Enhancement of imaging depth of two-photon microscopy using pinholes: Analytical simulation and experiments

doi:10.1364/OE.20.020605

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Enhancement of imaging depth of two-photon microscopy using pinholes: Analytical simulation and experiments

Woosub Song, Jihoon Lee, and Hyuk-Sang Kwon »View Author Affiliations

Optics Express, Vol. 20, Issue 18, pp. 20605-20622 (2012)
http://dx.doi.org/10.1364/OE.20.020605

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Abstract

Achieving a greater imaging depth with two-photon fluorescence microscopy (TPFM) is mainly limited by out-of-focus fluorescence generated from both ballistic and scattered light excitation. We report on an improved signal-to-noise ratio (SNR) in a highly scattering medium as demonstrated by analytical simulation and experiments for TPFM. Our technique is based on out-of-focus rejection using a confocal pinhole. We improved the SNR by introducing the pinhole in the collection beam path. Using the radiative transfer theory and the ray-optics approach, we analyzed the effects of different sizes of pinholes on the generation of the fluorescent signal in the TPFM system. The analytical simulation was evaluated by comparing its results with the experimental results in a scattering medium. In a combined confocal pinhole and two-photon microscopy system, the imaging depth limit of approximately 5 scattering mean free paths (MFP) was found to have improved to 6.2 MFP.

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

S. Paddock, “Tech.Sight. Optical sectioning--slices of life,” Science 295(5558), 1319–1321 (2002). [CrossRef] [PubMed]

–4

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003). [CrossRef] [PubMed]

]. The most widely used optical sectioning microscopies for fluorescence imaging are confocal fluorescence microscopy (CFM) and two-photon excitation fluorescence microscopy (TPFM) [5

J. G. White and W. B. Amos, “Confocal microscopy comes of age,” Nature 328(6126), 183–184 (1987). [CrossRef]

, 6

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990). [CrossRef] [PubMed]

]. 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

V. V. Tuchin, Tissue optics: Light scattering methods and instruments for medical diagnosis (Springer-Verlag, 2006).

], 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

J. B. Pawley, Handbook of Biological Confocal Microscopy (Springer, 2006).

]. 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

P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A 23(12), 3139–3149 (2006). [CrossRef] [PubMed]

] and experimentally [10

P. Theer, M. T. Hasan, and W. Denk, “Two-photon imaging to a depth of 1000 microm in living brains by use of a Ti:Al₂O₃ regenerative amplifier,” Opt. Lett. 28(12), 1022–1024 (2003). [CrossRef] [PubMed]

]. 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

Y. L. Grand, A. Leray, T. Guilbert, and C. Odin, “Non-descanned versus descanned epifluorescence collection in two-photon microscopy: Experiments and Monte Carlo simulations,” Opt. Commun. 281(21), 5480–5486 (2008).

] or using supplementary fiber-optic light collection systems [12

C. J. Engelbrecht, W. Göbel, and F. Helmchen, “Enhanced fluorescence signal in nonlinear microscopy through supplementary fiber-optic light collection,” Opt. Express 17(8), 6421–6435 (2009). [CrossRef] [PubMed]

]. 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

R. Gauderon, P. B. Lukins, and C. J. Sheppard, “Effect of a confocal pinhole in two-photon microscopy,” Microsc. Res. Tech. 47(3), 210–214 (1999). [CrossRef] [PubMed]

]. 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

A. Leray, K. Lillis, and J. Mertz, “Enhanced background rejection in thick tissue with differential-aberration two-photon microscopy,” Biophys. J. 94(4), 1449–1458 (2008). [CrossRef] [PubMed]

] 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

A. Leray, C. Odin, and Y. L. Grand, “Out-of-focus fluorescence collection in two-photon microscopy of scattering media,” Opt. Commun. 281(24), 6139–6144 (2008). [CrossRef]

]. 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

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE IS 5, 102–111 (1989).

