Application of orbital angular momentum to simultaneous determination of tilt and lateral displacement of a misaligned laser beam

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1 Lin et al. Vol. 7, No. 10/October 010/J. Opt. Soc. Am. A 337 Application of orbital angular momentum to simultaneous determination of tilt and lateral displacement of a misaligned laser beam J. Lin, 1, X.-C. Yuan,, * Mingzhou Chen, 3 and J. C. Dainty 3 1 School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore , Singapore Institute of Modern Optics, Key Laboratory of Optoelectronic Information Science and Technology, Ministry of Education of China, Nankai University, Tianjin , China 3 Applied Optics Group, School of Physics, National University of Ireland, Galway, University Road, Galway, Ireland Current address: School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 0138, USA *Corresponding author: xcyuan@nankai.edu.cn Received July 15, 010; revised September, 010; accepted September 3, 010; posted September 3, 010 (Doc. ID 13170); published September 9, 010 We present an analysis of the combined effects of tilt and lateral displacement on the orbital angular momentum spectrum of a laser beam. Our theory explains the symmetries and properties of the spectrum under the influence of misalignments. We apply the theory to establish a reliable and efficient method for determining and subsequently eliminating tilt and lateral displacement. An improved technique for obtaining the orbital angular momentum spectrum employing Laguerre Gaussian modes is proposed. Finally, a numerical experiment is carried out to verify the method. 010 Optical Society of America OCIS codes: , INTRODUCTION In a seminal paper, Allen et al. [1] revealed that photons can possess orbital angular momentum (OAM) in addition to spin angular momentum (SAM). Unlike SAM, which arises from polarization states [], OAM was first defined for photons in optical vortices [1], which are closely related to Laguerre Gaussian (LG) modes found in the solution of the paraxial wave equation in cylindrical coordinates [3]. The phase of an optical vortex varies in a screw-like manner along the beam s axis of propagation. This property enables the beam to interact with particles through a transfer of angular momentum []. Recently, applications of OAM have grown in importance and have emerged in fields as diverse as biophysics [5], microfluidics [6], and astronomy [7 9]. The greatest excitement, however, is being generated in quantum information processing. As the OAM state of an individual photon is characterized by its topological charge n, which in principle can take any integer number, it offers a multipledimensional orthogonal quantum state space [10,11]. This potentially gives rise to high-density information storage [1]. Although experimental measurement of a beam s OAM spectrum the weight of each OAM state of a beam is not difficult, it is frequently corrupted by misalignment [13]. Misalignment is equivalent to taking a reference axis other than the propagation direction of the beam. Owing to the extrinsic nature of OAM, the OAM spectrum is sensitive to even a small amount of misalignment [1]. In practice, misalignment is corrected by adjusting the measurement device until the signal of the target OAM state reaches the maximum in the presence of a single OAM state. Usually, a measurement setup has several degrees of freedom, which makes such a trial-and-error method cumbersome and often leads to non-ideal correction results [15]. Here, we carry out an analysis of the effects of misalignments on the OAM spectrum. Based on the analysis, an efficient and reliable method of misalignment correction is proposed. The method uses information inherent in the OAM weights to determine the misalignment parameters of the beam. The obtained parameters can in turn be used to accurately adjust the measurement apparatus or alternatively be used to filter out the artifacts caused by the misalignment in the post-processing. We also point out limitations of existing methods for measuring OAM states and propose an improved method, which makes use of LG modes. Finally, an experiment is designed to verify our method.. ANALYTICAL EXPRESSION FOR OAM WEIGHTS For the following calculations in this paper, TEM 00, the fundamental transverse mode, is used. Higher-order transverse modes would give similar results. In general, a laser beam is subject to two types of misalignment with respect to a reference axis: lateral displacement and tilt [1]. Figure 1 schematizes the orientation of the misaligned beam in a reference coordinate system. Under the condition of a small tilting angle, the complex amplitude of the TEM 00 laser beam can be expressed in polar coordinates as /10/ /$ Optical Society of America

