Characterization of the spatiotemporal evolution of ultrashort optical pulses using FROG holography

Nikhil Mehta; Chuan Yang; Yong Xu; Zhiwen Liu

doi:10.1364/OE.22.011099

1. Introduction

Accurate characterization of ultrashort optical laser field is important for many applications [1–9]. Typically, it is assumed implicitly, that an ultrashort optical field $E (\vec{r}, t)$ can be separated into a spatial distribution multiplied by an independent temporal evolution function. Common ultrashort pulse characterization techniques [10,11] (e.g., frequency resolved optical gating (FROG), spectral phase interferometry for direct electric-field reconstruction (SPIDER) [12]) make use of this assumption to obtain the pulse width of a “whole” beam. However, the above assumption is invalid if the spatial distribution of optical field is not the same for all frequency components, i.e., $\tilde{E} (\vec{r}, ω) \neq X (\vec{r}) Ψ (ω)$ where $\tilde{E} (\vec{r}, ω) = F .T . [E (\vec{r}, t)] = \int E (\vec{r}, t) e^{i ω t} d t$ and F.T. stands for Fourier transform. For example, coupled spatiotemporal evolution of the ultrashort optical field can occur at the focal point of an objective due to diffraction and chromatic aberration. Multiple scattering of coherent pulsed light through scattering media may also exhibit complex wavelength dependent speckle pattern [13]. For plasmonic systems with both nanoscale modal distribution (i.e., hot spots) as well as a large plasmonic resonance, this assumption is clearly invalid. Yet, plasmonic devices are extremely important for nanoscale focusing of optical field and coherent control of photons in nanometer femtosecond spatiotemporal scale [14]. Therefore, it is clearly important to develop a technique that can unambiguously characterize the spatiotemporal evolution of ultrashort optical pulses.

Spatiotemporal characterization of ultrashort pulses has been demonstrated by spatial-spectral interference [SSI] [15,16]. SEA TADPOLE [17] and STARFISH [18,19] are experimentally simpler implementations of SSI technique using optical fibers. Dorrer et al. [20] reported a technique combining SPIDER with spatial shearing interferometry for measuring both the spectral and spatial phase. STRIPED FISH [21] is a holographic method for single shot measurement of three-dimensional spatiotemporal electric field by recording spatially separated array of quasi-monochromatic holograms. NSOM probes may be used with SEA TADPOLE [22] or other techniques to improve spatial resolution. However, existing NSOM probes have limitations as they require converting local non-propagating field into radiating field traveling in free space or in the NSOM fiber, which could potentially lead to perturbation of the local field itself and hence inaccuracy for measurement at nanoscale. Nano-FROG [23] achieves high spatial resolution using individual second harmonic (SH) nanocrystal clusters dispersed on a substrate; however the presence of the substrate inhibits near field measurements. Multi-pulse Interferometric FROG (MI-FROG) which combines FROG and spectral interferometry is capable of single shot characterization of electric field consisting of multiple ultrashort pulses and obtaining the relative phase and the temporal offset between the pulses [24]. Previously, our group has developed a new class of Second HARmonic nano-Probes (SHARP), consisting of second harmonic nanocrystals attached to carbon nanotubes which are in turn attached to tapered optical fibers [25]. Using SHARP-based collinear FROG (cFROG) measurements, our group has demonstrated the capability to characterize ultrashort optical fields. However, since the pulse profiles at different locations are independently characterized, the relative phase relationship between different locations is undetermined. Critical information such as phase velocity and group velocity therefore cannot be retrieved.

Here we propose a holographic collinear FROG (HcFROG) technique to measure the spatiotemporal evolution of the ultrashort optical field through a combination of FROG and spectral holography. We use a fixed reference field at the SH frequency to record a spectral hologram of the cFROG trace, which provides a phase coherent ‘link’ across multiple locations to enable the measurement of the relative phase relationship between different positions, albeit at the expense of slightly increased system complexity. In the following, we discuss the theory of FROG holography and present a proof-of-concept demonstration by characterizing the temporal-propagation dependence of femtosecond pulses.

2. Theory and pulse retrieval using FROG holography

Our technique of HcFROG is designed for spatiotemporal characterization of the ultrashort optical field. To do this, we propose to interferometrically detect the cFROG signal by using a reference beam at the second harmonic frequency as a local oscillator. Let us consider the field $E (\vec{r}, t) = \tilde{E} (\vec{r}, t) \exp^{- i ω_{0} t}$ at location $\vec{r}$ , where $\tilde{E} (\vec{r}, t) = \sqrt{I (\vec{r}, t)} \exp^{- i ϕ (\vec{r}, t)}$ is the complex amplitude, $I (\vec{r}, t)$ is the intensity, and $ϕ (\vec{r}, t)$ is the phase, and $ω_{0}$ is the center angular frequency. Also, consider the SH reference pulse $E_{R}^{S H G} (t) = {\tilde{E}}_{R}^{S H G} (t) \exp^{- i 2 ω_{0} t},$ where ${\tilde{E}}_{R}^{S H G} (t)$ is its complex amplitude. The corresponding HcFROG trace is given by:

