Correction: Efficient communication channel for smart contact lens with resonant magnetoquasistatic coupling

The rapid rise of wearable electronics has reshaped how we consume personal and global information Datta and Sen (2023); Sen and Datta (2024); Sen et al. (2023). Smart eyewear, from connected spectacles Iqbal and Campbell (2023); Meta (2023) to emerging smart contact lenses (SCLs) Yao et al. (2024), pushes that trend to its logical extreme by embedding augmented reality (AR) graphics, virtual reality (VR) content, and continuous health sensing directly in the user's line of sight. Although today's smart glasses have matured enough to function as true "on-the-go" computers and lightweight AR/VR headsets, they are still relatively bulky and require frequent battery recharging. SCLs promise a far more discreet alternative but introduce an even tougher challenge: every sensor, processor, and wireless transceiver must be squeezed into a mm-scale, flexible form factor that rests comfortably on the eye. Related work has also explored communication and powering aspects more broadly. Shaker et al. (2023) survey several communication techniques for contact lenses, while Mirzajani et al. (2022) review external, in vivo, and hybrid energy-harvesting methods. Although such techniques offer promising routes for continuous powering, their implementation in a soft, mm-scale lens remains challenging, and many demonstrated systems still require proximity to an RF energy source Park et al. (2018). Misalignment effects in RF coil systems for implantables have been analyzed in Soma et al. (1987), but those studies typically consider coils of comparable size and relatively short separations.Figure 1a summarizes the distance-dependent design space for electromagnetically coupled systems.Strongly coupled systems within a few cm are standard for wireless power transfer applications and high-efficiency short-range communication. Radiative systems operate over much larger distances but at the cost of higher power consumption and more stringent antenna and regulatory constraints. Weakly coupled systems remain in the near-field regime with lower coupling strength, yet they can still support moderate power transfer or broader-band communication than tightly tuned resonant systems, enabling higher data rates De Quinto et al. (2024). Magnetically coupled communication has been demonstrated in prior work to support increased bandwidth, continuous implantable telemetry, and improved channel capacity in weakly coupled and misaligned near-field biomedical systems (Shaw et al. (2025a); Shaw and Sen (2025); Shaw et al. (2025b)).In this work, we extend the concept of weakly coupled electromagnetic systems and explore a magneto-quasi-static (MQS) resonant link between a wearable transmitter (Tx), embedded in a necklace, and a receiver (Rx) coil integrated into an SCL. The proposed geometry pairs a 15 cm multi-turn Tx coil worn as a necklace with an 8 mm peripheral Rx coil on the lens rim. In the 20-30 MHz band, these coils remain electrically small and the fields are predominantly magnetic. Because ocular and facial tissues have relative permeability close to that of free space (µ r ≈ 1), human tissue is largely "magnetically transparent" in this regime, and the link is primarily governed by coil geometry and relative position rather than tissue composition. Leveraging these properties, we show that a 26.8 MHz resonant necklace-lens link can sustain a ∼5 Mbps channel over a 1 MHz bandwidth at an axial distance of 15 cm and lateral offsets exceeding 9 cm, even under substantial misalignment between the necklace and the eye.Compared to RF, UWB, and optical links, the proposed resonant MQS approach occupies an intermediate point in the design space: it operates at low carrier frequencies that naturally ease ocular SAR constraints, uses simple mm-scale coils compatible with hydrogel-based lenses Yao et al. (2025), and provides a wideband, misalignment tolerant channel suitable for continuous AR/VR and sensing data streams, at the cost of being confined to the near-field regime.While powering the contact lens remains critical to its functionality, the present study focuses exclusively on the communication challenge. We assume that the lens is equipped with either a small battery or an energy-harvesting solution Griffith (2022); Chatterjee et al. (2023), and leave the detailed powerdelivery architecture to future work. Through detailed 3D finite-element simulations combined with careful analysis of coil sizing, placement, and orientation, we present an electromagnetic and information-theoretic proof-of-concept of the MQS channel between a necklace Tx and an SCL Rx coil.By demonstrating a robust, high-capacity link using resonant MQS coupling with strongly misaligned coils of dissimilar dimensions, this work establishes the foundational viability of necklace-lens MQS communication and motivates future research on power efficient transceiver integrated circuits (ICs), ocular compatible coil fabrication, and full system co-integration for AR, VR, and health-monitoring applications.An alternating current loop induces an electromotive force in a nearby conductor via its magnetic field.Its strength and the induced flux fall with coil size, separation, misalignment, and the surrounding medium.Greene's refined near-zone model for small circular loops Greene (1967) and Simpson's exact solution for any circular loop Simpson et al. (2001) capture these dependencies, accounting for