Controllable Spatial Array of Bessel-like Beams with Independent Axial Intensity Distributions for Laser Microprocessing

Transcription

1 JLMN-Journal of Laser Micro/Nanoengineering Vol. 3, No. 3, 08 Controllable Satial Array of Bessel-like Beams with Indeendent Axial Intensity Distributions for Laser Microrocessing Sergej Orlov, Alfonsas Juršėnas, Justas Baltrukonis and Vytautas Jukna Center for Physical Sciences and Technology, Industrial laboratory for hotonic technologies, Saulėtekio av. 3, Vilnius, Lithuania Bessel beams generated via axicons are widely used for various alications like otical tweeers or laser microfabrication of transarent materials. The secific intensity rofile having high asect ratio of beam width and length in turn generates high asect ratio void that resembles a needle. In contrast to commonly generated Bessel beam that has a fixed axial intensity distribution. We resent a novel method to engineer an otical needle that can have an arbitrary axial intensity distribution via suerosition of different cone angle Bessel beams. We analytically describe satial sectra of an otical needle having arbitrary axial intensity distribution. We also demonstrate a suerosition of indeendent otical needles and analye the hysical limitations to observe well searated otical needles as they are influence by mutual interference of the individual beams. In order to verify our theoretical and numerical results we generate controllable satial arrays of individual beams with various numbers and satial searations by altering a sectrum of incoming laser beam via satial light modulator. Lastly, we numerically examine distortions caused by roagation through lanar air-dielectric interface and show comensation method by aroriately modifying sectral masks. DOI: 0.96/jlmn Keywords: diffraction, otical engineering, satial light modulator, Bessel beam, otical needle, focal line, translation of Bessel beams. Introduction In many alications of laser microfabrication and otical traing it is advantageous to use laser beams with long deth of focus and narrow transverse intensity distribution []. One examle of such otical field is a nondiffracting Bessel beam, which exhibits such features in the so-called Bessel one []. These beams are usually generated by axicons and are widely used in such alications as laser micromachining [4-6] and otical tweeers [7]. However due to fixed bell shaed axial intensity distribution these beams are only a single tye of the otical needle family [3]. In ractical alications it is imortant to eliminate aberrations caused by lanar dielectric material interface (e.g. focusing from air into the volume of bulk material) [8]. Thus, a numerical investigation of the roblem with demonstration how the aberration may be eliminated is of a ractical imortance. It was shown in Ref. [9] that the suerosition of eroorder Bessel beams with secific axicon angles and comlex amlitudes can be used to create a redefined axial intensity distribution which is more ractical than axial intensity attern obtained by conventional conical lens. In this work we introduce a methodology of roducing arallel Bessel-like otical needles with controllable individual axial intensity attern. We resent also an exerimental imlementation of such beams using a satial light modulator. Lastly we dive into the roblem of the aberrations introduced by a lanar interface and show, how one could comensate axial intensity distortions due to focusing through air-dielectric interface. The elimination satial aberrations caused by lanar dielectric material interface (e.g. focusing from air into the the bulk material) are very imortant for ractical alications as it not only imacts the laser energy deosition efficiency [8] but also the axial intensity distribution of the otical needle. Thus, a numerical investigation of the roblem with demonstration how the aberration may be eliminated is of a ractical imortance.. Axial intensity control in an otical needle and translation of otical needles In this section we resent the theoretical basis for creation of arrays of arallel Bessel-like otical needle beams with controlled axial intensity attern. Here we use ideal Bessel beams as a basis functions. They are obtained from an angular sectrum described by a Dirac s delta function [0]. These nondiffracting beams are well enough aroximations of exerimentally observed intensity distributions near the focal oint of the Fourier lens.. Axial intensity control in an otical needle Ideal nondiffracting Bessel beam is a solution of scalar Helmholt equation in circular cylinder coordinates ψ( ρφ,, ) = J ( k ρ)ex(imφ + i k ), () m ρ where ψ - is electric field, J m - the m - th order Bessel function, ρφ,, - cylindrical coordinates, k ρ, k - radial 34

