Integrating Optics and Parametrically-Resonant Micro-Scanner Design for Large Working Distance Implantable Microscopy

Abstract

This article examines interdependent design of an optical path and a microelectromechanical system (MEMS) scanning mirror for a miniature, implantable fluorescence microscope with large working distance (WD). Linearized and numerical ray analyses are used to approximately decouple optical and mechanical functions during design. We then maximize scan rate in the scenario of high-NA focusing with a specified WD and field-of-view (FOV). To do so, dynamic rotational analysis is combined with a novel model for expected failure voltage of parametrically-resonant electrostatic MEMS scanning mirrors. Mirrors parameters are set to optimize mirror speed within constraints fixed by optical specifications, while compatible optical path is selected for small objective diameter. A prototype instrument achieving sub-cellular resolution up to approximately $500\mathrm{x}500\mu {\mathrm{m}}^2$ FOV at up to $300\mu \mathrm{m}$ WD is validated on imaging targets and excised mouse brain tissue.

I. Introduction

Nueroscientists increasingly use high-resolution imaging of neural activity in small mammals to study brain function and disease. This is most often done in vivo using large benchtop microscopes through a cranial window, which permits periodic imaging sessions with an animal’s head fixed in space. However, new development of implantable microscopes as small, fiber-coupled instruments allows imaging while animals, typically mice, move comparatively freely within enclosed spaces [1], [2], [3], [4]. Neuron activity is captured via fluorescence imaging, using modalities including widefield [5], [6], confocal [7], or two-photon [8], [9], [10] fluorescence imaging. Resulting microendoscopes range from comparatively low-cost, low-resolution but open-source systems [11] to commercial instruments [12].

A major limitation of existing implantable microscopes is that it is very challenging to achieve both high resolution and large field-of-view (FOV), particularly at large working distance (WD). Distal scanning of a focused laser can expand implantable microscope capabilities, with light scanned across the FOV using microelectromechanical system (MEMS) scanning mirrors and/or adaptive lenses [13], [14]. Some distal scanning also supports three-axis imaging [15], [16], including at large imaging depth through three-photon excitation [17]. Nonetheless, increasing imaging depth has required significant sacrifice in resolution or FOV in existing instruments, with limited size and/or speed of readily available micro-mirrors acting to constrain the optical design space.

Here, we examine the challenge of increasing implantable microscope performance at large WD as an interdependent optics and micro-actuation design problem, Fig. 1. The microactuation technology is a parametrically-resonant MEMS two-axis (or biresonant) mirror driven by electrostatic actuation, which uses nonlinear dynamics to generate substantial scan angles at high frequency [18], [19]. Parametrically resonant electrostatic mirrors have recently been implemented in endomicroscopy tools [20], [21], and have low fabrication complexity facilitating rapid prototyping and customization. However, operating voltage and resulting scan amplitude are constrained by the limit at which electrostatic comb fingers are pulled together by displacement in a direction other than that of desired motion (“off-axis displacement”), forming an electrical short. Increasing scanner size to improve optical system performance entails tradeoffs in robustness against off-axis movement, active drive amplitude, and resonant frequency. We have previously presented a simple model for comb finger bending during parametrically resonant scanning in reflectance microscopy [25]. However, full interaction between mirror and comb drive compliance, and implications for multi-photon microscopy are new to this article.

Fig. 1.

Summary of design process and resulting implantable microscope (NA: numerical aperture; WD: working distance).

This article thus describes joint optimization of optics and an electrostatic, parametrically resonant scanning mirror for an implantable two-photon microscope. Specific contributions include examination of relationships between optical performance and micromirror requirements that allow certain optimization goals to be separated between mechanical and optical functions; Lissajous scan analysis motivating a focus on fast-axis frequency; a new model for estimating maximum operating voltage of elliptical electrostatic mirrors; predicted mirror amplitude trends in fast-and slow-axes; and implications for optimal mirror design on desirable optics layout. A prototype instrument demonstrates large FOV at large WD, compared to other implantable microscopes from literature [22], [23], [24], as given in Table I.

TABLE I.

Working Distance (WD) of Implantable Two-Photon Microscopes Based on MEMS Scanning Mirrors, and Associated Field-of-View (FOV) and Resolution

	Max WD ($\mu \mathrm{m}$)	FOV ($\mu \mathrm{m}\mathrm{x}\mu \mathrm{m}$)	Axial Resolution ($\mu \mathrm{m}$)	Lateral Resolution ($\mu \mathrm{m}$)
Ozbay et al. [22]	150	240 x 240	10	2.6
Zong et al. [23]	180	420 x 420	12	2.0
Zong et al. [24]	240	420 x 420	16	1.5
This article	300	500 x 500	6	1.5

II. Analytical Methods

The basic function of a fiber-coupled microscope based on distal scanning is to reflect an excitation laser off a scanning mirror, then reconstruct an image from returning light intensity versus laser position. Imaging at depth and/or high resolution requires focusing the laser with additional lenses. Microscope design can be done to optimize any of several image collection metrics, i.e., WD, FOV, lateral or axial resolution, or frame rate. For several common optical modalities (single axis confocal microscopy, multi-photon microscopy), resolution is determined primarily by numerical aperture (NA) of the focusing optics, which may become the objective or a constraint. The conceptual design problem becomes

$$\underset{{\text{Θ}}_1,{\text{Θ}}_2}{\text{max}}J({\text{Θ}}_1,{\text{Θ}}_2)\text{s.t.}g({\text{Θ}}_1,{\text{Θ}}_2)<0$$ (1)

where $J$ is the selected imaging metric to be optimized, ${\mathrm{\Theta }}_1$ and ${\mathrm{\Theta }}_2$ are vectors of design parameters for the scanner and optics, respectively, and $g$ is a constraint function placed by other optical specifications and/or limits on fabrication and assembly.

The problem in (1) is a multi-variable optimization problem and might be addressed by various methods. However, as we will develop over proceeding analysis, for an objective of maximizing frame rate at a specified WD, FOV, and NA, scanner optimization is largely decoupled from optical design; a secondary objective may then be optimized for among feasible optical designs, such as maximum element diameter, objective diameter, path length, etc. Optimization becomes

$$\underset{{\text{Θ}}_1}{\text{max}}{\omega }_{n,x}({\text{Θ}}_1)\text{s.t.}\{\begin{array}{l}{r}_b\cdot {\theta }_{x,\text{max}}>k_{\text{opt}}\cdot \text{NA}\cdot \text{FOV}\\ g_{\text{mech}}({\text{Θ}}_1)<0\end{array}$$ (2)

$$\underset{{\text{Θ}}_2}{\text{min}}{J}_2({\text{Θ}}_2)\text{s.t.}{g}_{\text{opt}}({\text{Θ}}_2)<0$$ (3)

where (2) maximizes natural frequency of the scanner’s fast-axis (here, the $x$-axis), ${\omega }_{n,x}$, while ensuring that the product of the scanner’s minor radius, $r_b$ and maximum scan angle in the fast axis, ${\theta }_{x,\text{max}}$, exceeds a performance bound, which is linearly proportional to the product of NA and FOV, and satisfying other mechanical constraints, $g_{\text{mech}}$. Proportionality constant $k_{\text{opt}}$ depends on factors such as refractive index of the object. Among optical designs achieving a target NA and FOV, a secondary property of the optical path, $J_2$, may be optimized in (3) subject to remaining optical design, $g_{\text{opt}}$.

