We numerically and experimentally explored generation and regulation of subwavelength multiple focal spots produced by tight focusing patterned vector optical fields (PVOFs). We presented a modified Richard-Wolf diffraction integration method suitable for the tight focusing of the PVOFs. By tailoring the spatial geometry and the polarization distributions of the PVOFs, simulations show that the diverse spatial configurations of subwavelength multiple focal spots can be achieved. To verify our idea, we experimentally generated the theoretically calculated examples of femtosecond PVOFs, then tightly focused them on the surface of the crystalline silicon wafers, and finally characterized the morphologies of modified surfaces. The SEM (scanning electronic microscopy) images confirmed that the experimental results are in good agreement with the simulations. Based on the diverse controlling degrees of freedom of PVOFs, the resultant subwavelength focal fields are flexible and powerful in parallel processing, optical manipulation and so on.
© 2013 Optical Society of America
Recently, vector optical fields (VOFs), as a kind of novel optical fields with space-variant distribution of states of polarization (SoPs), have attracted great attention due to their novel properties [1–17]. The unique properties of the cylindrical VOFs (e.g., radially and azimuthally polarized ones) are to produce a far-field focal spot beyond diffraction limit [2–4] and to form a strong longitudinal electric component  under the tight focusing condition, which leads to important applications, such as high-resolution microscopy  and particle trapping . Due to the diversity of spatial SoP distributions of the VOFs, micro-drilling or micro-machining based on the ultrashort VOFs has been studied, showing different morphology of microholes, different micro-machining quality and different surface microstructures [14–17].
To improve the micromachining efficiency, parallel processing using multiple beams generated by a spatial light modulator (SLM) [18, 19] or multi-beam interference [20–22] have been used to fabricate multi-microholes, and two- and three-dimensional microstructures. However, these methods are less flexible and all the beams are scalar fields with space-invariant SoP distribution. To utilize the VOF, it is of great importance to control both the intensity and SoP distributions in its focal plane under tight focusing condition. For a single VOF with its center located at the axis of the optical path in the pupil plane, it is easy to give the focal intensity distribution by the Richard-Wolf diffraction integration method [1, 23, 24].
Multiple individual VOFs can flexibly design patterned vector optical fields (PVOFs), in particular, which are easily created with aid of the SLM . The diversity of individual VOFs in both their spatial arrangement and their SoP distribution give PVOFs a large advantage in applications. The focused PVOFs will certainly result in various focal patterns, which are useful in fabricating multi-microholes and three-dimension microstructures. Based on the Huygens-Fresnel principle, we deduced the intensity distribution at the focal plane for weakly focused PVOFs and fabricated multi-microholes with various patterns on silicon .
In this article, we demonstrate the realization of subwavelength multiple focal spots by tightly focused PVOFs composed of multiple individual VOFs. To perform this task, we first develop a modified Richard-Wolf diffraction integration method and then simulate the tight focusing properties of the PVOFs. Finally, we carry out experiments for fabricating the multihole microstructures on the silicon surface to verify our simulation results. We confirm that the experimental results are in good agreement with the simulation results. In particular,the spot sizes,geometric configuration and local SoPs of the multiple focal spots of the focused PVOF can be engineered by tailoring the spatial arrangement and the SoPs of the multiple individual VOFs forming the PVOF.
