Chromatic aberration in the beam deflector and its compensation
Beam steering with LC leverages its anisotropic characteristics to modulate the phase of light as it travels through a specified path, enabling precise control over the accumulated phase. This is commonly implemented using a matrix of LC pixels, each inducing a local phase shift between 0 and 2Ï€, which controls the phase front to direct the beam in a chosen direction through reflection or transmission mode. Despite the benefits of the reflection mode, such as accommodating a broader range of materials and achieving strong phase shifts over shorter paths, we chose to fabricate a beam deflector (BD) of transmission type using indium tin oxide (ITO) electrodes on glass substrates. This approach allows for easier alignment of light sources and detectors in a linear setup, making it more suitable for integration into compact systems.
The BD is composed of ITO electrodes configured in a one-dimensional (1D) architecture, which is illustrated in Fig. 1a. The surface facing the nematic LC layer of the upper glass substrate is coated with a planar ITO layer, working as the grounding electrode. On the other side, the surface facing the nematic LC layer of the lower glass substrate is coated with strip ITO electrodes, acting as the driving electrode. The strip electrodes on the lower substrate are in the direction of the y-axis, and the inner surfaces of both substrates are coated with polyimide films and rubbed in the x-axis. The optical axis of the LC is initially aligned perpendicular to the strip electrode pattern with a small pretilt angle in the absence of an applied voltage. The BD is driven by applying 9 V to the grounded upper planar ITO electrode and 0–18 V to the lower strip ITO electrodes.
Fig. 1
Principle and design of the beam deflector (BD) device with polarization-dependent phase modulation. a Schematic of the BD device. The upper glass substrate is coated with a planar indium tin oxide (ITO) layer, while the lower glass substrate is coated with strip ITO electrodes along with the y-axis. The LC directors are initially aligned homogeneously along the rubbing direction, the x-axis, which is perpendicular to the strip electrodes. b Selective response with the linear polarization direction of polarization dependence of BD and chromatic aberration depending on the difference in wavelength. c Different phase profiles of 2Ï€ phase modulation for RGB wavelengths. d Concept of achromatic beam deflector (ABD) by exploiting the polarization state of light. Color-selective retarder (CSR)R, G, and B: CSR for red, green, and blue, BDR, G, and B: BD for red, green, and blue
In the voltage-on state, the voltage across the strip ITO electrodes and the planar ITO electrode generates spatially nonuniform electric fields, and the LC directors are then reorientated by them. The local tilt angle θ of the LC directors increases with stronger electric fields between the upper and lower electrodes, indicating that θ is minimized when the potential difference between the upper and lower substrates is low and maximized when the potential difference is high. When a linearly polarized light beam is incident parallel to the rubbing direction, the effective refractive index experiences72
$${n}_{{eff}}\left(\theta \right)={{n}_{o}{n}_{e}/({{n}_{o}}^{2}{\sin }^{2}\theta +{{n}_{e}}^{2}{\cos }^{2}\theta )}^{1/2}$$
(1)
where \({n}_{e}\) and \({n}_{o}\) are the extraordinary and ordinary refractive indices of the LC material, respectively.
By calculating the final effective refractive index (neff) distribution, the difference of phase retardation of BD is determined. The accumulated phase retardation φ, resulting from the maximum difference between the refractive indices of the incident light beam, is calculated by73,74
$$\Delta \varphi =\frac{2\pi \cdot \Delta n\cdot d\,}{\lambda },\Delta {{n}}={{{n}}}_{{{e}}}-{{{n}}}_{{{o}}}$$
(2)
where λ is the wavelength of the light, d is the thickness of the LC layer, and birefringence Δn is defined as the difference between the extraordinary (ne) and ordinary (no) refractive indices.
To achieve a minimum of 2Ï€ modulation across all wavelengths, the thickness of the LC layer was determined by the longest wavelength, red, as the reference. The linear phase profile assigned to the BD should be wrapped by \(2{\rm{\pi }}\) phase, because the dynamic range of the optical path length difference induced by the LC is limited by its thickness. Although various shapes of optical prisms can be achieved through phase retardation equation as shown above, we implemented a sawtooth phase profile to steer light in a single direction through BD, making it suitable for display applications.
In order to distinguish between (H)-type and (V)-type BD, (H)-type BD is defined as having a vertical electrode pattern to deflect light horizontally (i.e., in the left-right direction), whereas (V)-type BD is defined as having a horizontal electrode pattern to deflect light vertically (i.e., up and down). Phase retardation occurs when the optical axis of the LC is parallel to the direction of linear polarization, the polarization state parallel to the director of the LC can be deflected by the (H)-type BD whereas the orthogonal polarization transmits the (H)-type BD without deflection as shown in Fig. 1b.
