simulation of propagation and transformation of thz bessel beams with orbital angular momentum

Available online at www.sciencedirect.com ScienceDirect Physics Procedia 84 (2016) 175 – 183 International Conference "Synchrotron and Free electron laser Radiation: generation and application", SFR-2016, 4-8 July 2016, Novosibirsk, Russia Simulation of propagation and transformation of THz Bessel beams with orbital angular momentum Yulia Choporovaa,b*,Boris Knyazeva,b, Mikhail Mitkova, Natalya Osintsevaa,c, Vladimir Pavelyevd a Budker Institute of Nuclear Physics, Novosibirsk, 630090, Russia b Novosibirsk State University, Novosibirsk, 630090 Russia c Novosibirsk State Technical University, Novosibirsk, 630073 Russia d Samara University, Samara, 443086 Russia Abstract Recently, terahertz Bessel beams with angular orbital momentum (“vortex beams”) with topological charges l r1 and l r2 were generated for the first time using radiation of the Novosibirsk free electron laser (NovoFEL) and silicon binary phase axicons (Knyazev et al., Phys Rev Letters, vol 115, Art 163901, 2015) Such beams are prospective for application in wireless communication and remote sensing In present paper, numerical modelling of generation and transformation of vortex beams based on the scalar diffraction theory has been performed It was shown that the Bessel beams with the diameters of the first ring of 1.7 and 3.2 mm for topological charges ±1 and ±2, respectively, propagate at a distance up to 160 mm without dispersion Calculation showed that the propagation distance can be increased by reducing of the radiation wavelength or using a telescopic system In the first case, the propagation distance grows up inversely proportional to the wavelength, whereas, in the latter case the propagation distance increases as a square of a ratio of the telescope lenses foci Modelling of the passing of the vortex Bessel beams through a random phase screen and amplitude obstacles showed the self-healing ability of the beams Even if an obstacle with a diameter of 10 mm blocks several central rings of Bessel beam, it reconstructs itself after passing a length of about 100 mm Results of the simulations are in a good agreement with the experimental data, when the latter exist © 2016 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license © 2016 The Authors Published by Elsevier B.V (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility oforganizing the organizing committee of SFR-2016 Peer-review under responsibility of the committee of SFR-2016 Keywords:Terahertz radiation, Bessel beams, beam with orbital angular momentum * Yu Yu Choporova Tel.: +7-913-383-4346; E-mail address:yu.yu.choporova@inp.nsk.su 1875-3892 © 2016 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of SFR-2016 doi:10.1016/j.phpro.2016.11.031 176 Yulia Choporova et al / Physics Procedia 84 (2016) 175 – 183 Introduction Beams with orbital angular momentum (OAM), or “vortex beams,” since 1992 [Allen et al (1992)]] drawn significant attention These beams can potentially transform many fields of optics and be applied to designing new kinds of devices In the visible spectral range, they have been utilized, for example, in optical tweezers [Moura et al (2015)], optical vortex coronograph [Foo et al (2005)], and communication systems [Krenn et al (2014)] In the latter case, an additional advantage of vortex beams is the possibility of OAM multiplexing Many kinds of amplitude and phase elements (all of them can be classified as “holograms”) were already applied to vortex beam formation: spiral phase plates [Oemrawsingh et al (2004), Sueda et al (2004), Turnbull et al (1996), Tyson et al (2008)], a deformable mirror [Heckenberg et al (1992)], a spatial light modulator [Jesacher et al (2008)], or a spiral Fresnel zone plate [Weibin et al (2009] For completeness, it has to be noted that the application of DOEs to terahertz radiation manipulation and transformation was considered in [Agafonov et al (2015), Agafonov et al (2013), Knyazev et al (2010), Siemion et al (2011), Siemion et al (2012), Sypek et al (2012), Walsby et al (2007)] Among the vortex beams, Bessel beams (BB) [McGloin et al (2005)] are of special interest because of their ability to propagate some distance without divergence, which is potentially beneficial for BB employment in communication systems and remote sensing Recent publication [Soifer (2012)] showed that wireless communication systems