Beam Shaping Optical Lens Designs for Diffraction-Free Bessel Beams

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Research Article: Open Access verse spreading. As the use of laser in materials pro- trans- to due [1] applications processing important industrially many for short too is beam Gaussian of DOF The factor. important very keep is values) intensity their peak the and size beam focal the which over waist beam the (from distance the or (DOF) field of depth the processing, materials laser for Further, applications. microfabrication for tive attrac- very is size spot small The [3]. NA aperture can be obtained using a lens with a given numerical can be shown that a focal spot size as small as 1 the of Gaussian beam provides a divergence small focused spot. It low the [1-8] processing materials for particular, In processing. optical in beam used shape common most the is beam laser Gaussian Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, in provided theoriginalauthorand sourcearecredited. reproduction and distribution, use, unrestricted permits which License, Attribution Commons Copyright: Cherri andHabib. Accepted: May27,2019 and Petroleum,KuwaitUniversity,P.O.Box5969,Safat13060, Kuwait,Tel:+965-2498-7376 author: *Corresponding Introduction Electrical EngineeringDepartment,CollegeofandPetroleum,KuwaitUniversity, Abdallah KCherri Citation: The fundamental mode (TEM mode fundamental The CherriAK,Habib MK(2019)BeamShapingOptical LensDesignsforDiffraction-Free BesselBeams.IntJOpt Photonic Eng4:017 © 2019 Cherri AK, et al. This is an open-access article distributed under the terms of the Creative the of terms the under distributed article open-access an is This al. et AK, Cherri 2019 © VIBGYOR Beam ShapingOpticalLensDesignsfor Beam shapingmethods,One-elementandtwo-elementrefractingsystem,Lensdesign Keywords will producesmallersystemlengththantheone-elementdesign. design two-element the that demonstrated be will It discussed. be will length system the optical and beam the of power the as such parameters design Various crossings. zeros its avoid and beam Bessel the of lobe main the within be to it confine and distribution beam uniform input the take lenses designed The derived. easily be can equations surfaces lens’ the where be demonstrated will system refracting two-lens and One-lens Beams. Bessel diffraction-free to beams uniform transforming for lenses optical design to applied is technique shaping Beam Abstract IntJOptPhotonicEng2019,4:017 Diffraction-Free BesselBeams ; Published: Abdallah K Cherri, Electrical Engineering Department, College of Engineering of Department, College Engineering K Cherri,Electrical Abdallah * andMahmoudKHabib May29,2019 00 )

of the cavity of cavity the of μm Optic andPhotonic Engineering International uh sg o mco n fmoeod aes to process a broad variety lasers of materials such as metals, femtosecond and micro of usage much in recently, appear, efforts researchers’ The tions. applica new many to in beams Bessel using attempt researchers attracts This property propagation. non-diffracting wave during lobe central in the spreading minimal a with DOF extended have beam shape. other any over advantage an has beam Gaussian where process the in available energy of amount the reducing of expense the at comes This range. distance good a over focusing beam laser accurate the non-diffracting Bessel beams [9], which have an as such range), Rayleigh micron-sized short, its to (due beams laser non-Gaussian other exploring in cessing increases, researchers increase their efforts esl em, n otat o asin beams, Gaussian to contrast in beams, Bessel Journal of ISSN: 2631-5092 - 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. that both the input and the output surfaces slope i o The parameter α sets the first zeros in the Bessel must be equal to each other, i.e. beams that are approximately 4, 2.4, and 1.2 for

dyi/ dr i = dy o / dr = tanθθii = tan io (3) α = 0.6, 1, and 2, respectively. The designer may r Furthermore, using Snell’s law at the input lens select a specific beam spot size o for a particular beam and obtain the size of the input beam ri. For (sinθii = nsinθri) along with trigonometric identities, one can obtain: instance, for Bessel beam with α = 0.6, the input power will be delivered to a spot size of r = 2 (50% n− cos θθ− o ( ii ri ) α r f′ = ( rrio− )  (4) of central lob width). Thus, for = 0.6, o = 2 and sin()θθii− ri for a selected value for k (say k = 1), we can easily

And determine the required value of the input beam ri =

1 = 0.6 0.9 = 1 = 2 0.8

0.7

0.6

0.5 Intensity 0.4

0.3

0.2

0.1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Output radial distance (cm): ro Figure 2: Intensity of zero-order Bessel functions for different α.

