1. Introduction

With the use of optical sectioning microscopy for fluorescence imaging, the domain of biomedical imaging has shifted from the understanding of two-dimensional cell function to understanding three-dimensional cellular interactions and tissue in vivo [1–4]. The most widely used optical sectioning microscopies for fluorescence imaging are confocal fluorescence microscopy (CFM) and two-photon excitation fluorescence microscopy (TPFM) [5, 6]. CFM uses a pinhole in an optically conjugated plane in front of the detector to reject out-of-focus fluorescence, allowing thin optical sectioning within thick samples. In TPFM, use of a pinhole is not necessary, because only fluorescence from two-photon excitation light that is confined to a small region around the focal volume contributes to the image formation. In addition, TPFM has more advantages than CFM in in vivo tissue imaging owing to its nonlinear process. The near-infrared excitation light employed in TPFM has significantly lower scattering and absorption in the specimen [7], which facilitates deeper tissue penetration and minimizes photo-damage to biological specimens. Thus, TPFM is compatible with in vivo imaging. Lower scattering can provide depth-resolved imaging of optically transparent tissue up to a few hundreds of micrometers. There is approximately a threefold increase in imaging depth for TPFM compared with CFM mainly by reduced scattering using near-infrared excitation [8]. Because near-infrared light for excitation is spectrally well separated from visible fluorescence emission, the former allows high sensitivity imaging by eliminating the contamination of the fluorescence signal by excitation light. Furthermore, being spectrally well separated ensures that the excitation light and the scattering-induced effects, such as Raman scattering, are rejected easily without filtering out any of the fluorescence photons, resulting in a higher signal-to-noise ratio (SNR).

Although TPFM has many advantages, it has limitations in achieving greater imaging depth. The primary limiting factor for this is the scattering process that degrades the excitation power at the focal volume. Degradation of excitation intensity could be compensated for by exponentially increasing the excitation laser power. However, as the imaging depth increases, the out-of-focus fluorescence background generated near the surface of the sample by high excitation light plays a more significant role in the detected signal, resulting in a lower SNR. This fundamental limitation has already been described analytically [9] and experimentally [10]. The imaging depth of two-photon microscopy is still limited to less than 5 scattering mean free paths (MFP), even when utilizing higher energy regenerative amplified pulses [10]. Considerable research efforts have been performed to improve imaging depth such as, using a high numerical aperture (NA) with large field-of-view objectives to increase fluorescence collection efficiency from scattering tissue [11] or using supplementary fiber-optic light collection systems [12]. However, all of these efforts involve the use of a modified microscope with external detectors and a large diameter tube lens or an additional fluorescence collection system with sophisticated optical fibers, which make the systems complicated.

In this study, we develop a simple alternative approach to reject out-of-focus background using a confocal pinhole. Studies have previously been conducted on the pinhole effect using TPFM. The resolution of TPFM corresponding to various sizes of pinholes has been demonstrated by Gauderon and associates [13]. They have shown the improvement of the SNR by using finite-sized pinhole experimentally, however they only focus on the improvement of resolution resulting in improvement of SNR, they did not fully demonstrate the effect of pinhole corresponding to the changes in scattering coefficient and imaging depth. Differential Aberration technique [14] is similar method in terms of pinhole usage improving SNR by rejecting out-of-focus fluorescence; deformable mirror was used to alter phase profile of excitation laser in the illumination path and achieve background rejection by subtracting of aberrated images from unaberrated images. However, the system includes complex process and needs more time to obtain two images; with and without a DM-induced aberration image and post data processing to get the improved reconstruction image. Furthermore, because of using typical TPEM collection structure, the fundamental limitation of imaging depth is still set at MFP of 5 at the best configuration. Recently, the Grand group at Universite de Rennes demonstrated the role of the collection efficiency in the SNR in TPFM by Monte Carlo simulations with a comparison of the descanned and non-descanned collection modalities [15]. However, Monte Carlo simulations have some drawbacks. They are computationally intensive and have a problem of getting good statistics when the point of interest is located far from the point of entry of the light, especially when the scattering is high [16,17]. Even with the large number of photons used in previous work (~10¹⁰), the descanned profile appears noisy as a consequence of the low collection efficiency and may fail to obtain the possible gain near the focus. Furthermore, the overall computation time for simulating three SNR profiles was approximately a week [15].