,17

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995). [CrossRef] [PubMed]

]. 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

A. Leray, C. Odin, and Y. L. Grand, “Out-of-focus fluorescence collection in two-photon microscopy of scattering media,” Opt. Commun. 281(24), 6139–6144 (2008). [CrossRef]

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

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profile of an ultrashort pulsed Gaussian beam,” IEEE J. Quant. Electron. 33(4), 566–573 (1997). [CrossRef]

–20

X. Fu, H. Guo, W. Hu, and S. Yu, “Spatial nonparaxial correction of the ultrashort pulsed beam propagation in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5), 056611 (2002). [CrossRef] [PubMed]

] 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

I (r, z, t) = P_{0}^{2} \frac{w_{0}^{2}}{w {(z)}^{2}} \exp (- \frac{2 {(t - \frac{r^{2}}{2 c R (z)} - \frac{z}{c})}^{2}}{τ_{0}^{2}}) \exp (- 2 (\frac{r^{2}}{w {(z)}^{2}} + \frac{r^{4}}{w {(z)}^{4}} \frac{1}{{(ω_{0} τ_{0})}^{2}})),

(1)

where P₀ is a constant value. w₀ = 0.61λ/θ_NA is the diffraction-limited beam waist at the focal plane, which is different from the Gaussian beam waist (w0 = λ/nπθ_NA), where θ_NA = sin⁻¹(NA/n); w(z) = w₀(1 + (z/z_r)²)^0.5 is beam waist at a different depth z and R(z) = z + z_r²/z is the radius of curvature where z_r is the Rayleigh range, r = (x² + y²)^0.5 is the distance from the optical axis, c is the speed of light, τ₀ is the pulse width, and ω₀ is the carrier frequency. We use the diffraction-limited beam waist instead of the Gaussian beam waist, because it is a better estimate for the focal spot size. For example, when the wavelength is 810 nm, NA is 1 and the refractive index is 1.33, the Gaussian beam waist is approximately 230 nm whereas the diffraction-limited beam waist is approximately 580 nm. Note that in Eq. (1), the expression of the intensity distribution includes both the spatial wavefront of the pulse and the temporal shapes of the pulse on propagation in Gaussian form with unchanged pulse duration. This is importantly different from the CW Gaussian beam approximation.

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

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990). [CrossRef] [PubMed]

, 21

C. Xu and W. W. Webb, “Measurement of two-photon excitation cross sections of molecular fluorophores with data from 690 to 1050 nm,” J. Opt. Soc. Am. B 13(3), 481–491 (1996). [CrossRef]

]

Q (r, z, t) \propto \int_{v} \int_{- \infty}^{\infty} q (r, t) d t d V, q (r, t) = \frac{δ C (r, t) I^{2} (r, t)}{h c λ} \frac{1}{τ_{0} F^{2}},

(2)

where q(r,t) is the spatial density that depends on the two-photon absorption cross section (δ), the spatial and temporal concentration of the fluorophore (C(r,t)), the plank constant (h), the excitation wavelength (λ), the pulse repetition rate (F), and the spatial and temporal intensity distribution of the excited pulse (I²(r, t)). When a focused Gaussian beam propagates through a highly scattering medium, such as biological tissues, the light deviates from its original beam path owing to scattering. Thus, light scattering is expected to increase as the amount of light penetrating into the tissue increases, which incrementally contributes to the generation of out-of-focus background TPEF. The total intensity with the scattering coefficient μ_s becomes the sum of ballistic (I_b) and scattered light intensity (I_s) distributions, and is given by

I_{t} (r, z, t) = I_{b} (r, z, t) + I_{s} (r, z, t),

(3)

Where

I_{b} (r, z, t) = I (r, z, t) \exp (- μ_{s} z) .

(4)

As the imaging depth increases, the ballistic intensity distribution decreases exponentially and the scattered light intensity simultaneously increases, which corresponds to a decrease in a portion of the ballistic intensity.

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

P. Theer, “On the fundamental imaging-depth limit in two-photon microscopy,” Dissertation, University of Heidelberg (2004).

]

I_{s} (r, z, t) = P_{0}^{2} \frac{1 - \exp (- μ_{s} z)}{π w_{s} {(z)}^{2}} \exp (- \frac{2 r^{2}}{w_{s} {(z)}^{2}}) \exp (- \frac{2 t^{2}}{τ_{s} {(z)}^{2}}),

(5)

where τ_s vis the scattered temporal pulse width and w_s is the transverse spatial beam waist, which are calculated in the small-angle diffusion approximation [23

J. W. McLean, J. D. Freeman, and R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37(21), 4701–4711 (1998). [CrossRef] [PubMed]

]. To determine the temporal and spatial distributions of the scattered ultra-short-pulse light propagation, the scattered spatio-temporal intensity distribution I_s was convolved with the laser pulse profile at the front lens of the microscope (f(t)) and written as [24

A. K. Dunn, V. P. Wallace, M. Coleno, M. W. Berns, and B. J. Tromberg, “Influence of optical properties on two-photon fluorescence imaging in turbid samples,” Appl. Opt. 39(7), 1194–1201 (2000). [CrossRef] [PubMed]

]

I'_{s} (r, z, t) = \int_{- \infty}^{\infty} I_{s} (r, z, t') f (t - t') d t' .

