Liquid Lens Imaging Receiver

• Liquid Lens-Based Imaging Receiver is an adaptive optical front end that uses a tunable lens to reshape incident optical power onto sensor arrays.
• It employs both mechanical and electrowetting-based designs to adjust focal length and orientation, thereby mitigating spot overlap and channel correlation in MIMO VLC systems.
• This architecture improves indoor VLC performance by reducing BER and outage probability while accommodating user mobility and receiver misalignment.

A liquid lens-based imaging receiver is an adaptive optical front end in which a tunable lens is placed above a photodiode (PD) array or focal-plane array and controlled to reshape how incident optical power is mapped onto sensor elements. In indoor visible light communication (VLC), this architecture has been used to reduce spatial correlation in multiple-input multiple-output (MIMO) channels, to compensate random receiver orientation and user mobility, and to improve bit-error rate (BER) or outage probability. Reported realizations include a tunable liquid convex lens with controllable focal length and orientation angles, an electrowetting cuboid tunable liquid lens (TLL) whose planar liquid interface is steered by voltage, and an electrically actuated liquid-crystal (LC) metasurface lens that can replace a conventional droplet or deformable-membrane lens in a fully solid-state autofocus receiver (Palitharathna et al., 13 Mar 2025, Palitharathna et al., 20 May 2025, Bosch et al., 2021).

1. Receiver configurations and physical realizations

In the MIMO VLC formulation, a ceiling-mounted array of $N_t$ LEDs illuminates a mobile receiver carrying an imaging front end composed of a tunable liquid convex lens and an $N_r$-element PD array. The lens center is located a fixed distance $d_{\rm len}$ above the PD plane. The receiver therefore differs from a conventional fixed-lens imaging receiver by introducing three controllable degrees of freedom: focal length $f$, azimuth $\theta_L$, and polar tilt $\phi_L$. Changing $f$ varies the effective aperture radius $r=k_\eta f$ and lens area $A_L=\pi r^2$, while $(\theta_L,\phi_L)$ rotate the lens normal and steer the optical axis so that different LEDs can be mapped onto different PD subsets (Palitharathna et al., 13 Mar 2025).

The liquid convex lens realization described for MIMO VLC is mechanically tuned by applying a vertical force on an annular ring, and its orientation is adjusted by tilting the ring with magnetic actuators. This makes the lens an adaptive imaging element rather than a passive concentrator. Reported optimization and simulation ranges include $N_r$0 cm and $N_r$1, with the lens mounted $N_r$2 cm above the PD plane in a $N_r$3 m room model (Palitharathna et al., 7 Aug 2025).

A second realization is an electrowetting cuboid TLL. Here the lens is a cuboid of dimensions $N_r$4, partially filled with an optically transparent liquid, with the four vertical walls coated by dielectric and hydrophobic layers. The liquid-vapor interface remains planar, and the contact angles at the walls are voltage-controlled. Typical dimensions are $N_r$5 mm and $N_r$6 mm, with $N_r$7, dielectric permittivity $N_r$8, and interfacial tension $N_r$9 N/m. The resulting surface tilts $d_{\rm len}$0 can be set continuously in a range $d_{\rm len}$1, approximately $d_{\rm len}$2 for typical lens sizes and voltages up to $d_{\rm len}$3 V (Palitharathna et al., 20 May 2025).

A common simplification is to equate liquid-lens receivers with focal-length-only droplet deformation. The reported literature is broader. Bosch et al. demonstrated an LC-embedded dielectric metasurface lens that can serve as an ultrathin, solid-state varifocal element in a liquid-lens-based imaging receiver by replacing the conventional droplet or deformable-membrane lens with a flat LC-encapsulated semiconductor metasurface. Its core element is a 300-nm-thick amorphous silicon meta-atom layer on 150 nm of indium-tin-oxide (ITO) coated fused silica, enclosed in a $d_{\rm len}$4m LC cell filled with Merck E7 and driven by an AC voltage $d_{\rm len}$5 with $d_{\rm len}$6 kHz (Bosch et al., 2021).

