Optical Intelligent Reflecting Surfaces

• OIRS is a technology comprising engineered surfaces that programmatically control visible/infrared light to mitigate line-of-sight issues and enhance indoor positioning.
• The article details the use of Lambertian channel models, ML/RML distance estimators, and IWLS algorithms to achieve cm-level accuracy with single-LED and single-photodiode setups.
• Adaptive beam steering through sequential OIRS activation minimizes multipath interference, ensuring robust optical localization and improved spatial coverage.

Optical Intelligent Reflecting Surfaces (OIRSs) comprise engineered surfaces that manipulate visible or infrared light by programmable means, typically to enhance or extend wireless communication capabilities. In visible light communication (VLC) and optical wireless communication (OWC) systems, OIRS technology is leveraged to overcome line-of-sight (LoS) blockages, facilitate spatial diversity, and enable advanced localization tasks. This article presents a detailed account of OIRS design and operation, with a specific focus on their role in single-LED, single-photodiode (PD) indoor positioning as established in recent research (Pugliese et al., 2 Jul 2025).

1. Geometric and Channel Model for OIRS-Enabled VLC Positioning

The fundamental system comprises a single anchor LED at a known position, a low-cost PD at an unknown floor location, and multiple spatially distributed single-element OIRSs mounted on the room's walls. Each OIRS element is characterized by its 3D position $\mathbf w_n$, orientation (horizontal yaw $\alpha_n$ and vertical pitch $\beta_n$), and a binary activation state $a_n$ (active or deactivated).

The direct LoS optical channel from LED to PD obeys the Lambertian model:

$$h = \frac{A\,T\,G\,(m+1)}{2\pi\,d^2}\, \cos^m\theta\, \cos\varphi,$$

where $A$ is the PD area, $T$ is the optical filter gain, $G$ is the concentrator gain, $d$ is the Euclidean distance from LED to PD, $\theta$ and $\varphi$ are the LED–PD irradiance and incidence angles, and $m$ is the Lambertian order.

Each OIRS-enabled NLoS path consists of a reflection point $\mathbf r_n$ on the $n$th OIRS, with:

$$h_n = \rho\,\frac{A\,T\,G\,(m+1)}{2\pi\,(s_n + d_n)^2} \cos^m\theta_n\,\cos\varphi_n,$$

where $\rho$ is the OIRS reflectivity, $s_n = \|\mathbf q - \mathbf r_n\|$ is the LED-to-OIRS distance, and $d_n = \|\mathbf r_n - \mathbf u\|$ is the OIRS-to-PD distance. All path angles and distances are determined by the OIRS orientation and the estimated PD position.

The system is configured such that only one OIRS is active at a time, avoiding multi-path interference and allowing for clear measurement of each NLoS component.

2. Activation Protocol and Measurement Strategy

The OIRS activation strategy proceeds sequentially. For each measurement epoch:

• All OIRS elements are initially deactivated.
• The LoS signal is sampled ($K$ times), yielding a direct measurement of the LED–PD channel.
• Each OIRS is activated in turn (setting $a_n=1$), while all others remain inactive. The PD records $K_n$ samples for the current OIRS configuration, effectively isolating the NLoS component via the $n$th OIRS.
• Subtracting the estimated LoS signal from each OIRS measurement cleanly extracts the single-reflection gain.

This protocol ensures that the statistical independence assumption holds for subsequent inference procedures.

3. Maximum Likelihood Estimation of Distances

The direct LoS distance $d$ is estimated from noisy signal amplitude measurements $\mu_{0,k}$. The signal statistics are:

$$\mu_{0,k} = R\,p\,h(d) + \eta_{0,k}, \quad \eta_{0,k} \sim \mathcal N(0, \sigma_0^2(d)),$$

where $R$ is the responsivity, $p$ the emitted optical power, and noise variance $\sigma_0^2(d)$ accumulates shot and thermal noise contributions.

The log-likelihood for $K$ samples allows for a closed-form ML estimator:

$$\hat d_{\text{ML}} = \left[\frac{\xi(\sqrt{(a+bS_1)^2 + b^2(S_2 - S_1^2) + \frac{b^4}{4} + a + \frac{b^2}{2}})}{2aS_1 + bS_2 - ab}\right]^{1/(m+3)},$$

with $S_1 = \frac{1}{K} \sum_k \mu_{0,k}$ and $S_2 = \frac{1}{K} \sum_k \mu_{0,k}^2$.

