Correcting vortex splitting in higher order vortex beams Richard Neo,1 Shiaw Juen Tan,2,3 Xavier Zambrana-Puyalto,2,3 Sergio Leon-Saval,1 Joss Bland-Hawthorn,1 and Gabriel Molina-Terriza2,3,∗ 1 Institute
of Photonics and Optical Science (IPOS), School of Physics, University of Sydney, NSW 2006, Australia 2 ARC Centre of Excellence for Engineered Quantum Systems (EQuS), Macquarie University, NSW 2109, Australia 3 Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia ∗ [email protected]
Abstract: We demonstrate a general method for the first order compensation of singularity splitting in a vortex beam at a single plane. By superimposing multiple forked holograms on the SLM used to generate the vortex beam, we are able to compensate vortex splitting and generate beams with desired phase singularities of order = 0, 1, 2, and 3 in one plane. We then extend this method by application of a radial phase, in order to simultaneously compensate the observed vortex splitting at two planes (near and far field) for an = 2 beam. © 2014 Optical Society of America OCIS codes: (080.4865) Optical vortices; (090.1000) Aberration compensation; (070.6120) Spatial light modulators; (230.6120) Spatial light modulators.
References and links 1. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). 2. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003). 3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). 4. G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94(4), 040501 (2005). 5. M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597(2), 1266 (2003). 6. C. Barbieri, D. Dravins, T. Occhipinti, F. Tamburini, G. Naletto, V. Da Deppo, S. Fornasier, M. D’Onofrio, R. A. E. Fosbury, R. Nilsson, and H. Uthas, “Astronomical applications of quantum optics for extremely large telescopes,” J. Mod. Opt. 54(2–3), 191–197 (2007). 7. F. Tamburini, B. Thid´e, G. Molina-Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7(3), 195–197 (2011). 8. N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492, 883–922 (2008). 9. N. M. Elias, “Photon orbital angular momentum and torque metrics for single telescopes and interferometers,” Astron. Astrophys. 541, 101 (2012). 10. G. Foo, M. P. David, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005). 11. X. Zambrana-Puyalto, X. Vidal, and G. Molina-Terriza, “Excitation of single multipolar modes with engineered cylindrically symmetric fields,” Opt. Express 20(22), 24536–24544 (2012). 12. I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159(1), 99–117 (1999). 13. F. Ricci, W. L¨offler, and M. P. van Exter, “Instability of higher-order optical vortices analyzed with a multipinhole interferometer,” Opt. Express 20(20), 22961–22975 (2012). 14. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9920
15. M. R. Dennis, “Rows of optical vortices from elliptically perturbing a high-order beam,” Opt. Lett. 31(9), 1325– 1327 (2006). 16. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. 20(8), 1635–1643 (2003). 17. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241(4), 237–247 (2004). 18. A. Wada, T. Ohtani, Y. Miyamoto, and M. Takeda, “Propagation analysis of the Laguerre-Gaussian beam with astigmatism,” J. Opt. Soc. Am. 22(12), 2746–2755 (2005). 19. I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103(5), 422–428 (1993). 20. C. H. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express 14(15), 6604–6612 (2006). 21. J. Carpenter, B. Thomsen, and T. Wilkinson, “Mode division multiplexing of modes with the same azimuthal index,” IEEE Photon. Technol. Lett. 24(21), 1969–1972 (2012). 22. 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(1), 013601 (2002). 23. R. Bowman, V. D’Ambrosio, E. Rubino, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, “Optimisation of a low cost SLM for diffraction efficiency and ghost order diffraction,” Eur. Phys. J. Spec. Top. 199(1), 149–158 (2011). 24. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). 25. A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of optical vortices,” Opt. Express 19(7), 6182–6190 (2011). 26. A. Kumar, P. Vaity, J. Bhatt, and R. P. Singh, “Stability of higher order optical vortices produced by spatial light modulators,” J. Mod. Opt. 60, 1696–1700 (2013).
1.
Introduction
The control of the angular momentum of a paraxial beam using its spatial properties has stimulated substantial research with applications in many areas of physics for example: spatial mode multiplexing for telecommunications [1], optical spanners [2], quantum optics [3, 4] and astronomy [5–9]. For many of these applications (e.g. vortex coronography [10]) it is required to control the phase singularities, also called optical vortices, of the paraxial field. The control of phase singularities in a paraxial beam is motivated because in this regime, the phase singularities of the beam are directly associated with the density of orbital angular momentum (OAM) that the beam carries. Outside the paraxial approximation, the split of the total angular momentum into orbital and spin components becomes problematic. Nevertheless, in some applications the control of the phase singularities in the paraxial regime determines the local structure of a focused beam. This is of paramount importance to control the interaction of light and matter at the nanoscale [11]. Paraxial beams with a well defined value of OAM, , have a cylindrically symmetric intensity profile. They present an optical vortex of charge at the centre, i.e. the phase of the beam twists around the centre of the beam by 2π radians. Thus, the intensity at the position of the singularity is zero. Unfortunately, implementing beams with higher order singularities is challenging. It has been observed that, due to any uncontrolled breaking of the symmetry, higher order phase singularities split into sets of lower order singularities [12, 13]. Typically, the splitting results in || singularities of order ±1 depending on the sign of . The splitting of the central null of a vortex beam suggests coherent interference occurring between multiple co-propagating modes, either due to noise or imperfections of the set-up. Clearly, the resulting field will be a superposition of different OAM modes [14–18]. In this article we show a very simple and effective technique in order to compensate for such effects when producing angular momentum states. In order to compensate for the interference, we generate multiple collinear beams each with some orbital angular momentum. Proper control of the weighting and relative phases using a phase-only spatial light modulator allows us to generate optical beams with stable high order phase singularities. This has previously been demonstrated using non-collinear beams for the case = 2 [19]. The method of producing sev#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9921
eral interfering copropagating modes is also used in the generation of “interference vortices” in particle trapping [20]. By tuning both the relative amplitude and phase of these beams, we can combine them such that they destructively interfere at a certain plane with the highest order OAM impurities in our original beam, thus increasing the purity of the generation of the desired OAM state. Other more complex ways of correcting this effect rely on aberration compensation methods based on fitting successive Zernike polynomials to each order of wavefront corrections [21]. 2.
Theoretical discussion
To understand the method proposed in this paper, let us first consider a paraxial mode with well defined OAM of order : (1) U (ρ , φ , z) = C (ρ , z) × exp(iφ ) where (ρ , φ , z) represent the cylindrical space coordinates and C (ρ , z) represents the radial profile. As we are only interested in the spatial properties of the field, this model would be valid for any polarization as long as it remains constant across the beam. An arbitrary beam (E) can be written as a superposition of OAM modes [22]: ∞
E = C (ρ , z) exp(iφ ) + ∑ Cn (ρ , z) exp(i(n φ + ϑn )),
(2)
n=1
where in this case we are singling out the component in which we are interested. Cn , n and ϑn are the amplitude, OAM and relative phase of each OAM mode respectively. The additional n terms can be thought of as undesired interference components. Experimentally, when producing a mode with a well defined , the contribution from the other unwanted modes decays very quickly with increasing values of | − n |. Let us now assume that the term with 1 is the dominant aberration, and hence we can disregard the higher orders. Our method for compensation is to introduce an additional collinear beam with OAM , weighting C and phase ϑ , allowing us to approximate E from Eq. (2) as E : (3) E ≈ C(ρ , z) exp(iφ ) +C1 (ρ , z) exp(i (1 φ + ϑ1 ) +C (ρ , z) exp(i φ + ϑ ) Thus by matching C (ρ , z) = C1 (ρ , z), = 1 and ϑ = ϑ1 + π we can remove the dominant aberration in our beam by destructive interference. In this paper we use a phase-only spatial light modulator [23] to generate the third term for compensation in Eq. (3). In this situation, the ability to control the radial profile C (ρ , z) is rather limited, which as we will see will impose some restrictions to our ability to fully compensate the vortex beam. Future approaches could make use of pseudo-amplitude modulation using a phase-only SLM as demonstrated in [24]. 2.1.
