymmetrical cell zone division method of the DPF. The 2D cell zone division method of the DPF is neither rotationally symmetrical about the center of the filter nor symmetrical about the z-axis. However, the deformations of these two spots are so slight, which indicates that the asymmetry of the generated focal field introduced by the proposed DPF can be neglected.

According to the principle of the DPF to achieve multifocal arrays (as shown in Sec. 2), if the distance between two adjacent focal spots is set too small, these two focal spots will affect each other. It may cause the generated focal field to deviate significantly from the desired multifocal array. Therefore, to ensure generating the desired multifocal array, the 3D position shifts Δx, Δy and Δz for each focal spot should be appropriately designed such that two adjacent focal spots can be completely separated. As shown in Fig. 2(d), the minimums of the normalized intensity profile that are closest to the point z = 0 locate at z = ± 1.2λ₀. To make two adjacent focal spots completely separated along the z-axis, the axial distance between these two focal spots should be larger than 2.4λ₀, which is twice of the distance from the intensity peak of the focal spot to the nearest minimum. Therefore, under the calculation parameters in this work, the minimal axial distance between focal spots generated by the RP beam is about 2.4λ₀.

3.2 Transversal 2D multifocal arrays in the focal plane

To further reveal the ability of the proposed DPF to achieve multifocal arrays, the transversal 2D subwavelength multifocal arrays generated by the proposed DPF are presented. A DPF with N = 16 designed to achieve a multifocal array with M = 4 by a RP beam in the focal plane is show in Fig. 3(a1). The designed 3D position shifts for the four focal spots are Δx = [2λ₀, 2λ₀, −2λ₀, −2λ₀], Δy = [2λ₀, −2λ₀, −2λ₀, 2λ₀] and Δz = 0. The modulation factor F_j was not introduced into this DPF, because the uniformity of the four generated focal spots shown in Fig. 3(a2) has already been 100.0%. The normalized intensity profiles along the two dotted lines crossing at the point (−2λ₀, −2λ₀) in Fig. 3(a2) are shown in Fig. 3(a3). The FWHMs of the intensity profile along the x-direction are 0.52λ₀ and 0.53λ₀ for the spots locating at x = −2λ₀ and 2λ₀, respectively. The FWHMs of the intensity profile along the y-direction are 0.53λ₀ and 0.52λ₀ for the spots locating at y = −2λ₀ and 2λ₀, respectively. The differences between these FWHMs are minimal, which indicate that the intensity profiles along the x- and y-directions shown in Fig. 3(a3) are consistent and the shapes of the four generated focal spots are also uniform. Along the normalized intensity profiles shown in Fig. 3(a3), after moving 0.8λ₀ away from the position x = y = −2λ₀, the first minimum is reached. Therefore, under the calculation parameters in this work, the minimal lateral distance between focal spots generated by the RP beam is 1.6λ₀.

Fig. 3 Generated 2D multifocal arrays with M = 4 in the focal plane. (a1) Phase distribution of the DPF with N = 16. (a2) Multifocal array generated by a RP beam and the DPF in (a1). (a3) Normalized intensity profiles along x- and y-directions denoted by dotted lines in (a2). (b1) Phase distribution of the DPF with N = 26. (b2) Multifocal array with the controllable polarization in each focal spot generated by an AP beam and the DPF in (b1). Double arrows indicate the polarization direction in each spot. (b3) Normalized intensity profiles along x- and y-directions denoted by dotted lines in (b2).