(6)

Using Eqs. (4) and (6), we plotted numerically calculated spatial distributions of the ballistic (left) and the scattered (right) intensities normalized by each maximum along the incremental 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, as shown in Fig. 1 , where the timescales of the ballistic and scattered light distributions are different to each other: −1 ps to 3 ps and −6 ps to 6 ps, respectively. Because the light intensity distributions for scattered light of propagation depth z are overlapped, we ignored the on-axis term z/c in Eq. (4) and considered only the relative delay term with respect to that of the on-axis pulses to evaluate the magnitude of the delay for plot. Most of the previous studies did not consider the spatial and temporal wavefront of the pulse that pass through the lens and simply assume it as flat in shape. However, the actual Gaussian pulse has the spatial and temporal shape on propagation as shown in the Fig. 1.

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.

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:

\tan^{- 1} (\frac{- r_{o} - r}{z}) \leq θ (r, z) \leq \tan^{- 1} (\frac{r_{o} - r}{z}),

(7)

where r_o is the radius of the OFA. As the fraction of light that reaches the detector through it, the angle of the fluorescence light that passes through and is refracted by the objective should be smaller than the angular acceptance range θ_p which can be determined by the pinhole size r_p and the microscope system magnification M:

\tan^{- 1} (\frac{r - f_{o} \tan (θ_{p})}{f_{o} - z}) \leq θ (r, z) \leq \tan^{- 1} (\frac{r + f_{o} \tan (θ_{p})}{f_{o} - z}),

(8)

where θ_p = tan⁻¹(r_p/Mf_o) and M = f_t/f_o, f_o and f_t being the focal lengths of the objective and tube lens of the microscope, respectively. These two constraints make the acceptance angle maximum at just above the focal point and this angle gradually decreases away from focal point (Fig. 2). It can be collected only when the angles obtained from Eq. (8) are between the ranges of angles specified from Eq. (7). Equations (7)-(8) explain why the collection efficiency for above and below the focal plane is reduced; especially when the acceptance angle above the focal point θ(0, z₁) is smaller than the maximum acceptance angle θ(0, f) = θ_NA. The conventional two-photon microscopes can be thought of as a large-sized pinhole system because the limited size of the detector acts as a relatively large pinhole. Owing to the fixed size of the detector, the large field of view with low magnification is essential for increasing the angular acceptance range and the TPEF collection of the microscopy system.

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.

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

η_{n s} (0, z) = \frac{Ω_{o n - a x i s} (z)}{4 π} = \frac{1}{2} (1 - \cos (θ (0, z))),

(9)

where θ is the previously calculated acceptance angle. For an off-axis source, an orthogonal projection of the OFA (Fig. 3 (a) , green ellipsoid) is relevant, which scales with the cosine of the increasing off-axis angle γ. This can be calculated by substituting the acceptable area radius r_a which is the portion of OFA (indicated as blue and red circle in Fig. 3(a)) determined by previously calculated acceptance angles using Eqs. (7)-(8) with the minor axis radius γ_r = r_acos(γ) and the off-axis angle γ = (θ₁- θ₂)/2, where θ₁ and θ₂ are the acceptance angles. Approximating the solid angle of the ellipse to those of a circle with the same area (S in Fig. 3 (b)) gives the off-axis collection efficiency as

η_{n s} (r, z) = \frac{Ω_{m} (z)}{4 π} = \frac{1}{2} (1 - \cos (arc \tan (\frac{r_{m}}{R}))) = \frac{1}{2} (1 - \cos (arc \tan (\frac{\sqrt{r_{a} r_{γ}} \cos γ}{z}))),

(10)

where Ω_m(r, z) is the modified solid angle, r_m is the identical circle radius (Fig. 3(b) yellow circle), and R is the distance from the source to the illuminated center of the OFA.

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).

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

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001). [CrossRef] [PubMed]

]. 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.

