1. Introduction

Over the last 30 years, Second Harmonic Generation (SHG) microscopy has been gradually confirmed as a valuable method to characterize non-centrosymmetric structures, in particular certain biological tissues [1]. This type of scanning microscopy uses a near-infrared laser for excitation that limits scattering inside tissues, thus allowing a high penetration depth [2]. The 3D confinement of the focal volume of excitation also enables a sub-micron spatial resolution [3]. SHG is endogenous, with no need of marker or staining which makes it minimally invasive. Being a parametric process that converts two exciting photons into a single photon at twice the frequency, it also does not involve any electronic excitation. It is therefore free from photobleaching, and greatly reduces phototoxicity compared to other two-photon processes, such as two-photon excited fluorescence (2PEF) [4]. Secondly, it is a coherent process which involves a relationship between the excitation and the converted SHG in terms of phase: this restricts the generation to non-centrosymmetric media and makes it intrinsically background-free. Thus, SHG microscopy has been shown to be very sensitive and highly specific to a wide variety of structural proteins [1,5] or carbohydrates [6]. It was also used to characterize ferroelectrics [7,8], especially to reveal phase transitions [9]. Magnetic structure of materials that can be centrosymmetric in their crystallographic structure but exhibiting an electrical quadrupole or a magnetic order (thus breaking the time-inversion symmetry) [10,11], as well as strongly correlated materials [12] were moreover studied by SHG.

The coherence of the SHG process also conserves the phase of the excitation field. This means that not only the intensity of the signal carries information, but also its phase, which adds an additional mode of contrast. Unfortunately, this phase information cannot be extracted with a standard SHG microscope, since it would require a detector with a bandwidth higher than the optical frequency (∼300THz), which is several orders of magnitude over the limit of usual detectors (few GHz). Interferometry is the usual indirect way of retrieving the phase information, and the so-called Interferometric-SHG (I-SHG) was first applied to characterize ferroelectrics [8]. It was later transferred to the imaging of biological tissues like muscle [13,14], tendon [15] and cartilage [16]. Since the acquisition speed was quite low (∼1-2h to acquire a standard I-SHG map), the technique was further improved, and confirmed its compatibility with standard scanning microscopy set-ups, featuring: (1) a laser-scanning system to enable frame acquisition within seconds [17], and (2) a femtosecond excitation that provides a high signal-to-noise ratio (SNR) and a reduced exposition to laser light [18]. These improvements recently allowed us to probe dynamical processes like the evolution of the polarity of microtubules during mitosis [19]. However, this frame phase-shifting I-SHG measurement implied the acquisition of 9 interferograms, with a dead-time of ∼1sec between them to shift the relative phase. In this case, it was additionally necessary to manually adjust the polarization and the depth of focus to follow the 3D movement of the microtubules. Altogether, the temporal resolution was limited to 45sec, even if each single interferogram was recorded in 2sec [14]. This is more than twice the theoretical time of 9×2 = 18sec [14], and very close to the acceptable limit of 1min – the duration of some phases of the mitosis – after which the microtubular spindles change their arrangement and loose their centrosymmetry. Moreover, the fact that subsequent interferograms are separated by a few seconds can lower the precision of the phase measurement, especially in moving samples. In addition, laser-scanning I-SHG comes with trade-offs compared to sample-scanning, which e.g. enables scans of large areas without any mosaic reconstruction [13,20].

In this work, we address and optimize these time - and imaging quality - affecting issues by implementing single-scan I-SHG (1S-ISHG) using a kHz electro-optic phase-scanner. We show that its fastest mode considerably increases the speed of I-SHG imaging by more than one order of magnitude with only a slight loss in precision. If required, a precision equivalent to standard I-SHG can be achieved with a slow mode of 1S-ISHG that is still twice as fast, and the different pixels of the interferograms are separated in time by only 200µs maximum, instead of a few seconds. Finally, we apply this technique to image samples exhibiting a relatively low signal-to-noise ratio (SNR): mice-tail tendon and horse’s meniscus from knee joint. The fastest mode of the phase-scanner leads to a noisier and less accurate I-SHG map in these bio-samples, but still enables full phase retrieval. Altogether, the increased speed is promising for future probing of relative polarity in dynamical processes and large-scale studies of various tissues or crystals, notably the aforementioned study of phase transition in condensed matter [21]. The improved accuracy of the slow mode also opens up the possibility of dealing with samples that exhibit a low SHG signal.

2. Material and methods

2.1 General principle

Single-scan I-SHG capitalizes on the MHz speed of electronic acquisition cards to split each pixel of a standard image in many subpixels, thus acquiring all interferograms almost simultaneously (by contrast to successive acquisitions of many images) and providing an increase of speed more than one order of magnitude compared to previously reported I-SHG measurements.