In this article, we optimize for largest frame rate and smallest objective lens diameter at $\mathrm{W}\mathrm{D}=300\mu \mathrm{m},\mathrm{F}\mathrm{O}\mathrm{V}=500\times 500\mu {\mathrm{m}}^2$, and $\mathrm{N}\mathrm{A}>0.45$, to meet a target lateral resolution $<1.5\mu \mathrm{m}$ during multiphoton microscopy with 920–960 nm excitation. These specifications are motivated by a desire to perform fluorescent protein-labeled imaging below the first layer of the brain in mice with sufficient FOV and resolution to capture neuron activation in locomotion and sensory tasks, for example as done with a benchtop microscope in [26]. This would significantly improve on existing implantable microscopy systems. Mechanical constraints include minimum electrostatic finger gap, device thickness, and feature size, while optical constraints include minimum component spacing for assembly and maximum permissible component diameter.

We next introduce ray transfer (see Section II-A) and ray trace (see Section II-B) methods used to establish the proportionality between mechanical and optical performance and to generate candidate optical designs; discuss frame rate under Lissajous scanning motivating a focus on fast axis dynamics (see Section II-C); introduce a dynamic model for parametrically-resonant electrostatic scan mirrors with an elliptical reflector (see Section II-D); and introduce a novel model for electrostatic failure of the scan mirror to estimate maximum scan angle as a function of reflector size (see Section II-E).

A. Linearized Ray Transfer Models

The nominal beam path for a scanned, fiber-coupled multi-photon microscope is to place the scan mirror at a ${45}^{\circ }$ angle to the beam path between the fiber and objective, with additional lenses dictating the beam diameter, and sensitivity to mirror rotation, at those locations. By rotating the mirror about two axes, the laser can be scanned over a 2D surface to generate an image. A schematic view of this architecture, with one or more lenses both before and after the mirror, is shown in Fig. 2(a).

Fig. 2.

(a) Conceptual beam path from optical fiber to objective, using MEMS rotational mirror for beam scan. (b) Generic lens sequence illustrated on fast-axis plane.

Optical properties depend on lens number, sequence, properties, and spacing, and were assessed using a both simple, linear beam path analysis and high-fidelity ray trace simulation. Rapid, linearized analysis of many hypothetical lens and mirror configurations was conducted using ray transfer matrices under small angle, paraxial approximation. Fig. 2(b) shows a conceptual beam path projected along the axis parallel to the mirror surface through a sequence of $N$ lenses. Ray radius and angle at the objective, $r_N$ and ${\varphi }_N$, respectively, are propagated through elements using linear matrices. For example, for a ray at a source with radius $r_0$ and angle ${\varphi }_0$, the ray radius and angle exiting the lens prior to the mirror (labeled lens $M$) is given by

$$\left[\begin{array}{l}{r}_M\hfill \\ {\varphi }_M\hfill \end{array}\right]=\left[\begin{array}{ll}{A}_M\hfill & B_M\hfill \\ C_M\hfill & D_M\hfill \end{array}\right]\dots \left[\begin{array}{ll}{A}_1\hfill & B_1\hfill \\ C_1\hfill & D_1\hfill \end{array}\right]\left[\begin{array}{l}{r}_0\hfill \\ {\varphi }_0\hfill \end{array}\right]$$ (4)

where $A_n,B_n,C_n$, and $D_n$ summarize ray diffraction through lens $n$. Each ray trace is parameterized lens type (spherical, GRIN, or flat pane), distance from the preceding lens, $L_n$, and certain lens-specific parameters: refractive index; radius of curvature; and nominal thickness for a spherical lens; nominal refractive index, lens length, and gradient constant for a GRIN lens; and refractive index and pane thickness for a pane.

The ray transfer progression from the mirror is given by

$$\left[\begin{array}{l}{r}_N\hfill \\ {\varphi }_N\hfill \end{array}\right]=\left[\begin{array}{ll}{A}_N\hfill & B_N\hfill \\ C_N\hfill & D_N\hfill \end{array}\right]\left[\begin{array}{ll}{A}_{N-1}\hfill & B_{N-1}\hfill \\ C_{N-1}\hfill & D_{N-1}\hfill \end{array}\right]\cdots \left[\begin{array}{ll}{A}_{M+1}\hfill & B_{M+1}\hfill \\ C_{M+1}\hfill & D_{M+1}\hfill \end{array}\right]\times \left[\begin{array}{c}{r}_M+d_M{\varphi }_M\\ {\varphi }_M+2{\theta }_x\end{array}\right]$$ (5)

where the mirror is offset by distance $d_M$ from lens $M$ and may rotate with a tilt angle ${\theta }_x$.

During preliminary optical path generation, a Monte Carlo simulation generated random lens and mirror sequences with 2 to 5 lenses present, with the mirror situated between at least one optical element on each side, i.e., $2\le M<N<5$. The beam edge at the objective, $r_{N,\text{edge}}$ and ${\varphi }_{N,\text{edge}}$, was set by target WD and NA of the system,

$$\text{NA}=\eta \text{sin}{\varphi }_{N,\text{edge}},$$ (6)

$$r_{N,\text{edge}}=\frac{\text{WD}}{\text{tan}({\varphi }_{N,\text{edge}})}$$ (7)

where $\eta$ is the refractive index of the imaging medium (taken to be approximately 1.38 for mouse cortex). The simplified beam geometry can then be mapped backwards through a candidate design to estimate required mirror radius by inverting $\mathrm{N}\mathrm{A}=\eta \mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_{N,\text{edge}}$ the ray trace matrices with ${\theta }_x=0$. To ensure fiber compatibility, lens 1 parameters were chosen to match the emission radius and angle of the fiber, approximated by fiber NA in an analogous manner to the objective. FOV was estimated by evaluating (2) as a function of mirror angle, and finding the new beam edge intersection, at radial position $r_{\text{FOV}}$.

B. Ray Trace Simulation

A subset of candidate optical designs was analyzed in depth using optical simulation software (Zemax OpticStudio). Commercially-available lens models with diameter 2 to 6 mm were considered (Thorlabs, Edmund Optics, Lightpath Technologies). Two adjustments to optical path analysis were made: spherical lenses were replaced with aspherical lenses to reduce spherical aberration, and a collection path was modeled, taking a dichroic mirror to be located where space permitted in a lens sequence. Lens spacing was adjusted to fix WD at the desired depth of $300\mu \mathrm{m}(\pm 50\mu \mathrm{m})$. Effective NA was calculated from the solid angle of the rays at the objective, while FOV was calculated at the maximum angle of the mirror for which the ray trace remained within all lenses (in contrast to ray transfer models where lenses could be made as large as possible). Fig. 3 shows a sample ray trace simulation for one lens sequence at its maximum FOV. The example in Fig. 3 is a low-NA design, but this allows the limit on beam steering imposed by small diameter optics to be clearly seen.

