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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 10 — Oct. 5, 2012
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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.

© 2012 OSA

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

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

4

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

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

, 6

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

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

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

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

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

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]

]. 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

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

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 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

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

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

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)vq(r,t)dtdV,q(r,t)=δC(r,t)I2(r,t)hcλ1τ0F2,
(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 μs becomes the sum of ballistic (Ib) and scattered light intensity (Is) distributions, and is given by
It(r,z,t)=Ib(r,z,t)+Is(r,z,t),
(3)
Where
Ib(r,z,t)=I(r,z,t)exp(μsz).
(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.

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
Fig. 2 Schematic diagram of TPEF collection path with pinhole. The figure only shows the on-axis terms. fo and ft are the focal length of the objective and tube lens, ro and rt 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, θp is the angular acceptable range according to the radius of pinhole rp, and θ is the maximum acceptance angle at each depth.
. 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 ro 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:
tan1(rorz)θ(r,z)tan1(rorz),
(7)
where ro 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 θp which can be determined by the pinhole size rp and the microscope system magnification M:
tan1(rfotan(θp)foz)θ(r,z)tan1(r+fotan(θp)foz),
(8)
where θp = tan−1(rp/Mfo) and M = ft/fo, fo and ft 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) = θNA. 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.

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

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]

]. If the fluorescence source is positioned on-axis (r = 0), the collection efficiency η is given by
ηns(0,z)=Ωonaxis(z)4π=12(1cos(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)
Fig. 3 (a) A TPEF collection from an arbitrarily positioned fluorescence source. The solid angle Ω depends on the acceptable area radius ra, 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 (ra rγ = rm2).
, 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 ra 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 γr = racos(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
ηns(r,z)=Ωm(z)4π=12(1cos(arctan(rmR)))=12(1cos(arctan(rarγcosγz))),
(10)
where Ωm(r, z) is the modified solid angle, rm 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.

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
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.
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

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.

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

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.

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

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]

]. 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 μsz are shown in Fig. 9
Fig. 9 Signal-to-noise ratio simulated with 20 μm (blue square) and non-pinhole (red circle) pinholes as a function of scattering MFP μsz. 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 corresponding excitation profile Fexc 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).

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

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.

Appendix

The free-space propagation of an electromagnetic pulse is governed by the wave equation
(21c22t2)E(r,z,t)=0,
(18)
where r=e^xx+e^yy are the transverse coordinates and e^x, 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,
[2+2t2+2ik(ω)z]U˜(r,z,ω)=0,
(19)
where k=ω/c is the wave number and [2=2/x2+2/y2is the transverse Laplacian; the time Fourier transform of the electric field is expressed as
E(r,z,t)=12πU˜(r,z,ω)exp(iωτ)dω.
(20)
By invoking the paraxial approximation in the temporal frequency domain, i.e.,
|zU˜(r,z,ω)||k(ω)U˜(r,z,ω)|,
(21a)
|2z2U˜(r,z,ω)||k(ω)zU˜(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.,
[2+2ik(ω)z]U˜(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 U˜(r,z,ω) is the lowest solution of a Gaussian beam, which satisfies Eq. (22), i.e.,
U˜(r,z,ω)=iz0q(z)exp(ikr22q(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

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

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]

] as
E(r,z,τ)=iz0q(z)P(τ'),P(τ')=12πP(ω)exp(iω(r22qcτ))dω
(24)
where τ'is the reduced time τ'=τr2/2cq(z), and P(τ') is the complex representation of the pulse. In [19

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(τ2τ02){exp(iω0t)iIm[exp(iω0τ)erfc(τ02+iττ0)]},
(25)
where ω0 is the carrier frequency. Now, only considering the real term of Eq. (25) that is the same results as that in [18

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
E(r,z,τ)=P0w0w(z)exp((tr22cR(z)zc)2τ02)exp(r2w(z)2+r4w(z)41(ω0τ0)2)×exp(iω0(tzcr22cR(z))(12r2w(z)21ω0τ0))exp(iξ(z)),
(26)
where w0 is the beam waist at the focal plane, w(z) = w0(1 + (z/zr)2)0.5 is beam waist, R(z) = z + zr2/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)
I(r,z,t)=|E(r,z,t)|2=P0w02w(z)2exp(2(tr22cR(z)zc)2τ02)exp(2r2w(z)2+2r4w(z)41(ω0τ0)2).
(27)
Figure 10
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.
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).

Figure 11
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.
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.

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

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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

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  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. Express17(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. E58(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. B13(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. Methods111(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. Express19(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]

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