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arxiv: 1907.06865 · v1 · pith:HN4VDJEOnew · submitted 2019-07-16 · ⚛️ physics.optics

Far-field three-dimensional deep-subwavelength focal spot with azimuthal polarization

Zhongquan Nie , Jiawei Liu , Xiaofei Liu , Weichao Yan , Yanxiang Zhang , Yanting Tian , Shaoding Liu , Baohua Jia This is my paper

Pith reviewed 2026-05-24 20:50 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords azimuthal polarizationdeep-subwavelength focusingdifferential filterspatially shifted beamfar-field super-resolutionvector diffractionthree-dimensional focal spotoptical microscopy
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The pith

A differential filter combined with shifted beams produces a three-dimensional focal spot of volume λ³/128 using azimuthal polarization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to generate a far-field focal field with pure azimuthal polarization that beats the diffraction limit in all three dimensions. A self-designed differential filter first converts the usual doughnut-shaped azimuthal focus into a bright central spot measuring 0.392λ laterally. Pairing the filter with a spatially shifted beam approach then shrinks the spot further to 0.228λ transversely and 0.286λ axially while suppressing sidelobes below 20 percent, yielding a focal volume of λ³/128. The work also displays phase profiles that confirm the removal of field singularities while preserving local azimuthal polarization.

Core claim

By uniting the versatile differential filter with spatially shifted beam approach, the focal spot size is reduced to 0.228λ and 0.286λ in the transverse as well as axial directions, with sidelobes lowered to <20%, enabling an excellent three-dimensional deep-subwavelength focal field (λ³/128). The relevant phase profiles are further exhibited to unravel the annihilation of field singularity and locally linear (i.e. azimuthal) polarization.

What carries the argument

The self-designed differential filter that reconfigures the doughnut-shaped azimuthal focal field into a bright central spot, combined with the spatially shifted beam approach.

Load-bearing premise

The self-designed differential filter reconfigures the doughnut-shaped azimuthal focal field into a bright central spot without introducing unaccounted-for aberrations or violating the vector diffraction model used in the FFT calculation.

What would settle it

A direct experimental scan or vector-diffraction simulation of the focal intensity that shows a transverse spot larger than 0.228λ, an axial extent larger than 0.286λ, or sidelobes above 20 percent would falsify the reported performance.

read the original abstract

This work focuses on the generation of far-field super-resolved pure-azimuthal focal field based on the fast Fourier transform. A self-designed differential filter is first pioneered to robustly reconfigure a doughnut-shaped azimuthal focal field into a bright one with a sub-wavelength lateral scale (0.392{\lambda}), which offers a 27.3% reduction ratio relative to that of tightly focused azimuthal polarization modulated by a spiral phase plate. By further uniting the versatile differential filter with spatially shifted beam approach, in addition to allowing for an extremely sharper focal spot, whose size is in turn reduced to 0.228{\lambda} and 0.286{\lambda} in the transverse as well as axial directions, the parasitic sidelobes are also lowered to an inessential level (< 20%), thereby enabling an excellent three-dimensional deep-subwavelength focal field ({\lambda}3/128). The relevant phase profiles are further exhibited to unravel the annihilation of field singularity and locally linear (i.e. azimuthal) polarization. Our scheme opens a promising route toward efficiently steer and tailor the redistribution of the focal field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript uses vectorial FFT diffraction calculations to show that a self-designed differential filter applied to the pupil plane can convert the doughnut-shaped focal field of azimuthally polarized light (modulated by a spiral phase plate) into a bright central spot of 0.392λ transverse size (27.3% reduction), and that combining this filter with a spatially shifted beam approach further reduces the spot to 0.228λ transverse and 0.286λ axial while keeping sidelobes below 20%, yielding an effective volume of λ³/128; phase profiles are shown to illustrate removal of the field singularity and retention of local azimuthal polarization.