Fig. 3.

Sample ray trace illustrating constraints on mirror angle imposed by small optics. This example projects collimated light through three aspherical lenses and a transparent pane.

C. Lissajous Frame Rates

While optical modeling captures WD, FOV, and NA, frame rate depends on mechanical scan speed. Two-axis, parametrically-resonant MEMS scanners generate images using a Lissajous scan pattern. Lissajous frame rate is a function of scan axis frequencies, relative phase, and acceptable image fill factor [27]. Fortunately, scanner designs can be adapted to provide advantageous frequency ratios between fast- and slow-axes. Fig. 4 shows representative frame rate trends as a function of fast-axis frequency (2–8 kHz), assuming that an ideal slow-axis frequency can be selected within a limited distance from a nominal slow-axis frequency ($\pm 5\mathrm{\%}$). Best slow-axis frequency and phase were evaluated numerically in MATLAB for each fast-axis and nominal slow-axis target, the latter varying from 20% to 100% of the fast-axis frequency, to meet a minimum fill-factor requirement of 90% over $500\times 500$ pixel image.

Fig. 4.

Simulated frame rate versus fast-axis frequency (Hz) and slow-axis/fast-axis frequency ration ($f_y/f_x$), for Lissajous scan with $500\mathrm{x}500$ pixels and 90% fill factor, assuming that slow axis frequency can be adjusted over a smaller frequency range ($\pm 5\mathrm{\%}$ of nominal target) for best rate at given fill factor.

Importantly, feasible frame rate is significantly more sensitive to fast axis frequency than relative slow axis frequency. Quantitatively, an incremental improvement to fast-axis frequency had approximately three times the benefit for frame rate of an incremental improvement to slow-axis frequency. Alternate assumptions for fill factor and pixel count affect the exact ratio, but based on this analysis we prioritize the analysis of fast axis frequency and angle in ensuing analysis.

D. Mirror Dynamics

To match mirror motion capabilities to those desirable from ray transfer matrix and ray trace simulations, feasible mirror tilt angle was projected for elliptical, parametrically-resonant, comb-driven scanning mirrors. Dynamics of parametrically-resonant mirrors having rectangular geometries for rotational and axial vibration have been previously studied in [18] and [28], and are expanded upon in this article. Rotational mirror vibration about the $x$- (fast-) axis can be modeled as mass spring damper system driven by nonlinear capacitive forcing

$$I_x{\ddot{\theta }}_x+b_x{\dot{\theta }}_x+k_x{\theta }_x=\frac{1}{2}\frac{\partial C({\theta }_x)}{\partial {\theta }_x}{V}_x^2$$ (8)

where $I_x$ is rotational inertia, $b_x$ is rotational damping, $k_x$ is the rotational spring constant of a supporting torsion beam, $C$ is total capacitance of the comb finger array as a function of displacement, and $V_x$ is driving voltage.

To estimate scanner amplitude versus mirror size and natural frequency, rotational inertia is calculated based on an elliptical reflector with minor radius $r_b$, and major radius $r_a=r_b\sqrt{}2$, as the mirror is placed at a ${45}^{\circ }$ angle to the beam path. All structures are assumed to be formed from a single silicon layer, with thickness $t_{Si}$, including a prismatic torsion spring. The spring’s stiffness is tuned to a target resonant frequency, ${\omega }_{n,x}=\sqrt{k_x/I_x}$.

For a given mirror size and target tilt angle, other design parameters (torsion spring dimensions, finger dimensions) were selected to maximize tilt angle based on models for dynamic displacement and feasible voltage limits. To generate a capacitance model, $C\left({\theta }_x\right)$, for a candidate mirror geometry, comb fingers are assumed to be evenly spaced about the reflector. Each finger attached to the reflector is placed between two comb fingers attached to a stator (i.e., the stationary frame of the scanner or a gimbal), with overlap length $L_f$, nominal gap $g_0$, and finger width $w_f$, as shown in Fig. 5(a). The total count of reflector fingers on each side is denoted by $J$, with $2J$ total fingers per mirror.

Fig. 5.

(a) Schematic electrode layout for uniform length, uniform width comb fingers surrounding an elliptical reflector. (b) Individual finger tilt angle conditions for estimation of capacitance versus tilt angle (about $x$-axis). (i) neutral position; (ii) trapezoidal overlap; (iii) triangular overlap; and (iv) no overlap. (b) Resulting numerical model for capacitance versus angle can be closely approximated by a Gaussian curve (example shown: 1.3 mm radius mirror with $235\mu \mathrm{m}$ long comb fingers, $7\mu \mathrm{m}$ wide with $5\mu \mathrm{m}$ gap. (c) Maximum dimensionless amplitude versus damping ratio and dimensionless forcing.

Due to the progressive arrangement of fingers around the elliptical mirror edge, this model tends to produce a total capacitance resembling a Gaussian curve with respect to out-of-plane rotation, in the form

$$C({\theta }_x)=2\sum _{j=1}^J{C}_j({\theta }_x)\approx C_0{e}^{\raisebox{1ex}{$-{\theta }_x^2$}\!\left/ \!\raisebox{-1ex}{${\theta }_r^2$}\right.}.$$ (9)

In the summation in (9), each reflector finger is modeled as having a capacitance approximated by the overlap between it and adjacent stator fingers in the plane of the stators, illustrated geometrically in Fig. 5a(ii)–(iv) and calculated as

$$C_j({\theta }_x)\approx \{\begin{array}{ll}2{ϵ}_0\frac{L_f}{g_0}{t}_{Si},\hfill & {\theta }_x=0\hfill \\ 2{ϵ}_0\frac{L_f}{g_0}\frac{2t_{Si}-(2r_{y,j}+L_f)\text{sin}({\theta }_x)}{2},\hfill & 0<|\text{sin}({\theta }_x)|\le \frac{t_{Si}}{r_{y,j}+L_f}\hfill \\ 2{ϵ}_0\frac{1}{g_0}\frac{\frac{t_{Si}}{\text{tan}({\theta }_x)}(2t_{Si}-\text{sin}({\theta }_x)r_{y,j})}{2},\hfill & \frac{t_{Si}}{r_{y,j}+L_f}<|\text{sin}({\theta }_x)|\le \frac{t_{Si}}{r_{y,j}}\hfill \\ 0,\hfill & \frac{t_{Si}}{r_{y,j}}<|\text{sin}({\theta }_x)|\hfill \end{array}$$ (10)

In (10), the first entry represents complete finger overlap in the mirror’s neutral position [see Fig. 5(a.i)]; the second entry an approximately trapezoidal overlap at small angles [see Fig. 5(b.ii)]; the third a triangular overlap as larger angles [see Fig. 5(c.iii)]; and the fourth a negligible capacitance when the reflector fingers pass fully out of the plane of the stator [see Fig. 5(d.iv)]. This formulation assumes that finger thickness (i.e., $t_{Si}$) is much larger than the finger and gap widths, such that fringe effects are small. Fig. 5(b) shows the numerical and Gaussian approximation for a sample finger arrangement.

