Compact demultiplexing of optical vortices by means

Compact demultiplexing of optical vortices by means of diffractive transformation optics G. Ruffato*a,b, M. Massaria,b and F. Romanatoa,b,c University of Padova, Department of Physics and Astronomy ‘G. Galilei’, via Marzolo 8, 35131 Padova, Italy; bLaNN, Laboratory for Nanofabrication of Nanodevices, Corso Stati Uniti 4, 35127 Padova, Italy; cCNR-INFM TASC IOM National Laboratory, S.S. 14 Km 163.5, 34012 Basovizza, Trieste, Italy a

*[email protected]

ABSTRACT Orbital angular momentum (OAM) states of light have been recently considered in new mode-division multiplexing techniques in order to increase the bandwidth of today’s optical networks. Many optical architectures have been presented and exploited in order to sort the different OAM channels. Here we present a diffractive version of the sorting technique based on log-pol optical transformation and we further improve the miniaturization level by integrating the two components into a single diffractive optical element. Samples have been fabricated with high-resolution electronbeam lithography and characterized in the optical range. The presented design is promising for integration into nextgeneration optical platforms performing optical processing of OAM modes, for applications both in free-space and optical fibers. Keywords: orbital angular momentum of light, optical vortices, transformation optics, diffractive optics, mode-division multiplexing, electron beam lithography.

1. INTRODUCTION Since the seminal paper of Allen and coworkers in 1992 [1], optical beams carrying orbital angular momentum (OAM) of light have known increasing attention with disruptive applications in a wide range of fields [2]: particle trapping and tweezing [3,4], phase contrast microscopy [5], stimulated emission depletion (STED) microscopy [6], astronomical coronagraphy [7], quantum-key distribution [8] and telecommunications [9, 10]. In this last field in particular, the fact that OAM beams offer a potentially unbounded state space has made them advantageous for increasing the amount of information that can be encoded into a single-photon. Thus the exploitation of this novel degree of freedom offers the possibility to enhance information capacity and spectral efficiency of today’s networks both in the radio and optical domains. A crucial point of an optical link based on OAM-mode division multiplexing (OAM-MDM) is represented by the sorting technique exploited in order to discriminate channels with different OAM content. Several solutions have been presented and described in literature: interferometric methods [11], optical transformations [12-19], time-division technique [20], integrated silicon photonics [21], coherent detection [22], diffractive optics [23-25]. Diffractive optical elements (DOE) appear as the most effective implementation technique for the realization of passive and lossless, compact and cheap optical devices for integrated (de)multiplexing applications [26]. Diffractive OAMmode analyzers have been widely presented for the analysis of OAM beam superposition. Basically, they are calculated as linear superposition of many fork-holograms differing in OAM singularity and carrier spatial frequency [23-25]. The far-field of such optical elements exhibits bright peaks at prescribed positions, whose intensity is proportional to the weight of the corresponding OAM channel in the incident beam. However the efficiency is inversely proportional to the number of states being processed, which makes this solution inefficient especially for single-photon applications. An alternative sorting technique exhibiting higher efficiency is represented by log-pol transformation optics. This system transforms the azimuthal phase gradient of an OAM beam into a linear phase gradient, i.e. a tilted plane wave, which is finally focused on the detector with a lens. Thanks to Fourier transform properties, the linear position of the far-field spot is related to the linear phase gradient, i.e. to the OAM content of the input beam. This procedure requires at least two

Laser Beam Shaping XVII, edited by Andrew Forbes, Todd E. Lizotte, Proc. of SPIE Vol. 9950, 995003 © 2016 SPIE · CCC code: 0277-786X/16/$18 · doi: 10.1117/12.2237765

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optical elements: the unwrapper, performing the conformal mapping which transforms the input OAM ring into a linear intensity distribution, and the phase-corrector, which adjusts the phase delay of different points due to different propagation distances and finally provides a linear phase gradient in output. In its first realization, the two elements were implemented with spatial light modulators [12]. At a later stage [13, 14], they were replaced by two freeform refractive optical components, exhibiting higher efficiency, though not a small size. More recently, the two optical elements have been realized in a diffractive version [19]. In this work, we apply this demultiplexing technique for the OAM sorting of perfect vortices and we further improve the miniaturization level by integrating the two components into a single optical device. Moreover, the integration simplifies the alignment procedure since the two optical elements are centered and parallel to each other by design.

