2.1.

Selection of MEMS DM and Zernike Modes for Test

Since speed is critical for active scanning and intrascan focus control and aberration correction, liquid crystal spatial light modulators that can achieve frame rates up to a few hundred Hertz are too slow and were not considered. On the other hand, commercially available MEMS DMs have electronic update rates of hundreds of kHz and published mechanical response times of tens of $\mu \mathrm{s}$, making them candidates for this application. Furthermore, previous publications^8,10,15 demonstrate that MEMS DMs are capable of correcting sample-induced aberration in a variety of tissue types up to imaging depths of a few hundred microns. To choose the proper MEMS DM for the active/adaptive microscope, we start with simulations of the systematic aberrations of the active/adaptive microscope in Zemax. This gives us a guideline for the selection of the MEMS DMs to be used in the active/adaptive microscope, and the Zernike modes that we will dynamically test. That simulation is described below.

There is another consideration in choosing the MEMS DM, which is the type of mirror. There are three main types of MEMS DM that are commercially available, which are the segmented mirror with only piston motion, the segmented mirror with piston/tip/tilt motion, and the continuous face sheet mirror surface. We are interested to know whether a segmented mirror is capable of creating the shapes that meet our prescription with a residual error that is small enough to achieve diffraction-limited imaging. Using Fourier optics analysis to compute the aberrated point spread function (PSF), we found that a segmented, piston-only array with $12\times 12$ actuators like the BMC segmented multi-DM cannot reach a Strehl of 0.8 (our criteria for diffraction-limited imaging) when compensating the aberrations we expect to see. However, a continuous face sheet mirror, or a segmented mirror with piston/tip/tilt, such as the IrisAO PTT111 DM, provides sufficiently small residual error for a diffraction-limited PSF. Segmented, piston-only mirrors with a larger number of actuators such as the BMC kilo-DM might achieve our criteria, but at greater cost and complexity, and were not evaluated.

Candidate mirrors we considered for the “tweeter,” therefore, included the tip/tilt/piston segmented IrisAO PTT111 DM with a $200\text{-}\mu \mathrm{s}$ mechanical response time and an $8\text{-}\mu \mathrm{m}$ mechanical stroke, and continuous face sheet mirrors such as the Alpao DM69 with an $800\text{-}\mu \mathrm{s}$ settling time and $30\mu \mathrm{m}$ surface stroke, the Mirao 52-e with a $25\text{-}\mu \mathrm{m}$ stroke (unspecified settling time), and the BMC multi-DM with a $60\text{-}\mu \mathrm{s}$ response time and $1.5\mu \mathrm{m}$ stroke. For the “woofer,” we considered the Alpao and Mirao mirrors, and the Revibro Optics 4-zone DM with a published settling time of less than $200\mu \mathrm{s}$. Using response time as the primary discriminator, with the cost and computational burden associated with a higher actuator count mirrors as a secondary consideration, we selected the BMC multi-DM continuous face sheet mirror to evaluate for the “tweeter” and the Revibro Optics DM to evaluate for the “woofer.”

The Revibro DM, shown in Fig. 3, is based on a design described previously by Moghimi et al.¹² It comprises a stretched metalized membrane, suspended over four concentric annular electrodes, and includes vertical channels through the backplate to control air damping. The device described in this paper has 3 rings of 16 holes per ring (Model No. M00120T), but other configurations can allow for more or less air damping.

Fig. 3

(a) Photomicrograph showing the electrodes and air hole pattern under the membrane and (b) top view of the mirror membrane.

It is worth noting that while the published technical data on the mechanical response of the MEMS DMs provide an estimate of the static and dynamic performance that is possible, it does not provide quantitative information on residual error that may be expected when driving to an actual prescription on the DM, or, just as important, how quickly the residual error will settle when switching from one prescription to another. Our testing directly measures these important performance measures.

