3.2.1.

Simulated effects of phase distortions on laser pulses

Based on the above-mentioned theoretical principles that govern pulse duration, we first simulated the pulse distortion effects of GDD and TOD. Laser pulses at the FTL display an ideal pulse shape, where the instantaneous wavelength is fixed and the temporal intensity envelope is minimized [Figs. 1(a) and 1(e)]. Pulses with GDD show a change in the instantaneous wavelength as a function of time and a broadened temporal intensity envelope [Figs. 1(b) and 1(f)]. This phase distortion can be compensated with pulse compressors consisting of prism or grating pairs to approach the FTL. However, pulses with TOD show an asymmetric beating of the electric field and modulated temporal intensity envelope [Figs. 1(c) and 1(g) and Figs. 1(d) and 1(h)]. Such asymmetric distortions are more challenging to correct. Furthermore, TOD can be difficult to detect and is impossible to quantify when using intensity autocorrelations since they cannot uniquely determine the laser electric field or the intensity profile of laser pulses.¹⁸ For example, the simulated intensity autocorrelations of Gaussian laser pulses with TOD could be incorrectly interpreted as pulses with no underlying phase distortions but a slightly different underlying pulse shape [Figs. 1(i)–1(l)]. Techniques such as SHG-FROG are designed to quantify phase distortions missed by autocorrelation (see Sec. 2). Specifically, the SHG spectrogram can be used to reconstruct the electric field and intensity profile of the underlying laser pulse [Figs. 1(m)–1(p)]. Ideal laser pulses at the FTL and laser pulses with GDD have SHG spectrograms with an elliptical shape, where an increase in temporal width is indicative of pulse broadening due to GDD [Figs. 1(m) and 1(n)]. Most critically, in the presence of TOD, the SHG spectrogram has an hourglass shape, where adding GDD leads to an asymmetric broadening of the hourglass shape [Figs. 1(o) and 1(p)].

3.2.2.

Comprehensive measurement of laser pulse properties

To directly measure the phase in our laser pulses, we built an SHG-FROG¹⁴ [see Sec. 2 and Fig. S1 in Supplementary Material]. The SHG-FROG measurements in Figs. 2(a) and 2(b) show a spectrogram and temporal pulse shape qualitatively consistent with the presence of GDD and TOD as observed in the simulations shown in Figs. 1(h) and 1(p), respectively. To quantify the amount of GDD and TOD, the measured spectral phase can be fit with a polynomial expanded about the central frequency [dashed black line in Fig. 2(c) and see Sec. 2 and Eq. (1)]. The second-order term is used to calculate the GDD, which for our laser system was $\sim -590{\mathrm{fs}}^2$. The third-order term is used to calculate the TOD, which for our laser was $\sim \mathrm{22,750}{\mathrm{fs}}^3$. Because the goal was to estimate the amount of TOD in the laser source, we performed these measurements at the laser head [see Fig. S1 in in Supplementary Material] rather than after passing through the table optics, microscope, and objective lens, which would add more TOD.¹⁹

3.3.1.

Effect of laser TOD on obtaining the shortest laser pulses

We added and removed GDD from a model Gaussian laser pulse containing TOD in the amounts measured from our laser source and compared these profiles to a model Gaussian pulse with no TOD [Fig. 2(d)]. When all GDD is removed from the laser pulse with TOD, it can only be compressed to $\sim 43\mathrm{fs}$, as opposed to $\sim 31\mathrm{fs}$ if there was no TOD. Furthermore, in the presence of TOD, the duration of this pulse from autocorrelations deviates strongly from the actual pulse duration [compare open and filled red data points in Fig. 2(d)]. Whereas, in the absence of TOD, the pulse durations would otherwise match identically [see blue data points in Fig. 2(d)]. Thus, the ability to compress laser pulses by conventional methods and the accuracy of the intensity autocorrelation in determining pulse duration both deteriorate as a function of TOD. In essence, this deterioration is due to the asymmetric redistribution of energy in the laser pulse [see Fig. S2 in Supplementary Material].

3.3.2.

Effects of laser TOD on fluorophore brightness and heating of the sample

The leftover asymmetric redistribution of energy due to TOD (described above) will limit the achievable peak intensity and therefore imaging brightness in three-photon imaging. Laser pulses with our measured TOD versus no TOD (both with the same pulse energy) were analytically modeled and the case with TOD shows a decrease in peak intensity by $\sim 32\%$ when compared to the case with no TOD [Fig. 2(e)]. In the instance of actual three-photon imaging, where laser pulses traverse through the rest of the path (table optics and microscope), the pulses would accumulate even more TOD, which would further limit their peak intensity even after all GDD was removed [see Fig. 3]. This implies that laser pulses with large TOD will need to operate at a higher average power to achieve the same brightness, when compared to laser pulses without TOD.