2 338 J. Opt. Soc. Am. A/ Vol. 7, No. 10/ October 010 Lin et al. n0), the integrals of Eq. () lead to C n = n u n! exp 0 0 u n. Fig. 1. Schematic illustration of a misaligned Gaussian beam with lateral displacement and tilt. The dashed line depicts the propagation direction of the beam. z-axis is taken as the reference axis. Er,;r 0, 0,, = 1 r + r 0 exp exp r 0r cos 0 expikr sin cos, where r 0 and 0 describe the beam s lateral displacement; and describe its tilt. k=/ is the wave number with being the wavelength, and is the waist of the laser beam. / is a normalization factor. The scalar wavefield can be represented by the superposition of spiral harmonics: Er,= a n rexpin, where a n r= 1 0 Er,exp ind. A general expression of the weight of the OAM state n is then given by C n =A 0 0 a n r rdr, with a normalization constant A 0 [16]. If the reference axis is chosen to be the z-axis for Er, defined in Eq. (1), the weight becomes C n r 0, 0,, =0 0 0 Er,;r 0, 0,,exp indrdr 0 Er,;r 0, 0,, rddr An analytical expression can be found for Eq. (): u 0 + v 0 C n u 0,v 0, = exp u 0 + v 0 +u 0 v 0 cos u 0 + v 0 ±u 0 v 0 sin. 1 n I n u 0 + v 0 +u 0 v 0 cos, 3 where u 0 =r 0 / and v 0 =k sin. The angles 0 and always appear as a difference and can thus be replaced with = 0. The sign in front of the term u 0 v 0 sin is positive if n0, and negative if n0. Here, I n x is the nth-order modified Bessel function of the first kind. For the special case where the denominator of the second term is zero (u 0 =v 0, =3/, n0, or if u 0 =v 0, =/, This occurs when the lateral displacement OP (see Fig. 1) is perpendicular to the direction of tilt and they are of the same normalized magnitude. Figure shows how the weight of the OAM state +1 varies with the three misalignment parameters u 0, v 0, and. As the misalignment increases, the weight allocated to the state increases from zero to a maximum value, and then declines. The global maximum of is reached when u 0 =v 0 =1.1 and =3/ as shown in Fig. (e). Figure 3 illustrates the OAM spectra distorted by misalignments as predicted by Eq. (3). It is interesting to note that Eq. (3) remains invariant if the variables u 0 and v 0 are interchanged, which explains the symmetry observed in Fig.. Equation (3) also explains the similar broadening effect on the OAM spectra shown in Figs. 3(a) and 3(b) when only one type of misalignment is present in the optical system. In this case, Eq. (3) reduces to C n = exp I n, where represents either u 0 or v 0 for the lateraldisplacement- or tilt-only case accordingly. In the lateraldisplacement-only case, i.e., =u 0, Eq. (5) matches Eq. (A6) in a previous publication [17]. Equation (3) also indicates that the misalignment parameters can be determined from information inherent in the OAM spectrum. We first note that the expression in Eq. (3) differs only by a sign change in the denominator for n= 1 and n=+1. Thus defining a relative signal =C 1 /C +1 will significantly reduce the complexity of the expression: u 0,v 0, = C 1 u 0 + v 0 +u 0 v 0 sin = C +1 u 0 + v 0 u 0 v 0 sin. 6 Equation (6) can be recast in Cartesian coordinates. We define lateral displacement as x 0 =r 0 cos 0 and y 0 =r 0 sin 0 so that Eq. (6) becomes k w 1 x 0 + y 0 sin k sin y 0 cos x 0 sin =0. 7 In order to determine the values of misalignment parameters x 0, y 0,, and from the relative weight, additional lateral displacement in the x-axis is introduced. Hence, we have k w 1 x y 0 sin k sin y 0 cos x 0 + sin =0. 8 The equation can be made linear by the following substitutions: z 1 =x 0 +y 0 +k sin /, z =k sin y 0 cos x 0 sin, and z 3 =k sin sin. As a result, Eq. (8) becomes

3 Lin et al. Vol. 7, No. 10/October 010/J. Opt. Soc. Am. A 339 Fig.. (Color online) Weight C +1 of the OAM state +1 as a function of normalized lateral displacement and tilt (u 0 =r 0 / and v 0 =k sin ) evolves with the relative azimuthal angle. Fig. 3. (Color online) OAM spectra subject to various misalignments. (a) Tilt only (u 0 =0, v 0 =1, and = 0.5); (b) lateral-displacement only (u 0 =1.5, v 0 =0, and = 0.9); (c) combination of tilt and lateral displacement (u 0 =1.5, v 0 =1, and = 0.9).