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Furthermore, the proposed DPF is able to achieve multifocal arrays with the controllable polarization in each focal spot. The principle of creating a single focal spot with the controllable polarization is modulating the incident AP beam with a π-phase-step filter [27,30]. The π-phase-step makes the incident AP beam constructively interfere at the geometric focus and achieve an elliptically shaped focal spot with the linear polarization. The linear polarization direction of the elliptically shaped focal spot is parallel to the π-phase-step line. Therefore, the polarization direction of the focal spot can be altered by varying the orientation of the π-phase-step line. Moreover, the linear polarization direction of the elliptically shaped focal spot is also parallel to the long axis of the elliptically shaped focal spot. Thus, the polarization direction of the focal spot can be distinguished by the elliptical long axis of the focal spot. For a DPF to achieve a multifocal array with the controllable polarization, the phase of the DPF is composed of two independent parts. One part is the phase to control the positions of the focal spots according to Eq. (5). The other part is π-phase-step filters to control the polarizations for each focal spot. For the j-th focal spot in the multifocal array, all the cell zones in the DPF for this focal spot are modulated by a π-phase-step filter. The π-phase-step line for this filter passes through the center of the DPF and the x-direction coincides with the orientation of the π-phase-step line after rotating Δψ anticlockwise.

To achieve a controllable polarized multifocal array by an AP beam, a DPF with N = 26 and M = 4 was designed as shown in Fig. 3(b1). The designed 3D position shifts for the four focal spots were Δx = [2λ₀, 2λ₀, −2λ₀, −2λ₀], Δy = [2λ₀, −2λ₀, −2λ₀, 2λ₀] and Δz = 0. The designed linear polarization directions of the elliptically shaped focal spots were Δψ = [0, π /4, π /2, 3π /4]. The modulation factor F_j was not introduced into this DPF. The four focal spots in the focal plane generated by the DPF and the AP beam are shown in Fig. 3(b2), with a uniformity of 99.0%. The elliptical shapes of each focal spot are clearly shown in Fig. 3(b2) and the polarization directions of each focal spot are consistent with the designed object. If one is interested in the detailed polarization properties in the entire focal plane, the Stokes parameters can be used as the analysis tool [10]. The normalized intensity profiles along the two dotted lines crossing at the point (−2λ₀, −2λ₀) in Fig. 3(b2) are also plotted, as shown in Fig. 3(b3). In Fig. 3(b3), the two side lobes of the x-directional intensity profile for the spot locating at the point of (−2λ₀, −2λ₀) are not symmetrical about the main lobe. This asymmetry is caused by the asymmetry of the DPF. However, the main lobe of the x-directional intensity profile at x = 2λ₀ nearly completely coincides with the main lobe of the y-directional intensity profile at y = 2λ₀, as shown in Fig. 3(b3). Therefore, the shapes of the main lobes for these four generated focal spots are still uniform for the asymmetrical DPF. These results demonstrate that the proposed DPF can achieve transversal 2D subwavelength multifocal arrays in the focal plane with the controllable position and polarization in each focal spot.

Moreover, along the x-directional normalized intensity profile shown in Fig. 3(b3), after moving 0.9λ₀ away from the position x = −2λ₀, the second minimum is reached. Therefore, perpendicular to the design linear polarization direction, the minimal lateral distance between focal spots generated by the AP beam should be 1.8λ₀ in this work. The second minimum was used here because the side lobe of the x-directional normalized intensity profile shown in Fig. 3(b3) should also be included in the entire focal spot, to avoid the significant interactions between side lobes. Along the y-directional normalized intensity profile shown in Fig. 3(b3), after moving 0.7λ₀ away from the position y = −2λ₀, the first minimum is reached. Therefore, parallel to the design linear polarization direction, the minimal lateral distance between focal spots generated by the AP beam should be 1.4λ₀ in this work.

To further validate the feasibility of the DPF to generate 2D multifocal arrays, DPFs for multifocal arrays with M = 5, 6 and 7 generated by RP and AP beams were designed, as shown in Fig. 4. The design parameters of the DPFs and the uniformities of the generated multifocal arrays are tabulated in Table 1. The results shown in Fig. 4 indicate that the multifocal arrays with the prescribed position and polarization in each focal spot are generated by the designed DPFs. Moreover, the lowest uniformity of the generated multifocal arrays still reaches 93.9%. It indicates that these multifocal arrays have uniform intensities. These results further verify the feasibility of the proposed DPF to generate 2D multifocal arrays.

