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Abstract
The manufacturing and characterization of a large-size 72-electrode liquid crystal-based reconfigurable spiral phase plate (SPP) is presented. The SPP is addressed by a custom-made driver with 72 independent channels, which allows for the generation of any arbitrary integer or fractional optical vortex beam with topological charges ranging from –24 to +24. The 25 mm diameter device is fabricated using direct laser writing, leading to a fill factor over 99%. The device performance and flexibility exceed previous transparent reconfigurable SPP in terms of size, tuning range, and fill factor. The device and the light path have been simulated using the angular spectrum propagation method, showing excellent correspondence.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the work of Nye and Berry in 1974 [1], optical vortices (OV) have attracted increasing interest owed to their vast range of applications. These stretch from particle manipulation [2,3], exoplanet observation [4], optical imaging [5,6], and lenses [7] to optical communications [8,9] and quantum communications [10,11]. An optical vortex has a phase singularity in its center that leads to a point of zero intensity in the optical field. In 1992, Allen et al. [12] studied the propagation of OVs on vortex beams, i.e., paraxial light beams carrying orbital angular momentum (OAM). In more recent years, fractional vortex beams have attracted attention due to their unusual properties, that enhance previously existing applications while also covering new ones. These range from advanced particle sorting [13], optical communications [14], and optical imaging [15] to quantum optics [16].
Vortex beams (VB) are characterized by an azimuthally varying phase variation that results in a propagating helical wavefront [17]. The phase profile can be mathematically described by exp(j$l$θ), where l is the topological charge and θ the azimuthal angle about the beam axis. At the central axis of the VB, all phases ranging from 0 to 2π⋅$l$ coexist, leading to a phase singularity with a zero-intensity point surrounded by a ring of intensity whose radius increases with the absolute value of l [18]. The topological charge describes the number of entwined helices in a wavelength of the wave front, which is related to the carried OAM, of $l{\hbar}$, per photon travelling in the beam (Fig. 1(a) and (c)) [12,19].
Fig. 1. Propagating wavefronts for five arbitrary topological charges: $l$=1, 1.5, 2, 3 and -2. In the integer topologies (a), (c), (d), and (e), continuous intertwined helically propagating wavefronts are highlighted. In (c), a single phase-discontinuity of 4π (2λ) is introduced by the SPP. In (d) and (e), multiple phase-discontinuities of 2π are introduced. The π phase discontinuity in (b) prevents the creation of helically propagating wavefronts. The π discontinuity leads to a radial dark opening, as illustrated in below figures.
Generation of VBs can be achieved through different methods like sub-wavelength gratings [20], meta-materials [21–23], annular gratings [24] or q-plates [25]. However, the simplest vortex-producing devices are spiral phase plates (SPP). An SPP introduces a continuously varying azimuthal phase delay in the range of 0 to 2π⋅$l$. In practice, due to fabrication limitations, usually discrete SPP having a finite number of phase levels instead of a continuous variation of the phase delay are manufactured [26,27].
Spatial Light Modulators (SLM) are a flexible method of generating optical vortices. SLMs are composed by a matrix of independently addressed LC pixels. The phase profile is generated by defining the switching state of the independent pixels of the LC such that it resembles any desired phase profile [28].
Due to the limitations of transparent SLMs (fill factor, cost, aliasing) several devices specifically designed for the generation of SPPs have been developed. These are based on LC cells with discrete or interconnected electrode patterns [29,30]. By being specifically designed for the generation of SPPs, the interpixel area is reduced and aliasing avoided.
This work covers the design, manufacturing, and characterization of a transparent reconfigurable dynamic multi-level SPP designed to minimize interpixel area based on liquid crystal cells. The device is, to the knowledge of the authors record breaking, in terms of number of independent electrodes, and is furthermore improved in respect to a previously reported device [29], with the outstanding additional ability to arbitrarily generate both fractional as well as integer vortex beams. The design is proven to allow optimization of manufacturing costs and scalability while maintaining its simplicity and low manufacturing costs.
2. Theory
The electric field of a paraxial vortex light beam at the source plane can be given by:
where $A(\textrm{r} )$ is the amplitude component, l is the topological charge and $\theta $ is the azimuthal angle [31] (Fig. 1(a)).Thus, a planar Gaussian beam with finite width incident on an SPP becomes:
where ${w_0}$ is the waist radius of the Gaussian beam.VBs with integer topological charge show a continuous phase evolution with constructive phase wrapping between successive wavefronts. However, when l takes fractional values, the continuous phase wrapping can no longer occur. This introduces a phase discontinuity with destructive interference between neighboring sections of the propagating light beam (Fig. 1 (b)).
