J. Infrared Milli Terahz Waves manuscript No.(will be inserted by the editor)

Q-plate for the generation of terahertz cylindrical vectorbeams fabricated by 3D-printing

A. I. Hernandez-Serrano · E.Castro-Camus · D. Lopez-Mago

Accepted: 05-May-2017, This is a Pre-print. visit http://link.springer.com/journal/10762for the final version

Abstract Accepted for publication at Journal of Infrared Millimeterand teraWe present the design, fabrication and characterization of a q-plate with con-tinuous birefringence variation at terahertz frequencies. This q-plate was fabri-cated by three-dimensional printing and is a simple solution for the generationof cylindrical vector beams. This device can find a number of applications infuture terahertz technologies such as telecommunications.

Keywords Vector Beams · Terahertz · Form birrefringence · Singular optics ·Q-plate

PACS 42.25.Ja · 42.25.Lc · 87.50.U-

1 Introduction

A q-plate is a birefringent wave plate with a space-dependent orientation ofthe ordinary and extraordinary axes in the transverse plane [1,2]. The struc-ture of such axes’ distribution is characterized by a number, known as thetopological charge q. This device is mainly used in optics for the generation ofstructured light, which are, inhomogeneous light beams containing particular

A. I. Hernandez-SerranoCentro de Investigaciones en Optica A.C., Loma del Bosque 115, Lomas del Campestre,Leon, Guanajuato 37150, Mexico

E. Castro-CamusCentro de Investigaciones en Optica A.C., Loma del Bosque 115, Lomas del Campestre,Leon, Guanajuato 37150, Mexico

D. Lopez-MagoPhotonics and Mathematical Optics Group, Tecnologico de Monterrey, Monterrey 64849,MexicoE-mail: dlopezmago@itesm.mx

2 A. I. Hernandez-Serrano et al.

phase or polarization singularities, with interesting properties and applicationsin microscopy, particle manipulation, quantum information processing, amongothers [3–5].

Recently, two exciting technologies are experiencing significant progress:terahertz devices and 3D printers. Terahertz radiation is now used in industrialquality control [6], characterization of materials [7] and free-space communica-tions [8]. A 3D printer has become an apparatus able to rapidly fabricate com-plicated geometrical objects at affordable prices. Fortunately, some polymers,that can be used with a 3D printer, are transparent in the THz band, whichallows the fabrication of THz devices, such as lenses [9,10], waveguides[11,12],and gratings [13,14] and form birefringent wave-plates [15]. Furthermore, aTHz vortex generation using 3D printed devices was demonstrated[16]. Theusual way to generate this THz vectorial beams is using a segmented quartzplate [17].

In this work, we present, to the best of our knowledge, the first 3D-printedcontinuous q-plate for the generation of THz cylindrical vector beams, whichare electromagnetic beams with nonuniform transverse polarization patterns[18] based in the form birefringence effect [19]. These beams can be used, forinstance, to efficiently couple radiation to metallic wave guides [20]. The 3Dprinting material was polystyrene, which is a highly transparent material inthe THz regime. The polarization structure was characterized by using a THztime-domain imaging system (THz-TDI). The results provide a simple andflexible fabrication method for the generation of structured THz beams.

2 Mathematical description

Consider a q-plate having a homogeneous phase retardation of π (half-wave)for light traveling in the longitudinal z direction. The plate has a transverselyinhomogeneous birefringent axes distribution o(x, y), lying in the xy plane.Consider that α(x, y) is the angle between o(x, y) and the x axis. Let α(x, y)be given by the equation [1,2]

α(r, ϕ) = qϕ+ α0, (1)

where r, ϕ are the polar coordinates in the transverse plane, q and α0 areconstants. In order to visualize the effect of this q-plate, consider a horizontallypolarized plane wave Ein = E0x impinging on the q-plate at normal incidence.The Jones matrix of a q-plate is given by

H =

[cos 2α sin 2αsin 2α − cos 2α

], (2)

for the basis of linear polarizations (x, y). In this basis, the q-plate with q = 1/2and α0 = 0 transforms the incident beam into

Eout =

[cosϕ sinϕsinϕ − cosϕ

] [E0

0

]= E0

[cosϕsinϕ

], (3)

Terahertz q-plate fabricated by 3D-printing 3

which can be expressed in the form Eout = E0 cosϕx + E0 sinϕy = E0r.Therefore, the outgoing wave shows a radial polarization distribution. It isstraightforward to show that if the incident field is vertically polarized, theoutgoing wave will show an azimuthal polarization mode Eout = E0ϕ.

Beams with radial or azimuthal polarization distribution are of interest insingular optics and find applications in tight focusing and waveguide coupling.Their intensity pattern has characteristic annular shape, showing a centralsingularity in the electric field distribution.