Fig. 4 Transversal 2D multifocal arrays in the focal plane. (a1)-(c1) Phase distributions of the DPFs for achieving multifocal arrays with M = 5, 6 and 7 generated by RP beam, respectively. (a2)-(c2) Multifocal arrays generated by RP beam and the DPFs in (a1)-(c1), respectively. (d1)-(f1) Phase distributions of the DPFs for achieving multifocal arrays with M = 5, 6 and 7 generated by AP beam, respectively. (d2)-(f2) Multifocal arrays with controllable polarization in each focal spot generated by AP beam and the DPFs in (e1)-(f1), respectively. Double arrows indicate the polarization direction of each spot. The ranges of x- and y-coordinates in (a2)-(c2) and (d2)-(f2) are from −3λ₀ to 3λ₀. Scale bar is λ₀.

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Table 1. Design Parameters and Uniformities of the Generated Multifocal Arrays

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Moreover, from the generated transversal 2D multifocal arrays, it can be found that the uniformity of the multifocal array declines if one focal spot locates at the geometric focus of the lens. Compared with the multifocal array with M = 6 generated by the RP beam, the multifocal array with M = 7 generated by the RP beam only adds a focal spot at the geometric focus of the lens. However, the uniformity decreases from 100.0% to 96.4%, even though a modulation factor F₂ = 0.962 is introduced to improve the uniformity of the multifocal array with M = 7. For the multifocal arrays with M = 6 and 7 generated by the AP beam, the situation is identical. The uniformity decreases from 94.5% to 94.3%. For the multifocal arrays with M = 4 and 5 generated by the RP beam respectively shown in Fig. 3(a2) and 4(a2), the 3D position shift for each focal spot is different. However, when a focal spot in the multifocal array with M = 5 locates at the geometric focus of the lens, the uniformity decreases from 100.0% to 93.9% for the multifocal arrays with M = 4 and 5 generated by the RP beam. For the multifocal arrays with M = 4 and 5 generated by AP beam, the uniformity degradation still exists and the uniformity decreases from 99.0% to 95.3%. These results indicate that the uniformity of the transversal 2D multifocal array will decline when a focal spot locates at the geometric focus of the lens. The reason of this phenomenon can be derived from Eq. (5). For a focal spot located at the geometric focus of the lens, its 3D position shifts are Δx = Δy = Δz = 0. According to Eq. (5), the phase to control the position for this focal spot is a constant zero, while the phases to control the positions for other focal spots are nonzero variables in the DPF. Thus, for a 2D multifocal array in the focal plane, only when a focal spot located at the geometric focus of the lens is desired, the designed DPF will have some zero phase zones. Therefore, the uniformity of the 2D multifocal array will decline when a focal spot locates at the geometric focus of the lens. We introduce the modulation factor F_j into the DPF, which can improve the uniformities for the transversal 2D multifocal arrays with a focal spot locating at the geometric focus of the lens. The detailed effects of the modulation factor F_j are discussed in Sec. 4.2.

3.3 Spatial 3D multifocal arrays

To achieve a multifocal array with more foci, a filter with more cell zones is needed. For a one-dimensionally divided filter, the central angle or radial width of each cell zone will become fairly small when more focal spots are desired. In the proposed DPF, the cell zones are two-dimensionally divided. When the central angles or radial widths of the cell zones are the same, there are more cell zones in the proposed DPF. Therefore, the proposed DPF is easier to achieve multifocal arrays with more focal spots. Using the DPF, 3D multifocal arrays composed of more foci can also be expected. Here, we designed a DPF with N = 380 as shown in Fig. 5(a1) to achieve a 3D multifocal array with M = 15 by a RP beam. In three parallel planes, the generated multifocal array forms a pattern composed of three letters HIT. In the plane z = −10λ₀, the first 7 focal spots form the letter of H. The focal spots numbered from 8th to 10th form the letter of I in the focal plane of the lens. The remaining 5 focal spots form the letter of T in the plane z = 10λ₀. The 8th focal spot locates at the geometric focus of the lens. Therefore, a modulation factor F₄ = 0.960 was introduced to improve the intensity uniformity. The detailed design parameters are shown in Table 2. The multifocal array generated by the designed DPF and the RP beam is shown in Fig. 5(a2) which has an intensity uniformity of 92.5%. The positions of the focal spots are consistent with the design object and a letter pattern of HIT is clearly visible. Moreover, a DPF with N = 359 was designed for an incident AP beam, as shown in Fig. 5(b1). This DPF was used to create a 3D multifocal array with the controllable polarization in each focal spot and the multifocal array could also form a letter pattern of HIT. The designed linear polarization directions of the 3D multifocal array are listed in Table 2. The multifocal array generated by the AP beam and the DPF with N = 359 is shown in Fig. 5(b2). A modulation factor F₃ = 0.928 was introduced and the uniformity of the generated multifocal array was 93.6%. The positions and polarization directions of the generated focal spots also coincide with the design object. These results demonstrate that 3D multifocal arrays can be generated by the proposed DPF.