2. THEORY AND DESIGN The conversion of OAM states into transverse momentum states can be performed through a log-pol optical transformation by using a sequence of two optical elements (Figure 1): the unwrapper and the phase-corrector. The former performs the conformal mapping of a position (x, y) in the input plane to a position (u,v) in the output plane, where v=a arctan(y/x) and u=-a ln(r/b), being r=(x2+y2)1/2, a and b design parameters. Its phase function Ω1 is as it follows:

1  x, y  

 x2  y 2 2 a  y  y arctan    x ln    f1  b x  

 x2  y 2    x  2a   

(1)

where a and b are two free parameters which determine the scaling and position of the transformed beam respectively. The parameter a takes the value d/2π, ensuring that the azimuthal angle range (0, 2π) is mapped onto the full width d of the second element. The parameter b is optimized for the particular physical dimensions of the sorter and can be chosen independently. The phase-corrector phase function Ω2 is given by:

2  u, v   

2 ab  u v exp    cos     u   v  f1  a a

(2)

where (, β) are fixed carrier spatial frequencies which control the position of the signal on the screen in far-field. Then a lens with focal length f2 is inserted after the phase-corrector element in order to focus the transformed beam onto a specified lateral position, which moves proportionally to the OAM content ℓ according to:

s 

f2 2 a

(3)

Alternatively, the focusing quadratic term can be integrated in the phase-corrector element as well. The same setup has been demonstrated to work as multiplexer, in reverse [17, 18]. azimuthal phase gradient

U

1

I I I

I

I I

input OAM beam

linear phase gradient

OAM sorter

I

unwrapper

phase -corrector

V al lens

far -field

Figure 1. Scheme of classical OAM sorting based on log-pol transformation optics. The input azimuthal phase gradient is transformed into a linear phase gradient with a sequence of two optical elements, i.e. unwrapper and phase corrector. Then the beam is Fourier-transformed by a lens and mapped onto a liner array in far-field.


The stable propagation of OAM modes along ring fibers has highlighted the need to excite and manipulate annular intensity distributions with fixed radius and width, regardless of their OAM content [27, 28]. However, conventionallygenerated OAM beams, e.g. Laguerre-Gaussian or Kummer beams, exhibit a ring diameter that increases with the topological charge and therefore are rather incompatible with applications involving ring fiber cross-sections or limitedsize optical elements. The so-called perfect vortices appear as the best solution in order to excite optical vortices whose ring dimensions can be adjusted independently of the OAM content [29, 30]. These beams allow the transportation of a helical phase-front, together with the confinement of the electromagnetic field within a ring of controlled radius and width. The corresponding annular field profile Eℓ at a fixed propagation distance and for a specific OAM ℓ can be approximated by:

  r  RV 2  E  r ,   ei  exp    RV2  

(4)

where (r, ϑ) are polar coordinates, RV and ΔRV define radius and width of the intensity annulus respectively. In our case of interest, the choice of perfect vortices for OAM encoding leaves the central region of the optical element unemployed, since the device acts merely on the zone with non-null incident field. Therefore, the non-illuminated central region can be exploited for integrating the second element designed for phase correction. In this novel configuration, after passing the outer unwrapping zone the beam is back-reflected by a mirror and goes through the optical element again across the inner central region for phase correction (Figure 2). Moreover, the optical device is realized as a diffractive optics, allowing further miniaturization especially if a shorter focus length is required. The whole DOE phase pattern Ω DOE is the composition of the two optical elements, according to:

DOE  1    1   2   1   

(5)

being ρ1 the radius of the central zone, Θ the Heaviside function (Θ(x)=1 for x>0, Θ(x)=0 otherwise), provided the condition ρ1>πa is satisfied. In addition, this solution significantly simplifies alignment operations, since unwrapper and phase-corrector result automatically centered and parallel to each other by design.

inner zone

input OAM beam

(2nd

incidence)

phase corrector

mirror

outer zone

DOE

(15t

incidence)

unwrapper

Figure 2. Scheme of the DOE working principle. The input optical vortex illuminates the outer unwrapping region, the beam is then back-reflected by a mirror and crosses the optical element again going through the inner central region which performs the phase correction.

3. FABRICATION Phase-only diffractive optical elements are fabricated as surface-relief patterns of pixels on a transparent substrate. This 3-D structure can be realized by shaping a layer of transparent material, imposing a direct proportionality between the thickness of the material and the phase delay of each pixel. Electron beam lithography (EBL) is the ideal technique in order to fabricate 3D profiles with high resolution [31-33]. By modulating the local dose distribution, a different dissolution rate in the exposed polymer is induced, giving rise to different resist thicknesses after the development process. Here, the DOE patterns were written on a polymethylmethacrylate (PMMA) resist layer with a JBX-6300FS JEOL EBL machine, 12 MHz, 5 nm resolution, working at 100 keV with a current of 100 pA. The substrate used for the fabrication is glass, coated with an ITO layer with conductivity of 8-12 Ω in order to ensure both transparency and a


