2.1.

Direct-Write Scanning System

The direct-write system we employ consists of an HeCd laser (325 nm, Kimmon Koha), an objective lens ($40\times$, Newport) to focus the laser beam to a small spot with approximately Gaussian profile, a polarization modulator, and an XY translation stage, as described more fully in Ref. 19. Figure 1(a) shows the ideal focused spot intensity profile, with a beam waist of $w_0$ typically in the range of several microns. The polarization modulator sets the orientation angle of the uniform linear polarization of the spot. We implemented this using a Pockels cell (Conoptics) and a quarter-waveplate (CVI Laser Optics). While it is, in principle, possible to also adjust the laser spot’s power and size during a scan, in this work, we ensure they do not vary. In one case we analyze below, we also employ a shutter (SH05, Thorlabs) to selectively strobe the writing beam during the scan.

Fig. 1

Spatial features relevant to the direct-writing of polarization gratings (PGs): (a) assumed intensity profile of writing beam/spot, with a Gaussian beam diameter $2w_0$, circularly symmetric; (b) perpendicular scan path; (c) parallel scan path, where small black lines indicate the flip region; (d) shuttered parallel scan path. In (b) to (d), the pink contour indicates the scan path and black arrows indicate the local orientation of the writing beam’s polarization.

2.2.

PG Materials and General Processing

We record the scanned polarization pattern within a layer of linear photopolymerizable polymer (LPP),^21,22 which has been coated onto a glass substrate. After exposure, we coat one or more polymerizeable LC layers in order to realize the birefringent PG pattern. We note that as long as the thickness of each LC layer is small compared to the PG pitch, subsequent LC layers will have an alignment pattern practically identical to the layers underneath.²³ A detailed protocol will be described in Sec. 3.4.

Note that the essential principle is that the in-plane orientation angle of the LC nematic director at each position follows the fluence-weighted average of the multiple exposures of different linear polarizations at that position, as described more fully in Ref. 19. Therefore, throughout this work, we intentionally overlap neighboring scans in order to convert the discrete scan contours into continuous LC orientation profiles. However, this is limited by the fact that the local net polarization purity, i.e., the local net degree of linear polarization (DOLP), degrades if the overlap becomes too large and results in weak LPP anchoring strength and poor LC alignment. This DOLP is not identical to the typical DOLP of a single polarization state. It is the time-averaged net degree of linear polarization at each position and is derived in Ref. 19.

2.3.

PG Figures of Merit

While the quality of a PG can be quantified using many possible performance metrics, here we will use polarizing optical micrographs and transmission measurements. We define the diffraction efficiency as ${\eta }_m=P_m/P_{\text{tot}}$, where $P_m$ is the power in the $m$’th far-field diffraction order, and $P_{\text{tot}}$ is the total power in the output hemisphere (measured as the transmission through a reference substrate). We, therefore, define haze, i.e., the fraction of light scattered outside the diffraction orders, as ${\eta }_h=1-\sum _{}^{}{P}_m/P_{\text{tot}}$. Additionally, we define the polarization contrast as ${\eta }_{+1}/{\eta }_{-1}$. For all our optical measurements, we input a circularly polarized probe laser, which is selected to maximize diffraction in the $m=+1$ order and minimize diffraction into the $m=-1$ order. Because we find that essentially every type of deviation from the ideal PG anisotropy, in simulation and/or experiment, causes some detectable degradation in efficiency, haze, or polarization contrast, we will use these metrics as the primary figures of merit in this work.

3.1.

Perpendicular Scan

The first path comprises scan lines that are perpendicular to the grating vector,¹⁹ shown in Fig. 1(b). The difference in orientation $\mathrm{\Phi }$ between adjacent lines is $\pi l/\mathrm{\Lambda }$ rad. Although this pattern consists of a discrete number of polarization angles, the LC orientation angle smoothly varies from one scan line to the next due to the averaging properties of the LPP and LC materials.