In this study, we derived ultra-short pulse light excitation and corresponding two-photon excitation fluorescence (TPEF) collection for a scattering tissue model, and successfully demonstrated the improvement in imaging depth of two-photon microscopy with the confocal pinhole. Firstly, we introduced the analytical model of free-space ultra-short-pulse Gaussian beam propagation for spatiotemporal ballistic light intensity distributions and then applied this model to determine the scattered light intensity distributions. In isolation from the excitation process, the fluorescence collection on both focus and out-of-focus through a scattering sample with pinholes of various sizes was then derived. Finally, we demonstrated that the SNR is improved for the TPFM system with the pinhole in a highly scattering tissue mimicking phantom materials with a proof-of-concept experiment.

2. Ultra-short-pulse Gaussian light excitation

We derived the analytical models of pulse propagation using the free-space ultra-short–pulse light propagation developed in previous studies [18–20] and utilized the analytical model developed by Theer et al. to compute the TPEF distribution [10]. Details of the derivation are in the Appendix. The intensity distribution from the lowest solution known as the Gaussian beam is in the form of

The number of fluorescence photons Q generated by two-photon excitation of the fluorophore is proportional to the square of the excitation intensity and is defined by [6, 21]

While the ballistic contribution can be determined accurately, consideration of the scattered light contribution is quite difficult. Assuming that forward scattering is dominant, which is valid for most biological tissues, the two-photon absorption can be ignored in comparison to scattering. The scattered light contribution I_s can be determined by solving the radiative transfer equation by introducing small-angle diffusion approximation, and can be written as [22]

 

Fig. 1 Simulated spatial distribution of ballistic (left) and scattered (right) light intensity along the depth z (z = 0, 0.4, 0.8, 1.2, 1.6, 2, and 2.4 mm) for λ = 810 nm, μ_s = 25 cm⁻¹, NA = 0.9, τ₀ = 140 fs, and g = 0.9. The timescales of ballistic and scattered cases are different, −1 ps to 3 ps and −6 ps to 6 ps, respectively.

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3. TPEF collection with pinholes of various sizes

In order to analyze the effect of the size of the pinhole on the generation and detection of the fluorescent signal in the TPFM system, we first have to analytically consider the TPEF collection path from the scattering sample to the microscope collection path. While two-photon excitation by short-pulsed Gaussian light in a highly scattering medium is a time-dependent problem and nonlinear process, TPEF collection is a time-independent problem and linear process. Thus, the collection process can be considered separately with excitation; only two considerations were used to determine the collection efficiency in the microscope detection system. One is the fraction of fluorescence light that reaches the objective front aperture (OFA) of the microscope, which is given by the solid angle of the OFA according to the generated fluorescence source position. The other is the fraction of the light detected by the microscope system, which is determined by the range of acceptance solid angle and which depends on the size of the pinhole. Firstly, we derive an analytical expression for the on-axis TPEF collection efficiency η corresponding to the incremental axial distance z from the OFA for the non-scattering medium. Then, we expand our model to off-axis TPEF collection efficiency according to off-axis position r and apply our model to a scattering medium case.

The schematic diagram of the TPEF collection path with a pinhole for a transparent (non-scattering) medium is illustrated in Fig. 2 . We assume that the point-like fluorophore is excited and emits fluorescence isotropically through the medium and may be located arbitrarily relative to the objective, but it will show only the on-axis term. We need to take into account both the solid angle covered by the OFA and the fraction of light that reaches the detector through it. Both factors depend on the OFA radius r_o and the position of the fluorescence source relative to the objective that is positioned, such as the distance z from the OFA and off-axis position r. Regarding the solid angle covered by the OFA, the fluorescence light in front of the objective lens should be in the area of the OFA, and it can simply be derived by the so-called “ray optics” approach:

 

Fig. 2 Schematic diagram of TPEF collection path with pinhole. The figure only shows the on-axis terms. f_o and f_t are the focal length of the objective and tube lens, r_o and r_t are the radii of the objective and tube lens, wd is the working distance of the objective lens, ro is the objective front aperture radius, θ_p is the angular acceptable range according to the radius of pinhole r_p, and θ is the maximum acceptance angle at each depth.