Hologram generation
Our aim is to remove the undesired modes by using a slightly modified pattern in the SLM. We achieved this by modifying the phase hologram to generate a second collinear diffracting beam with . This second beam is weighted by a real constant α with a phase relative to the first beam of ϑ . We are then free to tune α and ϑ to destructively interfere with the dominant unwanted OAM mode 1 as outlined previously. The total phase function displayed on the SLM ΨSLM is given below: (4) ΨSLM = arg exp(i[φ ]) + α exp(i[ φ + ϑ ]) where α and ϑ are the relative amplitude and phase of the mode with OAM of corresponding to 1 and is the desired OAM of the hologram. ΨSLM is then weighted by the amplitude of #206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9922
the combined exponentials within the arg operator in Eq. (4) in order to approximate intensity modulation with our phase-only SLM. We combine this phase hologram with a grating term to separate out our desired beam into the first diffraction order. The beam after the SLM in the first diffraction order therefore has the following form: sin(φ ) + α sin( φ + ϑ ) (5) E(ρ , φ , z) = G(ρ , z) exp i arctan cos(φ ) + α cos( φ + ϑ ) where G(ρ , z) is the Gaussian function. Using a phase-only SLM we can only access the phase of the field. Thus the OAM spectrum of the output of the SLM will not be exactly a mode with OAM plus the OAM of the new collinear beam . The complex amplitude of each OAM mode can be calculated from the following expression: C (ρ , z) =
2π 0
E(ρ , φ , z) exp(−iφ )d φ
(6)
Figures 1(b) and 1(c) show the original and modified hologram patterns with the following parameters: = 2, = 0, α = 0.19 and ϑ = 4.5 radians whereas Fig. 1(a) gives the OAM spectrum of the beam after acquiring the phase from the modified hologram shown in Fig. 1(c). These values of , α and ϑ of which exemplify a modified hologram in Fig. 1(c) are actually the optimal ones for the correction found after applying the method described in Sec.5. As demonstrated in Fig. 1(a), additional L-G modes with other OAM ( = 4 in this case) are inevitably generated from our hologram.
Fig. 1. (a) OAM spectrum calculated for = 2, = 0, α = 0.19, ϑ = 4.5 radians. (b) Phase hologram for beam of = 2. (c) Phase hologram for beam of = 2, and and weighting and relative phase corresponding to (a). The center of both holograms in (b) and (c) are expanded to highlight the differences.
3.
Experimental setup
The experimental setup for compensation of the OAM impurities in a beam is shown in Fig. 2. The source of the initial OAM beam was a 632 nm HeNe (Thorlabs) which was expanded by the arrangement of lenses L1 and L2 in order to maximize illumination of the pixels on the spatial light modulator (SLM). A half-wave plate (HWP) aligned the polarization of the field incident on the SLM. The angle between the incident beam and the SLM was minimized. A Cambridge Correlators SDE1024 SLM displayed the desired phase-only hologram (see Eq. (4)) to produce diffracting beams with non-zero OAM, separated from the zeroth order. An iris (SF1 ) selected only the first diffraction order. The phase contrast of the SLM (grey level to output phase) was varied [23] in order to maximise the diffraction efficiency into the first diffraction order.
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9923
Fig. 2. Experimental setup used to generate a beam with arbitrary OAM, , and subsequently compensate the observed separation of nulls in the near and far-field. The lens system consisting of lenses L3 and L4 magnifies and images the far-field spot onto the CCD. A typical image is given by inset (a). Lenses L3 and L5 magnify and image the near-field i.e. the plane at the SLM onto the CCD. A typical image is shown in inset (b). Note that the entire beam profile in (b) cannot be observed with the CCD. The focal lengths of the lenses are as follows: L1 = 25.4 mm, L2 = 75 mm, L3 = 500 mm, L4 = 35 mm, L5 = 125 mm. HWP = Half wave plate, SLM = spatial light modulator, BS = beam splitter, ND = neutral density filter, SF = spatial filter, CCD = charge-coupled device, HeNe = Helium-Neon laser.
To simultaneously observe both the near-field and far-field profile of the vortex beam, a 50:50 beam splitter (BS) separated the beam into two paths. We define the near-field as the plane just after the SLM display, while the far-field is the Fourier transform of the near-field plane. Lenses L3 and L4 in the upper path imaged the Fourier transform of the beam after the SLM onto the CCD. Inset (a) in Fig. 2 is a typical image of the far-field distribution for a beam generated with OAM = 2. Lenses L3 and L5 in the lower path in Fig. 2 combined to image the beam just after the SLM onto the CCD. This imaging system had a total magnification of 2.89 in order to sufficiently resolve the central null in the near-field image. Figure 2(b) is a typical image of the near-field intensity distribution of the same beam in Fig. 2(a). It should be noted that with the total magnification of the L3 , L5 imaging system, the entire near-field image was larger than the pixel display of the CCD. SF2 was used to select only the central region of the near-field image. 4.
Vortex splitting
Using the setup in Fig. 2, we generated beams using phase holograms constructed from the phase of beams with OAM = 0, 1, 2 and 3. The near- and far-field intensity distributions of the resulting beams were imaged to observe the effect of the aberrations specific to our system for beams of varying OAM. These results are summarized in Fig. 3. By observing the effect of the aberrations on the structure of the beam for different , we were able to deduce the of the first order correction to apply (Eq. (3)) for each beam of OAM in Section 5. Figures 3(a)–3(d) shows the far-field intensity profile of beams with an OAM of = 0, 1, 2, and 3 generated with the above setup (Fig. 2) in the absence of any compensation. The same beam was imaged in the near-field for = 0, 1, 2 and 3 in Figs. 3(e)–3(h). Figure 3 clearly shows the influence of aberrations on vortex beams with various OAM ranging from = 0-3. For = 0 in Fig. 3(a), aberrations in the beam have distorted the intensity profile into an ellipse. Observing the uncompensated = 1 beam in Fig. 3(b), the annular intensity distribution (dark red) lacks circular symmetry with the two lobes of high intensity separated by a line of low intensity resulting in a central null which is elliptical. We can tell by direct observation that
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9924
Fig. 3. (a)-(d) Far-field images of beams with OAM = 0, 1, 2 and 3 in the absence of any compensation. The configuration of lenses for these images was L3 and L4 (Fig. 2). (e)(f) Near-field images of the same beams are given for = 0, 1, 2 and 3. The configuration used for these images was L3 and L5 (see Fig. 2). From these images (a)-(d) and (e)-(h), we can see that the effect of aberrations on a vortex beam is to induce a splitting of the central singularity and a deformation of the intensity profile. This is consistent with the existence of another mode with 1 = − 2. In all images a logarithmic intensity scale is used. The physical scale used in all images is the same 490px×490px (3mm×3mm).