Because $\mathrm{\Phi }$ is constant for each scan along the $y$ dimension, the beam scanning speed $S$ is limited principally by the translation stages. The total time $T$ of each scan line has two parts: the time $t_y$ required to traverse a single line of height $Y$ and the time $t_r$ required to reverse direction and begin the next scan line, such that $T=t_y+t_r$. If we assume the speed within each line is approximately constant, then $t_y=Y/S$. The time to reverse direction depends additionally on both the maximum acceleration $A$ and jerk time $t_J$ (i.e., the maximum time required to change acceleration), such that $t_r=t_J+2S/A$. In our translation stages, $A=1000\mathrm{mm}/\mathrm{s}$ and $t_J=50\mathrm{ms}$. Notice that the speed $S$ affects these times in opposite ways: faster speeds decrease $t_y$ but increase $t_r$, and vice versa. To find the optimal scan speed needed to minimize time $T$, we find $\partial T/\partial S=0$ and solve for $S$. This leads to an optimal speed of $S_{\text{opt}}=\sqrt{YA/2}$. In our experience, this leads to typical speeds in the range from 10 to $150\mathrm{mm}/\mathrm{s}$ depending on $Y$.

3.2.

Parallel Scan

The second path comprises scan lines that are parallel to the grating vector, as shown in Fig. 1(c), where $\mathrm{\Phi }$ changes continuously within each line. A similar approach was used in Ref. 20, where a line-shaped beam was scanned while the polarization continuously varied. In these cases, the primary limitation is the speed of polarization cycling (in degrees/second) or, alternatively, the polarization modulation frequency (in Hertz). One option is a rotating half-waveplate,²⁰ but this is limited to a few hundred deg/second at best. As an alternative, the polarization modulator configuration (i.e., including a Pockels cell) we employ allows for rapid and continuous change of polarization orientation (up to $900,000\mathrm{deg}/\mathrm{s}$). However, its range is strictly limited to $\pm 120\mathrm{deg}$. This requires flipping the polarization modulator’s setting within every PG period from one extreme to the other (e.g., from $+90$ to $-85\mathrm{deg}$), thereby quickly traversing through all the polarization states between those extremes. We call the LPP region exposed by these incorrect polarizations the flip region.

The flip region is highly undesirable, as the incorrect polarizations result in a nonlinear orientation profile and/or degrade the LPP anchoring strength. The width of the flip region relative to the period can be estimated as $W_f=S/(f_p\mathrm{\Lambda })$, where $f_p$ is the polarization modulation frequency. In our system, the maximum rate the polarization can be modulated consistently is 5 kHz. The scan speed can, therefore, be found as $S=W_f{f}_p\mathrm{\Lambda }$.

3.3.

Shuttered Parallel Scan

The third path is identical to the parallel scan, except that the flip region is eliminated. To remove the flip region, the path is scanned twice [Fig. 1(d)], with the flip region on each scan adjusted to opposite sides of each period, and a shutter is used to block the laser illumination during each the flip region. The downside is that $S$ is now limited by the shutter modulation frequency $f_s$. In our implementation, the speed may be found as $S=\mathrm{\Lambda }{f}_s/10$.

3.4.

Experimental Comparison

We fabricated PGs using each of the three scan patterns on different regions of the same substrate. All three scans had period $\mathrm{\Lambda }=40\mu \mathrm{m}$, line spacing $l=8\mu \mathrm{m}$, a recording beam with diameter $2w_0=15\mu \mathrm{m}$, and a speed and power resulting in a spatially averaged fluence of $0.4\mathrm{J}/{\mathrm{cm}}^2$. The parallel scan had $W_f=10\%$, and for the shuttered parallel scan, $f_s=10\mathrm{Hz}$.

