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Abstract
We present dynamic time-resolved measurements of a multi-pixel analog liquid crystal phase modulator driven at a 1 kHz frame rate. A heterodyne interferometer is used to interrogate two pixels independently and simultaneously, to deconvolve phase modulation with a wide bandwidth. The root mean squared optical phase error within a 30 Hz to 25 kHz bandwidth is <0.5° and the crosstalk rejection is 50 dB. Measurements are shown for a custom-built device with a flexoelectro-optic chiral nematic liquid crystal. However, the technique is applicable to many different types of optical phase modulators and spatial light modulators.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Spatial light modulators (SLMs) have become essential components for a number of applications, by allowing the phase and/or amplitude of an optical beam to be spatially modified in two dimensions. For example, they are used as beam-steering components [1,2] for use in light detection and ranging (LiDAR) [3] as well as for free-space communications [4]. They are used to correct for aberration in high resolution microscopy [5], and laser micromachining [6]. They also find application in display technology [7], for example laser projection. A further use is for quantum computing applications [8]. The specification of SLMs to meet these demands is continually evolving, particularly towards ever faster frame rates and obtaining phase accuracy at these speeds. As the switching speeds become faster, the rise time resulting from the physical switching process can become significant, leading to phase errors not observable under static conditions. Furthermore, the error may vary depending on each different phase transition. It is therefore important to have a way of characterizing spatial light modulators under dynamic operation at high frame rates.
Liquid crystal on silicon (LCoS) has become a dominant technology for SLMs [9]. These devices operate in reflection by incorporating an active liquid crystal (LC) layer on top of a silicon backplane. The backplane allows an electric field to be applied across the active LC layer, such that it is subdivided into an array of individually addressable pixels. Various LC modes can be used such as ferroelectric for binary phase modulation and nematic for analog phase modulation. Recently, we demonstrated a phase modulator based on the flexoelectro-optic effect in chiral nematic LCs, which is capable of analog 0–2π modulation at 1 kHz frame rates [10]. One of the issues with LC devices is that they can exhibit a number of dynamic effects which manifest in addition to the desired electro-optic effect. For example, charge build up on the electrodes can result in an ionic screening, which acts against the applied electric field, causing the response to decrease over the duration of a pulse. This is generally more pronounced when there are slower transitions, such as when the phase is held at a particular value for a long period of time. We have also observed an effect in flexoelectro-optic devices whereby there is an asymmetry in the response depending on the field polarity. This effect can result in a phase error which depends on the pattern of the electric field previously applied (a memory effect) [11]. Even though it is possible to characterize the intrinsic LC material by itself, often there are additional components such as quarter-wave plates and polarizers which will have tolerances, resulting in additional deviation from the theoretical characteristic. Hence it is desirable to be able to make a direct measurement of the phase response of the complete optical phase modulator.
Phase modulation devices are typically characterized statically, for example by using a Twyman-Green interferometer to measure the shift in interference fringes [12,13]. SLMs may be characterized using a Young’s double slit technique [14,15], by observing the interference between two pixels or by imposing a diffraction grating on the SLM [16]. Another method of characterizing an SLM is to use a polarimetric technique [17,18]. However, these static techniques have the drawback that they are not able to resolve the spurious dynamic effects. Furthermore, during the time taken to record the individual fringe patterns at each phase setting, environmental drift in the interferometer can result in significant measurement uncertainty. There have been very few reports of time-resolved measurements of optical phase, and only at low frame rates with limited resolution. For example, Lizana et al. [19] analyzed the zero and first diffraction orders when a binary phase grating was applied to an SLM, to measure phase fluctuation when the device was held at a constant phase. More recently, Bennis et al. [20] measured the Young’s double slit interference fringes with a camera at a 25 Hz switching rate.
In this paper, we dynamically measure the time-resolved phase of a multi-pixel analog LC optical phase modulator operating at faster frame rates of 1 kHz with high resolution. The technique is demonstrated on a custom-fabricated flexoelectro-optic LC phase-only device, but it is equally applicable to any type of free-space optical phase modulator.
