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Abstract
Volume imaging based on a fast focus-tunable lens (FTL) allows three-dimensional (3D) observation within milliseconds by extending the depth-of-field (DOF) with sub-micrometer transverse resolution on optical sectioning microscopes. However, the previously published DOF extensions were neither axially uniform nor fit with theoretical prediction. In this work, complete theoretical treatments of focus extension with confocal and various multiphoton microscopes are established to correctly explain the previous results. Moreover, by correctly placing the FTL and properly adjusting incident beam diameter, a uniform DOF is achieved in which the actual extension nicely agrees with the theory. Our work not only provides a theoretical platform for volumetric imaging with FTL but also demonstrates the optimized imaging condition.
© 2017 Optical Society of America
1. Introduction
It is well known that microscopy techniques with optical sectioning capabilities are indispensable for observing the 3D structure of cells or thick tissues,. Two milestones for optical sectioning microscopy are confocal and multiphoton techniques. Confocal laser scanning microscopy (CLSM), which was invented by Marvin Minsky for neuron study in 1955 [1], achieves optical sectioning by rejecting out-of-focus emission with a pinhole, and CLSM has been proven to be a powerful tool for cellular observation during the last few decades [2]. Multiphoton microscopy (MPM), introduced by Watt W. Webb in 1990, provides optical sectioning by the nonlinearity of excitation, and is now popular for living tissue observation due to its advantages of low phototoxicity and deep penetration [3]. Both of these techniques provide much higher axial contrast and signal-to-noise ratios than conventional wide-field light microscopy. However, optical sectioning also limits the acquisition of axial information at different depths. A full 3D structure is typically obtained via axial displacement of either the objective or the sample with a slow translational stage [4]. Thus, the collection time of a complete 3D volume is on the order of minutes, significantly impeding the observation of fast 3D dynamics inside biological tissues.
To obtain 3D information at a higher speed while maintaining the advantage of optical sectioning, several solutions have been proposed, including random 3D point scanning using acousto-optic deflectors [5], 3D line scanning by moving the objective with resonant piezoelectric actuators [6], and extending the depth-of-field (DOF) using tunable devices [7–15]. The first two techniques increase imaging speed by partially sampling the whole volume. Without continuously monitoring the whole sample structure, they require excellent spatial stability of samples, which is very difficult for in vivo experiments. Besides, while complete 3D images can be acquired by these two methods, they take minutes to finish.
On the other hand, several tunable optical devices have been adopted to extend the DOF, including variable-focus liquid lens [7, 8], spatial light modulators [9–11], or acoustic focus-tunable lens (FTL) [12–17]. Their common underlying principle is to rapidly translate the focus along axial direction. Nevertheless, the focus translation speeds of liquid lenses and liquid-crystal-based spatial light modulators are relatively slow (~kHz) due to the inertia of the liquid and intermolecular interactions within liquid crystal, thus significantly limiting the capability of fast 3D image acquisition. The acoustic FTL provides much faster axial movement of focus (~MHz) since it is based on acoustic resonance and has been proven to extend the DOF on both wide-field [12] and laser scanning microscopes [13–15], as well as thin light-sheet generation [16]. In the wide-field mode, the DOF extension is up to 300-μm long with a 10 × objective, which corresponds well with thin lens imaging theory. In light-sheet generation application, the focus extension also fits with the simulation [16, 17]. However, when combined with a laser-scanning microscope, the DOF only extends to less than 20 μm [13, 14], which is much smaller than the theoretical prediction. In spite of a double-pass scheme to achieve an extension larger than 100 μm [15], a thorough quantitative analysis of the acoustic FTL imaging properties is still desirable for advanced applications.
In this work, we theoretically and experimentally analyze the DOF extension of FTL-based CLSM and MPM. We found that the position of the FTL critically affects the signal collection efficiency and DOF extension length. By correctly placing the FTL, the DOF extension of a laser scanning imaging system can be much larger than previous reports. In addition, the uniformity of the extended DOF is governed by the working frequency and incident beam size. By considering the interplay of these parameters, the optimized condition, i.e., uniform DOF extension, for FTL-based optical sectioning microscopy techniques is obtained. Our work provides a solid foundation for future volume imaging applications in biological tissues.