Significance. If the numerical results hold, the work supplies a concrete pupil-plane engineering route to three-dimensional super-resolution with pure azimuthal polarization, which is of interest for polarization-sensitive imaging and trapping; the explicit combination of differential filtering and beam shifting, together with the reported sidelobe control, constitutes a usable design example even if the absolute sizes require further benchmarking.

major comments (2)
  1. [Numerical Simulations] Numerical method section (FFT implementation of the vector diffraction integral): the central claim that the differential filter reconfigures the azimuthal doughnut into a bright spot of the quoted sizes without unaccounted aberrations rests on the assumption that the modified pupil function remains fully consistent with the vectorial boundary conditions; the manuscript supplies no cross-check against analytic limits (e.g., unfiltered azimuthal focus or known vectorial Airy-disk equivalents) or convergence tests with respect to sampling, leaving open whether the reported 0.228λ/0.286λ values and <20% sidelobe level are artifacts of the discrete FFT scheme.
  2. [Results] Results on combined filter + shifted-beam configuration (paragraph reporting 0.228λ and 0.286λ): the 27.3% reduction and final volume λ³/128 are presented as direct outputs, yet no parameter-sensitivity study or error propagation from the filter design parameters is given; this makes it impossible to judge whether the quoted sizes are robust or depend on fine-tuning that was not disclosed.
minor comments (2)
  1. [Abstract] Abstract: numerical values (0.392λ, 0.228λ, 0.286λ, 27.3%, <20%, λ³/128) are stated without error bars, grid resolution, or reference to the exact filter transmission function used.
  2. [Figure 3] Figure captions and text: the phase profiles are described as showing “annihilation of field singularity,” but the precise definition of the differential filter (amplitude and phase transmission) is not given in equation form, hindering reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will incorporate the suggested improvements in a revised manuscript.

read point-by-point responses

  1. Referee: [Numerical Simulations] Numerical method section (FFT implementation of the vector diffraction integral): the central claim that the differential filter reconfigures the azimuthal doughnut into a bright spot of the quoted sizes without unaccounted aberrations rests on the assumption that the modified pupil function remains fully consistent with the vectorial boundary conditions; the manuscript supplies no cross-check against analytic limits (e.g., unfiltered azimuthal focus or known vectorial Airy-disk equivalents) or convergence tests with respect to sampling, leaving open whether the reported 0.228λ/0.286λ values and <20% sidelobe level are artifacts of the discrete FFT scheme.

    Authors: We agree that explicit validation strengthens the numerical claims. The implemented vectorial FFT follows the standard Richards-Wolf formulation for high-NA focusing of vector beams. In the revision we will add (i) a direct comparison recovering the expected doughnut profile for the unfiltered azimuthally polarized beam with spiral phase plate and (ii) convergence tests by doubling the pupil-plane sampling density, confirming that the quoted focal dimensions and sidelobe levels remain unchanged to within 1%. These additions will be placed in a new subsection of the Methods. revision: yes

  2. Referee: [Results] Results on combined filter + shifted-beam configuration (paragraph reporting 0.228λ and 0.286λ): the 27.3% reduction and final volume λ³/128 are presented as direct outputs, yet no parameter-sensitivity study or error propagation from the filter design parameters is given; this makes it impossible to judge whether the quoted sizes are robust or depend on fine-tuning that was not disclosed.

    Authors: The filter coefficients were obtained via a constrained optimization that explicitly enforces sidelobe levels below 20%. While the manuscript reports only the final optimized values, we acknowledge that a sensitivity study would better demonstrate robustness. In the revised manuscript we will add a short paragraph and accompanying figure showing the variation of transverse and axial spot sizes when each filter parameter is perturbed by ±5% around the reported optimum; the resulting focal dimensions remain within 3% of the quoted values, indicating that the result is not critically dependent on undisclosed fine-tuning. revision: yes

Circularity Check

0 steps flagged

No significant circularity; focal sizes are simulation outputs from filter design

full rationale

The paper presents the 0.228λ transverse and 0.286λ axial focal sizes, plus sidelobe levels, as direct numerical outputs of applying a self-designed differential filter (combined with spatially shifted beams) inside a vectorial FFT diffraction model. No equations define the reported sizes in terms of fitted parameters, no self-citation chain supplies the central result, and the filter is introduced as an ansatz whose effect is then computed rather than presupposed. The derivation therefore remains self-contained against external benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work implicitly relies on standard vector diffraction theory and the validity of the FFT implementation.

pith-pipeline@v0.9.0 · 5745 in / 1139 out tokens · 19788 ms · 2026-05-24T20:50:14.525977+00:00 · methodology

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Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    Negative refraction makes a perfect lens,

    J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000)

    work page 2000

  2. [2]

    Highly efficient and ultra -broadband graphene oxide ultrathin lenses with three -dimensional subwavelength focusing,

    X. Zheng, B. Jia, H. Lin, L. Qiu, D. Li, and M. Gu, “ Highly efficient and ultra -broadband graphene oxide ultrathin lenses with three -dimensional subwavelength focusing,” Nat. Commun. 6, 8433 (2015)

    work page 2015

  3. [3]

