Abstract

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

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2013

L. X. Yang, X. S. Xie, S. C. Wang, and J. Y. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett.38, 1331–1333 (2013).
[CrossRef] [PubMed]

K. Lou, S. X. Qian, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang., “Femtosecond Laser Processing by Using Patterned Vector Optical Fields,” Sci. Rep.3, 2281 (2013).
[PubMed]

2011

2010

2009

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1, 1–57 (2009).
[CrossRef]

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282, 3421–3425 (2009).
[CrossRef]

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

X. Jia, T. Q. Jia, L. E. Ding, P. X. Xiong, L. Deng, Z. R. Sun, Z. G. Wang, J. R. Qiu, and Z. Z. Xu, “Complex periodic micro/nanostructures on 6H-SiC crystal induced by the interference of three femtosecond laser beams,” Opt. Lett.34, 788–790 (2009).
[CrossRef] [PubMed]

2008

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2, 501–505 (2008).
[CrossRef]

2007

2005

2003

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interferenceof femtosecond pulses,” Appl. Phys. Lett.82, 2758–2760 (2003).
[CrossRef]

2002

2000

1997

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.78, 4713–4716 (1997).
[CrossRef]

1959

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A253, 358–379 (1959).
[CrossRef]

Ahmed, M. A.

Brown, T.

Chen, J.

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282, 3421–3425 (2009).
[CrossRef]

Chong, C. T.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2, 501–505 (2008).
[CrossRef]

Dearden, G.

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

Deng, L.

Ding, J. P.

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282, 3421–3425 (2009).
[CrossRef]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett.32, 3549–3551 (2007).
[CrossRef] [PubMed]

Ding, L. E.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

Edwardson, S. P.

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

Fan, Y. X.

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282, 3421–3425 (2009).
[CrossRef]

Fearon, E.

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

Golub, I.

Graf, T.

Gu, B.

Gu, M.

Guo, C. S.

Hao, X. A.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.78, 4713–4716 (1997).
[CrossRef]

Hnatovsky, C.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Hsu, C. C.

Jia, B. H.

Jia, T. Q.

Jia, X.

Juodkazis, S.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interferenceof femtosecond pulses,” Appl. Phys. Lett.82, 2758–2760 (2003).
[CrossRef]

Kalosha, V. P.

Kang, H.

Kitamura, K.

Kondo, T.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interferenceof femtosecond pulses,” Appl. Phys. Lett.82, 2758–2760 (2003).
[CrossRef]

Kraus, M.

Krolikowski, W.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Kuang, C. F.

Kuang, Z.

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.78, 4713–4716 (1997).
[CrossRef]

Lai, N. D.

Leger, J. R.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

Li, Y. N.

K. Lou, S. X. Qian, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang., “Femtosecond Laser Processing by Using Patterned Vector Optical Fields,” Sci. Rep.3, 2281 (2013).
[PubMed]

K. Lou, S. X. Qian, X. L. Wang, Y. N. Li, B. Gu, C. H. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express20, 120–127 (2011).
[CrossRef]

Li, Y. P.

Liang, W. P.

Lin, C. H.

Lin, H.

Lin, J. H.

Liu, D.

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

Liu, X.

Lou, K.

K. Lou, S. X. Qian, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang., “Femtosecond Laser Processing by Using Patterned Vector Optical Fields,” Sci. Rep.3, 2281 (2013).
[PubMed]

K. Lou, S. X. Qian, X. L. Wang, Y. N. Li, B. Gu, C. H. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express20, 120–127 (2011).
[CrossRef]

Lukyanchuk, B.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2, 501–505 (2008).
[CrossRef]

Matsuo, S.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interferenceof femtosecond pulses,” Appl. Phys. Lett.82, 2758–2760 (2003).
[CrossRef]

Michalowski, A.

Misawa, H.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interferenceof femtosecond pulses,” Appl. Phys. Lett.82, 2758–2760 (2003).
[CrossRef]

Mizeikis, V.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interferenceof femtosecond pulses,” Appl. Phys. Lett.82, 2758–2760 (2003).
[CrossRef]

Ni, W. J.

Noda, S.

Perrie, W.

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

Qian, S. X.

K. Lou, S. X. Qian, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang., “Femtosecond Laser Processing by Using Patterned Vector Optical Fields,” Sci. Rep.3, 2281 (2013).
[PubMed]

K. Lou, S. X. Qian, X. L. Wang, Y. N. Li, B. Gu, C. H. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express20, 120–127 (2011).
[CrossRef]

Qin, J. Q.

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282, 3421–3425 (2009).
[CrossRef]

Qiu, J. R.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

Ren, Z. C.

K. Lou, S. X. Qian, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang., “Femtosecond Laser Processing by Using Patterned Vector Optical Fields,” Sci. Rep.3, 2281 (2013).
[PubMed]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A253, 358–379 (1959).
[CrossRef]

Rode, A.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Sakai, K.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.78, 4713–4716 (1997).
[CrossRef]

Sharp, M.