2. Theory
2.1 FTL position vs. DOF extension
The principle of FTL is to excite an acoustic resonant wave at ultrasonic frequency inside the lens cavity, thus creating a periodic density and refractive index (RI) variation in both the time and space domains. The RI variation is approximated and described by
where r is the spatial coordinate along lens radius, t is the temporal coordinate, n0 is the static RI, nα is the maximum RI variation as the lens works, J0 is a Bessel function of the first kind, ω is the driving frequency, and v is the sound speed in the lens media [18]. As Eq. (1) shows, the RI inside FTL spatially follows a Bessel function and temporally follows a sinusoidal modulation. Spatially, the main lobe of the Bessel function is defined as the effective aperture (EA) that periodically changes between the patterns of a convex and a concave lens, and the side lobes are named as higher-order modes (HOM), as indicated in Fig. 1.
Fig. 1 Schematic figure of the simulation concept. Inset red curve of FTL is the RI profile. The diameter of the input Gaussian beam fits with the EA. Beam convergence (orange dot-dashed line) and divergence (green dotted line) after FTL and objective are exaggerated, to show the focus extension as Δz.
Figure 1 shows the scheme of our simulation. A collimated Gaussian laser beam comes from the left-hand side with its beam waist located at the entrance pupil of FTL. At this stage, the Gaussian beam is assumed to fill the EA, i.e., the 1/e2 diameter is the same as the EA size. The effect of HOM will be considered later. After the FTL, the beam periodically modulates between divergent (green dotted lines) and convergent (orange dot-dashed lines) conditions, resulting in focus shift of an objective that is placed a distance d after the FTL. Apparently, with the limited aperture of the objective, the distance between FTL and objective, d, is critical for the effective focus extension. However, in earlier works, the FTL was placed either far away from the objective [13, 15], or close to it [12, 14], but no clear reason was given. Therefore, our first analysis focuses on the effect of d.
Conventionally, the focus extension Δz is defined by the largest moving range of focus, and can be written as [12]:
where ftube is the focal length of microscope tube lens, δFTL is the refractive power of FTL, and M is the objective magnification. The equation suggests that Δz becomes longer as d increases. However, when considering optical sectioning microscopy techniques, the full-width-at-half-maximum (FWHM) of the focus extension, ΔzFWHM, should be a better figure-of-merit. Figure 2(a) manifested the comparison between Δz and ΔzFWHM by plotting the full intensity distributions of the extended focus versus axial position. The simulation is based on ray transfer matrix analysis of a Gaussian beam, and home-built code in the Matlab platform. The simulation parameters follow our experimental conditions: ftube = 180 mm, δFTL = ± 0.6 m−1, M = 20. In addition, the EA of FTL is 5.5 mm, and the back aperture of the objective is 12 mm. The green/orange/blue curves represent d = 3/29/160 cm, respectively.Fig. 2 (a) The axial excitation intensity distributions of three points at d = 3 (green), 29 (orange) and 160 cm (blue). Δz and ΔzFWHM are compared with the blue curve. (b) Comparison of Δz and ΔzFWHM as functions of d. Large ΔzFWHM and uniform intensity distribution are obtained when d is as small as possible.
The difference between ΔzFWHM and Δz is highlighted with the blue curve, which shows very large Δz and agrees well with Eq. (2). Nevertheless, in reality, the meaningful extension for optical sectioning microscopy techniques should be determined by the ΔzFWHM, which is relatively small when the FTL is far from the objective.
A more complete comparison of Δz and ΔzFWHM versus d is given in Fig. 2(b), with the green/orange/blue cases in Fig. 2(a) marked by I/II/III, respectively. Interestingly, around the point of the blue dashed arrow, which corresponds to the blue curve in Fig. 2(a) (case III), Δz approaches infinity when d is equal to the focal length of the FTL (i.e., inverse of δFTL). However, ΔzFWHM is less than 10 μm, since the intensity drops quickly in the blue curve of Fig. 2(a). The reason for this substantial intensity variation is due to the strong variation of the excitation beam diameter at the objective pupil plane, thus affecting the effective NA as well as the focal intensity. The effect is more prominent when the FTL is far from the objective, as shown in the region of d > 30 cm in Fig. 2(b).