Fig. 5 Generated 3D multifocal arrays forming the letter pattern of HIT. (a1) and (a2) Phase distribution of the DPF and the corresponding 3D multifocal array generated by a RP beam. (b1) and (b2) Phase distribution of the DPF and the corresponding 3D multifocal array with the controllable polarization in each focal spot generated by an AP beam. Double arrows indicate the polarization direction in each spot.

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Table 2. Design Parameters for 3D Multifocal Arrays Forming the Letter Pattern of HIT

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4. Discussions

4.1 Effects of the design parameter N

There are two design parameters N and F_j, which determine the structure of the proposed DPF according to the desired multifocal array. To reveal the effects of the parameter N, uniformities as functions of N for the multifocal arrays with M = 6 and 7 are shown in Fig. 6(a) and 6(b), respectively. The designed 3D position shifts and linear polarization directions for the multifocal arrays with M = 6 and 7 are shown in Table 1. Moreover, uniformity curves of the annulus filter [30] and the fan-shaped filter [29] changing along with N under the same condition are also plotted in Fig. 6 as reference. The parameter N determines the number of the cell zones for all the three filters. For the uniformity curves of the DPF shown in Fig. 6, the modulation factor F_j is always one which means the each cell zone of the DPF has the same central angle.

Fig. 6 Uniformities as functions of the parameter N for three filters. (a) and (b) Uniformities of the generated multifocal arrays with M = 6 and 7, respectively. The designed 3D position shifts and linear polarization directions for the multifocal arrays with M = 6 and 7 are shown in Table 1.

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The fluctuations of the uniformity curve for the fan-shaped filter are the most violent as shown in Fig. 6. Conversely, there are almost no fluctuations in the uniformity curve for the annulus filter. The fan-shaped filter can achieve higher uniformities than the annulus filter with the increasing of N. For the DPF, the fluctuations of its uniformity curves are smaller than the fan-shaped filter and larger than the annulus filter, except the curve of the DPF for the multifocal array with M = 6 generated by a RP beam shown in Fig. 6(a). Moreover, the uniformities generated by the DPF also become larger than the annulus filter with the increasing of N, sometimes even larger than the fan-shaped filter. These results demonstrate that the proposed DPF can more stably achieve multifocal arrays with higher uniformities. This is because of the 2D division method in polar coordinates for the cell zones, which makes the DPF merging the advantages of the fan-shaped filter and the annulus filter. Additionally, the uniformity curve of the DPF for the multifocal array with M = 6 generated by a RP beam is always one as shown in Fig. 6(a), because of the designed 3D position shifts for the multifocal array and the incident radial polarization.