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The solid angle Ω(r, z) is given by S/R², where S is the area of a surface that typically has an ellipsoid shape and R is the distance of the source from the illuminated center of the OFA [12]. If the fluorescence source is positioned on-axis (r = 0), the collection efficiency η is given by

 

Fig. 3 (a) A TPEF collection from an arbitrarily positioned fluorescence source. The solid angle Ω depends on the acceptable area radius r_a, fluorescence distance R, and the off-axis angle γ. (b) A solid angle of the ellipse (left) and its identical solid angle of the circle (right). We assume that the areas of the ellipse and circle are identical; the solid angle is also the same (r_a r_γ = r_m2).

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Using Eqs. (7)-(10), we calculated the spatial distribution of the collection efficiency according to the different pinhole sizes in the non-scattering medium. Figure 4 shows the comparison of spatially distributed collection efficiencies computed for pinhole sizes of 20 and 2500 μm (non-pinhole), which are equivalent to 4 and 500 μm, respectively, in the objective space according to a source position with a magnification of 5 system for NA of 0.9 and focusing depth of 2000 μm. The size of non-pinhole case represents about 80% of the 20xobjective field of view [25]. The figure shows function of pinhole as rejecting out-of-focus fluorescence that enables thin optical sectioning capability within thick samples.

 

Fig. 4 Spatial distribution for collection efficiency of 20-μm pinhole (left) and non-pinhole (2500-μm)(right) in non-scattering medium corresponding to an arbitrarily positioned fluorescence source. Data were plotted on a semi-logarithmic scale.

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So far in the previous paragraph we only considered the non-scattering case, which is relevant to the ballistic light of the scattering medium explained before. However, an actual collection process involves both ballistic and scattered TPEF together, and most of the TPEF light reaching the OFA has been scattered. Thus, calculations of TPEF excitation and collection in which both ballistic and scattered TPEF coexist are complicated processes and have thus far depended mainly on numerical simulations such as Monte Carlo methods. However, numerical simulation is computationally intensive, requiring many photons in order to obtain reasonable results, i.e., In case of obtaining simulation result of fluorescent signal propagated from a depth of 8.5 scattering MFP lengths, huge amount of photons are needed because the probability of non-scattered photons to reach the OFA is less than 0.003%. Some efforts have been made to describe the epi-collection properties analytically, and the analytical scattering model developed in a previous study was used for our analysis [26] to describe the in-focus TPEF in scattering media. There were two approximations made for our collection efficiency simulation. First, fluorophores are distributed randomly throughout the sample, and fluorescence is generated and acted as if it were a point like source. Second, in the emission path, if fluorescence photon experience scattering event, it will enter OFA isotropically. As a matter of fact, he steady-state TPEF density distribution G(r,z) in the medium can be calculated by the method of images [26, 27] and is given by

 

Fig. 5 Spatially distributed total collection efficiency of 20-μm pinhole (left) and non-pinhole (right) in scattering medium. The scattering coefficient at the fluorescence wavelength is 42 cm⁻¹. Data were plotted on a semi-logarithmic scale.

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Fig. 6 Axial profiles of the collection efficiency of 20-μm (blue) and non-pinhole (red) for a focusing depth set at 2000 um. The scattering mean free path MFP at excitation and TPEF wavelength are also reported on the upper x-axis. The corresponding ratios of non-pinhole and 20 um pinhole profiles are also provided (dotted line). The scales are the same for individual collection efficiency profiles and a ratio.

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According to [10], the spatially distributed TPEF intensity collected by the microscope from the scattering medium is proportional to both the square of the excitation intensity, and collection efficiency η and can be calculated by multiplying them. Integration over the radial coordinate then reveals the collected fluorescence F_s from each plane normal to the optical axis z and is given by

 

Fig. 7 Semi-logarithmic representation of excitation (black line) and TPEF collection of 20-μm pinhole (red line) and non-pinhole (blue line) pinholes for the same excitation condition as that for Fig. 1 and at μ_f = 42cm⁻¹ the fluorescence wavelength of 585 nm. The corresponding ratio of the 20-μm and non-pinhole profile is also provided for comparison (dotted green line). Data were normalized by excitation intensity at the surface (z = 0).

