Creating Airy beams employing a transmissive spatial. light modulator

Transcription

1 Creating Airy beams employing a transmissive spatial Abstract light modulator Tatiana Latychevskaia 1,*, Daniel Schachtler 1 and Hans-Werner Fink 1 1 Physics Institute, University o Zurich, Winterthurerstrasse 19, 857 Zurich, Switzerland tatiana@physik.uzh.ch We present a detailed study o two novel methods or shaping the light optical waveront by employing a transmissive spatial light modulator (SLM). Conventionally, optical Airy beams are created by employing SLMs in the so-called all phase mode. In the irst method, a numerically simulated lens phase distribution is loaded directly onto the SLM, together with the cubic phase distribution. An Airy beam is generated at the ocal plane o the numerical lens. We provide or the irst time a quantitative properties o the ormed Airy beam. We derive the ormula or delection o the intensity maximum o the so ormed Airy beam, which is dierent to the quadratic delection typical o Airy beams. We cross-validate the derived ormula by both simulations and experiment. The second method is based on the act that a system consisting o a transmissive SLM sandwiched between two polarisers can create a transmission unction with negative values. This observation alone has the potential or various other waveront modulations where the transmission unction requires negative values. As an example or this method, we demonstrate that a waveront can be modulated by passing the SLM system with transmission unction with negative values by loading an Airy unction distribution directly onto SLM. Since the Airy unction is a real-valued unction but also with negative values, an Airy beam can be generated by direct transer o the Airy unction distribution onto such an SLM system. In this way, an Airy beam is generated immediately behind the SLM. As both new methods do not employ a physical lens, the two setups are more compact than conventional setups or creating Airy beams. We compare the perormance o the two novel methods and the properties o the created Airy beams. 1. Introduction In 1979, Berry and Balazs predicted the possibility o a wave propagating with acceleration and without diraction [1] The wave packet distribution is ound by solving the Schrödinger equation and described by an Airy unction. Although Berry and Balazs considered waves o particles o mass m, the irst experimental demonstration o Airy beams was reported or photons by Siviloglou et al. in 7 [-3]. Optical Airy beams have unusual properties: they are non-diractive over long distance and can sel-heal during the propagation when a part o the beam is blocked at some plane [4-5]. It has been shown that the propagation characteristics o Airy beams can be described under the travelling-wave approach analogous to that used or non-diracting Bessel beams based on the notion that Airy unctions are, in act, Bessel unctions o ractional order 1/3 [6]. The optimal conditions or generating Airy beams and their propagation properties have already been investigated [7-9]. Optical Airy beams have ound a number o applications, as or example or optical micromanipulation [1], optical trapping [11-14], and super-resolution imaging [15]. In 13, electron Airy beams were generated by diraction o electrons through a nanoscale hologram [16]. 1.1 Airy unction distributed wave The analogy between electron and light optical waves arises rom the similarity between the Schrödinger equation and the Helmholtz equation in the paraxial approximation ik, x z where k is the magnitude o the wave vector and x and z are coordinates in transverse and propagation direction, respectively. The solution to Eq. (1) is given by the expression: (1) b z ib z b z x, z Ai b x exp x, 4k k 6k where b has units o m -1. The delection o the main maximum in x -direction is ound as ()