1.68 and the minimum value of f’ = 0.37. Similarly, yo(ro). Note that in the design procedure, selecting for 95% of the central lobe (ro = 3.8), ri is found to design parameters should take into consideration be 2.08. that the system length to be as small as possible [1+ (dy / dr )2 ] 3/2 On the other hand, Eq. (11) shows that k has , and the radii of curvature, d22 y/ dr for the effects on the value of ri. At the same time, the value of f’ in Eq. (4) depends on r . Further, Eq. (5) lens surfaces to be large. In the following subsec- i tions, we will present designs for one- and two-el- depends on both f’ and ri. As a result, varying k will eventually vary the minimum value of the length of ement system for different parameters values of k, the system and the lens input and output surfaces. α, and (f’ or c’). These parameters will change the shape of the lens and the size of the designed sys- For example, for α = 0.6, ro = 2, k = 0.5; then ri = 0.84 and the minimum value of f’ = 1.33. tem. The second step in the beam shaping design pro- One-element designs cedure is to specify a value of the system length (f’, For the one-element design, we will present four c’, or D), which determine the numerical values for designs to demonstrate the effects of parameters the surfaces slope dyi/dri and dyo/dro from Eqs. (5) k and α. For the first two designs, we choose f’ = and (9). The third step in the design procedure is to 1.35 cm (D = 2.61), k = 1, ro = 2 (50% of central lob use a curve-fitting routine to determine mathemat- width), for α = 1 and α = 0.6, as shown in Figure 3 ical expressions for these slopes from their numer- and Figure 4. In these figures, when α decreases, ical values. Finally, by integrating the mathemati- then the input radial distance ri increases from ri = cal expressions of these slopes, we can obtain the 1.24 to ri = 1.68. Further, for α = 1, it is found that mathematical expressions of the surfaces yi(ri) and the minimum value for f’ is 0.87 which corresponds

(a) (b)

Figure 3: One-element design for uniform to Bessel for f’ = 1.35, D = 2.61, k = 1, α = 1: a) and b) The input and the output surfaces; c) and d) The input and the output radius of curvature of the surfaces.

(a) (b)

Figure 4: One-element design for uniform to Bessel for f’ = 1.35, D = 2.61, k = 1, α = 0.6: a) and b) The input and the output surfaces; c) and d) The input and the output radius of curvature of the surfaces.

(a) (b)

Figure 5: One-element design for uniform to Bessel for f’ = 1.35, D = 2.61, k = 0.6, α = 1: a) and b) The input and the output surfaces; c) and d) The input and the output radius of curvature of the surfaces.

(a) (b)

Figure 6: One-element design for uniform to Bessel for f’ = 1.35, D = 2.61, k = 0.5, α = 0.6: a) and b) The input and the output surfaces; c) and d) The input and the output radius of curvature of the surfaces.

Table 1: Various design parameters for one-element lens design. α = 0.6 α = 1 α = 2 k 0.5 1 2 0.5 1 2 0.5 1 2

ro 2 1.2 0.6 (50%) of central lobe ri 0.84 1.68 3.36 0.5 1.0 2.0 0.25 0.50 1.0

ro 3.8 2.28 1.14 (95%) of central lobe ri 1.04 2.08 4.16 0.62 1.24 2.48 0.31 0.62 1.24

(a) (b)

(c) (d) Figure 7: Two-element design for uniform to Bessel for c’ = 0.5, D = 0.96, k = 1, α = 1: a) and b) The input and the output surfaces; c) and d) The input and the output radius of curvature of the surfaces. to a minimum length distance D = 1.68 cm; while both 50% and 95% central lobe widths. α f’ for = 0.6, the minimum value for is 0.37 which As an illustration of the derivation of the math- D corresponds to a minimum length distance = 0.71 ematical expressions of the lens surfaces y (r ) and cm. i i yo(ro), we present the derivation of the design of

Next, for the second two designs, we selected f’ Figure 3. We choose f’ = 1.35 cm, k = 1, ro = 2, α = 1.