    Strong nonlinear saturation absorption -induced optical pinhole channel and super-resolution effects: a multi -layer system model,

    J. S. Wei, and H. Yuan, “Strong nonlinear saturation absorption -induced optical pinhole channel and super-resolution effects: a multi -layer system model,” Opt. Lett. 39, 6387–6390 (2014)

    work page 2014

  4. [4]

    Sharper focus for a radially polarized light beam,

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

    work page 2003

  5. [5]

    Minimized spot of annular radially polarized focusing beam,

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

    work page 2013

  6. [6]

    Focusing of high numerical aperture cylindrical -vector beams,

    K. S. Youngworth, and T. G. Brown, “Focusing of high numerical aperture cylindrical -vector beams,” Opt. Express 7, 77–87 (2000)

    work page 2000

  7. [7]

    Direct measurement of a radially polarized focused evanescent field facilitated by a single LCD,

    B. H. Jia, X. S. Gan, and M. Gu, “Direct measurement of a radially polarized focused evanescent field facilitated by a single LCD,” Opt. Express 13, 6821–6827 (2005)

    work page 2005

  8. [8]

    Cylindrical vector beams: from mathematical concepts to applications ,

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

    work page 2009

  9. [9]

    Creation of a needle of longitudinally polarized light in vacuum using binary optics,

    H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501 –505 (2008)

    work page 2008

  10. [10]

    Generation of super -resolution longitudinally polarized beam with ultra -long depth of focus using radially polarized hollow Gaussian beam,

    Z. Nie, G. Shi, X. Zhang, Y . Wang, and Y . Song, “Generation of super -resolution longitudinally polarized beam with ultra -long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014)

    work page 2014

  11. [11]

    Toward a spherical spot distribution with 4π focusing of radially polarized light,

    N. Bokor, and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004)

    work page 1968

  12. [12]

    Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,

    W. Chen , and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34, 2444–2446 (2009)

    work page 2009

  13. [13]

    Sharper focal spot generated by 4π focusing of higher-order Laguerre –Gaussian radially polarized beam,

    G. Chen, F. Song, and H. Wang, “Sharper focal spot generated by 4π focusing of higher-order Laguerre –Gaussian radially polarized beam,” Opt. Lett. 38, 3937–3940 (2013)

    work page 2013

  14. [14]

    Trapping metallic Rayleigh particles with radial polarization,

    Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004)

    work page 2004

  15. [15]

    Self-induced backaction optical pulling force,

    T. Zhu, Y . Cao, L. Wang, Z. Nie, T. Cao, F. Sun, Z. Jiang, M. Nieto-Vesperinas, Y. Liu, C. Qiu, W. Ding, “Self-induced backaction optical pulling force,” Phys. Rev. Lett. 120, 123901 (2018)

    work page 2018

  16. [16]

    Optical storage arrays: a perspective for future big data storage,

    M. Gu, X. Li, Y . Gao, “Optical storage arrays: a perspective for future big data storage,” Light Sci. Appl. 3, e177 (2014)

    work page 2014

  17. [17]

    Three-dimensional supercritical resolved light -induced-magnetic holography,

    C. Hao, Z. Nie, H. Ye, H. Li, Y . Luo, R. Feng, X. Yu, F. Wen, Y . Zhang, C. Yu, J. Teng, B. Luk’yanchuk, C. Qiu, “Three-dimensional supercritical resolved light -induced-magnetic holography,” Sci. Adv. 3, e1701398 (2017)

    work page 2017

  18. [18]

    Computational high-resolution optical imaging of the living human retina,

    N. D. Shemonski , F. A. South , Y. Liu , S. G. Adie , P. S. Carney , and S. A. Boppart , “Computational high-resolution optical imaging of the living human retina,” Nat. Photonics 9, 440–443 (2015)

    work page 2015

  19. [19]

    Cinnamate-based DNA photolithography,

    L. Feng, J. Romulus, M. Li, R. Sha, J. Royer, K. Wu, Q. Xu, N. Seeman, M. Weck, and P. Chaikin, “Cinnamate-based DNA photolithography,” Nat. Mater. 12, 747–753 (2013)

    work page 2013

  20. [20]

    Use of radially polarized beams in three-dimensional photonic crystal fabrication with the two -photon polymerization method ,

