2.1.1.

Resolution of reflectometry

Reflection intensity sensing. Reflectometers are limited both in the minimum value of $R$ that they can sense (termed the sensitivity) and in the minimum value of $\mathrm{\Delta }R$ that they can sense (termed the resolution). In our device, the baseline power reflected at the boundaries between $n_1$ and $n_2$ will be much greater than the sensitivity of the reflectometer. Thus, the ability of the reflectometer to measure a change in the index of refraction is limited by its resolution, which is in turn fundamentally limited by photon shot noise. For the simple case of a time-domain reflectometer, the number of photons registered at the detector due to a reflector of magnitude $R_0$ is given by

In all that follows, we will assume a bandwidth of 1 kHz, a quantum efficiency of 1, and a free-space wavelength $\lambda =1550\mathrm{nm}$. Note that a higher bandwidth would be required to see the detailed shapes of individual action potentials, as may be required for spike sorting. For $P=100\mathrm{mW}$, $\mathrm{QE}=1$, $\mathrm{BW}=1\mathrm{kHz}$, and $\mathrm{\Delta }{n}_0={10}^{-4}$, corresponding to $R_0=2.16\times {10}^{-10}$ ($-96.6\mathrm{dB}$), we then have a shot noise limited resolution of 0.01 dB, similar to existing reflectometers. More generally, for a signal $\mathrm{\Delta }R$ to be sensed on top of a signal $R_0$, we must have

Fig. 2

Minimum resolvable value of $\mathrm{\Delta }n$. (a) A schematic example of the expected output trace. Black color is the baseline reflection registered by the device; orange color is the reflection measured when the neuron fires. Spatial resolution is exaggerated for illustration. (b) The minimum change in the index of refraction of the optical fiber that can be sensed by an ideal, shot noise-limited reflectometer is shown as a function of the laser power $P$ for a 1-kHz bandwidth, index of refraction of 3.36, quantum efficiency close to 1, and 1550-nm wavelength.

So far, we have discussed the detection of changes in the refractive index via the modulation of reflectivity at each interface. An alternative strategy to detecting changes in the refractive index that accompany voltage signals is to measure the phase of the reflected light. This phase measurement can be performed with the identical Fourier-domain reflectometry scheme as for the amplitude-based measurement.

Spatial resolution. Other noise sources will also impact resolution in a realistic case, including laser power or phase noise and photodetector/amplifier/ADC noise. In particular, optical phase noise associated with the laser is limiting in current optical frequency-domain reflectometry (OFDR) systems;¹⁹ Littman–Metcalf external cavity tunable lasers, with narrow linewidths and low phase noise, can be swept at 1-kHz repetition rates over an optical frequency range of several THz, leading to an OFDR “spatial” resolution of roughly $20\mu \mathrm{m}$, which conveniently aligns with the average spacing between neurons in the cortex.

Repetition rate. Current commercial reflectometers achieve roughly 12-Hz repetition rates over 8.5 m. This corresponds to a measurement time of 1 ms for any given 10 cm segment of fiber, so using a similar device we anticipate that it would be possible to sense reflections along the length of a 10-cm fiber with a repetition rate of 1 kHz using frequency-domain reflectometers. In an OFDR system, the scan rate is limited by the frequency of laser wavelength scanning, the range of the scan determines the resolution, and the wavelength resolution of the scan and of the detector determines the scan range. Swept-source OCT constitutes demonstration of swept-source interferometry at a bandwidth of many kHz.²⁰

2.2.1.

Free carrier dispersion effect

The design shown in Fig. 1(b) consists of an extended multilayer semiconductor waveguide on a biased metal substrate, surrounded on three sides by insulation and on the fourth side by brain tissue or extracellular fluid. The “inner conductor” and “core” layers are weak, transparent conductors which function as resistive layers between the brain and the biased reference line. Throughout, we will assume that the core is made of n-doped InP, due to its large free-carrier dispersion effect,²³ although other core materials are possible (see Sec. 4). Both above and below the core, there are $\sim 5\mathrm{nm}$ thick layers [Fig. 1(c)] in which the material alternates along the length of the fiber between the core material and a nonconductive material. The nonconductive material is chosen to have a refractive index that differs from that of the InP core by 0.01. At the boundaries between the alternating regions, there is an effective change in the index of refraction of $\mathrm{\Delta }{n}_0\sim {10}^{-4}$, giving rise to a reflection $R_0\sim 2.16\times {10}^{-10}$ as per Eq. (2). This value of $R_0$ is chosen to avoid significant attenuation over the length of the fiber. Note that the sensitivity is independent of $\mathrm{\Delta }{n}_0$ as long as we remain in the linear regime of Eq. (3). The alternating regions are $20\mu \mathrm{m}$ in length, with randomness introduced on the order of $1\mu \mathrm{m}$ to avoid the formation of strong peaks in the reflectivity with wavelength due to interference. The effective spatial resolution in this design is then limited by the linear density of sensing sites, which are spaced at $40\mu \mathrm{m}$ from center to center, rather than by the underlying $20\text{-}\mu \mathrm{m}$ spatial resolution of the reflectometer.