To know which Zernike modes will be most important in order to compensate systematic aberrations, we performed a Zemax simulation of an optical system that comprised the MEMS mirrors, relays, scan lens, tube lens, and a representative $25\times$ 1.0 numerical aperture (NA) water immersion objective lens selected from the patent literature.¹⁶ The MEMS mirrors are modeled as Zernike fringe surfaces, with the Zernike coefficients treated as variables for optimization. The focus control mirror is capable of a stroke of $15\mu \mathrm{m}$, which corresponds to an $80\text{-}\mu \mathrm{m}$ focus adjustment using $\mathrm{NA}=1.0$ in an aqueous sample, using the equation ¹⁷

The dominant aberrations of the optical system that the DMs must correct are coma $Z_{3,1}$ and spherical aberration $Z_{4,0}$ [we use the American National Standards Institute (ANSI) $Z_{n,m}$ indexing scheme for radial order $n$ and azimuthal frequency $m$], arising from the fact that the active focus control makes the objective lens operate at a focus location other than its optimum working distance. Other off-axis aberrations also come into play when the galvo scanners steer the beam to larger scanning angles, with appreciable contributions from both the relay optics and the objective lens. The simulation results show significant astigmatism $Z_{2,2}$, secondary astigmatism $Z_{4,2}$, secondary coma $Z_{5,1}$, trefoil $Z_{3,3}$, and secondary spherical aberration $Z_{6,0}$. The range of simulated aberration coefficients are summarized in Table 1. Note that we are using the “unnormalized” polynomials for our Zernike basis, given explicitly here:

Table 1

The dominant aberrations over a 500  μm field of view and full 80  μm focus range, from our Zemax model. The numbering of the Zernike terms follows the ANSI Zn,m indexing scheme for radial order n and azimuthal frequency m.

The higher-order aberrations are within the deflection range that can be compensated by the Boston Micromachines multi-DM with a $1.5\text{-}\mu \mathrm{m}$ stroke operating as the “tweeter,” with defocus and spherical aberration handled by the Revibro Optics mirror as the “woofer.” Based on these simulations, we chose to characterize astigmatism, coma, and trefoil on the BMC multi-DM in order to include both lower-order and higher-order shapes for comparison of the dynamic responses. For completeness, we also characterized primary spherical aberration using that mirror. Defocus and primary spherical aberration were characterized on the Revibro Optics DM.

2.2.

Characterization by Stroboscopic Phase Shift Interferometry Synchronized to Mirror Movement

Stroboscopic interferometry can effectively freeze high-speed motion when the light pulses are sufficiently short and synchronized with the periodic movement of the MEMS DMs. In this case, exposure time is controlled by the laser pulse width rather than the integration time of the camera. The MEMS DM is driven with a periodic waveform. With proper delay, the pulsed laser diode illuminates the DM only at one specific phase of the periodic motion. The pulse width of the illumination is set short enough so as to maintain high fringe contrast for accurate phase unwrapping during data processing.

We have constructed a custom phase-shifting Michelson interferometer for dynamic testing (Fig. 4). The sync hardware includes four function generators (two SRS DS345, HP 33120A, Sony Tektronix AFG320), which are capable of being controlled remotely. In order to ensure that the illumination takes place at the right phase of each periodic cycle, arbitrary waveforms for delays and laser pulse triggering are created and called through the control software. The control software is developed under the Labview environment, which interfaces to all the hardware including the function generators, a pulsed laser driver (DEI PCX-7410), camera, and piezo-electric stage.

Fig. 4

The system setup for dynamic testing using stroboscopic phase-shift interferometry.

To control the BMC multi-DM, we use the high-speed X-driver with X-CL™ PCIe Interface Card. An SMA connector on the PCIe card allows external triggering to synchronize the movement of the DM and the strobe pulse. The 140 actuators on the BMC DM are set digitally, within the MATLAB environment, by assigning a matrix of values from 0 to 65535 mapped to a voltage range from 0 to 300 V. On the other hand, the Revibro DM is driven entirely in analog. For this, a two-channel function generator (Sony Tektronix AFG320) and a two-channel high speed high voltage amplifier (Trek 603 Piezo Driver/Power Amplifier) are coupled to provide the voltage needed for the electrostatic actuation of the metal coated polymer membrane on the Revibro DM. Although this mirror has four concentric actuation electrodes, potentially allowing fine control over higher-order spherical aberration (secondary spherical aberration $Z_{6,0}$ and tertiary spherical aberration $Z_{8,0}$), for dynamic testing, we connected the central two and outer two electrodes together to form a two-zone mirror. The two-zone control was sufficient to modulate primary spherical aberration independently from the focus setting.

2.3.

