Temporal focusing is a simple approach for achieving tight, optically sectioned excitation in nonlinear microscopy and multiphoton photo-manipulation. Key applications and advantages of temporal focusing involve propagation through scattering media, but the progressive broadening of the temporal focus has not been characterized. By combining a detailed geometrical optics model with Monte-Carlo scattering simulations we introduce and validate a simulation strategy for predicting temporal focusing characteristics in scattering and non-scattering media. The broadening of the temporal focus width with increasing depth in brain tissue is studied using both simulations and experiments for several key optical geometries, and an analytical approximation is found for the dependence of this broadening on the microscope’s parameters in a transparent medium. Our results indicate that a multiphoton temporal focus has radically different broadening characteristics in deep tissue than those of a spatial focus.
©2011 Optical Society of America
CorrectionsHod Dana and Shy Shoham, "Numerical evaluation of temporal focusing characteristics in transparent and scattering media: erratum," Opt. Express 20, 28281-28281 (2012)
Temporal focusing (TF) nonlinear microscopy [1–4] enables optically-sectioned excitation of a thin plane inside a three dimensional (3D) volume without the need to spatially focus the light beam. TF thus allows to benefit from the unique capacity of nonlinear optical methods to achieve microscopic sectioning deep inside scattering biological tissue  (or other scattering media) without being constrained to near-diffraction-limited excitation spots that have to be scanned across regions of interest. In imaging applications, this allows scan-less illumination of relatively large light spots while obtaining optically sectioned images in conventional [1,2,6] and super-resolution  fluorescence, and harmonic generation  microscopy. Moreover, the ability TF provides to decouple the lateral and axial focal dimensions, that are typically jointly determined by the numerical aperture (∝NA−1 and ∝NA−2 respectively for a focused Gaussian beam), also renders it a powerful method for shaping multiphoton light-matter interaction geometries as in bulk micromachining [8,9] and optogenetic neural stimulation [10–12].
Temporal focusing is typically achieved by introducing a diffraction grating into the optical path of a wide-field illumination or image projection system. This configuration leads to a pulse propagation geometry in which the excitation probability is maximal in the image plane, where the spatially segregated paths of the different spectral components in the pulse meet again, and the pulse thus reaches its minimum width (and duration). The elementary sectioning characteristics of temporal focusing systems have been derived and analyzed using basic geometrical optics considerations , as well as a more detailed Fresnel-diffraction models [3,4,13,14]. These analyses were limited to the treatment of cases in which TF light propagates through a non-scattering medium, even though key potential TF applications will certainly involve propagation through scattering biological media. Moreover, the predictions of previous analyses remain incompletely validated and it will probably prove challenging to directly extend them to more complex, practical TF geometries and setups, which optionally also include cylindrical lenses , a second diffraction grating , spatial light modulators (SLM)  or a phase contrast filter .
In this paper, we present the construction, analysis and validation of a combined geometrical optics – Monte Carlo model which quantitatively describes TF phenomena inside transparent or scattering media (e.g., biological tissue). The model evaluates and sums the contributions of individual spectral elements, enables varying the optical system characteristics with relative ease and is readily adaptable to complex geometries because the scattering effects are accounted for using Monte-Carlo simulations. [Note that as in previous TF studies, these simulations only capture the illumination’s sectioning, and not the related problem in imaging applications of emitted light scattering]. In section 2, we describe the computational model and the experimental setup used to test its accuracy. In section 3, we experimentally validate that the model accurately predicts TF performance in scattering biological and non-biological media, and use it to analyze (and find an approximate analytical fit to) the dependence of TF performance on optical magnification, NA, pulse duration, and spot size in a transparent media. We then use the model to examine the predicted performance characteristics in deep brain tissue of a recently introduced cell-matching multiphoton photo-stimulation TF system . We conclude in section 4, with a discussion of the paper’s main findings and limitations.
2.1. Model and simulations
In this section, we present a geometrical optics model for the propagation of a temporally-focused light beam, which analyzes the propagation of individual spectral elements from the objective's front aperture to the focal plane and sums these individual contributions (a conceptually similar decomposition was used in Ref. .). The effect of light scattering is introduced by convolving the light intensities with appropriate scattering kernels computed by a time-resolved Monte-Carlo simulation .
