1. Introduction

Scattering of light propagating through inhomogeneous media is a major limitation in many optical applications, ranging from microscopy and nanosurgery to astronomy. When ultrashort pulses are used, scattering also induces temporal distortions in addition to the spatial ones [1,2], hindering their use in nonlinear processes. Understanding and correcting the spatiotemporal distortions of ultrashort pulses is crucial when multiphoton processes are involved, e.g. in nonlinear microscopy and quantum coherent control experiments [3–5], as these require maintaining a tight spatiotemporal focus. Recently, it was shown that the spatiotemporal distortions induced on ultrashort pulses by thick, multiply-scattering media can be corrected by applying only spatial manipulation of the input beam via. wavefront shaping [6, 7], and that temporal distortions can be locally corrected by temporal shaping [8].

Here we consider the spatiotemporal distortions caused by focusing through a thin scattering layer. This scenario is relevant to experiments in which the sample can be considered as highly scattering and diffusive but the optical path differences are shorter than the transform limited (TL) pulse duration, e.g. a diffusing surface. We show that even such a simple medium can induce severe temporal distortions upon propagation. We analyze these distortions and show that the spread of the temporal distortions depend on the distance from the optical axis, and that the pulse is temporally undistorted at the original focusing point, but with a speckled spatial profile. This phenomena, which was introduced in [9] for a diffuser located at the back focal plane of a lens, applies here as well when focusing through a scattering medium. We experimentally measure and demonstrate the correction of these distortions using wavefront shaping [6], by focusing and temporally compressing a pre-chirped ultrashort pulse.

2. Spatiotemporal distortions regimes

When focusing an ultrashort pulse through a scattering medium one can distinguish between two regimes of spatiotemporal distortions (Fig. 1). The first is when the delay between different internal paths inside the medium (Thouless time, T_Thouless) is longer than the TL pulse duration. This corresponds to severe temporal distortions and broadening at every point behind the medium, as result of the multiple-scattering induced path delay spread inside the medium (Fig. 1(b)). The second regime is when T_Thouless is much shorter than the TL pulse duration. In this case, which is applicable to surface scattering and thin samples, although no temporal distortions are present at the output facet of the medium, the pulse may be temporally distorted after further propagation to the focus, as result of geometrical path delay spread (Fig. 1(c)) [10].

 

Fig. 1 Numerical simulation of spatiotemporal distortions when focusing an ultrashort pulse: (a) Free-space focusing: the pulse is perfectly focused in both space and time. (b) Focusing through a thick scattering sample: the pulse is distorted in both space and time, resulting in a spatiotemporal speckle.(c) Focusing through a scattering surface: the wavefront is distorted, producing a spatiotemporal speckle field which is still temporally transform-limited (TL) near the focal spot [9], inset: zoom-in on the spatiotemporal profile near the optical axis, showing a TL pulse with speckled spatial profile.

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To illustrate the differences between the spatiotemporal distortions in these two cases, we present in (Figs. 1(b–c)) results of numerical simulations for a 100 fs pulse which is focused with an f= 16mm lens through a scattering medium and an illustration of the ideal free-space focusing case (Fig. 1(a)). The numerical simulation for the pulse evolution uses the fresnel-huygens propagator under the paraxial approximation for each wavelength in the pulse bandwidth separately and then sums them up to get the full spatiotemporal profile. When T_Thouless ≫ T_TL, the pulse propagates through 9 surface scatterers separated by 1mm, which yields temporal distortions at every point behind the medium, forming a spatiotemporal speckle field (Fig. 1(b)). In the case of T_Thouless ≪ T_TL (Fig. 1(c)), the pulse is scattered from a single surface scatterer, although spatial speckles are apparent right after the scattering surface, no temporal distortions are induced at the output facet of the thin medium, as expected. However, as the scattering distorts the focused spherical wavefront, the pulse at each point behind the surface propagates in multiple directions. The result is that at the focal plane, temporal distortions are dependent on the distance from the optical axis and at the focus, on the optical axis, the pulse does not suffer from any temporal distortions, and is TL (Fig. 1(c) inset). This scenario is the focus of this work and is analyzed in the next sections.