The diffraction angle, \({\theta }_{\lambda }\) is determined by both the wavelength of light, \(\lambda\) and the period of the diffraction medium, \(p\) as following. \({\theta }_{\lambda }={\sin }^{-1}\left(\frac{\lambda }{p}\right)\). As described in Eq. (3), the steering angle is determined by the channel number of the prism and the cell size. Steering angle θstr can be derived as follows:
$${\theta }_{{str}}={\sin }^{-1}\frac{\lambda }{n\cdot p}\,,n=\frac{m}{i}(0 < {{i}}\le 360)$$
(3)
where λ is the wavelength of the light, n is the total channel number for one unit prism that is defined by m/i, m is the total number of channels, and i is the total number of unit prisms. The electrode pitch p is defined as 2 μm, consisting of an electrode width of 1.5 μm and an inter-electrode spacing of 0.5 μm. In Eq. (3), the diffraction angle is proportional to the wavelength, and the maximum steering angles are calculated as 6.60°, 7.72°, and 9.17° for 460 nm, 520 nm, and 638 nm, respectively. From the viewpoint of the whole device, considering all three wavelengths, the maximum diffraction angle should be determined by blue, which has the shortest wavelength among them.
Since BD has the common period \(p\), the diffraction angle differs for each wavelength, which causes the chromatic aberration of a single BD. The structure of a BD using 2\({\rm{\pi }}\) phase wrapping distorted phase distributions due to the phase discontinuity is described in Fig. 1c. The beam steering device utilized in this study features ITO electrodes with a 2 μm pitch and generates blazed phase profiles to electrically steer light in a desired direction. To direct light in a single direction when the three RGB wavelengths are incident simultaneously, the optical phase prism must be designed to optimize the wavelength-dependent diffraction angles. For instance, to direct all RGB light at an angle of 3.29°, the optical phase profile must repeat every 11.09 μm for red, 9.04 μm for green, and 8 μm for blue. For a BD with a 2 μm pitch electrode, a 2π modulation unit prism should be generated using 5.55, 4.52, and 4 electrodes for RGB, respectively. However, as the electrode can only represent integers and not decimals, phase discontinuities arise. The 2π modulation blazed phase profile is repeated several times across 720 channels, enabling the creation of a prism compatible with each RGB wavelength by fine-tuning the start and end values of each unit prism using an optimized algorithm.
The structure that selectively diffracts light of three wavelengths of RGB was described previously, and compensation for chromatic aberration will be explained as follows. Figure 1d shows the schematic for the 1D ABD system with three dynamic phased arrays. Each of the three BD is configured to deflect only the target wavelength out of the three incident lights, depending on their polarization. The color-selective retarders (CSRs) are required to control the polarization states independently among three wavelengths. It works as a half-wave retarder with 45° fast optic axis for designed wavelength while the other wavelengths undergo no retardation (or multiple of full-wave retardation). Thus, CSR only switches the polarization state of the target wavelength between two orthogonal polarization states. By switching between the orthogonal polarization states, RGB light can be controlled independently to arbitrary angles. It includes the case for the same diffraction angle for three different colors, which enables ABD. For the target wavelength, which is distinguished by the polarization state, the BD sets the appropriate phase profile with period of \({p}_{\lambda }=\frac{\lambda }{\sin \theta }\) where \(\lambda\) is the target wavelength and \(\theta\) is the diffraction angle. The 1D ABD beam steering can be used in a promising platform to achieve arbitrary 1D beam shaping at high speed, continuously tunable true-time delay lines in phased array antennas, a light focus in 1D to act as a cylindrical lens, and waveguide-integrated devices75,76,77,78,79.
2D achromatic beam deflector
For complete two-dimensional (2D) control of RGB light, additional beam steering is required in the vertical direction as well as the existing horizontal direction, which demands the use of additional (V)-type BDs. 2D beam steering is effectively realized by integrating two orthogonally oriented 1D BDs with intersecting electrode configurations. For optimal efficiency, the BD requires the input beam to be linearly polarized along the initial alignment of the LC. Thus, a 90° polarization rotator must be placed between the (H)-type and (V)-type BDs, and the (V)-type BDs are followed by the half-wave plate (HWP) after each (H)-type BD as shown in Fig. 2. It can be further extended the degree of freedom (DoF) so that one can construct the dual channel of 2D ABD, which independently controls two beams of RGB light, simultaneously. This can be achieved by launching the initial polarization state as the superposition of two orthogonal states |V〉+H〉| as shown in Fig. 3.
