## Abstract

We study experimentally the effective duty cycle of galvanometer-based scanners (GSs) with regard to three main parameters of the scanning process: theoretical/imposed duty cycle (of the input signal), scan frequency, and scan amplitude. Sawtooth and triangular input signals for the device are considered. The effects of the mechanical inertia of the oscillatory element of the GS are analyzed and their consequences are discussed in the context of optical coherence tomography (OCT) imaging. When the theoretical duty cycle and the scan amplitude are increased to the limit, the saturation of the device is demonstrated for a useful range of scan frequencies by direct measurement of the position of the galvomirror. Investigations of OCT imaging of large samples also validate this saturation, as examplified by the gaps/blurred portions obtained between neighboring images when using both triangular and sawtooth scanning at high scan frequencies. For this latter aspect, the necessary overlap between neighboring B-scans, and therefore between the corresponding volumetric reconstructions of the sample, are evaluated and implemented with regard to the same parameters of the scanning process. OCT images that are free of these artifacts are thus obtained.

© 2015 Optical Society of America

## 1. Introduction

Galvanometer-based scanners (GSs) are the most utilized devices for lateral scanning in various biomedical imaging techniques, including optical coherence tomography (OCT) [1,2], confocal microscopy (CM), and combinations of different imaging modalities [3]. Two-dimensional (2D) scanning of the sample, referred to as raster scanning, is usually achieved in such systems with dual-axis double GSs [4,5] or, more rarely with a combination of a polygon mirror (PM) scanner for the fast axis, and a one-axis GS to position precisely the line scan on the slow axis [6,7]. Two-dimensional GS systems have also been employed in association with Risley prisms [8,9] for volumetric (3D) scanning [10,11] or for time-included (4D) scanning in the context of real time *in vivo* imaging [12].

The use of unidimensional (1D) GSs has been also reported in various applications including generating the modulation necessary in *en face* time domain (TD) OCT with both on-axis and off-axis GSs [13], scanning transmissive delay line for Fourier domain (FD) OCT [14], phase or frequency modulation [15], and handheld scanning probes for spectral domain (SD) OCT [16,17].

The extensive use of GSs in high-end applications like biomedical imaging is related to their numerous advantages [18]: precision positioning, a large range of scan frequencies and velocities, compact constructive solutions, and accurate positioning of their mirror on the oscillatory axis. The latter two aspects are considered advantageous compared to PM scanners [4,5,19–21]. However, GSs provide lower scan frequencies and velocities compared to PMs and resonant scanners; the latter are usually built as microelectromechanical systems (MEMS). Nevertheless, the adjustable frequency of the GSs with regard to the fixed frequency of resonant devices is an advantage. Thus, different input signals can be employed for GSs whereas MEMS are usually driven with sinusoidal functions. As we have shown in previous work [22], triangular scanning provides artifact-free OCT images (with appropriate scan parameters), whereas sinusoidal scanning always introduces significant distortion of images. Such aspects have to be compensated in postprocessing in order to obtain distortion-free images in the case of sinusoidal scanning with MEMS [23].

GS technology is mature in terms of constructive and functional aspects [21]: motors, mobile elements, bearings, position sensors, heatsinkers, control structures, and testing. Some issues still remain unsolved, for instance the optimal functions necessary to obtain a maximum duty cycle. We have demonstrated theoretically in this respect that, for a maximum time efficiency of the scanning process (i.e., a maximum duty cycle of the GS), one has to use—for symmetrical/triangular scanning—linear plus parabolic scanning functions [24], while in the literature the linear plus sinusoidal functions have been previously considered best from this point of view [21].

However, the biomedical imaging community does not usually rely on custom-made input signals for the GSs but on common input signals like sawtooth, purely triangular (i.e., without nonlinear portions), and sinusoidal. In previous work, we have investigated experimentally these different scanning signals and their effects on OCT imaging [22]. Rules of thumb have been proposed to allow for an optimal use of these scanning devices in OCT setups, and some of these aspects will be briefly discussed at the beginning of this study. We extended in the present work these investigations from 1D GSs to 2D dual-axis GSs in OCT applications. All OCT systems that employ GSs for lateral scanning are targeted by these studies. Thus, while in the previous study an FD-OCT system was used to explore the characteristics of 1D galvoscanning, the effect of the mechanical inertia of the GS oscillatory element on imaging has also been taken into account (based on the work in [22]) in Doppler OCT to obtain optimized images of the retina with backstitched B-scans [25].

The remainder of this paper is structured as follows: Section 2 discusses the response of the device to two different input signals (triangular and sawtooth) for the 1D GSs. The effect of the different input parameters on the scanning functions is analyzed. Particularly, the relationship between the theoretical duty cycle and the effective duty cycle determined experimentally is defined and studied. In Section 3, a mathematical model is developed for both the scanning functions and the effective duty cycle. Section 4 presents the custom Gabor domain optical coherence microscopy (GD-OCM) system that was utilized for experimental validation, while Section 5 points out the effect of triangular and sawtooth functions on the images. The necessary overlap of the individual scans when imaging larger samples is evaluated and applied to both scanning regimes in order to obtain artifact-free images. The effective duty cycles obtained with galvoscanning in the context of OCT imaging are finally compared.

## 2. Effective Duty Cycle of a 1D GS

A schematic illustration of the operating principle of a GS is shown in Fig. 1. The device is electrically driven with a nonresonant oscillatory element (magnet or coil [21]). It usually operates in a closed loop with different control structures designed to minimize the response time of the device and maximize its precision [26]. Thus, error corrections, including in relation to temperature variations, are addressed through appropriate feedback loops.