Differentiating the capacitance model in (6)–(7) gives an equation of motion of the form

$$I_x{\ddot{\theta }}_x+b_x{\dot{\theta }}_x+k_x{\theta }_x=-\frac{C_0}{{\theta }_r^2}{\theta }_x{e}^{\raisebox{1ex}{$-{\theta }_x^2$}\!\left/ \!\raisebox{-1ex}{${\theta }_r^2$}\right.}{V}_x^2.$$ (11)

In [28], a parametrically-resonant system described by the model in (11) can be reduced to a dimensionless system

$$\frac{d^2\chi }{d{\tau }^2}+2\zeta \frac{d\chi }{d\tau }+\chi =-\eta \chi e^{-{\chi }^2}u$$ (12)

where $\chi$ is the dimensionless displacement, $\chi ={\theta }_x/{\theta }_r,\tau$ is dimensionless time $\tau ={\omega }_{n}t,u$ defines a periodic signal with amplitude $0\le u\le 1$ (i.e., a sine or square wave), $\zeta$ is damping ratio, $\zeta =c_x/\left(2\sqrt{k_x{J}_x}\right)$, and $\eta$ is a dimensionless forcing amplitude, given by

$$\eta =\frac{C_0{V}_x^2}{{\theta }_r^2{k}_x}.$$ (13)

Amplitude versus frequency can be parameterized by just $\zeta$ and $\eta$ for a given periodic input (i.e., sine or square wave), with maximum resonant amplitude denoted ${\chi }_{\text{max}}$. This allows rapid evaluation of candidate mirror geometries for maximum rotational amplitude, ${\theta }_{x,\text{max}}={\theta }_r{\chi }_{\text{max}}$, once $k_x,C_0$, and ${\theta }_r$ are estimated from mirror geometry and target operating frequency. However, to fully calculate (13), a maximum operating voltage must be established for $V_x$, which is described as follows.

E. Voltage Limit Modeling

In many electrostatically-actuated devices, maximum operating voltage is bounded by the pull-in voltage of adjacent comb fingers, when electrostatic attraction between rotor and stator fingers exceeds the restoring force of any supporting flexure or spring. This voltage may also be influenced by deformation of the fingers themselves. In a mirror structure supported by a silicon flexure, there are multiple possible rotor displacements that might lead to pull-in. However, the most common failure location of earlier elliptical mirrors observed by the authors was electrical shorting at fingers farthest from the torsion springs, corresponding to contact due to unwanted in-plane rotation of the reflector, i.e., ${\theta }_z$ motion in mirror axes. Fig. 6(a) shows an earlier device (following [29]) where shorting occurs at longest comb fingers near reflector center.

Fig. 6.

(a) Preceding scan mirror with finger contact and short. (b) Finger deformation model. (c) Sample in-plane rotational moment versus angle at different voltages; $r_a=1.8\mathrm{m}\mathrm{m},L_f=240\mu \mathrm{m},t_{Si}=50\mu \mathrm{m}$, and $g_0=5\mu \mathrm{m}$.

Based on this, we propose a model for in-plane stability of an elliptical, electrostatically-actuated micromirror. The mirror is assumed to be in stable equilibrium at rest while the in-plane rotational stiffness, $k_z$ is greater than the rate of increase of inplane moment, $M_z$, with in-plane rotation of the reflector, ${\theta }_z$, or

$$k_z>{\frac{\partial M_z}{\partial {\theta }_z}|}_{{\theta }_z=0}.$$ (14)

In other words, if there is an in-plane rotation of the reflector, attractive forces on either side of the comb fingers become imbalanced, and a net in-plane moment occurs. Pull-in will only occur if this in-plane moment rises more quickly than the restoring force of the spring. Unfortunately, exact in-plane moment depends on both the new finger positions and finger bending, which is difficult to solve exactly.

Instead, we construct a set of bounds on finger deformation and contribution to $M_z$ from each finger to establish a conservative estimate for maximum operating voltage. Using just one side of the mirror for analysis, as shown in Fig. 6(b), in-plane moment is defined as arising from parallel-plate electrostatic attraction from differential elements along each finger. Over the entire mirror, the approximate moment is

$$\begin{gathered} M_z({\theta }_z)\approx 2\sum _{j=1}^J{\int }_0^{L_f+r_{x,j}\mathit{sin}({\theta }_z)}(-d{F}_{j-}(s)\cdot (r_{y,j}+s) \\ +d{F}_{+}(s)\cdot (r_{y,j}+s))ds \end{gathered}$$ (15)

for sufficiently small angles that cosine contributions from inplane rotation can be neglected. $s=0$ is set to the tip of the stator fingers, so that changes in overlapping length from mirror rotation can be added to upper bound of integrand. $d{F}_{j-}(s)$ and $d{F}_{j+}(s)$ are incremental forces (per unit length) between rotor finger $j$ and the preceding and following stator fingers, respectively. Approximating incremental forces with a parallel plates model for electrostatic attraction gives

$$d{F}_{j-}(s)=\frac{1}{2}{ϵ}_0\frac{t_{Si}}{g_{j-}{(s)}^2}$$ (16)

$$d{F}_{j+}(s)=\frac{1}{2}{ϵ}_0\frac{t_{Si}}{g_{j+}{(s)}^2}$$ (17)

$$g_{j-}(s)\approx g_0+\text{sin}({\theta }_z)\cdot (r_{y,j}+s)+u_{r,j}(s)+u_{s,j-1}(s)$$ (18)

$$g_{j+}(s)\approx g_0-\text{sin}({\theta }_z)\cdot (r_{y,j}+s)-u_{r,j}(s)-u_{s,j}(s).$$ (19)

We then calculate finger deformation based on a hypothetical uniformly distributed force between each set of fingers, $d{F}_{j+}^{\mathrm{*}}$ and $d{F}_{j-}^{\mathrm{*}}$, where $d{F}_{j+}^{\mathrm{*}}>d{F}_{+}\left(L_f+r_{x,j}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_z\right)\right)$ and $d{F}_{j-}^{\mathrm{*}}<d{F}_{-}\left(L_f+r_{x,j}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_z\right)\right)$. Using these limits, one may define a displacement at the tip of each finger, $u_{r,j}^{\mathrm{*}}$ and $u_{s,j}^{\mathrm{*}}$, that will exceed the true deformation of each finger

$$\begin{gathered} u_{r,j}^{\text{*}}\equiv \frac{{(L_f+r_{x,j}\text{sin}({\theta }_z))}^4}{8E{I}_{f,zz}}\left(d{F}_{j+}^{\text{*}}-d{F}_{j-}^{\text{*}}\right) \\ \ge u_{r,j}(L_f+r_{x,j}\text{sin}({\theta }_z)). \end{gathered}$$ (20)

