Controlling orbital angular momentum using forked polarization gratings

Transcription

1 Controlling orbital angular momentum using forked polarization gratings Yanming Li, Jihwan Kim and Michael J. Escuti Dept. Electrical and Computer Engineering, North Carolina University, Raleigh, NC USA ABSTRACT We examine a novel method to control the orbital angular momentum (OAM) of lightwaves using forked polarization gratings (FPGs). We significantly improve the fabrication of FPGs and achieve substantially higher quality and efficiency than prior work. This is obtained by recording the hologram of two orthogonally polarized beams with phase singularities introduced by q-plates. As a single compact thin-film optical element, an FPG can control the OAM state of light with higher efficiency and better flexibility than current methods, which usually involve many bulky optical elements and are limited to lower OAM states. Our simulations confirm that FPGs work as polarization-controlled OAM state ladder operators that raise or lower the OAM states (charge l) of incident lightwaves, by the topological charge (l g ) on the FPGs, to new OAM states (charge l ± l g ). This feature allows us to generate, detect, and modify the OAM state with an arbitrary and controllable charge. An important application of FPGs are the essential state controlling elements in quantum systems based on OAM eigenstates, which may enable extreme high capacity quantum computation and communication. Keywords: orbital angular momentum, polarization gratings 1. INTRODUCTION It has long been recognized that lightwaves carry spin angular momentum (SAM) that corresponds to the polarization states of the photons. During the last two decades, research has revealed that lightwaves can carry orbital angular momentum (OAM) as well, which depends on the characteristics of the phase distribution of the lightwaves. 1 It was first suggested by Allen s group and demonstrated by experiments, that the lightwaves with azimuthal angle-dependent phase term exp(ilφ) (l integer) carry OAM of l per photon. Due to the phase singularity at their center, OAM lightbeams have many novel properties, including helical phase front, annular intensity profile, and spiral phase distribution at the cross section. 2 The most popular lightbeams with OAM are Laguerre-Gaussian beams and high order Bessel beams. This property of lightwaves deepens our understanding of light and broadens the application range. The main prospective utility includes particle manipulation 3 and quantum communication, 4 6 however, the generation, manipulation, and detection of the OAM states is still fairly challenging in most cases. The OAM property of light offers us an extra degree of freedom when dealing with light beams. Unlike spin, which has only two eigenstates (up and down, with corresponding eigenvalues l and l ), OAM has theoretically infinite eigenstates denoted by singularity of phase distribution l (also called topological charge), each with corresponding eigenvalue l. Thus, whereas the base-2 quantum system run with qubits, this base-d system can run with qudits, which allows much higher capacity of quantum computation and communication. 6 However, the idea of quantum communication using OAM of light still faces challenges. A prime one being that no method has yet been found that can generate and distinguish the OAM states efficiently using compact optical element(s). Polarization gratings (PGs) are a category of diffraction gratings that are formed in anisotropic materials, and function by periodically affecting the polarization state of the wavefront passing through them, often via the Pancharatnam-Berry phase effect. 7 9 In one sub-class, researchers have fabricated very high quality PGs by spatially aligning birefringent (liquid crystal) materials These PGs are thin and lightweight, operate with extremely high efficiency, and can be made either fixed or switchable. Because of these features, PGs Further author information: Send correspondence to Michael Escuti, mjescuti@ncsu.edu Laser Beam Shaping XI, edited by Andrew Forbes, Todd E. Lizotte, Proc. of SPIE Vol. 7789, 77890F 2010 SPIE CCC code: X/10/$18 doi: / Proc. of SPIE Vol F-1