2.2 Experimental set-up

320fs pulses at 810nm and 80MHz repetition rate were used for SHG, altogether with galvanometric (galvos) mirrors from Cambridge Technology (model 6220H) coupled with achromatic doublets as scan lens and tube lens. This maximizes the throughput at 405nm and the scanned field-of-view, while minimizing the amount of dispersive material in the excitation and reference beams such that only a small amount of calcite is needed to compensate for the group delay (see [2] for explanations). Figure 1(a) shows the full set-up used for the different modes of I-SHG: an air immersion objective (UplanSApo 20X, NA 0.75, Olympus, Japan) was used for illumination and the SHG emission was then collected with a condenser (NA = 0.55). The (measured) focal volume of excitation is 1×1×4µm³: the deviation from the theoretical values (0.4×0.4×1.9µm³, see [22]) comes from an imperfect collimation and underfilling of the back pupil of the objective, as well as from a reduced performance of the objective lens. Scanning and signal acquisition were synchronized using a custom-written Python program for stability and control. The average power on the sample was adjusted to 25mW, corresponding to 0.31 nJ/pulse. Raw data visualization was performed with FIJI-ImageJ (NIH, [23]), and image processing with a custom-made GUI in MATLAB that allows batch processing of the data and rendering.

 

Fig. 1. (a) Schematic view of the set-up used for I-SHG. The standard phase-shifter (a rotatable glass plate) can be by-passed to have the common-path interferometer pass in the phase-scanner (EOM). The beams are sent to galvos for standard laser-scanning (P2). They can also be by-passed using plane mirrors to directly send the beams into the objective (P1). The scan is then performed by a motorized stage (stage-scanning). The flat mirrors are represented by grey rectangles, with dotted outline if put on a flip mount. HWP1&2: half-wave plate at 810nm, full wave-plate at 405nm. HWP3: dual half-wave plate at 810nm and 405nm, QWP: quarter-wave plate at same wavelengths. (b) Simplified version of the ramp waveform used for the phase scanner, and its synchronization with the X and Y scanning of the scanning motor. (c) Zoom of (b) on two voltage ramps, showing how standard pixels are divided into N subpixels that correspond to N interferograms. (d) Description of the hierarchy between the motor (master) which triggers the timing of the EOM and the acquisition card (slaves).

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The setup is designed in a modular way to choose between different pathways for image acquisition: P1 for sample-scanning with a motorized stage and P2 for laser-scanning using galvos (Fig. 1(a)). In this article, we will now refer to “stage-scanning” for the sample-scanning, and to “galvos-scanning” for the laser-scanning method. For a complete description of the I-SHG method and set-up we refer to [24] for galvos-scanning and to [13] for stage-scanning. Both galvos and stage use unidirectional raster scanning, which scans the X direction at every step of the Y direction (see Fig. 1(b)).

In our previous study [2], we reported a 40 times speed improvement for laser-scanning, compared to stage-scanning I-SHG. However, the stage-scanning was based on a non-optimized LabView code, which we have later improved by controlling the scan with a Python software. This enabled a flexible choice of every scan parameter, to further optimize the scan duration. E.g. using a smooth scanning profile, we could reduce the imaging time by a factor of 1.4. If the resolution of the scan is set above 0.5µm, a faster velocity profile [25] can be employed which reduces the duration by another factor of 5. Since the acceleration of the stage is limited to 2000mm/s² [25], each scan size has its optimal pixel exposure time, related to the optimal speed calculated with the accelerations and decelerations at each line of the scan. The pixel exposure time can still be set to a different value, at the cost of a slight increase of acquisition time. Overall, taking into account the moving time of the static phase-shifter, we calculated that the scan time in I-SHG can be reduced by a factor of 8 to 35 using galvos-scanning compared to an optimized stage-scanning.

We also introduce here different modes for I-SHG: single-scan I-SHG (1S-ISHG) utilizing a phase-scanner, unlike conventional I-SHG that is based on a rotatable glass plate (see [2]). For general I-SHG phase extraction and data treatment, we refer to [14]. The special case of contrast enhancement achieved with 30 phase-shifts is detailed in the Appendix A.1.

2.3 Phase scanner

The phase scanner (developed in collaboration with Axis Photonique Inc, Canada) is a system that includes a transverse electro-optic phase modulator (EOM), a high-voltage kHz driver, its control electronics and software interface. The EOM uses a 3×3×35 mm³ RTP crystal (Y-cut, custom made by Raicol Crystals Ltd, see [26] for details). Such crystals are known for their stability and have already been utilized for high precision interferometers [27]. The required voltage to apply a π phase-shift between a pulse at 810nm (the wavelength of excitation) and its SHG at 405nm is ∼250V (measured). The high-voltage driver is set to work from 0 to 1300V, providing up to 5π phase-shift (see Appendix A.1 for the need of 5π). Linear voltage ramps (i.e. a sawtooth waveform) are generated to continuously change the phase-shift inside every pixel of the image (performing an intra-pixels phase scan), during the scans in X direction (see Fig. 1(b)): pixel time (i.e. ramp, see Fig. 1(c)) can be set to 20, 200 or 2000µs. These pixels form the standard microscopy image and are noted “SHG pixel”, while the subpixels of the interferograms are noted “I-SHG pixel” (Figs. 1(b) and 1(c), see also section 2.3). Because the scan is unidirectional and the acceleration of the moving part is limited to a finite value, there is a flyback time between the X scans [28] and some moving time for the Y direction in parallel. This leads to an overall latency time at the beginning of each image line (see Fig. 1(b)). The voltage ramps (phase scans) are thus re-synchronized at each line of the image to account for this latency, using a trigger between the EOM and both the acquisition card and the scanning motor (master/slave configuration, see Fig. 1(d)).