2 JLMN-Journal of Laser Micro/Nanoengineering Vol. 3, No. 3, 08 and axial wavenumbers resectively and m is a toological charge of the Bessel beam [0]. Any solution of scalar Helmholt equation can be reresented as a 3D integral containing lane waves with different wave vectors, which can be further reduced to a D integral, if the fields are axisymmetric, see [0]. This twodimensional field reresentation is based on Fourier-Bessel transform and can be rewritten for our uroses by changing the integration variables from k ρ to k. The axial comonent of the wave vector will enter into the integral as a Fourier transform. This aroach to the engineering of axial rofiles is discussed in great detail in Ref [9]. By enforcing the radial coordinate to be ero, one ends u in a Fourier series (with resect to axial coordinate ) using a suerosition of Bessel beams (), where a Fourier integral is defined as Ψ = () r A( K ) (; r K ) dk. () + k 0 ψ We assume, that k = k0 + K, where k 0 is a carrier wave vector and Ak ( )- comlex amlitude of the satial sectra (i.e. of the each individual Bessel beam comonent). The function ψ for the case m = 0 has values on axis ψ = ex(i k ), therefore the Eq. () on-axis will be an exression for the Fourier sectrum of a selected axial intensity distribution Ψ (0, ) = f ( ), and the term k 0 is used to shift the sectrum to ositive k values in order to restrict to forward roagating waves only = π. (3) + i A( k ) ( ) k f e d Thus, a continuous suerosition of Bessel beams Ψ() r defined in () with satial sectrum (3) can exhibit roerties of axial intensity similar to those defined by a function f( ).. Translation of an otical needle In order to control transverse osition of an otical needle we use addition theorem of Bessel beams [0] i nφ n ρρ m ρρ n+ m ρρ m= i m ( φ φ ) J ( k ) e = J ( k ) J ( k ) e, (4) of translated Bessel beam is a sum of many terms with Dirac delta functions m imφ k imφ i e δ( ksin θ k ) ρ ˆ ( kρ, k) Jm( kρ ) e. m= kρ ψ φ ρ = (5) In this way the Bessel beam with origin at shifted oint O may be exanded as a suerosition of Bessel beams in the unshifted origin O (see Fig.). Let us assume, we would like to have a number =,, P of indeendent arallel otical needles each with its own axial rofile and osition ( x, y ) in the transverse lane. We use the suerosition rincile and exress the resulting satial sectrum as a sum here Ψˆ ( k, k ) A ( K ) ( k, k ; x, y ) dk. (6) ˆ x y = + k 0 ψ x y = π is the Fourier transform f, and + i A ( ) ( ) k k f e d of the individual axial intensity rofile ( ) ψˆ ( k, k ; x, y ) is a Bessel beam s satial sectrum, when x y it is shifted in the transverse lane to the oint ( x, y ). 3. Comensation of the aberration due to the lanar interface In order to analye aberrations caused by air-dielectric interface we extend our scalar beam descrition to a vector one by assuming that the beam is x-olaried V( θ, φ) = x Ψˆ ( θ, φ) (7) here θ is the incidence angle on the air-dielectric interface, ˆΨ is the angular sectrum as defined in (7). We note also that Bessel beams roduced by conventional conical lens do not suffer sherical aberration as the beams contain single angles of incidence. The refraction only changes the Bessel angles but do not create aberrations as for a Gaussian beam. In our case, we have a continuous integral of individual Bessel beams with different angles, so the resulting beam exeriences aberrations due to the lanar interface. where, ρ Fig. Bessel function translation. ρ - cylindrical coordinates of first and second coordinate systems resectively and ϕ and ϕ are aimuthal angles of first and second coordinate systems. The sectrum of individual Bessel beam is Dirac delta function multilied by aimuthal hase term, therefore the sectrum Fig. Princial scheme of focusing trough air-dielectric interface. Here L focusing lens, n, n refractive indices, Einc - incident field focused by the lens L, θ, θ- focusing angles. According to [8,0] the transmitted field in dielectric medium E t can be exressed as the following integral over focusing angles π θmax s E t = { ( )[ (, ) ] ( )[ (, ) ] t θ V θ φ φ φ + t θ V θ φ θ θ } 0 0 (8) i( kxx+ kyy+ k) e sinθ cos θ dθ dφ, 35