The use of 3D-printing opens the possibility to create complex terahertzquasi-optical components that incorporate form-induced birefringence [21].Most q-plates reported to date divide the area of the plate in a finite numberof sections. In each section a birefringent material or structure is placed at agiven orientation, this way the polarization of the beam propagating throughthe q-plate is modified in each section, producing an electromagnetic modewhich has a polarization that is a discretized version of the desired mode. Inour case, the possibility of introducing form-birefringence on the 3D-printedcomponent, allows us to continuously vary the direction of the birefringence.The so-called streamlines of the q-plate define the local direction of the slow orfast axis (there is only a global phase difference between the resulting beamsof both cases).

For a q-plate with q = 1/2 and α0 = 0 the optical axis α is given byα = ϕ/2 where ϕ is the polar angle in cylindrical coordinates. We can definea unit vector field describing the optical axis distribution as

u = cos(ϕ/2)x+ sin(ϕ/2)y. (4)

In order to obtain the streamlines generated by this vector field, we have tosolve the differential equation y′(x) = ux/uy, where (ux, uy) are the Cartesiancomponents of u. Using Eq. (4) we have

dy

dx=

uy

ux=

sin(ϕ/2)

cos(ϕ/2)= tan(ϕ/2). (5)

We use the identity tan(ϕ/2) = sinϕ/(1 + cosϕ) and change to Cartesiancoordinates with sinϕ = y/(x2 + y2)1/2 and cosϕ = x/(x2 + y2)1/2. Theresulting differential equation is

dy

dx=

y√x2 + y2 + x

, (6)

which has solutions of the form

y(x) = ±eC/2√eC + 2x, (7)

where C is a real constant that characterizes a particular streamline. With theinitial condition y(0) = y0, we obtain

y(x) = ±√y0√y0 + 2x. (8)

Equation (8) is the one that we use for the fabrication of the q-plate. Theresulting pattern is shown in Fig. 1(a).

4 A. I. Hernandez-Serrano et al.

Optical Table

y-stage

x-stage

Receiver Emitter

Rotatory

holder

Q-plate

(a) (b)

(c)

(d)

y

Fig. 1 (color online) Fabrication and experimental setup. (a) Theoretical q-plate pattern.The tangent to the streamlines indicates the direction of the optical axis. (b) Fabricated q-plate using a 3D printer. The plate has transverse dimensions 40×40 mm2 with a thicknessof 5mm. Figures (c) and (d) show the actual experimental setup. The operating frequencyis 150GHz (0.15THz). The scanning system has a step size of 1mm.

3 Fabrication and results

We used a fused deposition modeling (FDM) Prusa i3 3D printer for thefabrication of the q-plate. The 3D printer heats the plastic and deposits itlayer-by-layer with a thickness of about 400µm. The transverse resolution isdetermined by the diameter of the nozzle, which in our case has a diameterof 400µm. Structures with discontinuous features are difficult to print withreasonable quality, because it implies stopping and starting the flow of plasticalong the printing trajectories which, in turn, introduces imperfection in theprinted object. Therefore, we decided to print the optical axis streamlines asdescribed in the previous section. In this way the structure could be printedwith a continuous injection of material producing a soft and continuous tra-jectory.

For the generation of cylindrical vector beams, we fabricated the q-platewith q = 1/2, which is shown in Fig. 1b. In order to describe the interactionbetween the device and the radiation we used the effective medium theory,

Terahertz q-plate fabricated by 3D-printing 5

given that the wavelength of the THz radiation (150GHz = 2mm) is largerthan the q-plate features (400µm).

In a strip line structure (as locally in the q−plate), the effective refractiveindex of the structure for radiation with polarization parallel and perpendic-ular to the strip lines is given by [19]

n2e = n2

∥ +1

3

[Λ

λπf1f2(n

21 − n2

2)

]2, (9)

n2o = n2

⊥ +

[Λ

λπf1f2

(1

n21

− 1

n22

)n∥n

3⊥

]2, (10)

respectively, where Λ is the period of the strip lines, λ is the wavelength, f1and f2 the volumetric fraction of the constituent materials and n1 and n2 arethe refractive indices of the polymer, Bendlay in this case (n1 = 1.5, measuredby transmission TDS) and air (n2 = 1), respectively. For λ ≫ Λ, the theorysimplifies to ne = n∥ and no = n⊥, where

n2∥ = f1n

21 + f2n

22, (11)

n2⊥ =

(f1n21

+f2n22

)−1

. (12)