Regardless of the complexity of the control module of the GS, the mechanical inertia does play a role in the functioning of the device. Thus, the galvomirror (together with the mobile element of the motor) cannot stop and turn instantaneously. As shown in [22], this is the reason why the scanning function of the GS (i.e., its output signal represented by the current angular position of the galvomirror) will actually differ from the input signal that usually has an ideal triangular or sawtooth profile [Fig. 2(a)]. This effect becomes significant when increasing the scan frequency (${f}_{s}$) above 100 Hz, especially at scan amplitudes (${\theta}_{m}$) above 1 V, yielding increased nonlinearity in the scanning function. Thus, as ${f}_{s}$ and ${\theta}_{m}$ are increased, the device will be less and less capable of following the input signal provided, as we further describe in this section for sawtooth input signals. A detailed study on the phenomena involved for different scanning regimes, as a consequence of the mechanical inertia, has already been reported in [22]. We then investigated three most common input signals of GSs (i.e., sawtooth, triangular, and sinusoidal) and demonstrated that sinusoidal scanning, although convenient from a mechanical point of view as it allows for a smoother functioning of the device, produces strong distortions in OCT images even at lower frequencies.

Triangular scanning is the best choice, while sawtooth scanning has the flyback portion that needs to be discarded when imaging larger samples [25]. However, as previously pointed out, even triangular and sawtooth scanning regimes are affected by an increasing nonlinearity of the output signal at high frequencies and amplitudes. This nonlinearity produces distortions at the margins of the scanned field in OCT imaging. Note that these distortions can be compensated in postprocessing, as usually done with sinusoidal scans, for which equally spaced pixels of the final OCT images are thus provided [23]. However, postprocessing of the images is a process one may want to avoid, especially when large amounts of data are involved and when real-time imaging is required.

Another parameter that has to be taken into account is the duty cycle, defined as the temporal scan efficiency, i.e., the ratio between the “active” time ${t}_{a}$ (the time interval used to scan the sample with constant velocity) and the time period $T$. For the scanning regimes, the theoretical or ideal duty cycle ${\eta}_{t}$ that is imposed and characterizes the input signals of the GS, is defined as [4]

The output signals of the GS (referred to as the scanning functions) will be characterized by an effective duty cycle (Fig. 3) whose value is lower than theoretically predicted because of nonlinearities. Indeed, the active time ${t}_{a}^{\prime}$ that characterizes the linear portion of the function is smaller than ${t}_{a}$ (Fig. 4), and the effective duty cycle $\eta $ is given as [21]

In this study, three theoretical duty cycles ${\eta}_{t}$ are considered for the sawtooth function: 50%, 75%, and 87.5%. The latter value is approximately the maximum duty cycle a signal generator may provide. The 50% value of ${\eta}_{t}$ corresponds to a triangular input signal in which only a unidirectional scan is used. The case of both forward and backward scanning within one cycle with a triangular input signal corresponds to a triangular scan with ${\eta}_{t}$ equals 100%. In this case, the effective duty cycle $\eta $ also varies from 100% to 70% with regard to the scan frequency and amplitude (Fig. 10 in [22]). In OCT, two B-scans per scan cycle may be utilized for image averaging or imaging at high speed.

For the systematic study of the scanning functions in Figs. 2 and 3, the 50% value is the necessary lower limit of ${\eta}_{t}$ for sawtooth input signals.

The scan frequency ${f}_{s}$ was varied (Figs. 2 and 3) from 10 Hz, which represents point-by-point scanning, up to 500 Hz, while intermediate values of interest for GSs and for OCT applications are considered. However, while most OCT systems operate at 2–200 Hz frame rate (assuming a number of A-scans in each frame of 500 and a typical A-scan rate of 1–100 kHz), scan frequencies of around 300 Hz have also been reported. The scan amplitude ${\theta}_{m}$ ranges from 0.2 to 3.2 V, which correspond to about 0.76–12.16 deg (mechanical angle of the oscillation of the galvomirror [5]) or 1.52–24.32 deg (optical angle of the reflected laser beam). This range of voltage is common for OCT investigations and is considered for the GSs, but the entire range was utilized only for lower scan frequencies as it can be remarked from Fig. 2(d), by example. This is because, as previously documented (Fig. 7 in [22]), there is a characteristic scan amplitude limit of a GS for each scan frequency and for each type of input signal; over this scan amplitude limit the GS system loses stability. Thus, different figures are required for the various GS voltages that are set by the signal generator of the GS.

The input signals are presented in Fig. 2(a), while output signals are shown in Figs. 2(b) and 2(c); the latter were obtained by directly reading the GS output voltage.

The output signals, which are proportional to the scanning angles, represent accurately the scanning locations on a sample surface when the equivalent output scan angles are multiplied by the focal length of the optics for a F-Theta scan lens used in its designed specification. If the lens is not an F-Theta scan lens or is not used according to its designed specification or suffers from optical distortion, the optical mapping must be established to relate the equivalent output scan angles to the scanning locations on the sample.

