What Happens If Fringing Fields Losses And Change In Propogatin

I. Introduction

Liquid crystal (LC)-based spatial light modulators (SLMs) are cogitating or transmissive components with a large number of electrically addressable pixels, that tin locally and dynamically command the phase or the polarisation of light [1–15]. Wavefront shaping of a coherent beam past stage modulation is required in many speedily developing applications including holographic optical trapping, holographic displays, solid-state low-cal detection and ranging (LIDAR), virtual and augmented reality, etc. For phase modulation, the light that is incident on the SLM is polarised in the aeroplane parallel to the pretilt of the LC [ane–11]. Electrically addressed reorientation of the LC is besides used in LC microdisplays based on amplitude modulation [xvi–19]. In this example, the polarisation of the incident calorie-free makes an angle of 45 degrees with the pretilt airplane. The polarisation state of the light is modified and a polariser is used to obtain modulation of the aamplitude. Two decades ago the evolution of the liquid crystal on silicon (LCOS) technology was driven by the successful employ of amplitude modulation in cinema projectors and rear projection TVs. Nowadays the engineering science is reviving in the form of phase modulating SLMs for structured low-cal applications [1]. Customer demands are becoming more and more stringent, requiring SLMs with ever increasing resolution and switching speed [1,5–7]. However, with the decrease in pixel dimensions, the cross-talk between neighbouring pixels becomes more important and the phase retardation within the pixel area is increasingly not-uniform [ii]. Peculiarly in driving patterns with strong spatial variations, the inhomogeneity in each pixel tin can lead to a phase blueprint that strongly deviates from the expected one. Moreover, the interaction between neighbouring pixels at dissimilar voltages can pb to an unwanted modify in the polarisation, instead of the intended phase modulation.

We here consider vertically aligned nematic (VAN) SLMs with a small pretilt (θ_P = three°) towards the +y management every bit illustrated in Figure 1. For phase modulation, lite polarised along the y-axis is incident perpendicularly onto the SLM (along the z-axis). As long as the LC managing director remains in the yz-plane (the airplane spanned by the pretilt and the substrate normal), the polarisation country of the light beam is unchanged and merely the phase is modulated. Yet, when the voltage departure between neighbouring pixels introduces a twist of the LC managing director out of the yz-aeroplane, also the polarisation state of the beam is affected. Polarisation change is an important loss factor for stage modulating SLMs and should be carefully evaluated to optimise the efficiency in enervating applications.

Effigy 1. (Colour online) Top view (xy-plane) of the VAN SLM with rows of equal voltage (a) and columns of equal voltage (b). The simulated unit of measurement cell is enlarged in (c). Vertical cross-section (yz-plane) (d) with schematic presentation of the electrical field lines for varying pixel voltages along the y-axis

VAN SLMs use LC with negative dielectric anisotropy (Δε < 0) on top of pixels on a silicon backplane. Upon application of an electric field, the LC preferentially orients the director in the plane perpendicular to the electric field lines. This is substantially different from LC with a positive dielectric anisotropy where the unique direction, parallel to the electric field lines, is preferred [sixteen]. The incertitude for the manager in LC with Δε < 0 to rotate in a particular management, can to some extent, be lifted past the pretilt in the anchoring at the surface. All the same, in this work we demonstrate that an out-of-plane twist (out of the yz-airplane) of the managing director can occur betwixt neighbouring pixels in VAN SLMs, besides when at that place is a voltage variation forth the y-axis (case (a) in Figure 1). An existing hypothesis explains that, close to these transitions where the fringe-field is inclined in the same direction equally the pretilt, a so-called reverse tilt zone can exist where the LC molecules tilt towards the – y axis (opposite to the rest of the pixel) [18–20]. We here prove that another deformation of the LC director occurs, where conflicting torques due to the anchoring pretilt and the fringe-field close to the interpixel gap atomic number 82 to an out-of-plane reorientation of the director. In this way, the total energy, containing elastic and electric free energy terms, tin exist minimised. This out-of-plane reorientation of the manager has far-reaching implications for the proper operation of the SLM device: it changes the phase delay, the polarisation of the reflected lite and the switching dynamics. 2 possible out-of-plane reorientation directions (towards +x or – ten direction) exist and result in a bistability or tedious switching, depending on the driving history of the SLM.

In this work, the voltage-induced managing director reorientation in VAN SLMs is studied for different voltage combinations betwixt neighbouring pixels, based on experiments and numerical simulations. The simulation results for the LC director are reported in department 2 and the false optical transmission is compared to microscopy images in section III. The effect on the far-field diffraction characteristics is commented upon in section Iv and the switching between binary column gratings and row gratings is discussed in section Five.