For each OIRS-mediated NLoS path, residual analysis gives:

$$\chi_{n,k} = \mu_{n,k} - \bar\mu_0(\hat d)\approx R p h_n(d_n) + \eta_{n,k},$$

where the dependence of $\sigma_n^2(d_n)$ on $d_n$ can support either full ML (via 1D grid search) or a relaxed ML (RML) estimator. The RML estimator admits a closed form as the real root of a depressed cubic.

4. Iterative Weighted Least Squares (IWLS) Positioning

After collecting $d$ and $\{d_n\}$, positioning reduces to solving nonlinear equations of squared distances between the PD and the reference points (LED, OIRS centers). This is cast as a weighted least squares problem:

$$\min_{\mathbf u} (\hat{\mathbf d}^2 - \mathbf f(\mathbf u))^T \mathbf W (\hat{\mathbf d}^2 - \mathbf f(\mathbf u)),$$

where weighting is set by the inverse Cramér-Rao lower bound (CRLB) for each distance estimate. The IWLS update iteratively linearizes the mapping $\mathbf f(\mathbf u)$ and converges rapidly (within 2–3 iterations) to the global optimum in practice.

5. Adaptive OIRS Beam Steering Algorithm

Once a PD position estimate $\hat{\mathbf u}$ is available, each OIRS adapts its orientation to maximize received signal strength. The element's surface normal is:

$$\mathbf o_n = \frac{\frac{\mathbf q - \mathbf w_n}{\|\mathbf q - \mathbf w_n\|} + \frac{\hat{\mathbf u} - \mathbf w_n}{\|\hat{\mathbf u} - \mathbf w_n\|}}{\left\|\frac{\mathbf q - \mathbf w_n}{\|\cdot\|} + \frac{\hat{\mathbf u} - \mathbf w_n}{\|\cdot\|}\right\|},$$

which geometrically bisects the incident (LED-to-OIRS) and reflected (OIRS-to-PD) directions.

Mechanical steering angles are then extracted as:

$$\beta_n = \arcsin(\mathbf o_n^T \mathbf e_3), \quad \alpha_n = \arcsin\left(\frac{\mathbf o_n^T \mathbf e_1}{\cos\beta_n}\right).$$

This procedure is repeated in the IWLS loop to continually refine both the position estimate and OIRS alignment.

6. Theoretical Bounds: Cramér-Rao Analysis

CRLBs are derived both for individual distance estimates (distance error bound, DEB) and for the overall position estimate (position error bound, PEB):

• For distance estimation, the Fisher information includes dependences on channel gain, noise statistics, and signal-to-noise ratio.
• The DEB and subsequently the PEB (via the chain rule and Jacobian transformation) quantify the minimal achievable variance for unbiased estimators, serving as benchmarks for algorithmic performance.

7. Simulation-Based Performance and System Insights

Extensive simulations under realistic noise and OIRS misalignment conditions establish that:

• LoS and NLoS distance estimators (ML, RML) achieve the DEB with only a few samples ($K\leq 5$) at modest SNR ($\approx10$–15 dB).
• Positioning via IWLS rapidly converges to within a few mm of the PEB, even for moderate OIRS misalignments and limited number of global iterations.
• As the number of OIRSs increases, both the PEB and actual root mean square error (RMSE) decrease, and volumetric coverage (fraction of room area where accurate positioning is attained) rises, indicating smooth performance scaling with network densification.

8. Technical and Practical Implications

The distributed single-element OIRS paradigm represents a robust, low-cost, and scalable solution for indirect indoor localization under minimal infrastructure. Closed-form ML/RML distance estimators and a principled IWLS fusion algorithm provide theoretical guarantees of accuracy, subject only to the system geometry and noise conditions. Adaptive beam steering with OIRSs can be achieved with standard mechanical actuation, and the modular activation protocol circumvents both multipath interference and complex channel estimation. This approach supports optical indoor positioning at cm-level accuracy without RF or multiple-LED infrastructure, thus expanding the operational envelope of VLC-based smart environments (Pugliese et al., 2 Jul 2025).