The modeled light propagation scheme of a basic TF setup is illustrated in Fig. 1 (a two-dimensional coordinate system provides a sufficient description for modeling wide-field TF propagation). A delta pulse beam, tilted by an angle α' with respect to the optical axis (z axis), propagates in the direction α', and impinges upon a diffraction grating. A collimating lens (of focal length f1) and a focusing lens (objective, focal length f2) arranged in a 4f configuration image the grating onto the latter’s focal plane with a magnification factor of . After undergoing diffraction by the grating, the collection of rays belonging to each spectral component propagates along a different optical path; near the focal plane, each is imaged onto a line (a plane in 3D) tilted at an angle α relative to the optical axis but propagating at another relative angle, β.
To obtain the relation between α and β we observe first that according to Fermat's principle, all impinging spectral components scan the focal plane simultaneously with a scanning speed  of . On the other hand, from elementary geometrical considerations we obtain that a spectral component which propagates in direction β, tilted by an angle α, scans the focal plane with scanning speed of . Equating these two expressions, we get:
To compute the suggested model and simulate the TF performance in a non-scattering medium we allocated a satisfactory number of spectral elements on the front aperture of the focusing lens (21-57 elements): β values were equally spaced between -βmax and βmax, determined by the focusing lens’s numerical aperture, and the laser Gaussian spectral profile was introduced by assigning different weights to each spectral component, α values for every element were computed according to Eq. (1). To optimize the TF optical sectioning properties while maintaining a high laser power transmission efficiency we chose to model the case where the Gaussian beam's 1/e diameter equals the objective's aperture diameter. A more excessive filling would result in better optical sectioning compared to the presented results, while underfilling of the objective's back aperture would result in a lower effective NA and worsen the optical sectioning. This assumption holds across all the presented model calculations, and allows treating the NA independently of the system's other parameters. Experimentally, this means that when the magnification, the beam diameter or the spectral bandwidth (i.e., pulse duration) is changed, the objective's aperture filling must be compensated.
Each spectral component scans the entire focal plane length l (the laser beam diameter divided by M) and, therefore, it is modeled as a segment with length . The time delays between different spectral components were set in a way that the center of all spectral components would overlap with the center of the TF plane at a specific time. At each time step all the components are moved forward towards their respective β direction and their intensities summed. The model results for a specific laser source were finally obtained by convolving the light intensity distribution computed for a delta pulse beam and the laser pulse’s temporal profile and then squaring the computed intensity to obtain a signal proportional to the fluorescence resulting from two-photon TF excitation.
To introduce scattering effects into the model, we computed scattering kernels, in advance, for various scattering depths using a time-resolved Monte-Carlo simulation  for a medium with a scattering mean free path of 200μm, g = 0.9 and negligible absorption. Upon entering the scattering medium, the different spectral elements’ intensity distributions are convolved with the matching scattering kernels. Since each spectral component has a different orientation as it enters the scattering medium, we rotated the matching scattering kernel by the same angle to precisely simulate the scattering directions.
2.2. Experimental methods
Our TF experimental setup for scanningless illumination is illustrated in Fig. 2 . The experimental setup is based on an upright microscope for illuminating the sample from above and an inverted imaging system for imaging the sample from below without encountering scattering effects on the emitted light. The excitation source was an amplified 800nm ultrafast laser (RegA 9000, pumped and seeded by a Vitesse duo; Coherent), providing 15-150mW of average power at the sample plane at a 150KHz repetition rate (0.1-1µJ/pulse). After passing through a beam expander and an electro-optic modulator (Conoptics), the beam hits a diffraction grating (1200 grooves/mm, Newport Corporation) with . An f = 200mm tube lens (Nikon) was used as a collimating lens and two interchangeable objective lenses (Nikon 60x, NA = 1 and Nikon 40x, NA = 0.8) as focusing lenses for illuminating TF planes with diameters of 50µm or 75µm, respectively. A scattering tissue phantom (described below) was placed near the objective's focal plane. To measure the fluorescent light intensity on the opposite side of the sample we used a second objective lens (Leica 40x NA = 0.8), an imaging lens and a 768x576 pixels CCD camera (UEye 2220SE-M, IDS). The scattering sample and the second objective lens were mounted onto two micromanipulators (MP-285 and MP-225 respectively, Sutter), which were used to move the sample and the detection system to controlled distances from the TF plane with 1 µm steps.