To quantify the effect of TOD on imaging brightness we calculated the expected three-photon signal photons per laser pulse at a range of TOD values and normalized this value to the expected three-photon signal photons per laser pulse for a laser pulse with no TOD [see Sec. 2 and Eq. (2)]. This fractional photons per pulse shows a decrease to less than half the original brightness using our measured value of $\mathrm{22,750}{\mathrm{fs}}^3$, and progressively less brightness as TOD is further increased [Fig. 4]. Using a higher average laser power to recover this loss in brightness comes with additional detriments of potential heat damage, which will limit the duration of imaging and/or the penetration depth. Three-photon microscopy users attempting to maximize fluorescence brightness and minimize collateral heating^20–22 will benefit most from systems that have laser pulses with minimal third (and higher)-order phase distortions.

The detrimental effects of these distortions can be partially mitigated by utilizing laser sources without an excessive bandwidth (i.e., only enough bandwidth to reach the desired FTL). Although somewhat counter-intuitive, at large values of TOD, some brightness may be recovered by utilizing a slightly narrower bandwidth. However, choosing a narrower bandwidth comes with the cost of increasing the FTL. To quantify the effect of TOD on imaging brightness at a range of laser bandwidths, we calculated the expected three-photon signal photons per pulse over a range of TOD values and laser bandwidths using Eq. (2). We normalized these values to the expected three-photon signal photons per pulse of laser pulses with the zero TOD and the broadest bandwidth investigated [Fig. S3 in Supplementary Material]. It remains to be seen whether increasing bandwidth beyond this value will be beneficial due to the presence of non-optimal wavelength components and other practical constraints.¹⁷ These results suggest that if a three-photon laser source has a high amount of TOD, imaging quality could be improved most by utilizing a larger bandwidth, and reducing TOD with additional optics in the light path that can compensate specific phase profiles, e.g., with chirped mirrors.

Assuming phase distortions such as TOD are minimized such that the most efficient fluorescence excitation can be achieved, end-users employing three-photon imaging in their laboratories must still contend with pulse-to-pulse intensity fluctuations that may exist in the excitation laser pulses. Therefore, in the following section we propose a new technique for evaluating the energy and intensity stability of laser pulses over timescales relevant for three-photon imaging.

3.4.1.

Imaging at the microscope

Several types of fluorescence artefacts on timescales relevant for imaging may be present in laser systems used for three-photon microscopy. Ideally, three-photon microscopy images of a uniformly fluorescent slide should present a field of view with relatively uniform brightness [Fig. 5(a)]. However, when artefacts on the order of $\sim 10$ to $100\mu \mathrm{s}$ are present, they can manifest as short dark streaks in the microscope images [Fig. 5(b)]. Alternatively, when artefacts on the order of $\sim 10$ to 100 ms are present in the microscope images, they can manifest as horizontal bright and dark banding [Fig. 5(c)]. The laser source used to generate each of the images in Fig. 5 was within specification with respect to RMS% average power stability (RMS% $<2\%$). However, RMS% power stability measurements use thermopile sensors, which typically have a time resolution of $\sim 1\mathrm{s}$. Thus, these measurements give no information on variations in pulse-to-pulse energy or intensity over the time scale of microseconds to milliseconds, which are the critical timescales for three-photon imaging.

Fig. 5

Three-photon microscopy images of a uniform fluorescent slide under normal conditions and in the presence of brightness artefacts. (a) Uniformly illuminated microscope image when the laser excitation source was optimized. In this and subsequent panels, minimum and maximum values of fluorescence in the image is scaled to $\pm 95\%$ of the mean such that red represents signals that are higher than the mean, white are at the mean, and blue are below the mean. (b) Image with intermittent microsecond-scale streaks of reduced brightness (dark blue pixels) followed by a very short rebound (red pixels). (c) Image with slower millisecond-scale temporal modulation of brightness. All images are $256\times 256\mathrm{pixels}$, where each pixel represents the combined fluorescence signal from three laser pulses, i.e., an integration time or pixel dwell time of $3\mu \mathrm{s}$ from a laser with a repetition rate of 1 MHz. Scale bar $100\mu \mathrm{m}$.

3.4.2.

Quantitative strategy to monitor laser pulse-to-pulse stability

To test the pulse-to-pulse intensity stability of the laser, we developed a technique called deep-memory diode imaging (DMDI) (see Sec. 2). In DMDI, pulsed laser light is projected onto a photodiode and the photodiode signal is recorded by a high sampling rate oscilloscope with a large buffer to capture the pulse signal peaks at a high temporal resolution. To match the order of photo-absorption process in three-photon imaging, a tight-focusing element (e.g., a microscope objective) can be used to focus the laser onto a photodiode with an appropriately chosen band-gap that disallows one- and two-photon excitation [e.g., GaP; see Figs. S4(a) and S4(b) in Supplementary Material]. Like three-photon excited fluorescence, the photodiode signal recorded in this way is sensitive to peak pulse intensity and not just to total pulse energy. Alternatively, to simply capture the stability of pulse energy, a photodiode sensitive to one-photon excitation (e.g., InGaAs) may be used, which obviates the need for any focusing elements [Fig. S4(c) in Supplementary Material]. To localize the source of any laser instability, the one- or three-photon photodiode being used in DMDI can be placed at different locations in the optical path, such as under the microscope objective [Fig. S4(a) in Supplementary Material] or at the laser head [Figs. S4(b) and S4(c) in Supplementary Material]. Due to the use of high sampling rates and large buffer memory, DMDI can record relatively long and uninterrupted segments of pulse trains without aliasing of the signal [see Sec. 2 and Fig. S5 in Supplementary Material]. Therefore, DMDI offers a versatile and robust method to measure the laser pulse stability relevant for three-photon imaging.