2. Time-resolved phase measurement system
Homodyne Michelson interferometers can be used to measure optical phase, however these can require stabilization at quadrature, have limited dynamic range, and be affected by low frequency laser noise. Heterodyne interferometers have previously been used to measure wide dynamic range acoustic signals with optical fiber sensors [21,22], as well as to measure the polarization properties (retardance and angle of optic-axis) of LC material at switching rates of up to 100 Hz [23,24]. Here a heterodyne interferometer is used to measure the dynamic optical phase of a general optical phase modulator, independent of its type. To obtain sufficient bandwidth to allow switching speeds in excess of 1 kHz, a quadrature demodulation scheme with digital signal processing is used. A schematic of the time-resolved phase measurement system is shown in Fig. 1(a). Light from a helium neon (He-Ne) laser (JDS Uniphase 1125P) at 632.8 nm is split into two paths with beam splitter BS1. The first path passes through an acousto-optic frequency shifter (AOFS, AOFS1). The light then passes via beam splitter BS2 onto Pixel 1 of the device under test (DUT) and back via BS2 onto photodetector PD1 (Thorlabs PDA55). The second path from BS1 passes through the second AOFS (AOFS2) and then via two mirrors (M1, M2) and BS2 onto PD1. The mirrors M1 and M2 are adjusted so that the two beams at PD1 are co-linear using a camera in place of PD1 (by gradually increasing the fringe spacing until there is a single spot). The two AOFSs (Gooch and Housego I-FS040-2S2E-3-OL3) are driven by two RF signal generators (Wavetek 395) at 40.0 MHz (AOFS1) and 39.8 MHz (AOFS2). These are frequency locked to ensure that there is a constant frequency offset of 200 kHz between them. An arbitrary function generator (AFG, Wavetak 195) is used to generate voltage signals, which were amplified with a 10× voltage amplifier (FLC Electronics F10AD) and applied to the DUT. The signals from the photodetectors are digitally sampled with a data acquisition module (Keysight U2331A). The AFG and the data acquisition module take an external clock reference frequency from the primary RF signal generator, so that everything is in synchronization. All the test equipment is remotely controlled using MATLAB running on a PC.
Fig. 1. (a) Schematic of the time-resolved phase measurement system. He-Ne: helium-neon laser, BS: beam splitter, AOFS: acousto-optic frequency shifter, LDBS: lateral displacement beam splitter, M: mirror, DM: D-mirror (‘D’ shaped), PD: photo-detector, RF GEN: RF generator, AFG: arbitrary function generator, VA: voltage amplifier, DAQ: data acquisition module. (b) Implementation of the LDBS using a beam splitter, two mirrors and a D-mirror.
Each AOFS has the effect of shifting the optical frequency by the RF frequency applied to it. Hence the two AOFSs result in a net optical frequency shift between the signal and reference beams by an amount equal to the difference in frequencies of the two RF generators. At the detector, the signal and reference beams interfere, giving rise to a carrier frequency at 200 kHz. Any optical phase difference imparted between the signal and reference beams results in a shift in the phase of this carrier frequency. Hence, by demodulating the carrier phase modulation, the dynamic phase response of the DUT is obtained. Very high phase accuracy can be obtained since the phase is measured relative to the clock reference of the RF signal generator. In principle, a more conventional Michelson interferometer configuration could be used. In such a configuration, the two AOFSs should be placed in the reference arm before the reference mirror and in the signal arm before the DUT. The beam would then pass through each AOFS twice, such that the carrier frequency is twice the frequency separation of the two RF sources. However, it was problematic trying to implement this configuration because the frequency-shifted diffracted beam comes out at an angle. This made it difficult to direct each beam back through its respective AOFS aperture to generate collinear interference signals. In contrast, the configuration used has an additional benefit in that it allows the signal and reference beams to be split into two parallel beams, allowing simultaneous phase measurement of two pixels.