Above the core, there is an insulating layer that serves both as cladding, and as a capacitor over which most of the voltage will drop. The capacitor layer must be thick enough to serve as effective optical cladding, while also having a high capacitance. To satisfy these constraints, a material like barium titanate, strontium titanate, or calcium copper titanate may be preferred. We set this layer’s thickness to $\sim 1\mu \mathrm{m}$. Clearly, the titanate layer must have lower refractive index than the core to act as a cladding. Although the optical properties of the titanate layer depend on its preparation, the band gap of a single crystal of barium titanate occurs at 3.2 eV,²⁴ and the refractive index of barium titanate is $\sim 2.4$ for $\lambda =600\mathrm{nm}$,²⁵ so it is safe to assume $n<2.4$ for $\lambda =1.5\mu \mathrm{m}$. Above the capacitor layer, there are alternating regions of metal and insulator, with the insulating regions coinciding with the alternating layers in the waveguide core. The metal regions provide the electrical interface to the brain and serve to define the sensing locations.

The InP core and inner conductor are doped and biased appropriately to allow most of the voltage to drop over the capacitor layer while maintaining low levels of optical attenuation, for example, $\sim {10}^{17}{\mathrm{cm}}^{-3}$. Other major materials requirements on the inner conductor are that it should ideally form an ohmic contact with both the InP core and the metal reference layer, and that its refractive index needs to be smaller than that of the n-doped InP, which is around 3.17 at $\lambda =1.5\mu \mathrm{m}$. Potential materials candidates then include type III-V semiconductors with lower refractive indices, such as GaP, or II-VI semiconductors, such as ZnSe or CdS. These have lower refractive indices at 3.05, 2.45, and 2.30, respectively. These can be expitaxially grown on InP or vice versa due to the small lattice mismatch,²⁶ and their conductivities can be tuned by doping. On the other hand, it would be important to prevent the formation of a rectifying junction at the semiconductor–semiconductor interface, the existence of which would depend on the band mismatch and doping levels. It might be possible to lower the junction barrier by, for example, minimizing the band gap difference between the two adjacent semiconductors. In the below analysis, we will assume that all junctions can be made ohmic. Note that the inner conductor is chosen to be thick enough to prevent optical attenuation due to the metal substrate (although there are other possible methods to reduce attenuation due to the metal, e.g., by removing the metal from the region directly under the waveguide, as in Ref. 13), and the metal substrate is chosen thick enough to provide a high-fidelity biased reference throughout the fiber.

The design relies on the free-carrier dispersion effect (also known as the plasma dispersion effect): the index of refraction within the InP core changes due to the accumulation of charge carriers in the InP when a voltage is applied across the capacitor layer.^13,23,27 Many current integrated semiconductor electro-optic modulators are based on the free-carrier dispersion effect.^12,13 In addition to the free-carrier effect, there exist other modalities of electro-optic modulation, such as the linear electro-optic (Pockels) effect,²⁸ the quadratic electro-optic effect (Kerr)²⁹ and the Stark effect.³⁰ All of these effects would benefit from reducing the thickness $d$ of the insulator layer to create a large electric field $V/d$.³¹ However, the free-carrier effect uniquely depends on the “charge,” rather than the field, and can thus be amplified further by increasing the relative permittivity of the capacitor layer. In short, we need a large capacitor, which can be achieved by reducing the thickness and increasing the relative permittivity. For a material with a suitably large value of ${\epsilon }_r/d$, the change in refractive index due to the free-carrier effect will be much larger than the changes that can be obtained via the other electro-optic effects. Although we focus on the free-carrier effect here, it should be noted that novel electro-optic materials, such as potassium tantalate niobate,³² with extremely high electrooptic coefficients compared to standard electrooptic materials like lithium niobate, could also potentially make possible designs based on the Pockels or Kerr effects.