in the terahertz (THz) range had a number of advantages comparing to the systems operating in the optical and radiofrequency ranges Wireless technologies below 0.1 THz are not able to support Tb/ps links because of capacity limitations, whereas free space optical communication systems in the infrared and visible ranges have practical limitations because of the eye safety limits, the impact of atmospheric effects on the signal propagation (fog, rain, aerosols), high diffuse reflection losses, and the impact of misalignment between transmitter and receiver [Beijersbergen et al (1994)] To date, only few publications were devoted to vortex beams in the THz range First experimental generation of THz vortex Bessel beams was described recently in paper [Choporova et al (2015)], in which silicon diffractive optical elements (DOEs), binary axicons with spiral configuration of the phase patterns [Knyazev et al (2015)], were applied for the transformation of a high-power Gaussian beam of Novosibirsk free electron laser [Kulipanov et al (2015)] into the quasi-Bessel ones The vortex beams with topological charges l r1 and l r2 were generated at a wave length of 141 Pm In these experiments the beams propagated without diffraction a distance of about 150 mm This value is not sufficient even for in-door communication systems, and the further experimental studies of THz beam propagation are under consideration In this paper we consider methods, which allow extending “non-diffractive” propagation length of vortex Bessel beams formed with binary axicons Numerical calculations were performed using the scalar diffraction theory In addition, we investigate self-healing of THz Bessel vortex beams passed through random phase screen, or a nontransparent large-size obstacle Computer simulation The computer simulation has been performed using Matlab software for studying beams properties Numerical modelling of generation and transformation of vortex beams based on the scalar diffraction theory has been performed Based on Huygens-Fresnel principle, each point of wave front is a secondary wave source (Fig 1) The diffractive optical element was illuminated by the simulated NovoFEL Gaussian beam Novosibirsk free electron laser generates monochromatic radiation which wavelength can be smoothly tuned within the range of 5-240 Pm The beam has a Gaussian shape with beam waist of 12 mm Distribution of electro-magnetic field U ([ ,K ) at the distance z behind the DOE with transmittance function E ( x, y) can be found through the Fresnel approximation: U ([ ,K ) § · ˜ exp ikz ³³ E ( x, y) exp ăă i k êô([  x)2 (K  y )2 ằ áádxdy ẳạ iO z â 2z (1) 177 Yulia Choporova et al / Physics Procedia 84 (2016) 175 – 183 This expression can be calculated via two-dimensional Fourier transform of exp ik ( x2  y ) / z ˜ E ( x, y ) , and via two-dimensional convolution of E ( x, y) and a function exp ik ( x  y ) / z Let’s consider the difference 2 between these two methods If we rewritten Ex in the form of Fourier transformation: Fig.1 a) Illustration of Huygens-Fresnel principle Diffraction on a diffractive optical element can be calculated using Fresnel-Kirchhoff integral Each point in image plane is found by summing the contributions from each point of DOE; b) DOE phase distribution (black color – phase 2πN, white color – π(2N+1), where N – integer) with topological charge l r1, l r2 exp ikz U ([ ,K ) iO z Đ k à exp ă i ([  K ) ¸ u z â (2) Đ k Ã Đ k à u exp ă i ( x  y ) E ( x, y )exp ă i ([ x  K y ) ¸ dxdy z z © ¹ © ¹ Complex distribution of field U ([ ,K ) in the real-image plane can be calculated by the two-dimensional Fourier transform of exp ik ( x  y ) / z ˜ E ( x, y ) , and its multiplication by a complex factor exp ikz ˜ exp ik ([ K ) / z / iO z involving discrete Fourier transform transformation Hence the intensity can be found as amplitude of complex value: I ([ ,K ) U ([ ,K ) The phase can be calculated: )([ ,K ) ª Im U ([ ,K ) º » Re U ([ ,K ) ẳ arctg ô (3) (4) Although Fourier transform is a rapid method for calculating of electromagnetic field in any plane behind the DOE, spatial frequencies u and v in the image plane in this case will depend inversely on the distance z according to the Fourier transform as 178 Yulia Choporova et al / Physics Procedia 84 (2016) 175 – 183 k u z [ k v z K (5) Thus, increasing of the distance would cause the restriction of the angular spectrum and thus, reduce the reconstructed image that