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4. Measurement of TPEF collection

The purpose of the experiments was to validate our analytical model with the two-photon microscope system with a pinhole by measuring the TPEF collection. The measurement was performed with a custom-built two-photon microscope setup with a femtosecond Ti:sapphire laser (Chameleon Ultra II, Coherent), which provides an average power of up to 3.8 W, an 80-MHz repetition rate, and a 140-fs pulse duration at a center wavelength of 810 nm. The laser beam was expanded to a 9 mm beam diameter with a pair of achromatic doublets (f10 and f75, Thorlabs), and the excitation intensity was adjusted with a zero-order half-wave plate (WPH05M-830, Thorlabs), calcite polarizer (GL5-B, Thorlabs), and variable neutral density filter (50FS04DV.4); it was controlled up to an average power of 500 mW in front of the laser source to prevent optical damage of components and filtered by a bandpass filter (810DF40, Omega Optical Inc.) to reject unwanted visible light from the excitation source. The excitation and collection were performed by a water-immersion objective lens with a long working distance and high NA (XLUMPLFLN 20 × /0.95 W, Olympus). The TPEF detection system was composed of a dichroic mirror (685dcxru, Chroma), a collection lens (f45 achromatic doublet, Thorlabs) that focuses to the pinhole (P20S, P50S, P150S, Thorlabs), a filter set (ET670sp, Chroma and FES0650, Thorlabs) to reject unwanted infrared light between a pair of secondary collection lens (ACL2520, Thorlabs), and an electron-multiplying charge-coupled device (EMCCD, LUCA DL-658M-TIL, Andor). Each set of 128 images at 20-μm incremental step sizes of fluorescent data was acquired by a PC (Intel Core i7 CPU, 2.8 GHz, 4 GB RAM) and all CCDs were operated by a custom-written program based on the LabView (Version 8.2, National Instruments) driver supplied by Andor. Post data processing and all simulations were performed with Matlab (Version 2009b, MathWorks).

A specific scattering sample was prepared with distilled water containing 1-μm-diameter non-fluorescent polystyrene microspheres (Polyscience Inc.) as the tissue-like optical phantom (Fig. 2 scattering medium). The concentration of the solution was 1.63 × 10⁹ microspheres/ml, which is, according to Mie theory [28], comparable to the calculated scattering coefficient of 20 cm⁻¹ and the anisotropy factor of 0.9 at a center wavelength of 810 nm (34 cm⁻¹ and 0.93 at 585 nm). The experiment was performed with the average power kept less than 500 mW to prevent potential damage to the system. Samples with higher scattering coefficients were also prepared; however, experiments using them were not conducted, because more than 500 mW of average power is needed to compensate for the decrease in excitation intensity in reaching the focal volume, which decreases exponentially with increasing MFP. The sample was inserted between the OFA and a specially prepared thin fluorescent slide moving along the optical axis z. A 10 mm10 mm square shape and a 20 μm high well was fabricated on the slide glass by electron beam lithography (Fig. 2 thin fluorescent slide) and filled with 8 μM quantum dot solution (Qdot 585, Invitrogen). The scattering sample and quantum dot solution were sonicated before mounting to avoid aggregation and inhomogeneity, and the edges of the cover slip were sealed to the slide with nail polish to minimize evaporation.

