2.1.

Efficiency of Temporal Focusing Through an Optical Diffuser

In this section we reinvestigate the temporal focusing effect through an optical diffuser. Specifically, we compute the variation of the pulse width along the optical axis based on geometrical optics. An axial resolution comparable to conventional confocal microscopy is derived.

Figure 1 depicts the schematic of a typical temporal focusing setup.⁹ An optical diffuser (DF) was used to transform the incoming ultrafast laser beam into a plane of point sources. These point sources were then projected onto the image plane (IP) of an infinity-corrected microscope, through the diffuser lens (L _D) and the objective (L _O). The resulting fluorescence was imaged onto a CCD camera through an imaging lens (L _I).

As discussed by Oron, the elongation of pulse width at an out-of-focus point q ^′ at a distance z away from the IP can be estimated through the maximal difference of pulse arrival times from the point sources within a cone of angle θ from the IP (Fig. 1 inset).⁹ Here, θ can be determined by the divergence angle of L_O, θ ≈ NA/n (Fig. 1 inset).⁹ To estimate the difference of pulse arrival times resulting from the geometry of the setup, we first considered the case where the DF is approximated as a flat plane of point sources that simultaneously generate ultrafast pulses. Using the lens formula and paraxial approximation, the elongation of the pulse width Δ [TeX:] \documentclass[12pt]{minimal}\begin{document}$\it t_G$\end{document} ${\mathit{t}}_{\mathit{G}}$ at the point q ^′

can be estimated as

We next take into account the surface roughness of DF and estimate how such roughness leads to a randomness of arrival times. To proceed, we consider a simple case where the roughness is represented by a step function with a height discrepancy Δh on the diffuser surface [Fig. 2a]. The time delay of a pulse p1 induced by Δh is simply Δh/C _Glass − Δh/C _Air = (n _Glass − n _Air) · (Δh/C ₀) ≈ 0.5 Δh / C ₀. Further, the time delay caused by the roughness in a region A _r of radius r on the diffuser surface is projected into a region [TeX:] \documentclass[12pt]{minimal}\begin{document}$A_r^{\prime }$\end{document} $A_r^{\prime }$ of radius r′ = f _O/f _D r on the image plane [Fig. 2b]. The maximal time delay Δt _RD within [TeX:] \documentclass[12pt]{minimal}\begin{document}$A_{r^{\prime }}$\end{document} $A_{r^{\prime }}$ may thus be approximated as

Fig. 2

Random time delay generated by the surface roughness of an optical diffuser. (a) The time delay of the laser pulses results from a discrepancy of height, Δh, on the diffuser surface. (b) A more general case for (a). Here, Δh corresponds to the maximal surface height discrepancy (i.e., the peak-to-valley difference) within an area of radius r (the corresponding area on the IP is of radius r f _O/f _D). (c) The schematic illustration of Δh under different scales of r. Upper: when r → 0, Δh → 0. Lower: when r ≫ D, Δh ≈ D.

Figure 3 shows the numerical results of τ_eff(z) for the cases of three different sample objectives commonly used for biomedical microscopy. Consistent with a previous report,⁹ we find that the contribution of Δt _G to τ_eff(z) is negligible when z is within a few Rayleigh lengths (z _R). Nevertheless, in this small z regime, Δt _RD in Eq. 6 can lead to a significant elongation of pulse width. In particular, for the small z regions where α f _D/f _O · NA/n · z < 0.5 D, Eq. 6 can be simplified as

Fig. 3

Effective pulse width and two-photon excitation strength as a function of z under different sample objectives. The numerical results were obtained from Eq. 6. Notice that Eq. 8 predicts [TeX:] \documentclass[12pt]{minimal}\begin{document}$\overline{z}^*\approx$\end{document} ${\overline{z}}^{*}\approx$ 3.53, 2.21, and 1.62 for these objectives, respectively, which are comparable with the numerical results. To estimate the normalized strength of two-photon excitation S _2p, the pulse energy was set as a constant and the inverse of τ_eff was used to represent S _2p [Eq. 12]. The horizontal (distance) and vertical (τ_eff) axes are expressed in units of Rayleigh length and τ₀, respectively. Parameters: f _D = 160 mm, D = 100 μm, α = 1, d = 200 mm, λ = 800 nm, and τ₀ = 100 fs. Sample objective 10×: NA = 0.3, f _O = 16 mm, n = 1. Sample objective 40×: NA = 0.75, f _O = 4 mm, n = 1. Sample objective 60×: NA = 1.1, f _O = 2.67 mm, n = 1.33 (water immersion).