Fig. 2
Structure of two-dimensional ABD. CSR R and B: CSR for red and blue wavelength, BDH R, G, and B: (H)-type BD for red, green, and blue, BDV R, G, and B: (V)-type BDs for red, green, and blue wavelength. Six BDs are used to control three colors in two dimensions. Note that there are no CSRs for green wavelength, since the required polarization state after each deflection differs for each step due to the arrangement of (H)-type and (V)-type BDs
Fig. 3
Structure of the dual channel 2D ABD. a Three BD stages are placed in sequence for each red, green, and blue wavelength. The deflected light components are highlighted by the pink dashed circles after each BD. b Actual experimental setup of the ABD with dynamic phased arrays and the prototype includes optical components (CSR, HWP, Polarization plates), BDs and control boards, power connectors, and R, G, B laser sources
Figure 3 shows each layer of the unit BD to explain how each unit BD selectively deflects color among the overall structure of the 2D ABD with dual channels described above. The unit BD consists of four BDs for the channel #1/#2 and (H)-/(V)-type, three CSRs for 22.5° and 45°, and two HWPs. Also, each of the two (H)-type and (V)-type BD is arranged to correspond to the channel #1 and #2, respectively.
Initially, the beam is incident on the first CSR with a 22.5° optical axis with horizontal polarization using a linear polarizer. When the optical axis passes through 22.5° CSR for red, the polarization of the red wavelength is rotated by 45°, and the rest of the green and blue wavelengths remain horizontal without changing the polarization. For red beam polarized at 45°, its vertical (90°) component is used as a beam directed to the channel #1, while the horizontal (0°) component is used to the channel #2 as they pass through the following 4 BDs. The vertical (90°) component of the channel #1 beam reacts with the LC in the first (V)-type BD and is vertically deflected. Before entering the second (H)-type BD, the polarization of all beams rotates by 90°, passing through the HWP with a 45° optical axis. Then, the red beam of channel #1 is deflected horizontally in the second (H)-type BD. The CSR for red with a 45° optical axis switches the polarization of the red wavelength between horizontal and vertical polarization. In this case, the previously controlled red beam for the channel #1 has the same vertical polarization state as the rest of the green and blue beams, and only the red beam for the channel #2 has horizontal polarization which is perpendicular to the polarization of remaining 5 beams. The third (H)-type BD deflects only the red beam for the channel #2. Next, the polarization of the entire beam is rotated by 90° with the HWP having an optical axis of 45°, and then the vertically polarized light is incident on the fourth (V)-type BD. At this time, the red beam of the channel #2 is deflected in the vertical direction, and control of both beams of the red wavelength is terminated. Then, the 22.5° CSR was used to make the polarization state of the red beams of both channels to ±45° so that their power remains the same as they pass through the final polarizer.
The overall process for the unit BD stage can be expressed as the product of Jones matrices for each optical element as below
$$J={J}_{{POL}}\cdot {J}_{{CSR}22.5}\cdot {J}_{{BDV}}\cdot {J}_{{HWP}45}\cdot {J}_{{BDH}}\cdot {J}_{{CSR}45}\cdot {J}_{{BDH}}\cdot {J}_{{HWP}45}\cdot {J}_{{BDV}}\cdot {J}_{{CSR}22.5}$$
$$J=\frac{1}{2}\left[\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right]{{{for}}\; {{target}}\; {{wavelength}}}$$
$$J=\left[\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right]{{{for}}\; {{non}}-{{target}}\; {{wavelength}}}$$
The details of polarization evolution among optical elements in the achromatic BD using Jones formalism are available in Supplementary Note 1. The beams passing through the unit BD for red are then incident on the unit BD for green and blue consecutively, which are composed in the same way as that of red.
The intensity loss of each optical component was measured, and the unit BD for red was measured to have an efficiency of ~31% relative to the incident light (Supplementary Note 2). It is attributed to depolarization occurring within the LC layer, the retardation films, and reflections at the glass surface. Depolarization can partially result from phase differences between polarization components caused by the misalignment of LC molecules due to irregularities in the rubbing process. Additionally, it can also be induced by light scattering from spacer balls used to maintain the uniform thickness of the LC cell. This can be accomplished by minimizing alignment defects through methods such as precise rubbing techniques or photoalignment, utilizing high-purity LC materials, and employing optimized anchoring layers to mitigate depolarization effects. Additionally, applying anti-reflection coatings and index-matching layers can help reduce multiple reflections and the associated polarization changes.
Table 1 shows the sequence of the entire operation until the light of three wavelengths of RGB passes through each component and finally exits. 2D ABD with dual channels consists of a total of 12 individual BDs, which control each of the two pairs of RGB beams in desired directions by deflecting them vertically/horizontally. The key to this structure is that while the target beam assigned to the target individual BD is deflected, the remaining 5 beams must maintain orthogonal polarization with respect to the target beam so that they are not deflected by the corresponding BD. For this reason, not only a HWP that simultaneously rotates polarized light in the entire wavelength band, but also CSRs with different retardation for each wavelength are required. The described structure is also compatible with circularly polarized and structured light. To enable this functionality, a quarter-wave plate (QWP) is positioned at the input to convert circular polarization into linear polarization. At the output, another QWP is added to transform the resulting linear polarization back into circular polarization. Also, structured light can be controlled in the same manner, provided its polarization is spatially uniform, whether it is linearly or circularly polarized.
Table 1 Structure of 2D ABD with dual channels for RGB wavelengths