The polarization of the fundamental beam is vertical and set parallel to the γ-axis (i.e. the electrode axis) of the RTP crystal, while the polarization of the reference SHG beam is set horizontal. This configuration leads to a different electro-optic modulation for the two beams, and thus to a relative phase-shift. Instead of intra-pixels, the phase-shift can also be changed frame-by-frame using a rotating glass plate (see Fig. 1(a)), according to the conventional I-SHG reported previously [13,24,29].

2.4 Buffer synchronization and filling of the interferograms

Figure 1(c) zooms on two ramps of Fig. 1(b): the “SHG” pixels are divided into N subpixels, which will serve to construct the N interferograms. There is also a dead-time (grey) between ramps due to the electronics, which is discarded (exaggeratedly illustrated for clarity): the complete process of subpixels extraction is described in A.2 in the Appendix, with Table 2 in particular showing the different exposure times of the interferograms, for the different modes.

While the exposure time in multiphoton microscopy is usually around 5-20µs, the 20µs 1S-ISHG phase-scanning needs to cope with less than 1µs per subpixel (Table 2). This is compensated by the fact that in I-SHG, the phase contrast is computed by taking successive interferogram frames, effectively leading to a much higher number of detected photons compared to those collected during a subpixel time (down to 0.6µs, see Table 2). It is also worth noting that a true pixel integrator synchronized with the laser pulse rate could be used to improve the detection sensitivity [17] without increasing the exposure time, since it avoids correlation between adjacent pixels and lowers the image noise. To further improve this aspect, a digital photon counter could also be implemented instead of the analog PMT, but this would require advanced synchronization, with an external clock at the subpixels frequency to extract the different interferograms (not implemented here).

2.5 Correction of the interpulse delay

The reference and excitation beams are collinear (common-path interferometer), and thus travel through the same amount of optical medium. The 1^st order normal dispersion of the different optics (microscope objective, lenses, RTP crystal) leads to group velocity mismatch (GVM, or temporal walk-off) between the pulses of these two beams. This resulting group delay is here compensated using a calcite prism pair (“delay compensator”, see [16]). Each optical configuration of the interferometer leads to a different group delay: they are all reported in Table 1.

Table 1. Group delays for different configurations in I-SHG, at 810nm fundamental wavelength.

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The last column of Table 1 indicates the difference of group delay dispersion (GDD) between the two pulses of the interferometer, which plays a significant role on the observed image contrast. A different GDD will imply a different frequency chirp on the two pulses. Because their central frequency is also different, one pulse (the reference SHG) will inevitably be wider in time than the other: the non-overlapping part of the reference pulse will thus not interfere, leading to a decrease of the interference contrast C (see Appendix A.3 for the equation). With the EOM’s crystal, the chirp difference is increased even more by the additional calcite amount needed for the GVM correction. For pulse durations of 120fs (10nm bandwidth), GDD values above 20000fs² would lead to a too strong temporal broadening. We measured a reduction in interference contrast C to 1.5:1 with 120fs pulses (not shown here), compared to a contrast C of 2.8:1 for 300fs pulses. For this reason, pulses at 320fs were preferred, which limits the difference in duration between the fundamental and the reference SHG beam to around ∼100fs, for both the stage and galvos configurations. The high GDD difference between the 405 and 810nm beams (∼14000fs²) is mainly due to the highly dispersive RTP crystal: this problem could be reduced by using a KD*P crystal instead. We have calculated that a two times longer KD*P crystal would have the same half-wave voltage of 260V, with a GDD difference of only ∼4000fs². This could theoretically enable the use of shorter pulses, close to 120fs.

In our configuration, to efficiently correct the chirp difference between both beams, a pulse compressor providing a certain amount of negative GDD at 810nm and twice this amount at 405nm is required. As conventional compression systems (prism or grating pair, chirped mirrors…) do not provide such feature, two different compressors for each wavelength 405 and 810nm would be required. This would however change the common-path geometry of the interferometer which ensures a good stability and ease of alignment. To nevertheless keep an optimal level of signal at the focus of the microscope, the GDD of the fundamental beam has been pre-compensated before the interferometer using a chirped mirrors pair (used with 32 bounces, −500fs² per bounce, 0°, Ultrafast Innovations GmbH).