frequency, propagation delay, and finite radii. As illustrated in Fig. 1b, the resulting field lines emerge perpendicular to each coil's plane, extend axially through the center, and curve back near the conductors, even when the Tx and Rx coils are laterally offset.For a weakly coupled pair, path loss given in [1] depends on the coupling factor (k) between the coils; separation distance [2]; alignment: axial (d z ), lateral (d x , d y ), and angular (θ); and operating frequency (ω). The properties of the coil, such as its inductance (L T x , L Rx ), quality factor (Q T x , Q Rx ), and parasitic capacitances, contribute to the coupling factor between the coils.P L = f (L T x , L Rx , r, d x , d y , d z , θ, k, Q T x , Q Rx , ω),(1)r = d 2 x + d 2 y + d 2 z ,(2)Greater dimensional mismatch lowers [3] and thus the mutual inductance [4]:k = M √ L T x L Rx ,(3)Since k is determined by the mutual inductance M , it ultimately depends on each coil's self-inductance Waters et al. (2014). For flat spiral coils, L is given by the modified Wheeler expression Hussain and Woo (2021), varying with the inner/outer diameters, conductor width, and number of turns.The arbitrarily aligned coil model of Golestani and Moghaddam (2018) provides the mutual inductance and hence the coupling coefficient for the misaligned link, in simplified formM = N Rx ϕ T x I T x = N T x ϕ Rx I Rx (4)where ϕ T x is the flux from the Tx that threads the Rx coil given by [5]ϕ T x = S Rx B T x . dS Rx (5)where S Rx is the Rx coil area and B T x = µH T x is the Tx's flux density. Closed-form expressions for M that incorporate coil position, angular misalignment, frequency, geometry, separation, and medium are detailed in Golestani and Moghaddam (2018).The voltage transfer ratio between the Tx and the Rx can be calculated using [6] The total power at the Rx(P T otal ) is a function of [7]V Rx V T x = jωM R Load (jωL T x + R Source )(jωL Rx + R Load ) + ω 2 M 2 (6) Detailed Circuit Diagram V Source C Tx C Rx R Source R Load L Tx L Rx V Rx Receiver Coil Transmitter Coil R Load R Source V Source Magnetic Coupling C Rx C Tx R Source C Tx L Tx L Rx C Rx R LoadP T otal = f (k, Q 1 , Q 2 , V IN , I IN )(7)Coil Dimensions: The number of turns, coil radius, wire diameter, and wire spacing change the coupling coefficient between the Tx and the Rx. Similar-sized coils have higher coupling efficiency than coil systems with widely varying diameters. As the number of coil turns increases, the coupling between the coils increases, but further increases lead to reduced efficiency due to increased parasitic capacitance between the wires. As the coil's radius decreases, the concentration of magnetic field lines decreases, leading to lower coupling. Variations in the coil radius and the number of turns change the inductance, series resistance, and parasitic capacitance of the inductor, thereby affecting the overall impedance and the quality factor of the circuit.Alignment: Link efficiency peaks when coils are similar in size and perfectly coaxial. In practice, the necklace Tx tilts with the chest, while the lens mounted Rx sits far away, fixed to the eye's axis. These unavoidable lateral, axial, and angular offsets limit coupling. The following sections quantify how such misalignments and required coil sizes affect performance and demonstrate that the proposed MQS link retains adequate margin under typical head and body motion.Source and Load Resistances: In Figure 2, R Source represents the effective internal resistance of the transmitter side, which includes the resistance of the voltage source as well as any series resistance from the transmitter coil and its driver circuitry. It governs the current supplied to the transmitter coil; hence, lowering R Source increases the coil current, strengthens the magnetic field, and enhances the power transfer to the receiver, albeit at the cost of higher source power consumption. On the receiver side, R Load denotes the effective resistance of the connected the receiver coil the C Rx . the or that energy from the induced R Load to a higher voltage the receiver coil for a given induced current, which is for communication a larger voltage is it the current at the 2 the resonant MQS The transmitter a L T x , tuned to the operating frequency by a series C T x and by a 1 V V Source through a source R Source that the coil The receiver uses a spiral L Rx with a C Rx to T x C T x = L Rx C Rx The induced voltage is the R Load as V Rx peaks from each coil's parasitic capacitance turns the spiral into a short necklace-lens link is to operate in the MHz (MQS) For the coil 15 cm necklace and separation 15 the = ≈ 26.8 ≈ in free this the field remains in the near field, is and the coils are electrically the coupling is predominantly magnetic and standard MQS are practice, we the and for a the coil and surrounding are in the the of the link at 26.8 26.8 MHz is a it is the of of concept implementation within the 20-30 MHz carrier in this MHz with similar coil geometry and and operating well the of the is to the tissue link and SAR in this coupled system is the Tx coil is embedded in a necklace and the Rx spiral is integrated the of the contact lens to a The coil are to the necklace and contact lens The number of turns of the coils is fixed such that coupling the parasitic capacitance and the quality factor of the predominantly and tissues have magnetic permeability close to that of free space (µ r ≈ the magnetic field and mutual coupling are weakly by the or detailed of