The system can be used to interrogate two pixels independently by inserting the components represented by dashed outlines in Fig. 1(a). A lateral displacement beam splitter (LDBS) is used to generate two parallel beams and consists of a beam splitter, two mirrors and a D-mirror, the configuration of which is shown in Fig. 1(b). The light for the second pixel follows the dashed path in Fig. 1, reflecting off Pixel 2 on the DUT. The interference signal for Pixel 2 is tapped off with a D-mirror (DM) and detected on PD2 (Thorlabs PDA55). Interrogating 2 pixels allows crosstalk effects to be measured in the DUT and it also allows common-mode environmental noise to be equalized, such as laser phase noise and the absolute phase of the DUT. The bandwidth is only limited by the sampling rate of the data acquisition module (2 MS/s). This limits how high the RF carrier frequency can be to prevent aliasing, which in turn dictates how much bandwidth there is either side for data. However, for the 1 kHz devices being tested, the bandwidth chosen of 25 kHz was more than sufficient, and it is desirable to limit the bandwidth to reduce system noise.
3. Signal processing
Digital signal processing is used to recover the time-resolved phase from the sampled data obtained from the acquisition module. A block of data is captured and then processed off-line in MATLAB. However, it would be possible to implement the signal processing in real-time if required, by choosing to use only causal filtering functions. The MATLAB program code to perform the signal processing, together with raw data files acquired from the data acquisition module are provided in Code 1 Ref. [25] of the supplementary information. The received signal, Vin, consists of a phase modulated carrier (fc = 200 kHz) and the phase can be recovered using quadrature demodulation. The signal processing elements are shown in Fig. 2. First a bandpass digital filter centered at 200 kHz is used to isolate the carrier and modulation sidebands whilst rejecting extraneous noise. This signal is multiplied by cos(2πfct) and low pass filtered with an image reject filter to generate an in-phase signal (I). Similarly, the bandpass filtered input signal is multiplied by ‒sin(2πfct) and low pass filtered to generate a Quadrature (Q) signal. The time dependent phase, ϕ(t) is then recovered as
Fig. 2. Digital signal processing elements used to demodulate the optical phase. MATLAB scripts to implement this signal processing, together with raw data from the data acquisition module are included in Code 1 Ref. [25] of the supplementary information.
The bandpass filter, image rejection filters and the anti-alias filter are implemented as finite impulse response (FIR) tapped delay line filters for a linear phase response. They are optimal filters with an equiripple response and an order of 251, designed using the Parks-McClellan ‘firpm’ function in MATLAB. Both have over 100 dB out-of-band rejection and care was taken to make the transition region smooth to prevent ringing on the pulse edges. The response of the bandpass and image rejection filters are shown in Fig. 3(a) and Fig. 3(b), respectively. The bandpass filter has a 3 dB bandwidth of 91 kHz centered on 200 kHz and the image rejection filters have a low pass response with a 3 dB bandwidth of 45 kHz. The anti-alias filter has a 3 dB bandwidth of 25 kHz. The high pass filter consists of a first order infinite impulse response (IIR) filter which is applied in both forward and backwards propagation directions on the acquired block of data to ensure zero phase distortion. The 3 dB bandwidth of both passes combined is 30 Hz.
Fig. 3. Digital signal processing spectra. (a) Digital bandpass filter. (b) Digital image reject low pass filter, (c) Spectrum of received signal in units of dB relative to 1 V rms (dBV). (d) Spectrum after bandpass filter (dBV). (e) Spectrum after multiplying filtered input signal by local oscillator (dBV). (f) Spectrum of demodulated optical phase in units of dB relative to 1 rad rms (dBrad). See Data File 1, Data File 2 and Data File 3 for the underlying data.
Figure 3(c) shows a spectrum of the unfiltered input signal and Fig. 3(d) shows the spectrum after passing through the bandpass filter, leaving just the 200 kHz carrier and the phase-modulated sidebands. The spectrum after the filtered input signal has been multiplied by cos (2πfct) is shown in Fig. 3(e). This results in a lower sideband at baseband and an upper image sideband at 400 kHz. The unwanted upper image is filtered out with the image reject filter to generate the in-phase signal, I. Figure 3(f) shows the spectrum of the signal after demodulation with the full signal processing chain shown in Fig. 2.