2. Numerical simulation of the director configuration

A finite chemical element Q-tensor model is used to simulate the director configuration in an SLM with a jail cell thickness d of 3 µm, a pixel pitch Λ of 4 µm and an electrode gap of 0.3 µm between neighbouring pixels [21,22]. A pixel pitch of 4 µm is typical for state of the art loftier-resolution phase modulators and a 3 µm thick layer thickness is common for phase modulators working in the visual wavelength range [2,3,sixteen,20]. Details of the simulation method, adult at University College London, can be found in previous references [21–26]. Potent anchoring is assumed at the alignment surfaces (fixed directors at z = 0 and z = d) and periodic purlieus atmospheric condition are applied in the lateral directions (Figure 1). The effect of an applied root-hateful-squared voltage on the pixel electrodes is simulated, without taking into business relationship multiplexed sequences of high and low voltages that are typically used in applications. The LC parameters (dielectric anisotropy Δε, refractive alphabetize contrast Δn and elastic constants K_xi, K₂₂ and Thousand₃₃ ), anchoring conditions (pretilt bending θ_P ) and prison cell thickness (Table ane) are roughly estimated based on comparing between experimental results and simulations (uniform voltage vs. retardation curve), and mutual values reported in the literature [2,10,20]. The experimentally measured SLM supports phase modulation gratings with UHD resolution (3840 x 2160), has a cogitating back contact, is designed for green light and has a maximum experimentally measured double laissez passer phase retardation Γ = (iv π Δdue north d)/λ ≈ 2.55 π for λ = 532 nm. Details nigh the type of LC textile used in the SLM are non disclosed by the supplier. Although the exact selection for the simulation parameters slightly influences the results, the general conclusions and insights in the VAN SLM behaviour remain valid. Similar results were obtained for somewhat adjusted prison cell thickness, pretilt angle, dielectric anisotropy, refractive index dissimilarity and rubberband constants.

Table 1. Simulation parameters

For simplicity, binary (on/off) gratings are considered and the equilibrium director configuration is imitation for four dissimilar voltage combinations (V_low, 5_high) at the bottom pixel electrodes: (1.75 V, 2.2 Five) (one.75 V, ii.35 V), (1.75 Five, two.65 V) and (2.35 5, 3.half dozen Five). The counter electrode at the elevation surface is grounded (0 V) and all pixel voltages are in a higher place the Fréedericksz threshold. For uniform driving of the SLM (with the same voltage applied to all pixels), the voltages i.75 V, ii.ii V, ii.35 V, two.65 5 and iii.vi V stand for to a double pass (propagating back and forth through the LC layer) stage retardation Γ= (four π Δn d)/λ for light-green calorie-free λ = 532 nm of respectively 0.xiv π, 0.77 π, 1.0 π, 1.37 π and ii.07 π. We focus on binary gratings with ii neighbouring rows (or columns) of pixels at a high voltage V_high and two rows (or columns) of neighbouring pixels at a low voltage V_low as shown in Effigy 1 (a) (or Figure 1 (b)).The false and experimentally measured gratings have a flow corresponding to four times the pixel pitch fourΛ = 16 µm.

Figure 2 shows the simulated equilibrium director configuration for gratings with rows or columns of equal voltages, for the four different voltage combinations (Five_low, 5_high). Remarkable results for the managing director configuration are found in the gratings with rows of equal voltage (Figure ii Row-1, Row-2, Row-three, Row-iv). When the voltage difference between neighbouring electrodes in these gratings is big plenty -for all cases, except for (1.75 V, ii.2 Five) in the simulated examples- the director rotates out of the yz-plane (Figure 2 (a)), near the pixel edge where the voltage goes from high to low in the direction of the pretilt. At this transition, corresponding to the upper edge of the rows with V_loftier applied in Effigy 2, the fringe-field is inclined in the same direction as the pretilt. This out-of-plane twist can occur in two equivalent directions (towards +x or – 10) and but ane of the two possible solutions is shown in Figure two. An asymmetric phase retardation profile is expected, with the upward and downward transition between pixels at V_high and V_depression beingness non-equivalent because of the presence of a non-cipher pretilt. In the VAN SLM configuration with negative dielectric anisotropy, the retardation profile in these gratings is not only asymmetric, also an out-of-aeroplane twist occurs nearly one of the ii voltage transitions.