the intensity modulation observed in the Fig. 3(b) is due to interference between components with = 1 with 1 = −1, where 1 denotes the undesired OAM in the beam with the highest weighting. Considering the two OAM modes above of phase φ and 1 φ , destructive interference occurs whenever φ − (−φ ) = nπ (for odd n), resulting in the line of low intensity. φ is defined in Section 2 as the azimuthal angle in polar coordinates. Calculations also show that the elliptical intensity profile in Fig. 3(a) is consistent with interference between the desired = 0 mode and a component of 1 = −2. A comparison of calculated and experimental far-field intensity profiles for a pure mode with = 0 and a mode consisting of ( = 0) and (1 = −2) is given in Fig. 4. The intensity profile in Fig. 4 was calculated for the following expression: G(r, φ )e0φ + 0.19G(r, φ )e−2φ where G(r, φ ) is a Gaussian profile in polar coordinates (r, φ ) and we have set α1 = 0.19, = 0 and 1 = −2. The value of α1 was chosen based on the value of α determined experimentally after applying the method described in Sec. 5. We neglect the phase difference ϑ1 as it only contributes a rotation to the elliptical profile in Fig. 4(b). In our calculation we also assume that the functional form of the field amplitude for the 1 component is identical to within a scaling constant to the electric field amplitude of the = 0 component. Our observations of the compensated near-field beam profile in the following section show that this is not necessarily the case. It is the difference in field profiles between the and 1 components which likely results in the differences between the calculated and experimental intensity profiles, however from our calculations we can see that an elliptical intensity profile associated with a = 0 Gaussian beam is qualitatively consistent with the existence of an additional 1 =-2 component. In Figs. 3(c) and 3(d) the splitting of the central null into || separate nulls is apparent. For
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9925
Fig. 4. Calculated far-field images of a beam with OAM = 0 (a) and = 0 + 1 = −2 (b) and experimentally measured far-field images of our = 0 beam uncompensated (c) and with compensation (d). Calculated images have identical scaling in z. In all images a logarithmic scale is used.
= 2, the central vortex has been split into two in the far-field in Fig. 3(c). This can be shown to be consistent with interference between a = 2 and 1 = 0 mode. Let us first recognize that the expression which gives rise to the singularity rm exp imφ (the rm term is contained within the function C(ρ , z) in Eq. (2)), may be written as (x + iy)m . We can then express the electric field at some plane due to our combination of = 2 and 1 = 0 modes as (x + iy)2 + A to within some factor, where A is a normalized constant. By finding the intersections of lines of zero real and imaginary components for this expression we find the corresponding singularities in the field. From this analysis it can be seen that the addition of the constant A due to the 1 = 0 mode results in a splitting of the central = 2 singularity. For the case of the uncompensated = 3 beam in the far-field in Fig. 3(d), the nulls are distributed along a horizontal line while there is still a null at the center of the beam. This central null of the total field implies that the = 3 component and the unknown 1 component must both possess a singularity at their centre and hence 1 = 0. We also know that singularities occur at positions at which the electric field amplitude is zero, and so we can associate the position of singularities with regions of destructive interference between the and 1 modes. Thus in Fig. 3(d), destructive interference must be occurring along the line of singularities. From our previous analysis of Fig. 3(b), we know that a single straight line of destructive interference is a consequence of interference between = 1 and = −1 components. In the case of Fig. 3(d), = 3, and hence 1 = 1 in order to maintain the relative phase difference for a straight line of singularities such that nulls occur at φ = n2π . This can also be verified by finding the zero crossing of the real and imaginary components in the expression (x + iy)3 + A(x + iy). The intensity distribution for = 2 and 3 are both elliptical. A beam with OAM of = 4 was also generated and observed that the central vortex split along a horizontal line of 4 separate vortices, however this data was not included in this paper. Comparing the desired OAM and the highest interfering OAM component 1 deduced from Figs. 3(a)–3(d), the relationship between the two is 1 = − 2. This relationship is independent of the value of imposed by the SLM on the input beam. Applying different values of will shift the maximum in the OAM spectrum to the appropriate value, while the non-zero OAM bandwidth arising from aberrations along the optical path remains unaffected. From this we conclude that the highest contributing OAM impurities have OAM values of 1 = −2. The question arises from where does this interference component originate? Let us consider #206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9926
the interference mode and consider the linear distribution of vortices for = 3 in the far-field shown in Fig. 3(d). This symmetry breaking implies a perturbation which lacks cylindrical symmetry. In fact rows of singularities have been shown theoretically [15] and experimentally [25] to arise from both small elliptical perturbations to and astigmatic transformations of LG modes respectively. The origin of the elliptical perturbation is possibly due to a non-normal angle of incidence between the output of the HeNe and the SLM [26]. Under the effect of both types of perturbations, the original LG mode is best approximated as either Ince-Gauss, or HermiteLaguerre-Gauss modes respectively [15] which have singularities distributed along a horizontal line. The typical near-field distribution of the vortex beams just after the SLM are shown in Figs. 3(e)–3(h). Figures 3(e)–3(h) indicate that in the near-field, just after the plane of the SLM, vortex splitting is occurring. It is interesting to note that in the near-field, the distribution of nulls for the case = 3 has symmetry in the azimuthal coordinate of the plane, compared to its far-field distribution. In order to resolve the structure associated with the core of the beam in the near-field, the imaging system (L3 and L5 in Fig. 2) magnified the beam profile in the plane of the SLM. 5.
Compensation
Fig. 5. To construct the phase hologram used to generate two collinear beams with OAM = 2 and = 0, we calculate the complex amplitude of each individual beam, add them and extract the resulting phase profile (ψSLM ). Parameters α and ϑ control the relative weighting and phase between the beams. When illuminated with a Gaussian beam, the output is a = 0 and a = 2 beam. Although not additive, displayed are the holograms associated with the = 0 component, = 2 component and the final hologram used to create the two collinear beams.
We apply compensation to the output of the SLM by holographically generating an additional collinear beam with a weighting (α ), relative phase (ϑ ) and orbital angular momentum ( ). A schematic describing the construction of phase-holograms for the generation of multiple beams is given in Fig. 5. We now describe in detail the method for finding the optimum parameters to correct for the splitting of the central null. Once the value of 1 has been deduced (1 = − 2), we incorporate a hologram of a beam with OAM = 1 into the original hologram (as in Fig. 5). Specifically we performed this for a desired OAM of = 2. We then fix the weighting (α ) at some arbitrary value, and vary the phase ϑ , relative to the original hologram while looking at the effect on the singularities in the far-field beam profile. We observe the rotation of the vortices in the far-field as the angles at which destructive interference occur are shifted by ϑ /( − 1 ) rad. When the distance between the vortices has been minimized, the phase (ϑ ) is then fixed, while the weighting (α ) is varied, again aiming to minimize the distance between the vortices for the
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9927
far-field image. The minimum vortex separations were qualitatively selected by comparing the CCD images at different α and ϑ . This method was sufficient for the purposes of the paper, but more precise algorithms can be implemented by calculating the separation distances between the vortices. This process is iterated as necessary until the separated nulls have been collapsed back to a single central null. At this point the collinear beam generated by the SLM ( ) and the 1 component of the original beam are out of phase by π radians, and destructive interference has been maximized at the observation plane of the CCD by having matched α of the additional collinear beam to the α1 of the unwanted interference component. These compensation parameters are valid for all values of imposed by the SLM. These are however, only a first order correction. For larger values of , higher order corrections must be applied (i.e. additional holograms with ϑn+1 , αn+1 , n+1 ).