2 are recognized as having the potential as good optical OAM operators. Choi and coworkers have investigated two possible PG structures as OAM beam generators and have shown mixed experiment results. 16 Thefirstis a reflective, reverse twisted nematic PG, which diffracts a plane wave into multiple orders with different OAM charges. This structure is polarization-insensitive, but since the input power is distributed into infinite diffraction orders, this configuration has limited relevance in applications. The second configuration is a transmissive PG made by two adjacent diffraction orders from a Computer Generated Hologram (CGH) (details in Sec. 2.3). They demonstrated that the samples they made can convert the plane wave into an OAM beam, and the OAM mode is sensitive to the polarization handedness. However, they omit the important discussion of how their structure would modify a lightwave that is already carrying OAM. Moreover, their reported data suggests very low diffraction efficiency and high scattering losses. We believe that there is room to improve; with our method and fabrication technology, we aim to find an optimum FPG (Forked Polarization Grating) with the potential to achieve OAM modulation with 100% efficiency, by polarization-controlled diffraction into a single order. The object of our work is to explore a new method to fabricate high quality FPGs, and use FPGs to modify the OAM states of lightwaves. Fig. 1 shows the schematic function of a FPG. It can either raise or lower the OAM charge of the input lightwave according to its polarization state. In the fabrication process, we utilize q-plates (details in Sec. 2.2) as the essential elements to introduce singularities into lightwaves. Due to its crucial role in this method, we will discuss q-plate fabrication as well. Figure 1. A charge l = +2 OAM beam propagating through a charge l g = +1 FPG is diffracted to different direction according to polarization. The +1st order diffraction is right circular polarized, with OAM charge l = +1; whereas the -1st order diffraction is left circular polarized, with OAM charge l = BACKGROUND 2.1 Polarization Gratings (PG) Polarization gratings (PGs) locally modify the polarization state of transmitted light, which is achieved by spatially varying birefringence and/or dichroism. Among many possible types and creation methods, in this work, we utilize the circular PGs created by polarization holography and recorded on polymer or LC materials. 17 The local optical axis orientation at a point (x, y, z) in a PG follows α = πx/λ+α 0. This forms a periodic structure in the x dimension, which is homogeneous in the other two dimensions. In our work, we use nematic LC materials with positive birefringence, and take α 0 = π/2, as shown in Fig. 2(a,b). The the transfer matrix is Proc. of SPIE Vol F-2

3 (a) y 0 0.5Λ Λ 1.5Λ 2Λ x Off-State (b) glass ITO (c) z x Nematic LC glass On-State photoalignment layers ITO (d) (e) (f) Off-State m=-1 m=+1 On-State d V>V th y x z m=0 = Right Handed Circular Polarization (RCP), = Left Handed Circular Polarization (LCP) Figure 2. Structure and polarization controlled diffraction of a switchable PG: OFF-state (a) periodic structure in x- dimension, (b) top view, (d,e) polarization-dependent diffraction; and ON-state (c) structure, and (f) transmission. then T ( x) =cos ζ [ ] cos 2α sin 2α 2 I+isinζ =cos ζ [ ] cos 2πx 2 sin 2α cos 2α 2 I+isinζ Λ sin 2πx Λ 2 sin 2πx Λ cos 2πx (1) Λ where ζ is the relative phase retardation due to the birefringence. The electric field of the m th order diffraction is D m = 1 Λ T (x)e in e i 2π Λ mx dx (2) Λ 0 D m =Γ m E in, Γ m = 1 T (x)e i 2π Λ mx dx (3) Λ 0 [ ] Doing the integral in Equation (3), we get Γ 0 =cosζi, Γ ±1 = 1 2 sin ζ i 1 2,andΓ 1 i m =0(m 0, ±1). Thus, a PG has only three orders of diffraction, and the relative intensity depends on the polarization of the input lightwave. If we carefully select the PG thickness so that the effective retardation is half wave, then 100% diffraction efficiency into the first order(s) can be achieved, as shown in Fig. 2(d,e). Particularly, liquid crystal PGs can be made switchable by external applied voltage (Fig. 2(c,f)). 2.2 Q-plates A q-plate is a birefringent plate that introduces a homogeneous half-wave retardation to the light propagating through, with inhomogeneous optical axis in the transverse plane (examples shown in Fig. 3). It can work as an OAM generator for circular polarized light. The local in-plane optical axis follows α = qφ + α 0, under half-wave retardation condition, omitting the imaginary unit, [ ] cos 2α sin 2α T ( x) = (4) sin 2α cos 2α ( ) 1 for instance, if incident light is right/left circular polarized (LCP/RCP) E in = E 0, then output is ±i ( ) E out = T ( x)e in = E 0 e i2α 1 (5) i Λ Proc. of SPIE Vol F-3