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

Sheppard, C.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2, 501–505 (2008).
[CrossRef]

Shi, L. P.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2, 501–505 (2008).
[CrossRef]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.78, 4713–4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.78, 4713–4716 (1997).
[CrossRef]

Shvedov, V.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

Sun, Z. R.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.78, 4713–4716 (1997).
[CrossRef]

Tu, C. H.

K. Lou, S. X. Qian, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang., “Femtosecond Laser Processing by Using Patterned Vector Optical Fields,” Sci. Rep.3, 2281 (2013).
[PubMed]

K. Lou, S. X. Qian, X. L. Wang, Y. N. Li, B. Gu, C. H. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express20, 120–127 (2011).
[CrossRef]

Voss, A.

Wang, H. F.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2, 501–505 (2008).
[CrossRef]

Wang, H. T.

Wang, S. C.

Wang, T. T.

Wang, X. L.

Wang, Z. G.

Wang., H. T.

K. Lou, S. X. Qian, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang., “Femtosecond Laser Processing by Using Patterned Vector Optical Fields,” Sci. Rep.3, 2281 (2013).
[PubMed]

Watkins, K.

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

Weber, R.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A253, 358–379 (1959).
[CrossRef]

Xie, X. S.

Xiong, P. X.

Xu, Z. Z.

Yang, L. X.

Youngworth, K.

Zhan, Q.

Zhang, Y. L.

Zhao, Y. Q.

Zhou, J. Y.

Adv. Opt. Photon.

Appl. Phys. Lett.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interferenceof femtosecond pulses,” Appl. Phys. Lett.82, 2758–2760 (2003).
[CrossRef]

Appl. Surf. Sci.

Z. Kuang, D. Liu, W. Perrie, S. P. Edwardson, M. Sharp, E. Fearon, G. Dearden, and K. Watkins, “Fast parallel diffractive multi-beam femtosecond laser surface micro-structuring,” Appl. Surf. Sci.255, 6582–6588 (2009).
[CrossRef]

Nat. Photonics

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2, 501–505 (2008).
[CrossRef]

Opt. Commun.

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282, 3421–3425 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett.106, 123901 (2011).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003).
[CrossRef] [PubMed]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.78, 4713–4716 (1997).
[CrossRef]

Proc. Roy. Soc. A

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A253, 358–379 (1959).
[CrossRef]

Sci. Rep.

K. Lou, S. X. Qian, Z. C. Ren, C. H. Tu, Y. N. Li, and H. T. Wang., “Femtosecond Laser Processing by Using Patterned Vector Optical Fields,” Sci. Rep.3, 2281 (2013).
[PubMed]

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the PVOF in the input plane and the coordinate systems.

Fig. 2
Fig. 2

Three PVOFs in the input plane and their tight focusing behaviors. First column shows the configurations of PVOFs, where the arrows indicate the SoPs of individual VOFs. Second column shows the total intensity distributions of the focal fields. Third, fourth and fifth columns show the intensity distributions of the r-, ϕ- and z-components, respectively. Top, middle and bottom rows correspond to PVOFs composed of circularly polarized vortices, linearly polarized vortices and azimuthally polarized vector fields, respectively. All the images from second to fifth columns have the same dimensions of 10 × 10 λ2.

Fig. 3
Fig. 3

Five PVOFs with different NA in the input plane and the simulated tight focusing behaviors. (a) NA = 0.27, (b) NA = 0.45, (c) NA = 0.63, (d) NA = 0.81, and (e) NA = 0.99. The simulation parameters are mij = 1, φ0ij = π/2, and β = π/3. Top row shows the arrangement of the PVOFs. Middle row shows the focal fields in the focal plane (xy-plane). Bottom row shows the intensity distributions in the vicinity of focus in the yz-plane. All the images in middle and bottom rows have the same dimension of 20 × 20 λ2.

Fig. 4
Fig. 4

Five PVOFs with the same NA and different n in the input plane and the simulated tight focusing behaviors. (a) n = 1, (b) n = 2, (c) n = 3, (d) n = 4, and (e) n = 5. The simulation parameters are mij = 1, β = π/3, φ0ij = π/2, and NA = 0.85. Top row shows the arrangement of the PVOFs. Middle row shows the focal fields in the focal plane (xy-plane). Bottom row shows the intensity distributions in the vicinity of focus in the yz-plane. All the images in middle and bottom rows have the same dimension of 24 × 24λ2.

Fig. 5
Fig. 5

(a) Dependence of the spot size on NA for the situation of Fig. 3. (b) Dependence of the interval between a pair of antipodal focal spots on n for the situation of Fig. 4.

Fig. 6
Fig. 6

Influence of the SoP configuration and the lattice symmetry of the PVOFs on the tight focusing fields. Top and bottom rows correspond to the square (β = π/2) and trigonal (β = π/3) lattice symmetry, respectively. (a) Arrangements of PVOFs with different lattice symmetry. (b)–(e) Tight focusing fields of PVOFs with different SoP configurations, (b) φ0ijπ/2, (c) φ0ij = (−1)iπ/2, (d) φ0ij = (−1)jπ/2, and (e) φ0ij = (−1)i(−1)jπ/2. The simulation parameters mij = 1, n = 5, and NA = 0.85. All the images in (b)–(e) columns have the same dimension of 24 × 24 λ2.