By choosing d = 29 cm (orange curve in Fig. 2(a), case II), the intensity at the negative side of the focus extension drops to half of the maximal intensity, so Fig. 2(b) shows that the excitation ΔzFWHM becomes similar to Δz (orange arrow). Nevertheless, in the orange curve of Fig. 2(a), the excitation intensity distribution is not flat axially. To obtain uniform axial intensity distribution, it is necessary to incorporate the FTL into the objective, i.e., d = 0. In reality, the minimal distance the FTL can be placed is ~3 cm before the objective (case I), and the corresponding axial intensity profile is shown by the green curve (d = 3 cm) in Fig. 2(a). It has nearly the same ΔzFWHM as the orange curve, while the intensity profile is much more uniform, which is preferred when considering optical sectioning microscopy techniques.
2.2 FTL position vs. signal collection efficiency
In the above analysis, we only considered the effect of the FTL position on excitation intensity distribution. For image formation, the effect on signal collection with FTL is certainly of equal importance, but surprisingly, it has not yet been considered in the literature. In this section, detailed analysis on collection efficiencies of epi-detected fluorescence signal in the case of CLSM and MPM are given. We will show that the key factor is whether or not the epi-fluorescence passes through the same FTL along collection path in CLSM and descanned MPM schemes. For the non-descanned detection (NDD) MPM case, the main factor is either signal passing through the FTL or using a large-area detector.
In the simulation, the ray transfer matrix is again adopted to calculate how much signal percentage is collected by the pinhole (CLSM and descanned MPM) and collection area of photo-detector (NDD MPM). For the simulation of CLSM, whose setup is schematically shown in the inset below Fig. 3(a), the parameters follow the design of our experimental setup (FV300, Olympus, Japan), i.e., the important features being a confocal pinhole with 1 Airy Unit (AU) diameter placed 70 cm from the back aperture of the objective, and a photo-detector that detects all signals after the pinhole. The “Pass” and “Not pass” insets below present the scenarios where the epi-fluorescence does and does not pass through the FTL, respectively. The green curve of Fig. 3(a) shows the variation of collection efficiency versus focus tuning position when the FTL is placed 3 cm before the objective with the signal passing through the FTL (the green FTL in the “Pass” setup). Due to the time reversibility of light propagation, the same collection efficiency (~86%) is expected throughout the focus tuning range, assuming no chromatic aberration and no loss due to the optical elements. However, when the FTL position is changed to 160 cm and the signal is not allowed to pass through the FTL (blue-dashed box in the “Not pass” setup), the collection efficiency drops off quickly at positions other than the inherent focal plane (δFTL = 0), since the signal is largely rejected by the confocal pinhole, as shown by the blue curve in Fig. 3(a). Therefore, in CLSM applications, the epi-collected signal should be allowed to pass through the FTL to maximize collection efficiency.
Fig. 3 Collection efficiency versus focus tuning. (a) CLSM setups, (b) MPM descanned setups. The two simulation setups with different FTL positions marked by green and blue colors are shown below the collection efficiency figures. The left-bottom (Pass) and the right-bottom (Not pass) represents the case of signal passing and not passing through FTL, respectively. The color of FTL corresponds to the color of the individual collection efficiency curve. Green: cases of signal pass pass through FTL when d = 3 cm; Blue: d = 160 cm with signal not passing through FTL. PD: photo-detector; XY: scanner; P/NP: signal pass/not pass through FTL.