2.6 Calibration of the polarization

The polarizations of the fundamental and reference SHG beam are controlled by a HWP and a QWP placed before the microscope, both designed to work at 405 and 810nm (see Fig. 1(a)). They are carefully calibrated with a modified version [30] of the routine developed in [31]. This enables to control the orientation of both input polarizations, to ensure that they are maximally linear and parallel to each other, even if the waveplates are placed before the input of the commercial microscope.

2.7 Sample preparation

Tendon samples were obtained from an 8 weeks old male C57/Bl6 mice, whose tail was harvested and fixed, and later embedded in OCT compound to be cut using a cryostat (Leica). The meniscus was banked from a previous study: it consists of a medial meniscus from the knee joint of an adult horse, where a slice orthogonal to the circumferential direction [32] was cut in the body part (middle one). It was then placed in 10% formaldehyde for 2h and later transferred to Ethylenediaminetetraacetic acid (EDTA) 20% for 2 weeks prior to paraffin embedding and subsequent sectioning (see [33] for details). For both samples, a five-µm thick section was cut, and then placed on a microscope slide (1mm thick) covered by a coverslip (#1.5H, Thorlabs).

2.7 Phase maps and polar histogram for I-SHG

The values of orientation φ obtained in I-SHG are circular, so their histograms are better represented with polar plots. To clearly see the differences between images and the distribution of φ, its values are not fitted to separate between positive and negative polarities as usual, but rather represented using a circular HSV (Hue, Saturation, Value) colormap.

3. Results

3.1 1S-ISHG in Periodically Poled Lithium Niobate (PPLN)

We first validated our method in a model sample, Periodically Poled Lithium Niobate (PPLN, see [14] for details about the structure). Figure 2 shows the phase maps obtained with the different I-SHG methods, the first line showing the acquisitions with stage-scanning, and the second line the acquisitions with galvos-scanning. It is worth noting that different zones of PPLN with different domain size are imaged in each case, which explains the difference in size of observable patterns.

 

Fig. 2. Map of the relative phase of PPLN obtained by I-SHG, with the different methods. First line: stage-scanning (50×40µm², 0.2µm step), second line: galvos-scanning (50×50µm², 0.125µm step). (a)&(e) 1S-ISHG with the 20µs phase scan, (b)&(f) with the 200µs phase scan, (c)&(g) with the 2000µs phase scan. (d)&(h) Standard I-SHG with the standard phase-shifter (a rotatable glass plate). Polar histograms of a croped part of the phase images (to have similar peaks height, for clarity) are inserted below each map (see section 2.7), and the upper part is in white for visibility. The precision on Δφ are ≈ ±0.003π, and ± 0.005π for the peak widths δ. Acquisition times are: 70sec (a) and 5sec (e) ; 70sec (b) and 65sec (f) ; 156sec = 2.6min (c) and 450sec = 7.5min (g) ; 18×(68 + 1) = 1242sec = 21min (d) and 18×(5 + 1) = 110sec∼2min (h). Scale-bars: 10µm.

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Despite having different number of interferograms and exposure times, the phase distributions (indicated as a histogram below each sub-plot) of the 200µs (Figs. 2(b) and 2(f)) and 2000µs phase scan modes (Figs. 2(c) and (g)) are similar to the standard I-SHG (Figs. 2(d) and 2(h)): two peaks at –π/2 and π/2, with a distribution width of ≈ 0.060 ± 0.004π and 0.100π±0.004π. The phase distribution is much wider with the 20µs phase scan (peaks are spaced by 0.92π with a width of 0.118 ± 0.005π and 0.165 ± 0.05π), because the number of photons detected for each interferogram is lower (see Eq. (a5) in the Appendix). Also, with 20µs phase scan, the temporal fluctuations of the laser source are not averaged out because the pixel time of the interferograms corresponds to a rate over 500 kHz (see Appendix A.2), resulting in a signal variation from line to line (Figs. 2(a) and 2(e)). This might also broaden the phase histogram.

3.2 Phase correction

Because the ramp time calculated by the acquisition process is precise only at ${{\tau}_\textrm{R}}$=0.2µs (see A.2 in the Appendix), there might be a small residual difference compared to the real duration of the phase scan fixed by the phase-scanner. To correct this effect, a calibration of the phase-shift can be performed, even if the stage-scanning mode is normally calibration-free with standard I-SHG [13]. Galvos I-SHG requires in any case a calibration to correct the aberration effect of the objective [24]: this is performed via I-SHG on a homogeneous sample like a quartz plate [16], which corrects for both effects. Since it is alignment-dependent, this calibration must be repeated for every change in scan size and every set of experiment to be accurate. The need for a phase correction due to the phase-scanner however only depends on the timing parameters (duration of the voltage ramps, hardware-coded timings), which are supposed to be constant. Thus, these calibrations for stage-scanning need to be performed only once for each setting.