Figure 2. (Colour online) Simulated director configuration for gratings with columns of equal voltage (Col) and rows of equal voltage (Row), with dissimilar applied voltages V_low and 5_high. Indices 1, two, 3 and 4 stand for voltage combination (V_depression, 5_high) = (1.75 Five, two.two V) (1.75 V, 2.35 V), (one.75 Five, 2.65 Five) and (2.35 V, 3.6 V). For all gratings the midplane cross-section (z = d/ii) is shown in (a) with a colour representation for the twist angle φ. A cross-department for fixed y (b) and for fixed 10 (c) with the colour representing the tilt bending θ for column or row gratings

When a voltage is applied to the device in the initial 0 V state (all directors along the preferred pretilt direction), the field in the middle of the pixel is practically parallel to the z-axis, and the fringe fields well-nigh the horizontal edges of the pixels are tilted in the yz-plane (Figure ane (d)). In the region where the tilt of the fringe field is larger than the pretilt of the director, the torque is in the positive x-direction, while in the rest of the pixel the torque is in the negative x-direction. The large elastic free energy resulting from a tilt in the contrary directions (creating a so called reverse tilt zone [18–xx],) tin be reduced by introducing an out-of-plane twist (out of the yz-plane) in the director configurations. Equally tin can exist seen in Effigy 2, a larger voltage divergence between neighbouring electrodes leads to a larger twist and a larger expanse with twist. It is clear that the region with an out-of-plane twist can extend over a large distance and disrupt the expected pixel response.

A director reorientation exterior the yz-aeroplane also occurs for gratings with columns of equal voltage, as shown in Figure 2 Col-1, Col-2, Col-3 and Col-iv, merely the origin and the consequences of this effect are different. The director twist is now symmetric for the two edges of the columns (to the left and to the right of the columns with Five_high) and there is no threshold voltage for out-of-plane twist reorientation (Figure 2 (a)). The fringe electrical fields are pointing away from the centre of the column with high voltage, and the LC director tilts towards the middle of this cavalcade, which is expected for a material with negative Δε. In this case in that location is no slow switching dynamics and at that place are no competing domains. Out-of-plane reorientation occurs with the same unique twist direction along the total grating edge.

Iii. Optical transmission: simulation and experiment

Based on the simulated director configuration (every bit shown in Effigy 2), the optical transmission for different orientations of the polariser is simulated with the help of the Jones Calculus for λ = 532 nm. 100% reflectivity of the back contact is assumed and both the incident and the reflected calorie-free laissez passer through the same polariser. The resulting optical transmission (in reflection mode) at dissimilar driving levels is compared with experimental microscopy images in Effigy 3. In the experiment, one polariser is placed on tiptop of the reflective SLM, and the input light and the reflected light are passing through the same polariser. An expanse of 8 × 8 pixels is shown (with e'er two pixel-wide columns or rows at a voltage V_high and ii pixel-wide columns or rows at Five_low). Observations with the polariser oriented along the y-axis are used to place out-of-plane reorientation effects. When the LC director reorients solely in the yz-airplane, by irresolute its tilt angle but not its twist angle, the transmission should remain high in this case. Any out-of-plane reorientation of the director induces a change in the polarisation state when the light is propagating through the LC layer and will result in a lower transmission past the analyser. Also observations with the polariser oriented at 45° are shown, which roughly allow to monitor the induced stage retardation (although the out-of-aeroplane twist also has an effect on the transmission). There is very good agreement between the experimental results and the simulation results in Effigy 3, which clearly indicates that the LC configuration in the SLM (as discussed in section Two) is correctly simulated for dissimilar voltage combinations.

Figure 3. (Colour online) Simulated and experimentally measured optical microscopy images for eight × viii pixels, for gratings with columns (Col) and rows (Row) with equal voltages (alternating V_low and V_hig _h) for two orientations of the polariser (along the y-centrality and forth the xy-diagonal). Simulations are performed for λ = 532 nm and a green filter was used in the experiments

IV. Simulation of the far-field diffraction characteristics

Proper agreement of the director configuration for different voltage combinations on the pixels is very important for the apply of SLMs in applied applications. Pixel crosstalk and director twist virtually the edges tin can strongly modify the desired device response. When the SLM is used to deflect a light beam, the incident calorie-free is polarised along the y-axis, parallel to the pretilt plane. Due to the out-of-plane twist of the manager, not only the phase delay of the light beam will exist modulated but also the polarisation state will exist changed. Subsequently reflection, in many applications the low-cal is passing through a polariser that is parallel to the polarisation management of the input beam. Out-of-plane reorientation of the manager therefore constitutes a loss due to the change in the polarisation. Moreover, in gratings with rows of equal voltage and variable voltage in the pretilt (y) direction (Figure ane (a)), the out-of-plane reorientation near 1 edge induces an asymmetry between the +1st and −1st diffraction order.