Figure 8 shows the measured axial TPEF intensity distribution for different sizes of pinholes under conditions described in the previous section. Background noise produced from the electronics and parasitic light from the microscope environment were also measured and used for compensation. The data was obtained with 100 images with 30 fps at each specific depth and average value of 100 data at each specific depth was used to plot. To compare with the simulation results, measurement data were normalized by the value of surface intensity with a 20-μm pinhole. Considering a thickness of 20 μm for the fluorescent slide, simulation data were summed axially for 20 μm and normalized by the local maximum value near the surface, and then compared with the measured fluorescence signal. We did not overfill the back aperture of the objective, because the experimental setup was fixed to the general microscope, which typically has an objective with back aperture of 9 mm. In the excitation simulation, we changed the NA to 0.5 accounting for a mismatch between the beam expansion of 9 mm and the objective rear aperture diameter of 17 mm, which lowers the effective NA. We detected the unwanted source which comes from reflected excitation light by replacing the fluorescent slide with a non-fluorescent one and used for compensation. The experimental data was mostly well matched to the analytical simulation, except for small discrepancy near the surface and long penetration depth including the focus region. Whereas the measured data stay constant near the surface and decays, those of simulation profiles indicate increase slightly up to some depth in the sample possibly due to our assumption on isotropic photon scattering at the surface. In near the focus, due to imperfectly rejected laser source which considered as a noise, the detected fluorescent signal level has higher values than those of the simulation. Meanwhile, at the focus, the measured intensity is lower than that of the simulation. A similar disagreement was also observed in lower scattering samples. This can be explained by the mismatch of the NA of the experiment and simulation or an aberration of the setup originating from a mismatch in the refractive indices [30]. As the size of pinhole is decreased, TPEF power profile data appears noisier as a consequence of the low collection efficiency of the corresponding pinhole. In simulation results, the SNR is 5.2, 1.34 and 0.5 at pinhole size at 20, 50, 150 μm, respectively while SNR of 3.1, 1.3 and 0.3 were found from experimental results. Although experimental and simulation results are slightly different, the both data confirms general tendency that SNR gets bigger as the pinhole size become smaller.

 

Fig. 8 Semi-logarithmic plot of measured and simulated axial TPEF intensity according to 20-, 50- and 150-μm-diameter pinholes (blue asterisk, red christcross, and black cross, respectively). Measured data were normalized by the surface intensity of the 20-μm pinhole and simulation data were normalized by the surface maximum value of the 20-μm pinhole. The excitation simulation was run with NA 0.5 and axial TPEF intensity was summed with 20 μm to compensate for the 20-μm thickness of the experiment.

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5. Enhancement of depth limit

In order to determine the effect of pinhole size, we calculated the SNR by integrating the collected TPEF perifocal volume over a range of 1/e² with the focal peak values (in-focus signal) and another range (out-of-focus background). The imaging depth limit is defined as the depth at which the ratio of the in-focus signal to the out-of-focus background noise becomes 1 [10]. Simulation results of the SNR for different sizes of the pinhole (20 μm and non-pinhole) and the SNR of excitation as a function of scattering MFP μ_sz are shown in Fig. 9 . The corresponding excitation profile F_exc obtained by subtracting η(r,z) in Eq. (17) is also shown for comparison. The simulation was performed under ideal conditions similar to those for Fig. 1, except for various excitation scattering coefficient equivalents to the scattering MFPs of 1, 2, 3, 4, 5, and 6 (1.68, 3.4, 5, 6.8, 8.4, and 10 at the fluorescence wavelength of 585 nm).

 

Fig. 9 Signal-to-noise ratio simulated with 20 μm (blue square) and non-pinhole (red circle) pinholes as a function of scattering MFP μ_sz. The corresponding SNR for excitation is also plotted for comparison (black triangle). The constraint of imaging depth is assumed to be fallen at SNR = 1

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The SNR of excitation is slightly lower than the one in previous studies [10,15], because we used the diffraction-limited beam waist rather than the Gaussian beam waist. For MFP < 3, the axial TPEF power increases monotonically from the surface to the focus, the contribution of scattered light to the generation of out-of-focus noise is insignificant, and most background noise is created by ballistic out-of-focus light; thus, the SNR shows a saturated tendency. For MFP > 3, the background noise is dominated and mainly generated by the scattered light; thus, the SNR decreases and pinhole cases have a steeper slope than the excitation SNR owing to the limited field-of-view angle. Thus, the small pinhole case has a steeper slope owing to its smaller field-of-view angle. The data clearly show that the small-sized pinhole has a better SNR because the ballistic fluorescence emitted from the focal volume, despite being small in portion, contributed to the collected signal and that the out-of-focus background, which was relatively large in portion, was rejected. The small-sized pinhole can increase the imaging depth theoretically up to 6.2 MFP by using the constraint of imaging depth. Certainly, using a finite-sized pinhole reduces the collection efficiency. If the total excitation light that reach the focusing area due to scattering of sample is not enough to generate fluorescent signal for detecting, adding a pinhole may cause a problem for proper signal detection. However, if there is no pinhole, no matter how increase excitation power even utilizing higher energy regenerative amplified pulse, the imaging depth is fundamentally limited to MFP 5 due to the out-of-focus background, on the other hand, pinhole can improve fundamental limitation to more than 6 MFP.