4. Flexoelectro-optic optical phase modulator
For our tests, we used a multi-pixel version of our previously reported optical phase modulator [10], to allow a very fast switching speed together with analog control. The schematic of the optical phase modulator device is shown in Fig. 4(a). The device operates in reflection, such that input light passes through a polarizer, a quarter wave plate, a LC layer, a second quarter waveplate, after which it is reflected off a mirror and back through the same components. The LC layer consists of a chiral nematic LC aligned in the uniform lying helix geometry. The layer effectively acts as an optical waveplate, with an optic-axis in a plane normal to the propagation of the incident light. When an electric field is applied across this layer, the flexoelectro-optic effect results in a tilt in the angle of the optic-axis within the plane of the device [26]. If the thickness of the LC layer is chosen such that it is nominally equivalent to a half-waveplate, the configuration results in the reflected light being phase modulated. By using an LC material which exhibits a large variation in tilt angle of ±45° with applied field, it is possible for the device to achieve full analog phase control between 0–2π. An advantage of the flexoelectro-optic effect is that it can occur on very fast submillisecond timescales, allowing switching frequencies exceeding 1 kHz.
Fig. 4. (a) Flexoelectro-optic optical phase modulator configuration. (Pol.: Polarizer, QWP: quarter waveplate). The LC is a 4.88 µm cell filled with CBC7CB and 3 wt.% BDH1281 (b) Polarizing microscope images of the LC cell which has been divided into 4 pixels (numbered 1–4). (i) with a voltage applied to Pixel 1; (ii) with an voltage applied to Pixel 2. The applied voltage was a ±10 V, 1 kHz square-wave.
We constructed a benchtop 4-pixel spatial light modulator, based on this configuration. The chiral nematic device used consisted of a mixture of 4′,4‴-(heptane-1,7-diyl)bis(([1,1′-biphenyl]-4-carbo-nitrile)) (CBC7CB, synthesized in-house) and the chiral dopant BDH1281 (Merck). In order to create multiple pixels, a custom-made cell was fabricated from two 25×25×1.1 mm glass substrates coated in 10 nm thick indium tin oxide (ITO) (Merck Sigma-Aldrich 703176). A regeneratively amplified Ti:Sapphire laser (Spectra-Physics Solstice) was used to ablate the ITO on one substrate to create electrically isolated pixels. The laser wavelength was 790 nm, the pulse width was 250 fs and the repetition rate was 1 kHz. The pulse energy was adjusted using a polarizer preceded by a rotatable half waveplate. A 0.1 NA ×4 objective (Microtek) was used to focus onto the substrate surface. Two straight lines were inscribed at right-angles to divide it up into 4 individual pixels, each nominally a quarter of the available area. One track was inscribed at a pulse energy of 1.2 µJ and 0.25 mm/s writing speed, whilst the other was inscribed with 0.8 µJ pulses and a speed of 0.5 mm/s. The track widths were less than 10 µm thick and the electrical resistance between adjacent pixels was measured and found to be open-circuit. The nonpatterned substrate was cut smaller to enable access for electrical contacting.
To fabricate the alignment layer, the glass substrates were cleaned using acetone and an ultrasonic bath for 20 minutes. A solution of 0.5 wt% polyvinyl alcohol in deionised water was stirred for 24 hours at 80°C and spun-coated onto the glass substrates at 1500 rpm for 30 s. This deposited a layer between 10–100 nm thick. The substrates were then mechanically rubbed using a rubbing machine for 5 times in the same direction. A cell was constructed from the two substrates using 5 µm spacer beads mixed with Norland 61 adhesive around the edges. A clamp was applied and adjusted until a single large fringe could be observed and then the cell was cured with ultraviolet light for 3 minutes. The alignment layers on the inside had opposing rubbing directions. The cell thickness was measured to be 4.88 µm from the Fabry-Perot fringes observed on a UV-VIS spectrometer (Agilent 8454), which was close to the target 5 µm required for a half-waveplate with CBC7CB. The cell was capillary filled with a mixture of CBC7CB with 3wt.% chiral dopant BDH1281 at 123°C. Electrical connections were made to the 4 pixels and the common ground with indium solder.
The 4-pixel LC cell was observed on a polarizing microscope (Olympus BX51) and images are shown in Fig. 4(b), with the pixels numbered 1 to 4. A ±10V, 1 kHz square-wave was applied to different pixels individually to test them. Figure 4(b)(i) shows an image of the cell with the voltage applied to Pixel 1, while Fig. 4(b)(ii) shows an image when the voltage is applied to Pixel 2. When the voltage is applied, the resulting field causes the LC within that pixel to align, changing its color.