Accompanied by a corresponding bound on stator finger deformations

$$\begin{gathered} u_{s,j}^{\text{*}}\equiv \frac{{(L_f+r_{x,j}\text{sin}({\theta }_z))}^4}{8E{I}_{f,zz}}\left(d{F}_{j+}^{\text{*}}-d{F}_{(j+1)-}^{\text{*}}\right) \\ \ge u_{s,j}(L_f+r_{x,j}\text{sin}({\theta }_z)). \end{gathered}$$ (21)

If we define two reference differential forces, $d{F}_{j+}^{\mathrm{*}}$ and $d{F}_{j-}^{\mathrm{*}}$,

$$d{F}_{j+}^{\text{*}}\equiv \frac{1}{2}{ϵ}_0\frac{t_{Si}}{{\left(g_0-\text{sin}({\theta }_z)(r_{y,j}+L_f+r_{x,j}\text{sin}({\theta }_z))-u_{r,j}^{\text{*}}-u_{s,j-1}^{\text{*}}\right)}^2}$$ (22)

$$d{F}_{j-}^{\text{*}}\equiv \frac{1}{2}{ϵ}_0\frac{t_{Si}}{{\left(g_0+\text{sin}({\theta }_z)(r_{y,j}+L_f+r_{x,j}\text{sin}({\theta }_z))+u_{r,j}^{\text{*}}+u_{s,j}^{\text{*}}\right)}^2},$$ (23)

then since $u_{r,j}^{\mathrm{*}}$ and $u_{s,j}^{\mathrm{*}}$ are upper bounds on finger deformation, $d{F}_{j+}^{\mathrm{*}}$ and $d{F}_{j-}^{\mathrm{*}}$ will satisfy the assumptions in (17) and (18).

Together, (20), (21), (22), and (23) define a system of conservative nonlinear differential equations for finger deformation, permitting a conservative estimate for in-plane actuation moment

$$M_z^{\text{*}}({\theta }_z)\approx 2\sum _{j=1}^J{\int }_0^{L_f+r_{x,j}\text{sin}({\theta }_z)}\left(-d{F}_{j-}^{\text{*}}\cdot (r_{y,j}+s)+d{F}_{j+}^{\text{*}}\cdot (r_{y,j}+s)\right)ds.$$ (24)

Fig. 6(c) shows a sample progression of in-plane actuation moment versus reflector rotation angle, at progressively increasing voltages. Unsurprisingly, the estimated in-plane moment increases nonlinearly with both voltage and tilt-angle, reflecting the capacitive forcing function and gap-closing geometry. In-plane moment profiles are compared to in-plane rotational stiffness of a candidate mirror’s spring geometry to provide a conservative estimate of the maximum voltage permissible while operating the mirror. There may also be dynamic operating modes that avoid pull-in, but these can be difficult to implement without risk of error due to unmodeled dynamics, and increase complexity of mirror control.

III. Integration and Domain Optimization

Using methods from Section II, a sequence of optics and mirror design decisions were made, to establish conditions on mechanical constraints from optical simulation then select for desired mechanical (high frequency) and optical (small objective) goals. Design parameters were optimized within bounds on dimensions summarized in Tables II and III, respectively, with resulting mirror dimensions also given in Table III.

TABLE II.

Lens Parameter / Selection Bounds

Spherical lenses	Lens thickness (mm)	1.0–5.0
	Radius of curvature (mm)	2.0–20.0
	Refractive index	1.4
GRIN lenses	Lens length (mm)	2.0–5.0
	Refractive index	1.2–2.0
	Gradient constant (/mm)	0.3–1.6
Panes	Pane thickness (mm)	0.1–1.0
	Refractive index	1.4
Aspherical lenses	Lens diameter (mm)	1.0–5.0
	Lens thickness (mm)	1.0–3.0

TABLE III.

Mirror Parameters

		Bounds	Selected
Silicon thickness ($\mu \mathrm{m}$)	$t_{Si}$	20–40	40
Finger width ($\mu \mathrm{m}$)	$w_f$	5–15	5
Finger length ($\mu \mathrm{m}$)	$L_f$	100–250	170 (at center) 210 (maximum)
Gap width ($\mu \mathrm{m}$)	$g_0$	5–10	5
Torsion spring width ($\mu \mathrm{m}$)	$w_s$	20–100	35

The key result of optical modeling is that the product of NA and FOV is nearly proportional to the product of reflector size and tilt-angle, Fig. 7. In the idealized, linear ray transfer matrix, variability appears due to neglect of lens thickness variation and small angle assumptions. Greater variation is observed in ray trace simulation, in which non-idealities and exact geometry / lens fill factor can be considered. Physically, a beam diverging from a smaller reflector must be folded more dramatically at the objective, meaning that larger mirror rotations are required.

Fig. 7.

Target numerical aperture (NA) and field-of-view (FOV) of a miniaturized microscope correlate closely with required reflector size and tilt angle for a micromirror performing laser scanning at distal optics (dimension and tilt angle of minor axis shown).

This trade-off is not unique to this application. For example, an idealized lens free scan, $\mathrm{F}\mathrm{O}\mathrm{V}\approx 4{\theta }_{\text{x,max}}$. WD, with mirror size $\left(2r_b\right)\approx \mathrm{W}\mathrm{D}\cdot \mathrm{s}\mathrm{i}\mathrm{n}\left({\varphi }_{\text{N,edge}}\right)$, predicts a proportionality for $r_b\cdot {\theta }_{x,\text{max}}$ to $\mathrm{F}\mathrm{O}\mathrm{V}\cdot \mathrm{N}\mathrm{A}$ of $0.0143\mathrm{m}\mathrm{m}{-}^{\circ }/\mu \mathrm{m}$. A pixel-based approach as by [30] for projection using a micro-mirror estimates $N_x$ pixels by $N_x\approx 4{\theta }_{x,\mathrm{m}\mathrm{a}\mathrm{x}}\left(2r_b\right)/\left(\mathrm{\lambda }{K}_T\right)$, where $\mathrm{\lambda }$ is wavelength and $K_T$ is a factor for Gaussian clipping; matching FOV to nominal resolution, i.e., $\mathrm{F}\mathrm{O}\mathrm{V}\approx {\mathrm{N}}_{\mathrm{x}}\cdot 0.61\mathrm{\lambda }/\mathrm{N}\mathrm{A}$, predicts a proportionality of $0.0135\mathrm{m}\mathrm{m}{-}^{\circ }/\mu \mathrm{m}$. By methods above, proportionality from ray trace modeling ($0.0158\mathrm{m}\mathrm{m}{-}^{\circ }/\mu \mathrm{m}$) and full simulation ($0.021\mathrm{m}\mathrm{m}{-}^{\circ }/\mu \mathrm{m}$) are only modestly larger, affected by varied refractive index among elements and underfilling when simulating discrete off-the-shelf lenses.