2. Optical modeling and imaging-channel formation

For the liquid convex-lens MIMO receiver, the end-to-end channel between LED $d_{\rm len}$7 and PD $d_{\rm len}$8 is modeled as

$d_{\rm len}$9

where $f$0 is the Lambertian free-space gain from LED $f$1 to the lens aperture center and $f$2 is the imaging gain from the lens to pixel $f$3. The line-of-sight term is written as

$f$4

with $f$5, $f$6, $f$7 the LED-lens-center distance, $f$8 the irradiance angle, and $f$9 the incidence angle onto the lens. The imaging term is obtained by tracing each LED vertex through the lens using geometric optics and Snell’s law, constructing a spot polygon of area $\theta_L$0, and computing

$\theta_L$1

The spot area is evaluated with the Shoelace formula, and the overlap term captures how much of the refracted spot reaches a given PD (Palitharathna et al., 13 Mar 2025).

This formulation makes spatial correlation a geometric consequence of spot overlap on the PD plane. When several LED spots overlap strongly, the columns of the MIMO channel matrix $\theta_L$2 become highly correlated. By jointly adjusting $\theta_L$3, the lens re-shapes and re-orients the spot pattern, steering spots away from each other and increasing the pairwise distances $\theta_L$4 that appear in BER bounds (Palitharathna et al., 7 Aug 2025).

The electrowetting TLL formulation uses a related but distinct optical model. The wall contact angles satisfy the Young-Lippmann relation

$\theta_L$5

with opposing-wall constraints $\theta_L$6 and $\theta_L$7. The liquid-surface tilts are defined by

$\theta_L$8

The standard Lambertian model is then extended by refraction at the liquid surface. After Snell-law steering, the effective incident angle at the PD becomes $\theta_L$9, and the composite gain is

$\phi_L$0

Writing $\phi_L$1 isolates the lens-dependent term as an effective lens gain (Palitharathna et al., 20 May 2025).

3. Control objectives and adjustment schemes

The central optimization problem in the MIMO formulation is BER minimization over the liquid-lens degrees of freedom. Using a pairwise-error-probability-based upper bound, the problem is posed as

$\phi_L$2

subject to

$\phi_L$3

The corresponding problem $\phi_L$4 is non-convex. Exhaustive search or mixed monotonic programming is possible but costly for online use, which motivates approximate and low-complexity control schemes (Palitharathna et al., 13 Mar 2025, Palitharathna et al., 7 Aug 2025).

Scheme	Controlled quantities	Defining rule
PBML	$\phi_L$5	CNN extracts spatial features, LSTM/BiLSTM predicts user state, FC layers estimate optimal lens parameters
CLS	$\phi_L$6	Steer toward the closest LED $\phi_L$7
BSR	$\phi_L$8	Maximize $\phi_L$9, equivalently $f$0
VULO	$f$1 or $f$2	Force the liquid-surface or lens normal toward global $f$3

The prediction-based blockwise ML (PBML) architecture divides the task into three separately trained blocks. Block 1 takes a $f$4 received-power image $f$5, applies two convolution-ReLU-maxpool stages, flattens the resulting features, and uses fully connected layers to estimate $f$6. Block 2 takes the last $f$7 such estimates and uses an LSTM layer, a RepeatVector, a BiLSTM layer, and a time-distributed dense layer to predict the next user position and orientation. Block 3 maps the predicted state to $f$8 through a fully connected cascade. Training uses a blockwise mean-squared-error loss with labels from exhaustive search, and stochastic gradient descent updates $f$9 (Palitharathna et al., 13 Mar 2025).

The closest LED selection (CLS) scheme replaces ML inference with analytic steering. It identifies the nearest LED $r=k_\eta f$0, computes $r=k_\eta f$1 from closed-form expressions, and then sets $r=k_\eta f$2 to align the focus with the spot distance. Its stated complexity is $r=k_\eta f$3, and no artificial neural network inference is required (Palitharathna et al., 13 Mar 2025).

For the electrowetting TLL, the best signal reception (BSR) scheme seeks $r=k_\eta f$4, equivalently $r=k_\eta f$5, by numerical root-finding over feasible $r=k_\eta f$6. If no solution exists in the allowed tilt interval, the control falls back to a feasible boundary. The vertically upward lens orientation (VULO) scheme is lower-complexity: it forces the liquid-surface normal to global $r=k_\eta f$7, yielding $r=k_\eta f$8 and $r=k_\eta f$9 in the formulation of the paper. The trade-off reported for these schemes is explicit: BSR provides the best outage performance, whereas VULO is simpler but loses approximately $A_L=\pi r^2$0 dB SNR at high $A_L=\pi r^2$1 (Palitharathna et al., 20 May 2025).