2 and the total delection: 3 bz xmax ( z) xmax ( z) xmax () 4k (3) 3 bz rmax ( z) xmax ( z) ymax ( z). 4k (4). Creation o Airy beams by Fourier transorm o cubic phase distribution.1 Two-dimensional Airy unction and its Fourier transorm It is a property o an analytical Airy unction that its Fourier transorm is a complex-valued unction with a cubic phase distribution: 3 i Ai, exp i( x y) d d b exp b x 3 y 3 b b 3, (5) here b is a constant that has units o m -1, and ( xy, ) and (, ) are the coordinates in real space and the Fourier domain, respectively. Equation (5) can be rewritten in the orm o an inverse Fourier transorm, where by replacing and, we obtain: i exp b x y exp i( x y) dxd y Ai,. 3 b b b (6) The dierence between the Airy unction in (, ) and (, ) coordinates is that the ormer appears as triangle distribution in the bottom-let quarter o (, ) plane while the latter appears as triangle distribution in the top-right quarter o (, ) plane.. Obtaining Airy wave by Fourier transorm o cubic phase distribution In most experimental setups or creating optical Airy beams, a relective spatial light modulator (SLM) that provides phase modulation only is employed [-5, 7]. In this way, the cubic phase distribution is transerred onto the SLM, the SLM is illuminated with a plane wave, a lens is placed into the beam and an Airy beam is generated in the BFPL. This arrangement o optical elements, and in particular the lens whose ocal length is as long as 1 m, requires a certain length o the setup. In addition, it requires a special relective SLM that can modulate only the phase o an incident waveront. Instead o employing a physical lens, we propose loading a lens phase distribution directly onto the SLM together with the cubic phase distribution. Such an approach has been experimentally demonstrated in [17], but no quantitative analysis on the properties o the ormed Airy beams was presented. The total phase distribution loaded onto the SLM is given by: 1,. (7) xslm yslm b1 xslm yslm xslm yslm The waveront in the BFPL is obtained by orward propagation as:

3 i i i U ( x, y ) exp b1 xslm yslm exp xslm yslm 3 i exp x xslm y yslm dxslmdyslm i i 3 i exp exp 3 3 i x y b x y exp x x y y dx dy 3 i i x y exp x y Ai,, b 1 b1 b1 1 SLM SLM SLM SLM where we used Eq. (5). By comparing the argument o the Airy unction in Eq. (8) to that in Eq. (), we obtain the ollowing relation 1. b b1 (9) and an Airy beam is generated at the BFPL, where the waveront distribution is given by Eq. (8). The exponential actor in ront o the Airy unction distribution in Eq. (8) does not play any role when the intensity distribution is measured at the BFPL. However, this actor aects the Airy beam propagation properties. (8).3 Airy beam propagation and delection In general, an Airy beam propagation or a distance z rom plane 3 X x, y to plane, XY is calculated as i i U ( X, Y ) Ai bx, by exp x X y Y d x d y z z i 3 b b b z b b z exp z exp iz X exp iz Y 3 4k 4k Ai b bz 4k 3 bz, b Y. 4k With the additional exponential actor, present in ront o the Airy unction in Eq. (8), the waveront propagated rom the BFPL is calculated as (1)

4 i i i U ( X, Y ) exp x y Ai bx, by exp x X y Y d x d y z z i i c( z) z z z exp 1 X Y i c( z) c( z) Aib x, b y exp x X y Y d x c( z) z z Ai b X z 6 c( z) i c( z) i 3 b exp 1 X Y exp c( z) z z z b c( z) b c ( z) b c( z) b c ( z) exp ic( z) X exp ic( z) Y z 4k z 4k 3 3 cz ( ) b c ( z ) cz ( ) ( ), b c z b Y. 4k z 4k where we introduced The total delection is given by: d y (11) (1) c() z z zc() z z Rz ( ), 3 4 z k b1 where we substituted b rom Eq. (9). The total delection given by Eq. (13) exhibits zc( z) z / z dependency on z-distance, which is dierent rom delection or a conventional Airy beam given by Eq. (4) as z. The maximum o the intensity as a unction o z-distance is given by the constant actor in Eq. (11): c() z max I X, Y. z z The delection and the intensity as unctions o z-distance are veriied below by both simulation and experiment. (13) (14).4 Experimental setup The experiment setup is based on the OptiXplorer Educational Kit. It consists o the source o laser light with a wavelength o 65 nm and a transmissive SLM that employs twisted nematic liquid crystals (TN-LC) and has pixels with pixel pitch SLM = 3 m. The SLM is sandwiched between two rotatable linear polarisers: Polariser P1 SLM Polariser P (P1-SLM-P). Because the laser beam intensity distribution has Gaussian distribution orm with a non-uniorm ull width at hal maximum, the laser source was rotated such that the ormed Airy beam had symmetrical intensity distribution. The intensity distribution was imaged behind P1-SLM-P system on a screen made o semitransparent paper and captured by a 1-bit CCD camera. The screen and the camera were bound together and placed on an optical rail or an easy acquisition at dierent z- distances. The overall setting o the optical scheme is shown in Fig. 1.