Based on the simulated results for the optical transmission with the Jones calculus (section III), far-field diffraction characteristics for linearly polarised calorie-free were found with the help of a Fourier transform. Equally mentioned earlier, 100% reflectivity of the back contact is causeless and both the incident and the reflected/diffracted light pass through the same polariser. Tabular array 2 summarises the simulated intensities for the −i^st order, 0 club and +1^st club diffraction (for λ = 532 nm) for gratings with rows or columns of equal voltage, assuming only y-polarised light is detected. The loss fraction, absorbed past the polariser, is also given. A visual representation of the data in Table ii is shown in Figure 4.

Figure iv. (Colour online) Visual representation of the diffraction efficiencies reported in Tabular array two. Simulated diffraction efficiencies for gratings with 2 pixel columns or rows with equal voltage, for λ = 532 nm and different drive levels (polariser along y). Col, Row and Theo respectively correspond column gratings, row gratings, and according to the theory (with constant stage retardation in each pixel). Indices one, 2, iii and 4 stand up for voltage combination (5_depression, 5_loftier) = (1.75 Five, two.2 Five) (i.75 V, two.35 5), (1.75 V, ii.65 Five) and (2.35 V, three.6 Five). The theoretical stage difference, based on the retardation for compatible driving, in situation i, ii, iii and 4 is respectively 0.63 π, 0.86 π, ane.23 π and 1.07 π

Table 2. Simulated diffraction efficiencies for gratings with 2 pixel columns or rows with equal voltage, for λ = 532 nm and different bulldoze levels (polariser along y). Absolute values (w.r.t. the incident lite) are reported, neglecting losses at the glass interfaces and back contact. Col, Row and Theo respectively stand for cavalcade gratings, row gratings, and co-ordinate to the theory (with constant phase retardation in each pixel). Indices 1, two, iii and 4 represent voltage combination (5_depression, V_high) = (1.75 V, 2.2 V) (1.75 V, 2.35 V), (ane.75 V, 2.65 V) and (2.35 V, 3.6 5). The theoretical phase departure, based on the retardation for uniform driving, in instance 1, 2, 3 and 4 is 0.63 π, 0.86 π, ane.23 π and 1.07 π

The diffraction efficiencies are compared to the theoretically calculated diffraction efficiencies based on an idealised (pixelated) phase profile with abiding phase over the area of the high and low voltage pixels, without modify in polarisation. For this comparison, the phase for the pixels is fixed at the value obtained from uniform driving of the SLM (e.g. 1.75 Five corresponds to Γ = 0.14 π etc.). The deviations between the theoretically expected diffraction efficiency and the simulated efficiency tin be explained by two effects. The consequence of pixel crosstalk in SLMs is well known: neighbouring pixels are influencing each other and the stage contour is smoothed and non uniform within the pixel expanse. This effect becomes stronger with decreasing pixel size, increasing SLM thickness and increasing voltage difference between neighbouring pixels. The other event is the out-of-plane twist of the director: this introduces loss by reducing the y-polarised component, which may likewise alter the distribution of light over the unlike diffraction orders.

For gratings with columns of equal voltage, symmetric diffraction with equal powers in the +i^st and −ane order is observed for all driving levels (Table two, Effigy 4). The out-of-airplane twist betwixt pixels at unlike driving levels introduces a not-zero loss past changing the polarisation. Such a loss is also observed for gratings with rows of equal voltage, when the voltage difference between neighbouring pixels is sufficiently big (for all cases, except for (V_depression, 5_high) = (i.75 V, two.2 V)). The results in Table 2 (Effigy 4) illustrate that the diffraction for the gratings with rows of equal voltage is disproportionate, with a much higher efficiency for diffraction in the −1^st society than in the 1^st order. Symmetry breaking happens in these gratings because of the non-goose egg pretilt, and a certain disproportion tin be expected even when the director reorients just in the yz-plane (as in the example with (V_low, V_high) = (one.75 5, two.two 5)). The out-of-plane reorientation of the director at the elevation side of the rows with Five_high applied (from V_high to Five_low along +y) makes the diffraction deviate much more form the theoretical prediction (Figure four). The results for the binary grating with (V_depression, V_high) = (1.75 V, ii.65 V) illustrate this: 53% of the calorie-free is diffracted in the −1st diffraction order while simply iv% is diffracted in the +1st social club. This example makes clear that detailed noesis of the manager configuration in the SLM is crucial to understand the performance of the device. Theoretical approximations, that only take into business relationship smoothing of the phase retardation contour and asymmetry due to the pretilt, are not able to properly estimate the diffraction, if out-of-airplane reorientation effects are neglected.