2. Focal fields of tightly focused PVOFs
Here we focus only on the situations that the PVOFs are composed of multiple identical VOFs, in particular, all the identical VOFs are cylindrical VOFs. Before exploring the PVOFs, it is very useful that we should first pay attention to a single VOF. In the input plane with a polar coordinate system (ρ, φ), a cylindrical VOF can be written as [1, 11, 13]11, 13, 14]. When the center of the individual VOF described by Eq. (1) is located at the axis of the focusing optical system, by using the Richard-Wolf diffraction integral formula under the tight focusing condition, its focal field E(r,ϕ,z) in a cylindrical coordinate system (r,ϕ,z) can be written as [1, 2]
In the present article, we study the tight focusing behavior of the PVOFs composed of multiple identical VOFs which are arranged regularly. The arrangement of the PVOF in the input plane with a Cartesian coordinate system (x, y) and a corresponding polar system (ρ, φ) is schematically shown in Fig. 1. To clearly describe the position of every individual VOF and the symmetry of the PVOF, it is convenient to introduce another skew coordinate system (ξ, η), which is in fact the lattice coordinate system. The origins of the two coordinate systems, and the axis x and the axis ξ coincide. The symmetry of the PVOF is determined by the included angle β between the two axes ξ and η. dξ and dη are the lattice constants along the ξ and η axes, respectively. The position of a certain individual VOF [which is defined by two indices (i, j), where i and j are integers] can be characterized by (Xij, Yij) in the Cartesian coordinate system (x, y) and by (ξi, ηj) in the lattice coordinate system (ξ, η). ξi and ηj can be written as
To describe the (i, j) individual VOF of the PVOF, we build a local polar coordinate system (ρij, φij) with its origin located at (Xij, Yij) or (ξi, ηj), which is the center of the individual VOF, as shown in Fig. 1. So that, as Eq. (1), the (i, j) individual VOF can be written at in the local coordinate system as
Inasmuch as the center of the (i, j) individual VOF deviates from the axis of the optical system, we cannot directly use the Richard-Wolf integral formula Eq. (3) to calculate the focal field near the focal plane. Based on the transformation relations of ρ and φ with ρij and φij below, we can obtain the contribution of the (i, j) individual VOF to the total focal field under the tight focusing condition, as follows
3. Simulation results
To manipulate the focal field distribution of the tightly focused PVOF near the focal plane, we use the parameters, such as θij, dξ, dη, ρ0, β, mij, φ0ij, and NA. It should be pointed out that NA is effective numerical aperture of the PVOF and is defined as NA = ρmax/f, where ρmax is the maximum radius of the PVOF. In our simulations, all the individual VOFs are closely arranged (i.e., dξ = dη = d0 = 2ρ0), thus the relation ρmax = (2n + 1)ρ0 determines the number n (or the radius ρ0) of the individual VOFs forming the POVF for a given ρ0 (or a given n). For example, for a focal lens with NA = 0.85, then ρ0 is determined to be ρ0 = 0.077 f for the given n = 5. Here our simulations focus on the PVOFs with two kinds of symmetries (triangular and square), corresponding to β = π/3 and β = π/2, respectively.
To implement the numerical simulations, by using our modified Richard-Wolf integral formula, for a given NA and a chosen lattice symmetry (or β), we should first determine n or ρ0 and then give the position of every individual VOF. Thus we can obtain the expression of the (i, j) individual VOF of the input PVOF as Eq. (7), and then determine the lattice arrangement of the input PVOF as Eq. (8). With the aid of Eqs. (9) and (10), we can simulate the focal field generated by the (i, j) individual VOF. In the last step, we give the total focal field by the coherent superposition of the focal fields of all individual VOFs, as Eq. (11). In addition, to accurately and time-savingly simulate the focal fields, we should carefully determine the most suitable increments in radius and angle for an individual VOF, because the oversized increments will result in the low accuracy while the undersized increments will be time-consuming.
For comparison, we first consider three simple situations: three PVOFs which are composed of multiple individual VOFs that are (i) circularly polarized vortices, (ii) linearly polarized vortices, and (iii) azimuthally polarized vector fields, respectively. They are expressed as followsFigure 2 shows the simulation results for the above situations. The first column of Fig. 2 shows the configurations of the three input PVOFs in the input plane. The other four columns of Fig. 2 show the intensity distributions of the focal fields of the tightly focused PVOFs in the focal plane. Clearly, the total intensities in the focal plane for all three PVOFs have six subwavelength spots which exhibit a hexagonal arrangement, as shown in the second column of Fig. 2. The focal spots are circular, elliptical, and gibbous moon-shaped for these three PVOFs, respectively.
The focal spots of the input PVOF composed of the circularly polarized vortices have the same size of 0.62λ (full width at half maximum, FWHM). For the PVOF composed of the linearly polarized vortices, the elliptical focal spots with their long axes orientated along the direction of linear polarization have the long and short axes of 0.73λ and 0.56λ (FWHM), respectively. The gibbous-shaped focal spots for the PVOF composed of the azimuthally polarized vector fields exhibit a six-fold rotation symmetry. Evidently, the symmetry of the focal pattern depends on the symmetry of the SoPs of the individual VOFs forming the PVOF. From the third to fifth columns, we can see that the contribution of the r-component (or ϕ- or z-components) to the total focal field depends on the SoP symmetry of the PVOF. For instance, the six spots of the r-component (or ϕ- or z-components) have the identical contribution to the total focal field, for the PVOF composed of the circularly polarized vortices. The two spots of the r-component and the four spots of the ϕ-components have the dominant contribution to the total focal field, while the z-component has the identical contribution to the total focal field (although small), for the PVOF composed of the linearly polarized vortices. The r-component has no contribution while the ϕ-component has the dominant and identical contribution and the z-component has a small and identical contribution, to the total focal field, for the PVOF composed of azimuthally polarized vector fields.