Next, we explore the collection efficiency of FTL with MPM, which exhibits intrinsic optical sectioning capability. Two different MPM schemes, descanned and NDD, are considered in Figs. 3(b) and 4, respectively, with their schematic setups given below each panel in the “Pass” and “Not pass” insets. In the descanned MPM case, the same parameters of CLSM are adopted, but the pinhole size is increased five-fold to remove the confocal filtering. When the FTL is close to the objective (3 cm, Fig. 3(b), the green FTL in the “Pass” setup), and the epi-fluorescence passes through it, nearly 100% collection efficiency is obtained throughout the whole focus tuning range, as shown by the green curve in Fig. 3(b). The improved collection efficiency in comparison to CLSM is due to the larger pinhole size. Now, if the signal is not allowed to pass through FTL (blue-dashed box in Fig. 3(b), the “Not pass” setup), the blue curve of Fig. 3(b) shows that the collection efficiency is still high around the center of focus extension, but decays rapidly as focus moves 10 μm away from inherent focal plane, because the spot size exceeds the detection region. Moreover, the blue curve of Fig. 3(b) shows that the collection efficiency becomes asymmetric due to the asymmetry of focus extension (Fig. 2(a), the blue curve). Overall, the main difference between the CLSM and descanned MPM schemes is the increase of collection efficiency due to larger pinhole size.
Fig. 4 Collection efficiency versus focus tuning in MPM NDD schemes with different collection area photo-detectors. (a) small-area detector. (b) large-area detector. Different color curves represent different cases with the FTL position and detailed setups shown below. Green/red: cases of signal pass/not pass through FTL when d = 3 cm; Blue: d = 160 cm with signal not passing through FTL. S/L: small/large-area detector.
Figure 4 considers the case of NDD MPM, which is commonly implemented in MPM to further increase collection efficiency. In the NDD simulation scheme, a photo-detector is placed 20 cm from the back aperture of the objective, with a collection lens, but no pinhole, before the detector. Unlike the descanned CLSM and MPM cases, the NDD MPM setup allows placing the FTL close to the objective without signal passing through it, as the red-dashed box in the “Not pass” condition shows. Two parameters are considered here: the signal path (pass P or not pass NP) and the detector size (large L or small S). Once again, the best scenario, i.e., 100% collection efficiency maintained throughout the focus tuning range, is obtained when signal passes through FTL (the green FTL in the “Pass” setup of Fig. 4), as shown by the green curves in Figs. 4(a) and 4(b), regardless of the size of detector.
For the “Not pass” cases, if a large-area detector is used (such as the H7422-40 photomultiplier tube from Hamamatsu with 5-mm diameter), regardless of the FTL position with respect to the objective, the collection efficiency is kept high throughout the focus tuning range, as given by the overlapping red and blue curves in Fig. 4(b). On the other hand, if a small-area detector is used (such as the S8664-02K avalanche photodiode from Hamamatsu with 200-μm diameter), the red and blue curves in Fig. 4(a) indicate that the collection efficiency drops around the endpoints because spot size exceeds the collection area. If only a limited depth-of-field is required, say ± 20 μm, the small-detector setup could provide acceptable detection uniformity.
Note that the red dashed setup, i.e., placing the FTL 3 cm before the objective, may not be practical for accommodating an additional dichroic mirror. Nevertheless, if the signal is not allowed to pass through FTL and a small-area detector is used, even with a very short distance between the FTL and the objective, the signal collection efficiency drops quickly. In reality, if the distance between the FTL and the objective increases, not only the collection efficiency but also the excitation uniformity would further degrade. In summary, signals should be allowed to pass through FTL in CLSM and descanned MPM applications in order to maximize the collection efficiency. For NDD MPM, detection is optimized by allowing the signal to pass through the FTL or by using a large area detector.
2.3 DOF extension combining excitation and detection
To obtain the overall DOF extension, both excitation and detection should be considered equally. By combining the excitation DOF in Fig. 2(a) and the collection efficiencies in Fig. 3 and Fig. 4 (via direct multiplication and normalization), the effective signal intensity distributions along the axial direction for both CLSM and MPM (squared excitation intensity is considered for two-photon fluorescence case) are shown in Fig. 5. In Fig. 5(a), the green curve shows the DOF effective axial intensity under the conditions that FTL is 3 cm before the objective and signal is allowed to pass through it. As a result, the ΔzFWHM is the same as the green curve in Fig. 2(a) since the collection efficiency of the pinhole is maximized. If the signal is not allowed to pass through FTL, the ΔzFWHM significantly decreases as the blue curve indicates. Therefore, in the confocal case, the FTL should be placed as close to the objective as possible, and the signal should pass through the FTL.