Figure 3(a) shows one example of such a calibration: since the voltage is scanned along the X direction, the phase correction remains constant along Y and consists in a linear variation in X only. In comparison, a galvo scanning uses a 2D parabola fit (Fig. 3(b)). The phase could also be calibrated by the measured quartz phase itself as in our previous work [2], but we found that correcting by a fit instead avoids the influence of imperfections or noise that might be present in the calibration phase map of quartz.

 

Fig. 3. (a) Example of a calibration in quartz for a stage-scanning of 40×10µm², step 0.2µm, 20µs phase scan (3D dot plot), along with a linear surface fit of the calibration, color-coded in phase. (b) Same with a galvos-scanning 50×50µm², step 0.125µm and 200µs phase scan: the fit is a 2D parabola.

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For more details and an example about the correction of the aberrations induced by the galvos-scanning across the imaging objective, we refer to [2].

3.3 1S-ISHG in biological tissues

Figure 4 refers to the I-SHG maps in tendon using galvos-scanning, after correction by the calibration map acquired in a quartz plate. As previously reported [15,24], there are two peaks in the phase distribution spaced by π. The bi-Gaussian fitting of the phase histograms for standard I-SHG scan leads to FWHM distribution’s around 0.34 ± 0.01π, slightly better than the 0.37 ± 0.01π measured for 1S-ISHG using a 2000µs or 200µs phase-scan, and in good agreement with our value previously reported [15]. This increases to 0.42 ± 0.015π for 20µs phase scans: it shows well that a diminution of exposure time (and thus of the number of photons) used to reconstruct the phase map is translated into a higher uncertainty on the phase determination, as the imaged area remains rigorously constant for each case.

 

Fig. 4. I-SHG phase-map of mice-tail tendon (50×50µm², 0.125µm step). 1S-ISHG with: (a) 20µs phase scan acquired in 5sec, (b) 200µs phase scan acquired in 65sec, (c) 2000µs phase scan acquired in 450sec = 7.5min. (d) Standard I-SHG acquired in 18×(5 + 1) = 110sec∼2min. The precision on Δφ are ± 0.01π, and 0.015π for the peak widths. Scale-bars: 10µm.

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To show the full benefits of the stage-scanning mode, i.e. the ability to scan a large ROI at once without any need for mosaic reconstruction, we performed I-SHG measurements in a 0.5×0.1mm² ROI in meniscus. This scan size is 5 times higher (in X) than the maximum acceptable scan that can be performed with galvos-scanning, while ensuring a sufficient I-SHG contrast C over the whole ROI in this biological sample with low SNR. As can be seen in Fig. 5, the relative polarity of this tissue is again characterized by two π phase-shifted peaks. The 20µs mode (Fig. 5(a)) is acquired in 25sec only, but needs a smoothing average to display a correct phase map. The 2000µs mode (Fig. 5(c)) shows very similar patterns and phase distribution compared to the reference (Fig. 5(d)), with the benefit of an acquisition time divided by 32/8 = 4. Finally the 200µs mode (Fig. 5(b)) shows a result closer to reference than the 20µs one, but still exhibit some slight phase errors. This confirms that large exposure times are required to retrieve with full accuracy the phase from such a low SNR sample.

 

Fig. 5. I-SHG phase-map of the central part of an adult horse meniscus (500×100µm², 0.5µm step) by stage-scanning. 1S-ISHG with : (a) 20µs phase scan acquired in 25sec, (b) 200µs phase scan acquired in 126sec, (c) 2000µs phase scan acquired in 495sec = 8.3min. (d) Standard I-SHG with the standard phase-shifter acquired in 18×(106 + 1) = 1926sec∼32min. The precision on Δφ are ± 0.01π, and 0.01π for the peak widths. Scale-bars: 50µm.

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The interferometric contrast (or “amplitude”) that provides an enhanced intensity map of the sample can also be extracted from the acquired interferograms (see [14]), in parallel of the phase (not shown here).

4. Discussion

4.1 Subpixels shift and synchronization

Because the phase-shift is changed while the motor is scanning, there might be a spatial shift between the interferogram pixels, especially between the first and the last ones. However, if the Nyquist criterion is respected - and the step size of the image is set below the lateral resolution of the microscope (here ∼1µm) - this effect becomes negligible. If the parameters do not match this criterion, the measurement will be distorted by a so-called motion artifact [34]. This artifact however would show up identically in any sample (such as the calibration one) and can thus be compensated by the correction described in section 3.2. It is worth noting that a motion artifact can also exist in conventional I-SHG as the laser beam or the sample can move during different phase-shifts, as described earlier. A single-scan approach reduces this artifact. Notably, a scan free of any motion artifact would require a piezo whose driving voltage is increased at every image pixels (no acceleration or deceleration, only steps), but the response time would need to follow the ∼ 50kHz pixel rate while piezos are usually limited to under kHz.