Based on our target $500\mathrm{\mu }\mathrm{m}$ FOV and 0.45 NA, feasible micro-mirrors must achieve a minor axis and tilt-angle product $>4\mathrm{m}\mathrm{m}{-}^{\circ }$ following ray transfer models, and $>5.5\mathrm{m}\mathrm{m}{-}^{\circ }$ after detailed ray trace simulations. Mirror parameters were then evaluated for minor radius $r_b$, and system resonant frequency, $f_r$ scenarios. This meant that under an assumption of prismatic torsion springs, spring width, $w_s$ and spring length, $L_s$ are not independent, being required to satisfy

$$k_x=2\frac{G_{Si}\beta t_{Si}{w}_s^3}{L_s^3}={\omega }_r^2{J}_x$$ (25)

where $G_{Si}$ is the shear modulus of silicon, $\beta$ is a constant for rectangular beam torsion accounting for aspect ratio, ${\omega }_r$ is target resonant frequency in rad/s (${\omega }_r=2\pi f_r$), and rotational inertia $J_x$ is estimated as that of a silicon ellipse with thickness $t_{Si}$.

Maximum tilt angle was then optimized over $w_f,L_f,g_0$, and $t_{Si}$ for various values of $r_b$ and ${\omega }_r$. Sequential gradient searches were performed from randomly generated initial parameters. Fig. 8(a) shows trends for minor axis / tilt angle product versus elliptical reflector minor axis projected following the Section II-C and II-D. Largest mirror amplitudes are projected for low frequencies, due to comparatively low spring stiffness about the x-axis, while increasing distance between torsion springs produces proportionally higher stiffness against in-plane rotation. However, eventually the increasing distance from the axis of rotation to the fingers reduces finger overlap at even small rotation angles, limiting useful actuation torque.

Fig. 8.

Maximum minor radius—tilt angle product projected for elliptical, parametrically-resonant micro-mirrors after optimization with torsion springs for representative frequencies of (a) inner axis (2–12 kHz, rotation about major axis) and (b) outer axis (1–4 kHz, rotation about minor axis). Among larger fast-axis mirrors ($>0.8\mathrm{m}\mathrm{m}$ minor axis), highest amplitudes are predicted for comparatively low-frequency designs having high compliance in out-of-plane torsion, despite reduced stiffness of individual springs against in-plane torsion. For smaller mirrors (<0.6 mm minor axis) there are competing tradeoffs between reduced moment arm of electrostatic torque versus stiffness of supporting torsion spring. Slow-axis mirrors follow broadly similar trends but cannot achieve comparable frequencies.

The outcome of this analysis is that mirrors having the largest resonant frequencies at sufficient size and displacement are anticipated near 1 mm minor radius, with $5^{\circ }$ to $6^{\circ }$ mechanical scan angle. Fig. 8(b) also shows results of a repeated analysis with fingers and torsion springs placed on the orthogonal axes to represent likely slow-axis performance if implementing rotation about the long axis of a larger mirror plus gimbal. For a likely slow axis minor radius near 1.5 mm, comparable scan angles are also projected as feasible near 1 kHz.

Examples of underlying component trends in fast axis functionality are shown in Fig. 9: Maximum voltage increases with radius due to larger in-plane stiffness as the distance from the center of the reflector to the torsion spring increases, as well as with increased spring stiffness (higher $f_r$). Meanwhile, the permissible in-plane rotation angle before finger collision decreases at larger radii, eventually constraining feasible voltage. Fig. 10 shows individual parametric trends with fast axis minor radius and frequency, after applying a moving average to smooth numerical variation. In brief, optimal finger lengths are less sensitive to mirror size than spring width and thickness, which are ideally increased to reduce relative off-axis rotation. Lower scanning frequencies permit somewhat longer fingers as their compliance is less significant than that of a softer spring design. Naturally, higher frequencies and larger mirrors require larger spring cross-section, both width and thickness, with very low frequency mirrors again depending heavily on spring design to avoid in-plane rotation. Spring length is dominated by the frequency selection and mirror size.

Fig. 9.

Trends in sample scanner parameters that optimize tilt angle versus reflector minor radius and resonant frequency. (a) Voltage versus size and finger length show local optima in voltage limits versus reflector radius due to tradeoffs between an increase in in-plane mirror stiffness versus a decrease in in-plane mirror rotation without comb-finger collision. (b) Radius increase is accompanied by reduction in tilt angles over which substantial finger overlap is maintained (described by ${\theta }_r$), with little influence from other design parameters.

Fig. 10.

Trends in mirror design parameters maximizing anticipated scan angle versus mirror minor radius and target resonance. (a) Finger length. (b) Spring width. (c) Device thickness. (d) Spring length.

Finally, optical designs were sorted to select for small objective radius, Fig. 11(a). As might be expected, inclusion of a GRIN lens is beneficial for keeping a small objective radius, keeping the beam constrained within a small diameter while producing a high NA output. Fig. 11(b) shows sample high-performing configurations selected from 2- and 4-lens designs with a 1 mm mirror minor radius. Using more lenses generally (but not always) increases optical path length, but with smaller individual elements. This advantage becomes progressively smaller for each additional lens (i.e., Lens 5+). Use of 2 lenses selected for a half-pitch GRIN lens as or immediately before the objective, while for 4-lenses a quarter-pitch lens was preferable.

Fig. 11.

Sample design trends from ray transfer matrices. (a) Required objective versus reflector diameter indicates that smaller-diameter objectives can be used as lens number increases and with inclusion of a GRIN lens to fold the beam path. (b) Sample designs generated with 2 versus 4 lenses and sorted by objective diameter give more compact lateral dimensions with use of additional lenses, though at increase in total optical path length. Note that ray radii are shown only at entry and exit of lenses; beam profiles within the GRIN lens (typically the final or near final element) are not graphed.

This analysis indicated that the best optical designs would use a GRIN lens following the MEMS mirror (alone or compound objective), accompanied by at least one additional lens to adapt collimated or a low NA beam at the reflector to desirable incident angles at the GRIN lens. In the final realization, shown in Fig. 12(a), the MEMS mirror is followed by a scan lens followed by quarter-pitch GRIN lens combined with an aspheric plano-convex lens. Functionally, this arrangement resembles the use of a half-pitch GRIN lens as a method for relaying a focal point through a small diameter aperture, but with reduced length and radius while the aspheric plano-convex lens compensates for spherical aberration of the GRIN lens. In simulation, a nominal $\pm 5^{\circ }$ deflection in mechanical scan angle of the MEMS mirror results in a FOV of $450\mu \mathrm{m}\times 520\mu \mathrm{m}$, over which range the simulated spot size and point spread function from ZEMAX simulation indicates diffraction limited resolution. Fig. 12(a) illustrates the motion of the focal point with MEMS mirror tilt, while resulting spot size is illustrated in Fig. 12(b) and FOV in Fig. 12(c).