Fig. 5 Extended DOF effective axial intensity combined excitation with detection. (a) and (b) are CLSM and MPM descanned cases, respectively, while (c) and (d) are small- and large-area detector used in MPM NDD cases, respectively. d = 3 cm or 160 cm is marked in parenthesis.
In Fig. 5(b), the descanned MPM exhibits larger ΔzFWHM than CLSM when the signal does not pass through FTL since a larger pinhole is used (blue curve). However, only when signal is allowed to pass through FTL, the ΔzFWHM is maximized, as shown by the green curve. Therefore, the FTL should be placed as close to the objective as possible and the signal should pass through it in the same manner as the confocal case.
If the FTL is used in the NDD mode of MPM, both the FTL passage and detector size affect the ΔzFWHM, as shown in Fig. 4. To optimize the ΔzFWHM, as given by the green and red curves in Fig. 5(c) and in Fig. 5(d) (which are overlapped), either the signal needs to pass through the FTL, or a large-area detector must be used. When neither is incorporated, the ΔzFWHM is smaller due to signal loss at the endpoints, as shown by the red curve in Fig. 5(c). For example, placing the FTL 160 cm from the objective results in a much smaller ΔzFWHM, regardless of the detector size, due to the greatly reduced ΔzFWHM in excitation, as shown by the blue curve of Fig. 2(a). In brief, placing the FTL as close to the objective as possible, allowing the signal pass through the FTL, and using a large-area detector are the best strategies in the NDD MPM case.
From the above analysis, the FTL should be placed as close to the objective as possible in order to achieve maximal DOF extension with uniform axial intensity profile in all three different imaging schemes. In the mean time, the signal should be allowed to pass through the FTL in order to maximize the collection efficiency and obtain the largest ΔzFWHM.
2.4 DOF extension combining HOM and temporal resonance
In the above analysis, nearly-flat excitation and collection patterns are achieved with FTL when its temporal characteristics are not considered. However, if the temporally sinusoidal modulation is included in the simulation, the focus stays longer at the two endpoints of the DOF extension than at the center. Therefore, the average intensity at endpoints is stronger than at the center, resulting in a dumbbell-like axial intensity distribution as shown in Fig. 6(a) and observed in [16]. Although signal post-processing can be used to compensate this uneven axial intensity, we found that, by expanding the incident beam size to utilize the HOM of FTL, better uniform extension can be achieved.
Fig. 6 Simulated axial profile of focus extension when the beam size is (a) 0.8, (b) 2, and (c) 1.4 times of the EA, respectively. All these simulations follow the refractive index in Eq. (1).
In section 2.1–2.3, the incident beam size is limited to within the EA, i.e., the FTL acts as a simple convex/concave lens. If the incident beam is larger than EA, the portions that are refracted by the HOM region should focus to a different focal plane since HOM has smaller RI variation than EA (see inset of Fig. 1). That is, the HOM region helps to disperse the intensity distribution to the center of extension and thus lowers the intensity at the endpoints. To simulate this phenomenon, a Bessel-distributed phase is added onto a plane wave, then focused onto the focal plane of objective. Complete treatment can be found in [19]. The effect is shown in Fig. 6(b), where the incident beam is set as 2 times EA. Apparently, the incident beam is too large so that most of the intensity is concentrated into the center region, resulting in significantly reduced FWHM of extension. By choosing the beam size as 1.4 times EA, Fig. 6(c) demonstrates a more uniform DOF extension, but the extension seems to be less than Fig. 6(a). However, in the following section, we will experimentally show that, when beam size is properly selected, uniform and long extension can be achieved simultaneously.
Based on our simulation in section 2, the largest DOF extension (ΔzFWHM) is achieved when the FTL is as close as possible to the objective, and the epi-detected signal should pass through the same FTL to maximize the collection efficiency. To obtain uniform intensity distribution, the incident beam size should be 1.4 times EA. In the following section, we will verify the simulation results by optimizing the focus extension in CLSM.