2.2.

Efficiency of Multiphoton Excitation at Low Repetition Rate

To solve the limitation of low frame rate, we next examine how the repetition rates of pulsed lasers influence the efficiency of two-photon excitation (at constant average power). In short, we find that a 10⁵-fold increase in signal-to-noise ratio (SNR) is obtained by lowering the repetition rate from 100 MHz to 1 kHz, thus providing a signal level comparable to that of conventional multiphoton and epifluorescence microscopy.

For simplicity, we consider a two-photon excitation process and estimate the light intensity required for wide-field two-photon excitation. The fluorescence signal obtained from a single laser pulse at a single pixel is

4.1.

Axial Resolution of Plane-Projection Multiphoton Microscopy is Comparable to Conventional Confocal Microscopy

Figure 4 shows the axial resolution of the optical setup depicted in Fig. 1. The axial resolution was determined by the FWHM of the fluorescence signal. With [TeX:] \documentclass[12pt]{minimal}\begin{document}$M_{L_O}$\end{document} $M_{L_O}$ = 60, NA = 1.4, n = 1.5, the axial resolution was found to be ∼2 μm, and the corresponding [TeX:] \documentclass[12pt]{minimal}\begin{document}$\overline{z}^*\approx 3$\end{document} ${\overline{z}}^{*}\approx 3$ . This is comparable to the axial resolution of an optimized conventional confocal microscope, which has [TeX:] \documentclass[12pt]{minimal}\begin{document}$\overline{z}^* \approx 1$\end{document} ${\overline{z}}^{*}\approx 1$ . Note that it should be possible to obtain an axial resolution of [TeX:] \documentclass[12pt]{minimal}\begin{document}$\overline{z}^* \approx 1$\end{document} ${\overline{z}}^{*}\approx 1$ by optimizing the microscope design, as we described in Sec. 5.

4.2.

Frame Rate of Plane-Projection Multiphoton Microscopy is Comparable to Conventional Epifluorescence Microscopy for Live-Tissue Imaging

To demonstrate that PPMP microscopy has the capability of imaging live tissues at high frame rate, we performed optical sectioning of live, three-dimensional MCF-10A cell clusters having hemispherical shapes (Fig. 5). The images were then used to reconstruct the 3D view (Fig. 6 and Video 1). Here, the exposure time was set at 200 ms, equivalent to 5 fps, which is 10 times faster than the conventional multiphoton microscope we also used to image the same sample, and is 150 times faster than a temporal focusing setup using a nonamplified 75-MHz Ti:sapphire oscillator to image cells stained with (much brighter) fluorescent dye.⁹ Such an exposure time also lies within the same order of magnitude of that typically used in conventional epifluorescence microscopy (100 ms), which we also used to image the same sample (Fig. 5, lower panel) through the same CCD camera with a mercury vapor lamp (X-Cite^® 120Q, Lumen Dynamics, attenuated by OD 2 to prevent significant photobleaching). Figure 5 demonstrates resolution of the boundaries of cell nuclei along the z axis (i.e., the optical axis). Nuclei in the PPMP image appear oval in structure, resembling the normal shape of cell nuclei. In contrast, the lateral view obtained from epifluorescence microscopy (Fig. 5, lower panel) shows distortion of the proper cell nuclear shape, due to the spreading of the out-of-focus signal in an epifluorescence microscope. These results suggest that PPMP microscopy can obtain high-frame-rate optical sectioning on live tissues.

4.3.

Inhomogeneity of the Illumination Field Can be Reduced by Rotating the Diffuser

In this study, we found that conventional diffusers can cause a significant inhomogeneity of the light intensity in the illumination field, i.e., bright spots [Fig. 7a]. Although optical diffusers are known for generating speckle when used with temporally and spatially coherent light sources,¹⁰ we believe that the observed field inhomogeneity is not from speckles, but rather is an effect of the diffuser used.

