OptRot: Rotational Techniques in ML & Optics

• OptRot is a multidisciplinary concept that employs rotational symmetry and optimization across machine learning, optical physics, and materials science to improve performance.
• It uses advanced mathematical frameworks such as optimization over Stiefel manifolds and Berry phase analysis to reduce weight outliers in neural networks and enable lossless polarization conversion in magneto-optics.
• Applications span high-fidelity quantization for large language models, dynamic structured light manipulation, first-principles optical rotatory power computation, and enhanced 3D reconstruction in optofluidic diffraction tomography.

OptRot refers to several rigorous methodologies and physical phenomena across modern machine learning, optical physics, and condensed matter theory, unified by the concept of rotation—whether in weight matrices, electromagnetic fields, or quantum mechanical operators. In post-training quantization, OptRot denotes a class of rotations that provably mitigate outlier magnitude in LLM weights, yielding tighter quantization bounds and improved accuracy. In optics, OptRot spans coherent perfect rotation (a lossless two-port Faraday resonance converting polarization) and structured-light "rotatum," where optical vortices acquire a quadratic chirp in orbital angular momentum along the propagation path. In material science, the optical rotatory power (OptRot) of crystals is now computed through first-principles modern theory, accounting for both electric-dipole and electric-quadrupole contributions. OptRot additionally appears as a reconstruction strategy in optofluidic single-cell diffraction tomography, where precise angular calibration via rotation about tilted axes enables three-dimensional refractive-index mapping. Across these domains, OptRot encapsulates the fusion of rotational symmetry with data-driven, coherent, or first-principles optimization.

1. OptRot in Quantization of LLMs

OptRot provides a mechanism for mitigating weight outliers in large-scale neural networks during post-training quantization, particularly GPTQ (Greedy Per-Row Quantization) (Gadhikar et al., 30 Dec 2025). Outliers—extreme values in weight or activation tensors—expand the quantization grid, causing high mean-squared reconstruction error,

$$\mathcal{L}(W,\hat W;H) = \mathbb{E}_{x\sim p(x)}\|W x - \hat W x\|^2 = \mathrm{tr}\left((\hat W - W) H (\hat W - W)^\top\right)$$

where $H$ is the input activation covariance.

OptRot exploits the invariance of linear layers in transformer-based architectures to orthogonal (basis-changing) rotations, learning fusible orthogonal matrices $R$,

$$\ell_{\mathrm{rot}}(W;R) = \sum_{i,j}\left[(R^\top W)_{ij}\right]^4$$

which minimizes the element-wise fourth power (proxy for $\|W\|_\infty^2$), reducing high kurtosis and compressing outlier magnitudes. The optimization occurs over rotations on the Stiefel manifold $O(n)$ by Cayley-SGD,

$$R \leftarrow (I - \frac{\eta}{2}A)^{-1} (I + \frac{\eta}{2}A) R$$

where $A$ is a skew-symmetric matrix derived from the gradient of the rotation loss.

Rotations are integrated and fused into the weights before quantization, preserving the original function while allowing more effective representation in low-bit formats. Empirically, OptRot outperforms Hadamard (QuaRot), SpinQuant, and OSTQuant for weight quantization in Llama-3.2-1B, achieving lower WikiText perplexity in the W4A8 regime (GPTQ weights + 8-bit activations). In more aggressive W4A4 (4-bit) quantization, exclusive focus on weight outlier reduction is less effective due to activation-dominated error, revealing a tradeoff. The data-dependent extension OptRot$^+$ further incorporates activation covariance information to tighten quantization bounds but at increased computational cost (Gadhikar et al., 30 Dec 2025).

2. Coherent Perfect Rotation in Magneto-Optics

Coherent Perfect Rotation (CPR) is the direct analogue to coherent perfect absorption (CPA) within conservative magneto-optics (Crescimanno et al., 2012). CPR manifests when two counter-propagating, coherent beams (e.g., both $x$-polarized) input into a slab are entirely converted into the orthogonal polarization (e.g., $y$-polarized) with zero outgoing $x$-polarized fields. This phenomenon occurs exclusively in time-reversal-odd (Faraday-type) systems, not reciprocal optical activity.

The theoretical framework is built upon a 4×4 transfer matrix $M_4$ acting on field vectors, incorporating block submatrices $M$ and $C$ with power conservation and time-reversal symmetry constraints. For Faraday rotation,

$$(n_1 + 1/n_1)S_1C_2 - (n_2 + 1/n_2)S_2C_1 = \pm [(n_1 - 1/n_1)S_1 - (n_2 - 1/n_2)S_2]$$

where $n_{1,2}$ are the refractive indices of circularly polarized modes, $S_{1,2}$ their respective propagation-phase sines, and $C_{1,2}$ cosines. Only with non-reciprocal (time-odd) mixing terms can all boundary conditions be simultaneously satisfied, leading to the two-port CPR resonance that is both phase-sensitive and lossless (Crescimanno et al., 2012).

Device designs with high-Verdet materials (e.g., terbium-gallium-garnet) and phase-locked input allow precise resonance tuning. CPR supports ultra-sensitive magneto-optical sensors by exploiting sharp extinction responses to external magnetic perturbations.