would be shown on experimental data below On the other hand expression (1) can be rewritten as two-dimensional convolution of E ( x, y) and a function exp ik ( x  y ) / z F 1 > S ( Z ) ˜ S ( Z )@ f f 2S f 1 f 2S f ³ S ( u ) 2S ³ S ( Z )exp iZ( t  u )d Zdu f ³ S ( Z ) ˜ S ( Z )exp iZt dZ f f f ³ S ( Z ) ³ S ( u )exp iuZ du ˜ exp iZt d Z f f (6) ³ S ( u )S ( t  u )du f In this case distribution of electro-magnetic field at the distance z can be performed through direct calculation of the Fourier transform of DOE transmission function E ( x, y) , multiplying by the Fourier transform of exp ik ( x  y ) / z and then performing the inverse Fourier transform This method requires three times more matrix operations: performing of two Fourier transforms and then inverse Fourier transform, but spatial frequencies are no longer depending on the distance z u [ v K (7) The size of output image is equal to the size of the real imaging deviсe for vortex beam investigation at NovoFEL facility Microbolometer focal plane array (MFPA) with 16.32x12.24 mm physical size and 51x51 Pm pixel size was used as a detector Results Recent reports have shown that beams with OAM can be used for data transmission in RF links [Tamagnone et al (2012), Tamburini et al (2012), Tamburini et al (2013)] For potential application in communication systems, in our paper vortex diffractive elements was manufactured as a combination of axicon and azimuthal phase plate which transforms Gaussian beam into Bessel beam with spiral phase Additional advantages of the Bessel beams with OAM are the increase of the information capacity thanks to an additional degree of freedom, with orbital quantum number l 0, r 1, r , and the ability of self-reconstruction after propagation through obstacles The first unique property of Bessel beams is non-diffraction, which means that the beam doesn’t change its intensity distribution for some distance Simulation of longitudinal section of the beam is shown in Fig The beam starts forming at the 100 mm and keeps its intensity distribution up to 260 mm Thus, the propagation distance is about 160 mm The cross section of beam represents donut-like intensity distribution with the size of the first ring of 1.7 and 3.2 mm for vortex with ℓ =1 and ℓ =2 respectively Absorption of THz radiation in the air is excluding 179 Yulia Choporova et al / Physics Procedia 84 (2016) 175 – 183 Fig Simulation of longitudinal section of the beam with O0=141 μm at distance L=10÷460 mm for topological charge a) ℓ=+1; b) ℓ=+2 Input picture represents the amplitude distribution at the distance L=100, 150, 250 mm For the beams formed with a binary axicon, the region, in which the quasi-Bessel beam exists, is limited by the expression zmin R / - , where R is an axicon radius, - O0 / p is a diffraction angle of the binary axicon with a period p, and O0 is the wavelength for which the axicon was designed Obviously, when the axicon radius increases and decreases the angle of diffraction, the distance at which the beam exists, significantly grows up This result can be achieved in several ways, increasing R, increasing circular grating period, or using wavelengths equal to O0 / n , where n is an integer (Fig 3) As can be seen, propagation distance increase inversely proportional to the radiation wavelength by and times for O / and O / , respectively Fig Increasing of propagation distance by reducing the wavelength Simulation of longitudinal section of beam with O0 141, 70 and 47 P m at distances L=10÷900 mm for topological charge ℓ=+1 To increase the propagation distance, we have also used an optical system shown in Fig Focal distances of the telescopic lenses were f1 =75 mm and f2=250 mm The distance between lenses is equal to sum of focal distances of the lenses f1 and f2 Using the matrix method one can obtain two parameters which are defining the propagation distance of the beam: diffractive angle and axicon radii 180 Yulia Choporova et al / Physics Procedia 84 (2016) 175 – 183 Fig Increase of propagation distance by using telescopic system M Đ ă ă ă f â 0à áuĐ1 áá ăâ Đ f1  f à ă u ăă  â f1 0à áá Đ f2 ă ă f1 ă ă â à f1  f ¸ ¸ f1 ¸  ¸ f2 The coordinates were recalculated: Đ f2 Ã Đ f2 à ă  r  ( f1  f )- ă  f r Đ rc Ã Đ r à ă f1 á|ă ă cá M uă ă ă f1 f1 â- â-  ă ă -á f2 © ¹ © f2 ¹ After passing the telescopic system, Bessel beam radii grew up by a factor of f / f1 distance increased by a factor f / f1 (8) (9) 3.5 , and the propagation , i e., 1.5 meters It is easy to calculate, that, if increase both the input beam and the axicon radii by times and simultaneously increase by times the axicon period, the propagation distance grows up by 10 times, which in our case corresponds to 1600 mm Another beam property is self-healing (Fig 5) Propagation of laser beams through turbulent atmosphere has important impacts in many applications such as free-space optical communications, remote sensing, Laser Radar, Light Detection and Ranging (LIDAR) Numerical calculations of the intensity and phase distribution are performed It is assumed that an obstacle is kept at the plane, where Bessel beam is totally formed (in our case 130 mm behind the DOEs) The size of the obstacle is represented in Fig e, l First, we calculated passage of the beam through inhomogeneous phase screen The random phase screen is generated using the method of filtering white Gaussian noise A 512x512 field of pseudorandom complex numbers A+jB are generated with Matlab The results are shown for distances of 10, 50 and 100 mm After passage through random phase screen the beam recover its intensity and phase distribution even at the distance of 10mm Yulia Choporova et al / Physics Procedia 84 (2016) 175 – 183 181 Fig Self-healing ability Simulation of passage the beam through random phase a) The initial Bessel vortex beam with ℓ=+1 at a distance L=130 mm from DOE; e) random phase screen; (b, c, d) intensity distribution after screen at the distance of 10, 50 and 100 mm, respectively; (f, g, h) phase distribution of Bessel vortex beam behind the disturbance plane at the distances of 10, 50 and 100 mm, respectively; i) amplitude obstacle diameter mm (j, k, l) intensity distribution behind the obstacle at the distance of 10, 50 and 100 mm, respectively The results of vortex beam propagation after hindrance are demonstrated for the distances of 10, 50 and 100 mm Self-healing of the vortex Bessel beams after amplitude obstacles can be observed for obstacles sizes up to DOEs radii The reconstruction distance is inversely proportional to the size of obstacle Summary Numerical modelling of generation and transformation of the vortex beams based on scalar diffraction theory has been performed in this paper The results of simulations shown in Figs 2, 4, are in a good agreement with the experimental results described in papers [Choporova et al (2015), Knyazev 2015), Volodkin et al (2016)] Shapes of the beams calculated using the Fresnel–Kirchhoff integral are in an excellent agreement with the experimentally observed beam shapes The diameters of the first ring are of 1.7 and 3.2 mm for vortex beams with topological charges ±1 and ±2, respectively In our simulations, as well as in the experiments published in [Knyazev et al (2015)], the spatial distribution of the beam intensity not change throughout the distance L=100260 mm for both beams By numerical modelling, it was shown that, if use the same axicon, the propagation distance increased inversely proportional to the radiation wavelength The use of a telescopic system also increases the propagation distance After passing the telescopic system, Bessel beam rings radii grows up by a factor of a ratio of lens focuses, whereas the propagation distance increases by a square of a telescope magnification The simulation results have demonstrated that, the use of a telescopic system with magnification factor of 3.25, the propagation distance grows up by 10 times that in our case equal to 1600 mm This unique property of propagating at a longdistance without dispersion, as well as, the self-healing ability of vortex Bessel beams after passage through a 182 Yulia Choporova et al / Physics Procedia 84 (2016) 175 – 183 random phase screen and amplitude obstacles, can be effectively applied to the radioscopy of the objects partially transparent to the terahertz radiation, to wireless communication and remote sensing Acknowledgements Diffractive optical elements were designed and fabricated under support of the Ministry of Education and Science of the Russian Federation (project 1879); study of vortex beam characteristics was supported by RFBR grant 15-0206444 The authors thank the Russian Science Foundation grant 14-50-00080 for the support of the development and assembly of the dedicated terahertz focusing system (beamline) The modelling was carried out for the experiments which were 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