3. Experiments
3.1 Setup
The FTL-based CLSM setup is shown in Fig. 7. An inverted microscope (IX71, Olympus, Japan) with a confocal scanning unit (FV300, Olympus, Japan) and a 20 × /0.7 NA objective (UAPON 340, Olympus, Japan) was used. The FTL (TAG 2.0, TAG Optics Inc.) was placed with distance d = 3 cm before the objective. Both excitation and epi-collected signal photons passed through it. The maximum δFTL was ~0.6 m−1, and the FTL driving frequency was ~140 kHz with 5.5-mm EA diameter. The excitation source was a 532-nm diode laser. To characterize DOF extension and to avoid chromatic aberration, light scattering from 80-nm diameter gold nanoparticles was imaged. On the other hand, to demonstrate the volumetric imaging capability, 15-μm-diameter green fluorescence beads immersed in agarose were used.
Fig. 7 Experimental setup. BS: beamsplitter; IM: inverted microscope. PD: photodiode, XY: scanning unit, FTL: focus tunable lens.
3.2 Extended DOF measurements
Figure 8 shows our measured DOF extension. When the incident beam size before FTL is 4.5 mm (beam size/EA ~0.8), a dumbbell-like intensity distribution is obtained in Fig. 8(a), corresponding well to Fig. 6(a). When the beam size increases to 8 mm (beam size/EA ~1.4), a uniform DOF extension is found in Fig. 8(b). Very interestingly, the experimental extension provides similar uniformity as the simulation in Fig. 6(c), but the FWHM is much larger. The underlying reason is that the simulation is based on a Bessel distribution of refractive index (see Eq. (1)), which is merely an approximation to real index distribution [18]. Therefore, although the simulation qualitatively predicts the existence of uniform extension, it does not quantitatively describe the experimental results.
Fig. 8 DOF extension characterized by an 80-nm gold nanoparticle. (a)-(c) are axial intensity distributions with corresponding xz images. The incident beam sizes in front of the FTL are (a) ~4.5, (b) ~8, and (c) ~11 mm, respectively, and the most uniform distribution is found in (b). The arrowheads in (a)-(c) mark one endpoint that is influenced by reflection from glass/water interface on the coverslip. The arrow in (b) points the center of extended DOF with the xy image in (d) while the xy image of the endpoint in (e) is indicated as the text in (b). Scale bar in (a)-(c): 10 μm, (d) and (e): 1 μm.
By further increasing the beam size to 11 mm (beam size/EA ~2), when most HOM regions are included, the center intensity becomes stronger than the endpoints, as shown in Fig. 8(c), which is well predicted by the simulation in Fig. 5(b). The reason that the shape of the axial intensity profile in Fig. 8(c) is not as narrow as that in Fig. 5(b) is because of the reflection from glass/water interface of the coverslip, the positions of which are indicated by the arrowheads in Figs. 8(a)–8(c).
In Figs. 8(d) and 8(e), the lateral profile of the center and the endpoint of the uniformly extended DOF are presented, showing FWHMs of 0.74 and 0.8 μm, respectively. The variation is less than 10% throughout the DOF extension and is thus good enough for most cellular imaging applications. With the 8-mm beam size and the 12-mm objective back aperture, the effective NA of the objective reduces from 0.7 to 0.54, and the corresponding theoretical lateral resolution should be ~0.6 μm. To utilize the full NA of the objective, an objective with back aperture equal to 1.4 times EA would be ideal. The reason that the measured FWHM is slightly larger than theoretical prediction might be due to the additional aberration caused by the FTL.
3.3 Volumetric imaging with 3D dispersed fluorescence sample
To demonstrate the image performance by the uniformly extended DOF, several fluorescent beads spreading over an axial depth of 100 μm are mapped. Figure 9(a) shows the optically sectioned images acquired by conventional CLSM, whose DOF is about 1 μm. Therefore, at least 100 sections are necessary to cover the whole volume, typically requiring a few minutes. On the other hand, by using the uniformly extended DOF (~8 mm beam size in use, FTL driving at ~140 kHz), the three beads are imaged simultaneously within a single acquisition. As shown in Fig. 9(b), comparable contrasts are found between the CLSM projection and the extended DOF images with slightly lower intensity for the left and right beads as they are at the top and the bottom of the extended DOF (Fig. 8(b), the top and bottom of the extended DOF has lower intensity than the center); however, the collection time of the extended DOF is less than one second, i.e., two orders of magnitude faster than CLSM.