Mirror Training

With a phase-shifting Michelson interferometer, we are able to capture the surface height of the DMs. For analysis, the surface height is fit to 55 spatially orthogonal aberration modes (the unnormalized Zernike modes, up to order $n=10$), with the surface deflection S reconstructed according to $S={\sum }_{n,m}{a}_{n,m}{Z}_{n,m}(\rho ,\theta )$. Since the commercially available DMs do not come with a mapping between surface shapes and control voltages, the first task was to train the mirror to the target shapes we elected to test.

An iterative algorithm was used to find the control voltages that produced a single Zernike mode of desired amplitude while suppressing all other terms. In brief, the algorithm is as follows. A target surface shape was calculated for the desired mode. The mirror surface deflection was then measured using the interferometer, and the height of a region near the center of each actuator zone was compared to the target for that location. The control voltage $V_i$ is updated according to $V_{i,n+1}^2=V_{i,n}^2+\alpha {\epsilon }_{i,n}$, where ${\epsilon }_{i,n}$ is the error between the target height and the measured height and $V_{i,n}$ is the control voltage, for the $i$’th zone and the $n$’th iteration. The gain parameter $\alpha$ is chosen to facilitate convergence of the algorithm. We work with the square of the voltage, since the electrostatic force varies as the square of the applied voltage. It should be noted that because the BMC mirror and the Revibro mirror are electrostatically actuated they can be deflected only in one direction. On the other hand, the Zernike modes exhibit both positive and negative displacements. We, therefore, bias the mirror with an initial deflection, with the Zernike mode superimposed on that initial shape. For the BMC mirror, this initial deflection is an “active flat” surface with each actuator deflected to 50% of the rated $1.5\mu \mathrm{m}$ maximum deflection. For the Revibro mirror, the initial deflection is a midrange defocus setting that is purely parabolic in shape.

We were successful in training each of the test modes, with residual contribution from all other modes less than 20 nm rms. Height maps from these trained shapes on the BMC multi-DM are shown in Fig. 5 (all mode amplitudes in the figure are 400 nm). Similarly, control voltages were determined to train both defocus and primary spherical aberration on the Revibro mirror.

Fig. 5

(a) Standard Zernike shapes trained on the BMC multi-DM using a closed loop feedback training process. From left to right, astigmatism, coma, defocus, primary spherical, and trefoil. The amplitude $a_{n,m}$ for these standard Zernike shapes is 400 nm. The residual rms error on these shapes is 5, 19, 7, 20, and 12 nm for astigmatism, coma, defocus, primary spherical, and trefoil, respectively. (b) 300 nm primary spherical aberration superimposed on $3\mu \mathrm{m}$ defocus trained on the Revibro DM. Scale bars show height in nm.

2.4.

Description of Specific Tests Performed

2.4.1.

Zernike mode frequency response

For continuous intrascan correction of aberrations such as coma or astigmatism, the time dependence of any particular wavefront shape will be nearly periodic, synchronized to the fast-scan mirror. In order to quantify the capability of the DM, we chose to measure a sinusoidal steady-state frequency response for specific aberration modes. The mode amplitude is varied sinusoidally at a particular frequency, and the mirror response is measured in terms of amplitude and phase delay of that specific mode relative to the driving waveform. These are plotted versus frequency using a Bode style plot.

As shown in Fig. 6, a sampled sinusoidal signal at frequencies ranging from near DC to several kHz drives the MEMS DM. For the BMC DM, a pulse triggers a new voltage map to be assigned to the 140 actuators, with 12 shapes per period comprising the stepwise sinusoidal trajectory. For example, in the figure, the mirror is driven to 12 different amplitudes of trefoil, with the amplitude following a sinusoidal time dependence. Synchronously, the phase-shifting interferometer strobe is progressively set to six different sampling delays, generating six different height maps. Each height map is decomposed into Zernike modes, and the measured amplitude of the mode of interest is then curve fit to find the magnitude and the phase relative to the driving phase. Using the X-driver and high speed camera card (X-CL™ PCIe Interface Card) we can update the BMC voltage map at a maximum speed of 400 kHz. This sets a maximum sinusoidal test frequency of $400/12=33\mathrm{kHz}$. Our tests included a maximum sinusoid frequency of 12.5 kHz.