6. Conclusions

In this report, we fully derived ultra-short-pulse light excitation and demonstrated both analytically and experimentally TPEF collection with finite-sized pinholes and non-pinhole. The placement of a pinhole, which results in decrease in a small portion of the ballistic in-focus signal compared with a large increase in the amount of rejected background noise mainly occurring at the surface contributed to the improvement in the SNR, thereby leading to an increase in penetration depth, the nature of which was characterized by an analytical simulation. Although pinholes function not only to reject out-of-focus signals but also to focus scattered signals due to their finite aperture thus significant decreases collection efficiency about 10⁻⁴ than non-pinhole case, pinhole can improve fundamental depth limitation more than 6 MFP. Furthermore, if the fluorescence signal at specific depth gets larger or enhanced by method such as applying optical clearing technique [31], the significant enhancement of depth imaging can be possible with use of a finite-sized pinhole. Measured experimental results were mostly well matched to the analytical simulation under applied experimental conditions. Our results showed that the analytical framework complement well with the more commonly used Monte Carlo approaches and simply introducing a pinhole in the collection path in a combined confocal pinhole and two-photon microscopy system enhanced imaging depth on account of an increase in the SNR up to 6.2 scattering MFP.

Appendix

The free-space propagation of an electromagnetic pulse is governed by the wave equation

(18)$$({\nabla }^2-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2})E(\overrightarrow{r},z,t)=0,$$

where $\overrightarrow{r}={\widehat{e}}_{x}x+{\widehat{e}}_{y}y$ are the transverse coordinates and ${\widehat{e}}_x$, ${\widehat{e}}_y$ are the unit vectors in the x and y directions, respectively. Adopting the comoving frame coordinates, i.e., τ = t-z/c and z = z, and taking time Fourier transform, both in the temporal frequency domain,

(19)$$[{\nabla }_{\perp }{}^2+\frac{{\partial }^2}{\partial t^2}+2ik(\omega )\frac{\partial }{\partial z}]\tilde{U}(\overrightarrow{r},z,\omega )=0,$$

where $k=\omega /c$ is the wave number and $[{\nabla }_{\perp }{}^2={\partial }^2/\partial x^2+{\partial }^2/\partial y^2$is the transverse Laplacian; the time Fourier transform of the electric field is expressed as

(20)$$E(\overrightarrow{r},z,t)=\frac{1}{\sqrt{2\pi }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\tilde{U}(\overrightarrow{r},z,\omega )\exp (-i\omega \tau )d\omega }.$$

By invoking the paraxial approximation in the temporal frequency domain, i.e.,

(21a)$$\left|\frac{\partial }{\partial z}\tilde{U}(\overrightarrow{r},z,\omega )\right|\ll \left|k(\omega )\tilde{U}(\overrightarrow{r},z,\omega )\right|,$$

(21b)$$\left|\frac{{\partial }^2}{\partial z^2}\tilde{U}(\overrightarrow{r},z,\omega )\right|\ll \left|k(\omega )\frac{\partial }{\partial z}\tilde{U}(\overrightarrow{r},z,\omega )\right|,$$

which indicates that the paraxial approximation is satisfied for each frequency component ($\omega$), the paraxial propagation equation in the temporal solution for the transverse components can be derived, i.e.,

(22)$$[{\nabla }_{\perp }{}^2+2ik(\omega )\frac{\partial }{\partial z}]\tilde{U}(\overrightarrow{r},z,\omega )=0,$$

The analytical solution for the paraxial beam can be derived only for some specific cases, such as a Gaussian beam. Suppose $\tilde{U}(\overrightarrow{r},z,\omega )$ is the lowest solution of a Gaussian beam, which satisfies Eq. (22), i.e.,

(23)$$\tilde{U}(\overrightarrow{r},z,\omega )=\frac{i{z}_0}{q(z)}\exp (-ik\frac{r^2}{2q(z)})P(\omega ),$$

where q(z) is the so-called complex beam parameters q(z) = z + iz₀, z₀ is the Rayleigh range, and $P(\omega )$ is the complex representation of the initial on-axis spectral distribution of the pulse. The time domain pulsed field can be derived from the inverse Fourier transform of Eq. (20); then, a pulsed Gaussian beam can be obtained [18–20] as