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

E. Beaurepaire and J. Mertz, “Epifluorescence collection in two-photon microscopy,” Appl. Opt. 41(25), 5376–5382 (2002). [CrossRef] [PubMed]

] 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

E. Beaurepaire and J. Mertz, “Epifluorescence collection in two-photon microscopy,” Appl. Opt. 41(25), 5376–5382 (2002). [CrossRef] [PubMed]

, 27

J. D. Jackson, “Classical Electrodynamics,” John Wiley & Sons, Inc., New York, (1999).

] and is given by

G (r, z) = \frac{Q}{4 π D} (\frac{1}{\sqrt{r^{2} + {(z_{0} - z)}^{2}}} - \frac{1}{\sqrt{r^{2} + {(z_{0} + z)}^{2}}}),

(11)

where Q is the number of TPEF photons calculated by Eq. (2), D is the effective diffusion coefficient, and z₀ is the axial position of the source fluorescence. The TPEF surface density distribution F(r,z) can be calculated by the gradient of the density distribution at the surface and is simply given by

F (r, z) = {D \frac{\partial}{\partial z} G (r, z) |}_{z = 0} = \frac{Q}{2 π} \frac{z_{0}}{{(r^{2} + z_{0}^{2})}^{3 / 2}} .

(12)

Using Eq. (10), we calculated the non-scattering collection efficiency according to the pinhole size at the top surface (OFA), where no scattering occurs. Assuming that the light hitting the top surface has lost its memory of direction (isotropic escape) and only has a forward direction portion, we can apply this equation to the scattered TPEF case given by

F_{p} (r, z_{0}) = {D \frac{\partial}{\partial z} G (r, z) \frac{Ω_{m} (r, z = 0)}{2 π} |}_{z = 0} = \frac{Q}{2 π} \frac{z_{0}}{{(r^{2} + z_{0}^{2})}^{3 / 2}} \frac{Ω_{m} (r, 0)}{2 π},

(13)

where Ω_m(r, 0) is the previously calculated solid angle at the top surface. Then, the on-axis collection efficiency of the scattered TPEF can be calculated by integration of Eq. (13) multiplied by the fraction permitted by the angular acceptance of the objective lens

η_{n s} (0, z_{0}) = \frac{Ω (0, z_{0})}{4 π} \int F_{p} (r, z_{0}) 2 π r d r,

(14)

where Ω(0, z₀) = 2π(1-cos(θ_z) is the on-axis solid angle according to the source depth and θ_z = tan⁻¹(r_o/z₀) is the angular acceptance. The off-axis scattered TPEF collection efficiency is similar to that of the non-scattering case, except for the acceptable area of the OFA. In the non-scattering case, the acceptable area is limited by the acceptance angle, which is typically smaller than the OFA area; however, in the scattering case, the acceptable area is fixed to the OFA area. The solid angle Ω(0, z₀) depends on the OFA radius r_o, the source distance R, and the off-axis angle γ. Furthermore, assuming that the solid angle of the ellipse is identical to the area of the circle, we can obtain the off-axis scattered TPEF collection efficiency as

\begin{array}{l} η_{s} (r, z_{0}) = \frac{Ω (r, z_{0})}{4 π} \int F_{p} (r, z_{0}) 2 π r d r \\ = \frac{1}{2} (1 - \cos (\tan^{- 1} (\frac{r_{s m}}{z_{0}} \cos γ))) \int F_{p} (r, z_{0}) 2 π r d r, \end{array}

(15)

where

r_{s m} = \sqrt{r_{o} r_{γ}}

is the scattering case with an identical circle radius,

r_{s m} = r_{o} \cos γ

, and

γ = \tan^{- 1} (r / z_{0})

. Finally, the total TPEF collection efficiency in a scattering medium is the sum of ballistic (η_ns) and scattered (η_s) components multiplied by their portion:

η (r, z) = η_{n s} \exp (- μ_{f} R) + η_{s} (1 - \exp (- μ_{f} R)) .

(16)

Figure 5 shows a comparison of spatially distributed total collection efficiencies of 20-μm pinhole and non-pinhole calculated for the scattering coefficient of 42 cm⁻¹ calculated by Mie theory [28

C. Matzler, Matlab Functions for Mie Scattering and Absorption (Institut fur Angewandte Physik: University of Heidelberg 2004).