Scattering tissue phantoms  were prepared from a mixture of 968 µl of 0.5% agar (Acumedia Manufacturers Inc) and water with 32 µl of non-fluorescent 1 µm beads (Thermo Scientific). The phantom’s scattering mean free path was measured to be 200 µm by inserting a few 1 µm fluorescent beads (Thermo Scientific) and measuring the two-photon signal attenuation for different phantom depths, using standard two photon laser scanning microscopy (TPLSM). These measurements were fit with an exponential function and were used to construct a calibration graph, which was subsequently used for estimating the thickness of individual scattering phantom slabs (this depth estimation method was validated in previous studies [18,19]). The anisotropy factor of these beads was calculated using Mie theory  to be approximately 0.9 for 800 nm incident light, The phantom was placed on top of a fluorescent sample of 10 µm fluorescent beads (Thermo Scientific), and images were serially collected for different distances from the TF plane (Fig. 2c).
Coronal cortical brain slices of varying thickness were prepared from 15 to 25 days-old Sprague Dawley rats according to a standard surgical and preparation procedure  which was approved by the institutional ethics committee. The slices were similarly placed on top of a fluorescent sample of 10 µm fluorescent beads. The typical scattering mean free path and anisotropy factor for near infrared wavelengths in rat cerebral cortex are also approximately 200µm and 0.9 respectively [5,22,23].
A pulse duration of ~200fs was measured at the laser’s output using an autocorrelator (PulseCheck, APE). Pulse duration at the TF focal plane (after passing through all of the optical components) was estimated to be 325fs by fitting the (scatter-free) TF optical sectioning measurements to model predictions for different pulse durations (see Fig. 2d).
3.1 Model validation
To examine the model’s accuracy we tested its predictions of TF’s central characteristic – the optical sectioning width – in scattering and non-scattering media. Optical sectioning was experimentally measured by axially scanning across the focal plane 10 µm fluorescent beads that were located underneath biological (rat cortical slices) and non-biological scattering media of various thicknesses (see section 2.2 for details). Results of these measurements and model predictions (τ = 325fs) for two different optical setup parameters (different objective lenses) are shown in Fig. 3 . The underlying optical sectioning FWHM (i.e., without the beads – data not shown) increases from 7.25µm to 52µm for the 60x objective and from 16.5µm to 93µm for the 40x objective.
3.2 Dependence on optical parameters in non-scattering media
The results shown in Fig. 3 demonstrate that picking a specific objective changes not only the illuminated cross-section, but also the axial thickness of the TF focal volume. Therefore, an optical setup's NA and magnification could potentially be used to control the characteristics of TF setups: optical sectioning, illuminated volume dimensions, efficient use of laser source power or robustness to scattering. In order to investigate the magnification’s role, we simulated a situation where a constant-size TF plane is illuminated by setups with different magnifications. This scenario may be realized by adding a beam expander before the diffraction grating, which changes the beam diameter so that the ratio of beam diameter to magnification remains constant. The results, shown in Fig. 4 , highlight the fact that the magnification has a significant effect on the TF plane’s width.
Following the simulations above we simulated a more common situation, in which the beam diameter on the grating remained unchanged, and therefore any change in the system's magnification causes a respective change in the TF plane’s diameter. The results are similar to the previous situation and are shown in Fig. 5a . We also checked the effect of changing the beam diameter at the diffraction grating’s surface - this diameter and the system's magnification together define the TF plane’s diameter. By changing the beam diameter while maintaining a constant magnification, a cost-effective solution may be achieved for controlling TF performance. Since the optical sectioning capability of TF, from a geometrical standpoint, is derived from the partial overlap of propagating spectral beamlets, one may expect to attain a better optical sectioning as the illuminated plane becomes smaller. The simulation results for a constant magnification are shown in Fig. 5b. Laser pulse duration also has an effect on TF optical sectioning performance . The predicted, nearly-linear dependence of the optical sectioning FWHM on pulse duration is shown in Fig. 5c.
Arguably, the most interesting question is the importance of using a high NA objective. In contrast to standard scanning techniques, the importance of high NA objectives as a part of TF setup is less clear. Simulation results for TF performance using a constant magnification and different NA objectives are shown in Fig. 5d for scattering and non-scattering media.
3.3 Analytical approximation
In agreement with previous analyses [6,13,15], our model’s results for the TF axial cross-section in a non-scattering medium (see e.g., Fig. 4) are well-fit across a very wide range of parameters by a square-root of a Lorentz-Cauchy function:
3.4 Single-cell excitation optics
As mentioned in the introduction, TF optical systems were recently applied in the development of single-cell optogenetic multiphoton photo-stimulation systems. The experimental setup presented in Ref. . used a combination of a pulse duration of 140fs, very high magnification (180) and a high NA (1.2) to illuminate a 6μm diameter TF plane with a measured axial width (FWHM) of ≈1.6μm. The possibility of using this system for deep-tissue stimulation is an exciting but as yet unexplored possibility. To study the performance of this system at various depths we studied the model’s behavior inside a brain-like scattering medium with a Henyey-Greenstein phase function (g = 0.9).