4 30 J. Opt. Soc. Am. A/ Vol. 7, No. 10/ October 010 Lin et al. 1 x z z 1+z =0. 9 Now a linear system of four equations can be constructed by varying : x z = 1 1 3, 3 z 1 z 10 where i=0,1,,3 is the measured set of signals corresponding to the set of lateral displacements i=0,1,,3. The first extra lateral displacement is set to zero, i.e., 0 =0. The lateral displacement x 0 in the x-axis is given by the solution to Eq. (10). The other three misalignment parameters can also be found by having another set of equations: y 0 + k sin / = z 1 x 0, k sin y 0 cos = x 0 z 3 + z, k sin sin = z To avoid the multiple solutions for y 0 in Eq. (11), additional displacement can be introduced in the y-axis to determine y 0 with a process analogous to the one carried out above. We also note that since Eq. (3) contains information of how the misalignment affects the OAM spectrum, a program can be written to filter out the components of the OAM due to misalignment, thus offering a way of retrieving the uncorrupted signal without readjusting the experimental setup. 3. IMPROVED EXPERIMENTAL SETUP FOR MEASURING OAM WEIGHTS OAM weights are not only needed for the misalignment calculations but also crucial for determining the OAM spectrum. These weights can be obtained in the process of measuring the topological charge n of a vortex beam [18 0]. In a widely used method, a conjugate spiral phase is used to convert the vortex beam into a plane wave/gaussian beam, of which the far-field distribution is a bright dot in the center. This can easily be distinguished from the doughnut shape of the far-field distribution of a vortex beam. However, this approach might not work in certain cases, for instance, a vortex beam with the complex amplitude profile J 0 k r rexpin, where J 0 x is the zeroth-order Bessel function of the first kind. The farfield distribution of the beam after passing through a conjugate phase mask exp in is given by the Hankel transform 0 J 0 k r rj 0 rrdr= D k r /k r, with D x being the Dirac delta function. This represents a ring in the focal plane. As a result, the matching phase mask of the form exp in cannot be used to convert such a beam into a bright dot in the center of the far-field. Furthermore, if the vortex beam is misaligned, components such as those with the amplitude profile J 0 k r rexpin cannot be ruled out generally. To avoid this, we suggest that one adopts a -f configuration as shown in Fig.. Consisting of two Fourier transforms, the setup is essentially an optical correlator that utilizes a matched filter in the Fourier plane to measure the cross-correlation of the incoming beam and the target mode [1]. The complex amplitude distribution in the output plane can be described by vx,y = g,h + x, + ydd, 1 where g and h denote the incoming beam and the target mode, respectively. The transmittance function of the matched filter placed in the Fourier plane is given by the Fourier transform of h x, y, where denotes the complex conjugate. If the incoming beam is identical to the target mode, the intensity at the origin of the output plane is unity under normalization. However, there exists no matched filter that can directly provide the weight of a specific OAM state in the output plane. One way to solve this problem is to decompose the incoming field into twodimensional orthogonal basis, for example, LG modes LG p,n : E LG p,n r, = p! 1 n + p! r nl p n r exp r expin, 13 where L n p is the associated Laguerre polynomials. The weight of each mode can then be measured by a corresponding matched filter that is the conjugate of the target LG mode []. The weight of an OAM state C n is then given by the sum of all the modes with the same azimuthal index n C n = p=0 C LG p,n. However, we shall show in the following that for the relative weights, only two LG modes are needed. First, the weight of a particular LG mode in the misaligned beam is calculated as Fig.. (Color online) Experiment schematic to measure the weight of OAM states. The point detectors are used to measure the strength of the cross-correlation intensity at various locations, which is equivalent to lateral shift of the input.