4 Figure 3. Anisotropic structure of q-plates: (a)q =+1/2; (b)q =+1;(c)q = +2. Arrows illustrate the space-varying in-plane local optical axis. Figure 4. Forked CGHs with modulation represented by grayscale: (a) charge l g = 1; (b) charge l g =2. which is right/left circular polarized (RCP/LCP) wave with an azimuthal-angle-dependent relative phase retardation e i2α = e i2qφ. That is, a q-charge q-plate can introduce a charge ±2q phase singularity to the beam propagating through it. 2.3 Forked Conventional (Isotropic) Gratings A widely studied optical element to generate a helical wavefront is an isotropic diffraction grating with singularities. These gratings are special isotropic gratings with well-designed fork-shape dislocations at their center (see examples in Fig. 4). The forked patterns are usually calculated by computer and then made as Computer Generated Holograms (CGHs) with either transmittance modulation 18, 19 or blazed phase modulation. 20 These forked CGHs can be characterized by their topological charge at the dislocation. If a single grating stripe splits into l g +1 branches, then we call it a charge l g grating. (in Fig. 4, (a) l g =+1,(b)l g =+2). A CGH diffracts the input lightbeam into infinite orders of diffraction as a conventional diffraction grating does; and the m th order diffracted beam acquires a phase singularity of charge l = m l g. Here, both l g and m can be positive or negative. 3. FORKED POLARIZATION (ANISOTROPIC) GRATING Whereas a normal 1D PG has periodic anisotropy along one dimension and is uniform along the others, the forked PG we utilize in this work has similar properties, but has a fork-shape dislocation at the center, like the Forked CGHs. With both periodic anisotropy over one dimension and a singularity at the center, FPGs are like a combination of PGs and q-plates, or a polarization-modulating version of Forked CGHs. Thus, it has the potential to achieve polarization controlled OAM state operation. The local optical axis orientation can be expressed by α = πx/λ+qφ + π/2 = πx/λ+ 1 2 lφ + π/2, q= 1 2 l. Because a q-plate modifies the OAM state by ±2q, a±l ladder operator can be expressed by a term of 1 2 lφ. Fig. 5 shows two examples of FPGs with arrows indicating the local optical axis, which is also the local nematic director in liquid crystal FPGs. Proc. of SPIE Vol F-4

5 Figure 5. Anisotropic structure of (a) charge l g = +1 FPG and (b) charge l g = +2 FPG. Arrows illustrate the local optical axis at each spacial point. In a half-wave retardation condition, we notate γ = α, then [ ] cos γ sin γ ˆT ( x) = = 1 sin γ cos γ 2 eiγ Ŝ e iγ Ŝ + (6) where Ŝ± are ladder operators, Ŝ ± χ ( ) = 2 χ (±), χ (±) are the two spin eigenstates, corresponding to right/left circular polarization states. This is essentially the same as Eq. (4) for q-plates; writing them into the operator form allows us to observe its function more clearly. If the incident light is RCP/LCP, with singularity charge l 0, E in = e il0φ χ (±), then the electric field immediately after the FPG is, ˆT ( x)e il0φ χ (±) = e i(l0±l)φ e i 2π Λ x χ ( ) (7) For low charge, the FPG still has an overall 1D periodic anisotropy structure; the singularity has little effect on diffraction behavior. So we can use the PG formula, assuming light diffracted to m th -orders, then D m = C e i(l0±l)φ e i 2π Λ x e i 2π Λ mx da χ ( ) (8) doing the integral, we get D m = C δ m±1 e i(l0±l)φ χ ( ) (9) that is, if the input light is RCP, then the diffraction has non-zero power for only m = 1 order, and the diffracted beam is LCP, with OAM charge l out = l 0 + l ; if the input light is LCP, then the diffraction has non-zero power for only m = +1 order, and the diffracted beam is RCP, with OAM charge l out = l 0 l. Since FPGs have the same advantages as PGs they are thin and can be made switchable we can stack switchable FPGs with different charges, so that the complete set is able to modify light OAM state within a large range by applied voltages. Furthermore, if we stack one more fixed 1D PG, the set can modify OAM states solely without changing the propagation direction or polarization handedness. Proc. of SPIE Vol F-5