The LPP/LC processing was as follows, which is optimized to achieve a half-wave retardation for 633 nm wavelength. The photoalignment material ROP-108 (Rolic) was coated (1500 rpm, 30 s) on clean glass substrates, followed by baking on a hotplate (1 min, 115 deg). After exposing this LPP, a 3:1 solution of the polymerizeable LC solution RMS03-001C (Merck) and propylene glycol monomethyl ether acetate was coated (1200 rpm, 75 s). The LC layer was dried and cured with blanket UV within a dry nitrogen environment (1 min, 365 nm at $2\mathrm{mW}/{\mathrm{cm}}^2$). Finally, another LC layer was coated using the same processing conditions as the first LC layer.

Polarizing optical micrographs of each result are shown in Fig. 2. We observe that the perpendicular scan manifests the most regular and smooth texture, implying the best alignment. The parallel scan result includes many LC disclination defects, resulting from poor LPP anchoring strength caused by the polarizing flipping. The shuttered parallel scan appears to be nearly free from these defects and shows almost the same quality as the perpendicular scan.

Fig. 2

Polarizing optical micrographs of the PGs written with three scan paths: (a) perpendicular scan, (b) parallel scan, and (c) shuttered parallel scan. All scans had $\mathrm{\Lambda }=40\mu \mathrm{m}$, $l=8\mu \mathrm{m}$, $2w_0=15\mu \mathrm{m}$, and a speed and power resulting in a spatially averaged fluence of $400\mathrm{mJ}/{\mathrm{cm}}^2$. Different shades correspond to different alignment orientations.¹⁹ Note that each period includes two dark fringes because the intensity observed in this view is the same for $\mathrm{\Phi }(x)$ and $\mathrm{\Phi }(x)+90\mathrm{deg}$.

It is now important to compare the scan times of these two scan paths. As a hypothetical case, consider a $3\times 3{\mathrm{cm}}^2$ area with $\mathrm{\Lambda }=10\mu \mathrm{m}$ and $l=2\mu \mathrm{m}$: the perpendicular scan would take $~2.5\mathrm{h}$ (at $S=122\mathrm{mm}/\mathrm{s}$), while the shuttered parallel scan would take over 1.5 years (since $S=0.01\mathrm{mm}/\mathrm{s}$)! Clearly, only the perpendicular scan is feasible. Even the regular parallel scan would take over one day (at $S=5\mathrm{mm}/\mathrm{s}$).

We conclude that while the shuttered parallel scan may be usable in some situations, the perpendicular scan pattern is much more practical and does not appear to have any disadvantages. We use the perpendicular scan pattern for the remainder of this paper.

4.1.

Parameter Space Normalization

We begin with the assumption that the LC alignment at any point is governed solely by the exposure of the LPP directly beneath it. While it is clearly very simplistic and neglects the influence of the LC itself and hence, will not be valid when the grating period approaches the scale of the LC thickness, it enables the development of a valid first-order predictive model.

4.2.

Scan Parameter Assessment

We assess any given scan parameter combination ($B$, $L$, and $F_{\text{avg}}$) by examining the resulting PG first-order efficiency, in either simulation or experiment. The first step is to use the mathematical formalism given in Ref. 19 to predict the LC response, which is characterized by the orientation angle $\angle \mathbf{A}(x)$ and alignment confidence $|\mathbf{A}(x)|$. We set $\mathrm{\Phi }(x)=\angle \mathbf{A}(x)$ since the LPP we use here aligns parallel to the direction of the exposing linear polarization; furthermore, note that a zero pretilt is always assumed. The alignment confidence is bounded between 0 and 1 and represents our confidence that the LC will align along $\angle \mathbf{A}$. It is calculated from $F(x)$ and $\mathrm{DOLP}(x)$, as described in Ref. 19. Physically, variations in $|\mathbf{A}|$ may correlate with variations in the alignment strength of LPP, out-of-plane tilt, LC order parameter, and LC scattering. However, in this study, we are interested only in scan patterns that result in $|\mathbf{A}|$ as close to unity as possible, since this correlates to strong, in-plane anchoring without haze.