Several conclusions can be drawn from the multiparametric analysis performed in Figs. 2 and 3:

- (i) At low scan frequencies (roughly up to 50 Hz), the position of the galvomirror can still follow the input signal provided to the device, except at the highest ${\eta}_{t}$ where the nonlinearity begins to manifest after the flyback portion, as shown in Fig. 2(b3).
- (iii) From Fig. 2(c) a comparison can be made between the nonlinearity of the stop-and-turn portion of the maximum amplitude (after the active scan) and that of the minimum amplitude (after the flyback portion). These two portions are equal in time (i.e., the curves are symmetrical) for the triangular scan, whereas they are longer after the flyback for the sawtooth function. This observation is consistent with the fact that the flyback is performed at a higher speed, therefore the device needs more time to stop-and-turn. This phenomenon is even more significant for a higher ${\eta}_{t}$.
- (iv) As a consequence of the latter aspect, the effective active time ${t}_{a}^{\prime}$ and the corresponding effective duty cycle $\eta $ decrease accordingly. An evaluation of the effective duty cycle $\eta $ with regard to the scan frequency ${f}_{s}$ and scan amplitudes ${\theta}_{m}$ was performed for each value of ${\eta}_{t}$ and reported in Fig. 2(d). Results show that the effective duty cycle decreases with the increasing of frequency and amplitude.
- (v) A “migration” of the peaks of the output signals of the GS occurs for high values of ${f}_{s}$—Figs. 2(c2) and 2(c3). Thus, ${\eta}_{t}$ decreases even more because of this effect of switching from sawtooth to triangular functions at high ${f}_{s}$ and ${\theta}_{m}$, an effect that was already demonstrated in [22].
- (vi) While the $\eta ({f}_{s})$ graphs in Fig. 2(d) start at the same value of $\eta $ equal to ${\eta}_{t}$ (for small ${f}_{s}$ and ${\theta}_{m}$), a decrease in $\eta $ is produced, as pointed out previously, with the increase in ${f}_{s}$. This decrease in $\eta $ is more significant for larger samples to be imaged (i.e., for higher ${\theta}_{m}$). As expected, when ${\theta}_{m}$ gets too large, there is a limit value of ${f}_{s}$ that can still be employed. The systematic study of this aspect provided the limit possible scan amplitudes for each ${f}_{s}$ [22]; such limitations are illustrated in Figs. 3(e) and 3(f) by the lower reachable values of ${\theta}_{m}$ at the highest ${f}_{s}$ considered.
- (vii) The spread in the values of $\eta $ [corresponding graphs in Fig. 2(d)] increases with ${\eta}_{t}$. One may conclude that the GS is following the input signals with increasing difficulty for a higher ${\eta}_{t}$, even if the same ${f}_{s}$ and ${\theta}_{m}$ parameters are set.
- (viii) The previous remark has suggested a study of $\eta $ with regard to ${\eta}_{t}$, while maintaining ${f}_{s}$ and ${\theta}_{m}$ constant. This study, shown in Fig. 3, reveals a most peculiar aspect that should be taken into account by GSs users: while pushing the scanning device to the limit in terms of ${\eta}_{t}$, the effect may be opposite to the expectation when high ${f}_{s}$ and ${\theta}_{m}$ are employed. Thus, for higher ${f}_{s}$ (i.e., from around 200 Hz), a bending of the $\eta ({\eta}_{t})$ graphs is produced at the higher values of ${\theta}_{m}$. While only three measurements were acquired per graph, which is not sufficient to precisely locate the peak of each curve, Figs. 3(d)–3(f) show that there is a turning point to the effective duty cycle as a function of the theoretical duty cycle. Specifically, there will be a different position of the peak of the $\eta ({\eta}_{t})$ graphs with regard to the specific GS type, but there will always be a turning/bending point, for which a maximum in $\eta $ is reached, with the remark that this happens at sufficiently high values of the scan amplitude (i.e., towards the amplitude limit where, for a specific scan frequency, the GS oscillation loses stability).
This phenomenon is more significant at intermediate values of ${f}_{s}$ (i.e., 200 Hz). At even higher values (e.g., towards 500 Hz), all the values of $\eta $ drop and, although the phenomenon shown is still present, it becomes less significant. Thus, for ${f}_{s}>300\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Hz}$, one may say that, while increasing ${\eta}_{t}$, $\eta $ remains roughly constant; a saturation of the scanning regime is reached. However, the same practical conclusion remains valid for GS users: one must not push the device to the limit, as an increase in ${\eta}_{t}$ over 75% does not provide the results expected, while it only has a negative impact on the device from a mechanical point of view.

- (ix) As an overall tendency extracted from the study in Fig. 3, one may say that the $\eta ({\eta}_{t})$ function starts with an identity, regardless of ${\theta}_{m}$ for point-by-point scan [i.e., for low ${f}_{s}$, Fig. 3(a)]. The graphs of these functions spread progresivelly for higher ${f}_{s}$, with a maximum gradient of the spread around 200 Hz, then they get closer again at higher ${f}_{s}$, because of the drop of all the values of $\eta $, for the now smaller interval of possible scan amplitudes.

## 3. Mathematical Model of the Effective Duty Cycle

From the experimental study performed (Fig. 2), the output signals/scanning functions (i.e., the current positions of the galvomirror) for both the triangular and for the sawtooth regime have been derived. Figure 4 presents the mathematical model of the phenomenon described above, where output functions are shown in contrast with the input signals.