4.2 Effects of the modulation factor

For a DPF without the modulation factor F_j, N is the unique design parameter and the uniformity of the multifocal array generated by this DPF can only be improved by increasing N. However, it should be noticed that the uniformity will not always be improved by enlarging N. As shown in Fig. 6(b), the uniformity will not be significantly improved after N is larger than 50. Therefore, the modulation factor F_j was proposed as another design parameter of the DPF to improve the uniformity even when the parameter N was not very large. Theoretically, for a DPF designed to achieve any multifocal array, the modulation factor F_j can be employed into the DPF. However, it is unnecessary to apply the modulation factor F_j into every DPF under the practical circumstance. The results in Sec. 3 have indicated that a DPF without the modulation factor F_j can also achieve an enough high uniformity, when a 2D transversal multifocal array without a focal spot locating at the geometric focus or a 1D axial multifocal array is desired. By contrast, when one spot of a 2D multifocal array in the focal plane locates at the geometric focus of the lens, the uniformity of the 2D multifocal array will decline. For a 3D multifocal array, when a focal spot locates at the geometric focus of the lens, the uniformity reduction still exists. The reason is that the 3D multifocal array includes a 2D multifocal array in the focal plane with a focal spot locating at the geometric focus of the lens. Therefore, the modulation factor F_j is necessary for achieving the 2D or 3D multifocal arrays with a focal spot locating at the geometric focus of the lens.

To quantitatively reveal the effects of the modulation factor F_j, uniformity curves of the multifocal array with M = 7 for different modulation factors are shown in Fig. 7. The multifocal array with M = 7 is a 2D multifocal array in the focal plane, which includes a spot locating at the geometric focus. The designed 3D position shifts and linear polarization directions are shown in the last row of Table 1. The ordinal number j of F_j is determined by the ordinal number of the focal spot with the highest intensity in the multifocal array generated by the DPF without the modulation factor F_j. As show in Fig. 7(a), the uniformity curve with F_j = 0.95 is below the uniformity curve with F_j = 1 when N is larger than 30. Therefore, F_j is not always beneficial to improve the uniformity. To improve the uniformity, an appropriate F_j should be determined by parameter sweeping. When F_j = 0.962, the uniformity curve in Fig. 7(a) has larger fluctuations and higher peaks than the uniformity curve with F_j = 1. This result indicates that the uniformity of the multifocal array with M = 7 generated by RP beam is improved after the modulation factor F_j = 0.962 is introduced. In Fig. 7(b), the introduction of F_j = 0.956 raises the uniformity curve than F_j = 1, instead of increasing the fluctuations of the curve. Thus, the introduction of F_j = 0.956 improves the uniformity of this multifocal array with the controllable polarization in each focal spot. These results demonstrated the feasibility of the proposed modulation factor F_j to improve the uniformity of a desired multifocal array.

Fig. 7 Uniformities of the multifocal array with M = 7 for different modulation factors. (a) Uniformities of the multifocal array generated by RP beam and the ordinal number j of the modulation factor F_j. (b) Uniformities of the multifocal array generated by AP beam and the ordinal number j of the modulation factor F_j. The designed 3D position shifts and linear polarization directions are shown in the last row of Table 1.

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5. Conclusion

A non-iterative method using a DPF was proposed to achieve multifocal arrays by cylindrical vector beams. The proposed DPF was designed based on analytic formulas and two-dimensionally divided in polar coordinates, along the radial and azimuthal directions. Due to the 2D cell zone division method, the proposed DPF can more stably achieve multifocal arrays with higher uniformities. Moreover, a modulation factor F_j was first proposed to improve the intensity uniformity of the generated multifocal array. The modulation factor F_j can reduce the total area of the cell zones for the j-th focal spot in the DPF, and then improves the uniformity of the multifocal array. Using the proposed DPF, the axial 1D, transversal 2D and spatial 3D subwavelength multifocal arrays were generated by cylindrical vector beam. Meanwhile, the polarization and spatial position of each focal spot in the generated multifocal arrays are controllable. This work provides a method to achieve multifocal arrays by cylindrical vector beams, especially for the multifocal arrays with the controllable position and polarization. The proposed DPF and the generated subwavelength multifocal arrays can be applied in polarization-multiplexed data storage, polarization-sensitive nanophotonic devices and parallel direct laser writing.

Funding

National Natural Science Foundation of China (NSFC) (61675056, 51475111); Natural Science Foundation of Heilongjiang Province (F2017010).