3.4.3.

Linking photodiode measurements from the laser to microscopy images

We first used DMDI to monitor the pulse-to-pulse intensity stability of the laser by placing the three-photon photodiode (GaP) under the objective lens in our microscope [Fig. S4(a) in Supplementary Material]. Since the microscope images in Fig. 5 use a $3\text{-}\mu \mathrm{s}$ integration time (and therefore each pixel represents the fluorescence signal of three laser pulses from a laser operating at 1 MHz repetition rate), the sum of three successive photodiode signal peaks was taken as a proxy for the brightness of one imaging pixel. This summed peak voltage signal was measured under conditions of artefact-free imaging and during imaging that had short and long-timescale artefacts. Plotting the time courses of summed peak voltage signals under these three conditions revealed an underlying fluctuation in pulse intensities or lack thereof [Figs. 6(a), 6(c), and 6(e)]. We then rearranged this linear array of summed peak voltages into a $256\times 256$ matrix to resemble a three-photon image [Figs. 6(b), 6(d), and 6(f)]. Whether the image is generated in the traditional manner with the aid of a microscope or by DMDI, the same phenomenological fluctuations are clearly identified. This is readily demonstrated in comparing Fig. 5(b) with Fig. 6(d), and Fig. 5(c) with Fig. 6(f), which were obtained when using laser sources that had 10 to $100\mu \mathrm{s}$ fluctuations, and 10 to 100 ms fluctuations, respectively.

Fig. 6

Direct measures of laser pulse intensity fluctuations using a photodiode. (a) and (b) Normal condition with stable laser pulses. (c) and (d) Microsecond-scale fluctuations of laser pulses with clear streaks of reduced brightness. (e) and (f) Millisecond scale wave-like fluctuations. For each of the three conditions, to mimic a pixel integration time of $3\mu \mathrm{s}$ that was used during imaging (as shown in Fig. 5), here the photodiode data are arranged as peak voltage signals, which were summed from every three successive pulses (left column). Then the same voltage signals are shown as $256\times 256\mathrm{pixel}$ images to visualize laser pulse intensity fluctuations in a manner akin to a microscope image (right column). The time axis in (c) and (e) are adjusted to reveal the relevant fluctuation patterns. The $y$ axis and color bar in each case are scaled to 35% of the mean voltage.

3.4.4.

DMDI in laser diagnosis and repair

Beyond attributing the imaging artefacts to pulse intensity fluctuations, DMDI aided our laser manufacturer in improving pulse-to-pulse stability. To this end, we performed DMDI at the laser head using a one-photon photodiode, revealing that the 10- to $100\text{-}\mu \mathrm{s}$ intensity fluctuations stemmed from fluctuations in pulse energy originating from the laser source itself [see Fig. S6 in Supplementary Material]. The slower 10- to 100-ms intensity fluctuations may originate from several sources in addition to the laser, e.g., we found that air currents introduced in the beam path through small gaps ($<1\mathrm{mm}$) in beam-shielding tubes can contribute to these fluctuations.

3.4.5.

More sensitive statistic to describe laser pulse stability

Conventional statistics (e.g., RMS%, see Sec. 2) on data collected either with thermopile sensors or photodiodes used in DMDI may fail to capture potential pulse-to-pulse instabilities. For example, the RMS% of the signal peaks of the pulse trains in all three conditions shown in Fig. 6 was 3% to 6%. We found that another simple statistic is more sensitive to large departures from the mean, i.e., $\mathrm{\Delta }\mu =100\times (S_{\max }-S_{\min })/\overline{S}$, where $S_{\max }$ is the largest signal peak, $S_{\min }$ is the smallest signal peak, and $\overline{S}$ is the average signal peak over the entire time course of the measurement. When calculated on the photodiode pulse train data over three recording periods, this “min-max” statistic ($\mathrm{\Delta }\mu$) was $42.5\pm 1.4\%$ (for the stable condition) and $76.9\pm 2.4\%$ (for the 10 to $100\mu \mathrm{s}$ “streak” artefacts). Statistics presented here are mean ± standard deviation. While RMS% measurements can capture large-scale oscillations in the amplitude variability of pulse trains, the min-max statistic better captures the momentary 10- to $100\text{-}\mu \mathrm{s}$ deviations from the mean. In addition to providing an intuitive snapshot of laser performance communicated through the deep-memory diode image, computing the min-max statistic for data collected by DMDI provides a direct quantitative measurement of pulse-to-pulse stability over conventional methods such as RMS%.