3. Geometric temporal distortions - theoretical and experimental analysis

In this section we analyze the temporal delay spread at the focal plane as result of geometrical path lengths differences induced on a pulse focused through a scattering surface. Consider the case of a lens with a given numerical aperture (NA) focusing an ultrashort pulse through a thin scattering surface, placed at distance z from the lens focal plane as presented in Fig. 2(a). We define r_s as the transverse distance from the optical axis at the scatterer plane (A scattering point), and r_o as the transverse distance at the observation plane (the measurement point). The distance between these two points is $L\left(r_s,r_o\right)=\sqrt{z^2+{\left(r_s-r_o\right)}^2}$. Because prior to hitting the scatterer the pulse had passed through a focusing lens it has a spherical wavefront, which provides a position dependent delay at the scatterer plane. This extra optical path length relative to the path on the optical axis is given by $L_{\mathit{lens}}\left(r_s\right)=z-\sqrt{z^2+r_s^2}$. Hence, the total geometric delay between any point on the scatterer, r_s, to any point at the focal plane r_o is the sum of these two delays divided by the speed of light:

 

Fig. 2 (a) Geometrical sketch of focusing through a scattering surface (e.g. a diffuser). r_s is a scattering point at the diffuser plane and r_o is the distance from the optical axis at the observation plane. (b) Plot of the estimated pulse duration τ_pulse at the focal plane as function of r_o, as expected from Eq. (4). (c) The result at the focal plane as seen by the naked eye: A broadband super-continuum source is focused through a diffusive piece of scotch-tape and a screen is placed at the focal plane. Near the optical axis, at the center of the image, the pulse is temporally undistorted, and the speckle contrast is high. Farther from center, the pulse is distorted, the different spectral contributions do not overlap and the speckle contrast lowers [11], as is visible by the ‘smeared’ speckle pattern [9].

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The maximal temporal spread at the point r_o is the difference between the maximal and minimal delays from the different points on the surface that scatter light to r_o. Since τ_delay is monotonic in r_s, and assuming that the surface scatters from each r_s to all r_o (a surface with a wide scattering angle), the full temporal spread is determined by the spot size on the scatterer plane. This spot size is bounded by the NA of the objective −ztanφ ≤ r_s ≤ ztanφ. Therefore, the maximal geometrical spread is will be given by substituting the spot limitation to Eq. (1).

For r_o ≪ z Eq. (2) can be simplified to linear dependence of r_o

The above estimate is for the additional temporal delay spread induced by the geometry of the system. For estimating the resulting pulse temporal envelope for a finite input pulse duration one needs to convolute the input pulse with the system’s response function. Assuming the input pulse is a gaussian pulse with width given by a standard deviation (std) of τ₀, while the system’s response function has a Gaussian envelope with std of τ_spread, the output’s temporal variance is the sum of variances. This yields the formula for estimating the pulse’s duration at the output:

In order to experimentally verify the spatiotemporal profile of the scattered spatiotemporal distribution, the pulse temporal width should be measured at each point in space behind the medium. To achieve this we have used the experimental setup presented in [6] Fig. 3(a) with a 3M’s scotch tape as a diffuser (60μm thick with a scattering angle of about 5°). The phase only spatial light modulator (SLM) is placed arbitrary before the focusing lens. Scanning the delay of a Michelson interferometer placed before the medium while imaging (with a X20 objective and a 20cm lens) the two-photon fluorescence (2PF) from a 2PF screen placed at the lens’ focus, spatially resolved interferometric autocorrelation is measured on the entire field simultaneously. The intensity autocorrelation at each point is attained by averaging over the autocorrelation temporal fringes, by dithering the interferometer mirror at a faster rate than the camera integration time. A map of the $\frac{1}{\text{e}}$ width of the intensity autocorrelation is shown in Fig. 3(b). We have used this map to plot the radial dependence of pulse’s temporal width at the focal plane. This experimental radial dependence and a fit to the geometrical model prediction of Eq. (4) is plotted in Fig. 3(c), showing excellent agreement to the simple geometrical model. We note that the scattered spatiotemporal distribution highly resembles a spatiotemporal x-wave [9], where temporal distortions are drastic far from the optical axis. This distortion is apparent in the linear intensity measurement with a standard camera or with the naked eye. In such an observation, the intensity distribution at the focal spot will resemble an ’exploding speckle’ [9]. As a demonstration, we have taken a picture of the colorful speckles at the focal plane using a super continuum source which was focused through a piece of scotch-tape using a pocket color camera. Inspection of the image (Fig. 2(c)) reveals that the speckle contrast at the focus is very high (close to one), and gets lower farther from the optical axis, due to spectral dependent ’smearing’ of the speckles. This is an especially striking sight when a wide bandwidth source covering the visible spectrum is used (Fig. 2(c)). The temporal distortions can be estimated directly from the speckle contrast without requiring a nonlinear measurements [11].