Fig. 6. Far-field images of beams with OAM quantum numbers of = 0, 1, 2 and 3 (a)-(d) with compensation applied as detailed in the text. The deformation of the intensity profiles are decreased compared to Figs. 3(a) and 3(b). Vortex splitting associated with = 3 and 4 (c,d) has been reduced relative to Fig. 3. Near-field images of the same beams are given for = 0, 1, 2 and 3 (e)-(f). In all images a logarithmic scale is used.
5.1.
Post-compensation
In order to test our proposed method of compensation, we generated beams with OAM = 0, 1, 2 and 3 (using the same setup in Fig. 2), with the addition of the compensation terms mentioned above and compared them with our original beams from Fig. 3. The resulting far-field and near-field images are presented in Figs. 6(a)–6(d) and Figs. 6(e)–6(h) respectively for = 0–3. Comparing with Fig. 3, introduction of our proposed method has yielded improvements to the singularity splitting and the induced ellipticity due to aberrations in the far-field. For = 0 in Fig. 6(a), the intensity distribution has decreased in ellipticity compared to the uncompensated beam in Fig. 3(a). This is also the case for = 1, which presents a more even distribution of intensity over the beam profile relative to without compensation. This increase in symmetry is consistent with the removal of the unwanted = -2 and = -1 components present in the = 0 and = 1 beams respectively. The most pronounced improvements in the beam profile occur in Fig. 6(c) for = 2 where the two previously split singularities in Fig. 3(c) have been reformed into a central null as a result of the compensation. Comparing The far-field #206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9928
compensation of the = 3 beam shown in Fig. 6(d) resulted in a reduction in the separation of the 3 nulls compared with Fig. 3(d) to the extent that each singularity could no longer be fully resolved. For both = 2 and = 3, we successfully removed the singularity splitting due to aberration, by destructive interference of holographically generated and inherent 1 components at the plane of the CCD. The absence of circular symmetry in the singularities however, implies that there exist higher order OAM components that have not been accounted for. For = 1-3 in Figs. 6(b)–6(d), after the removal of the most significant unwanted OAM mode there still exists an asymmetry in the beam profile, with intensity concentrated on the lower right hand side of the singularity. This arises from a slight misalignment of the beam from the hologram. In fact this asymmetry can also be seen in the uncompensated beams in Fig. 3. Comparison of compensated with uncompensated intensity profiles in the near-field, Figs. 3(e)–3(h) and Figs. 6(e)–6(h) respectively, shows an increase in the separation of the nulls for = 2 and 3 upon addition of the collinear compensating beam, while the nulls in the = 3 case have rearranged along a straight line to resemble the uncompensated far-field profiles. This is a reflection of our inability to fully control the radial profile of the compensation term. Thus, in this case what we have achieved is C (δ ρ , zFF ) ≈ C1 (δ ρ , zFF ), with zFF the far field plane and δ ρ the radial points close to the center of the beam ρ = 0. However, in general C (ρ , z) = C1 (ρ , z). Even though for most applications this can be enough to probe the local properties of the higher order optical vortex, in the next section we provide a method to allow a better control of the radial profile of the compensating mode. 6.
Near-field compensation
While we have demonstrated control of the singularities of an optical field, reforming the split singularities in the far-field, this is apparently achieved at the cost of separating the nulls of the beam in the near-field. In the paraxial regime, the far-field intensity distribution F, is related to the near-field f , by a Fourier transform (and vice versa): F(kx , ky ) =
∞ ∞ −∞ −∞
f (x, y)ei(kx x+ky y) dxdy
(7)
Considering the centre of the beam at F(0, 0), Eq. (7) becomes: F(0, 0) =
∞ ∞ −∞ −∞
f (x, y)dxdy
(8)
which is the average of the field f (x, y). Thus by modulating the field f (x, y) such that it’s mean is zero, we can create a null at the origin F(kx , ky ). By manipulating the amplitude of the field such that the mean value of the field is zero we can force the center of the beam to have zero or a minimal amplitude to minimize the vortex splitting observed. We implement this by introducing an additional phase ψr (r) to the expression from which we calculate our holograms (Eq. (4)) in order to modulate the field of the compensation ( ) component generated by the SLM: (9) ΨSLM = arg exp(i[φ ]) + α exp(i[ φ + ϑ + ψr (r)]) Where ψr (r) is a step function in the radius of the beam varying from 0 to π radians: 0 if r ≤ r0 ψr = π if r > r0
#206010 - $15.00 USD (C) 2014 OSA
(10)
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9929
This radial phase modulation maintains the relative phase relationship among each term in Eq. (3) in the vicinity of the singularities, and hence preserves the effects of the compensation demonstrated in Section 5. By inverting the sign of the field within a radius of r0 we can find a r0 at which the mean value of the field is zero for the collinear beam. To perform this experimentally, we apply the far-field compensation using the parameters determined in the previous section for the case = 2, splitting the vortex singularities in the near-field. We then modify the hologram according to Eq. (9) and vary r0 , minimizing the separation of the singularities associated with the near-field profile of the beam.
Fig. 7. Far-field (a) and near-field (b) images of a beam with OAM = 2 in the absence of any compensation. Far-field (c) and near-field (d) images of an = 2 beam compensated by the removal of the 1 component by destructive interference with a collinear beam of . Far-field (e) and near-field (f) images of a = 2 beam with the same compensation as applied in (c),(d) but with a radial phase (Eq. (10)) applied to the = 0 compensation component. For the radial phase, r0 = 10 pixels (90 μ m). The result of this additional radial phase is that the separation of the vortices in the near-field has been reduced from (c) to (e). In all images a logarithmic scale is used.
The results of our radial phase modulation for the = 2 beam are described in Fig. 7 and compared with the previously compensated and uncompensated beams = 2 beams from. Figures 7(a) and 7(b) and Figs. 7(c) and 7(d) are control images of the far-field and near-field distribution of nulls without and with the far-field compensation from Figs. 3(c) and 3(g) and Figs. 6(c) and 6(g) respectively. The increase in vortex splitting between Figs. 7(a) and 7(c) (without and with compensation) is visible. From the near-field image, we observe that while there still exists a distortion in the intensity of the near-field (two lobes of higher intensity circling the central nulls), the distance between the nulls has been reduced to distances which appear comparable to the original vortex splitting occurring in the near-field. With this compensation, we are successful in reducing the splitting of the two nulls to below that of the original uncompensated beam in Fig. 7(e), while retaining the central singularity in the far-field.
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9930
7.
Conclusion
We propose a simple method for the compensation of mode impurities in a OAM beam generated by holographic mode conversion. By introducing an additional OAM mode and varying the relative phase and weighting, we can remove the highest order OAM impurity at a single plane by destructive interference. This is performed with a phase-only SLM, removing the intensity distribution associated with interference between multiple OAM modes for = 0 and 1, while collapsing the multiple split singularities associated with interference among higher order modes, = 2, 3. We achieved this by interference with a collinear beam of OAM = − 2. We then applied a phase with a radial step-function between 0 and π , and by varying the radius at which the step occurs, we were able to simultaneously reform the nulls in the near-field while retaining our compensation in the far field for OAM of = 2. Acknowledgments This work was supported by the Australian Research Council (ARC) Future Fellowship (FT1110924) scheme.