5 Lin et al. Vol. 7, No. 10/October 010/J. Opt. Soc. Am. A 31 C LG p,n =0 0 1 = p!n + p! 1 Er,;u 0,v 0,E LG p,n rddr 6p+3n u 0 + v 0 exp u 0 + v 0 +u 0 v 0 cos p+n, 1 u 0 + v 0 ±u 0 v 0 sin n where the sign in front of the term u 0 v 0 sin is positive if n0, and negative otherwise. Now, we can calculate the relative weights of the two LG modes with the same index p, yielding LG = LG C p, 1 u LG = 0 + v 0 +u 0 v 0 sin u 0 + v 0 u 0 v 0 sin, C p,+1 p = 0,1,,..., 15 which is identical to Eq. (6). Therefore, only two LG modes are needed for calculating the relative weights, which means that we can replace the ratio between the weights of two OAM states ( 1 and +1) used in the misalignment measurement with the ratio between the weights of two LG modes LG 0, 1 and LG 0,+1 to simplify the measurement greatly. At first glance, introducing additional lateral displacements in the x- and y-axes to the misaligned beam seems difficult to implement, because one needs to displace the incident beam in lateral directions without introducing any additional tilt. Nevertheless, the necessary extra lateral displacement can be readily realized by moving the point detector with the same amount of lateral displacement in the opposite direction [3] or by simply putting an extra photodetector there. This can be explained by the straightforward manipulation of the cross-correlation defined in Eq. (1) with a laterally shifted input: g x 0, y 0 h + x, + ydd = vx + x 0,y + y VERIFICATION OF THE ANALYTICAL METHOD A numerical experiment has been carried out to test the accuracy of our analytical method for misalignment determination. The misalignment parameters used in the simulation are x 0 =0.3 mm, y 0 = 0.1 mm, =.5 10 rad, and =1. rad. The wavelength of the light is 633 nm, the waist of the Gaussian beam is 0.1 mm, and the focal length of the two lenses is 500 mm. The OAM weights are measured using the method detailed in the previous section. Figure 5 shows the amplitude and phase distribution of the input beam, which is focused by the first lens onto the matched filter. To measure both modes, LG 0, 1 and LG 0,+1, the filter has to be adjusted accordingly. A second lens performs another Fourier transform and produces the optical field in the output plane. Seven point detectors are placed in the output plane, three of which are in the x-axis and three in the y-axis, and another one is placed at the origin of the output plane. The number of the detectors in each axis ensures that the two sets of Eqs. (10) and (11) will have unique solutions. The detectors give the following relative OAM weights: x =0.1910,0.3733, 0.97,0.636 and y =0.1910,0.306,0.691, After substituting the measurement data into Eqs. (10) and (11), and solving them, we have been able to retrieve the misalignment parameters of the input Gaussian beam with negligible numerical errors. Besides the relatively large misalignment parameters chosen here, we have also examined the cases in which the misalignment is minuscule (10 orders of magnitude below the present misalignment values), and the method remains accurate with negligible errors mostly attributable to numerical implementation. Certainly the actual sensitivity would eventually be limited by other factors such as the performance and the aperture size of detectors. It may also be noted that the proposed method would not be applicable in the absence of tilt, i.e., =0, since the ratio remains unity for any lateral displacement as indicated in Eq. (15); and, consequently, Eq. (10) becomes underdetermined. However, the method still works for the case when the lateral displacement is zero since nonzero virtual displacements are included in the measurement. Fortunately, there are plenty of methods available to measure the simple case of lateral misalignment once one obtains the relative weight as unity [,5]. 5. CONCLUSION In summary, we have formulated and successfully tested an analytical method to determine a laser beam s lateral and tilt misalignments by using the beam s OAM. Our method performs simultaneous extraction of all misalignment information from the OAM spectrum. Hence, we are able to analyze the combined effect of both lateral displacement and tilt. Our analysis also sheds what we believe to be new light on how the OAM spectrum is reshaped by the two types of misalignments. A notable observation is that even though lateral displacement and tilt appear as different misalignments, they both affect the OAM spectrum in the same way due to the symmetry of Eq. (3). This leads to the symmetry in the OAM spectrum, which until now has eluded clear theoretical explanation. Since OAM measurements lie at the core of many experiments involving vortex beams, this study will help the experimenter to better understand and remove the artifacts caused by system misalignment. With the help of the efficient and reliable method we have provided, these two types of misalignment in optical systems can be corrected simultaneously. This represents a conceptual advance over earlier methods, which have often treated the contributions from the two misalignments separately. Furthermore, because our analysis readily reveals the effects of misalignment artifacts, they can be filtered out without the need for adjusting the experimental setup. The method could also be applied to compensate errors in tip-tilt systems that are widely used in adaptive optics.