3. Optical Rotatum: Dynamic Topological Control in Structured Light

Optical rotatum denotes a class of beams wherein the local topological charge $\ell(z)$—the quantized orbital angular momentum per photon—varies quadratically (or polynomially) along the propagation axis $z$ (Dorrah et al., 2023). This introduces a nonzero second derivative,

$\frac{d^2\ell(z)}{dz^2} \neq 0$

yielding unique spatial phase and intensity structures. The field envelope is engineered such that

$\Psi(\rho, \phi, z) \sim e^{i\,\ell(z)\,\phi}$

with $\ell(z) = \alpha z^2$. The accompanying geometric Berry phase accumulated by the evolving mode family is

$\gamma_B(z) = -\pi\, [\ell(z) - \ell(0)]$

which in turn perturbs the longitudinal wavenumber,

$$\Delta k(z) = \frac{d\gamma_B}{dz} = -\pi\, \ell'(z)$$

modulating propagation constant and spatial frequency. Singularities in the beam trace logarithmic spiral patterns, $r(\phi) = r_0\, e^{b\phi}$, directly linked to the chirp rate $\alpha$.

Generation utilizes spatial light modulators with amplitude-to-phase encoding, while measurement extracts $\ell_{\mathrm{eff}}(z)$ via Poynting vector analysis along $z$.

Applications include depth-selective optical torque for light–matter interaction, depth multiplexing for free-space communications, and metrology through the beam's quadratic OAM mapping. Analogies span topological phases in condensed matter and quantum gases, where “rotatum” provides parallel control over winding numbers and phase singularities (Dorrah et al., 2023).

4. Optical Rotatory Power in Crystals: First-Principles Theory

Optical rotatory power (OR)—the degree of polarization plane rotation per unit length under transmission through a chiral crystal—is now rigorously computable for periodic band insulators (Desmarais et al., 2023). Modern theory decomposes orbital magnetization into local (LC) and itinerant (IC) circulations,

$$M_{\mathrm{LC}} = \frac{1}{2c} \mathrm{Re}\, \int_{\mathrm{BZ}} \sum_{i\in \mathrm{occ}} \Braket{ \nabla_k u_{i,k} \wedge (H_k/i) \nabla_k u_{i,k} }$$

and generalizes to non-local Hamiltonians via the Hermitized angular-momentum operator

$$\hat{\Lambda} = (r + i \nabla_k) \wedge (\nabla_r / i)$$

Rotatory strength for transitions $i \rightarrow a$ in direction $u$ is given by

$$R_{i\to a}^u(k) = [\, \Braket{\psi_{i,k}|\mathbf{p}|\psi_{a,k}} \wedge \Braket{\psi_{a,k}|(\hat{\Omega}_u+\hat{\Omega}_u^\dagger)/2\,\mathbf{p}|\psi_{i,k}} ]_u$$

with separation into electric-dipole–magnetic-dipole (DD) and electric-dipole–quadrupole (DQ) terms. DQ terms are as significant as DD for solids (as in $\alpha$-quartz), necessitating hybrid functionals and large AO bases. Linear-response DFT with orbital-relaxation yields rotatory angles in close agreement with experiment, contingent on accurate commutator corrections and gauge-invariance prescriptions (Desmarais et al., 2023).

5. OptRot in Optofluidic Diffraction Tomography

OptRot describes a reconstruction scheme in optical diffraction tomography (ODT) exploiting optofluidic rotation about arbitrarily tilted axes (Müller et al., 2018). Classical phase tomography is extended by modeling light diffraction through the Helmholtz equation,

$\nabla^2 u(r) + k_m^2 n^2(r) u(r) = 0$

with scattering potential $V(r) = k_m^2 [n^2(r) - n_m^2]$. After phase retrieval, the 3D Fourier diffraction theorem allows reconstruction:

$$\Delta n(r) = -i \frac{k_m}{2\pi} \int_{\theta} D_{\theta} \left\{ F_{2D}^{-1}[ \tilde{\phi}(k_\perp, \theta) |k_\perp| ] \right\} d\theta$$

where $D_\theta$ rotates back-propagated slices. When the rotation axis $\mathbf{r}_A$ is tilted by angles $(\alpha, \beta)$, reconstruction filters generalize to $|k_x v + k_y u|$, requiring calibration via diffraction-spot ellipse fitting in amplitude images.

The workflow—acquisition via dual-beam optical trapping and flow-induced rotation, tilt calibration via spot tracking, and slice-wise ODT reconstruction—enables high-fidelity, label-free 3D refractive index maps of individual cells, even under substantial tilt. Correction for tilt yields marked improvement in subcellular contrast (30–50%) and resolution (10–20%), enabling mechanobiological studies and advanced cell sorting (Müller et al., 2018).

6. Comparative Table: OptRot Domains and Mechanisms

OptRot Context	Physical/Algorithmic Mechanism	Key Impact/Metric
LLM Quantization (Gadhikar et al., 30 Dec 2025)	Fusible orthogonal rotation minimization	Lower GPTQ error, reduced outlier magnitude
Magneto-optics (Crescimanno et al., 2012)	CPR via Faraday rotation, 4×4 transfer matrix	Lossless polarization conversion, magneto-optical sensing
Structured Light (Dorrah et al., 2023)	Quadratic OAM chirp, Berry-phase shift	Tunable spatial structure, depth multiplexing
Crystalline OR (Desmarais et al., 2023)	Modern theory with hybrid DFT, DD & DQ decomposition	Accurate rotatory angles for solids
ODT–Optofluidics (Müller et al., 2018)	Tilted-axis calibration, ramp-modified back-projection	Improved 3D refractive index, cellular tomography

Across all contexts, OptRot mechanisms exploit underlying rotational symmetry or dynamic control to achieve improved quantization, mode conversion, phase control, or tomographic resolution, underpinned by mathematical frameworks from optimization over Stiefel manifolds to quantum geometric phase and reciprocal-space integrals.