The imaging precision could be improved by the use of a more sensitive detector, as discussed in section 2.3. Alternatively, one could utilize beam-splitters to obtain 2 interferograms with a phase separation of 180° at half the incident amplitude, that can be measured on 2 different detectors: some setups extend this scheme to the simultaneous recording of 4 interferograms on 4 detectors [35]. Such approaches could reduce the number of interferograms extracted from the phase-scan by a factor of 2 to 4, and thus avoid potential motion artifacts. However, the 2 to 4 gain in exposure time does not relate to a higher number of photons per interferogram, as the beam splitting process distributes the photons on the different detectors, and is subjected to potential losses. Also, such a set-up is very sensitive to the output polarizations of the two beams of the interferometer, as a slight tilt in polarization can unbalance the different paths of detection.

Importantly, some setups of polarization-resolved SHG (p-SHG) using an EOM have been reported, with the difference that they modulate the polarization rather than the phase. Stoller et al. [36] used an EOM which was - similarly to ours - driven by a kHz sawtooth waveform, but however required a lock-in detection. In our setup, the EOM acts as the master clock and trigger, which removes the need of lock-in detection. More recently, Tanaka et al. [34] and DeWalt et al. [37] showed another way to diminish motion artifacts by the use of an EOM to switch between two polarization states at each pixel, and a single-scan p-SHG by recording 3 polarizations on 3 different detectors using analyzers (no modulation of the polarization) has also been reported [38]. Yet, they are not applicable for phase-shifting.

4.2 Comparison with static phase-shifter I-SHG

In standard I-SHG with a static phase-shifter, the different interferograms used for phase reconstruction are separated in time by a few seconds, which limits the precision of the measurement for various reasons: (1) 3D position: the position of the sample in the focal volume may slightly vary because of external vibrations. For living samples like mitotic spindle exhibiting natural movements, this position is expected to vary even more (outside the focal volume) within a time-scale of a few seconds or below. Such change in depth position will directly affect the measured phase due to the variation of the Gouy phase-shift in the focal volume, as explained in our previous work [39]. Any drift in the two other dimensions (lateral plane) has to be corrected in the image treatment (automatically if the SNR is sufficient, otherwise manually) to avoid substantial errors on the final result. These problems do not apply to 1S-ISHG, because the interferogram pixels are temporally separated by at most the phase-scan time.

Additionally, phase-scans of 200 and 2000µs lead to a longer exposure time, and thus a higher risk to cause phototoxicity due to laser damage on the sample. To attenuate this problem, a kHz shutter could be used to block the irradiance during the flyback of the motor [28], which accounts for ∼20% of the galvos-scanning time.

Switching from stage to galvos-scanning allowed us in [24] to increase the I-SHG acquisition speed by 1 to 1.5 orders of magnitude (from the hour scale to the minute scale), see abscissa of Fig. 6. Here, going from standard phase-shifting to the fastest single-scan mode (20µs) further increases this speed by another order of magnitude (from the minute scale to the second scale, here a 20× factor). This comes with a trade-off in precision of the phase retrieval – measured with the average width of the phase distribution in PPLN of Fig. 2 – of ∼ ×2 as shown in the ordinates of Fig. 6: the phase peaks are approximately two times wider. However, 1S-ISHG is shown to be similar in precision to standard I-SHG for the 200 and 2000µs phase-scans, with an acquisition time more than 10 times shorter for stage-scanning, and still 2 times shorter for galvos-scanning. A phase-scan time between 20 and 200µs could also be implemented to ensure a precision close to the 200µs mode with a scan speed slightly lower than the 20µs one. Noteworthily, the 20µs 1S-ISHG mode still performs well at retrieving the relative polarity inside any of the presented samples, i.e. discriminating π phase-shifted zones within them, as was presented in our other works [13,15,16,24] with a Red (for minus sign) and Green (for plus sign) colormap instead of HSV.

 

Fig. 6. Comparison of image time in I-SHG for a 250×250µm², 0.5µm step scan (abscissa, in log-scale) versus the phase distribution width, averaged on the two phase peaks of the model sample PPLN of Fig. 2. Orange = static phase-shifter, green = 2000µs phase scan, blue = 200µs, violet = 20µs. The duration corresponding to stage-scanning (resp. galvos-scanning) are indicated by square (resp. triangle) markers. The duration of standard I-SHG with stage-scanning used in [24] is also indicated (1.5h).