Fig. 9 (a) 5 CLSM images of 3 fluorescent beads at different depths. (b) Top: average projection from 100 CLSM image of (a). Bottom: single scan image by uniformly extended DOF.
4. Discussion
In section 2, we provided a thorough theoretical analysis of DOF extension with a FTL, considering its position, signal transmission, temporal response, and HOM. The results are clearly observed in the experiment described section 3. One major contribution here is to realize a uniformly extended DOF across the whole extension length, which has not been reported by previous publications [12–17]. Based on our modeling, now we can examine the previously published results and explain why the DOF extension is much shorter than theoretical prediction when combining FTL with optical sectioning microscopy techniques. The comparisons are listed in Table 1. For example, in [14], FTL is combined with CLSM, and the DOF extension is less than 10 μm because the objective magnification is large (40 × ) and the δFTL is low. For NDD MPM applications [13, 15], both of them place the FTL far from the objective, and additionally, relay lenses are added between the FTL and the objective. The purpose of the relay lenses is to expand the beam size to fill the back aperture of the objective, but they also provide extra magnification power, thus reducing the DOF extension. As a result, less than 20 μm extension is obtained in [13]. In [15], the incident laser beam has to pass through the FTL twice to enhance the effective δFTL, plus a low magnification objective, and thus they can achieve an extension of greater than 100 μm, at the cost of requiring a more sophisticated setup.
Table 1. Comparisons of references using FTL for DOF extension.
In Fig. 9(b), only the z-projection image is given, without complete axial information. However, note that, unlike a true Bessel beam, which only gives projection images, the FTL-based method can provide fully recovered axial information with the aid of a high-speed data acquisition system, which has been previously demonstrated by [12], [14], and [15].
A final remark is the frequency at which the FTL is driven, which is critical to consider for high-speed 3D imaging applications. The FTL is a resonant component, and multiple resonances at different harmonics are expected. As mentioned in an earlier work reported by the FTL manufacturer, δFTL exhibits a quadratic dependency on driving frequency [20, 21] as the driving amplitude of the FTL is maintained, so a higher driving frequency allows for a longer DOF extension, as shown in Fig. 10(a). However, EA is directly related to the wavelength of the sound wave inside the FTL; driving frequency is inversely proportional to the size of EA, as shown in Fig. 10(b). Thus, higher driving frequencies result in smaller EA, and smaller beam sizes should be used to achieve uniform DOF extension, which leads to significantly reduced lateral resolution since the effective NA is reduced. Although using an objective with small aperture or placing magnification lens pairs between the FTL and objective can be an alternative method to fill objective aperture, the former reduces light collection efficiency, and the latter increases the total magnification power of the imaging system, which subsequently reduces the range of focus tuning [13, 15]. This tradeoff between extension length and lateral resolution needs to be carefully considered for individual applications.
Fig. 10 (a) ΔzFWHM vs FTL driving frequency. Individual resonant frequencies on the spec of FTL are square-dots marked. (b) EA vs driven frequency.
5. Summary
In summary, we have performed complete theoretical and experimental characterization for FTL-based CLSM with results compatible with MPM. The DOF extension and collection efficiency are optimized through correct placement of the FTL and correct ratio of beam size/EA. A uniform DOF extension is achieved for the first time. We also address the frequency dependence of usable beam size and DOF extension in theory. Our work provides a solid foundation of FTL-based optical sectioning microscopy techniques and paves the way to boost studies of fast 3D acquisition dynamics.
Funding
National Science Council (now Ministry of Science and Technology), Taiwan (MOST-106-2321-B-002-020, MOST-105-2628-M-002-010-MY4, MOST-104-2218-E007-022-MY2, and MOST-105-2633-B-007-001).
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