5 Fig. 1 Optical setup or creating optical Airy beams employing a transmissive SLM..5 Experimental results Experimentally created optical Airy beams are shown in Fig.. The intensity distributions were acquired at z distances ranging rom z = at the BFPL to z = 3 cm, with an increment o cm. For alignment purposes, at each z distance, a calibration intensity distribution whereby only the lens phase distribution is transerred to the SLM was recorded. Each calibration intensity distribution exhibited just a ocused spot, which provided a reerence or alignment. At each z-distance, two images were recorded: with a low and high exposure set at the camera. Also, or all experimental images presented in this work, a background image, i.e. with laser light blocked o, was recorded, and was subtracted rom the measured intensities. The two intensity distributions recorded at low and high exposures were recombined into one high-dynamic range image [18]. Figure (a) shows the measured intensity proiles o the Airy beam at three selected z distances. The intensity has a maximum at the BFPL where the maximum value is set to 1 a.u. The related simulated intensity distributions are shown in Fig. (b). In the simulation o the wave propagation we used the angular spectrum method (ASM) [19-]. The intensity o the Airy beam decreases during beam propagation, as is evident rom both experimental and simulated results. Figures (c) and (d) exhibit two-dimensional intensity distributions in the ( X, z) -plane obtained as ollows: at each z-distance, the coordinate Y where the maximum o the intensity is observed is deined, and at this coordinate, the one-dimensional distribution o intensity along the X-axis is extracted. As can be seen rom Figs. (e) and (), the position o the maximum o the Airy beam intensity ollows a theoretically predicted dependency, as described by Eq. (13). Figure (e) indicates a perect agreement between experimental, simulated and theoretically predeined positions o the intensity maxima, ollowing a ballistic trajectory. As predicted in theory and described by Eq. (14), the intensity o the main maximum decreases as a unction o z-distance in both the simulated and experimentally acquired images, as evident rom the plots in Fig. (). However, the experimentally measured intensity decreases slightly aster than the intensity in the simulated images. This can be explained by a slight divergence o the beam, which is addressed later.

6 Fig.. Experimental results o the creation o optical Airy beams employing a transmissive SLM with a numerical lens. (a) Experimental and (b) simulated distributions o intensity at selected z distances, with = 65 nm, =.8 m and b 1 = 117 m -1. (c) and (d) two-dimensional (X, z) intensity distributions. (e) The delection o the maximum o the intensity as a unction o z-distance as obtained rom experimentally measured and simulated intensity distributions, and as predicted by theory (Eq. (13)). () The relative intensity o the maximum o the intensity as a unction o z- distance as obtained rom experimentally measured and simulated intensity distributions, and as predicted by theory (Eq. (14)).

7 3. Airy beams generated by direct transer o an Airy unction distribution on a SLM 3.1 Obtaining real-valued distributions on a SLM The Airy-unction is in act a real-valued unction with negative values. Having transparency with a negative transmission unction values would allow an Airy beam to be obtained immediately ater the transparency. The P1-SLM-P system can be set into a coniguration that allows negative values or the transmission unction, as illustrated later in Fig. 3. We explain this notion below by both simulation and experiment. In this section we consider the polarization properties o P1-SLM-P system is more detail, because these properties allows or negative values o the transmission unction. The Jones matrix o the twisted nematic SLM can be written as [1]: i ig h ij W e, h ij ig (15) where cos cos sin sin h sin cos cos sin g sin cos j sin sin, (16) and, (17) where bireringes is the only parameter which depends on the applied voltage, = and= (±9 ) []. The dependency o the bireringes on the greyscale values (... 55) is not linear, but or simulations here we assume a linear dependency grayscale, which well approximates the simulated 55 transmission curves o the SLM when comparing them to the measured transmission curves presented in []. From Eq. (15) it ollows that there is always amplitude and phase modulation o the light when it passes through the SLM. However, the angles o the polarizers P1 and P ( 1 and, respectively) can be set to achieve amplitude mostly or phase mostly modulations o the incoming light. The incident wave has the linear polarization deined by the angle o the irst polarizer: E cos1, sin1 1 1 (18) the second polarizer is described by the Jones matrix: cos cos sin P, cos sin sin and the waveront ater the second polarizer can be described as: 1 1 (19) E,, =P W P1. ()