] for a 1-μm-diameter polystyrene microsphere at the fluorescence wavelength of 585 nm and the axial profile and its ratio are also plotted in semi-logarithmic scale in Fig. 6 . The overall collection efficiency of a finite-sized pinhole is lower than that of a non-pinhole about 4 orders of magnitude except for focus region because the pinhole rejects not only out-of-focus background but also useful in-focus TPEF. However, relatively improved collection efficiency at the focal point will lead to an improvement in the SNR because the collection efficiency at the focal point is more than approximately two orders of magnitude larger than those of the surface. Note that there are significant differences with previous work [15

A. Leray, C. Odin, and Y. L. Grand, “Out-of-focus fluorescence collection in two-photon microscopy of scattering media,” Opt. Commun. 281(24), 6139–6144 (2008). [CrossRef]

] with ours especially in terms of collection efficiency with finite-sized pinhole. In the previous work, the collection efficiency increases up to the first 400-500 μm from the surface and then decreases with no gain in the focus, while our results indicate that collection efficiency stays constant or slowly decreases for the first 100-200 μm and then decreases until significant gain in the focus region. Although Leray’s model have succeeded in accurately simulating photon propagation through scattering samples and the post-objective collection pathway, inaccurate simple lens approximations have been used to model the microscope objective [29

J. P. Zinter and M. J. Levene, “Maximizing fluorescence collection efficiency in multiphoton microscopy,” Opt. Express 19(16), 15348–15362 (2011). [CrossRef] [PubMed]

]. This approximation has led results of previous work to significantly deviate from ours.

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.

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.

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

F_{s} \propto \int η (r, z) I_{t} (r, z) 2 π r d r .

(17)

A semi-logarithmic representation of the TPEF excitation and collection of 20-μm pinhole and non-pinhole is shown in Fig. 7 . The corresponding ratio of the 20-μm pinhole and non-pinhole is also shown for comparison. Data were normalized by TPEF excitation intensity at the surface (z = 0). The simulation tends to show results very similar to those obtained in a previous work using a Monte Carlo simulation [15

A. Leray, C. Odin, and Y. L. Grand, “Out-of-focus fluorescence collection in two-photon microscopy of scattering media,” Opt. Commun. 281(24), 6139–6144 (2008). [CrossRef]

], the exception being the focus; this difference in focus is mainly due to the difference in the scattering coefficients (intralipid solution μ_f = 55 – 80cm⁻¹) and the previously explained limitations of objective lens approximations. Let’s consider excitation and emission process with pinhole placement. While the excitation intensities drop immediately from the surface, the TPEF power increases slightly up to some depth in the sample. This can be explained by collection efficiency plotted in Fig. 5 which shows sharp decrease near the surface in the lateral direction and slightly decrease at the first 100-200 μm and then steeply until in front of focus in the axial direction.

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).

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

C. Matzler, Matlab Functions for Mie Scattering and Absorption (Institut fur Angewandte Physik: University of Heidelberg 2004).

], 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

W. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169(3), 391–405 (1993). [CrossRef]

]. 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.

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

The SNR of excitation is slightly lower than the one in previous studies [10

,15

A. Leray, C. Odin, and Y. L. Grand, “Out-of-focus fluorescence collection in two-photon microscopy of scattering media,” Opt. Commun. 281(24), 6139–6144 (2008). [CrossRef]

], 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

S. G. Parra, T. H. Chia, J. P. Zinter, and M. J. Levene, “Multiphoton microscopy of cleared mouse organs,” J. Biomed. Opt. 15(3), 036017 (2010). [CrossRef] [PubMed]

], 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.

Appendices

Appendix

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

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

(18)

where

\vec{r} = {\hat{e}}_{x} x + {\hat{e}}_{y} y

are the transverse coordinates and

{\hat{e}}_{x}

{\hat{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,

[\nabla_{⊥}^{2} + \frac{\partial^{2}}{\partial t^{2}} + 2 i k (ω) \frac{\partial}{\partial z}] \tilde{U} (\vec{r}, z, ω) = 0,

(19)

where

k = ω / c

is the wave number and

[\nabla_{⊥}^{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

E (\vec{r}, z, t) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} \tilde{U} (\vec{r}, z, ω) \exp (- i ω τ) d ω .

(20)

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

| \frac{\partial}{\partial z} \tilde{U} (\vec{r}, z, ω) | ≪ | k (ω) \tilde{U} (\vec{r}, z, ω) |,

(21a)

| \frac{\partial^{2}}{\partial z^{2}} \tilde{U} (\vec{r}, z, ω) | ≪ | k (ω) \frac{\partial}{\partial z} \tilde{U} (\vec{r}, z, ω) |,

(21b)

which indicates that the paraxial approximation is satisfied for each frequency component (

ω

), the paraxial propagation equation in the temporal solution for the transverse components can be derived, i.e.,

[\nabla_{⊥}^{2} + 2 i k (ω) \frac{\partial}{\partial z}] \tilde{U} (\vec{r}, z, ω) = 0,