SPPs may be transparent discs with continuous azimuthal phase variation [26,27] spanning
Conventional homogeneously aligned positive nematic LCs are typically employed in such devices. These switch upon the application of an electrical field. A switching plane is defined by the direction of the applied field and the preferential alignment direction, introduced in the cell during the manufacturing process. Light polarized in this plane will experience a given refractive index depending on the switching degree of the LC, which in turn depends on the LC anchoring to the alignment layer, the elastic constants of the LC and the torque induced by the applied electrical field. In absence of field, the LC molecules will be in a relaxed state, in plane with the substrates, governed by the alignment layers, whereas in presence of field, the molecules will tend to orient along it. Through this reorientation, the effective refractive index (${n_{eff}}$) may be varied between the ordinary (${n_o}$) and the extraordinary (${n_e}$) refractive indices [37].
3. Materials and methods
The $4\;\mathrm{\mu}\textrm{m}$-thick liquid crystal-based device is made of 72 independent pie sliced sections. The LC cells used for the generation of SPPs consist of two parallel glass substrates ($1.1\;\textrm{mm}$ thickness) coated with transparent ITO electrodes. A $50 \times 45\;\textrm{m}{\textrm{m}^2}$ glass substrate is used for the patterned electrodes. A $50 \times 35\;\textrm{m}{\textrm{m}^2}$ glass substrate is used for the opposing ground plane. The patterned electrodes, shown in Fig. 2(a), cover a circular active area with $12.5\;\textrm{mm}$ radius. Each pie slice is connected to a contact pad measuring $0.25 \times 11\;\textrm{m}{\textrm{m}^2}$ with a horizontal pitch of $0.5\; \textrm{mm}$. The pattern was engraved using a direct laser writing (ablation) system (Lasing S.A.) equipped with a UV laser (Explorer Laser Spectrum Physics, Mountain View, US) using a back-scribing approach (i.e., ITO ablation is performed from the opposite face of the glass). The achieved electrode separation is approximately $2\,\mathrm{\mu}\textrm{m}$.
Fig. 2. Manufactured devices and their design. (a) Electrode pattern design with active area of $25\; mm$ diameter and 72 electrodes. In orange, the contact area for the connector to be attached. (b) Manufactured LC cell at ${45^o}$ between two crossed polarizers. Variation in color is a result of thickness inhomogeneity of the cell, which must be kept constant in the active area. The active linear polarization of the device, which is defined by the rubbing direction of the alignment layer is shown in (b).
Polyimide (PIA-2304 from Chisso LIXON aligner, Japan) was used as alignment layer to induce a homogeneous alignment of the LC molecules in the rubbing direction in the plane of the substrate surface. A positive dielectric anisotropy nematic LC (MDA-98–1602 Merck KGaA, Germany) is used.
The $4\;\mathrm{\mu}\textrm{m}$ thick homogeneous cell thickness was achieved using spherical spacers (HIPRESICA, Japan). The cell was filled with LC and sealed. The electrical connectors were glued to the contact pads using anisotropic conductive adhesive (Fig. 2(b)). The homogeneity of the cell can be appreciated by positioning the cell at a ${45^\textrm{o}}$ angle between crossed polarizers.
A custom-made driver [29] is used to generate the voltages needed for the independent phase delay levels that conform the phase profile. The driver controls the ${V_{RMS}}$ of 72 pulse-width modulation (PWM) square signals (one per independent electrode) by varying their duty cycle.
The relationship between the introduced phase delay and applied duty cycle of the PWM signal (i.e., the ${V_{RMS}}$) of the manufactured LC cells must be determined. This is done by applying a uniform electrical field to all electrodes simultaneously while measuring the phase delay induced by the LC cell. To measure the phase delay, the cell is situated between two crossed polarizers at 45° to the switching plane while the transmitted intensity is measured using a camera as previously described [29]. In this way, the transmitted intensity depends on the induced phase delay [29,32]:
Figure 3 shows the relationship between the duty cycle and the transmitted intensity. The variation of the duty cycle corresponds to a ${V_{RMS}}$ range of $0$ to $ 7.5\;{V_{RMS}}$. In this work the cell was calibrated and characterized using a He-Ne laser at 632.8 nm.
Fig. 3. Characterization results. Transmitted intensity as a function of the duty cycle of a PWM signal with ${V_{RMS}} = 15\; V$. Each color line represents the data received by the red pixels of different regions of the light beam. The inset shows the fitting results of the relationship between both $\delta /\pi $ (in orange with blue points) and the normalized intensity (in yellow with grey points) to the duty cycle of the PWM signal.