An appropriate bias voltage will be applied through the reference conductor to ensure that the InP core layer operates in accumulation. This is necessary in order to avoid depletion,³³ which would reduce the charge recruited to the surface of the capacitor for a given change in extracellular voltage, and thus reduce the sensitivity. Thus, we use the reference potential in the brain plus some fixed bias to achieve accumulation in the InP core along the waveguide. If needed, this bias could be achieved locally, but as long as the brain has no large voltage differences (e.g., $>1\mathrm{V}$), one global bias may be sufficient to allow the entire InP core to operate in accumulation.

Changes in the index of refraction in the free-carrier modulated region of the InP may be modeled as changes in the overall effective index of refraction of the fiber.³⁴ The magnitude of this effective change is given by weighting the magnitude of the change in the free-carrier modulated layer by the percentage of power contained in that layer, i.e.,

We will denote by $d$ the thickness of the capacitor layer, by $b$ the thickness of the layer of injected charge carriers in the InP, and by $a$ the remaining thickness of the InP layer. An order-of-magnitude approximation for $\eta$ is then given by

Upon applying a voltage across the capacitor layer, the density of charge carriers injected into the active layer inside the InP core, denoted by $\mathrm{\Delta }Q$, is simply given by the equation for a parallel plate capacitor:

For InP, the value of $C_e$ is given for $1.55\mu \mathrm{m}$ light by³⁵

Similar values are obtained for other semiconductors and other wavelengths.^23,35 To find the effective refractive index within the InP waveguide, we multiply Eq. (14) by the volume factor $1-\eta$ from Eq. (10). Assuming $b\ll a$ (i.e., that the injected charge layer is deeply subwavelength while the waveguide core thickness is on the same order as the wavelength), we find

2.2.2.

Effects of other capacitances

The brain–electrode interface also has a capacitance $C_{\mathrm{ext}}$ of $\sim 100\mu \mathrm{F}{\mathrm{cm}}^{-2}$,^36,37 which arises due to the presence of an electrical double-layer [see Fig. 1(d)].

In addition, there is a capacitance $C_Q$ due to the finite length scale of the charge distribution inside the semiconductor. This latter capacitance is given by $C_Q={\epsilon }_{\text{core}}{\epsilon }_{0}A/d_Q$, where ${\epsilon }_{\text{core}}$ is the relative permittivity of the core material, ${\epsilon }_0$ is the permittivity of free space, $A$ is the area of the sensing region, and $d_Q$ is the charge centroid. The core can be one of many semiconductor materials (e.g., Si, InP), leading to similar fundamental electrostatics. We performed semiconductor simulations using the Sentaurus TCAD device simulator (version K-2015.06, June 2015) to evaluate $C_Q$. We used Si as the simulated core material, because of the availability of accurate and readily available tools for Si electrostatics simulation. To calculate the charge centroid $d_Q$, we simulated the electrostatics of an interface between silicon $n$-doped to a level ${10}^{17}{\mathrm{cm}}^{-3}$ and a layer of oxide with relative permittivity of ${\epsilon }_r=5$ and thickness of $d=1\mathrm{nm}$. When the silicon was in accumulation, the charge centroid was found to be 2.4 nm. The charge centroid is expected to be similar for our setup provided the value of ${\epsilon }_r/d$ is similar for the capacitor layer, which it would be for a layer of barium titanate with ${\epsilon }_r=5000$ and $d=1000\mathrm{nm}$ (see below). It is more difficult to do these simulations for less common materials like InP, but we anticipate that the charge centroid for InP will be similar. Thus, in all following calculations, we will assume a capacitance of $4.5\mu \mathrm{F}{\mathrm{cm}}^{-2}$ for the interface between the core and capacitor layer.

The electrostatics simulations also showed that, although the total charge $\mathrm{\Delta }Q$ recruited to the capacitor layer surface upon application of a voltage is greater for higher doping levels, the relative change in charge ($\mathrm{\Delta }Q/Q$) is greater for lower doping levels. However, for $\mathrm{\Delta }Q\ll Q$, the sensitivity condition in Eq. (8) depends to a good approximation only on $\mathrm{\Delta }Q$, not on $\mathrm{\Delta }Q/Q$, so the sensitivity of the device is increased for higher doping levels.