- (a) For the triangular input signals [Fig. 4(a)], the nonlinearity that occurs for the stop-and-turn portions produces a decrease in the scan amplitude from the theoretical value ${\theta}_{m}$ to the effective value ${\theta}_{a}$ that characterizes the linear portion of $\theta (t)$. The angular velocity increases from its theoretical value to the effective value where $2\tau $ is the time corresponding to the nonlinear portion of the scanning function.
From the five portions considered in Fig. 4(a), the scanning function for the triangular scanning regime is solved in Appendix A and described as follows:

$$\theta (t)=\{\begin{array}{ll}\omega t,& t\in [0,T/4-\tau )\\ -\frac{\omega}{2\tau}{t}^{2}+\frac{\omega T}{4\tau}t+{\theta}_{m}-\frac{\omega {T}^{2}}{32\tau},& t\in [\frac{T}{4}-\tau ,\frac{T}{4}+\tau )\\ \omega (-t+T/2),& t\in [T/4+\tau ,3T/4-\tau )\\ \frac{\omega}{2\tau}{t}^{2}-\frac{3\omega T}{4\tau}t-{\theta}_{m}-\frac{9\omega {T}^{2}}{32\tau},& t\in [\frac{3T}{4}-\tau ,\frac{3T}{4}+\tau )\\ \omega (t-T),& t\in [3T/4+\tau ,T)\end{array}$$From Fig. 4(a), the effective duty cycle of the triangular scanning function can be described as

This effective duty cycle would double if a bidirectional scan was considered for the GS. By considering Eqs. (4) and (A9) derived in Appendix A, Eq. (6) becomes

where we used a notation similar to the one introduced in [24] with - (b) For the sawtooth input signals [Figs. 4(b1) and 4(b2)], the nonlinear stop-and-turn portions are characterized by two different effective amplitudes ${\theta}_{a}$ and ${\theta}_{a}^{\prime}$ that characterize the portions of the scanning function after the active scan and after the flyback, respectively. An assymetry of the time intervals of the different parts of the nonlinear portions also appears in comparison with the triangular scan in Fig. 4(a).

Two cases can be distinguished for sawtooth scanning:

- (b1) For low ${f}_{s}$ and ${\theta}_{m}$, the peaks of the $\theta (t)$ output/scanning function have the same positions as those of the input function [Fig. 4(b1)] and the output function is just beginning to curve due to inertia as ${f}_{s}$ increases, as shown in Fig. 2, but also in the detailed study of the functions in [22]. For low ${\theta}_{m}$ (i.e., 200–400 mV), the shape of the function $\theta (t)$ is preserved even at high ${f}_{s}$ [e.g., at 500 Hz, Figs. 2(c2) and 2(c3)]; the consequence is only the slight decrease in the effective duty cycle $\eta $ as ${f}_{s}$ increases, as shown in Fig. 3.
- (b2) For higher ${\theta}_{m}$, two migrations $\mathrm{\Delta}{t}_{1}$ and $\mathrm{\Delta}{t}_{2}$ of the two peaks per period $T$ of $\theta (t)$ occur [Fig. 4(b2)], as well as an increasing nonlinearity of the respective portions. This shift of the output signal of the GS is shown in Figs. 2(c2) and 2(c3) and it has also been documented in the media files provided in [22]. At a certain scan frequency ${f}_{s}$, for each ${\eta}_{t}$ the previous phenomenon leads to the complete shift of the sawtooth profile to the triangular one. The final profile is triangular with different nonlinear portions [Fig. 4(b2)], as determined experimentally in Fig. 2(c2), for example for ${\theta}_{m}$ equals 0.8 V at ${f}_{s}$ equals 500 Hz.

The mathematical discussion and the resulting model will be therefore different for these two cases:

- (b1) For the graph in Fig. 4(b1), the theoretical angular velocity of the active scan of time interval ${t}_{a}={\eta}_{t}\xb7T$ is given as while the effective angular velocity, for the effective active scan time ${t}_{a}^{\prime}=\eta \xb7T$ is given as The theoretical angular velocity of the flyback is given as while the effective angular velocity of the flyback is given as
The scanning function for the sawtooth input signal is solved in Appendix B for the five portions considered in Fig. 4(b1) and is described as follows:

$$\theta (t)=\{\begin{array}{ll}\omega t,& t\in [0,{\eta}_{t}T/2-\tau )\\ \frac{-\omega}{2{\tau}_{1}}{(t-\frac{{\eta}_{t}T}{2})}^{2}+{\theta}_{m},& t\in [\frac{{\eta}_{t}T}{2}-{\tau}_{1},\frac{{\eta}_{t}T}{2})\\ \frac{{\omega}_{f}}{2{\tau}_{1}^{\prime}}{(t-\frac{{\eta}_{t}T}{2})}^{2}+{\theta}_{m},& t\in [\frac{{\eta}_{t}T}{2},\frac{{\eta}_{t}T}{2}+{\tau}_{1}^{\prime})\\ {\omega}_{f}(t-{\eta}_{t}T/2-{\tau}_{1}^{\prime})+{\theta}_{a},& t\in [{\eta}_{t}T/2+{\tau}_{1}^{\prime},(1-{\eta}_{t}/2)T-{\tau}_{2})\\ \frac{-{\omega}_{f}}{2{\tau}_{2}}{[t-(1-\frac{{\eta}_{t}}{2})T]}^{2}-{\theta}_{m},& t\in [(1-\frac{{\eta}_{t}}{2})T-{\tau}_{2},(1-\frac{{\eta}_{t}}{2})T)\\ \frac{\omega}{2{\tau}_{2}^{\prime}}{[t-(1-\frac{{\eta}_{t}}{2})T]}^{2}-{\theta}_{m},& t\in [(1-\frac{{\eta}_{t}}{2})T,(1-\frac{{\eta}_{t}}{2})T+{\tau}_{2}^{\prime})\\ \omega (t-T),& t\in [(1-{\eta}_{t}/2)T+{\tau}_{2}^{\prime})\end{array}\mathrm{.}$$From Eqs. (B4) and (B28), the relationship between the time intervals of the nonlinear portions is obtained as