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9931
of Photonics and Optical Science (IPOS), School of Physics, University of Sydney, NSW 2006, Australia 2 ARC Centre of Excellence for Engineered Quantum Systems (EQuS), Macquarie University, NSW 2109, Australia 3 Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia ∗ [email protected]
Abstract: We demonstrate a general method for the first order compensation of singularity splitting in a vortex beam at a single plane. By superimposing multiple forked holograms on the SLM used to generate the vortex beam, we are able to compensate vortex splitting and generate beams with desired phase singularities of order = 0, 1, 2, and 3 in one plane. We then extend this method by application of a radial phase, in order to simultaneously compensate the observed vortex splitting at two planes (near and far field) for an = 2 beam. © 2014 Optical Society of America OCIS codes: (080.4865) Optical vortices; (090.1000) Aberration compensation; (070.6120) Spatial light modulators; (230.6120) Spatial light modulators.
References and links 1. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). 2. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003). 3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). 4. G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “Experimental quantum coin tossing,” Phys. Rev. Lett. 94(4), 040501 (2005). 5. M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597(2), 1266 (2003). 6. C. Barbieri, D. Dravins, T. Occhipinti, F. Tamburini, G. Naletto, V. Da Deppo, S. Fornasier, M. D’Onofrio, R. A. E. Fosbury, R. Nilsson, and H. Uthas, “Astronomical applications of quantum optics for extremely large telescopes,” J. Mod. Opt. 54(2–3), 191–197 (2007). 7. F. Tamburini, B. Thid´e, G. Molina-Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7(3), 195–197 (2011). 8. N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492, 883–922 (2008). 9. N. M. Elias, “Photon orbital angular momentum and torque metrics for single telescopes and interferometers,” Astron. Astrophys. 541, 101 (2012). 10. G. Foo, M. P. David, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005). 11. X. Zambrana-Puyalto, X. Vidal, and G. Molina-Terriza, “Excitation of single multipolar modes with engineered cylindrically symmetric fields,” Opt. Express 20(22), 24536–24544 (2012). 12. I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159(1), 99–117 (1999). 13. F. Ricci, W. L¨offler, and M. P. van Exter, “Instability of higher-order optical vortices analyzed with a multipinhole interferometer,” Opt. Express 20(20), 22961–22975 (2012). 14. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9920
15. M. R. Dennis, “Rows of optical vortices from elliptically perturbing a high-order beam,” Opt. Lett. 31(9), 1325– 1327 (2006). 16. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. 20(8), 1635–1643 (2003). 17. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241(4), 237–247 (2004). 18. A. Wada, T. Ohtani, Y. Miyamoto, and M. Takeda, “Propagation analysis of the Laguerre-Gaussian beam with astigmatism,” J. Opt. Soc. Am. 22(12), 2746–2755 (2005). 19. I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103(5), 422–428 (1993). 20. C. H. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express 14(15), 6604–6612 (2006). 21. J. Carpenter, B. Thomsen, and T. Wilkinson, “Mode division multiplexing of modes with the same azimuthal index,” IEEE Photon. Technol. Lett. 24(21), 1969–1972 (2012). 22. 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(1), 013601 (2002). 23. R. Bowman, V. D’Ambrosio, E. Rubino, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, “Optimisation of a low cost SLM for diffraction efficiency and ghost order diffraction,” Eur. Phys. J. Spec. Top. 199(1), 149–158 (2011). 24. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). 25. A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of optical vortices,” Opt. Express 19(7), 6182–6190 (2011). 26. A. Kumar, P. Vaity, J. Bhatt, and R. P. Singh, “Stability of higher order optical vortices produced by spatial light modulators,” J. Mod. Opt. 60, 1696–1700 (2013).
1.
Introduction
The control of the angular momentum of a paraxial beam using its spatial properties has stimulated substantial research with applications in many areas of physics for example: spatial mode multiplexing for telecommunications [1], optical spanners [2], quantum optics [3, 4] and astronomy [5–9]. For many of these applications (e.g. vortex coronography [10]) it is required to control the phase singularities, also called optical vortices, of the paraxial field. The control of phase singularities in a paraxial beam is motivated because in this regime, the phase singularities of the beam are directly associated with the density of orbital angular momentum (OAM) that the beam carries. Outside the paraxial approximation, the split of the total angular momentum into orbital and spin components becomes problematic. Nevertheless, in some applications the control of the phase singularities in the paraxial regime determines the local structure of a focused beam. This is of paramount importance to control the interaction of light and matter at the nanoscale [11]. Paraxial beams with a well defined value of OAM, , have a cylindrically symmetric intensity profile. They present an optical vortex of charge at the centre, i.e. the phase of the beam twists around the centre of the beam by 2π radians. Thus, the intensity at the position of the singularity is zero. Unfortunately, implementing beams with higher order singularities is challenging. It has been observed that, due to any uncontrolled breaking of the symmetry, higher order phase singularities split into sets of lower order singularities [12, 13]. Typically, the splitting results in || singularities of order ±1 depending on the sign of . The splitting of the central null of a vortex beam suggests coherent interference occurring between multiple co-propagating modes, either due to noise or imperfections of the set-up. Clearly, the resulting field will be a superposition of different OAM modes [14–18]. In this article we show a very simple and effective technique in order to compensate for such effects when producing angular momentum states. In order to compensate for the interference, we generate multiple collinear beams each with some orbital angular momentum. Proper control of the weighting and relative phases using a phase-only spatial light modulator allows us to generate optical beams with stable high order phase singularities. This has previously been demonstrated using non-collinear beams for the case = 2 [19]. The method of producing sev#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9921
eral interfering copropagating modes is also used in the generation of “interference vortices” in particle trapping [20]. By tuning both the relative amplitude and phase of these beams, we can combine them such that they destructively interfere at a certain plane with the highest order OAM impurities in our original beam, thus increasing the purity of the generation of the desired OAM state. Other more complex ways of correcting this effect rely on aberration compensation methods based on fitting successive Zernike polynomials to each order of wavefront corrections [21]. 2.
Theoretical discussion
To understand the method proposed in this paper, let us first consider a paraxial mode with well defined OAM of order : (1) U (ρ , φ , z) = C (ρ , z) × exp(iφ ) where (ρ , φ , z) represent the cylindrical space coordinates and C (ρ , z) represents the radial profile. As we are only interested in the spatial properties of the field, this model would be valid for any polarization as long as it remains constant across the beam. An arbitrary beam (E) can be written as a superposition of OAM modes [22]: ∞
E = C (ρ , z) exp(iφ ) + ∑ Cn (ρ , z) exp(i(n φ + ϑn )),
(2)
n=1
where in this case we are singling out the component in which we are interested. Cn , n and ϑn are the amplitude, OAM and relative phase of each OAM mode respectively. The additional n terms can be thought of as undesired interference components. Experimentally, when producing a mode with a well defined , the contribution from the other unwanted modes decays very quickly with increasing values of | − n |. Let us now assume that the term with 1 is the dominant aberration, and hence we can disregard the higher orders. Our method for compensation is to introduce an additional collinear beam with OAM , weighting C and phase ϑ , allowing us to approximate E from Eq. (2) as E : (3) E ≈ C(ρ , z) exp(iφ ) +C1 (ρ , z) exp(i (1 φ + ϑ1 ) +C (ρ , z) exp(i φ + ϑ ) Thus by matching C (ρ , z) = C1 (ρ , z), = 1 and ϑ = ϑ1 + π we can remove the dominant aberration in our beam by destructive interference. In this paper we use a phase-only spatial light modulator [23] to generate the third term for compensation in Eq. (3). In this situation, the ability to control the radial profile C (ρ , z) is rather limited, which as we will see will impose some restrictions to our ability to fully compensate the vortex beam. Future approaches could make use of pseudo-amplitude modulation using a phase-only SLM as demonstrated in [24]. 2.1.