To investigate the influence of the NA of the PVOF on the tight focusing field, we construct five different PVOFs with the same trigonal lattice, composed of identical azimuthally polarized VOFs. However, the five PVOFs have different NAs (NA = 0.27, 0.45, 0.63, 0.81 and 0.99), i.e., different numbers of the individual VOFs (n = 1 ∼ 5). The simulation results are shown in Fig. 3, where the top row shows the PVOFs in the input plane, and the middle and bottom rows show the intensity patterns in the xy-plane and the yz-plane of the tight focusing fields in the vicinity of focus. It can be seen that the NA does not change the positions of the focal spots, as shown in the second row. In contrast, the spot sizes decrease from 1.92λ to 0.47λ as NA increases from 0.27 to 0.99, as shown in the middle row of Fig. 3 and Fig. 5(a), which is easily understood with the help of the difference between the dual-beam interference and the multiple-beam interference. Correspondingly, the depth of focus decreases from 26.81λ to 1.39λ, as shown in the bottom row. In addition, the spot shape changes from circular, elliptical, to gibbous moon-shaped as NA increases.
To explore the influence of the number n (or the radius ρ0) of the multiple individual VOFs forming the PVOF on the tight focusing behavior when NA (or ρmax) is kept a constant, we construct five different PVOFs with the trigonal lattice, composed of a different number of azimuthally polarized VOFs. Figure 4 shows the simulation results when NA ≡ 0.85, where the parameters used for simulations are mij = 1, φ0ij = π/2 and β = π/3. We can see from Fig. 4 that the six focal spots exhibit a hexagonal arrangement, which is the same as Figs. 2 and 3, originating from the trigonal lattice symmetry of the PVOF. As shown in the middle row of Fig. 4, it is obvious that the sizes and shapes of the focal spots have no change for different number n of the individual VOFs. In addition, the depths of focus of the focal spots have almost no change with changing n. However, the interval between any pair of antipodal focal spots increases near linearly with n, as shown in the bottom row of Fig. 4 and Fig. 5(b), that is to say, the interval is inversely proportional to the radius ρ0 of the individual VOF. This phenomenon is easily understood as follows. Similar to the diffraction pattern of the crystal, as is well known, the interval between spots in the diffraction pattern in the reciprocal space increases linearly with the decrease of lattice constant of the crystal in the real space. For the tight focusing situation of the PVOFs, the arrangement of individual VOFs in the input plane is analogous to the atomic arrangement of the crystal in the real space, while the tight focusing pattern in the focal plane is analogous to the diffraction pattern of the crystal in the reciprocal space. Naturally, we can easily understand the phenomenon that the interval between the focal spots increases linearly as n increases, because the lattice constant of the PVOF decreases linearly.
From the simulation results shown in Figs. 2 – 5, the sizes of focal spots can be controlled by the NA (or the whole size of the PVOF), the interval between the focal spots can be manipulated by the number of the individual VOFs, and the shapes of focal spots depend on the SoPs of the individual VOFs. As is well known, the SoPs of the individual VOFs can be changed by varying the initial phase φ0ij and the topological charge mij, for example, the VOFs are radially and azimuthally polarized when φ0ij = 0 and π/2 for mij ≡ 1, respectively [1, 11, 13]. Having investigated the situation in which all the individual VOFs forming the PVOF have the same SoPs, we also find it very interesting to consider the situation in which the individual VOFs are allowed to possess different SoPs, implying that the SoP configuration of the PVOF another controlling degree of freedom can be used to manipulate the focal field. In addition, the parameter β, e.g. the lattice symmetry of the PVOF, can also be used to control the focal field.