A bulk optics implementation of the 4-pixel optical phase modulator was constructed on the optical bench at the required position within the interferometer. At the front was a Glan-Taylor polarizer, followed by a quarter waveplate and the 4-pixel LC cell. The cell was mounted on a hot-stage (Instec) which had apertures aligned to Pixel 1 and Pixel 2, allowing light to pass through the LC. The LC cell was aligned by heating it above the clearing temperature of 113°C and allowing it to slowly cool to 108°C under an applied ±26 V, 1 kHz square-wave signal. Behind the cell there was a quarter wave-plate, followed by a mirror. The rear mirror was adjusted so that the returned beam passed back through the aperture onto the beam splitter. The quarter waveplates were on rotation mounts and adjusted so that the first gave circularly polarized light, while the second maximized the reflected light back through the first polarizer.
5. Dynamic time-resolved measurements of optical phase
The dynamic phase measurement system was used to characterize the custom-built 4-pixel optical phase modulator. A series of voltage waveforms were applied to the DUT using the AFG and data were acquired from the data acquisition module. Figure 5(a) shows the measured phase against time when ±30V, 1 kHz square-wave signals are applied to Pixel 1 and Pixel 2 in antiphase. The optical phase of Pixel 1 switches by ±180.8° whilst Pixel 2 switches by ±176.6°. The 10–90% rise time is 97.5 µs, while the corresponding fall time is 85 µs. There is a slight reduction in the phase over the duration of the pulse, likely due to ionic screening and other effects we have previously reported on [11]. The information is useful as it allows designers of dynamic optics systems to determine the time window over which the actual phase value is within the desired limits. It could also allow the electrical drive signal applied to the device to be modified to compensate for these time-dependent effects. The optical phase measured when the same signals are applied to Pixel 1 and Pixel 2, but with a 90° phase offset is shown in Fig. 5(b). This is important as it shows that there does not appear to be any influence from one pixel to the other. Figure 5(c), on the other hand, shows results when Pixel 1 is driven by a 500 Hz square-wave with an amplitude linearly reducing over time and Pixel 2 is driven by a 1 kHz square-wave, with a constant phase offset between the two. Figure 5(d) is similar to Fig. 5(c), except there is a pseudo random signal instead of a ramp on Pixel 1. Both Fig. 5(c) and Fig. 5(d) show that the two channels are operating independently and are not impacting each other. However, the dynamic measurements reveal dynamic effects that are not observable from a static characterization. For example, the ramp in Fig. 5(c) has an asymmetry with higher amplitude for the positive polarity. The optical phase of the pseudo-random signal shows spurious dynamic effects leading to discrepancy in the actual phase imparted. It would be possible to measure SLMs with higher pixel densities by using an objective to focus onto the smaller pixel area. The measurement acquisition can be very fast as it is single-shot, requiring only a few milliseconds of data to be captured. Hence it would be possible to scan across a two-dimensional area of an SLM.
Fig. 5. Dynamic time-resolved phase response of the multi-pixel flexoelctro-optic liquid crystal analog phase modulator under different modulation patterns. The device is as per the configuration in Fig. 4(a) with a 4.88 µm LC layer of CBC7CB mixed with 3 wt.% BDH1281 at 108°C. The blue and red lines are the measured phase responses for Pixel 1 and Pixel 2, respectively. (a) A 1 kHz, ±30 V square-wave is applied to Pixel 1 and Pixel 2 in antiphase; (b) A 1 kHz, ±30 V square-wave is applied to Pixel 1 and Pixel 2 with a 90° phase offset; (c) A 500 Hz square-wave of decreasing amplitude applied to Pixel 1 and a 1 kHz, ±30 V square-wave is applied to Pixel 2; (d) a pseudo random 500 Hz multilevel signal applied to Pixel 1 and a 1 kHz, ±30 V square-wave is applied to Pixel 2. See Data File 4 and Data File 5 for the underlying data.