Hologram generation
Our aim is to remove the undesired modes by using a slightly modified pattern in the SLM. We achieved this by modifying the phase hologram to generate a second collinear diffracting beam with . This second beam is weighted by a real constant α with a phase relative to the first beam of ϑ . We are then free to tune α and ϑ to destructively interfere with the dominant unwanted OAM mode 1 as outlined previously. The total phase function displayed on the SLM ΨSLM is given below: (4) ΨSLM = arg exp(i[φ ]) + α exp(i[ φ + ϑ ]) where α and ϑ are the relative amplitude and phase of the mode with OAM of corresponding to 1 and is the desired OAM of the hologram. ΨSLM is then weighted by the amplitude of #206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9922
the combined exponentials within the arg operator in Eq. (4) in order to approximate intensity modulation with our phase-only SLM. We combine this phase hologram with a grating term to separate out our desired beam into the first diffraction order. The beam after the SLM in the first diffraction order therefore has the following form: sin(φ ) + α sin( φ + ϑ ) (5) E(ρ , φ , z) = G(ρ , z) exp i arctan cos(φ ) + α cos( φ + ϑ ) where G(ρ , z) is the Gaussian function. Using a phase-only SLM we can only access the phase of the field. Thus the OAM spectrum of the output of the SLM will not be exactly a mode with OAM plus the OAM of the new collinear beam . The complex amplitude of each OAM mode can be calculated from the following expression: C (ρ , z) =
2π 0
E(ρ , φ , z) exp(−iφ )d φ
(6)
Figures 1(b) and 1(c) show the original and modified hologram patterns with the following parameters: = 2, = 0, α = 0.19 and ϑ = 4.5 radians whereas Fig. 1(a) gives the OAM spectrum of the beam after acquiring the phase from the modified hologram shown in Fig. 1(c). These values of , α and ϑ of which exemplify a modified hologram in Fig. 1(c) are actually the optimal ones for the correction found after applying the method described in Sec.5. As demonstrated in Fig. 1(a), additional L-G modes with other OAM ( = 4 in this case) are inevitably generated from our hologram.
Fig. 1. (a) OAM spectrum calculated for = 2, = 0, α = 0.19, ϑ = 4.5 radians. (b) Phase hologram for beam of = 2. (c) Phase hologram for beam of = 2, and and weighting and relative phase corresponding to (a). The center of both holograms in (b) and (c) are expanded to highlight the differences.
3.
Experimental setup
The experimental setup for compensation of the OAM impurities in a beam is shown in Fig. 2. The source of the initial OAM beam was a 632 nm HeNe (Thorlabs) which was expanded by the arrangement of lenses L1 and L2 in order to maximize illumination of the pixels on the spatial light modulator (SLM). A half-wave plate (HWP) aligned the polarization of the field incident on the SLM. The angle between the incident beam and the SLM was minimized. A Cambridge Correlators SDE1024 SLM displayed the desired phase-only hologram (see Eq. (4)) to produce diffracting beams with non-zero OAM, separated from the zeroth order. An iris (SF1 ) selected only the first diffraction order. The phase contrast of the SLM (grey level to output phase) was varied [23] in order to maximise the diffraction efficiency into the first diffraction order.
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9923
Fig. 2. Experimental setup used to generate a beam with arbitrary OAM, , and subsequently compensate the observed separation of nulls in the near and far-field. The lens system consisting of lenses L3 and L4 magnifies and images the far-field spot onto the CCD. A typical image is given by inset (a). Lenses L3 and L5 magnify and image the near-field i.e. the plane at the SLM onto the CCD. A typical image is shown in inset (b). Note that the entire beam profile in (b) cannot be observed with the CCD. The focal lengths of the lenses are as follows: L1 = 25.4 mm, L2 = 75 mm, L3 = 500 mm, L4 = 35 mm, L5 = 125 mm. HWP = Half wave plate, SLM = spatial light modulator, BS = beam splitter, ND = neutral density filter, SF = spatial filter, CCD = charge-coupled device, HeNe = Helium-Neon laser.
To simultaneously observe both the near-field and far-field profile of the vortex beam, a 50:50 beam splitter (BS) separated the beam into two paths. We define the near-field as the plane just after the SLM display, while the far-field is the Fourier transform of the near-field plane. Lenses L3 and L4 in the upper path imaged the Fourier transform of the beam after the SLM onto the CCD. Inset (a) in Fig. 2 is a typical image of the far-field distribution for a beam generated with OAM = 2. Lenses L3 and L5 in the lower path in Fig. 2 combined to image the beam just after the SLM onto the CCD. This imaging system had a total magnification of 2.89 in order to sufficiently resolve the central null in the near-field image. Figure 2(b) is a typical image of the near-field intensity distribution of the same beam in Fig. 2(a). It should be noted that with the total magnification of the L3 , L5 imaging system, the entire near-field image was larger than the pixel display of the CCD. SF2 was used to select only the central region of the near-field image. 4.
Vortex splitting
Using the setup in Fig. 2, we generated beams using phase holograms constructed from the phase of beams with OAM = 0, 1, 2 and 3. The near- and far-field intensity distributions of the resulting beams were imaged to observe the effect of the aberrations specific to our system for beams of varying OAM. These results are summarized in Fig. 3. By observing the effect of the aberrations on the structure of the beam for different , we were able to deduce the of the first order correction to apply (Eq. (3)) for each beam of OAM in Section 5. Figures 3(a)–3(d) shows the far-field intensity profile of beams with an OAM of = 0, 1, 2, and 3 generated with the above setup (Fig. 2) in the absence of any compensation. The same beam was imaged in the near-field for = 0, 1, 2 and 3 in Figs. 3(e)–3(h). Figure 3 clearly shows the influence of aberrations on vortex beams with various OAM ranging from = 0-3. For = 0 in Fig. 3(a), aberrations in the beam have distorted the intensity profile into an ellipse. Observing the uncompensated = 1 beam in Fig. 3(b), the annular intensity distribution (dark red) lacks circular symmetry with the two lobes of high intensity separated by a line of low intensity resulting in a central null which is elliptical. We can tell by direct observation that
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9924
Fig. 3. (a)-(d) Far-field images of beams with OAM = 0, 1, 2 and 3 in the absence of any compensation. The configuration of lenses for these images was L3 and L4 (Fig. 2). (e)(f) Near-field images of the same beams are given for = 0, 1, 2 and 3. The configuration used for these images was L3 and L5 (see Fig. 2). From these images (a)-(d) and (e)-(h), we can see that the effect of aberrations on a vortex beam is to induce a splitting of the central singularity and a deformation of the intensity profile. This is consistent with the existence of another mode with 1 = − 2. In all images a logarithmic intensity scale is used. The physical scale used in all images is the same 490px×490px (3mm×3mm).