= 1.75 cm (D = 3.384), k = 0.5, ro = 2 (50% of central Substitutingf’ in EQ. (7) and using both (3) and (11), lob width), for α = 1 and α = 0.6, as shown in Figure we can obtain the numerical values for the surface

5 and Figure 6. Note that when k decreases by a slopes dyi dr and dyoo dr . The mathematical ex- factor of 2, then ri decreases by the same factor. pressions for these slopes are achieved using a However, the minimum values for f’ are changed polynomial curve-fitting routine to yield: dy to 1.58 (D = 3.01 cm) and 1.33 (D = 2.57 cm), for i = ( 4.3210)rrrr432−( 8.9124) +( 6.0735) −+( 1.3532) 0.0572 dr iiii(12a) α = 1 and 0.6, respectively. This may explain why i f’ = 1.75 cm was selected in our design. Table 1 dyo 432 = ( 0.0935)rrrrooo−( 0.1875) +( 0.3307) −−( 0.1099) 0 0.0073 (12b) dr summarizes these results and includes others the o designs parameters such as for α = 1 and k = 2 for Integrating the above expressions to provide

(a) (b)

Figure 8: Two-element design for uniform to Bessel for c’ = 0.5, D = 0.96, k = 1, α = 0.6: a) and b) The input and the output surfaces; c) and d) The input and the output radius of curvature of the surfaces.

(a) (b)

Figure 9: Two-element design for uniform to Bessel for c’ = 0.5, D = 0.96, k = 0.5, α = 1: a) and b) The input and the output surfaces; c) and d) The input and the output radius of curvature of the surfaces.

5432 still use the same value ofc’ = 0.5 cm (D = 0.96). The yrrrrrri( i) = ( 0.8642) iiiii−+−+( 2.2281) ( 2.0245) ( 0.6766) ( 0.0572) (13a) effect ofk is restricted to only changing the value of 5432 yroo( ) = ( 0.0187) ro−+−−( 0.0469) ro ( 0.1102) ro ( 0.0550) ro ( 0.0073) r0 (13b) ri. Again, the two-element design achieved smaller Two-element design system length than the one-element one. Howev- er, accurate alignment and precise separation be- Similarly, for the two-element design, Figure 7, tween the two lenses are essential to achieve the Figure 8, Figure 9 and Figure 10 illustrate the four desired output profile. designs to demonstrate the effect of parameters k and α. Note that, in contrary to Eq. (5), Eq. (9) does Conclusion not suffer from any singularity for the chosen glass One- and two-element lens designs for trans- with n = 1.5172. forming laser beam profiles are considered in this For the first two designs, we have chosen the work using optical refracting system. The lens-de- same design parameters as for the one-element sign is applied to transform circular-uniform beam to be focused on the central beam lobe of a Bes- design, i.e., k = 1, ro = 2 (50% of central lob width), sel beam profile. In laser material processing, ex- for α = 1 (ri = 1.24) and α = 0.6 (ri = 1.68). However, we have selected a much smaller system length c’ tending the DOF is very important. The minimal = 0.5 cm (D = 0.96 cm) compared to D = 2.61 cm. spreading of the central lobe of the Bessel beam These designs are shown in Figure 7 and Figure 8. during wave propagation makes extending the DOF In these figures, when α decreases, then the input possible. A procedure for obtaining the mathe- radial distance ri increases for a fixed k. matical expressions for the two aspheric surfaces is outlined. An example was provided to illustrate Next, to demonstrate the effect of k on the de- the procedure. In addition, few important param- sign, we have reduced k to 0.5 and kept the same parameters as the previous two designs. These eters such as the length of the system, the power designs are shown in Figure 9 and Figure 10. Since ratio of the beams, and the first zero crossing of

(a) (b)

Figure 10: Two-element design for uniform to Bessel for c’ = 0.5, D = 0.96, k = 0.5, α = 0.6: a) and b) The input and the output surfaces; c) and d) The input and the output radius of curvature of the surfaces.