4. BER behavior in MIMO VLC receivers

The detailed MIMO simulations use a $A_L=\pi r^2$2 m room, $A_L=\pi r^2$3, inter-LED spacing $A_L=\pi r^2$4 m, PD pitch $A_L=\pi r^2$5 mm, generalized spatial modulation with $A_L=\pi r^2$6, and 2-PAM intensity levels. Mobility follows a clothoid model with variance $A_L=\pi r^2$7, and orientation follows an AR(1) model with variance $A_L=\pi r^2$8. The PBML training set contains $A_L=\pi r^2$9 channel samples with a 70/10/20 train/validation/test split (Palitharathna et al., 13 Mar 2025).

An explicit steering example illustrates the geometric mechanism. With a fixed lens set to $(\theta_L,\phi_L)$0, $(\theta_L,\phi_L)$1, and $(\theta_L,\phi_L)$2 cm, one spot falls off the PD array and the others overlap, giving $(\theta_L,\phi_L)$3. After PBML adjustment to $(\theta_L,\phi_L)$4, $(\theta_L,\phi_L)$5, and $(\theta_L,\phi_L)$6 cm, all four spots are well separated and the BER becomes $(\theta_L,\phi_L)$7 (Palitharathna et al., 13 Mar 2025).

Across average SNR, the same study reports $(\theta_L,\phi_L)$8 at $(\theta_L,\phi_L)$9 dB for a receiver with no lens adjustment under random orientation, $N_r$00 for CLS, and $N_r$01 for PBML. The PBML result is only about $N_r$02 dB from the exhaustive-search optimum of approximately $N_r$03. As the input window $N_r$04 increases to $N_r$05, the prediction MSE for $N_r$06 falls from $N_r$07 to $N_r$08, and the reported PBML online inference time is approximately $N_r$09 ms (Palitharathna et al., 13 Mar 2025).

A related optimization study that emphasizes analytic schemes rather than PBML reports similar qualitative behavior. In that formulation, the static receiver BER saturates above $N_r$10, VULO yields modest gains, and CLS provides the strongest low-complexity improvement. At a random receiver orientation variance of $N_r$11, the BER is improved from $N_r$12 to $N_r$13 by employing the proposed liquid lens. The same study identifies inter-LED spacing optima of approximately $N_r$14 m for $N_r$15 and approximately $N_r$16 m for $N_r$17; very small spacing increases spot overlap, while very large spacing reduces edge-LED gains. Larger $N_r$18 and smaller PD pitch $N_r$19 further improve BER by capturing more spots (Palitharathna et al., 7 Aug 2025).

These results make the main design principle explicit: the advantage of the liquid lens is not only concentration gain but channel decorrelation through controllable spot placement. In the reported formulations, the lens is used to separate beams spatially on the PD plane so that the resulting MIMO channel is better conditioned for GSM and ML detection (Palitharathna et al., 13 Mar 2025).

5. Outage-probability analysis for mobile VLC

The electrowetting TLL literature addresses a different system objective: outage rather than BER. The received SNR is defined as

$N_r$20

and the outage probability is

$N_r$21

User position is modeled by a random-waypoint process in which $N_r$22 is random with a polynomial pdf over the room radius $N_r$23, while receiver orientation uses $N_r$24 and $N_r$25. For BSR, the choice $N_r$26 makes $N_r$27, and the resulting outage probability admits a three-case closed form in terms of the SNR threshold. For VULO, substituting the closed-form $N_r$28 into the channel model and applying Taylor expansion, product-and-convolution theorems, Gauss-Hermite quadrature, and Gauss hypergeometric identities yields a closed-form approximation for $N_r$29 (Palitharathna et al., 20 May 2025).

The reported numerical example uses $N_r$30, $N_r$31 m$N_r$32, $N_r$33 A/W, $N_r$34, $N_r$35 m, $N_r$36 m, $N_r$37, $N_r$38, $N_r$39, a transmit-power sweep $N_r$40 dBW, and $N_r$41. Under these conditions, at $N_r$42 dBW and $N_r$43, the outage probability drops from $N_r$44 for the no-lens receiver to $N_r$45 under BSR. At the same operating point, VULO yields $N_r$46, described as a $N_r$47 improvement over the fixed-lens receiver (Palitharathna et al., 20 May 2025).