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5. Conclusion

In this work, we have demonstrated, for the first time, the possibility of single-scan acquisition in I-SHG microscopy (1S-ISHG) based on an electro-optic phase-scanner. The scan of phase-shifts is performed for each pixel of the I-SHG image being acquired, which corresponds to a total acquisition of only a few seconds. For comparison, standard I-SHG implied the acquisition of 9 or more frames, which lasted few minutes or more: the speed is increased 20 times, while the phase precision is decreased by only a factor 2, which is still sufficient to accurately discriminate opposite polarities. This opens the possibility of measuring - in a minimally invasive way - the relative polarity in many biophysical systems undergoing fast dynamics, e.g. in microtubules in various stages of the mitosis, living bio-samples, or tissues submitted to dynamic traction assays, with an enhanced time resolution. It also opens I-SHG to large scale studies, where a large bank of samples need to be imaged and characterized in a reasonable time, for instance in biology [40]. Even at a precision equivalent to standard I-SHG, using a slower phase-scan, the corresponding imaging time is reduced by a factor of two. Lastly, a high exposure time can be used to characterize samples with low SNR, without the need to average several frames that are potentially exposed to 3D motion. 1S-ISHG can also scan large areas without mosaic reconstruction using stage-scanning, in a time which is more than one order of magnitude smaller than for standard I-SHG. As the phase-scanner is set to work as a slave of the scanning motor, and because the complete interferometer only consists of optics and electronics set separated from the microscope, this phase-scanner can be quite straightforwardly implemented into existing multiphoton imaging systems with only a slight modification in the code used to construct the images. This technology can also be applied to other SHG enhancements such as polarization-resolved SHG (p-SHG), resulting in a similar speed improvement.

Appendix

A.1 Phase extraction

The intensity measured at pixel i on an interferogram at phase-shift ${\delta _j}$ can be written as [14,41,42]:

(a0)$${I_i}({\delta _j}) = {A_{ij}} + {B_{ij}}{\mathop{\textrm {sinc}}\nolimits} (\Delta /2)\cos ({\varphi _i} + {\delta _j}) = {a_i} + {b_i}\cos ({\delta _j}) + {c_i}\sin ({\delta _j})$$

${{\varphi }_\textrm{i}}$ being the relative phase at pixel i and Δ the integration range of time where the phase-shift varies linearly, the background intensity ${a_i}$ and the modulation amplitude ${B_{ij}} = {B_i}$ depending on i only as they are assumed to not vary with phase-shift [42]. Also, the phase-shifts are regularly spaced: ${\delta _j} = j \times \delta $ where δ is constant. The contrast frames are obtained by subtracting two π phase-shifted interferograms:

(a1)$$\left\{ {\begin{array}{{c}} {D(0) = {I_i}({\delta_j}) - {I_i}({\delta_j} + \pi ) = 2{B_i}{\mathop{\textrm {sinc}}\nolimits} (\Delta /2)\cos ({\varphi_i} + {\delta_j})}\\ {D(\pi ) = {I_i}({\delta_j} + \pi ) - {I_i}({\delta_j} + 2\pi ) ={-} 2{B_i}{\mathop{\textrm {sinc}}\nolimits} (\Delta /2)\cos ({\varphi_i} + {\delta_j})}\\ {D(2\pi ) = {I_i}({\delta_j} + 2\pi ) - {I_i}({\delta_j} + 3\pi ) = 2{B_i}{\mathop{\textrm {sinc}}\nolimits} (\Delta /2)\cos ({\varphi_i} + {\delta_j})}\\ {D(3\pi ) = {I_i}({\delta_j} + 3\pi ) - {I_i}({\delta_j} + 4\pi ) ={-} 2{B_i}{\mathop{\textrm {sinc}}\nolimits} (\Delta /2)\cos ({\varphi_i} + {\delta_j})} \end{array}} \right.$$

This process can be iterated several times to increase even more the interferometric contrast ${\gamma _i}$:

(a2)$$\Rightarrow \left\{ {\begin{array}{{c}} {D(0) - D(\pi ) = 4{B_i}\,{\mathop{\textrm {sinc}}\nolimits} (\Delta /2)\cos ({\varphi_i} + {\delta_j})}\\ {D(3\pi ) - D(2\pi ) ={-} 4{B_i}\,{\mathop{\textrm {sinc}}\nolimits} (\Delta /2)\cos ({\varphi_i} + {\delta_j})} \end{array}} \right.$$

(a3)$$\Rightarrow [{D(0) - D(\pi )} ]- [{D(3\pi ) - D(2\pi )} ]= 8{B_i}\,{\mathop{\textrm {sinc}}\nolimits} (\Delta /2)\cos ({\varphi _i} + {\delta _j}) = {\gamma _i}\cos ({\varphi _i} + {\delta _j})$$

where ${\gamma _i} = 8{B_i}\,{\mathop{\textrm {sinc}}\nolimits} (\Delta /2)$. In the general case, if the subtraction is performed n times, N=(360/δ*(n + 2)/2) interferograms are needed, and ${\gamma _i}$ will be multiplied by 2ⁿ. In our example n = 3, so ${\gamma _i}$ is multiplied by 8, and N = 360/30*(3 + 2)/2 = 30 phase-shifts are required if the base phase-shift step is δ=30°.