After obtaining¹⁹ the vector field $\mathbf{A}(x)$, the PG diffraction behavior can be simulated by assuming the phase profile of PG is twice the LC orientation angle: $\delta (x)=2\mathrm{\Phi }(x)$. This follows from theory regarding the geometric phase (Pancharatnam-Berry phase).²⁴ With $\delta (x)$, the appropriate near-to-far-field transformation³ is applied to obtain the far-field diffraction orders ${\eta }_m$. The first-order diffraction efficiency ${\eta }_m$ is a particularly useful scalar number with which to assess a scan parameter set.

As an example of a poorly performing parameter set, consider $B=0.6$ and $L=0.4$. The intensity cross-section for all scan lines impacting a single period is given in Fig. 3(a). This sparse scan leads to the spatially varying fluence, $F$, shown in Fig. 3(b) and the $\mathrm{DOLP}(x)$ shown in Fig. 3(c). The associated $|\mathbf{A}(x)|$ and $\delta (x)$ are shown in Figs. 3(d) and 3(e). Finally, the diffraction efficiency for all orders is shown in Fig. 3(f). Now we can judge the effectiveness of this scan parameter combination. First, note that the diffraction efficiency is only 84%, and leakage may be seen in the other orders. Second, the low alignment confidence in portions of Fig. 3(d) implies that this region may have substantial disorder in the LC, which is likely to dramatically increase haze and reduce the diffraction efficiency even further than predicted in Fig. 3(f).

Fig. 3

Simulation of scan parameters leading to poor diffraction behavior, with $B=0.6$, $L=0.4$, and $F_{\text{avg}}=200\mathrm{mJ}/{\mathrm{cm}}^2$: (a) intensity cross-section for each scan line affecting a single period; (b) net fluence; (c) net $\mathrm{DOLP}(x)$; (d) alignment confidence $|\mathbf{A}(x)|$; (e) resulting phase $\delta (x)=2\mathrm{\Phi }(x)=2\angle \mathbf{A}(x)$. (f) Diffraction efficiencies, found by applying the near-to-far-field transformation (NTFFT) on (e).

Conversely, a much better parameter set is shown in Fig. 4, with $B=0.8$ and $L=0.2$. With this larger beam size and smaller line spacing [Fig. 4(a)], much more averaging of the orientation profile takes place. This results in a virtually constant local fluence [Fig. 4(b)], $\mathrm{DOLP}(x)$ [Fig. 4(c)], and $|\mathbf{A}(x)|$ [Fig. 4(d)]. Not surprisingly, this leads to a very linear phase profile [Fig. 4(e)], and high single-order diffraction efficiency [Fig. 4(f)], predicted as ${\eta }_{+1}>99.99\%$. Since $|\mathbf{A}|$ is close to unity everywhere, we choose to trust the calculated ${\eta }_{+1}$, and we judge this as an excellent set of scan parameters.

Fig. 4

Simulation of scan parameters leading to excellent diffraction behavior, with $B=0.8$, $L=0.2$, and $F_{\text{avg}}=200\mathrm{mJ}/{\mathrm{cm}}^2$: (a) intensity cross-section for each scan line affecting a single period; (b) net fluence; (c) net $\mathrm{DOLP}(x)$; (d) alignment confidence $|\mathbf{A}(x)|$; (e) resulting phase $\delta (x)=2\mathrm{\Phi }(x)=2\angle \mathbf{A}(x)$. (f) Diffraction efficiencies, found by applying NTFFT on (e).

4.3.

Parameter Space Analysis

To map the parameter space, we applied this analysis to a range of $L$ and $B$. In Fig. 5, we show minimum value of the predicted $|\mathbf{A}|$, and in Fig. 6, we show the predicted first-order efficiency. With these two graphs, we can now offer an explanation as to the different degradation mechanisms in our direct-write PGs and identify optimal scanning parameters.