8 The simulations are done as ollowing: at each pixel, its greyscale value is transormed into value. At each pixel, Eq. () is employed to simulate the waveront distribution ater the second polarizer. The transmission curve is simulated as T E / E E assuming the incident waveront 1 E1 1. Figure 3(a) shows the employed optical setup. The angle o the irst polariser P1 is set to achieve the maximum o the intensity, The intensity distributions are measured at a distance z = 1 cm behind the SLM. The experimentally acquired images are normalized by dividing with a background image, the image which is recoded when a uniorm image o greyscales = 55 is loaded onto the SLM. The two test images were studied. The irst test image is a cosine unction distribution cos x shown in cos x and has twice as many maxima as cos x. The Fig. 3(b). The intensity o this distribution is given by second test image is an Airy unction distribution shown in Fig. 3(c). Figure 3(d) exhibits the polarisers setting when amplitude mostly modulation is achieved: 1 33 and 15, and the related simulated transmission curve. In this setting, the two test images result in an intensity distribution that closely matches the original distributions. The test cosine distribution exhibits values ranging rom -1 to +1 shown in Fig. 3(b), but when it is transerred onto the SLM (i.e. the SLM alone, without 1 1 cos x taking the eect o the polarisers into account) it has only positive values, best described as. In both simulation and experiment, the intensity distribution ater the second polarizer is best described as 1 1 cos x, not cos x, see Fig. 3(e) (). The test Airy unction image results in an intensity distribution that closely matches the original distribution, see Fig. 3(g). Next, it is possible to adjust the angles o the polarisers P1 and P in such a way that the total transmission unction o the P1-SLM-P system will have negative values and the values o the cosine unction, or example, will range rom negative to positive values. Such a setting o the P1-SLM-P system and the related transmission curve are illustrated in Fig. 3(h). Should there be no negative values in the transmission unction, the measured intensity o the cosine pattern would always repeat the distribution best described as 1 1 cos x. However, in this setting o the polarizers, the measured intensity o the cosine unction resembles cos x, as shown in Fig. 3(i) (j). In a urther experimental study, we use the polarisers settings such as to allow or a transmission unction with negative values, i.e. the setting shown in Fig. 3(h). In this way, we obtain an Airy beam directly behind the P1-SLM-P system, as illustrated in Fig. 3(k), and we study the propagation properties o the ormed Airy beam.

9 Fig. 3. An illustration to the principle o generating Airy beams by direct transer o a real-valued Airy unction onto the SLM. (a) Scheme o the optical setup. (b) A test cosine unction distribution loaded onto the SLM. (c) An Airy unction distribution loaded onto the SLM. (d) (g) Results when the polarisers are set or an amplitude mostly modulation o light: 1 33and 15. (d) The simulated transmission response curve o the P1-SLM-P system. (e) Simulated and () measured intensity distribution when the test cosine distribution is loaded onto the SLM. (g) Measured intensity distribution when the Airy unction is loaded onto the SLM. (h) (k) Results when the polarisers angles are: 1 33 and 76. (h) The related transmission response curve o the P1-SLM-P system. (i) Simulated and (j) measured intensity distribution when the test cosine distribution loaded onto the SLM. (k) Measured intensity distribution when the Airy unction is loaded onto the SLM. All the experimental intensity distributions are measured at the distance z = 1 cm behind the SLM. At such a short distance rom the SLM, the intensity distributions exhibit higher-order images overlapping with the zero-order images. 3. Simulating a real-valued Airy unction distribution or the SLM With the notion that an Airy unction is obtained by the Fourier transorm o the cubic phase distribution, we deine the cubic phase distribution in reciprocal space (, ) as i exp b, 3 (1)