Fig. 12.

Selected optical and MEMS design, as simulated. (a) Lens sequence selected from off-the-shelf optics having approximately 1 mm minor radius reflector; a near-collimated beam (lens 1 not shown) strikes scanner, followed by an aspheric lens—GRIN lens—aspheric lens sequence. The mirror between the first aspheric lens and GRIN lens represents a dichroic mirror to separate collection light. (b) Ray trace showing lateral spot size within Airy disk. (c) Simulated translation of focal point across a 0.5 mm FOV with $\pm 5^{\circ }$ mechanical scan angle. (d) Mirror layout for micro-fabrication.

IV. Prototyping and Experimental Results

A. Mirror Fabrication and Testing

Several modifications to idealized mirror layout must be made for practical layout and fabrication. The placement of the lower-frequency, slow-axis degree-of-freedom using a gimbal requires space for the outer axis’s torsion spring, which disrupts nominal comb finger layout. Fig. 12(d) shows the comb finger locations after allocating space for slow-axis torsion springs and electrical connections to external bond pads. This reduces actuation torque but does allow higher voltages as the most vulnerable fingers are not present; some fingers farther from the center were also lengthened as gimbal space permitted.

Mirror fabrication was performed following [31], with the reflector, comb finger, and torsion spring geometry defined through photolithography and deep reactive ion etching (DRIE) of the silicon device layer of a silicon-on-insulator wafer. A two-step backside DRIE process is then performed to remove silicon behind the moving components of the scanner, followed by blanket coating of the device with a metal reflector layer.

Mirror dynamic response was measured with a photosensitive array. The mirror frequency response is shown in Fig. 13, in which characteristic nonlinear bifurcation dynamics of the parametrically-resonant system may be observed. Largest amplitudes are obtained during frequency down-sweep, with $\pm {11}^{\circ }$ optical scan ($\pm {5.5}^{\circ }$ mechanical scan angle), consistent with the simplified scanner model. Total fast-axis amplitude is very close to that predicted by simulation. Experimental resonant frequency is about $2\mathrm{\%}$ higher than simulation, which while significant is within fabrication tolerance given variation in silicon wafer layer thickness and silicon spring etch resolution ($10\mathrm{\%}$ typ.). Slow-axis response was closely matched to predicted frequency, Fig. 13(c), but full predicted amplitude was not achieved in all devices, due to another vibration mode arising close to the nominal slow-axis resonant frequency. While the completed scanner contains fewer comb fingers than the idealized model, this was offset by tolerance for slightly higher operating voltage than the analytical bound, thanks to conservative assumptions in the scanner failure model. Due to small device quantities, formal failure testing could not be performed. However, inadvertent failure was observed near 100 V in at least one device, compared to an estimated failure voltage of approximately 90 V.

Fig. 13.

Experimental mirror characterization. (a) Scanning mirror following fabrication. (b) Mirror frequency response at 80 Vpp, compared to simulated down-sweep for analytical model optimized at 4 kHz and $r_b=1\mathrm{m}\mathrm{m}$. (c) Slow-axis response, compared to modeled analytical maximum.

Table IV compares scanner performance to a selection of biresonant electrostatic mirrors from literature. Only significantly larger, slower mirrors achieve necessary angle- radius product, though higher frequencies can be achieved for much smaller reflectors. Some recent piezoelectric biresonant mirrors [32], [33] have also achieved comparable angle-radius product at higher frequencies, which while small in reflector size suggest piezoelectric forcing as a path to performance improvements, if at increased fabrication complexity and cost.

TABLE IV.

Sample Biresonent MEMS Scan Mirrors, $\sim 5\mathrm{M}\mathrm{m}$ Chip Size or Smaller, Ordered by Fast-Axis Frequency

		Fast-axis		Slow-axis
	reflector (mm x mm)	radius·±MSA (mm°)	freq. (Hz)	radius x · ±MSA (mm°)	freq. (Hz)
[34], 2017	3.6 x 3.6 1.2 x 1.2	14.3 3.7	0.32 2.6	14.3 3.7	0.32 2.6
This article	2 x 2.8	5.5	4	3.8	1
Duan et al. [35]	2 x 2	4	4.3	4.5	1
Hung et al. [36]	1 x 1	4.1	22	3.7	1.4
Wu and Lin [37]	0.5 x 0.5	1.6	31	0.6	4.9

B. System Integration

To perform imaging with the proposed optics and scanning mirror, a prototype instrument was assembled and integrated with a laser source and fluorescence collection hardware, shown in Fig. 14. An ultrafast laser source (MaiTai Spectra Physics) delivers a femtosecond pulse that is focused on a single mode fiber. The focused beam is then collimated and directed into the miniature optical assembly. The fluorescence signal from a specimen is collected through the dichroic mirror and transmitted to a photodetector. Imaging software then utilizes the scanned data to construct an image of the target.

Fig. 14.

System integration. (a) Femtosecond laser is directed to the miniature optics, and MEMS scanner; fluorescent signal from a specimen is collected via dichroic mirror (DM) and transmitted to a photomultiplier tube (PMT) with analog-to-digital converter (ADC), from which the MEMS drive signal and intensity data is used to reconstruct the collected image. (b) Miniature optical components were packaged within a 3D-printed frame shown as a solid model, with a modular chip base for the scanner. (c) Prototype with mouse cortex sample.

C. Optical Characterization

To evaluate the lateral resolution, the well-established knife edge test method was employed while operating the prototype imaging probe in reflectance mode. First, to establish lateral resolution, the 10%–90% intensity transition was measured across a sharp edge (border of a standard USAF 1951 resolution target). This transition corresponds to the widely recognized Rayleigh resolution criterion. Lateral resolution was measured experimentally as $1.5\mu \mathrm{m}$, consistent with the nominal spot size and resolution predicted by ray trace simulation and meeting design specifications. To evaluate axial resolution and working distance, a knife edge test was conducted in which reflected light intensity was collected while translating a reflective surface parallel to the focused beam. Axial resolution of the prototype optics was estimated at $6\mu \mathrm{m}$, again consistent with optical simulations and significantly improving on prior two-photon implantable imaging probes. Details on resolution characterization in reflectance mode follow [25].

Once installed in the instrument, two-axis scanning does entail a small truncation of the optical path as the input beam diameter exceeds the mirror dimension by $5–10\mu \mathrm{m}$, but this is not a significant influence on imaging compared to finite lens diameter as mentioned above. We note that biresonant mirrors may experience some reduction in scanning amplitude when operating both axes, due to spring stiffening or asymmetry [38]. However, the former is limited by the gimbal structure and the latter by symmetric layout; amplitude reductions were not observed during imaging tests. The most significant limiting aperture was the GRIN lens, specifically for mirror tilt greater than approximately $3^{\circ }$, resulting in poorer focus and reduced light collection at the edge of the image.