the and are by the coil geometry and their relative separation and is with prior studies on wireless power the human body is in the of the source magnetic field it as a conductor and the magneto-quasi-static field et al. In we a for the and tissues to the while necklace-lens separation and misalignment to for Tx flat spiral is the while a Rx spiral is embedded the contact lens rim. spiral is as an The Tx is by a 1 26.8 MHz through a series source and C T x C T x the link The Rx in a series C Rx and R Load , which V Rx is is in Figure 2, the flat spiral coil is in the form of to the the Rx spiral is as a coil that the contact lens as in Figure its radius, wire diameter, and spacing remain to the geometry of Figure In the necklace and lens coils are as with fixed primarily to capture the electromagnetic and link using an In practice, these coils can be using or on ocular as demonstrated in prior smart contact lens and coil et al. As resonant MQS coupling is largely determined by coil geometry, inductance, and series resistance, we comparable channel performance when with such compatible conductors, with in quality The necklace transmitter is to a the Rx coil is to the their relative with head and body study coils at turns and the Tx to quantify its on the capacity of the resonant communication channel is with the given by is the bandwidth of the resonant link, and is the voltage ratio at the = 2 1 + this work, we to a on the information of the necklace-lens MQS link under the following a fixed of over the bandwidth, by the of the between the Rx coil and the receiver with implementation and of The resulting capacity be as for the rather than data rates for a receiver source is the of the receiver that the Rx voltage is as in Fig. at with channel ≈ for = , and the as a design for the larger a the as by the curve in Fig. 1 summarizes the link As the quality are the bandwidth and the yet the is an in channel in to combined with a bandwidth to 1 the data from Mbps to to bandwidth Mbps data which is for or biomedical telemetry, while still a margin the under the practice, smart contact lens operate under stringent area and power constraints, which limit the and the For a in the as 2 , further the larger and hence higher current and power consumption. For low power operating from a V and ) power these the effective by several relative to the such as implementation and also and the data the even with a in effective the link in 1 that data rates at cm separation remain when the MQS link is combined with a receiver full of the Rx and link design is for future work. ≈ the link when the Tx spiral is directly from the source and with C T x that coils at 26.8 MHz the voltage by the more than the The bandwidth of the tuned link is ≈ but be to 1 MHz by lowering the coil Q to a we to to Mbps channel is it of of link and provides bandwidth to data AR/VR coils are to several to the of body tissues on the channel between the Tx and the Rx The coils are in the any human body the Tx coil is the head and body model of the while the Rx coil is to the tissues as in Figure the for the of is on of the Rx coil, Figure frequency for coils in and resonant coils in free and and and tissues with an to a The voltage is in resonant tissue The in Figure at the 26.8 MHz show magnetic field from the relative permeability of body tissues (µ r ≈ 1), which the MHz MQS field largely et al. The tissues weakly the magnetic field and introduce a relatively small of the magnetic coupling on the of and facial tissues the at 26.8 the human body is largely and that the link through in separation and rather than through of the this the relative magnetic permeability of tissues is (µ r ≈ 1), the tissues the magnetic the the tissue is at 26.8 and can lead to These the MQS the communication channel is to the surrounding ocular and tissues at the operating in the in the proposed for enabling efficient communication in SCLs with a resonant MQS coupled the coils be work the of such a system and the communication channel efficiency to AR/VR on contact The of the Rx coil is perpendicular to the the Tx coil is a necklace, its can due to body and its spacing from the from to at rather than the perpendicular axial distance and the lateral between the Tx and the Rx coil on their body and Figure the misalignment to for the in channel = = and Tx = are the and for the coil are for the proposed but they can still be to misalignment in coupled Tx from to the to to the the coil or perpendicular to the is within the to a of 8 For a circular necklace, an of has been simulations to demonstrate the of wireless power transfer between a necklace and a contact For the lateral of the the channel capacity is in when the Tx coil is at and It is that the remains over Mbps as in Figure The is mm to show in human for coil coupling between the axially aligned coils increases as the distance between the magnetic field lines to strongly when their are In Figure misalignments are into For = mm it is that the field lines coupled as the distance between the coils increases, while for = the The in the Tx is in this the channel is for with a of over the resistance and the the coil Load impedance at the Rx the voltage the coil for the source resistance R Source the current in the Tx coil, the magnetic field, and lowers the induced at the a R Source a higher V Rx and thus a larger Figure 8 this at the 26.8 MHz R Source from 1 to the voltage by The is higher transmitter power for a 1 V when R Source = operating point is as a