Fig. 5

Simulated minimum alignment confidence $|\mathbf{A}|$ as a function of the normalized line spacing $L$ and normalized beam size $B$. Note $L$ and $B$ were sampled in steps of 0.05 and 0.025, respectively, and $F_{\text{avg}}=200\mathrm{mJ}/{\mathrm{cm}}^2$ for all cases. Values of $|\mathbf{A}|$ noticeably lower than unity are associated with liquid crystal defects, incorrect retardation, and incorrect alignment orientation.

Fig. 6

Simulated first-order diffraction efficiency ${\eta }_{+1}$ as a function of the normalized line spacing $L$ and normalized beam size $B$. Note $L$ and $B$ were sampled in steps of 0.05 and 0.025, respectively, and $F_{\text{avg}}=200\mathrm{mJ}/{\mathrm{cm}}^2$ for all cases. For this simulation alone, we furthermore assumed $|\mathbf{A}|=1$.

Recall that whenever $|\mathbf{A}|$ is noticeably lower than unity, it indicates that at least some part of every period has poor alignment. This may occur if the local fluence or/and DOLP are too low. Consider the former (fluence): since we have fixed $F_{\text{avg}}$, the only way some positions receive a too-low fluence is when the beam size is small relative to the line spacing, i.e., when $O$ is low. In Fig. 5, this begins to occur at the left portion of the graph, where $O\simeq 1$. Now consider the latter (DOLP): as $L$ increases, the polarization difference between neighboring scan lines increases. Assuming $O$ remains constant, this results in a decreased DOLP for positions within a period. In Fig. 5, this occurs at the upper portion of the graph, beginning around $L=0.4$. Finally, note that if $B>1$, which means that the beam size is larger than the period, then the DOLP will be low, regardless of $L$. In Fig. 5, this occurs at the right portion of the graph, beginning around $B=1$.

It is important to note that even with high alignment confidence, ${\eta }_{+1}$ may still be low because the recorded $\delta (x)$ may not be linear enough. This results in parameters that lead to a nonlinear $\angle \mathbf{A}(x)$. To isolate this, we can simulate the phase pattern resulting from each point in the parameter space while assuming that $|\mathbf{A}|=1$. This is shown in the graph of Fig. 6, within which we observe these trends: diffraction efficiency is better for larger $B$ and/or lower $L$. Our understanding of this is that a larger $B$ results in a higher $O$ and a lower $L$ results in more scan lines per period, both of which result in a more linear profile.

4.4.

Optimal Scan Parameters

Our prime interest are those scan parameters that result in ${\eta }_{+1}>99\%$ for which we have high alignment confidence (i.e., $|\mathbf{A}|\sim 1$). This subset of the parameter space can be approximated as a triangle, as shown in Fig. 7, by combining Figs. 5 and 6. This is the range where $B\le 0.85$ and $O\ge 2.5$.

Fig. 7

Predicted optimal parameter space for direct-writing PGs with high efficiency. Parameters inside the white triangular region are predicted to have high efficiency. Since scans with larger $L$ result in a shorter scan time, we identify the point that offers minimum scan time while preserving a predicted efficiency of $>99.9\%$. Since scans with parameters furthest away from the edges of the optimal parameter space will be the most robust with regard to small variations in scan parameters, we identify the point that offers maximum alignment quality while preserving a predicted efficiency of $>99.9\%$.

If minimizing the total scan time is the primary concern, then choosing $B=0.85$ and $L=1/3$ is ideal. If maximizing alignment quality is the primary concern (i.e., reaching for the highest possible ${\eta }_{+1}$), then choosing $B=0.5$ and $L=0.05$ could be considered ideal. This parameter set is robust with regard to small variations in scan parameters and, so, may also be particularly useful when the beam size cannot be controlled precisely or when attempting small $\mathrm{\Lambda }$.