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Beam Shaping Optical Lens Designs for Diffraction-Free Bessel Beams

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Beam Shaping Optical Lens Designs for Diffraction-Free Bessel Beams InternationalJournal of VIBGYOR Optic and Photonic Engineering Beam Shaping Optical Lens Designs for Diffraction-Free Bessel Beams Research Article: Open Access ISSN: 2631-5092 Abdallah K Cherri* and Mahmoud K Habib Electrical Engineering Department, College of Engineering and Petroleum, Kuwait University, Kuwait Abstract Beam shaping technique is applied to design optical lenses for transforming uniform beams to diffraction-free Bessel Beams. One-lens and two-lens refracting system will be demonstrated where the lens’ surfaces equations can be easily derived. The designed lenses take the input uniform beam distribution and confine it to be within the main lobe of the Bessel beam and avoid its zeros crossings. Various design parameters such as the power of the beam and the optical system length will be discussed. It will be demonstrated that the two-element design will produce smaller system length than the one-element design. Keywords Beam shaping methods, One-element and two-element refracting system, Lens design Introduction cessing increases, researchers increase their efforts in exploring other non-Gaussian laser beams (due The fundamental mode (TEM ) of the cavity of 00 to its short, micron-sized Rayleigh range), such as Gaussian laser beam is the most common beam the non-diffracting Bessel beams [9], which have an shape used in optical processing. In particular, for accurate laser beam focusing over a good distance materials processing [1-8] the low divergence of range. This comes at the expense of reducing the the Gaussian beam provides a small focused spot. It amount of energy available in the process where can be shown that a focal spot size as small as 1 μm Gaussian beam has an advantage over any other can be obtained using a lens with a given numerical beam shape. aperture NA [3]. The small spot size is very attrac- tive for microfabrication applications. Further, for Bessel beams, in contrast to Gaussian beams, laser materials processing, the depth of field (DOF) have extended DOF with a minimal spreading in or the distance (from the beam waist over which the central lobe during wave propagation. This the focal beam size and the peak intensity keep non-diffracting property attracts researchers to their values) is very important factor. The DOF of attempt using Bessel beams in many new applica- Gaussian beam is too short for many industrially tions. The researchers’ efforts appear, recently, in important processing applications [1] due to trans- much usage of micro and femtosecond lasers to verse spreading. As the use of laser in materials pro- process a broad variety of materials such as metals, *Corresponding author: Abdallah K Cherri, Electrical Engineering Department, College of Engineering and Petroleum, Kuwait University, P. O. Box 5969, Safat 13060, Kuwait, Tel: +965-2498-7376 Accepted: May 27, 2019; Published: May 29, 2019 Copyright: © 2019 Cherri AK, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Cherri and Habib. Int J Opt Photonic Eng 2019, 4:017 Citation: Cherri AK, Habib MK (2019) Beam Shaping Optical Lens Designs for Diffraction-Free Bessel Beams. Int J Opt Photonic Eng 4:017 Cherri and Habib. Int J Opt Photonic Eng 2019, 4:017 ISSN: 2631-5092 | • Page 2 of 10 • glass, semiconductors, biological samples, etc with Recently, we have proposed the design of such re- very high degree of precision and reproducibility, fractive systems for generating various other laser and with minimized laser induced damage. Bessel beams profiles [24,25] such as: beams can be generated by many methods [10- (i) Annular-uniform-to-uniform; 15]. Lasers with high output powers are required for a number of applications, such as for materi- (ii) Annular-Gaussian-to-uniform; al processing (welding, cutting, drilling, soldering, (iii) Gaussian-to-uniform; marking, surface modification) [7,16], where many of these beams are generated in unstable resona- (iv) Gaussian to Annular-uniform; tor. For instance, using a ring-type reflective mirror (v) Annular-Gaussian to Annular-uniform; inside the resonator results in an output annular (vi) Annular-Gaussian to Gaussian; beam profile. In this paper, we further extend the previous On the other hand, beam shaping techniques works to design refractive systems to transform were proposed to convert laser beam profiles to uniform circular beam to diffraction-free circular other beams profiles [17,18]. Refractive optical Bessel beams using one-element and