(22)

The analytical solution for the paraxial beam can be derived only for some specific cases, such as a Gaussian beam. Suppose

\tilde{U} (\vec{r}, z, ω)

is the lowest solution of a Gaussian beam, which satisfies Eq. (22), i.e.,

\tilde{U} (\vec{r}, z, ω) = \frac{i z_{0}}{q (z)} \exp (- i k \frac{r^{2}}{2 q (z)}) P (ω),

(23)

where q(z) is the so-called complex beam parameters q(z) = z + iz₀, z₀ is the Rayleigh range, and

P (ω)

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

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profile of an ultrashort pulsed Gaussian beam,” IEEE J. Quant. Electron. 33(4), 566–573 (1997). [CrossRef]

–20

] as

E (\vec{r}, z, τ) = \frac{i z_{0}}{q (z)} P (τ'), P (τ') = \frac{1}{2 π} \int_{- \infty}^{\infty} P (ω) \exp (i ω (\frac{{\vec{r}}^{2}}{2 q c} - τ)) d ω

(24)

where

τ'

is the reduced time

τ' = τ - r^{2} / 2 c q (z)

, and

P (τ')

is the complex representation of the pulse. In [19

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58(1), 1086–1093 (1998). [CrossRef]

P (τ)

is given by

P (τ) = \exp (- \frac{τ^{2}}{τ_{0}^{2}}) {\exp (i ω_{0} t) - i Im [\exp (i ω_{0} τ) e r f c (\frac{τ_{0}}{2} + i \frac{τ}{τ_{0}})]},

(25)

where

ω_{0}

is the carrier frequency. Now, only considering the real term of Eq. (25) that is the same results as that in [18

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profile of an ultrashort pulsed Gaussian beam,” IEEE J. Quant. Electron. 33(4), 566–573 (1997). [CrossRef]

], we obtain

\begin{array}{l} E (\vec{r}, z, τ) = P_{0} \frac{w_{0}}{w (z)} \exp (- \frac{{(t - \frac{r^{2}}{2 c R (z)} - \frac{z}{c})}^{2}}{τ_{0}^{2}}) \exp (- \frac{r^{2}}{w {(z)}^{2}} + \frac{r^{4}}{w {(z)}^{4}} \frac{1}{{(ω_{0} τ_{0})}^{2}}) \\ \times \exp (i ω_{0} (t - \frac{z}{c} - \frac{r^{2}}{2 c R (z)}) (1 - \frac{2 r^{2}}{w {(z)}^{2}} \frac{1}{ω_{0} τ_{0}})) \exp (i ξ (z)), \end{array}

(26)

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

ξ (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)

\begin{array}{l} I (\vec{r}, z, t) = {| E (\vec{r}, z, t) |}^{2} \\ = P_{0} \frac{w_{0}^{2}}{w {(z)}^{2}} \exp (- \frac{2 {(t - \frac{r^{2}}{2 c R (z)} - \frac{z}{c})}^{2}}{τ_{0}^{2}}) \exp (- \frac{2 r^{2}}{w {(z)}^{2}} + \frac{2 r^{4}}{w {(z)}^{4}} \frac{1}{{(ω_{0} τ_{0})}^{2}}) . \end{array}

(27)

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.

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.

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.