Fig. 6

Stroboscopic imaging for frequency response. Twelve frames of 12 different amplitudes (trefoil shown in this figure) form a periodic sinusoidal movement on the BMC DM, and the submicrosecond light illumination pulses take place at 30 deg, 90 deg, 150 deg, 210 deg, 270 deg, and 330 deg in each cycle to capture the instantaneous movement of the BMC DM.

3.1.

BMC Multi-DM

3.1.1.

Frequency response

Figure 8 plots the magnitude and phase response for five different Zernike modes on the BMC multi-DM. The results show that astigmatism, defocus, and trefoil exhibit a 3-dB frequency near 6.5 kHz. Coma has a 3-dB frequency closer to 7 kHz, while spherical aberration has the largest 3 dB bandwidth near 10 kHz. All modes were driven with the same 400 nm amplitude, but the membrane shapes are different, as are the peak-to-peak deflections. Defocus, astigmatism, coma, and trefoil all have peak-to-peak displacements of the membrane equal to twice the Zernike amplitude (800 nm), while spherical aberration has a peak to peak displacement of 1.5 times the amplitude (600 nm). The higher 3-dB frequency of spherical aberration thus seems to correlate to a smaller peak-to-peak membrane deflection, but the damping due to air flow beneath the membrane is important, and its dependencies on mode shape and amplitude have not been separated here. It is noteworthy that the frequency response for all modes shows a smooth roll-off with no peaking, indicating that these mirrors are over-damped and exhibit no resonances at least up to 12.5 kHz.

Fig. 7

Stroboscopic imaging for a step response. The submicrosecond illumination pulses capture the instantaneous movement of the rising edge during a step from flat to the target amplitude of a specific Zernike mode (trefoil shown in this figure). By using proper delays, we evaluate the full evolution of the mirror shape.

Fig. 8

Frequency response for different standard Zernike shapes on the BMC multi-DM.

3.1.2.

Step response

Figure 9 shows step response measurements for five different Zernike modes. Each mode was driven from the active flat membrane position to the mode shape with a 400-nm amplitude. Rise times (10% to 90%) are calculated and tabulated in Table 2. Rise times measured for all five aberration modes ranged from 44 to $61\mu \mathrm{s}$. By computing the rms surface deviation from the final target shape, we can plot rms surface error versus time in an error relaxation plot. Although the step height for each mode has the same 400 nm amplitude, these are the nonnormalized Zernike modes and so exhibit a different rms deviation. That is reflected in the starting rms error value for each of the five modes. Note that the RMS error includes the error contribution from all modes, including the mode that is being stepped; the full transient membrane shape is accounted for. A meaningful metric is the error settling time, when the overall mirror rms error is less than $\lambda /28$, corresponding to Maréchal’s criterion for $\mathrm{Strehl}\ge 0.8$. With $\lambda =1.0\mu \mathrm{m}$, this limit is 36 nm rms, which is indicated by the solid red line in Fig. 9. Error settling time for all five of these mode steps is less than $42\mu \mathrm{s}$. The results are tabulated in Table 3.

Fig. 9

Step response for different tested Zernike modes at a step from 0 to 400 nm in amplitude on the BMC multi-DM (referred to the right side y-axis). The rms error is also plotted against time for each Zernike mode step (referred to the left y-axis). The solid line at 36 nm RMS error is the threshold to recover to 80% Strehl, according to the Marechal’s criterion ($\lambda =1.0\mu \mathrm{m}$).

Table 2

Summary of the rise time for the BMC multi-DM.

	Rise time (μs)
	100 nm	200 nm	300 nm	400 nm	500 nm	−400  nm
Astigmatism	41	54	56	61	57	55
Coma	57.6	41	42	45	43.6	45
Defocus	42	49	51	44	48	61
Spherical	42	37	36	44	37	39
Trefoil	61.5	53	53	52	52	52

Table 3

The time needed for the BMC MEMS DM to settle below 36 nm of RMS error to achieve diffraction-limited imaging.