Figure 1(d) shows an equivalent circuit diagram of the device, which includes the interfacial capacitance and the capacitance associated with the charge distribution. At $<1000\mathrm{Hz}$, and subject to appropriate materials choices (see Sec. 2.2.3), the impedance of the circuit is dominated by these three capacitors rather than by purely resistive elements of the circuit. For this reason, we may ignore the purely resistive elements and treat the capacitances as though they were in series. To a good approximation, therefore, the charge that accumulates on the surface of the insulating region in response to a voltage $\mathrm{\Delta }V$ across the entire device is given by

The capacitance of the insulating region is given by

The effective capacitance $C_{\mathrm{eff}}$ is shown in Fig. 3(a) as a function of the capacitance $C_{\mathrm{ins}}$ of the capacitor layer, assuming a surface capacitance per unit area^36,37 of $\sim 100\mu \mathrm{F}{\mathrm{cm}}^{-2}$ and a sensing length of $20\mu \mathrm{m}$. Note that the effective capacitance ceases to increase for values of $C_{\mathrm{ins}}/A\gg 4.5\mu \mathrm{F}{\mathrm{cm}}^{-2}$, because for these values, the capacitance is dominated by the core-capacitor interface.

Fig. 3

Properties of the design parametrized by $C_{\mathrm{ins}}/A$. (a) The effective capacitance $C_{\mathrm{eff}}/A$ given in Eq. (18) is shown as a function of $C_{\mathrm{ins}}/A$, assuming a capacitance of $100\mu \mathrm{F}{\mathrm{cm}}^{-2}$ at the surface of the device and a capacitance of $4.5\mu \mathrm{F}{\mathrm{cm}}^{-2}$ at the interface between the core and capacitor layers. (b) The minimum detectable change in voltage [obtained from Eqs. (8) and (16)] is shown as a function of $C_{\mathrm{ins}}/A$ for systems with (from top to bottom) a 10-mW laser (blue), a 30-mW laser (orange), and a 100-mW laser (green). The black dashed lines correspond to 50 and $100\mu \mathrm{V}$. To sense signals at the $50\mu \mathrm{V}$ level with a 100-mW laser, a capacitance on the order of $10\mu \mathrm{F}{\mathrm{cm}}^{-2}$ is necessary.

2.2.3.

Noise sources

A primary electrical constraint on the device is that the impedance at 1 kHz must be dominated by the capacitor layer. If the effective capacitance per unit area of the capacitor is $C_{\mathrm{eff}}=5\mu \mathrm{F}{\mathrm{cm}}^{-2}$, corresponding to a value $C_{\mathrm{ins}}=10\mu \mathrm{F}{\mathrm{cm}}^{-2}$, then the capacitance of a region with width $4\mu \mathrm{m}$ and length $20\mu \mathrm{m}$ is 4 pF, corresponding to an impedance of $40\mathrm{M}\mathrm{\Omega }$ at 1 kHz.

Assuming that the metal layer has a resistivity no greater than $100\mathrm{n}\mathrm{\Omega }\mathrm{m}$ ($10\times$ that of silver), if the metal layer is made at least 500 nm thick, it will have a resistance of $\mathrm{10,000}\mathrm{\Omega }$ along the entire length of the fiber. The resistance of the inner conductor and core will be negligible compared to the huge capacitive impedance, provided they are chosen to be semiconductors. Finally, we must consider the voltage noise on the recording site itself, i.e., the metal contact pad interfacing directly with the brain. The recording site is often modeled as a constant phase element³⁸ and noise contributions come from the real part of its impedance,² and are frequency dependent. We choose to write it in terms of parameters $G$ and $m$, with an impedance of $Z_e=1/G{(j\omega )}^{-m}$. The parameter $G$ reflects the conductivity of the material, and the parameter $m$ is often related to surface roughness and transport to the metal–electrolyte interface,³⁹ with typical parameters ranging from 0.5 to 0.9. We will assume $m=0.5$ for representing a rough surface, and a 1 kHz impedance magnitude of $0.1\mathrm{M}\mathrm{\Omega }$. Thus, the resistance of the device is dominated by the recording site, as opposed to the ground lead or other elements, and the above parameters amount to a total RMS noise over a 1 kHz band, found by integrating $4k_{B}T\mathrm{Re}(Z_e)\mathrm{d}f$ from $f=0\mathrm{Hz}$ to $f=1000\mathrm{Hz}$, of $\sim 1\mu \mathrm{V}$. This model agrees with what is found experimentally for similar sized electrode pads.³⁹