With the obtained data, a fit to a pseudo exponential function was done to cover a $2\mathrm{\pi }$ range of phase delays. The range of $\mathrm{\pi }$ to $3\mathrm{\pi }$ has been selected since it has been found that the phase delay as a function of the PWM duty cycle ($\textrm{dc}$) fits nicely the expression $\mathrm{\delta } = \textrm{A} \cdot {\textrm{e}^{\textrm{B} \cdot \textrm{dc}}} + \textrm{C}$. In the case shown, the obtained fitting parameters are $\textrm{A} = 10.57$, $\textrm{B} ={-} 4.16$ and $\textrm{C} = 1.93$. The agreement between the fitted curve and the measured data is shown in the inset of Fig. 4.
Fig. 4. Generation of vortex beams with arbitrary integer TC ($l$). Columns keep TC constant. (a)-(f) LC cell at ${45^o}$ between cross polarizers with white backlight. Individual phase delay levels ranging from π to 3π (counterclockwise) can be observed as different interference colors, images are of approximately 250 × 250 µm in size. (g)-(l) Far field intensity distribution with light polarized in the switching plane of the LC. Central singularity increases in size with higher absolute TC values and rotation direction depends on the sign of the TC. (m)-(r) Simulated intensity pattern considering 72 electrodes, setup elements and distances to better reproduce the obtained results.
This fitting curve is used to generate the phase delay levels needed to obtain SPPs with arbitrary topological charge. Topological charges in the range of $- 36$ to $36$ can be theoretically achieved using $2\pi $ phase wrapping. However, the higher topologies (positive or negative) employ less electrodes to generate each $2\mathrm{\pi }$ phase profile; hence the diffraction efficiency is reduced like in any conventional discrete linear diffraction grating [38]. The reduced diffraction efficiency when using less electrodes significantly hinders the purity of the produced VBs. It is concluded that only topological charges in the range of $- 24$ to $24$ can be achieved while maintaining some purity of the VB.
4. Results
By electronically addressing the 72 independent electrodes, the developed SPP phase profile is reconfigured, giving rise to both integer and fractional topological charges (TC). Thus, generating a discrete approximation to a continuously varying phase delay ranging over 2π⋅$l$, with l being an integer for VBs and a fractional value for Fractional Vortex Beams (FVBs).
The device performance with a selected set of integer topological charges is shown in Fig. 4. The device is, in all figures, addressed so that the first pixel is vertically located at 12 o'clock. For positive topological charges, this pixel will introduce a π phase retardation, and the retardation will increase in a counterclockwise manner. For negative topological charges, the first pixel will introduce a phase retardation close to 3π, and the retardation will decrease in a counterclockwise manner. The individual phase profiles applied to each of the 72 electrodes is defined by an equidistant sampling in phase of the 2π⋅$l$ phase range.
The first row of Fig. 4 shows the interference colors (described by the Michel-Levy Birefringence Chart), of the LC cell ranging from π to 3π, placed between crossed polarizers with the switching plane at ${45^\textrm{o}}$ to either of them, revealing the imprinted phase profile. The higher the topological charge, the smaller the number of electrodes used per $2\mathrm{\pi }$ range, thus reducing the diffraction efficiency. With topological charges of opposite sign, the phase delay levels are kept constant, but the phase delay pattern varies from left- to right-handed corresponding with positive and negative values of topological charge.
The second row of 4 shows the intensity distribution at a $50\;\textrm{cm}$ distance measured by a camera CMOS sensor. The topological charge may (e.g., 24) or may not (e.g., 7) be an integer divisor of the total number of electrodes. The beam passes a $6\;\textrm{mm}$ diameter diaphragm and the diffractive pattern thereof can be observed together with intensity non-uniformities. The diameter of the first order SPP induced diffraction ring increases with increasing absolute topological charge values. Due to the reduced diffraction efficiency as the TC increases, discontinuities in the first intensity ring start appearing when l is greater than ${\pm} 24$, as it can be seen for $l$ = 36 in Fig. 4. The difference between topological charges of opposite signs is reflected in the handedness of the pattern of the higher order discontinuous diffraction rings (Fig. 4(g) and 4(k)). The radial variations of the amplitude result from the optical setup, since the circular aperture of the diaphragm generates the Airy disk diffraction pattern.