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 10 of 10 • the Bessel function, are discussed. The radii of cur- 3959-3962. vature of the designed surfaces are high. This leads 12. Chattrapiban N, Rogers EA, Cofield D, Hill III WT, Roy to smooth and gradual change on the surfaces for R (2003) Generation of nondiffracting Bessel beams by relative ease for machining and fabrication. It can use of a spatial light modulator. Opt Lett 28: 2183-2185. be concluded that the two-element design will pro- 13. Kuntz KB, Braverman B, Youn SH, Lobino M, duce smaller system length than the one-element Pessina EM, et al. (2009) Spatial and temporal design. characterization of a Bessel beam produced using a References conical mirror. Phys Rev A 79: 043802. 1. Ion JC (2005) Laser processing of engineering materi- 14. Hwang C, Kim K, Lee B (2011) Bessel-like beam als: Principles, procedure, and industrial application. generation by superposing multiple Airy beams. Opt Elsevier Butterworth-Heinemann, Burlington, USA. Express 19: 7356-7364. 2. Narendra SPH, Dahotre B (2008) Laser fabrication and 15. Chu X, Sun Q, Wang J, Lü P, Xie W, et al. (2015) Gen- machining of materials. Springer, New York, USA. erating a Bessel-Gaussian beam for the application in optical engineering. Sci Rep 5. 3. Duocastella M, Arnold CB (2012) Bessel and annular beams for materials processing. Laser Photonics Rev 16. Paschotta R (2008) High-power lasers. In: The 6: 607-621. encyclopedia of laser physics and technology. (1st edn), Wiley-VCH. 4. Kohno M, Matsuoka Y (2004) Microfabrication and drilling using diffraction-free pulsed laser beam 17. Dickey FM, Lizotte TE (2017) Laser beam shaping generated with axicon lens. JSME International applications. (2nd edn), CRC Press publishing. Journal Series B 47: 497-500. 18. Shealy DL, Chao S (2002) Geometric Optics-based 5. Courvoisier F, Lacourt PA, Jacquot M, Bhuyan MK, design of laser beam shapers. Opt Eng 42: 3123-3138. Furfaro L, et al. (2009) Surface nanoprocessing with 19. Awwal AAS, Ahmed JU (2001) Refractive system nondiffracting femtosecond Bessel beams. Opt Lett for generation of high-power diffraction-free laser 34: 3163-3166. beams. Optics & Laser Technology 33: 97-102. 6. Zambon V, McCarthy N, Piche M (2008) Fabrication 20. Karim MA, Cherri AK, Awwal AAS, Basit A (1987) Re- of photonic devices directly written in glass using fracting system for annular laser beam transforma- ultrafast Bessel beams. Proc SPIE 7079: 70992J. tion. Appl Opt 26: 2446-2449. 7. X Tsampoula, V Garcés-Chávez, M Comrie, DJ 21. Thewes K, Karim MA, Awwal AAS (1991) Diffraction- Stevenson, B Agate, et al. (2007) Femtosecond free beam generation using refracting systems. Opt cellular transfection using a nondiffracting light Laser Technol 23: 105-108. beam. Appl Phys Lett 91: 053902-053904. 22. Arif M, Hossain NM, Awwal AAS, Islam MN (1998) Two- 8. Zhang G, Stoian R, Zhao W, Cheng G (2018) Femto- element refracting system for annular Gaussian-to- second laser Bessel beam welding of transparent to Bessel beam transformation. Appl Opt 37: 4206-4269. non-transparent materials with large focal-position tolerant zone. Optics Express 26: 917-926. 23. Arif M, Hossain MN, Awwal AAS, Islam MN (1998) Refracting system for annular Gaussian-to-Bessel 9. Durnin J (1987) Exact solutions for nondiffracting beam transformation. Appl Opt 37: 649-652. beams I: The scalar theory. J Opt Soc Am A 4: 651-654. 24. Cherri AK, Khachab NI, Habib MA (2017) Two- 10. Lu J-Y, Greenleaf JF (1992) Diffraction-limited beams element refracting system for Gaussian and annular- and their applications for ultrasonic imaging and tissue Gaussian beams transformation. International characterization. In: FL Lizzi, New developments in Journal of Photonics and Optical Technology 3: 5-10. ultrasonic transducers and transducer systems. Proc SPIE 1733: 92-119. 25. Cherri AK, Khachab NI, Habib MA (2017) Designs of Two-element optical refracting system to achieve 11. Turunen T, Vasara A, Friberg AT (1988) Holographic uniform laser beam profile. Journal of Applied generation of diffraction-free beams. Appl Opt 27: Mathematics and Physics 5: 2371-2385.

Citation: Cherri AK, Habib MK (2019) Beam Shaping Optical Lens Designs for Diffraction-Free Bessel Beams. Int J Opt Photonic Eng 4:017