Figure 6 shows the simulation results for the effect of the SoP configuration and the lattice symmetry of the PVOF on the focal field. The SoP configuration of the PVOF can be generated according to a certain arrangement rule of φ0ij, for β = π/2 (square lattice) in the top row and β = π/3 (trigonal lattice) in the bottom row. Figure 6(a) shows the lattice arrangements of the PVOFs. Figures 6(b)–6(e) show the tight focusing fields of the PVOFs generated by the rules of φ0ij = π/2, φ0ij = (−1)iπ/2, φ0ij = (−1)jπ/2, and φ0ij = (−1)i(−1)jπ/2, respectively, implying that the SoP of a certain individual VOF is determine by its position indices i and j. As shown in Fig. 6(b), in where all the individual VOFs have the same SoPs, the focal spots are arranged as a square and a hexagon for β = π/2 and β = π/3, respectively, which agree with their lattice symmetries. However, as shown in the top row of Figs. 6(c)–6(e), the focal field patterns exhibit a rectangle (orientated in the y and x directions) composed of six spots and a square composed of four spots, for β = π/2 (square lattice), respectively. As shown in the bottom row of Figs. 6(c)–6(e), the three focal field patterns exhibit a rhombus composed of four spots, with different orientations, for β = π/3 (trigonal lattuce). In addition, when φ0ij obeys the arrangement rule of φ0ij = (−1)i(−1)jπ/2, we find that the orientation angle of the long diagonal of the rhombus is π − β with respect to the +x direction, as shown Fig. 6(e). For example, the orientation angles are π/4 (π − π/2) for β = π/2 and 3π/4 (π − π/3) for β = π/3, respectively.
The influence of the SoP configuration and the lattice symmetry of the PVOFs on the numbers and positions of the focal spots can be understood as follows. The focal patterns of the PVOFs in the reciprocal space depend on the lattice symmetry and the unit cell of the PVOF in the real space. The positions of the focal spots depend on the lattice symmetry and the number of the focal spots is dominated by the unit cell. For instance, in the bottom row of Fig. 6, the PVOF in the case of Fig. 6(b) exhibits a triangular lattice, so that the focal spots should also exhibit a triangular arrangement, finally the focal pattern forms a hexagon composed of six spots due to the modulation of the focal pattern with a doughnut shape for the unit cell (an individual VOF). In the three cases of Figs. 6(c)–6(e), the PVOFs exhibit three rhombic lattices with different orientations, thus the focal patterns exhibit three rhombic rhombus with different orientations composed of four spots.
To furthermore prove the results shown in Fig. 6(e), we choose an arbitrary β when φ0ij obeys the rule of φ0ij = (−1)i(−1)jπ/2. As an example, Fig. 7(a) shows the arrangement of the PVOF for β = 75.5°. From the focal field patten shown in Fig. 7(b), the orientation angle of the rhombus composed of four spots is π − β = 104.5°, which is in good agreement with the conclusion in Fig. 6(e). To check the focus depth, we give the intensity distributions of two pairs of spots in the two diagonals in the two planes normal to the xy plane, where the two planes contain the short diagonal y = xtan(β/2) and the long diagonal y = xtan(π/2 +β/2), respectively, as shown in Figs. 7(c) and 7(d). The distances between the two pairs of spots in the short and long diagonals are 6.72λ and 15.44λ, respectively. The focal depth is ∼1.8λ. Based on the relation between the orientation angle of the rhombus and β, the shape of the rhombus composed of four spots can be flexibly designed by varying β.
4. Experimental results and discussions
To confirm the simulation results as mentioned above, we need to generate the required PVOFs, in order to process the surface of crystalline silicon wafers and then to indirectly characterize the focal fields. The experimental schematic is shown in Fig. 8. The laser source is a Ti:sapphire fs regenerative amplifier system (Coherent Inc.), which has a fundamental Gaussian mode, a central wavelength of 806 nm, a pulse duration of 35 fs, and a repetition rate of 1 kHz. A λ/2 wave plate and a broadband polarization beam splitter (PBS) are used to control the intensity of the laser pulses to avoid damaging of the SLM (Pluto NIR-II, Holoeye Inc.). A mechanical shutter is used to switch on/off the fs laser pulses. The linearly polarized fs laser pulses are incident into a fs vector field generation system (VFGS) [14, 16], which is similar to the generation system for the continuous-wave vector fields [11, 13]. The two points should be emphasized: (i) in this fs VFGS, two broadband λ/4 wave plates for 650–1100 nm and two achromatic thin lenses with a focal length of 300 mm are used to suppress the pulse broadening effect; (ii) the holographic grating displayed at the SLM is not a single grating but a grating array which is designed according to the simulation conditions previously described, as the PVOFs shown by the inset (a) of Fig. 8. The generated fs PVOF with a duration of ∼65 fs, as shown by the inset (b) of Fig. 8, is tightly focused by an objective with a NA = 0.85 and normally incident on the p-type crystalline silicon Si (100) wafer sample, which is fixed on the XYZ translation stage (P-562, PI Inc.). The sample is also illuminated by the white light of a fiber lamp and imaged by magnification on a CCD camera to observe in situ the sample surface. The focal field intensity distribution of the tightly focused fs PVOF and the surface morphology ablated by the fs PVOF are shown in the insets (c) and (d) of Fig. 8.
