Magnetically Controlled Optical Logic Gates Based on the Faraday Effect and the Three–Polarizer Paradox

Magnetically Controlled Optical Logic Gates Based on the Faraday Effect and the Three–Polarizer Paradox Ricardo Adonis Caraccioli Abrego∗Michael Joel Spilsbury Fuentes† Marco Antonio Reyes Pagoada‡ Universidad Nacional Autónoma de Honduras (UNAH) Campus Cortés, Honduras February 24, 2025 Abstract We propose and analyze a reconfigurable optical logic gate architecture that combines the Faraday rotation effect with the classical three–polarizer paradox. By dynamically rotating the polarization plane of light between two crossed polarizers, the system enables magnetic control of logical states, supporting AND, OR, NOT and multi–level logic operations. The approach suggests new pathways toward magneto–optical logic, optical memory elements, and programmable photonic/quantum circuits. Keywords: Magneto-optics, Optical logic, Faraday effect, Polarization, Photonic computing 1 Introduction Optical logic is a central component in advanced photonic computation, low–latency signal processing, and quantum information systems. A fundamental but counterintuitive phenomenon in polarization optics is the three–polarizer paradox, where inserting an intermediate polarizer between two crossed polarizers allows nonzero transmission. In this work, we extend this concept by introducing a magnetically controlled Faraday rotator as the intermediate element. Because the Faraday effect rotates the polarization proportionally to an applied magnetic field, it naturally enables reconfigurable logical states and analog/multi–level logic transitions. 2 Physical Principles 2.1 Three–Polarizer Transmission Two ideal polarizers oriented at 0◦and 90◦fully suppress transmission. If a rotation θoccurs between them, the output intensity follows the cascaded Malus law: Iout =I0cos2(θ) sin2(θ) = I0 4sin2(2θ).(1) ∗[email protected] †[email protected] ‡[email protected] 1 This provides a natural nonlinear transmission curve, with a maximum at θ= 45◦. 45 90 0.25 0.5 θ(◦) I/I0 I/I0=1 4sin2(2θ) Figure 1: Normalized transmitted intensity for the three–polarizer configuration, following Eq. (1). 2.2 Faraday Rotation In a Faraday medium, the plane of polarization rotates by θF=V BL, (2) where Vis the Verdet constant, Bthe axial magnetic field, and Lthe interaction length. Thus, the magnetic field acts as a tunable control parameter for optical transmission. 3 Proposed Architecture Figure 2 illustrates the proposed device: a linearly polarized laser passes through a first polarizer (P1), a Faraday rotator, and a crossed analyzer (P2). A photodiode measures the final intensity. Laser Polarizer (P1) Faraday Rotator Polarizer (P2) Photodiode Figure 2: Schematic of the magnetically controlled optical logic gate: linearly polarized laser, polarizer P1, Faraday rotator, crossed polarizer P2 and photodiode. 3.1 Logical Interpretation Logical states are defined as: •Logical 0: B= 0 ⇒θF= 0◦⇒Iout ≈0. •Logical 1: B=B45 such that θF= 45◦, yielding maximal transmission. •Analog / Multilevel: Continuous variation of Bproduces graded intensity levels between 0 and the maximum. 2 3.2 State Table Input Light Magnetic Field BRotation θFIout/I0 Present 0 0◦0 Present B45 45◦0.25 Present B90 90◦0 Absent Any — 0 Table 1: State table of the magnetically controlled optical logic gate. 4 Implementation of Logic Gates 4.1 NOT Gate A NOT operation can be implemented by interpreting the magnetic field as logical input while reading the optical intensity as output. For instance, if we define a high output (logical 1) as the absence of magnetic field and a low output (logical 0) when the field drives the system to maximum transmission, we can write: B= 0 ⇒1out, B =B45 ⇒0out. Alternative conventions (e.g. intensity inversion using an additional polarizer) can be adopted depending on the target logic family. 4.2 AND / OR Gates To implement two–input logic gates, we consider magnetic fields B1and B2representing the inputs. AND gate. A simple realization uses a single Faraday rotator with total magnetic field B=B1+B2. By choosing the individual input levels such that B1=B2=B45/2for logical 1 and 0for logical 0, one can set an intensity threshold Ith so that Iout > Ith only when B1=B2=B45/2, that is, only when both inputs are at logical 1. OR gate. For an OR operation, two parallel optical paths can be used, each with its own Faraday rotator and analyzer, generating normalized intensities I1and I2(each mapped to the interval [0,1] after suitable normalization and biasing). The total output intensity can be interpreted as Iout ≈I1+I2−I1I2, which acts as a smooth optical analogue of the logical OR operation (equal to 1 when either I1 or I2is 1, and 0 only when both are 0). Practical implementations can use beam combiners and suitable scaling to approximate this behavior in a robust way. 3 4.3 Optical Memory If the Faraday medium exhibits magnetic hysteresis (e.g. certain ferrimagnetic garnets), then a nonzero rotation can persist even after removing the external field, providing an optical storage mechanism. In such a configuration, a write pulse sets the rotation (and thus the logical state), while a weaker read beam probes the stored state without significantly disturbing it. 5 Discussion and Feasibility To estimate realistic operating parameters, consider terbium gallium garnet (TGG) with a Verdet constant of approximately V≈40 rad/(T ·m) near a wavelength of 1064 nm. For a target Faraday rotation of θF= 45◦=π/4and an interaction length L= 5 cm, the required magnetic field magnitude is B=θF V L ≈π/4 40 ×0.05 ≈0.39 T. This is within reach of compact electromagnets or integrated magneto–optical structures with permanent magnets. The extinction ratio of typical high–quality polarizers in a crossed configuration can exceed 30 dB, ensuring a well defined logical “zero” level. On the other hand, the maximum transmitted intensity at θF= 45◦is Iout =I0/4, which can be further amplified or normalized in subsequent optical or electronic stages. The switching speed of the device is primarily limited by the dynamics of the magnetic field, which can range from microseconds to milliseconds with conventional drivers. However, the use of high–speed current drivers, resonant magnetic structures, or magneto–optical resonators could significantly reduce the effective switching time, opening the door to faster optical logic and memory elements. 6 Conclusion We have presented a magnetically tunable optical logic gate based on the Faraday effect and the three–polarizer paradox. The device supports digital, analog and multi–level logic, and admits natural extensions toward more complex architectures. In particular, this scheme can be extended to: •Reconfigurable logic gates whose transfer function is tuned in situ via magnetic control. •Non–volatile optical memory elements using magneto–optical materials with pronounced hysteresis. •Analog optical computing blocks for matrix operations and weighted summation. •Hybrid interfaces between classical photonic processing and emerging quantum or spintronic platforms. These features illustrate a promising direction for magneto–optical computing and for the integration of logic, memory and processing within a unified photonic framework. 4 Acknowledgments The authors thank colleagues at UNAH Campus Cortés for helpful discussions on magneto–optical materials and photonic implementations. References [1] R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. 1, Addison– Wesley, 1964. [2] B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley, 2007. [3] A. Yariv, P. 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