3 JLMN-Journal of Laser Micro/Nanoengineering Vol. 3, No. 3, 08 here s t, t are Fresnel coefficients [8], θ, θ, φ - focusing angles as deicted in Fig. From the analysis of the integral (8) we note that the otical beam in the second medium is distorted due to s Snell s law and Fresnel s coefficients. If the ratio t / t of Fresnel s coefficients is close to the unity then we can correct distortions just by accounting for a change in angles due to Snell s law. This correction is done by substitution θ θ ( θ ) in the focusing integral kernel (8) π θmax s E t = { ( )[ (, ) ] ( )[ (, ) ] t θ V θ φ φ φ + t θ V θ φ θ θ } 0 0 (9) i( kxx+ kyy+ k) e sinθ cos θ dθ dφ, such change corresonds to the change in the satial sectrum. To illustrate this method of distortions comensation let us analye the focusing of a circle of arallel otical needles (see Fig 3) from air into dielectric medium with refractive index n =.5 rofiles. The satial sectra with angles adjusted so, that uon the entrance through the lanar interface, the Snell s law enforces roer axial shae of individual needles, see first column (or the left side) in Fig.. 4. Exerimental setu Verification of numerically simulated arrays of otical needles in the air were erformed using a hase-only satial light modulator (SLM) together with an otical set-u deicted schematically in Fig. 4. The linearly olaried beam of the wavelength of 53 nm were used for the exeriments. The beam is limited to only a small central art (8 mm diam.) of its exanded diameter using a diahragm to achieve a more uniform intensity over the matrix of the SLM. The beam slitter (BS) cube ensures that the incident angle of ero degree is achieved. A reflected beam has a hase, which was modified via hase delays induced by the SLM. The reflected beam undergoes a rescaling inside a 4f imaging system and is further Fourier transformed by a Fourier lens. Axial and transverse intensity rofiles are catured with an otical imaging system mounted on a linear translation stage. We note that we achieve a x7.4 transverse magnification and aroximately a x6 longitudinal magnification in our setu, when comared to numerically simulated beams. Fig. 4 Otical set u of the exeriment. CW laser, lenses (4f imaging and Fourier), objectives (40x and 0x), satial light modulator (PLUTOVIS-006-A, HOLOEYE Photonics AG), beam slitter BS and CCD camera. 5. Results and discussion Exeriments were carried out using a SLM to verify diffraction of such beams in the air and to examine our control caabilities while controlling a) individual needles and b) their arrays, as well as c) distortions caused by destructive interference between the individual needles inside an array. Fig. 3 Comarison of situations with comensation of distortions due to focusing trough air-dielectric interface ( n =, n =.5 ), see first column or index, and without, see second column or index. Here (a,, b,) are amlitude and hase distributions of satial sectra and c, are electric field intensity distributions. The lanar air-dielectric medium interface is located at = 0, the air is the region with < 0. Results of our numerical simulations are resented in Fig. 3. First of all, we observe that the resence of a lanar interface introduces two tyes of distortions (see the second column or the right side): the longitudinal one, due to the change in the hysical angles and a weaker transverse one, which can be noticed uon careful comarison of the beam 5. Axial intensity rofile of single needle Here we aim to create a constant ste-like axial rofile. For our exeriments we select the function f( ) of the axial rofile to be a suer-gaussian function N 0 f( ) = ex, (0) 0 /N where 0 = L/ [log( / )] is a arameter controlling the axial intensity full width at half maximum (FWHM), N - is the order of suer-gaussian function, L is the length at FWHM. This choice enables for smoother axial intensity rofiles (fluctuations around the desired ste-like due to the Gibbs 36

4 JLMN-Journal of Laser Micro/Nanoengineering Vol. 3, No. 3, 08 henomenon are smaller both in the numerical simulations and in the exeriment). Firstly, we examine exerimentally axial intensity rofiles described by the Eq. (0). The main aim here is to achieve a smooth intensity rofile over length L with stee edges at the beginning and the end. The shae of this function is strongly deendent on the order N of the suer- Gaussian function. For low N values (N=-4) the edge steeness is oor, but the axial rofile itself is rather smooth. When increasing the number N, we increase also the edge steeness but we lose the smoothness. Exerimental results for a single case of an otical needle with L= mm are deicted in Fig. 5 which are the best results (N = 7 ) which we have achieved. In this case we observe the most otimal balance between two cometing factors, thus, giving us a stee and even axial intensity rofile. We will further use suer-gaussian axial rofiles with N = 7. Fig. 5 A comarison of exerimentally measured (orange line) and numerically calculated (blue line) suer-gaussian (N = 7) axial intensity rofiles of an otical needle of the length L = mm generated with our otical set-u. 5. Arrays of otical needles Next, we are using in further exeriments the same beam arameters as in revious section. Here, we comare exerimentally and numerically obtained transverse rofiles of three arallel otical needles ositioned in one row with satial searation of ρ = 60 λ, see Fig.. We observe a good agreement between exerimental results and numerical simulation as the direct comarison shows only minor differences in transverse intensity rofiles (Fig. 6), which indicates both roer work of our exerimental set-u and the correctness of theoretical methods. Some interference of nearby beams occurs in both cases, which can be seen as distortion of tyical ring system around Bessel-like beams. All transverse rofiles are catured at the middle of the axial rofile which is set to be at the Fourier lane. Satial searation has a big influence on the formation of individual otical needles. As we try to bring them closer, by lowering the individual searation ρ, the destructive interference tends to increase and otical needles are no longer generated correctly. On the other hand, by increasing the searation distance ρ, distant needles lose some intensity due to limitations of our set-u, see Fig. 7, which also limits the ositioning caabilities of the needles using the method described here. Length of the array is also an imortant factor to consider as it also heavily affects the occurrence of destructive interference. As it can be seen from Fig.8, an array of individual needles with shorter individual lengths L shows nearly no interference between individual arts. As one could exect, longer individual otical needles in the array cause more destructive interference. This can be understood considering the fact, that the creation of individual Bessel one requires some volume for the lane waves lying on the cone to interfere and create an otical needle. The longer the needle the larger is the volume of this effective Bessel one. Furthermore, we demonstrate an ability to control the length of an individual otical needle inside the array searately. Here, we construct an array of four otical needles with different individual lengths (Fig.9). The influence of the individual length L can be also noticed in this exeriment, as the interference between adjacent elements increasingly gets stronger as adjacent otical needles gets longer (Fig. 9 b)). Fig. 7 Intensity distribution of exerimentally obtained arallel arrays of three otical needles. The axial length L of individual needle is L = mm, N = 7, the satial searation ρ between the needles in the transverse lane is deicted on the grah. Fig. 6 Intensity distribution of exerimentally obtained and numerical calculated arallel arrays of three otical needles. The axial length L of individual needle is L = mm, N = 7, the satial searation ρ between the needles in the transverse lane is ρ = 60 λ. Fig. 8 Intensity distribution of exerimentally obtained arallel arrays of three otical needles. The axial length L of individual needle is L = 0.4 mm and.4 mm, N = 7, the satial searation ρ between the needles in the transverse lane is ρ = 60 λ. 37