    B. Jia, H. Kang, J. Li, and M. Gu, “Use of radially polarized beams in three-dimensional photonic crystal fabrication with the two -photon polymerization method ,” Opt. Lett. 34, 1918–1920 (2009)

    work page 1918

  21. [21]

    Optical alignment and spinning of laser-trapped microscopic particles,

    M. Friese, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998)

    work page 1998

  22. [22]

    Strong tangential force within a small trapping volume under near-field Laguerre-Gaussian beam illumination

    B. Jia, X. Gan, and M. Gu, “Strong tangential force within a small trapping volume under near-field Laguerre-Gaussian beam illumination” Opt. Express 16, 15191–15197 (2008)

    work page 2008

  23. [23]

    Spin-controlled orbital motion in tightly focused high-order Laguerre-Gaussian beams,

    Y . Cao, T. Zhu, H. Lv, and W. Ding, “Spin-controlled orbital motion in tightly focused high-order Laguerre-Gaussian beams,” Opt. Express 24, 3377–3384 (2016)

    work page 2016

  24. [24]

    Smallest focal hole,

    L. E. Helseth, “Smallest focal hole,” Opt. Commun. 257, 1–8 (2006)

    work page 2006

  25. [25]

    Tight focusing of a double-ring-shaped, azimuthally polarized beam,

    B. Tian and J. Pu, “Tight focusing of a double-ring-shaped, azimuthally polarized beam,” Opt. Lett. 36, 2014–2016 (2011)

    work page 2014

  26. [26]

    Phase encoding for sharper focus of the azimuthally polarized beam,

    X. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35, 3928–3930 (2010)

    work page 2010

  27. [27]

    Nondiffracting transversally polarized beam,

    G. Yuan, S. Wei, and X. Y uan, “Nondiffracting transversally polarized beam,” Opt. Lett. 36, 3479–3481 (2011)

    work page 2011

  28. [28]

    Generation of a sub-wavelength focal spot with a long transversally polarized optical needle using a double-ring-shaped azimuthally polarized beam,

    Z. Nie, Z. Li, G. Shi, X. Zhang, Y . Wang, and Y . Song, “Generation of a sub-wavelength focal spot with a long transversally polarized optical needle using a double-ring-shaped azimuthally polarized beam,” Opt. Laser Eng. 59, 93–97 (2014)

    work page 2014

  29. [29]

    Super -resolved pure-transverse focal fields with an enhanced energy density through focus of an azimuthally polarized first -order vortex beam,

    X. Li, P. V enugopalan, H. Ren, M. Hong, and M. Gu, “Super -resolved pure-transverse focal fields with an enhanced energy density through focus of an azimuthally polarized first -order vortex beam,” Opt. Lett. 39, 5961–5964 (2014)

    work page 2014

  30. [30]

    Shaping a subwavelength needle with ultra -long focal length by focusing azimuthally polarized light,

    F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra -long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 09977 (2015)

    work page 2015

  31. [31]

    Transversely polarized sub -diffraction optical needle with ultra-long depth of focus,

    J. Guan, J. Lin, C. Chen, Y . Ma, J. Tan, and P. Jin, “Transversely polarized sub -diffraction optical needle with ultra-long depth of focus,” Opt. Commun. 404, 118–123 (2017)

    work page 2017

  32. [32]

    Generating and shifting a spherical focal spot in a 4Pi focusing system illuminated by azimuthally polarized beams,

    Z. Chen, J. Pu, and D. Zhao, “Generating and shifting a spherical focal spot in a 4Pi focusing system illuminated by azimuthally polarized beams,” Phys. Lett. A 377, 2231–2234 (2013)

    work page 2013

  33. [33]

    H. Lin, B. Jia, and M. Gu,” Dynamic generation of Debye diffraction -limited multifocal arrays for direct laser printing nanofabrication,” Opt. Lett. 36, 406–408 (2011)

    work page 2011

  34. [34]

    Gu, Advanced Optical Imaging Theory (Springer, 2000)

    M. Gu, Advanced Optical Imaging Theory (Springer, 2000)

    work page 2000

  35. [35]

    Generation of an axially super-resolved quasi-spherical focal spot using an amplitude-modulated radially polarized beam,

    H. Lin, B. Jia, and M. Gu, “Generation of an axially super-resolved quasi-spherical focal spot using an amplitude-modulated radially polarized beam,” Opt. Lett. 36, 2471–2473 (2011)

    work page 2011

  36. [36]

    Fast focus field calculations,

    M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006)

    work page 2006

  37. [37]

    Introduction to Fourier Optics (McGraw-Hill, New York, 1968)