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M	Designed 3D position shifts and linear polarization directions	N	F_j	Uniformity
5	Δx = [0, 2λ₀, 0, −2λ₀, 0], Δy = [0, 0, 2λ₀, 0, −2λ₀], Δz = 0,Δψ = [π /4, π /4, 3π /4, π /4, 3π /4] (only for AP)	RP: 25 AP: 35	F₂ = 0.978 F₅ = 0.928	93.9% 95.3%
6	Δx = 2λ₀cos(t·π /3), Δy = 2λ₀sin(t·π /3), Δz = 0, t = 0, 1, 2, …, 5, Δψ = t·π /3 (only for AP)	RP: 31 AP: 33	−− −−	00.0% 94.5%
7	Δx = [0, 2λ₀cos(t·π /3)], Δy = [0, 2λ₀sin(t·π /3)], Δz = 0, t = 0, 1, 2, …, 5, Δψ = [π /2, t·π /3] (only for AP)	RP: 44 AP: 40	F₂ = 0.962 F₃ = 0.956	96.4% 94.3%

	Letter H	Letter I	Letter T
Δx	[0, 2λ₀, 2λ₀, −2λ₀, −2λ₀, −2λ₀, 2λ₀]	[0, 0, 0]	[0, 0, 0, 2λ₀, −2λ₀]
Δy	[0, 0, 2.5λ₀, 2.5λ₀, 0, −2.5λ₀, −2.5λ₀]	[0, 2.5λ₀, −2.5λ₀]	[0, −2.5λ₀, 2.5λ₀, 2.5λ₀, 2.5λ₀]
Δz	10λ₀	0	−10λ₀
Δψ (only for AP)	[π /2, 0, π /4, 3π /4, 0, π /4, 3π /4]	[0, π /2, π /2]	[0, 0, π /2, π /4, 3π /4]

M	Designed 3D position shifts and linear polarization directions	N	F_j	Uniformity
5	Δx = [0, 2λ₀, 0, −2λ₀, 0], Δy = [0, 0, 2λ₀, 0, −2λ₀], Δz = 0,Δψ = [π /4, π /4, 3π /4, π /4, 3π /4] (only for AP)	RP: 25 AP: 35	F₂ = 0.978 F₅ = 0.928	93.9% 95.3%
6	Δx = 2λ₀cos(t·π /3), Δy = 2λ₀sin(t·π /3), Δz = 0, t = 0, 1, 2, …, 5, Δψ = t·π /3 (only for AP)	RP: 31 AP: 33	−− −−	00.0% 94.5%
7	Δx = [0, 2λ₀cos(t·π /3)], Δy = [0, 2λ₀sin(t·π /3)], Δz = 0, t = 0, 1, 2, …, 5, Δψ = [π /2, t·π /3] (only for AP)	RP: 44 AP: 40	F₂ = 0.962 F₃ = 0.956	96.4% 94.3%

	Letter H	Letter I	Letter T
Δx	[0, 2λ₀, 2λ₀, −2λ₀, −2λ₀, −2λ₀, 2λ₀]	[0, 0, 0]	[0, 0, 0, 2λ₀, −2λ₀]
Δy	[0, 0, 2.5λ₀, 2.5λ₀, 0, −2.5λ₀, −2.5λ₀]	[0, 2.5λ₀, −2.5λ₀]	[0, −2.5λ₀, 2.5λ₀, 2.5λ₀, 2.5λ₀]
Δz	10λ₀	0	−10λ₀
Δψ (only for AP)	[π /2, 0, π /4, 3π /4, 0, π /4, 3π /4]	[0, π /2, π /2]	[0, 0, π /2, π /4, 3π /4]

Non-iterative dartboard phase filter for achieving multifocal arrays by cylindrical vector beams

Abstract

1. Introduction

2. Theory and method

2.1 Principle of moving the 3D position of the focal spot

2.2 Dartboard phase filter

3. Results

3.1 Axial 1D multifocal array

3.2 Transversal 2D multifocal arrays in the focal plane

3.3 Spatial 3D multifocal arrays

4. Discussions

4.1 Effects of the design parameter N

4.2 Effects of the modulation factor

5. Conclusion

Funding

References

Cited By

Figures (7)

Tables (2)

Equations (7)

Optics Express