One way to examine the physical basis of the field inhomogeneity is to compare the images of a (homogeneous) fluorescent film with the diffuser at the focus and at a position slightly out of focus (i.e., with the diffuser translated along the optical axis by a short distance). If the bright spots are speckles, slightly defocusing the diffuser will only rearrange the location of the speckles, while the statistical properties such as the mean and the variation of brightness will remain the same.¹⁰ The results of this experiment are shown in Fig. 7. After the diffuser was translated ∼3 mm from its in-focus position, the bright spots appearing in the in-focus image [Fig. 7a] were found to remain in place and become dimmer and blurred [Fig. 7b]. These measurements suggest that the observed bright spots are not from the speckle effect, but rather from the randomness of the scattering structures distributed on the diffuser surface.

The observed field inhomogeneity also leads to inhomogeneous sectioning across the field of view; the level of which can be measured by imaging a homogeneous dye film, then separating the field of view into several areas and comparing the FWHMs of their intensity profiles. In our setup, the standard deviation of the FWHMs was found to be ∼0.3 μm. One way to reduce this inhomogeneity is through the use of multiple diffusers. However, each diffuser would generate a certain level of time delay and thus contribute to pulse broadening. At the image plane, the telescoping can only recover the pulse width at the input to the last diffuser (after it has already been broadened by the first diffuser). Thus, having multiple tandem diffusers can reduce the axial resolution of the optical sectioning. One way to mitigate this effect is by proper ordering of the diffusers. As mentioned in Sec. 2, diffusers with larger D and α induce longer random time delay. Therefore, in such multiple-diffuser setups, the diffuser with smaller D and α should be placed before the one with larger D and α. As an alternative solution, we have chosen to simply rotate the diffuser. By rotating the optical diffuser during the acquisition of a single frame, the inhomogeneities in the illumination field are averaged out. This effect is demonstrated in Fig. 8.

5.1.

Optimization and Limit of Axial Resolution

Here we discuss the further optimization of PPMP microscopy, and the axial resolution limit that can ultimately be obtained, as derived within the realm of geometrical optics. Equation 8 suggests that [TeX:] \documentclass[12pt]{minimal}\begin{document}$\overline{z}^*$\end{document} ${\overline{z}}^{*}$ can be further reduced by using an objective with a higher magnification and NA (which often exhibits a smaller f _O NA/n), as shown in Fig. 3. Likewise, increasing f _D, α, or reducing τ₀ leads to smaller [TeX:] \documentclass[12pt]{minimal}\begin{document}$\overline{z}^*$\end{document} ${\overline{z}}^{*}$ . We stress that these estimations are derived based on geometrical optics, and so may not be valid in the extreme case where [TeX:] \documentclass[12pt]{minimal}\begin{document}$\overline{z}^* < 1$\end{document} ${\overline{z}}^{*}<1$ , in which case the optimal axial resolution of our temporal focusing setup would be the same as that of a laser scanning microscope.⁹

A further fundamental advantage of diffuser-based temporal focusing over blazed grating approaches is that the diffuser-based technique can achieve the axial resolution of a point-scan setup, whereas (single) grating-based temporal focusing is limited to that of a line-scan setup. The difference arises from the way in which the time delays are generated. For optical diffusers, the time delay results from the surface roughness of the diffusers, which creates a 2D spatial profile for the randomness of the time delay. In contrast, the time delay in grating-based temporal focusing is created by the one-dimensional scan of the laser pulses on the grating surface; such a mechanism restricts the time delay to be one-dimensional. This restriction has been overcome by using two orthogonally aligned gratings.¹¹ In such a setup, the two gratings must differ in groove density sufficiently, such that the scanning of the laser pulse can be well separated in two orthogonal dimensions. Such a design increases the complexity of the apparatus and will likely require multiple pairs of gratings when multiple/tunable excitation wavelengths are used.

From Eq. 4, the spread, or distribution, of arrival times produced from the surface roughness of a diffuser is upper bounded by the factor D. This suggests that diffusers with larger D should be used to ensure a sufficiently large spread of arrival times. The roughness of the diffuser surface, however, leads in turn to roughness of the image plane, D ^′. Using the thin lens formula, we estimate D ^′ to be (f _O/f _D)² D. This suggests that D ^′ can be negligible if f _D ≫ f _O. Thus, with a proper arrangement of parameters, the roughness of the image plane can be reduced below one Rayleigh length, while the surface roughness of the diffuser is sufficiently large to create temporal focusing.

5.2.