Efforts to reduce the recording site impedance are only needed for adjusting the noise influence of the recording site itself. Even an unplated gold surface will be sufficient here, because instead of $0.1\mathrm{M}\mathrm{\Omega }$ for an electroplated surface, we will have $\mathrm{Re}(Z)=1\mathrm{M}\mathrm{\Omega }$, with a resulting noise of $\sim 4\mu \mathrm{V}$ RMS instead of the $1\mu \mathrm{V}$ RMS calculated above. Only if $C_{\mathrm{eff}}$ was increased dramatically (e.g., to the equivalent impedance of $\sim 1\mathrm{M}\mathrm{\Omega }$ at 1 kHz), would efforts be needed to reduce the recording site impedance to prevent attenuation of the signal via the voltage divider. In any case, the voltage drops primarily over the capacitor layer and is not attenuated by resistors prior to the capacitor, and these electronic noise voltages are lower than the sensitivity of the device, which is limited by optical shot noise, and so can be neglected. Note that the impedances given here are also large enough for the input impedance of an implanted recording device.⁴⁰

Other forms of exogenous noise include mechanical bending of the fiber and thermo-optic effects, which may be particularly significant given the small width of the waveguide. However, these effects are expected to occur at a much lower frequency than the $\sim 1\mathrm{kHz}$ frequency content of spikes, and thus can be filtered out. Likewise, static or slowly changing bends (e.g., due to the heart beat) in the fiber can be subtracted off.

2.2.4.

Dynamic range

Local field potentials in the brain may vary by up to hundreds of millivolts, generating fields on the order of $1{\mathrm{kV}\mathrm{cm}}^{-1}$ across a $1\text{-}\mu \mathrm{m}$ capacitor layer. By contrast, the dielectric breakdown strength of barium titanate is roughly $10{\mathrm{kV}\mathrm{cm}}^{-1}$, so dielectric breakdown is unlikely to be an issue.⁴¹ On the other hand, the dynamic range of the device may be limited by the density of states in the core, and thus it will be necessary to adjust the bias of the device (using the conductive reference layer) in order to ensure that the device can function in accumulation. If the device is allowed to function in depletion, $C_Q$ will be much smaller than $C_{\mathrm{ins}}$, thus reducing the sensitivity. Similarly, operation in inversion will suffer from deep depletion effects.

2.2.5.

Tissue heating

When we send light down the fiber, some light power may dissipate into the tissue. Depending on the level of round-trip light attenuation in the waveguide, each probe will dissipate a fraction $f$ of the applied light power $P$. We next evaluate the acceptable level of such dissipation and how this constrains the device properties.

The human brain endogenously dissipates 25 W or $19\mathrm{mW}{\mathrm{mL}}^{-1}$. The blood perfusion rate⁴² of human brain gray matter and white matter is roughly $r_{\text{perfusion}}=35000{\mathrm{Wm}}^{-3}°{\mathrm{C}}^{-1}$. To avoid $>2°\mathrm{C}$ brain temperature rise, per the requirements laid out in Ref. 3 and elsewhere, we then require that each probe is surrounded by a perfusion volume of $V_{\text{perfusion}}\approx \frac{f·P}{r_{\text{perfusion}}·2°\mathrm{C}}$.

For a sense of scale, assuming that a 100-mW laser is used for the reflectometer, if $f\approx 50\%$ of this light power is dissipated into the tissue on a round-trip reflection, we then require a $250\mu \mathrm{L}$ perfusion volume, or a cylinder of radius 1.5 mm around each probe, assuming a 10-cm probe length. An attenuation of 50% over 20 cm corresponds to $\sim 3\mathrm{dB}$ over the length of the fiber, or $\sim 0.15\mathrm{dB}{\mathrm{cm}}^{-1}$, on the order of the intrinsic optical attenuation of silicon⁴³ or indium phosphide.⁴⁴ An additional potential source of tissue heating arises from transverse scattering of light at the interfaces between the successive waveguide segments of different refractive index. Using MEEP simulations to quantify the amount of light scattered out of the waveguide core, for adjacent segments with refractive indices of 3.409 and 3.41, we estimate that there will be a $3\times {10}^{-5}\%$ loss per boundary. With 500 boundaries per centimeter, this means a 0.015% loss per centimeter or 0.3% loss over a round trip in a 10-cm fiber. However, if $\mathrm{\Delta }{n}_0$ is $\sim {10}^{-4}$ instead, as discussed above, the amount of scattering generated this way is expected to be substantially reduced.

Attenuation due to bending is expected to be insignificant, with silicon-on-insulator waveguides reported to experience attenuation of only $\sim 0.1\mathrm{dB}$ per 90 deg turn at a radius of curvature of $1\mu \mathrm{m}$. Finally, to avoid transmitting any light into the brain tissue itself, a strong reflector can be placed at the end of the probe. Because the reflectometer has high spatial resolution, a large reflection from the end of the probe is not expected to interfere with the measurements.