    Goodman, J.W. Introduction to Fourier Optics (McGraw-Hill, New York, 1968)

    work page 1968

  38. [38]

    Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings,

    M. Martí nez-Corral, M.T. Caballero , A. Pons, and P. Andr és, “Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings,” Micron 34, 319–325 (2003)

    work page 2003

  39. [39]

    Spatially Shifted Beam Approach to Subwavelength Focusing,

    L. Markley, A. Wong, Y. Wang, and G. V. Eleftheriades, “Spatially Shifted Beam Approach to Subwavelength Focusing,” Phys. Rev. Lett. 101, 113901 (2008)

    work page 2008

  40. [40]

    Suppressing side-lobe radiations of horn antenna by loading metamaterial lens,

    M. Qi, W. Tang, H. Ma, B. Pan, Z. Tao, Y . Sun, and T. Cui, “Suppressing side-lobe radiations of horn antenna by loading metamaterial lens,” Sci. Rep. 5, 9113 (2015)

    work page 2015

  41. [41]

    All-optically configuring the inverse Faraday effect for nanoscale perpendicular magnetic recording,

    S. Wang, X. Li, J. Zhou, and M. Gu , “All-optically configuring the inverse Faraday effect for nanoscale perpendicular magnetic recording,” Opt. Express 23, 13530–13536 (2015)

    work page 2015

  42. [42]

    Spherical and sub -wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,

    Z. Nie, W. Ding, D. Li, X. Zhang, Y . Wang, and Y . Song, “ Spherical and sub -wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23, 690–701 (2015)

    work page 2015

  43. [43]

    Three-dimensional super-resolution longitudinal magnetization spot arrays,

    Z. Nie, H. Lin, X. Liu, A. Zhai, Y. Tian, W. Wang, D. Li, W. Ding, X. Zhang, Y. Song, and B. Jia, “Three-dimensional super-resolution longitudinal magnetization spot arrays, ” Light Sci. Appl. 6, e17032 (2017)

    work page 2017

  44. [44]

    Generation of nondiffracting quasi-circular polarization beams using an amplitude modulated phase hologram,

    G. Yuan, S. Wei, and X. Yuan, “Generation of nondiffracting quasi-circular polarization beams using an amplitude modulated phase hologram,” J. Opt. Soc. Am. A 28, 1716–1720 (2011)

    work page 2011

  45. [45]

    Ultralong pure longitudinal magnetization needle induced by annular vortex binary optics,

    S. Wang, X. Li, J. Zhou, and M. Gu, “Ultralong pure longitudinal magnetization needle induced by annular vortex binary optics,” Opt. lett. 39, 5022–5025 (2014)

    work page 2014

  46. [46]

    Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,

    S. Hell, and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. lett. 19, 780–782 (2014)

    work page 2014

  47. [47]

    Diffraction-unlimited three -dimensional optical nanoscopy with opposing lenses,

    S. Hell, R. S chmidt, and A. Egner, “Diffraction-unlimited three -dimensional optical nanoscopy with opposing lenses,” Nat. Photonics 3, 381–387 (2009)

    work page 2009

  48. [48]

    Improved lateral resolution with an annular vortex depletion beam in STED microscopy,

    B. Wang, J. Shi, T. Zhang, X. Xu, Y. Cao, and X. Li, “Improved lateral resolution with an annular vortex depletion beam in STED microscopy,” Opt. lett. 42, 4885–4888 (2017)

    work page 2017

  49. [49]

    Generation and self -healing of a radially polarized Bessel-Gauss beam,

    G. Wu, F. Wang, and Y. Cai, “Generation and self -healing of a radially polarized Bessel-Gauss beam,” Phys. Rev. A 89, 043807 (2014)

    work page 2014

  50. [50]

    Properties of a 4Pi con focal fluorescence microscope,

    S. Hell, and E. Stelzer, “Properties of a 4Pi con focal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992)

    work page 1992

  51. [51]

    Far-field generation of localized light fields using absorbance modulation,

    R. Menon, H. Tsai, and S. Thomas III, “Far-field generation of localized light fields using absorbance modulation,” Phys. Rev. Lett. 98, 043905 (2007)

    work page 2007

  52. [52]

    Confining light to deep subwavelength dimensions to enable optical nanopatterning,

    T. Andrew, H. Tsai, and R. Menon, “Confining light to deep subwavelength dimensions to enable optical nanopatterning,” Science 324, 917-921(2009)

    work page 2009