 

Fig. 3 Spatiotemporal characterization of the scattered field: (a) Experimental setup: utilizing a Michelson interferometer while imaging the two-photon fluorescence (2PF), spatially resolved autocorrelation is measured on the entire field simultaneously. (b) Map of the temporal $\frac{1}{\text{e}}$ width of the spatially resolved temporal autocorrelation; scale bar: 100μm. (c) Autocorrelation width as function of the distance from the optical axis: blue - experimental measurements, red - fit curve with two fit parameters according to Eq. (4).

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4. Correction of temporal distortion by wavefront shaping

In this section we show that the temporal distortions can be corrected using wavefront shaping [6] and even be used to compress chirped pulses. We apply a setup similar to that of [6] which is demonstrated in Fig. 3(a) where the SLM is imaged to the back-focal-plane of the focusing lens. As shown in [6, 7], this configuration allows temporal control using only spatial wavefront shaping, as each SLM pixel generates a different temporal speckle pattern behind the medium. Unlike the scenario of [6, 7], here the temporal smearing is caused by the geometrical path differences for an off-axis configuration, which envelope is described by Eq. (4) and plotted in Fig. 3(b). These geometrically induced abberations generate the spatiotemporal coupling [6], which is used for temporal control. The spectral resolution of such a basis is $\mathrm{\Delta }f=\frac{1}{{\tau }_{\mathit{spread}}}$ where τ_spead is found in Eq. (3) and the spectral span is determined by the bandwidth of the input pulse. It can be seen that in contrast to the case of multiple scattering medium [6], here the pulse can’t be temporally controlled on the focal spot since the spectral resolution is equal to the bandwidth. Therefore, temporal control can be achieved only due to geometrical elongation, far from the optical axis.

In order to demonstrate the ability to manipulate temporal degrees of freedom by utilizing an SLM and a diffuser, we chose to compress a temporally chirped initial pulse. An experimental indicator to a spatiotemporal focusing is the intensity of the 2PF signal [6]

The experimental setup is the same as the one presented in Fig. 3(a) with an addition of a 152mm glass slab to pre-chirp the pulse. The optimization algorithm maximizes a point on the 2pf screen which is off the optical axis. The pre-chirped pulse is the same as the one discussed in Fig. 3(b). After chirping the pulse one can see that the temporal width shown in Fig. 4(c) changes dramatically to a temporally long pulse all over the imaged area. We then chose a point on the screen at which the scattered temporal width is approximately the same as the temporal width of the chirped pulse for use as the 2PF feedback signal to the optimization algorithm. the optimization yielded the intensity profile shown in Fig. 4(b), where a the emergence of a focused spot at the right lower corner of the picture can be clearly seen (marked by arrow) in comparison to the un-optimized intensity profile Fig. 4(a). The temporal width of this optimized signal in plotted in Fig. 4(d) where it can be seen that the pulse was compressed almost to its TL duration in the optimized spot (marked by arrow).

 

Fig. 4 Experimental results: 2PF intensity profile of the chirped unoptimized scattered field (a) and the optimized chirped scattered field (b), where the arrow marks the optimized point. Temporal autocorrelation width of the chirped unoptimized scattered field (c) and the optimized chirped scattered field (d), where the arrow marks the optimized point. scale bars: 100μm. All the 2PF intensity profiles are normalized to 1. Inset: optimized SLM pattern.

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5. Summary

We have analyzed and measured the spatiotemporal distortions of an ultrashort pulse focused through a thin scattering surface. The fact that at the focal point the pulse remains temporally undistorted is of importance to nonlinear microscopy, as the pulse may remain short even deep inside scattering tissue. A similar result was observed with a 100fs pulse focused through 1mm thick brain tissue in [6]. Moreover, we have shown that surface scattering can couple the spatial and temporal degrees of freedom, and can be used to control the temporal profile of the pulse.