6 Figure 6. Intensity profile (top row) and phase distribution (bottom row) of the output light from q-plates: (a,d) q =+1/2, (b,e) q = +2, and (c,f) q = +3. input: Gaussian beam, wavelength: 325nm, beam width: 6mm, propagation distance: 20cm. 4. SIMULATIONS Our simulations are based on spatial-varying Jones Matrix calculation and discrete Fourier Transforms. Both intensity and phase spatial distributions are acquired. 4.1 Q-plates introduce singularity into lightwaves We simulated Gaussian beams with various waist propagating through a q-plate and plotted the intensity and phase distribution at several distances after the q-plate. As shown in Fig. 6, the singularity introduced by the q-plate is obvious in the phase distribution plots, but unnoticeable in the intensity profiles for large beams within a reasonable lab-scale propagation distance. This result suggests that for our laser beam ( 2cm width) used in experiments, we can adjust the q-plate position so that the intensity at the sample plane is uniform overall. Thus, the null range in the resultant grating is minimized. This is also an advantage of q-plates as singularity generators over CGHs. Because CGHs have multiple diffraction orders, to single out a specific order we need either optical element(s) or finite propagation distance. This limits the minimum distance between the CGH and the sample. After a certain propagation distance, the OAM beam becomes doughnut-shaped with a relative large zero-intensity center, which leads to poor alignment of the sample. 4.2 Recording FPGs by lightbeams with OAM The superposition of two orthogonal circular polarized lightwaves forms a PG. If one or both of them carry OAM, then it results in a FPG. The charge of the FPG is the difference between the OAM charges of the two writing beams, that is, l g = l 1 l 2. This rule is verified by our simulations, and makes our fabrication method flexible: a charge l g FPG can be written either by two charge ±l g /2 lightwaves (requiring two l g /4 q-plates), or achargel g lightwave (requiring one l g /2 q-plate) and a normal Gaussian beam or plane wave (special case of OAM charge l =0). 4.3 FPG control of input lightbeam The diffraction of a FPG is polarization sensitive, just as a 1D PG. Under half-wave retardation condition, the zero-order which would be present at the center is suppressed; all input light is diffracted into one or both of the the first orders according to its polarization. The OAM charges of these two beams are dependent on the OAM charge l in of incident wave and the topological charge l g of the FPG. As shown in Fig. 7, for the m =+1order diffraction, OAM charge l +1 = l in l g ;forthem = 1 order diffraction, OAM charge l 1 = l in + l g. Here, all of l g,l in,andl ±1 can be positive or negative. Proc. of SPIE Vol F-6

7 Figure 7. Simulated diffraction from FPG, showing (a-i) far field intensity and (j-l) phase at beam cross-section. Left column: charge l g =+1FPG,chargel = 0 input light (Gaussian); center columns: charge l g =+2FPG,chargel =0 input light (Gaussian); right column: charge l g =+2FPG,chargel = 1 input light. Input lightwave polarization: top row: linear, second row: left circular, third row: right circular. Bottom row shows the phase distributions of each diffracted beam above them. 5. EXPERIMENTS 5.1 Introduction We fabricate the FPGs by writing a polarization hologram into photo-alignment materials: Two coherent beams with orthogonal circular polarization from a HeCd laser (325nm) superpose with a small angle between them, creates a spatial polarization modulation with constant intensity. We inscribe this modulation into a thin layer of photo-sensitive polymer on a glass substrate, which aligns to the local electric field direction of the UV laser. Then we spin-coat a layer of reactive mesogens on it, which will polymerize following the alignment after curing. This processing is based on the procedure detailed in Ref. 17 To record FPGs, we make the two writing beams now carry OAM charges, which are introduced by the q-plate(s) in the optical path(s). Fig. 8(a) illustrates exposure, the key step in FPG fabrication. In the figure, the photo-sensitive polymer will align to form a FPG with charge l g =2 q. Note that the two writing beams are from the same laser and are separated by beam splitter with equal power. The HWP could be spared if the two beams are originally set to the same handedness polarization (Fig. 8(b)). It also could be substituted by another q-plate, in which case the result FPG will have charge l g =2 q 1 2 q 2,whereq 1, q 2 are the charges of the two q-plates. 5.2 Q-plate fabrication Since the singularities of the beams that write the FPGs are introduced by q-plates, the quality of the FPGs are highly dependent on the quality of the q-plates. We use the same materials described in Sec. 5.1 to record the space-varying anisotropy. The azimuthal-varying photo-alignment is realized by relative rotation between the sample and the light polarization orientation. 21, 22 The linear polarized lightbeam from HeCd laser is focused by a cylindrical lens into a slim strip on the sample (as shown in Fig. 9). By continuously rotating the cylindrical lens and the substrate at respective speeds during the exposure, we can modulate the local photo-alignment Proc. of SPIE Vol F-7