The same study reports an optimum LED height of approximately $N_r$48–$N_r$49 m depending on $N_r$50, due to a trade-off between beam divergence and coverage. It also states that liquid-lens steering effectively extends the field of view: a conventional PD with $N_r$51 can sustain up to $N_r$52 extra misalignment. Practical design guidelines include voltages up to approximately $N_r$53 V to obtain $N_r$54 for $N_r$55 mm, real-time BSR root-finding at approximately $N_r$56 ms on an embedded MCU, and a TLL stack that adds approximately $N_r$57 mm thickness and approximately $N_r$58 g weight, with control voltages multiplexed on existing QVGA-driver lines (Palitharathna et al., 20 May 2025).

6. Sensor integration, implementation constraints, and solid-state varifocal extensions

The imaging-receiver concept generalizes beyond droplet-like liquid lenses to LC-driven metasurfaces. In the LC-embedded dielectric metasurface device, a metalens that focuses to distance $N_r$59 imposes the phase profile

$N_r$60

which reduces to $N_r$61 for small numerical aperture. Voltage tuning reorients the LC director and changes the effective refractive index seen by the meta-atoms according to

$N_r$62

with $N_r$63 and $N_r$64 for E7. When the local meta-atom phase follows the ideal varifocal profile, the focal length obeys

$N_r$65

The meta-atom library is built from COMSOL finite-element unit-cell simulations, and lens synthesis uses Fresnel-zone discretization and angular-spectrum propagation (Bosch et al., 2021).

For a 500 $N_r$66m-wide lens at $N_r$67 nm, the numerical design shows continuous tuning from $N_r$68 mm to $N_r$69 mm, corresponding to $N_r$70, with focusing efficiency $N_r$71–$N_r$72. Because the LC is driven at $N_r$73 kHz, the maximum tuning speed is on the order of $N_r$74 ms and is limited by the viscosity of E7. The experimental demonstration uses a bifocal metalens with aperture approximately $N_r$75m, illuminated by a $N_r$76-polarized $N_r$77 nm laser. When $N_r$78 is switched between $N_r$79 V$N_r$80 and $N_r$81 V$N_r$82 at $N_r$83 kHz, the focus shifts from $N_r$84 mm in the off state to $N_r$85 mm in the on state. The reported switching contrast for Lens 2 reaches a $N_r$86 increase of intensity at $N_r$87 and a $N_r$88 decrease at $N_r$89. The focal-spot FWHM is $N_r$90m in the off state and $N_r$91m in the on state, approaching the diffraction-limited value of approximately $N_r$92m, with minimal off-axis aberrations over a $N_r$93 mm field (Bosch et al., 2021).

For receiver integration, the metasurface lens can be mounted directly in front of a CMOS or InGaAs focal-plane array. In the reported interpretation, its active $N_r$94 tuning replaces both mechanical focus and liquid-droplet shape change. The stated depth-of-field example is a $N_r$95 focal-shift range covering object distances from approximately $N_r$96 cm to infinity for a typical $N_r$97 mm lens, with autofocus achieved by adjusting $N_r$98 while monitoring contrast on the sensor at a sub-millisecond tuning timescale. The total device thickness is below $N_r$99m, low power is on the order of $d_{\rm len}$00W, and the comparison given for mechanical liquid lenses is a tuning speed of approximately $d_{\rm len}$01 ms and a footprint of approximately $d_{\rm len}$02 mm. Reported implementation constraints include maintaining homogeneous ITO conductivity with electrode resistance below $d_{\rm len}$03, compensating residual spherical or chromatic aberrations by achromatic metasurfaces or cascaded metasurface layers, and stabilizing temperature dependence with a built-in temperature sensor and feedback on $d_{\rm len}$04 (Bosch et al., 2021).

Taken together, the reported literature defines the liquid lens-based imaging receiver as an adaptive optical architecture in which focal length, interface orientation, or phase profile is actively controlled to preserve or improve the mapping from incident optical beams to sensor pixels. In VLC, that control is used primarily to reduce channel correlation and outage under mobility and receiver tilt; in solid-state metasurface variants, it is used to realize ultrathin autofocus behavior with millisecond-class electrical tuning (Palitharathna et al., 13 Mar 2025, Palitharathna et al., 20 May 2025, Bosch et al., 2021).