$$\frac{{\tau}_{1}^{\prime}}{{\tau}_{1}}=\frac{{\tau}_{2}}{{\tau}_{2}^{\prime}}=\frac{1-{\eta}_{t}}{{\eta}_{t}}.$$The effective duty cycle of the scanning function in this case is

which, taking into account Eqs. (10), (B14), and (B28), becomes - (b2) For the graph in Fig. 4(b2), as the sawtooth switches to triangular, the effective angular velocity becomes$$\omega =\frac{2{\theta}_{a}}{\mathrm{\Delta}}=\frac{2{\theta}_{a}^{\prime}}{{\mathrm{\Delta}}^{\prime}}=\frac{2({\theta}_{a}+{\theta}_{a}^{\prime})}{\mathrm{\Delta}+{\mathrm{\Delta}}^{\prime}},$$therefore while from Fig. 4(b2),

The scanning function for the sawtooth input signal is solved in Appendix C for the five portions considered in Fig. 4(b2) and is described as

In conclusion to this analysis, by comparing the effective duty cycle $\eta $ for (a) triangular scan, Eqs. (6), (b1) sawtooth scan, Eq. (15), and for (b2) sawtooth turning to a pseudo-triangular scan, Eq. (22), one can see the confirmation of the study in Figs. 3(d)–3(f): with regard to ${\eta}_{t}$, $\eta $ starts from immediately below 50% (from a lower value for a higher scan frequency), it increases for the “purely sawtooth scan” (b1), and then it decreases again—to a value closer to the triangular scan—for the case (b2), where the switch from sawtooth to triangular is completed. The latter situation occurs for a sufficiently high scan amplitude to have the above “switch” completed [i.e., the highest one in Figs. 3(d)–3(f)].

## 4. OCT System and Scan Protocols

The layout of the OCT system that was employed in this part of the experimental study is shown in Fig. 5. This system is a custom-designed GD-OCM setup that provides a high volumetric resolution of 2 μm (in depth as well as lateral resolution) across a field of view of $2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}\times 2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$, and an imaging depth of 1.6 mm [27]. The source is a super luminescent diode (SLD) with the center wavelength of 840 nm and a full width at half-maximum of 100 nm (BroadLighter D-840-HP-I, Superlum, Ireland). A custom spectrometer with a high-speed CMOS line camera (spl4096-70 km, Basler Inc.) was implemented to acquire the spectral information [28]. The system allows for a real-time visualization of the sample after the acquisition, using a multiple graphic processing units (GPU) recently implemented in the system [29]. The scanning probe is equipped with a dual-axis XY galvanometer (6215HSM40B, Cambridge Technology).

To investigate the phenomena described in Section 2 on OCT imaging, GD-OCM images were acquired at several scan frequencies ${f}_{s}$, from 50 to 600 Hz, including at around 300 Hz, where the differences between the scanning regimes that employ different theoretical duty cycles were easier to see. For contrast, two scan amplitudes ${\theta}_{m}$ were employed, a small one (200 mV) and a large one (3.2 V), with three theoretical duty cycles: 50%, 75%, and 90%. Note that the 50% duty cycle corresponds to a triangular scan.

In all the experiments, the line period of the camera was kept constant with ${T}_{c}$ equal 15 μs. For each scan frequency, the number of samples (${N}_{s}$) for one period of the scanner is given as

The camera is configured to acquire data only during the rising time of the scanner or equivalently the increase in voltage. Thus, the number of A-scans to acquire is set by the product of ${N}_{s}$ and the theoretical duty cycle. The fast axis (X) is following the protocol described above, while the slow axis (Y) moves step-by-step from one frame to another. The camera and the scanner are synchronized through an external TTL trigger.

## 5. OCT Image Distortion Due to the Inertia of the GS

In Sections 2 and 3, investigations show that the GS scanner may not exactly follow the input signal due to inertia. To further illustrate the inertia effect in OCT images, a sample with a regular grid structure is imaged using different scanning schemes. During the imaging process, different parts of the sample are imaged and then stitched together in order to obtain the entire image of the sample. Stitching of subimages is commonly done in many domains, including holography [30,31].

To the best of our knowledge, this aspect has been applied in OCT for transversal imaging of the retina with Doppler OCT [25], taking into account the results in [22]. The issue of the GS inertia discussed in Sections 2 and 3 needs to be further addressed in terms of numerical assesment, as imaging large samples has a multitude of applications, both medical and nonmedical—the latter, for example, in materials studies [32] or for art restauration [33].

For high scan frequencies and amplitudes (and especially when sawtooth scanning is used), the nonlinear portions of the scanning function shown in Fig. 2(c) are going to produce artifacts in the image of a regular stucture image—lines and even gaps for higher theoretical duty cycles, as shown in Fig. 6, in which results are shown for both scanning regimes: triangular and sawtooth. Several scan frequencies ${f}_{s}$ have been considered in the study, from 50 to 600 Hz. In Fig. 6, the images obtained for three values of ${f}_{s}$ are shown: a low value (100 Hz) in column 1, an intermediate value (300 Hz) in column 2, and the highest value considered in the study in Fig. 2 (i.e., 500 Hz) in column 3. The numbers of A-scans considered for half of each image are given in Table 1, for each ${f}_{s}$ and ${\eta}_{t}$. A number of 200 B-scans, with a step size of 3.65 μm, were taken for each 3D image.