References and links

1.	S. Paddock, “Tech.Sight. Optical sectioning--slices of life,” Science 295(5558), 1319–1321 (2002). [CrossRef] [PubMed]
2.	J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005). [CrossRef] [PubMed]
3.	F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005). [CrossRef] [PubMed]
4.	W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003). [CrossRef] [PubMed]
5.	J. G. White and W. B. Amos, “Confocal microscopy comes of age,” Nature 328(6126), 183–184 (1987). [CrossRef]
6.	W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990). [CrossRef] [PubMed]
7.	V. V. Tuchin, Tissue optics: Light scattering methods and instruments for medical diagnosis (Springer-Verlag, 2006).
8.	J. B. Pawley, Handbook of Biological Confocal Microscopy (Springer, 2006).
9.	P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A 23(12), 3139–3149 (2006). [CrossRef] [PubMed]
10.	P. Theer, M. T. Hasan, and W. Denk, “Two-photon imaging to a depth of 1000 microm in living brains by use of a Ti:Al₂O₃ regenerative amplifier,” Opt. Lett. 28(12), 1022–1024 (2003). [CrossRef] [PubMed]
11.	Y. L. Grand, A. Leray, T. Guilbert, and C. Odin, “Non-descanned versus descanned epifluorescence collection in two-photon microscopy: Experiments and Monte Carlo simulations,” Opt. Commun. 281(21), 5480–5486 (2008).
12.	C. J. Engelbrecht, W. Göbel, and F. Helmchen, “Enhanced fluorescence signal in nonlinear microscopy through supplementary fiber-optic light collection,” Opt. Express 17(8), 6421–6435 (2009). [CrossRef] [PubMed]
13.	R. Gauderon, P. B. Lukins, and C. J. Sheppard, “Effect of a confocal pinhole in two-photon microscopy,” Microsc. Res. Tech. 47(3), 210–214 (1999). [CrossRef] [PubMed]
14.	A. Leray, K. Lillis, and J. Mertz, “Enhanced background rejection in thick tissue with differential-aberration two-photon microscopy,” Biophys. J. 94(4), 1449–1458 (2008). [CrossRef] [PubMed]
15.	A. Leray, C. Odin, and Y. L. Grand, “Out-of-focus fluorescence collection in two-photon microscopy of scattering media,” Opt. Commun. 281(24), 6139–6144 (2008). [CrossRef]
16.	S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE IS 5, 102–111 (1989).
17.	L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995). [CrossRef] [PubMed]
18.	Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profile of an ultrashort pulsed Gaussian beam,” IEEE J. Quant. Electron. 33(4), 566–573 (1997). [CrossRef]
19.	M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58(1), 1086–1093 (1998). [CrossRef]
20.	X. Fu, H. Guo, W. Hu, and S. Yu, “Spatial nonparaxial correction of the ultrashort pulsed beam propagation in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5), 056611 (2002). [CrossRef] [PubMed]
21.	C. Xu and W. W. Webb, “Measurement of two-photon excitation cross sections of molecular fluorophores with data from 690 to 1050 nm,” J. Opt. Soc. Am. B 13(3), 481–491 (1996). [CrossRef]
22.	P. Theer, “On the fundamental imaging-depth limit in two-photon microscopy,” Dissertation, University of Heidelberg (2004).
23.	J. W. McLean, J. D. Freeman, and R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37(21), 4701–4711 (1998). [CrossRef] [PubMed]
24.	A. K. Dunn, V. P. Wallace, M. Coleno, M. W. Berns, and B. J. Tromberg, “Influence of optical properties on two-photon fluorescence imaging in turbid samples,” Appl. Opt. 39(7), 1194–1201 (2000). [CrossRef] [PubMed]
25.	M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001). [CrossRef] [PubMed]
26.	E. Beaurepaire and J. Mertz, “Epifluorescence collection in two-photon microscopy,” Appl. Opt. 41(25), 5376–5382 (2002). [CrossRef] [PubMed]
27.	J. D. Jackson, “Classical Electrodynamics,” John Wiley & Sons, Inc., New York, (1999).
28.	C. Matzler, Matlab Functions for Mie Scattering and Absorption (Institut fur Angewandte Physik: University of Heidelberg 2004).
29.	J. P. Zinter and M. J. Levene, “Maximizing fluorescence collection efficiency in multiphoton microscopy,” Opt. Express 19(16), 15348–15362 (2011). [CrossRef] [PubMed]
30.	W. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169(3), 391–405 (1993). [CrossRef]
31.	S. G. Parra, T. H. Chia, J. P. Zinter, and M. J. Levene, “Multiphoton microscopy of cleared mouse organs,” J. Biomed. Opt. 15(3), 036017 (2010). [CrossRef] [PubMed]

OCIS Codes
(110.2990) Imaging systems : Image formation theory
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(190.4180) Nonlinear optics : Multiphoton processes
(110.0113) Imaging systems : Imaging through turbid media
(180.4315) Microscopy : Nonlinear microscopy

ToC Category:
Microscopy

History
Original Manuscript: June 18, 2012
Revised Manuscript: August 13, 2012
Manuscript Accepted: August 15, 2012
Published: August 23, 2012

Virtual Issues
Vol. 7, Iss. 10 Virtual Journal for Biomedical Optics

Citation
Woosub Song, Jihoon Lee, and Hyuk-Sang Kwon, "Enhancement of imaging depth of two-photon microscopy using pinholes: Analytical simulation and experiments," Opt. Express 20, 20605-20622 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-18-20605

