Generation of ultra-long multiple optical tubes using annular Walsh function filters
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Generation of ultra-long multiple optical tubes using annular Walsh function filters D. Thiruarul1 · K. B. Rajesh1 · M. Lavanya2 · G. Mahadevan3 · Dhayalan Velauthapillai4 · Z. Jaroszewicz5 Received: 26 February 2020 / Accepted: 11 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The tight focusing properties of an azimuthally polarized Bessel Gaussian beam phase modulated by annular Walsh function filter is studied numerically by vector diffraction theory. It is observed that upon suitable optimization of order and annular obstruction ratio of an annular Walsh function filter, one can generate multiple sub wavelength scale optical tubes (optical holes) with super long focal depth. Such a focal system is usable for Nanolithography, particle trapping and transportation, as well as confocal and STED microscopy, microstructure fabrication etc. Keywords Walsh filter · Azimuthally polarized beam · Bessel Gaussian beam
1 Introduction In Askin et al. (1986) successfully trapped the microscopic particles using the gradient force of a strongly focused Gaussian beam. Recently, optical tweezers finds more applications in single molecule and cell studies (Choudhary et al. 2019), studying the elasticity of cell membrane (Nussenzveig 2018), red blood cells (Jinyong Lin et al. 2018), etc.,.In an optical tweezers setup, the conventional gradient force acting on the particle is proportional to ± ΔE2,where E denotes the electric field of the beam and ± sign indicates the refractive index difference between the surrounding medium (n1) and trapping particle (n2). If the refractive index of the particle is higher than that of the surrounding medium, normally the fundamental Gaussian beam having peak centered intensity profile is used to trap and * K. B. Rajesh [email protected] 1
Department of Physics, Chikkanna Government Arts College, Tiruppur, Tamilnadu, India
2
Department of Physics, PSGR Krishnammal College for Women, Coimbatore, Tamilnadu 641004, India
3
Department of Mathematics, Gandhigram Rural Institute (Deemed to be University), Dindigul, Tamil Nadu, India
4
Faculty of Engineering and Science, Western Norway University of Applied Sciences, 5063 Bergen, Norway
5
Department of Physical Optics, Institute of Applied Optics, Warsaw, Poland
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manipulate the particles. However, in the case of trapping low refractive index particles (1.e n2 0 , ⎪ sgn(y) = ⎨ 0, y = 0, ⎪ ⎩ −1, y < 0. The orthogonality condition implies that 1
∫
𝜓n𝜎 (𝜃)𝜓m𝜎 (𝜃)𝜃d𝜃 =
1 − 𝜎2 𝜐mn 2
𝜎
where 𝜐mn is the Kronecker delta defined as { 0, n ≠ m, 𝜐mn = 1, n = m. The order of Walsh function m is equal to zero crossings in the interval (σ,1) and location of the for functions 𝜓m𝜎 (𝜃),m = 0,1, …(N−1), is given by √ [(N − i)𝜎 2 + i × 𝛼, i = 1, 2, ..., (N − 1) 𝜃i = N The inner and outer radii of the annulus is θ0 = σ and θN = 1. Here i = 1, 2, ..., (N − 1) determines all zero crossing locations for paticular set of Walsh functions.
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