For low ${f}_{s}$, the images show no artifacts. For the two higher values of ${f}_{s}$, the nonlinear portions of the scan functions produced blurred and distorted parts of the images related to the loss in duty cycle from ${\eta}_{t}$ to $\eta $, as previously investigated. The number of distorted A-scans (Table 1) is proportional to the time intervals $\tau $ [Figs. 4(a) and 4(b2)] and ${\tau}_{1}$ [Fig. 4(b1)] for the respective cases. This is due to the fact that each A-scan is obtained in the OCT system in an identical time interval, set from the aquisition time of the camera, as pointed out in the previous section. In the same way, the total number of A-scans in an individual OCT image is proportional to half of the time period $T$ of the GS—as also set from the scanning algorithm of the system.

Taking into account the number of distorted A-scans, which is proportional to the nonlinear time portions of the time intervals in Fig. 4, the equations for the effective duty cycle $\eta $ deduced in Section 3 were used to obtain the effective duty cycles $\eta $ that result from the OCT measurements. These values are given as the ${\eta}_{\mathrm{OCT}}$ [%] row of Table 1, which are then compared with the effective duty cycles that have been obtained in Section 2 from direct voltage measurements of the position of the galvomirror for the same values of ${f}_{s}$, ${\theta}_{m}$, and ${\eta}_{t}$ as above. These latter values of ${\eta}_{\mathrm{GS}}$ were given in Figs. 3(c), 3(e), and 3(f).

The values of ${\eta}_{\mathrm{OCT}}$ are affected by measurement errors in $\tau $ and $T$ because both these time intervals are determined with a possible error equal at most to $Tc$, which is the line period of the camera. The equations for these errors, denoted as ${\epsilon}_{OCT}$ [%], were deduced for each of the equations of ${\eta}_{\mathrm{OCT}}$ corresponding to various scanning regimes in Appendix D, as the following: Eqs. (D5), (D7), (D9) provide ${\epsilon}_{\mathrm{OCT}}$ for Eqs. (6), (15), and (22), respectively. These ${\epsilon}_{OCT}$ errors are provided in Table 1. As it can be remarked, they are below 5% for each of the ${\eta}_{\mathrm{OCT}}$ values determined.

As the two sets of $\eta $ values (i.e., ${\eta}_{\mathrm{OCT}}$ and ${\eta}_{\mathrm{GS}}$) are compared in the last row of Table 1, the relative errors $\epsilon =({\eta}_{\mathrm{OCT}}-{\eta}_{\mathrm{GS}})/{\eta}_{\mathrm{GS}}$ between these values are calculated for each value of ${f}_{s}$ and ${\eta}_{t}$.

One may note from this comparison that the mathematical model obtained for the triangular scan is always valid and is the most precise. However, from the values in Table 1, the mathematical models developed for the sawtooth regimes are also applicable, as discussed in Section 3, for the case in Fig. 4(b1) for a low ${f}_{s}$ (i.e., 50 Hz) and for the case in Fig. 4(b2) for the highest ${f}_{s}$ considered (i.e., 500 Hz).

The limits of the mathematical modeling can be noticed from the intermediate case of ${f}_{s}$ equal 300 Hz and ${\eta}_{t}$ equal 75%: the two models developed cannot be applied, as the time intervals $\mathrm{\Delta}{t}_{1}$ and $\mathrm{\Delta}{t}_{2}$ pointed out in Fig. 4(b2) for the migrations of the peaks of the output function of the GS are not null (as for a low ${f}_{s}$) and they are not maximum (as for the highest ${f}_{s}$).

Another remark can be extracted from the experimental values of $\eta $ obtained with the OCT experiments: the saturation of the effective duty cycle can be clearly seen for the 300 and 500 Hz scan frequencies (as studied and demonstrated from direct measurement of the scanning function of the GS in Section 2), as seen from the saturated values of ${\eta}_{\mathrm{GS}}$ in Table 1.

## 6. Distortion Correction

From the results shown in Fig. 6, columns 1–3, the artifacts in the images were evaluated. The effective duty cycle ${\eta}_{\mathrm{OCT}}$ was then calculated by using the appropriate mathematical model from those developed in Section 3 for each scan parameter ${\eta}_{t}$, ${f}_{s}$ and a scan amplitude ${\theta}_{m}$ equal to 1.6 V. The equation used for each case is indicated in Table 1. A comparison is then made with the effective duty cycle $\eta $, also determined experimentally, but from the galvoscanning in Section 2. One may see that a good match is obtained for all the cases for which a model could be applied.

To correct the distortion in the images, the effective duty cycles need to be taken into account when designing the scanning protocol. In Fig. 7(a1), the nonlinear part of the GS scanning is given as

where $L$ is the scanning range determined by the scanner amplitude. The nonlinear scanning part is where distortion occurs. To correct for the distorted part in the imaging acquisition, the camera was configured to acquire data only in the linear scanning part of the scanner. The scanning voltage for the second subimage was then shifted by a value corresponding to $\mathrm{\Delta}l$ to compensate the field of view loss due to the removal of distortion, which is shown in Fig. 7(a2).OCT images were obtained to show the distortion correction. In Fig. 6, column 3, the frontal images of the structured sample are shown, for three scan frequencies: 100 Hz (c1), 300 Hz (c2), and 500 Hz (c3). Only the worst case scenario, with sawtooth scan and ${\eta}_{t}$ equals 90% is considered. The final images without the artifacts are shown, for each value of the ${f}_{s}$, in Fig. 6, column 4. The portions where the correction was applied are marked on each figure.