8 M (a) (b) BS UV LASER l=0 l=2q LPP Substrate q-plate HWP QWP l=0 l=0 M QWP q-plate sample = left circular polarization; = right circular polarization; HWP = half-wave plate; LPP = linear photopolymerizable polymer. BS = beam splitter; M = mirror; QWP = quater-wave plate Figure 8. Fabrication of FPG: (a) thin layer of photo-sensitive polymer is exposed to UV hologram wrote by orthogonal circular polarized lights with OAM charge, which in introduced by q-plates; (b) optical setup. Figure 9. Photograph of the optical setup to fabricate q-plates. direction on the substrate as a function of azimuthal location, following Eq. (4). A specific charge q-plate can be fabricated by setting the rotations according to q = ωs /(ωs ωl ), where ωs and ωl are the rotation speeds of the sample and the cylindrical lens, respectively. Some samples we made are shown in Fig. 10. Because they are essentially wave plates with space-varying optical axis, they are best viewed between two crossed polarizers with a backlight. Note that the intensity modulations in Fig. 10 are as expected for each q-plate. Every darkbright-dark transition indicates a π/2 variation of local optical axis, thus, the charge 0.5 q-plate has two intensity modulations, the charge 1 has four, and the same for the others. The quality of the q-plates fabricated by this method is mainly limited by the Rayleigh limit of the beam strip. The narrower beam strip we use, the finer modulation is inscribed. To lower this limit, we could optically tune the lightwave, such as optimizing beam divergence. Besides that, we propose an iterative method to fabricate high-charge q-plates. Assuming we have fabricated a high quality charge q plate, then a linear polarized lightwave propagating through it will acquire a space-varying rotation of polarization orientation, and become a radial or azimuthal beam. That is, the output lightwave is still linear polarized, but with a continuously varying orientation with respect to azimuthal location. If we record this variation by our photo-alignment substrate, Proc. of SPIE Vol F-8

9 (a) (b) (c) (d) (e) (f) (g) (h) Figure 10. Sample q-plates viewed between crossed polarizer. Charges from left to right, top to bottom: 0.5, 1, 2, 4; 8, 16, 32, 64. Figure 11. Polarizing optical microscope images (w/ quarter-wave plate) of fabricated FPGs: (a) charge lg = +1 fork, (b) charge lg = +2 fork, and (c) area far away from the singularity. it will result in a charge 2q plate. Iterating this recording, we can finally fabricate a very-high charge q-plate. Fig. 10(d-h) are sample q-plates made by this method. By making every iteration high quality, this method is capable of very high orders q-plate fabrication, which is only limited by the finest resolution of the materials. 5.3 FPG fabrication and characterization We adapt the PG fabrication setup described in Sec. 5.1 for FPG fabrication, by inserting a q-plate into the optical path of one writing beam (as shown in Fig. 8(b)). To avoid the low intensity center caused by the OAM singularity during light propagation, we minimize the distance between the q-plate and the sample. This result in a uniform intensity distribution to align the photo-sensitive polymer, leading to good alignment over the whole sample. This is crucial to FPG quality, since the center area is where the fork-shape singularity will be formed. Fig. 11 shows the polarization-microscope pictures of two FPGs. We made these two FPGs by using the q-plates shown in Fig. 10(a and b), respectively. As derived in Eq. (5), charge 0.5 q-plate introduces charge l = +1 OAM to the beam. Together with the other Gaussian beam (equivalent to OAM charge l = 0), it writes a charge lg = +1 FPG (Fig. 11(a)). In the case of charge 1 q-plate, the output beam carries charge l = +2 OAM, thus the resultant FPG has charge lg = +2 (Fig. 11(b)). In both cases, the pictures are centered at the singularity to show the variation. In sample areas other than the center, the alignment is uniform in one dimension with a period of 16 microns in the other dimension, just like a PG (shown in Fig. 11(c)). This is coherent with both theory and simulation. Proc. of SPIE Vol F-9