6 3 J. Opt. Soc. Am. A/ Vol. 7, No. 10/ October 010 Lin et al. Fig. 5. (Color online) Numerical simulation of the proposed experiment. (a) and (b) are the amplitude and phase distribution in the reference plane, respectively (the origin of the coordinates is denoted with + ); (c) and (d) are the two-dimensional amplitude and phase function of the matched filter designed to measure the weight of LG 0, 1 mode, respectively. The other filter for the measurement of LG 0,+1 has exactly the same amplitude function but a conjugate phase function. (e) and (f) are the intensity distributions of the output planes corresponding to the different matched filters for the measurement of LG 0, 1 and LG 0,+1 modes, respectively (the locations of the detectors are denoted with + and, while the cross also indicates the origin of the coordinates in the output plane). ACKNOWLEDGMENTS J. Lin acknowledges the fellowship support from the Singapore Millennium Foundation and the Agency for Science, Technology and Research (A*STAR), Singapore. X.-C. Yuan acknowledges the support from the National Natural Science Foundation of China (NSFC) under Grant No and the Ministry of Science and Technology of China under Grant no. 009DFA5300 for China-Singapore collaborations. This work was partially supported by the Science Foundation Ireland under Grant No. 07/IN.1/I906. The authors would like to thank Dr. Alexander Goncharov for helpful discussions and Dr. Baixi Su-Alexander, who has shared with us his enthusiasm for this project and shown wonderful insight. REFERENCES 1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Orbital angular-momentum of light and the transformation of Laguerre Gaussian laser modes, Phys. Rev. A 5, (199).. R. A. Beth, Mechanical detection and measurement of the angular momentum, Phys. Rev. 50, (1936). 3. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Taylor & Francis, 003).. H. He, M. E. J. Friese, N. R. Heckenberg, and H.

7 Lin et al. Vol. 7, No. 10/October 010/J. Opt. Soc. Am. A 33 Rubinsztein-Dunlop, Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity, Phys. Rev. Lett. 75, (1995). 5. D. G. Grier, A revolution in optical manipulation, Nature, (003). 6. K. Ladavac and D. G. Grier, Assembly of 3-dimensional structures using programmable holographic optical tweezers, Opt. Express 1, (00). 7. G. Foo, D. M. David, and G. A. Swartzlander, Optical vortex coronagraph, Opt. Lett. 30, (005). 8. B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, Utilization of photon orbital angular momentum in the low-frequency radio domain, Phys. Rev. Lett. 99, (007). 9. G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, Optical vortices with starlight, Astron. Astrophys. 88, (008). 10. D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 000). 11. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Entanglement of the orbital angular momentum states of photons, Nature 1, (001). 1. G. Molina-Terriza, J. P. Torres, and L. Torner, Twisted photons, Nat. Phys. 3, (007). 13. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas ko, S. M. Barnett, and S. Franke-Arnold, Free-space information transfer using light beams carrying orbital angular momentum, Opt. Express 1, (00). 1. M. V. Vasnetsov, V. A. Pas ko, and M. S. Soskin, Analysis of orbital angular momentum of a misaligned optical beam, New J. Phys. 7, 6 (005). 15. Y. Liu, C. Gao, X. Qi, and H. Weber, Orbital angular momentum (OAM) spectrum correction in free space optical communication, Opt. Express 16, (008). 16. G. Molina-Terriza, J. P. Torres, and L. Torner, Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum, Phys. Rev. Lett. 88, (001). 17. S. M. Barnett and R. Zambrini, Resolution in rotation measurements, J. Mod. Opt. 53, (006). 18. S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, How to observe high-dimensional two-photon entanglement with only two detectors, Phys. Rev. Lett. 9, (00). 19. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, Observation of the orbital angular momentum spectrum of a light beam, Opt. Lett. 8, (003). 0. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Measuring the orbital angular momentum of a single photon, Phys. Rev. Lett. 88, (00). 1. C. Scott, Introduction to Optics and Optical Imaging (IEEE, 1998).. V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, Light field decomposition in angular harmonics by means of diffractive optics, J. Mod. Opt. 5, (1998). 3. J. W. Goodman, Introduction to Fourier Optics (McGraw- Hill, 1996).. G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Method to measure off-axis displacements based on the analysis of the intensity distribution of a vortex beam, Phys. Rev. A 79, (009). 5. S. Cui and Y. C. Soh, Improved measurement accuracy of the quadrant detector through improvement of linearity index, Appl. Phys. Lett. 96, (010).