two-element system technique is among many successful and lenses. A procedure to derive the mathematical efficient methods that can achieve this conver- expressions of the input and output lens surfaces sion. This technique relies on geometric optics for is provided for beams transformations from uni- designing laser beam shapers. The designed opti- form to Bessel. The designed lenses take the input cal element functions as a field mapping of the in- beam distribution and confine it to be within the put beam distribution to provide a desired output main lobe of the Bessel beam and avoid its zeros beam profile. In the past one-element and two-el- crossings. It will be shown that for a given system ement refracting systems were reported to trans- specified with certain dimensions and power con- form annular-Gaussian to Bessel beam [19-23]. centration, then the design parameters can be θio θ io ro d d θro yo yo θro ro Air Glass D D θ ri θri θ ii y y i θ i ri ii ri Figure 1: The geometric set-up of the refraction system beam transformation: a) One-element design and b) Two-element design. Citation: Cherri AK, Habib MK (2019) Beam Shaping Optical Lens Designs for Diffraction-Free Bessel Beams. Int J Opt Photonic Eng 4:017 Cherri and Habib. Int J Opt Photonic Eng 2019, 4:017 ISSN: 2631-5092 | • Page 3 of 10 • changed to meet these specifications. Further, the 2 2− 1/2 designed surfaces of the lenses are smooth enough dyi// dr i = dy o dr o = n{( f '/ ( rio− r)) −+ n 1} (5) to be fabricated. It will be demonstrated that the Note that Eq. (5) imposes a minimum value on two-element design will produce smaller system the length f’ to avoid the singularity. On the other length than the one-element design. hand, similarly, for the two-element design, we can Design Considerations obtain the following equations: 22 Figure 1a and Figure 1b show schematics of half nyi+( r io − r) +( D −+ yi y o) + n( d − yo) = c (6) of the proposed axially symmetric refracting sys- θθ−− tem for uniform to Bessel beam transformation for sec()ri ii n c′ = ( rrio− ) (7) one-element and two-element, respectively. The tan()θθri− ii lens is made from glass with index of refraction n = dy/ dr = dy/ dr = −− tanθθ = tan (8) 1.5172. D is the distance between the input and the i i o o ii io 2 2− 1/2 output horizontal reference planes. Starting with dyi//{'/ dr i = dy o dr o = −( c( rio − r)) +− n 1} (9) an incident ray that hits the first surface of the lens Where c is a constant and c’ = c - nd - D. with an angle θii, it will be refracted by an angle θri. The refracted ray reaches the second lens surface Laser Beam Transformations with an angle θro and then it is refracted by an angle The first step in the procedure for designing the θio. Note the parallelism between the incident and refracting lens starts with establishing a relation- the exited ray. The lens surfaces are represented ship between ri and ro the radial distances of the by yi(ri) and yo(ro) functions of the radial distances lenses’ surfaces. The above-mentioned third con- ri and ro of the input and output surfaces, respec- dition of beam shaping can be used in this regard. tively. This condition is translated into a relationship be- Beam shaping method design imposes few con- tween the cross-sectional areas of the uniform and ditions that must be met to realize beam conver- the Bessel beams: r sion such as: a) All input rays that enter and leave π22o πρ 2 αρ ρ rki = ∫ 2 J0 ( ) d (10) the lens must travel the same optical path length; 0 Where J is the zero-order Bessel function of first b) All input rays that enter and leave the lens must 0 kind, r is the radius of the input circular beam, r is be parallel to each other’s; and c) The ratio of the i o input beam power to the output beam power must the radius of the output Bessel beam, k is a constant equal to a constant. that defines the amplitude, and α is a parameter that sets the width or diameter of the main lobe of For the one-element design, the first two design the Bessel beam as shown in Figure 2. The diameter conditions are translated into the following equa- of the central lobe is a main parameter that may tions: control the Rayleigh range [3] for Bessel beam and 22 the power in the central lobe. Eq. (10) leads to the yi+ nrr( io −) +( Dy −i + y o) +− dy o = f (1) following relationship: 2 tan(θθii− ri ) = tan( θθio − ro) =(r i − r o ) / ( D −+ yi y o ) (2) 2 22 rio = ( rkJrJr) [ 01(αα o) + ( o)] (11) Where f is a constant and f’ = f - d - D. Further, Eq. (11) expresses the relationship between four the input and the output rays parallelism dictates parameters in the lens design, i.e., r , r , k, and α.
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