V. Switching from column gratings to row gratings

Finally, simulations are performed to better sympathise the switching behaviour between the two types of gratings. Effigy 5 shows an intermediate director configuration (and respective polarised transmission images) that is obtained when switching from a grating with columns of equal voltage to a grating with rows of equal voltage, both with voltages V_low = 1.75 V and V_high = 2.65 V. As a result of the previously applied column gratings, two domains with different out-of-plane twist (towards +ten and – x) are formed when the row design is applied. This configuration evolves towards a configuration that is invariant along the ten-direction (Figure 2 Row-3) by preferred growth of the largest domain. Cheers to the symmetry in the structure, ii concluding configurations with opposite out-of-airplane twist can exist formed with equal probability. Experimentally we observed a slow growth of the domains until the final configuration is reached. The growth rate can depend on the driving history, the applied voltage combination (5_low, V_high), the number of pixels in the grating, etc. A detailed report of the switching speed for unlike SLMs (different thickness, different LC, etc.), dissimilar voltage combinations and different grating patterns is exterior the scope of this work, merely our experimental observations betoken that switching times well to a higher place 1 2d occur regularly. One of the master factors influencing the switching speed is the number of pixels in the grating. Switching between gratings with merely a limited number of pixels (due east.m. 1 or ii) in the grating period, is faster than switching betwixt gratings with more pixels in the grating catamenia.

Figure 5. (Colour online) Intermediate configurations (0.2 s after switching) with two domains with different out-of-airplane twist when switching from a column grating (Col-3) to a row grating (Row-3) (both with V_low = i.75 V, V_high = 2.65 V). (a) simulated managing director configuration; (b) false and experimentally measured transmission for two unlike orientations of the polariser (along the y-axis and forth the xy-diagonal)

VI. Determination

To conclude, in this commodity we experimentally measured and numerically simulated the LC director configuration in VAN SLMs for one-dimensional binary gratings with different driving voltages. The correspondence betwixt microscopy images and numerical simulations of the optical behaviour confirms the validity of the simulation approach. We have, to our knowledge for the first fourth dimension in VAN SLMs, demonstrated that an of import out-of-plane reorientation of the LC managing director occurs almost pixel edges that are perpendicular to the plane of the pretilt. This effect is asymmetric and only occurs at pixel edges where the fringe electrical field is inclined in the same direction as the pretilt. This has a huge affect on the operation principle of the device: strongly asymmetric diffraction is observed in binary gratings, and very slow residual switching occurs when a grating with strong voltage variations in the direction of the pretilt is applied. These tedious dynamics are related to the inherent bistability between ii configurations with an opposite out-of-aeroplane reorientation of the managing director. Initial switching, leading to changes in the phase retardation near the middle of the pixel, is fast merely residuum switching associated with the competition between different twist domains can exist wearisome. In an ideal SLM with pretilt direction in the yz-airplane, out-of-plane director reorientation towards the +ten-axis and -10-axis is as likely. In practice, a preference for ane configuration may be induced by non-idealities such every bit a small divergence of the pretilt management (at one or both substrates). It is very of import to take into account the effects discussed in this commodity for future apply of VAN SLMs in real-life applications.

Source: https://www.tandfonline.com/doi/full/10.1080/02678292.2021.1881831

Posted by: jacobsthadet.blogspot.com

What Happens If Fringing Fields Losses And Change In Propogatin

I. Introduction

Published online:

2. Numerical simulation of the director configuration

Published online:

Table 1. Simulation parameters

Published online:

Iii. Optical transmission: simulation and experiment

Published online:

IV. Simulation of the far-field diffraction characteristics

Published online:

Published online:

V. Switching from column gratings to row gratings

Published online:

VI. Determination

0 Response to "What Happens If Fringing Fields Losses And Change In Propogatin"

Post a Comment

Iklan Atas Artikel

Iklan Tengah Artikel 1

Iklan Tengah Artikel 2

Iklan Bawah Artikel