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References

S. Paddock, “Tech.Sight. Optical sectioning--slices of life,” Science295(5558), 1319–1321 (2002). [CrossRef] [PubMed]
J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods2(12), 920–931 (2005). [CrossRef] [PubMed]
F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2(12), 932–940 (2005). [CrossRef] [PubMed]
W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21(11), 1369–1377 (2003). [CrossRef] [PubMed]
J. G. White and W. B. Amos, “Confocal microscopy comes of age,” Nature328(6126), 183–184 (1987). [CrossRef]
W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248(4951), 73–76 (1990). [CrossRef] [PubMed]
V. V. Tuchin, Tissue optics: Light scattering methods and instruments for medical diagnosis (Springer-Verlag, 2006).
J. B. Pawley, Handbook of Biological Confocal Microscopy (Springer, 2006).
P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A23(12), 3139–3149 (2006). [CrossRef] [PubMed]
P. Theer, M. T. Hasan, and W. Denk, “Two-photon imaging to a depth of 1000 microm in living brains by use of a Ti:Al2O3 regenerative amplifier,” Opt. Lett.28(12), 1022–1024 (2003). [CrossRef] [PubMed]
Y. L. Grand, A. Leray, T. Guilbert, and C. Odin, “Non-descanned versus descanned epifluorescence collection in two-photon microscopy: Experiments and Monte Carlo simulations,” Opt. Commun.281(21), 5480–5486 (2008).
C. J. Engelbrecht, W. Göbel, and F. Helmchen, “Enhanced fluorescence signal in nonlinear microscopy through supplementary fiber-optic light collection,” Opt. Express17(8), 6421–6435 (2009). [CrossRef] [PubMed]
R. Gauderon, P. B. Lukins, and C. J. Sheppard, “Effect of a confocal pinhole in two-photon microscopy,” Microsc. Res. Tech.47(3), 210–214 (1999). [CrossRef] [PubMed]
A. Leray, K. Lillis, and J. Mertz, “Enhanced background rejection in thick tissue with differential-aberration two-photon microscopy,” Biophys. J.94(4), 1449–1458 (2008). [CrossRef] [PubMed]
A. Leray, C. Odin, and Y. L. Grand, “Out-of-focus fluorescence collection in two-photon microscopy of scattering media,” Opt. Commun.281(24), 6139–6144 (2008). [CrossRef]
S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE IS 5, 102–111 (1989).
L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed.47(2), 131–146 (1995). [CrossRef] [PubMed]
Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profile of an ultrashort pulsed Gaussian beam,” IEEE J. Quant. Electron.33(4), 566–573 (1997). [CrossRef]
M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E58(1), 1086–1093 (1998). [CrossRef]
X. Fu, H. Guo, W. Hu, and S. Yu, “Spatial nonparaxial correction of the ultrashort pulsed beam propagation in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(5), 056611 (2002). [CrossRef] [PubMed]
C. Xu and W. W. Webb, “Measurement of two-photon excitation cross sections of molecular fluorophores with data from 690 to 1050 nm,” J. Opt. Soc. Am. B13(3), 481–491 (1996). [CrossRef]
P. Theer, “On the fundamental imaging-depth limit in two-photon microscopy,” Dissertation, University of Heidelberg (2004).
J. W. McLean, J. D. Freeman, and R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt.37(21), 4701–4711 (1998). [CrossRef] [PubMed]
A. K. Dunn, V. P. Wallace, M. Coleno, M. W. Berns, and B. J. Tromberg, “Influence of optical properties on two-photon fluorescence imaging in turbid samples,” Appl. Opt.39(7), 1194–1201 (2000). [CrossRef] [PubMed]
M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods111(1), 29–37 (2001). [CrossRef] [PubMed]
E. Beaurepaire and J. Mertz, “Epifluorescence collection in two-photon microscopy,” Appl. Opt.41(25), 5376–5382 (2002). [CrossRef] [PubMed]
J. D. Jackson, “Classical Electrodynamics,” John Wiley & Sons, Inc., New York, (1999).
C. Matzler, Matlab Functions for Mie Scattering and Absorption (Institut fur Angewandte Physik: University of Heidelberg 2004).
J. P. Zinter and M. J. Levene, “Maximizing fluorescence collection efficiency in multiphoton microscopy,” Opt. Express19(16), 15348–15362 (2011). [CrossRef] [PubMed]
W. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc.169(3), 391–405 (1993). [CrossRef]
S. G. Parra, T. H. Chia, J. P. Zinter, and M. J. Levene, “Multiphoton microscopy of cleared mouse organs,” J. Biomed. Opt.15(3), 036017 (2010). [CrossRef] [PubMed]

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