5 JLMN-Journal of Laser Micro/Nanoengineering Vol. 3, No. 3, 08 rings which aear as we suerose multile beams to control the axial rofile. The rings also contains some intensity modulation which occurs as we create and suerose translated beams (see, Eq. (7)). The hase structure of the satial sectra is here rather comlicated, what can be seen as a manifestation of the comlexity of this rather simle structure while reroducing it with lane waves. Fig. 9 Intensity distribution of exerimentally obtained arallel arrays of four otical needles. Intensity rofiles are deicted in a) y and b) xy lanes. The individual lengths L in the array are L= 0. mm, 0.4 mm, 0.6 mm and 0.8 mm. Finally, we resent here our caability to form comlex satial structures by lacing individual otical needles in secific laces in the focal lane. A good examle would be a situation when we form a circle of eight searate needles of individual lengths L = 0.5 mm (Fig. 0). Shorter beams and right satial searation between allows us to minimie the destructive interference and form a structure with clearly exressed individual needles. Fig. 0 Three dimensional deiction of the exerimentally measured intensity distribution in the array of eight otical needles with individual length L = 0.5 mm. Otical needles are osition so that they create a circle. 6. Conclusion We have resented a flexible technique, which enables us to create exerimentally controlled arrays of arallel otical needles with indeendent axial intensity rofiles. We have analyed how the searation between individual otical needles interlays with the individual lengths of the otical needles. We show, that the destructive interference between adjacent needles is less ronounced when they are of different lengths. Our reliminary analysis shows, that this is caused by the fact, that otical needles of different lengths have different satial modulation in the Fourier sace. The distortion between the neighboring otical needles aears due to the satial overlaing of the beams. Therefore to achieve the best results it is advisable to searate the beams so that they have limited overla. The downside of this technique is limiting the smallest searation between the beams, or limits the needle length, or limits the beam width. Additionally we have introduced a ste-like axial intensity rofile described using a suer-gaussian function. This has enabled us to avoid roblems caused by the Gibb s henomenon, when the edges of a ste function exhibit very shar oscillations. We have found otimal arameter of the suer Gaussian function N=7, which ensures both nice smoothness of the rofile and sharness of the edges. Moreover, we have demonstrated the roof-of-concet imlementation of the technique, which allows for comensation of various distortions due to aberrations introduced by a lanar interface between air and dielectric. In conclusion, the method resented here allows creation of various satial intensity distributions in 3D which might be alicable for ossible secific microfabrication tasks or otical tweeing set-us. The imlementation of this aroach into high ower laser systems will require us to move away from the satial light modulator, as it cannot sustain high laser owers. However the satial light modulator can be a versatile device to rototye the geometrical hase element, which are known for their sustainability to high laser owers. Acknowledgments and Aendixes This research is/was funded by the Euroean Social Fund according to the activity Imrovement of researchers qualification by imlementing world-class R&D rojects of Measure No LMT-K-7. Fig. Amlitude and hase distributions of the satial sectra of a circular array containing eight otical needles, deicted in the Fig. 0. The satial sectra of such structure is deicted in Fig.. We see from the amlitude distribution of the satial sectra that it contains not only a single ring (as exected for Bessel-Gaussian beam) but a few more, not so intense References [] H. Misawa, and S. Juodkais: 3D Laser Microfabrication Princiles and Alications Edited by. (Wiley- VCH; 006). 38

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