With the standard I-SHG, the phase-shift is changed by discrete steps such that $\textrm{sinc}({\Delta /2} )= 1$. Considering that the relative phase is obtained by $\tan {\varphi _i} = {c_i}/{b_i}$, one has ${\varphi _i}({\Delta = 0} )= {\varphi _i}({\Delta \ne 0} )$: the relative phase is independent of $\Delta $. The interferometric contrast ${\gamma _i} = \sqrt {{b_i}^2 + {c_i}^2} $ implies ${\gamma _i}({\Delta \ne 0} )= {\gamma _i}({\Delta = 0} )\textrm{sinc}({\Delta /2} )$, i.e. just a constant multiplicative factor, because Δ (the ramp time) is constant.

A.2 Detail of the extraction of the interferograms from the buffer synchronized by the phase scans

In Fig. 7, we explain how the pixels of the interferograms are extracted from the buffers normally used to compute the pixels of the standard image. To measure the signal from the PMT (R6357, Hamamatsu Photonics, Japan), the signal is oversampled by an Analog-to-Digital acquisition card (PCI-6110, National Instruments, Austin, USA) at ${\textrm{f}_{\textrm{sample}}}$= several MHz (settable), which is much higher than the pixel rate ${\textrm{f}_{\textrm{pixel}}}$, usually in the range 1-100kHz. This oversampling allows to split each pixel into up to N=${\textrm{f}_{\textrm{sample}}}/{\textrm{f}_{\textrm{pixel}}}$ ∼ 5-1000 subpixels. The number N of subpixels is equal to the imposed number of interferograms. Between ramps, the high-voltage must be reset (see Fig. 1(c) or Fig. 7) which leads to dead-times (=${\textrm{T}_{\textrm{dead}}}$) that have to be taken into account such that:

(a4)$$\textrm{Pixel time} = \frac{1}{{{\textrm{f}_{\textrm{pixel}}}}} = {\textrm{T}_{\textrm{ramp}}} = {\textrm{T}_{\textrm{dead}}} + \mathop \sum \limits_{\textrm{j} = 1}^\textrm{N} {\textrm{T}_{\textrm{subpixel}}}(\textrm{i} )$$

 

Fig. 7. Detail of the filling of the interferogram arrays, from the acquired buffer. One ramp corresponds to one pixel of the standard image, but the oversampling of these pixels allows to split them into several subpixels. Samples from the buffer are averaged by packet and form each subpixel, i.e. the pixels of the N different interferograms. There are some dead-times between ramps indicated in grey, and a latency at the beginning of each line indicated by striations.

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Because the maximum rate of the acquisition card ${\textrm{f}_{\textrm{sample}}}$ is limited to 5MHz [43], the smallest accessible time interval is ${{\tau }_\textrm{R}}$ = 1/5 = 0.2µs. Also, the acquisition rate is always a sub-multiple of the 20MHz master time-base rate [43], which means this time resolution can be adjusted by steps down to $\Delta {{\tau}_\textrm{R}}$ = 1/20 = 0.05µs only. This implies that ${\textrm{T}_{\textrm{ramp}}}$, ${\textrm{T}_{\textrm{subpixel}}}\,(\textrm{i})$ and ${\textrm{T}_{\textrm{dead}}}$ have to be multiples of ${{\tau }_\textrm{R}}$, and can be adjusted by steps of $\Delta {{\tau }_\textrm{R}}$ only.

Table 2 shows the subpixel times (i.e. the pixel time of the I-SHG interferograms) corresponding to the different phase-scan modes: a certain number of samples from the buffer are averaged to compute the subpixels, and some other are discarded because they correspond to the dead-time between the ramps.

Table 2. Timing used for the different ramp times of the phase scanner: pixel time of the 30 interferograms, and corresponding number of samples used to form these pixels.

View Table | View all tables in this article

A.3 Equation of the precision of the phase measurement

If C is the level of interference contrast and N the number of interferograms, the precision ${\delta \varphi }$ of the phase calculated by our method (see [41]) is estimated by [44]:

(a5)$${\delta \varphi } \sim \frac{1}{{\textrm{C} \times \textrm{SNR}}}\sqrt {\frac{2}{\textrm{N}}}$$

The SNR is a sum of various terms: the electronical noise, the laser’s fluctuations and the photon shot noise $\textrm{SN}{\textrm{R}_{\textrm{shot}}} \sim \sqrt {{\textrm{n}_{\textrm{photons}}}} $. The number ${\textrm{n}_{\textrm{photons}}}$ of collected photons is also assumed proportional to the exposure time.

Funding

Canada Foundation for Innovation; Natural Sciences and Engineering Research Council of Canada; Fonds de Recherche du Québec - Nature et Technologies.

Acknowledgments

M.P. thanks the company Raicol Crystals for technical discussions, the CREATE/GRK program for scholarship, and Antoine Laramée for implementation help. M.P. also acknowledges Sheila Laverty from St-Hyacinthe veterinary school for meniscus samples, and Magali Millecamps & Laura Stone from McGill’s Edwards Centre for Research on Pain for tendon samples.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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