In order to calibrate the device, a series of square-waves at different voltage amplitudes were applied to Pixel 1, with Pixel 2 grounded. The differential optical phase between Pixel 1 and Pixel 2 was sampled in the middle of each pulse. This resulted in the calibration plot shown in Fig. 6, with the positive half of the voltage characteristic coming from the positive pulses and the negative half coming from the negative polarity pulses. We previously conducted a calibration of a very similar device [10], based on observing the position of interference fringes from camera images, so it consequently had low resolution. This was compounded by drift in the interferometer over the time needed to make the measurement. Comparing the two calibration plots, the previous assessment has considerably more measurement uncertainty. In contrast, the measurements here in Fig. 6 fall on a smooth curve. The increased accuracy reveals that the response has a sublinear characteristic at high voltage amplitudes. Obtaining a more accurate phase-voltage curve gives a better calibration of the device.
Fig. 6. Calibration plot of the optical phase of Pixel 1 (relative to grounded Pixel 2) as a function of voltage, which was extracted from the time-resolved phase measurement data. The device is as per the configuration in Fig. 4(a), with a 4.88 µm LC layer of CBC7CB mixed with 3 wt.% BDH1281 at 108°C. See Data File 6 for the underlying data.
6. Optical phase measurement error
6.1 Noise
The heterodyne interferometric measurement is inherently very accurate as it is effectively measuring the phase of a recovered carrier relative to a synthesized clock source which can be very accurate and stable (the Wavtek 395 has a 10 MHz reference clock with 30 ppm absolute accuracy). Furthermore, there are no components that are likely to cause systematic errors, only phase offsets which can be calibrated out. The dominant sources of error are therefore likely to be laser phase noise and optical path length changes, for example those caused by acoustic vibration. In order to mitigate noise, the upper bandwidth was limited to 25 kHz with the anti-alias filter. Additionally, the low frequency noise was limited to 30 Hz with the high pass IIR filter, in order to reduce the impact of low frequency laser phase noise. For many applications, it is the relative phase noise between pixels which is important rather than the absolute optical phase. Hence, a differential measurement was made between Pixel 1 and Pixel 2 to cancel common-mode noise such as laser phase noise and optical path length changes within the device.
A ±32.5V, 1 kHz square-wave was applied to Pixel 1, while Pixel 2 was grounded, and the optical phase of both Pixels was recorded. The differential phase was obtained by subtracting the optical phase of Pixel 2 from that of Pixel 1. The optical phase of each positive polarity pulse was sampled in the middle of the pulse. The optical phase error was determined by subtracting the mean optical phase of all the pulses (nominally 180°). The measurement results are shown in Fig. 7. Figure 7(a) shows the optical phase of Pixel 1 (without reference subtraction) with the high pass filter removed, showing that there is an intrinsic error of up to ±20°. However, by subtracting the Phase of Pixel 2, to give a differential measurement, much of this phase noise is cancelled, as shown in Fig. 7(b). Here there is a deviation of approximately ±3°. However, cancelling this phase noise now reveals a low frequency perturbation, likely due to the phase noise of the He-Ne laser. As this is a dynamic measurement on a 1 kHz square-wave, there is no need to have the measurement DC coupled, indeed LC devices will often not operate satisfactorily at DC due to ionic screening effects. Hence, by incorporating the 30 Hz high pass filter, the overall noise is reduced to approximately ±1.6°, as shown in Fig. 7(c). The higher noise at the beginning and end of the trace is due to the step response of the IIR filter used in both forwards and backwards directions. The reduction in noise due to the differential cancellation (with the high pass filter in place) is illustrated in Fig. 8. Figure 8(a) shows a histogram of the optical phase of Pixel 1 on its own (without reference subtraction), while Fig. 8(b) shows a histogram of the optical phase of Pixel 1 with the optical phase of Pixel 2 subtracted. There is a very clear reduction in optical phase error, with the root mean squared error reduced from 3.76° [Fig. 8(a)] to 0.50° [Fig. 8(b)].
Fig. 7. Experimentally measured optical phase error for the phase modulator device [configuration shown in Fig. 4(a)]. A 1 kHz ±32.5 V square-wave bipolar signal was applied to Pixel 1 and Pixel 2 was grounded. The optical phase error was determined by sampling the optical phase in the center of each positive polarity pulse and subtracting the mean across all pulses. (a) The optical phase error of Pixel 1 (with the high pass filter removed). (b) The optical phase error of Pixel 1 with the optical phase of Pixel 2 subtracted (high pass filter removed). (c) The optical phase error of Pixel 1 with the optical phase of Pixel 2 subtracted (with the high pass filter in place). See Data File 7 for the underlying data.