D. Imaging Results

Prototype two-photon imaging tests were first conducted using polystyrene microbeads coated with green or yellow fluorescent protein (GFP and YFP) dye. A sample image from GFP-dyed, $10\mu \mathrm{m}$ beads is shown in Fig. 15(a). Imaging shows clear discrimination and differentiation of individual beads, consistent with anticipated instrument resolution. It should be noted that maximum intensity of beads is obtained only in the center of the image due to slight curvature of the focal plane when using distal scanning. Ex-vivo tissue imaging was then performed using Thy1-YFP-H mouse brain samples containing neurons expressing YFP, as shown in Fig. 15(b). Individual neurons and connecting neurites could be resolved from the sample tissue, verifying feasibility of neuron imaging using the proposed optical design and scanning mirror. Total FOV exceeded $500\mu \mathrm{m}$ in $x$- and $y$-axes, again matching design specifications. Cortex images were collected at 960 nm excitation wavelength with 140 fs pulse width and 80 MHz repetition rate. Fig. 15 images were collected from among four imaging sessions with various samples based on availability, conducted over six weeks. Individual sessions ranged from 30 min to $>2\mathrm{h}$. Mirror characterization and imaging tests were conducted at room temperature and pressure, without hermetic packaging. While total lifetime of the MEMS scanner and prototype instrument has not been measured, this suggests good system repeatability and robustness.

Fig. 15.

Prototype two-photon microscope sample images. (a) $10\mu \mathrm{m}$ diameter polystyrene microbeads with green fluorescent protein (GFP) dye, excited at 920 nm, demonstrate image resolution sufficient to distinguish neuron-scale features. (b) Mouse neurons in excised mouse cortex labeled with yellow fluorescent protein (YFP), demonstrate imaging after estimated tissue penetration of $250\mu \mathrm{m}$, approximate $500\times 500\mu {\mathrm{m}}^2$ FOV.

V. Conclusion

Implantable two-photon microscopy has been used to achieve high-resolution neural imaging in moving mammals. However, maximum WD and FOV remain limited, in part due to constraints on laser scanning technologies. This article synthesizes optical modeling with MEMS scanning mirror dynamic analysis to create a two-photon implantable microscope design that extends comparatively large FOV imaging ($500\times 500\mu {\mathrm{m}}^2$) to a $\mathrm{W}\mathrm{D}>300\mu \mathrm{m}$. The resulting optical design uses a compound objective lens to maintain small optical spot size across the full FOV, while the parametrically-resonant scanning mirror provides $\pm {5.5}^{\circ }$ fast-axis scan angle at 4 kHz with a minimum reflector radius of 1 mm. To produce this design, a model for maximum voltage before pull-in failure of elliptical, electrostatically-driven scanning mirrors is proposed in order to estimate maximum operating voltages and nonlinear dynamic response.

For translation to imaging in moving mammals, the instrument probe must be enhanced to protect the scanning mirror and internal optics after implantation, and light collection efficiency through an optical fiber must be improved. Meanwhile, while the proposed mirror model provides reasonable predictions of fabricated micro-mirror performance, approximations made to simplify comb finger deformation modeling can cause voltage bounds to be underestimated. Nonetheless, proposed methods appear to provide useful bounds on micro-scanner capabilities and trends in scanner design that can be used to produce high quality two-photon microscopy with small MEMS and optical elements.

Biographies

Tayebeh Sahraeibelverdi received the B.S. degree in electronics from Islamic Azad University, Saveh, Iran, and the M.S. degree in micro-electromechanical system from the Malekashtar University of Technology, Tabriz, Iran and the Ph.D. degree from the University of Michigan, Ann Arbor, MI, USA, in 2023, where she focused on developing optical system design, microsystem integration, and MEMS scanner fabrication and characterization

Prior to that, during a Ph.D. program in electrical engineering with the University of Tabriz, she further specialized in photonic sensor design. She currently works as an Optical Engineer at Meta Reality Labs.

Ahmad Shirazi received the Ph.D. degree in integrative systems and design from the University of Michigan, Ann Arbor, MI, USA, in 2023.

He is currently an Imaging and Sensing Hardware Development Engineer with Apple, Cupertino, CA, USA. His research focused on the design and fabrication of novel micro- and small-scale imaging systems for medical applications.

Miki Lee received the B.S.E. degree in aerospace engineering and the M.E. degree in space engineering from the University of Michigan, Ann Arbor, MI, USA, in 2015 and 2016, respectively.

She was a Research Lab Specialist Associate with the Department of Internal Medicine, University of Michigan, and is currently a Systems Design Engineer with the Allen Institute. Her research interests include the imaging system development and MEMS sensor design.

Haijun Li received the Ph.D. degree in microelectronics and solid-state electronics from Jilin University, Changchun, China, in 2007.

He was a Post-doctoral with Nanyang Technology University, Singapore, from 2008 to 2011. He was a Senior Engineer with Hebei Semiconductor Research Institute, Shijiazhuang, China, from 1997 to 2008, and a Research Investigator with the University of Michigan, Ann Arbor, MI, USA, from 2011 to 2022. His research interests include MEMS, infrared detectors, and wafer-level packaging.

Sung Eun Kwon received the B.S degree in neuroscience from the Australian National University, Canberra, ACT, Australia, in 2004, and the Ph.D. degree in physiology from the University of Wisconsin, Madison, WI, USA, in 2011.

He was a Post-doctoral Fellow with Stanford University and Johns Hopkins prior to an appointment as an Assistant Professor of Cellular, Developmental, and Molecular Biology with the University of Michigan, Ann Arbor, MI, USA, from 2018-2023. He is currently a Senior Scientisted with AbbVie, North Chicago, IL, USA. His research seeks to develop novel therapeutic strategies for neurological disorders with sensory/motor abnormalities.

Thomas D. Wang received the B.S. degree in mathematics and physics from Harvey Mudd College, Claremont, CA, USA, in 1985, and the Ph.D. degree in medical engineering and medical physics from the Massachusetts Institute of Technology, Cambridge, MA, USA, in 1996.

He is currently a Professor of Medicine, Biomedical Engineering, and Mechanical Engineering with the University of Michigan, Ann Arbor, MI, USA, and the H. Marvin Pollard Collegiate Professor of Endoscopy Research. His research interests are in the field of biomedical optics, multiplexed detection, and molecular imaging.

Kenn R. Oldham (Member, IEEE) received the B.S. in mechanical engineering from Carnegie Mellon University, Pittsburgh, PA, USA, in 2000, and the Ph.D. degree in mechanical engineering from the University of California at Berkeley, Berkeley, CA, USA, in 2006.

He is currently a Professor of mechanical engineering with the University of Michigan. His research focuses on the intersection of control systems and microscale sensing and actuation, with applications in microrobotics, microscopy, and inertial, and physiological sensing.