the intensity modulation observed in the Fig. 3(b) is due to interference between components with = 1 with 1 = −1, where 1 denotes the undesired OAM in the beam with the highest weighting. Considering the two OAM modes above of phase φ and 1 φ , destructive interference occurs whenever φ − (−φ ) = nπ (for odd n), resulting in the line of low intensity. φ is defined in Section 2 as the azimuthal angle in polar coordinates. Calculations also show that the elliptical intensity profile in Fig. 3(a) is consistent with interference between the desired = 0 mode and a component of 1 = −2. A comparison of calculated and experimental far-field intensity profiles for a pure mode with = 0 and a mode consisting of ( = 0) and (1 = −2) is given in Fig. 4. The intensity profile in Fig. 4 was calculated for the following expression: G(r, φ )e0φ + 0.19G(r, φ )e−2φ where G(r, φ ) is a Gaussian profile in polar coordinates (r, φ ) and we have set α1 = 0.19, = 0 and 1 = −2. The value of α1 was chosen based on the value of α determined experimentally after applying the method described in Sec. 5. We neglect the phase difference ϑ1 as it only contributes a rotation to the elliptical profile in Fig. 4(b). In our calculation we also assume that the functional form of the field amplitude for the 1 component is identical to within a scaling constant to the electric field amplitude of the = 0 component. Our observations of the compensated near-field beam profile in the following section show that this is not necessarily the case. It is the difference in field profiles between the and 1 components which likely results in the differences between the calculated and experimental intensity profiles, however from our calculations we can see that an elliptical intensity profile associated with a = 0 Gaussian beam is qualitatively consistent with the existence of an additional 1 =-2 component. In Figs. 3(c) and 3(d) the splitting of the central null into || separate nulls is apparent. For
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9925
Fig. 4. Calculated far-field images of a beam with OAM = 0 (a) and = 0 + 1 = −2 (b) and experimentally measured far-field images of our = 0 beam uncompensated (c) and with compensation (d). Calculated images have identical scaling in z. In all images a logarithmic scale is used.
= 2, the central vortex has been split into two in the far-field in Fig. 3(c). This can be shown to be consistent with interference between a = 2 and 1 = 0 mode. Let us first recognize that the expression which gives rise to the singularity rm exp imφ (the rm term is contained within the function C(ρ , z) in Eq. (2)), may be written as (x + iy)m . We can then express the electric field at some plane due to our combination of = 2 and 1 = 0 modes as (x + iy)2 + A to within some factor, where A is a normalized constant. By finding the intersections of lines of zero real and imaginary components for this expression we find the corresponding singularities in the field. From this analysis it can be seen that the addition of the constant A due to the 1 = 0 mode results in a splitting of the central = 2 singularity. For the case of the uncompensated = 3 beam in the far-field in Fig. 3(d), the nulls are distributed along a horizontal line while there is still a null at the center of the beam. This central null of the total field implies that the = 3 component and the unknown 1 component must both possess a singularity at their centre and hence 1 = 0. We also know that singularities occur at positions at which the electric field amplitude is zero, and so we can associate the position of singularities with regions of destructive interference between the and 1 modes. Thus in Fig. 3(d), destructive interference must be occurring along the line of singularities. From our previous analysis of Fig. 3(b), we know that a single straight line of destructive interference is a consequence of interference between = 1 and = −1 components. In the case of Fig. 3(d), = 3, and hence 1 = 1 in order to maintain the relative phase difference for a straight line of singularities such that nulls occur at φ = n2π . This can also be verified by finding the zero crossing of the real and imaginary components in the expression (x + iy)3 + A(x + iy). The intensity distribution for = 2 and 3 are both elliptical. A beam with OAM of = 4 was also generated and observed that the central vortex split along a horizontal line of 4 separate vortices, however this data was not included in this paper. Comparing the desired OAM and the highest interfering OAM component 1 deduced from Figs. 3(a)–3(d), the relationship between the two is 1 = − 2. This relationship is independent of the value of imposed by the SLM on the input beam. Applying different values of will shift the maximum in the OAM spectrum to the appropriate value, while the non-zero OAM bandwidth arising from aberrations along the optical path remains unaffected. From this we conclude that the highest contributing OAM impurities have OAM values of 1 = −2. The question arises from where does this interference component originate? Let us consider #206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9926
the interference mode and consider the linear distribution of vortices for = 3 in the far-field shown in Fig. 3(d). This symmetry breaking implies a perturbation which lacks cylindrical symmetry. In fact rows of singularities have been shown theoretically [15] and experimentally [25] to arise from both small elliptical perturbations to and astigmatic transformations of LG modes respectively. The origin of the elliptical perturbation is possibly due to a non-normal angle of incidence between the output of the HeNe and the SLM [26]. Under the effect of both types of perturbations, the original LG mode is best approximated as either Ince-Gauss, or HermiteLaguerre-Gauss modes respectively [15] which have singularities distributed along a horizontal line. The typical near-field distribution of the vortex beams just after the SLM are shown in Figs. 3(e)–3(h). Figures 3(e)–3(h) indicate that in the near-field, just after the plane of the SLM, vortex splitting is occurring. It is interesting to note that in the near-field, the distribution of nulls for the case = 3 has symmetry in the azimuthal coordinate of the plane, compared to its far-field distribution. In order to resolve the structure associated with the core of the beam in the near-field, the imaging system (L3 and L5 in Fig. 2) magnified the beam profile in the plane of the SLM. 5.
Compensation
Fig. 5. To construct the phase hologram used to generate two collinear beams with OAM = 2 and = 0, we calculate the complex amplitude of each individual beam, add them and extract the resulting phase profile (ψSLM ). Parameters α and ϑ control the relative weighting and phase between the beams. When illuminated with a Gaussian beam, the output is a = 0 and a = 2 beam. Although not additive, displayed are the holograms associated with the = 0 component, = 2 component and the final hologram used to create the two collinear beams.
We apply compensation to the output of the SLM by holographically generating an additional collinear beam with a weighting (α ), relative phase (ϑ ) and orbital angular momentum ( ). A schematic describing the construction of phase-holograms for the generation of multiple beams is given in Fig. 5. We now describe in detail the method for finding the optimum parameters to correct for the splitting of the central null. Once the value of 1 has been deduced (1 = − 2), we incorporate a hologram of a beam with OAM = 1 into the original hologram (as in Fig. 5). Specifically we performed this for a desired OAM of = 2. We then fix the weighting (α ) at some arbitrary value, and vary the phase ϑ , relative to the original hologram while looking at the effect on the singularities in the far-field beam profile. We observe the rotation of the vortices in the far-field as the angles at which destructive interference occur are shifted by ϑ /( − 1 ) rad. When the distance between the vortices has been minimized, the phase (ϑ ) is then fixed, while the weighting (α ) is varied, again aiming to minimize the distance between the vortices for the
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9927
far-field image. The minimum vortex separations were qualitatively selected by comparing the CCD images at different α and ϑ . This method was sufficient for the purposes of the paper, but more precise algorithms can be implemented by calculating the separation distances between the vortices. This process is iterated as necessary until the separated nulls have been collapsed back to a single central null. At this point the collinear beam generated by the SLM ( ) and the 1 component of the original beam are out of phase by π radians, and destructive interference has been maximized at the observation plane of the CCD by having matched α of the additional collinear beam to the α1 of the unwanted interference component. These compensation parameters are valid for all values of imposed by the SLM. These are however, only a first order correction. For larger values of , higher order corrections must be applied (i.e. additional holograms with ϑn+1 , αn+1 , n+1 ).