Limitation of Frame Rate and Benefits of Low Repetition Rate

For live-tissue imaging, the frame rates of our setup are limited by the low SNR of fluorescent proteins expressed in living systems. Nevertheless, Eq. 12 suggests that SNR can be further enhanced by lowering the repetition rate while maintaining the average power of the laser. For example, the frame rate of our setup can be further increased by equipping our system with a pulsed laser of much lower repetition rate, e.g., 100 Hz. With such a low repetition rate, Eq. 12 suggests a 10-fold stronger SNR than what is presented in this study. This would lead to a frame rate of up to 50 fps, a rate sufficient to study most biological processes such as cell division, migration, and polarization.

This limiting frame rate is estimated for imaging the fluorescent proteins expressed in living systems. This limitation is relaxed, though, if the signals are derived from materials with strong fluorescence efficiency such as fluorescent dyes and nanoparticles. In such cases, we are able to achieve higher frame rates, as shown in the top panel of Fig. 9, which was obtained with an imaging speed 1000 times faster than a temporal focusing setup using a nonamplified 75 MHz Ti:sapphire oscillator and a similarly dyed sample.⁹ Furthermore, by using microscope objectives of low magnification, we can also achieve large fields of view. This capability for large field of view is demonstrated in the bottom panel of Fig. 9, where we imaged dye-stained MDCK cells with a 10× objective. The size of the field of view shown here is 0.6 × 0.6 mm² and the exposure time was 100 ms.

Our setup can achieve these larger fields of view with a relatively short exposure duration simply because the 1-kHz amplifier is very powerful; that is, because it is supplying its average power at a low repetition rate and low duty cycle and thus achieving a high peak power. To generate multiphoton excitation at the level required for imaging with reasonable frame rates, the peak intensity is commonly around or greater than 1 kW/μm².¹² Therefore, to excite an area up to 1 mm², one needs a light source with peak power greater than 10⁹ W. The maximal peak power of our amplifier is roughly 10¹¹ W, and is thus powerful enough to support a large field of view for most microscopy applications. It should be noted that in the original temporal focusing setup,⁹ a 140 × 140–μm² field of view was obtained with an average power of 30 mW and an exposure time of 30 s. This indicates that a 1-mm² field of view can be achieved with that instrument by using a low magnification objective and an average power of around 1.5 W, though the exposure time in such a setup could be slightly longer than 30 s because lower magnification objectives are typically less efficient in collecting light.

However, from a biologist's point of view, we would also like to point out that discussing the imaging speed for fixed biological samples stained with fluorescent dye is less important than the speed achievable for living systems. Once a sample is fixed, using an imaging time of either 3 h or 10 s would most likely provide the same level of details and information. On the other hand, for studies of dynamic biological processes, the imaging speed would determine the temporal resolution of the observations. To the best of our knowledge, this is the first report of imaging live cells expressing fluorescent protein by a temporal focusing microscope at a frame rate faster than 1 fps.

In addition to the enhancement of the signal level and frame rate, there are certain potential benefits provided by lowering the repetition rate from the megahertz to kilohertz regime. It has been reported that the use of low repetition rates (at the same optical power) can reduce photobleaching.^{13, 14} This is achieved through the avoidance of dark state conversion. Indeed, a 5- to 25-fold enhancement of total fluorescence yield, before detrimental effects from photobleaching, has been reported.¹³ Moreover, lowering the repetition rate is equivalent to providing the system a longer window of no excitation. This would allow slow processes such as heat dissipation to occur more efficiently, thus minimizing sample damage caused by a continuous accumulation of heat.⁴ As a result, even with a similar amount of thermal energy introduced by the excitation process, a sample excited at a lower repetition rate is less likely to be damaged by heat accumulation as compared to the use of a higher repetition rate.

5.3.

Potential Applications as Structured Light Microscopy

In principle, the inhomogeneity of the illumination field can be utilized for structured light microscopy.⁶ This could be particularly useful in applications where reasonable optical sectioning, as provided by temporal focusing, is not achievable. Examples include coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering microscopy, where picosecond pulses are generally required to obtain chemical specificity.^{15, 16} Based on Eq. 8, pulse widths of picosecond duration would greatly reduce the sectioning effect. Nevertheless, by using the inhomogeneity of the illumination field as a structured light source, it is possible to regain the optical sectioning of these systems, as demonstrated in a previous study.⁶ This allows one to integrate CARS with multiphoton excitation in a wide-field microscope simply by using an optical diffuser.