design in this its for receiver power and link energy efficiency are in and its for ocular are in Tx of the link can be by a of the necklace coil inductance L Tx , its series resistance R coil , the source resistance R , and the C Tx the operating ω = √ L Tx C Tx , the current is Tx = V R + R coil + Tx Tx ) V is the source the and the current = R + R coil + R R any series resistance due to and The quality factor of the Tx can be as = ω L Tx R + R coil + R the bandwidth is Tx ≈ f Q expressions that the source resistance R the coil current at thereby lowering the magnetic field and while also the Q and the a R increases and Q , leading to higher link but a the resistance R Load increases the voltage the receiver coil, thereby the As in Figure V Rx with R Load and to R Load this resistance further a R Load is in the to communication sections on the electromagnetic link and the transmitter power required to sustain a channel a operating point in a of the receiver power on the smart contact lens and the resulting link energy efficiency is receiver on the SCL the lens coil and a to the induced a simple and a small for and to low power and near-field in the MHz to typically total power in the to a few of for Mbps data we assume a SCL receiver power in the of Rx ≈ for the Mbps channel this to a receiver energy of = Rx R ≈ ≈ is compatible with for short-range In for the transmitter power in this the energy is = ≈ 2 at this operating the overall link energy efficiency is by the transmitter rather than the is to that is an design point to demonstrate of the MQS channel under misalignment and In practice, the power can be reduced by coil geometry and impedance and or power that the channel full design and detailed power of the SCL receiver are the of this work, but these show a receiver power in the to a few of on the contact lens is and the proposed MQS link can support such a receiver while margin to transmitter energy in necklace-lens link operates in the MQS regime at 26.8 the coupling is predominantly magnetic and the fields in tissue remain relatively In this frequency the relative permeability of ocular and facial tissues is that of free space (µ r ≈ 1), and the fields are largely As in the of and as well as an to a change in voltage and that a small of the power is in in the coils and rather than in the a on for the operating point in we consider a in which the is in a tissue in the In that the is calculated using = = = 1 a SAR limit of 2 over of tissue for head and to the on even under this the SAR is at a factor of the practice, the power near the be than the link substantial between the necklace Tx and the SCL and the near field that of the power is either in the field or in rather than in The be as an design point for link and channel a the required power can be reduced further by the coil geometry, and and by or power on channel be in future work, using and simulations to SAR 1 and and rise under These studies with SAR and for ocular and further the for continuous of smart contact support for the we a of the necklace-lens MQS link The Tx coil is a loop with an of cm and a conductor of The Rx coil is a spiral with an of 1 cm, spacing of and conductor of a show of the in and with a tissue of the space between the coils are tuned to near MHz using On the Tx side, a C Tx ≈ is in series with the On the Rx side, a C Rx ≈ is in series with the Rx As in Fig. the Tx path of a function by a voltage that the series of the Tx coil and C Tx . The Rx path of the Rx coil in series with C Rx and directly connected to a is at the and the frequency is from to MHz while the is at the Rx. is in Fig. from which a bandwidth of MHz MHz is a the Tx coil is by with to the plane, and the Rx coil is cm the Tx and laterally by these strongly misaligned the a near with an overall channel loss of at is with the link by simulations for comparable coil and misalignment, and that MQS coupled communication is at of MHz over even in Figure the channel separation d V for and the loss remains in the of to as d V increases from to the of a simple human tissue between the coils, using with properties close to those of human tissue at the operating the is between the Tx and Rx coils, the channel loss a few the frequency and for that the of has on the magnetic coupling at is with the magneto-quasi-static analysis and simulations which that tissues with r ≈ 1 are largely and that coupling is by the coil geometry and relative The resulting bandwidth, and channel are in Fig. These are as a integrated smart contact lens but they that the of this MQS coupling can sustain communication over distances with substantial misalignment and the of the link in the frequency work a (MQS) resonant link for data communication with a smart contact lens primarily on and and by a The proposed system pairs a 15 cm Tx coil with an 8 mm Rx coil on the lens a link of and a channel capacity of Mbps over a 1 MHz performance is in while 15 cm axial 9 cm lateral and angular Because human tissue the permeability of free space the link remain when are and the required power is on the of with prior SCL confined to cm these extend the communication simple with necklace and lens coils, near MHz and under misalignment and with a tissue that communication is at distances and that tissue has a on the channel these support the for the geometry and frequency the communication channel is to be the factor for AR/VR and sensing of the proposed approach with prior work on communication to smart contact lenses is in