The simulation FWHM in a non-scattering medium was calculated to be 1.25 µm, slightly better than the experimental value that was reported (≈1.6 µm). In a scattering medium this value remained almost constant up to a scattering depth of 400μm (Fig. 6a ). However, this parameter alone does not fully describe the excitation distribution, since large tails can be seen to develop outside the focal plane. In addition, the excitation peak decays exponentially with a decay parameter of ≈125μm (Fig. 6b). This decay rate is slower than the expected attenuation rate of a TPLSM signal in the same scattering medium (100μm), but is faster than the expected decay for the 60x wide-field illumination geometry described in section 3.1. The combination of exponential decay and loss of localization due to scattering sets the ultimate limit at which such a system is useful.
4. Discussion and conclusions
A new numerical approach for calculating planar TF performance in transparent and scattering media was introduced, studied and shown to agree well with experimental results, both in scattering phantoms and in ex-vivo brain samples. This model and its analytical approximation for non-scattering media (Eq. (3) are likely to prove useful for in-depth analysis and design in the rapidly growing field of TF-based optical systems. As noted above, our analysis focused on the illumination’s sectioning performance because of its universal importance to all kinds of TF optical systems, but a complete evaluation of the performance of imaging TF systems will clearly also require accounting for the scattering of the emitted light.
Our results suggest that TF axial sectioning deteriorates much faster with propagation depth than TPLSM. For example, there is more than a five-fold axial broadening over 500μm of propagation (2.5 scattering mean free paths) for both optical setups studied in Fig. 3. In related TPLSM setups the axial broadening would be on the order of 20% or even less [18,19,24] (in addition, TPLSM fluorescence imaging is relatively insensitive to the effects of scattering on the emitted photons). The narrower illumination spot used for single-cell excitation (section 3.4) displayed an intermediate-level of depth-dependent broadening, but a stronger depth-dependent distortion of the axial cross-section (Fig. 6a) than the other, lower-magnification & lower NA TF setups. The fundamental axial cross section of TF two-photon excitation was previously expressed [6,13,15] by a square-root of a Lorentz-Cauchy function (Eq. (2), which is consistent with our model results in a non-scattering media. Our simulations suggest that the axial broadening due to scattering is a function of both the optical setup’s and the medium’s parameters and that this functional fit rapidly breaks down, and is inaccurate at depths greater than 100μm (half a scattering mean free path - e.g., Fig. 6a). The peak attenuation rate due to scattering also depends on both the optical and the medium parameters. As seen in Fig. 6b, planar TF shows a significantly slower attenuation than TPLSM, which accelerates for the narrower illumination geometries used for single-cell excitation. Overall, these results highlight the differences between the behavior of TF and TPLSM as light propagates into a scattering medium, extending the clear differences between these methods’ behavior in transparent media. Fundamentally, TF has more degrees of freedom than TPLSM (see, e.g., Eq. (3), and can achieve tight optical sectioning even with a moderate NA microscope, by using higher magnification, shorter pulse duration or a smaller laser beam diameter. The complex behavior seen in the results of TF simulations suggests that full model calculations will be required in order to predict the performance of specific optical systems in a scattering medium.
Our approach is based on geometrical considerations and ray-tracing and, therefore, does not take into account any phase dependent effects, such as light diffraction. This limitation does not generally play a significant role for the prediction of optical sectioning, as argued previously  and as seen in the good agreement between the empirical measurements and the model’s predictions. Recently, methods for Monte-Carlo simulation of optical electrical field propagation were introduced, which allow such diffraction effects to be simulated [25–27]. Extending our approach along those lines may improve the model's precision and extend its applicability to include diffractive optical systems. Another assumption that was used in our model in order to strike a balance between TF optical sectioning and laser transmission, is that the objective back aperture is always filled by the incoming light beam (similar to the overfill of the objective back aperture typically used in TPLSM ). This assumption effectively couples the TF optical setup parameters, such as the NA, magnification, beam diameter, grating period and collimating lens focal distance, so that a change in a single parameter imposes changes in the others.
The authors wish to thank Suhail Matar, Lior Golan, and two anonymous reviewers for their helpful comments on the manuscript, and the financial support of the European Research Council (starting grant #211055) and the Israel Science Foundation (1248/06).
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