The third row of Fig. 4 presents the simulation results for the full optical setup. The simulations were developed using MATLAB as described in [39], which are founded on scalar diffraction theory [40] using the angular spectrum propagation method [41]. In this way, the simulations are in concordance with the obtained intensity measurements, showing a similar variation of the annular pattern size as well as the discretization patterns.
The novel version of the addressing software allows for the study of non-integer topological charges (Fig. 5). The first row shows the intensity patterns recorded of the most extreme non-integer, i.e., half TCs, along with their neighboring integer TCs. For fractional TC values an approximately radial dark split in the high-power intensity ring can be seen, its azimuthal position corresponds with phase profile discontinuity in the SPP, situated at 12 o'clock azimuthally. Opposite signed TC led to mirror imaged far field intensity distributions (Fig. 5(a) and Fig. 5(e)). The second row of Fig. 5 shows the simulation results for the same integer and fractional TCs.
Fig. 5. Generation of FVBs with arbitrary TC ($l$). Columns keep TC constant. (a)-(f) Intensity pattern with a single polarizer aligned with the device active linear polarization. Appearance of radial dark opening from phase discontinuity at fractional TCs. Rotation direction depends on the sign of the TC. (g)-(l) Simulated intensity pattern for the same TCs and considering the elements of the optical setup.
The evolution of far field intensity distribution with the fractionality is shown in Fig. 6. The first row shows the active device observed between crossed polarizers. The TCs presented are fractional values between $l = 1$ and $l = 2$. The appearance of the phase discontinuity at 12 o'clock can be observed in the micrographs in Fig. 6, first row which highlight the difference between the two kinds of phase discontinuities. In Fig. 6(b) and Fig. 6(f), the intensity variation is continuously evolving, oscillating between dark and bright areas azimuthally, since the 2$\mathrm{\pi }$ phase discontinuity caused by the phase wrapping (at 12 and at 12 and 6 o'clock respectively) doesn’t incur disruptive phase interference. Whereas in Fig. 6(c), (d) and (e) a clear intensity step is visible at 12 o'clock, especially in Fig. 6(d) ($l = 1.5$) where the extreme phase delay discontinuity from 2 $\mathrm{\pi }$ (dark) to $\mathrm{\pi }$ (bright) occurs. The phase discontinuity of $\mathrm{\pi }$ leads to the most pronounced radial dark opening (Fig. 6(j)).
Fig. 6. Generation of fractional vortex beams with fractional steps in TC where the appearance and elimination of the radial dark opening can be observed. (a)-(f) LC cell at ${45^o}$ between cross polarizers with $632.8\; nm$ laser source. Odd multiples of $\pi $ show maximum intensity whereas even multiples show minimum intensity. As the fractional part of the TC increases, the phase discontinuity becomes more apparent, reaching the extreme for $l = 1.5$. (g)-(l) Far field intensity pattern for polarized light aligned with the device’s active linear polarization. (m)-(r) Simulated intensity pattern produced considering the elements of the optical setup.
The second row of Fig. 6 has been captured in the same manner as in Figs. 4 and 5. The appearance and disappearance of the radial dark opening can be observed (between Fig. 6(i) and Fig. 6(k)). The third row of Fig. 6 shows the simulation results for the same topological charges.
5. Conclusion
A reconfigurable SPP generator with a record breaking 72 independent electrodes has been manufactured and characterized. The number of electrodes has been multiplied by a factor of 3 with respect to the previous version [29], improving the diffraction efficiency of the device significantly, especially for higher topological charges. The active area of the device has been increased to one inch diameter, which allows for its use with larger area and thus higher total power incident light. Additionally, the driver has been redesigned to allow for the generation of any arbitrary topological charge, both integer and fractional, being the first time that a specific SPP generator has been employed as such. The 14-bit driver allows for a topological charge that can be tuned in steps as small as 0.01. The device is also characterized by an extremely high fill factor of around $99.63\%$ with less than $1.8\; m{m^2}$ (36 cuts each $25\; mm$ long and $2\; \mu m$ wide) of interpixel space in a $490\;m{m^2}$ active area.
Funding
European Space Agency (4000133048/20/NL/KML); Horizon 2020 Framework Programme (G.A 101004462); Comunidad de Madrid (APOYO-JOVENES-21-9FOMOQ-22-0CNGFM, BEAGALINDO-21-QU81R4-7-0QQBF3, IND2020/TIC-1724, S2018/NMT-4326); Agencia Estatal de Investigación (PDC2021-121370-C21, PID2020-114172RB-C22, PLEC2022-009381, TSI-063000-2021-83).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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