Fig. 8. Histograms of optical phase error for the phase modulator device [configuration shown in Fig. 4(a)]. A1 kHz, ±32.5 V square-wave bipolar signal was applied to Pixel 1 and Pixel 2 was grounded. The optical phase error was determined by sampling the optical phase in the center of each positive polarity pulse and subtracting the mean. To avoid artefacts of the high pass filter, the first and last 100 ms of data were discarded. (a) Optical phase error of Pixel 1. (b) Optical phase error of Pixel 1 with optical phase of Pixel 2 subtracted. See Data File 8 for the underlying data.
6.2 Crosstalk
An important consideration is crosstalk between pixels. This can occur in LC SLMs from fringing fields affecting adjacent pixels. Other potential forms of crosstalk are due to insufficient electrical isolation between pixels, finite electrical isolation in the drive circuitry and electromagnetic pick up. There is very high electrical isolation between the pixels on our device as we used a femtosecond laser to inscribe breaks in the conductive surface, such that there was no measurable impedance between them.
To establish the crosstalk sensitivity, the device was driven with a ±32.5 V, 1 kHz square-wave signal on Pixel 1, with Pixel 2 grounded. Spectra of the two Pixels are shown in Fig. 9. Figure 9(a) shows the spectrum of Pixel 1, showing the fundamental frequency (1 kHz) to have a relative power of 14.97 dBrad. On the other hand, Fig. 9(b) shows the spectrum from Pixel 2, showing a relative power of −35.03 dBrad at 1 kHz. Hence, the total crosstalk of the device and the system is 50.0 dB. This demonstrates the very high electrical isolation provided by the ITO. We might expect that with a reduced pixel size, the proportionate pixel area affected by fringing fields from adjacent pixels would increase, resulting in higher crosstalk.
Fig. 9. Crosstalk measurements with ±32.5 V, 1 kHz square wave applied to Pixel 1 and Pixel 2 grounded. (a) Spectrum of optical phase of Pixel 1 in units of dB relative to 1 rad rms (dBrad). (b) Spectrum of optical phase of Pixel 2 (dBrad). See Data File 9 for the underlying data.
7. Analysis of multi-pixel flexoelectro-optic phase modulator
It is useful to examine the predicted performance of the actual LC device used in order to validate the results of the system and also gain further insight into the performance of this particular device. We previously developed a system to measure the time dependent variation in retardance and optic axis angle of birefringent devices [27] and used it to study large tilt angle LC crystal devices [11]. This system injects circularly polarized light into a DUT and the output is measured using a polarizer and a photodetector. By acquiring a series of synchronized time traces of the same modulation pattern at different polarizer angles, the retardance and optic-axis angle can be deconvolved. Using the system in Ref. [27], we measured the tilt-angle and birefringence of the intrinsic LC cell (i.e., without the mirror, quarter wave plates and polarizer) as a function of time. This is useful for predicting the phase modulation, since we expect the phase modulation change to be 4 × the change in the optic axis tilt angle, when it is used in the configuration shown in Fig. 4(a) [10].
The measured tilt-angle against time for a {x, −x, x, −x, x, x, −x, −x, x, −x, x, x, x, −x, −x, −x, x, −x} pattern, for voltage amplitudes of x in ten 3.25V increments up to 32.5V is shown in Fig. 10(a). The corresponding experimentally measured phase modulation for the optical phase modulator in configuration of Fig. 4(a), using the same LC cell, with the same voltage pattern applied to it is shown in Fig. 10(b). The phase modulation response is consistent with the tilt-angle data, showing the same slight asymmetry. It is clear that there is significantly more distortion than the simple square wave plots in Fig. 5(a) and (b). This is consistent with our earlier investigation of the time dependent polarization properties of this material.