Based on these equations it is possible to make different kinds of birefrin-gent devices, for example, half-waveplates. According to the theory we needa continuos orientation variation of a half-waveplate in order to produce acontinuous q-plate. Every local section shown in Fig. 1(a) is a half-waveplatewith a different orientation angle, necessary for the generation of the cylin-drical vector beam. The printed device is shown in Fig. 1b. The thickness ofthe structure is calculated using the birefringence of the structure. In partic-ular, the structure proposed in this work had a birefringence of 0.1. So thethickness of the structure is 5mm for a half-waveplate at 150GHz. In order tomeasure the polarization distribution of the beam after propagation throughthe q-plate, we used a terahertz time-domain imaging system. A schematic ofthe experimental setup is shown in Fig. 1. (d)

We used a Menlo system Tera K15 fiber-coupled spectrometer in trans-mission configuration, which shows a dynamic range of 59 dB (at 150GHz)and a frequency resolution of 3.57GHz. The radiation was produced and de-tected using two InGaAs photoconductive antennas. One TPX lens was usedto collect the radiation after propagating through the sample. The receiver wasmounted on two translational stages forming an optical system that collectsthe radiation at a given point on the q-plate that can be raster scanned acrossits surface. The photoconductive detector is only sensitive to the electric fieldin one linear polarization direction, therefore the detector had to be rotatedby 90◦ in order to obtain the two components of the terahertz electric field. Aterahertz waveform is measured on a 40mm by 40mm mesh with 1mm stepsin order to obtain the full position-dependent electric field of the beam afterpropagation through the q-plate.

6 A. I. Hernandez-Serrano et al.

-1

0

1

xy

radial

azimuthal

spiral

Ex Ey

E

B

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Fig. 2 (color online) Transverse amplitude measurements with THz-TDI. A terahertz pulsewith homogeneous linear polarization propagates through a q-plate oriented as shown in thetop row. Figures (a) and (b) show the electric field components Ex and Ey , respectively,when the incident field is oriented along x. It produces a radial polarization distribution,which is shown in (c). Figures (d), (e) and (f) show the same measurements when the incidentelectric field is polarized along y, which produces a vector field with azimuthal polarization.In general, if the polarization of the incident field forms an arbitrary angle with respect tox, the outgoing field will have spiral polarization distribution, which is shown in Figs. (g),(h), (i), where we used a field with polarization oriented at 45◦. Transverse dimensions of40× 40 mm.

Figure 2 shows the experimental measurements. In order to produce a radi-ally polarized beam, the incident polarization should be parallel to the singularstraight line of the q-plate. The experiment is set-up as shown in the top rowof Fig.2. Hence, a horizontally polarized beam is transformed into a “radiallypolarized” beam and a vertically polarized beam becomes an azimuthal vec-tor beam. The Figs. 2(a) and (b) show the 150GHz terahertz electric field Ex

and Ey components respectively when presented in the form of a vector field[Fig. 2(c)], it is easy to see that the electromagnetic mode shows radial polar-ization distribution. The slight asymmetry observed in Figs. 2(b) is caused byminor missmatch of the optical axis of the beam an that of the q-plate. FromEq. (3) we see that Ex ∝ cosϕ and Ey ∝ sinϕ, which shows good agreement

Terahertz q-plate fabricated by 3D-printing 7

with the experimental results. Analogous measurements were performed afterrotating the q-plate 90◦. The Ex and Ey components for these measurementsare shown in Figs. 2(d) and (e). The vector field formed in this case, presentedin Fig. 2(f), shows an azimuthal distribution. Given that λ = 2mm (150GHz),the q-plate structure is an effective medium since the structure features aresignificantly smaller (400µm), therefore the wavefront of the cylindrical beamshould not be affected by scattering.

Radially and azimuthally “polarized” beams are particular cases of a moregeneral class of vector beams, the “spirally polarized” ones [22]. If the incidentelectric field forms different angle with respect to the singular line of the q-plate, the field will emerge having spiral polarization distribution. Figures2(g), (h) and (i) show the resulting spiral polarization distribution when theincident electric field is oriented 45◦ with respect to x.

4 Conclusions

We have presented a simple fabrication method of a q-plate for the genera-tion of terahertz cylindrical vector beams. The q-plate was fabricated with acommercial 3D printer. The generated cylindrical beams were measured usingtime domain imaging. This device can be used to efficiently couple terahertzradiation into waveguides, as shown by Zheng, et al. [20], where they used seg-mented waveplates to generate radially polarized beams. Furthermore, otherq-plates can be fabricated at low cost for the generation of complex-structuredvector beams in order to study physical phenomena of interest in singularoptics, such as angular momentum and polarization topologies [23,24].

Acknowledgements We acknowledge the financial support from Consejo Nacional deCiencia y Tecnologıa (CONACYT) (Grants 257517, 255114, 252939).

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