10 where and are digitised as ollows: ii N / and jj N /, b is a parameter whose units are in metres, N is the number o pixels, Δ is the pixel size in reciprocal space, ii and jj are the pixel numbers. The Airy unction is obtained according to the transormation (similar to Eq. (6)): i xslm y SLM exp b exp i( xslm yslm ) dd Ai,, 3 b b b () where xslm, y SLM are real space coordinates in the SLM plane. From the SLM plane SLM, SLM, x y, the waveront is propagated to some plane xy. Comparing the argument o Airy unction in Eq. () to that in b 1/ b Eq. (), we obtain, and the total delection: z r ( z). (3) max 3 b 4k As ollows rom Eq. (1), which described propagation o Airy beam, the maximum o the intensity remains constant. A digital Fourier transorm o a cubic phase unction, as expressed by Eq. (), leads to the ollowing relation: i i / N, where Δ SLM is the pixel size in the real space, or example, the pixel size o the SLM SLM onto which the real part o the Airy unction will be transerred. From this relation the pixel size in reciprocal space is given by: 1 N The steps or simulating the real-part Airy unction distribution or the SLM are as ollows: (1) A complex-valued unction described by the distribution given by Eq. (1) is simulated, the pixel size is given by Eq. (4). () The Fourier transorm o (1) is calculated, which provides the two-dimensional Airy unction distribution. (3) The real part o () provides the distribution that is transerred onto the SLM. 3.3 Experimental results Experimentally measured intensities o optical Airy beams generated by direct transer o the Airy unction onto a transmissive SLM are shown in Fig. 4(a). The intensity distributions were acquired at z distances ranging rom z = 11 cm, which is the shortest distance one could place a screen ater the P1-SLM-P system, to z = 11 cm, with an increment o 5 cm. At each z distance, also a control ocused spot image was acquired or alignment as described above. Figures 4(a) and (b) show intensity proiles at three selected z distances. Figure 4(b) shows the related simulated intensity distributions. For the simulating the wave propagation we used the angular spectrum method (ASM) [19-]. In this experiment, we were able to study the beam propagation over 1 m, whereas in the previous experiment the propagation distance was 3 cm limited by the length o the setup and by the diraction properties o the created Airy beams. Figures 4(c) and (d) depict two-dimensional (x, z) intensity distributions obtained as ollows: at each z-distance, a one-dimensional distribution o intensity along the x-axis at the y coordinate where the maximum o the intensity is extracted. As can be seen rom Figs. 4(c) and (d), the position o the maximal intensity ollows a z -dependent trajectory as described by Eq. (3). The slight disagreement between theory and experiment is explained by the act that the beam used in the experiment was not ideally parallel, but slightly divergent. This slight divergence became notable only at a larger distance o propagation. A better agreement between the theory and experiment is achieved when in the simulation, an incident wave onto the SLM with an additional phase o i exp div is used which corresponds to a spherical wave originating at x SLM. y SLM SLM div (4) (5) z. The best match was achieved at div = m. With this extra actor, correcting or divergence o the incident wave, the position o the intensity maxima as a unction o z is shown as the magenta curve in Fig. 4(e).

11 The value o the intensity maximum as a unction o z-distance does not noticeably decrease, see Fig. 4(a) and (). It rather remains almost constant over a distance o 1 m. This represents a signiicant dierence between directly and conventionally generated Airy beams. In the case o a conventionally generated Airy beam, its intensity decreases noticeably as a unction o z-distance: see Fig. (). The measured intensity is also slightly disturbed by the superimposed signal rom the irst diraction order, as evident rom the intensity distributions shown in Fig. 4(a) at z = 6 cm. Fig. 4. Experimental results or an Airy beam generated by direct transer o the Airy unction onto a transmissive SLM. (a) Experimental and (b) related simulated distributions o intensity at selected z distances obtained at = 65 nm and b = 1 m. (c) and (d) two-dimensional (x, z) intensity distributions obtained by extracting at each z-distance, one-dimensional distribution o