In Fig. 7(b), the 3D reconstructions of the same large sample are shown. A number of 200 B-scans, with a step size of 3.65 μm were taken for each 3D image. The image obtained in Fig. 7(b1) for ${f}_{s}$ equals 500 Hz was corrected to remove the artifacts. The final image without the artifacts is shown in Fig. 7(b2).

## 7. Conclusions

We have investigated the output signals/scanning functions of galvanometer-based scanners (GSs)—when triangular and sawtooth input signals are considered. The profiles of the scanning functions (i.e., the current angular position of the galvomirror) were studied with regard to the main functional parameters of the process: scan frequency (${f}_{s}$), scan amplitude (${\theta}_{m}$), and theoretical/ideal duty cycle (${\eta}_{t}$) of the input signal—the latter ranging from 50% (for triangular scanning) to a maximum value of 90% (for sawtooth scanning).

One of the main issues of the scanning process is precisely related to the effective duty cycle not equalling the theoretical duty cycle ${\eta}_{t}$, with an alteration increasing as ${f}_{s}$ is increased—especially at higher ${\theta}_{m}$. The increased nonlinearity of the stop-and-turn portions of the scanning function thus decreased the duty cycle to an effective value $\eta $, which we studied with regard to ${f}_{s}$ and ${\theta}_{m}$. Thus, for the fast GS of the imaging system the triangular scanning was demonstrated to be better than sawtooth scanning, as it produced images that were distortion-free up to higher values of ${f}_{s}$ and ${\theta}_{m}$. However, this is valid only with OCT systems that utilize GSs that are programmed for bidirectional scan, as with the FD-OCT system we have used in [22].

It is often advantageous to use a unidirectional sawtooth scan, as the lateral resolution can thus be increased—more A-scans can be placed in the active time ${t}_{a}^{\prime}$ of the fast-axis GS. A tendency that GS users may have in this respect is to set the device to the limit in terms of the theoretical duty cycle, to take advantage of this feature. In this respect, a main finding of this study was that for sawtooth input signals a saturation of the scanning regime was reached. Although ${\eta}_{t}$ is set towards its maximum, the effective duty cycle (i.e., the time efficiency of the scanning process) did not increase further; in fact, it was shown to even decrease in some cases (Fig. 3). This situation occurred when ${f}_{s}$ was increased, especially at higher ${\theta}_{m}$, which is desired in order to have a field of view (FOV) that is as large as possible. Approximate limits of the parameters for this general phenomenon were indicated for ${f}_{s}$ roughly higher than 200 Hz (with differences that may appear with regard to the specific GS considered) and a ${\theta}_{m}$ close to the limit scan amplitude specific to the respective value of ${f}_{s}$.

Mathematical models for both the output signal/scanning function and the effective duty cycle were deduced, based on the experimental findings on the galvoscanning. Both (a) the triangular and (b) the sawtooth scanning regimes were considered. For the latter type of scan, two subcases had to be considered: (b1) for low ${f}_{s}$ and ${\theta}_{m}$, the peaks of the scanning function had the same positions as those of the input signal and the duty cycle was altered slightly due to inertia as ${f}_{s}$ increased; (b2) for higher ${\theta}_{m}$, migrations of the two peaks per period $T$ of the signals occurred, and at a certain ${f}_{s}$ (for each ${\eta}_{t}$) the sawtooth profile shifted completely to a triangular one with different nonlinear portions. This shift already manifested for ${f}_{s}$ higher then 300 Hz and it was final for ${f}_{s}$ higher then 500 Hz—for each ${\theta}_{m}$ close to the stability limit of the GS.

An OCT experiment was set to apply these findings for the optimization of the galvoscanning algorithms for Gabor domain optical coherence microscopy (GD-OCM). An in-house GD-OCM setup was used to image a regular multilayered structure for a larger FOV than the one provided by a single 2D scan—and 3D reconstruction—of the sample. Two individual scans were thus considered for a sufficiently high value of ${\theta}_{m}$ (i.e., 1.6 V, which means about 12 deg optically) and for a row of ${f}_{s}$ (from 50 to 600 Hz) in order to study the artifacts produced by scanning because of the nonlinear portions of the scanning functions and of the shifts of the sawtooth ones. The equations for the effective duty cycle $\eta $ were deduced for each case and they were used to obtain $\eta $ from the OCT measurements. These values were then compared in Table 1 with those obtained directly from the galvoscanning discussed in Section 2. A good agreement was obtained between the two sets of $\eta $ values.

The artifacts produced in the OCT imaging (i.e., deformations of the regular structure imaged) were then used to perform the necessary stitching of the two individually scanned portions. The results was an artifact-free collated image, even at high scan frequencies ${f}_{s}$ (i.e., up to 600 Hz).

Future work is envisioned with the optimized scanning algorithms on other OCT and OCM applications, both biomedical and industrial [32], including with handheld scanning probes equipped with galvoscanners [17].