Fig. 10. (a) Time-resolved tilt-angle data for the LC cell: a 4.88 µm LC layer of CBC7CB mixed with 3wt% BDH1281 at 108°C (similar to Fig. 3(b) in Ref. [11] which was for CBC7CB mixed with the chiral dopant R5011); (b) Time-resolved phase measurement for the optical phase modulator [configuration as per Fig. 4(a)] incorporating the same LC cell. For both (a) and (b), each colored line represents a measurement at a different amplitude voltage x in increasing steps of 3.25 V from 3.25 V to 32.5 V for a pattern {x, −x, x, −x, x, x, −x, −x, x, −x, x, x, x, −x, −x, −x, x, −x}. See Data File 10 and Data File 11 for the underlying data.
The measured polarization properties were then used to verify the dynamic phase modulation performance was consistent with that predicted from the measurements of retardance and tilt-angle. A Jones matrix model of the phase modulator in Fig. 4(a) was used, as described in Ref. [10]. By treating the LC layer as a waveplate, the angle of which rotates under an applied field, the output electric field vector, Eo, of light reflected from the optical phase modulator may be written as
Equation (2) was used to simulate the phase modulation of the device, by applying a vector of retardance values δ (t) and angle values θ(t), over time, taken from the time-resolved tilt angle data when an identical voltage waveform was applied to the same LC cell (without the additional components). An assumption in the model is that the quarter-wave plates were exactly 0.25λ and that the angles were perfectly aligned. However, the specification for the waveplates is ± λ/300, so we would not expect an exact correlation between the measured and simulated values. Figure 11 shows the simulated optical phase and the direct phase measurement data obtained with this system superimposed on top of each other, using the same pseudo-random input signal. Despite the uncertainty in the retardance and angle of the quarter waveplates, there is very good correlation between the two time domain waveforms which gives good confidence in the accuracy of the phase measurement system.
Fig. 11. Comparison of the phase measurement data to that predicted by simulations. Blue solid line: Experimentally measured phase data for Pixel 1 of the phase modulator device, driven with a pseudo-random voltage input [configuration as per Fig. 4(a) with a 4.88 µm LC layer of CBC7CB mixed with 3 wt.% BDH1281 at 108°C]. Red dashed line: Simulated phase obtained by acquiring the time-domain tilt-angle and birefringence measurement data for the same LC cell in isolation (without the polarizer, quarter-waveplates and mirror) and passing it through a Jones matrix model of the device. See Data File 12 and Data File 13 for the underlying data.
8. Conclusions
We have demonstrated a novel instrumentation system to measure dynamically the phase response of free space optical phase modulators. The system uses a heterodyne interferometer to measure directly the phase of light reflected from the optical phase modulator device, relative to a reference path. Acousto-optic frequency shifters are used to offset the optical frequency between the signal and reference paths. The resulting interference signals are demodulated using digital signal processing to recover the time dependent optical phase to high accuracy. The system works on blocks of non-repeating signals without time-averaging, enabling pulse-to-pulse effects to be observed. The root mean squared optical phase error was found to be 0.5° over a bandwidth between 30 Hz and 25 kHz. However, the bandwidth is not fundamentally limited, other than by the sampling rate of the data acquisition module used, so the system bandwidth could be extended as the switching speeds evolve. The crosstalk for the system and device was measured to be 50 dB.
Using the measurement system presented herein, we have characterized the dynamic performance of a flexoelectro-optic liquid crystal optical phase modulator with 0–2π analog phase range. The phase from two pixels were simultaneously measured as time-varying voltages were applied at a 1 kHz frame rate. This has allowed the finite rise time pattern dependent effects to be observed. The phase measurement results were also consistent with simulation results from a Jones matrix model of the device based on time domain measurements of the LC tilt angle and birefringence. The system is suitable for characterizing a range of free space optical phase modulators and has the potential for real-time operation.
Funding
Engineering and Physical Sciences Research Council (EP/M015726/1, EP/M016218/1, EP/M017923/1, EP/R004803/1, EP/T00326X/1); European Space Agency (4000125232/18/NL/AR/zk).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Code 1, Ref. [25], Data File 1, Data File 2, Data File 3, Data File 4, Data File 5, Data File 6, Data File 7, Data File 8, Data File 9, Data File 10, Data File 11, Data File 12, and Data File 13 of the Supplementary Information.
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