Fig. 6. Far-field images of beams with OAM quantum numbers of = 0, 1, 2 and 3 (a)-(d) with compensation applied as detailed in the text. The deformation of the intensity profiles are decreased compared to Figs. 3(a) and 3(b). Vortex splitting associated with = 3 and 4 (c,d) has been reduced relative to Fig. 3. Near-field images of the same beams are given for = 0, 1, 2 and 3 (e)-(f). In all images a logarithmic scale is used.
5.1.
Post-compensation
In order to test our proposed method of compensation, we generated beams with OAM = 0, 1, 2 and 3 (using the same setup in Fig. 2), with the addition of the compensation terms mentioned above and compared them with our original beams from Fig. 3. The resulting far-field and near-field images are presented in Figs. 6(a)–6(d) and Figs. 6(e)–6(h) respectively for = 0–3. Comparing with Fig. 3, introduction of our proposed method has yielded improvements to the singularity splitting and the induced ellipticity due to aberrations in the far-field. For = 0 in Fig. 6(a), the intensity distribution has decreased in ellipticity compared to the uncompensated beam in Fig. 3(a). This is also the case for = 1, which presents a more even distribution of intensity over the beam profile relative to without compensation. This increase in symmetry is consistent with the removal of the unwanted = -2 and = -1 components present in the = 0 and = 1 beams respectively. The most pronounced improvements in the beam profile occur in Fig. 6(c) for = 2 where the two previously split singularities in Fig. 3(c) have been reformed into a central null as a result of the compensation. Comparing The far-field #206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9928
compensation of the = 3 beam shown in Fig. 6(d) resulted in a reduction in the separation of the 3 nulls compared with Fig. 3(d) to the extent that each singularity could no longer be fully resolved. For both = 2 and = 3, we successfully removed the singularity splitting due to aberration, by destructive interference of holographically generated and inherent 1 components at the plane of the CCD. The absence of circular symmetry in the singularities however, implies that there exist higher order OAM components that have not been accounted for. For = 1-3 in Figs. 6(b)–6(d), after the removal of the most significant unwanted OAM mode there still exists an asymmetry in the beam profile, with intensity concentrated on the lower right hand side of the singularity. This arises from a slight misalignment of the beam from the hologram. In fact this asymmetry can also be seen in the uncompensated beams in Fig. 3. Comparison of compensated with uncompensated intensity profiles in the near-field, Figs. 3(e)–3(h) and Figs. 6(e)–6(h) respectively, shows an increase in the separation of the nulls for = 2 and 3 upon addition of the collinear compensating beam, while the nulls in the = 3 case have rearranged along a straight line to resemble the uncompensated far-field profiles. This is a reflection of our inability to fully control the radial profile of the compensation term. Thus, in this case what we have achieved is C (δ ρ , zFF ) ≈ C1 (δ ρ , zFF ), with zFF the far field plane and δ ρ the radial points close to the center of the beam ρ = 0. However, in general C (ρ , z) = C1 (ρ , z). Even though for most applications this can be enough to probe the local properties of the higher order optical vortex, in the next section we provide a method to allow a better control of the radial profile of the compensating mode. 6.
Near-field compensation
While we have demonstrated control of the singularities of an optical field, reforming the split singularities in the far-field, this is apparently achieved at the cost of separating the nulls of the beam in the near-field. In the paraxial regime, the far-field intensity distribution F, is related to the near-field f , by a Fourier transform (and vice versa): F(kx , ky ) =
∞ ∞ −∞ −∞
f (x, y)ei(kx x+ky y) dxdy
(7)
Considering the centre of the beam at F(0, 0), Eq. (7) becomes: F(0, 0) =
∞ ∞ −∞ −∞
f (x, y)dxdy
(8)
which is the average of the field f (x, y). Thus by modulating the field f (x, y) such that it’s mean is zero, we can create a null at the origin F(kx , ky ). By manipulating the amplitude of the field such that the mean value of the field is zero we can force the center of the beam to have zero or a minimal amplitude to minimize the vortex splitting observed. We implement this by introducing an additional phase ψr (r) to the expression from which we calculate our holograms (Eq. (4)) in order to modulate the field of the compensation ( ) component generated by the SLM: (9) ΨSLM = arg exp(i[φ ]) + α exp(i[ φ + ϑ + ψr (r)]) Where ψr (r) is a step function in the radius of the beam varying from 0 to π radians: 0 if r ≤ r0 ψr = π if r > r0
#206010 - $15.00 USD (C) 2014 OSA
(10)
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9929
This radial phase modulation maintains the relative phase relationship among each term in Eq. (3) in the vicinity of the singularities, and hence preserves the effects of the compensation demonstrated in Section 5. By inverting the sign of the field within a radius of r0 we can find a r0 at which the mean value of the field is zero for the collinear beam. To perform this experimentally, we apply the far-field compensation using the parameters determined in the previous section for the case = 2, splitting the vortex singularities in the near-field. We then modify the hologram according to Eq. (9) and vary r0 , minimizing the separation of the singularities associated with the near-field profile of the beam.
Fig. 7. Far-field (a) and near-field (b) images of a beam with OAM = 2 in the absence of any compensation. Far-field (c) and near-field (d) images of an = 2 beam compensated by the removal of the 1 component by destructive interference with a collinear beam of . Far-field (e) and near-field (f) images of a = 2 beam with the same compensation as applied in (c),(d) but with a radial phase (Eq. (10)) applied to the = 0 compensation component. For the radial phase, r0 = 10 pixels (90 μ m). The result of this additional radial phase is that the separation of the vortices in the near-field has been reduced from (c) to (e). In all images a logarithmic scale is used.
The results of our radial phase modulation for the = 2 beam are described in Fig. 7 and compared with the previously compensated and uncompensated beams = 2 beams from. Figures 7(a) and 7(b) and Figs. 7(c) and 7(d) are control images of the far-field and near-field distribution of nulls without and with the far-field compensation from Figs. 3(c) and 3(g) and Figs. 6(c) and 6(g) respectively. The increase in vortex splitting between Figs. 7(a) and 7(c) (without and with compensation) is visible. From the near-field image, we observe that while there still exists a distortion in the intensity of the near-field (two lobes of higher intensity circling the central nulls), the distance between the nulls has been reduced to distances which appear comparable to the original vortex splitting occurring in the near-field. With this compensation, we are successful in reducing the splitting of the two nulls to below that of the original uncompensated beam in Fig. 7(e), while retaining the central singularity in the far-field.
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9930
7.
Conclusion
We propose a simple method for the compensation of mode impurities in a OAM beam generated by holographic mode conversion. By introducing an additional OAM mode and varying the relative phase and weighting, we can remove the highest order OAM impurity at a single plane by destructive interference. This is performed with a phase-only SLM, removing the intensity distribution associated with interference between multiple OAM modes for = 0 and 1, while collapsing the multiple split singularities associated with interference among higher order modes, = 2, 3. We achieved this by interference with a collinear beam of OAM = − 2. We then applied a phase with a radial step-function between 0 and π , and by varying the radius at which the step occurs, we were able to simultaneously reform the nulls in the near-field while retaining our compensation in the far field for OAM of = 2. Acknowledgments This work was supported by the Australian Research Council (ARC) Future Fellowship (FT1110924) scheme.
#206010 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 6 Apr 2014; accepted 10 Apr 2014; published 17 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009920 | OPTICS EXPRESS 9931