	Settling time (μs)
	100 nm	200 nm	300 nm	400 nm	500 nm	−400  nm
Astigmatism	3.3	22	32	40	46	38
Coma	2	14	18	32	37.5	32
Defocus	9	30	38	40.5	48	52
Spherical	6	24	32	42	40	36
Trefoil	2	17	28	33.5	40	34

We also investigated the influence of mode amplitude on step response and error settling time. Figure 10 shows a family of step response curves for astigmatism with amplitudes ranging from 100 to 500 nm, and also for $-400\mathrm{nm}$. Rise times range from $40\mu \mathrm{s}$ for the smallest step to $63\mu \mathrm{s}$ for the 400-nm step. Error settling times show a particularly strong dependence on mode amplitude, since the threshold of 36 nm error is barely exceeded for the full amplitude 100 nm astigmatism step. Even the 500 nm astigmatism step settles to within the 0.8 Strehl limit within $46\mu \mathrm{s}$. Table 2 summarizes the rise times, and Table 3 summarizes the wavefront error settling times for all five tested modes for amplitudes from 100 to 500 nm and for $-400\mathrm{nm}$. Within this range of correction, all of the steps reach the threshold for diffraction-limited precision in less than $52\mu \mathrm{s}$.

Fig. 10

Step response for different amplitudes of astigmatism on BMC multi-DM and the RMS error is plotted against time for astigmatism on BMC multi-DM at its rising edge during a step response.

3.2.

Revibro Optics DM

3.2.1.

Step response

Figure 11 plots the response for five different defocus steps of the Revibro DM. The peak-to-peak deflection is twice the Zernike amplitude, so that 1- and $7.5\text{-}\mu \mathrm{m}$ steps correspond to mirror sags of 2 and $15\mu \mathrm{m}$, respectively. We see that the step response is somewhat underdamped with a slight overshoot, so that the error plots show some oscillation. The smaller step from 2.9 to $3.1\mu \mathrm{m}$ also shows a slight overshoot. Rise times (10% to 90%) range from 28 to $39\mu \mathrm{s}$ for these tests.

Fig. 11

Step response of active focus control with Revibro MEMS DM at 0 to 1, 0 to $3\mu \mathrm{m}$, 2.9 to $3.1\mu \mathrm{m}$, 0 to 5, and 0 to $7.5\mu \mathrm{m}$. The RMS error is also plotted against time at the rising edge of a step response for different amplitudes in focus control on the Revibro multi-DM.

Because of the ringing in the step response, the settling times are a bit longer for the larger focus steps, all of which require at least $76\mu \mathrm{s}$ (up to $226\mu \mathrm{s}$ for maximum deflection at $7.5\mu \mathrm{m}$) to reach a diffraction-limited shape. The smaller defocus step from 2.9 to $3.1\mu \mathrm{m}$ has an error that settles much faster in about $32\mu \mathrm{s}$. The rise times and settling times for defocus on the Revibro mirror are summarized in Table 4.

Table 4

Summary of dynamic performance for defocus on Revibro MEMS DM.

	Defocus on Revibro DM
	1  μm	3  μm	2.9 to 3.1  μm	5  μm	7.5  μm
Rise time ($\mu \mathrm{s}$)	28	39	32	37	58
Settling time ($\mu \mathrm{s}$)	82	76	32	138	226

Finally, we measured the step response for a change in spherical aberration $Z_{4,0}$ on the Revibro mirror. We characterized mode amplitudes of 200, 400, and 600 nm. These are superimposed on a defocus bias of $3\mu \mathrm{m}$ ($6\text{-}\mu \mathrm{m}$ mirror sag). The results are plotted in Fig. 12. More ringing is observed for these steps, showing that the spherical aberration shape is less damped in this free-standing membrane. The rise times are quite short, less than $15\mu \mathrm{s}$. The error settling times are variable, ranging from 13 to $45\mu \mathrm{s}$, affected by the oscillation in the step response. The rise times and error settling times for primary spherical aberration on the Revibro DM are summarized in Table 5.

Fig. 12

The step response for different amplitudes in spherical aberration correction on the Revibro MEMS DM. The RMS error (excluding defocus) during the step response is also plotted (referred to the $y$-axis on the left side).

Table 5

Summary on dynamic performance for primary spherical aberration on Revibro MEMS DM.

	Primary spherical on Revibro DM
	−100 to 100 nm	−200 to 200 nm	−300 to 300 nm
Rise time ($\mu \mathrm{s}$)	14	13.4	13.4
Settling time ($\mu \mathrm{s}$)	13	15	45