10 Figure 12. Experimental results of FPG diffraction: Measured intrinsic diffraction efficiency (I m/ I m) for input light with: (a) linear or unpolarized, (b) left circular, and (c) right circular polarization; images on top show the captured far field intensity distribution in each case. (d) a comprehensive view of the polarization-sensitive diffraction of FPGs. We began characterization by examining the diffractive properties of the experimental FPGs with a laser. We used a collimated beam from HeCd laser (UV, 325nm) as the normally incident light, and received the diffraction from the FPG by a fluorescence screen in the far field region. Images on the screen are then captured and shown in Fig. 12 (top row). Most transmitted light ( 94.5%) is diffracted to the lowest three orders, among which the zeroth order was much weaker than the first orders. Less than 1% of the light goes to higher orders, the remaining power ( 4.5%) is lost by scattering. The power distribution between the two first orders is dependent on the polarization of the incident light (Fig. 12). Right circular polarized light is mostly diffracted to the 1st order, whereas left circular polarized light is diffracted to the -1st order. When the incident light is linear polarized or unpolarized, which is equivalent to equally composed right and left circular polarization, the total power is equally distributed into the two first orders. Thus, the FPGs are demonstrated to have the same polarization sensitivity as PGs, as predicted by theory and simulation. Examining each of the diffracted beam, the zeroth order is weak but uniform in intensity, whereas for the two first orders, doughnut-shape intensity profiles are pronounced. The dark center of the beam at far field indicates the singularity due to the OAM charge the beam is carrying. To examine the charge of each individual diffracted beam, we set up a Mach-Zehnder interferometer with a HeNe laser (633nm) and captured the far field interference pattern. The objective beam is the diffraction from FPG, the reference beam is a Gaussian wave. Since the sample is optimized for a UV laser, it has obvious zeroth order diffraction with the HeNe laser. Fig. 13 shows the results for a OAM charge l = +1 input light and a charge l g = +2 FPG. The interference patterns show that the three orders of diffraction have different OAM charges. The 1st order (Fig. 13(c)) for example, the three-bifurcation-fork pattern indicates that the objective beam has a 3 2π phase variation in the azimuthal direction. Thus, the beam carries OAM of charge l =+3. In Proc. of SPIE Vol F-10

11 Figure 13. Interference pattern of the diffraction from a charge lg = +2 FPG and charge l = +1 input. The three orders of diffraction have different OAM charge: (a) -1st, charge l = 1; (b) 0th, charge l = +1; and (c) 1st, charge l = +3. Incident lightwave is linear polarized; the reference lightwave is a Gaussian beam. the same manner, -1st and 0th orders have OAM charge l = 1 and l = +1, respectively. This result perfectly matches the theory (Eq. 9) and simulation. 5.4 Discussion FPGs with good quality have been fabricated. Two key points make our fabrication approach successful. The first one is adapting the classic PG fabrication method for FPGs. As a result, our FPGs keep all of the good diffraction properties of PGs. The second one is using a q-plate to introduce OAM singularity. Unlike other OAM elements, such as forked CGHs, q-plates are transmissive thin wave plates, which can be integrated into the holographic setup with great flexibility and relative simplicity. The resulting FPGs exhibit all the properties predicted by theory and simulation. Power measurements of the diffraction orders showed that our current FPG samples have very good diffraction efficiencies ( 96%). We believe by adjusting the recording setup and fine-tuning the sample processing, it is possible to achieve diffraction efficiency > 99%. This mean that these FPGs could modulate the OAM state of the lightwave with nearly zero power loss. Now that high quality FPGs can be achieved, many configurations of FPGs with other elements can be investigated; for example, a 1D PG could be stacked with an FPG to compensate the polarization handedness and diffraction angle. To fully study the properties and potential of FPGs, further characterization must be done, such as diffracted beam quality analysis, mode purity detection, among others. 6. CONCLUSIONS We have proposed a polarization-controlled OAM state modulation method based on FPGs and have successful fabricated the two key optical elements: the q-plate and the FPG. The q-plates generate OAM singularity to circular polarized lightwave and offer high-quality writing beams for the fabrication of FPGs. We demonstrated that with our polarization hologram setup and photo-alignment technique we can make high-quality FPGs that modulate the light OAM state highly efficiently. Moreover, FPGs can be stacked or combined with other thin-film optical elements, such as 1D PGs, to achieve more powerful functions. With the comparative advantages, including simple processing, compact size and light weight, FPGs will greatly promote the developing of applications of light OAM, especially in carrying high capacity quantum information. REFERENCES [1] Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C., and Woerdman, J. P., Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes, Physical Review A 45(11), (1992). [2] Padgett, M., Courtial, J., and Allen, L., Light s orbital angular momentum, Physics Today 57(5), (2004). [3] Garces-Chavez, V., McGloin, D., Padgett, M. J., Dultz, W., Schmitzer, H., and Dholakia, K., Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle, Physical Review Letters 91(9), (2003). [4] Knill, E., Laflamme, R., and Milburn, G. J., A scheme for efficient quantum computation with linear optics, Nature 409(6816), (2001). Proc. of SPIE Vol F-11

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