good discharge during the exposure. Patterned samples were developed under slight agitation in a temperature-controlled developer bath for 60 s in a solution of deionized water: isopropyl alcohol (IPA) 3:7. For the considered wavelength of the laser beam used (λ=632.8nm), PMMA refractive index results nPMMA=1.489 from spectroscopic ellipsometry analysis (J.A. Woollam VASE, 0.3 nm spectral resolution, 0.005° angular resolution). The height dk of the pixel belonging to the kth layer is given by:

dk 

k 1  k  1, N nPMMA  1

, N 1

(6)

being N the number of phase levels. The fabricated DOE is a 1024 x 1024 matrix of pixels with N=64 phase levels (Figure 3), pixel side 1.562 nm. For the given laser wavelength and PMMA refractive index, we get: d1=0 nm, d64=1273.8 nm, step Δd=20.2 nm. (a)

50 µm

200 µm

50 µm

Figure 3. DOE samples for integrated transformation optics. 1024 x 1024 pixels, pixel area 1.562 x 1.562 μm2. (a) Focusing outer zone, parameters: a=70 μm, b=50 μm, f1=20 mm. (b) Focusing outer zone, carrier spatial frequency on the inner zone for image tilt, parameters: a=70 μm, b=50 μm, f1=20 mm, =β= 0.1 μm-1. (c-e) Details of sample in (b).

4. OPTICAL CHARACTERIZATION The characterization setup was mounted on an optical table (see scheme in figure 4). The Gaussian beam (λ= 632.8 nm, beam waist w0=240 μm, power 0.8 mW) emitted by a HeNe laser source (HNR008R, Thorlabs) is linearly polarized before illuminating the display of a LCoS spatial light modulator (PLUTO-NIR-010-A, Holoeye, 1920 x 1080 pixels, 8 m pixel size, 8-bit depth) for optical vortex generation. The reflected beam is collimated with a first lens of focal length f0=25 cm and a beam-splitter is used to analyze the field profile and its OAM content at the same time. The field profile is collected with a CCD camera (DCC1545M, Thorlabs, 1280x1024 pixels, 5.2 μm pixel size, monochrome, 8-bit depth). The beam illuminates the first outer zone of the DOE sample, mounted on a XY translation sample holder with micrometric drives. A mirror is placed on a micrometric translator (TADC-651, Optosigma) for back reflection. After passing through the DOE inner zone, the far-field is collected by a second CCD camera (1500M-GE, Thorlabs, 1392x1040 pixels, 6.45 μm pixel size, monochrome, 12-bit depth) placed at the back-focal plane of a lens of focal length f2=12.5 cm.


Figure 4. Scheme of the experimental setup. The laser beam is linearly polarized and illuminates a SLM display for perfect vortex generation. A first lens (L1) collimates the beam which is split for OAM beam analysis with the first camera (CCD1). The latter part of the beam illuminates the DOE mounted on a sample holder. The optical vortex impinges the DOE outer zone (unwrapper) and is reflected back with a mirror (M) upon the DOE inner region (phase corrector). The second lens (L2) performs Fourier-transform and the far-field is collected on the second CCD camera (CCD2).

The SLM implements a phase mask ΩSLM combining axicon and spiral phase functions with a quadratic-phase term for curvature correction:

SLM  r ,       r  k

r2 2R

(7)

where γ is the axicon parameter, k is the wavevector in air, and R is the curvature radius of the incident beam. If the SLM is located at a distance z from the laser source, the curvature term can be estimated as R=(z2+zR2)/z, being zR the Rayleigh range. At the back focal plane of the first lens, the axicon function forms a ring having the diameter RV, varying with the axicon parameter and the focal length f0 according to:

RV 

 f0 k2  2

.

(8)

The spiral phase term gives the vortex nature with topological charge ℓ to the ring beam. The ring-width ΔRV can be controlled by changing the incident beam radius w, according to ΔRV=2f0/(kw). In our case of interest, the ring width results around ΔRV=56 μm. For the axicon parameter γ =0.019μm-1, we get RV=480 μm. There is a crucial point to be considered during miniaturization. The sorting element is designed to work perfectly for collimated beams, in which all light rays arrive at normal incidence at the first unwrapping component. For noncollimated beams it is required that the angular deviation introduced by the unwrapper dominates over any deviation from normal incidence with which rays arrive at the first element. If the input vortex is a ring with radius RV and the separation between the two elements is f, the angular deviation introduced by the first element is approximately RV/f for small angles. Irrespective of their divergence, OAM beams are never plane waves, as there is an azimuthal component of the Poynting vector whose local skew angle is around ℓ/(kRV). It follows that for the sorter to work properly with beams of radius RV, it is required that ℓ/(kRV)