The amplitude transmission function of the holographic grating displayed on the SLM can be described by13] according to the above simulation conditions. We used them to ablate the crystalline silicon wafers and observe the morphology of the processed surface, and then to characterize the focal field of the PVOFs under the tight focusing condition.
According to the simulations in Fig. 4, the NA was set to 0.85 requiring the generated PVOFs to fill the aperture of the objective. The n value can be changed from 1 to 5, implying that the radii of the individual VOFs must be different for different n. We experimentally generated the PVOFs according to the simulation conditions in Fig. 4, the measured intensity distributions of the PVOFs are shown in Figs. 9(a)–9(e) and all the individual VOFs are azimuthally polarized. As shown in Figs. 9(f)–9(j), the intensity distributions behind a horizontal polarizer become the fan-like extinction patterns, owing to the cylindrical symmetry polarization distribution. From the measured intensity distribution, the patterns and the arrangements are the same as the designed PVOFs. Figures 9(k)–9(o) show the corresponding SEM images of the ablation results for the tightly focused PVOFs shown in Figs. 9(a)–9(e). Comparing the focal field patterns in the second row of Fig. 4 with the ablation morphology shown in Figs. 9(k)–9(o), we find that the experimental results are in good agreement with the simulation results. The shapes of the ablated holes are very similar to the shapes of the focal spots, and the long and short axes are about 0.7 μm and 0.5 μm, corresponding to 0.87λ and 0.62λ, respectively.
For additional comparison, we generated the PVOFs shown in Figs. 6 and 7 to ablate the surface of the wafer sample. The experimental results are shown in Figs. 10 and 11, respectively. As shown in Fig. 10, the shapes and arrangements of the fabricated holes are the same as the focal field patterns simulated in Fig. 6. Figure 11(a) shows the holographic grating displayed on the SLM, where β = 75.5°, corresponding to the simulation condition in Fig. 7. The morphology of the sample surface contains four holes arranged in a rhombus. The measured orientation angle π − β shown in Fig. 11(b) is ∼104.5°, in good agreement with the analytic result of π − β = 104.5°. Clearly, all experimental results agree with the simulation results, implying that the simulation method proposed here is correct and powerful.
We have numerically explored the tight focusing behavior of the PVOF composed of the individual VOFs, based on our modified Richard-Wolf diffraction integration formula. By tailoring the spatial arrangement and the SoP distribution of the individual vector optical field, the sub-wavelength multiple focal spots with different patterns could be easily realized. The size of the focal spots, the interval between focal spots, and the arrangement of focal spots could be manipulated by using the spatial configuration, numerical aperture, spatial symmetry, and the states of polarization of the PVOF. The focal spots with the arrangement of hexagon, rectangle and rhombus could be achieved under different conditions. To confirm the simulation results, we experimentally generated the PVOFs according to the simulation conditions. The tightly focused PVOFs were used to ablate the crystalline silicon wafer surface. Based on the SEM images of the ablated sample surface, we found that the experimental results agree with the intensity patterns and the sizes of the focal spots in the simulations. The tight focusing of selectable PVOFs provides an effective means for controlling tight focusing fields. These results may be useful and flexibly applied in many applications, such as micro-nano parallel fabrication and optical manipulation.
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 Nos. 11274183 and 11374166, the Natural Science Foundation of Tianjin under Grant No. 12JCYBJC10700, the Open Research Fund of State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences under Grant No. SKLST201206 and the National scientific instrument and equipment development project under Grant No. 2012YQ17004.
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