Fig. 7
Fig. 7

PVOF with β = 75.5°, mij = 1, n = 5, φ0ij = (−1)i(−1)jπ/2, and NA = 0.85 and its tight focusing property. (a) Arrangement of the PVOF. (b) Focal spots in the focal plane. (c) Intensity distribution in the vicinity of focus in the plane perpendicular to the xy plane and containing the diagonal y = xtan(β/2). (d) Intensity distribution in the vicinity of focus in the plane perpendicular to the xy plane and containing the diagonal y = xtan(π/2 +β/2). All the images in (b)–(d) columns have the same dimension of 24 × 24 λ2.

Fig. 8
Fig. 8

Generation system of PVOFs and the experimental schematic of the sample ablation. Insets: (a) holographic grating displayed on the SLM, (b) intensity distribution of the generated PVOF, (c) tight focusing field captured by CCD, and (d) silicon surface morphology ablated by the PVOF captured by CCD. BS–beam splitter.

Fig. 9
Fig. 9

(a)–(e) Measured intensity distributions of the generated PVOFs composed of azimuthally polarized vector fields with different n from 1 to 5, where all the conditions are the same as Fig. 4. (f)–(j) Measured intensity distributions behind a horizontal polarizer for the generated PVOFs shown in (a)–(e). (k)–(o) SEM images of the silicon surface ablated by the tight focusing fields of the generated PVOFs shown in (a)–(e), where all the images have the same dimension of 20 × 20 μm2.

Fig. 10
Fig. 10

SEM images of the silicon surface ablated by the tight focusing fields in Fig. 6. All the eight images have the same dimension of 20 × 20 μm2.

Fig. 11
Fig. 11

(a) Holographic grating displayed on the SLM with β = 75.5°, mij = 1, n = 5, and φ0ij = (−1)i(−1)jπ/2, (b) SEM image of the silicon surface ablated by the tight focusing field in Fig. 7, with a dimension of 20 × 20 μm2.

Equations (18)

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E 0 = A 0 circ ( ρ / ρ 0 ) [ cos δ e ^ ρ + e j θ sin δ e ^ φ ] ,
circ ( ρ / ρ 0 ) = { 1 ρ / ρ 0 1 0 otherwise ,
E ( r , ϕ , z ) = j k f A 0 2 π 0 2 π d φ 0 ρ 0 ( M ρ cos δ + M φ e j θ sin δ ) P ρ d ρ ,
M ρ ( r , ϕ , z ) = [ 1 ρ 2 / f 2 cos ( φ ϕ ) 1 ρ 2 / f 2 sin ( φ ϕ ) ρ / f ] ,
M φ ( r , ϕ , z ) = [ sin ( φ ϕ ) cos ( φ ϕ ) 0 ] ,
P ( r , ϕ , z ) = f 2 ( 1 ρ 2 / f 2 ) 1 / 4 e j k [ z 1 ρ 2 / f 2 + ( r ρ / f ) cos ( φ ϕ ) ] .
ξ i = i d ξ and η j = j d η ,
X i j = ξ i + η j cos β and Y i j = η j sin β .
E 0 i j = A 0 circ ( ρ i j ρ 0 ) ( cos δ i j e ^ ρ i j + e j θ i j sin δ i j e ^ φ i j ) ,
E 0 = i , j E 0 i j .
E i j ( r , ϕ , z ) = j k f A 0 2 π 0 2 π d φ i j 0 ρ 0 { M ρ [ cos ( δ i j + φ i j φ ) ( e j θ i j 1 ) sin δ i j sin ( φ i j φ ) ] + M φ [ sin ( δ i j + φ i j φ ) + ( e j θ i j 1 ) sin δ i j cos ( φ i j φ ) ] P ρ i j d ρ i j ,
φ = π + arctan ρ i j sin φ i j + Y i j ρ i j cos φ i j + X i j π 2 sgn ( ρ i j sin φ i j + Y i j ) [ 1 + sgn ( ρ i j cos φ i j + X i j ) ] ,
ρ = ρ i j 2 + X i j 2 + Y i j 2 + 2 ρ i j X i j cos φ i j + 2 ρ i j Y i j sin φ i j .
E ( r , ϕ , z ) = i , j E i j ( r , ϕ , z ) .
E 0 i j = A 0 2 circ ( ρ i j ρ 0 ) e j φ i j ( e ^ x i j j e ^ y i j ) = A 0 2 circ ( ρ i j ρ 0 ) ( e ^ ρ i j j e ^ φ i j ) ,
E 0 i j = A 0 circ ( ρ i j ρ 0 ) e j φ i j e ^ y i j = A 0 circ ( ρ i j ρ 0 ) e j φ i j ( sin φ i j e ^ ρ i j + cos φ i j e ^ φ i j ) ,
E 0 i j = A 0 circ ( ρ i j ρ 0 ) e ^ φ i j .
t ( x , y ) = 1 2 + i , j t i j = 1 2 + 1 2 i , j circ ( ρ i j ρ 0 ) cos ( 2 π x Λ + m i j φ i j + φ 0 i j ) ,

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