12 intensity along the x-axis at the y coordinate where maximal intensity is observed. (e) Delection o the main maximum o the intensity as a unction o z-distance obtained rom experimental and simulated intensity distributions, and as theoretically predicted by Eq. (3). The magenta curve indicates the delection o the main maximum o the intensity obtained rom simulations when the SLM is illuminated with a slightly divergent waveront. () Intensity o the main maximum o the intensity as a unction o z-distance obtained rom experimental and simulated intensity distributions, and as predicted by theory. 4. Conclusions We demonstrated two methods or creating Airy beams by employing a transmissive SLM. Both setups do not employ a physical lens and thus allow or a very compact design. In the irst method, we load the phase distribution o the lens onto the SLM together with the cubic phase distribution instead o using a physical lens. We derived the ormula or delection as a unction o the distance, which is dierent to the well-known quadratic dependency in the case o conventional Airy beams. The ormula is validated by simulated and experimental results. In the second method, we employed the properties o the polariser-slm-polariser system, which can deliver a transmission unction with negative values; thus the two-dimensional Airy unction distribution can directly be loaded onto the SLM. An Airy beam is thereore created directly ater the SLM system, giving the minimal possible length o the optical setup. The intensity o the Airy beam, generated by the second method, stays constant over an unusually large distance, we have measured it over a distance o 1 m. The second method oers the most compact setup or generating Airy beams. Furthermore, it has the potential or other waveront modulations where the transmission unction requires negative values. Reerences [1] M. V. Berry, and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, (1979). [] G. A. Siviloglou et al., "Observation o accelerating Airy beams," Phys. Rev. Lett. 99, 1391 (7). [3] G. A. Siviloglou, and D. N. Christodoulides, "Accelerating inite energy Airy beams," Opt. Lett. 3, (7). [4] J. Broky et al., "Sel-healing properties o optical Airy beams," Opt. Express 16, (8). [5] F. Zhuang et al., "Quantitative study on propagation and healing o Airy beams under experimental conditions," Opt. Lett. 4, (15). [6] J. Rogel-Salazar, H. A. Jimenez-Romero, and S. Chavez-Cerda, "Full characterization o Airy beams under physical principles," Phys. Rev. A 89, 387 (14). [7] G. A. Siviloglou et al., "Ballistic dynamics o Airy beams," Opt. Lett. 33, 7 9 (8). [8] J. E. Morris et al., "Realization o curved Bessel beams: propagation around obstructions," J. Opt. 1, 14 (1). [9] Y. Hu et al., "Optimal control o the ballistic motion o Airy beams," Opt. Lett. 35, 6 6 (1). [1] J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photon., (8). [11] D. N. Christodoulides, "Optical trapping riding along an Airy beam," Nature Photon., (8). [1] R. Cao et al., "Microabricated continuous cubic phase plate induced Airy beams or optical manipulation with high power eiciency," Appl. Phys. Lett. 99, 6116 (11). [13] P. Zhang et al., "Trapping and guiding microparticles with morphing autoocusing Airy beams," Opt. Lett. 36, (11). [14] Z. Zheng et al., "Optical trapping with ocused Airy beams," Appl. Optics 5, (11). [15] S. Jia, J. C. Vaughan, and X. W. Zhuang, "Isotropic three-dimensional super-resolution imaging with a selbending point spread unction," Nature Photon. 8, 3 36 (14). [16] N. Voloch-Bloch et al., "Generation o electron Airy beams," Nature 494, (13). [17] J. A. Davis et al., "Generation o accelerating Airy and accelerating parabolic beams using phase-only patterns," Appl. Optics 48, (9). [18] P. E. Debevec, and J. Malik, "Recovering high dynamic range radiance maps rom photographs," SIGGRAPH 13 (1997). [19] J. W. Goodman, Introduction to Fourier optics (Roberts & Company Publishers, 4). [] T. Latychevskaia, and H.-W. Fink, "Practical algorithms or simulation and reconstruction o digital in-line holograms," Appl. Optics 54, (15). [1] C. Soutar, and K. H. Lu, "Determination o the physical properties o an arbitrary twisted-nematic liquidcrystal cell," Opt. Eng. 33, (1994). [] Holoeye, "OptiXplorer Manual.8," (7).