(24)$$E(\overrightarrow{r},z,\tau )=\frac{i{z}_0}{q(z)}P(\tau \text{'}),P(\tau \text{'})=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}P(\omega )\exp (i\omega (\frac{{\overrightarrow{r}}^2}{2qc}-\tau ))d\omega }$$

where $\tau \text{'}$is the reduced time $\tau \text{'}=\tau -r^2/2cq(z)$, and $P(\tau \text{'})$ is the complex representation of the pulse. In [19], $P(\tau )$is given by

(25)$$P(\tau )=\exp (-\frac{{\tau }^2}{{\tau }_0^2})\{\exp (i{\omega }_{0}t)-i\mathrm{Im}[\exp (i{\omega }_0\tau )erfc(\frac{{\tau }_0}{2}+i\frac{\tau }{{\tau }_0})]\},$$

where ${\omega }_0$ is the carrier frequency. Now, only considering the real term of Eq. (25) that is the same results as that in [18], we obtain

(26)$$\begin{array}{l}E(\overrightarrow{r},z,\tau )=P_0\frac{w_0}{w(z)}\exp (-\frac{{(t-\frac{r^2}{2cR(z)}-\frac{z}{c})}^2}{{\tau }_0^2})\exp (-\frac{r^2}{w{(z)}^2}+\frac{r^4}{w{(z)}^4}\frac{1}{{({\omega }_0{\tau }_0)}^2})\\ \times \exp (i{\omega }_0(t-\frac{z}{c}-\frac{r^2}{2cR(z)})(1-\frac{2r^2}{w{(z)}^2}\frac{1}{{\omega }_0{\tau }_0}))\exp (i\xi (z)),\end{array}$$

where w₀ is the beam waist at the focal plane, w(z) = w₀(1 + (z/z_r)²)^0.5 is beam waist, R(z) = z + z_r²/z is the radius of curvature, and $\xi (z)$ is axially-varying phase shift. Thus, an explicit expression of Eq. (26) is devoted to describing the space-time propagation of a space-time Gaussian pulse. The intensity of the pulse is readily derived from the absolute square of the wave fields given by Eq. (26)

(27)$$\begin{array}{l}I(\overrightarrow{r},z,t)={\left|E(\overrightarrow{r},z,t)\right|}^2\\ =P_0\frac{w_0^2}{w{(z)}^2}\exp (-\frac{2{(t-\frac{r^2}{2cR(z)}-\frac{z}{c})}^2}{{\tau }_0^2})\exp (-\frac{2r^2}{w{(z)}^2}+\frac{2r^4}{w{(z)}^4}\frac{1}{{({\omega }_0{\tau }_0)}^2}).\end{array}$$

Figure 10 shows the space-time pulse shape as observed at the surface (z = 0). Note that the pulse shape is just bent spherically at each wing to the propagation direction while maintaining its implicit pulse duration and absolute pulse duration at each lateral position (off-axis) equal to its initial pulse width. r is the distance from the optical axis and t is the relative time. We ignore the on-axis term z/c and consider only the relative off-axis term r²/2cR(z).

 

Fig. 10 Space-time pulse shape observed at the surface z = 0. r is the distance from the optical axis and t is the relative time. λ₀ is 810 nm and τ₀ is 100 fs, NA is 1 and focus depth is 500 μm.

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Figure 11 illustrates the whole averaged pulse width (black) and each pulse width observed at each lateral r position (0, 200, 400, 600 μm), which clearly shows that each pulse width is the same despite their amplitudes being different. Note that normalized factors are different in that the former is normalized by its maximum value and the latter is normalized by a maximum value at r = 0.

 

Fig. 11 Averaged pulse width (black line) and each pulse width observed at each lateral r position (0, 200, 400, 600 μm) at the surface z = 0. The red line is for r = 0 μm, blue line is for r = 200 μm, pink line is for r = 400 μm, and the cyan one is for r = 600 μm from the optical axis.

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Acknowledgments

This work was supported by the BioImaging Research Center at GIST, and “Basic Research Projects in High-tech Industrial Technology” Project through a grant provided by GIST in 2012.

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