## Appendix A: Effective Duty Cycle of a Triangular Scanning Function [Fig. 4(a)]

A parabolic (second-order polynomial) equation is considered for the nonlinear portions of the triangular scanning function:

- (i) For $t\in [0,T/4-\tau )$, the equation of this portion of the scanning function is given as where $\omega $ represents the effective angular velocity of the galvomirror—Eq. (4).
- (ii) For $t\in [T/4-\tau ,T/4-\tau )$, the value of the scanning function can be evaluated at particular points to set the boundaries conditions as From Eqs. (A1), (A2), and (A4)–(A7), the coefficients are given as and a supplemental designing equation is also obtained as
- (iv) For $t\in [3T/4-\tau ,3T/4+\tau )$, the boundaries conditions of $\theta (t)$ are given as From Eqs. (A1), (A2), and (A11) to (A14), the coefficients are given as$$a=\omega /2\tau ,\phantom{\rule[-0.0ex]{2em}{0.0ex}}b=-3\omega T/4\tau ,\phantom{\rule[-0.0ex]{2em}{0.0ex}}c=-{\theta}_{m}-9\omega {T}^{2}/32\tau ,$$and Eq. (A9) is also obtained as a supplemental equation.

## Appendix B: Effective Duty Cycle of a Sawtooth Scanning Function at Low Values of the Scan Frequency [Fig. 4(b1)]

For the nonlinear portions of the scanning function, the same parabolic functions in Eq. (A1) will be considered.

- (ii) For $t\in [{\eta}_{t}T/2-{\tau}_{1},{\eta}_{t}T/2)$, the $a$, $b$, $c$ constants are obtained from the conditions given as [Fig. 4(b1)] which were written for the continuity and smoothness of the scanning function $\theta (t)$. From Eq. (A1), using Eqs. (B2)–(B5), the three coefficients are given as and a designing equation is also obtained as
- (iii) For $t\in [{\eta}_{t}T/2,{\eta}_{t}T/2+{\tau}_{1}^{\prime})$, the continuity and smoothness conditions are [Fig. 4(b1)] expressed as which were written for the continuity and smoothness of the scanning function $\theta (t)$. From Eq. (A1), using Eqs. (B8)–(B11), the three coefficients are then given as$$a=-{\omega}_{f}/2{\tau}_{1}^{\prime},\phantom{\rule[-0.0ex]{2em}{0.0ex}}b={\omega}_{f}{\eta}_{t}T/2{\tau}_{1}^{\prime},\phantom{\rule[-0.0ex]{2em}{0.0ex}}c={\theta}_{m}+{\omega}_{f}{\eta}_{t}^{2}{T}^{2}/8{\tau}_{1}^{\prime},$$and a designing equation is also obtained as From Eqs. (B7) and (B13), taking into account Eqs. (10) and (12), a relationship between the time intervals of the previous portion was obtained as
- (v) For $t\in [(1-{\eta}_{t}/2)T-{\tau}_{2}),(1-{\eta}_{t}/2)T)$ the continuity and smoothness conditions are [Fig. 4(b1)] given as and the coefficients of the functions are expressed as where a designing equation is also obtained as
- (vi) For $t\in [(1-{\eta}_{t}/2)T,(1-{\eta}_{t}/2)T+{\tau}_{2}^{\prime})$ the continuity and smoothness conditions are [Fig. 4(b1)] given as and the coefficients of the functions are expressed as where a designing equation is also obtained as
From Eqs. (B21) and (B27), taking into account Eqs. (10) and (12), a relationship between the time intervals of the previous portions is obtained as

- (vii) For $t\in [1-{\eta}_{t}/2)T+{\tau}_{2}^{\prime},T)$ Eq. (A16) is valid in this case as well.

## Appendix C: Effective Duty Cycle of a Sawtooth Scanning Function at High Values of the Scan Frequency [Fig. 4(b2)]

For the nonlinear portions of the scanning function, the same parabolic functions in Eq. (A1) will be considered.

- (i) For $t\in [0,\mathrm{\Delta})$, Eq. (B1) is valid in this case as well.
- (ii) For $t\in [\mathrm{\Delta},\mathrm{\Delta}+2\tau )$, the continuity and smoothness conditions of $\theta (t)$ are [Fig. 4(b2)] given as From Eqs. (A1), (A2), and (C1) to (C4), the coefficients are expressed as$$a=-\omega /2\tau ,\phantom{\rule[-0.0ex]{2em}{0.0ex}}b=\omega (\mathrm{\Delta}+\tau )/\tau ,\phantom{\rule[-0.0ex]{2em}{0.0ex}}c={\theta}_{m}-\omega {(\mathrm{\Delta}+\tau )}^{2}/2\tau ,$$and Eq. (A9) is also obtained as a supplemental equation.
- (iv) For $t\in [\mathrm{\Delta}+T/2-\tau ,T/2+\mathrm{\Delta}+3\tau )$, the continuity and smoothness conditions of $\theta (t)$ are [Fig. 4(b2)] given as From Eqs. (A1), (A2), and (C7)–(C10), the coefficients are expressed as and a designing equation is also obtained as
- (v) For $t\in [T/2+\mathrm{\Delta}+3\tau ,T)$ Eq. (A16) is valid in this case as well.

## Appendix D: Measuring Errors of the Effective Duty Cycle of Triangular and Sawtooth Scanning Functions

The effective duty cycle $\eta $ for triangular scanning functions is provided by Eq. (6), while for sawtooth scanning functions by Eqs. (15) or (22), as discussed in Section 3. The maximum measuring error of each of the $\tau $ and $T$ time intervals equals the line period of the camera ${T}_{c}$ (equals 15 μs, see Section 4). The absolute measuring error of $\eta $ is

From Eqs. (D2) and (D3), taking into account that $1/T={f}_{s}$, the absolute and relative errors are, respectively,

## FUNDING INFORMATION

II-VI Foundation; National Science Foundation (NSF) (ENG 1346453); Romanian National Authority for Scientific Research (PN-II-PT-PCCA-2011-3.2-1682).

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