I_{H c F R O G} (\vec{r}, ω, τ) = {| F .T . [κ {(E (\vec{r}, t) + E (\vec{r}, t - τ))}^{2} + E_{R}^{S H G} (t)] |}^{2}

where

κ

is a proportionality constant. Let

Ω = ω - 2 ω_{0}

. We define the following:

F .T . [κ {\tilde{E}}^{2} (\vec{r}, t)] = E_{S H G} (\vec{r}, Ω),

F .T . [2 κ \tilde{E} (\vec{r}, t) \tilde{E} (\vec{r}, t - τ)] = E_{F R O G} (\vec{r}, Ω, τ)

and

F .T . [{\tilde{E}}_{R}^{S H G} (t)] = E_{R} (Ω)

. Equation (1) can then be rewritten as

\begin{matrix} I_{H c F R O G} (\vec{r}, Ω, τ) = | E_{S H G} (\vec{r}, Ω) + E_{S H G} (\vec{r}, Ω) e^{i Ω τ} e^{i 2 ω_{0} τ} + \\ E_{F R O G} (\vec{r}, Ω, τ) e^{i ω_{0} τ} + E_{R} (Ω) |^{2} \end{matrix}

Figure 1(a) shows a simulated HcFROG trace with its characteristic fringe pattern along both dimensions. Its two dimensional F.T. in Fig. 1(b) clearly identifies the HcFROG term, i.e. the 2D F.T. of

H (\vec{r}, Ω, τ) = E_{F R O G} (\vec{r}, Ω, τ) e^{i ω_{0} τ} E_{R}^{*} (Ω)

, which can be digitally filtered.

Fig. 1 (a) Simulated HcFROG trace and (b) its 2D Fourier transform. The d.c. and HcFROG term are highlighted.

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We note that $H (\vec{r}, Ω, τ)$ is linear in $E_{F R O G} (\vec{r}, Ω, τ)$ . Since the reference field is fixed during the entire measurement, the ratio of $H (\vec{r}, Ω, τ)$ at ${\vec{r}}_{i}$ and ${\vec{r}}_{j}$ yields the relative FROG phasor

e^{i Δ Φ ({\vec{r}}_{i}, {\vec{r}}_{j})} = (e^{i Φ_{F R O G} ({\vec{r}}_{i}, Ω, τ)} / e^{i Φ_{F R O G} ({\vec{r}}_{j}, Ω, τ)})

where

Φ_{F R O G} (\vec{r}, Ω, τ)

denotes the phase of

E_{F R O G} (\vec{r}, Ω, τ)

at

\vec{r}

. Thus the common reference field need not be characterized prior to recording the HcFROG trace. We can set one location as the “pivot” location

{\vec{r}}_{p i v o t}

, and compute the relative FROG phasor

e^{i Δ Φ (\vec{r}, {\vec{r}}_{p i v o t})}

at every other location with respect to

{\vec{r}}_{p i v o t}

(Eq. (3)). The conventional iterative algorithm [26] is applied at

{\vec{r}}_{p i v o t}

to compute the time-domain FROG trace,

{\tilde{E}}_{s i g} ({\vec{r}}_{p i v o t}, t, τ) = \tilde{E} ({\vec{r}}_{p i v o t}, t) \tilde{E} ({\vec{r}}_{p i v o t}, t - τ)

from the recorded spectrogram via the 1D F.T. We normalize it by noting that

\int \tilde{E} (\vec{r}, t - τ) d τ = \int \tilde{E} (\vec{r}, t) d t = \sqrt{\iint {\tilde{E}}_{s i g} (\vec{r}, t, τ) d t d τ}

so that

\tilde{E} ({\vec{r}}_{p i v o t}, t) = \sqrt{I ({\vec{r}}_{p i v o t}, t)} \exp^{- i φ ({\vec{r}}_{p i v o t}, t)} = \frac{\int {\tilde{E}}_{s i g} ({\vec{r}}_{p i v o t}, t, τ) d τ}{\sqrt{\iint {\tilde{E}}_{s i g} ({\vec{r}}_{p i v o t}, t, τ) d t d τ}}

is used as the guess for next iteration, until convergence is reached. At every other

\vec{r}

_,the measured trace_{$I_{F R O G} (\vec{r}, Ω, τ)$ and the corresponding relative FROG phasor_{$e^{i Δ Φ (\vec{r}, {\vec{r}}_{p i v o t})}$ is utilized to obtain the complex spectrogram $\sqrt{I_{F R O G} (\vec{r}, Ω, τ)} e^{i Δ Φ (\vec{r}, {\vec{r}}_{p i v o t})} e^{i Φ_{F R O G} ({\vec{r}}_{p i v o t}, Ω, τ)}$ . As before, this is followed by 1D inverse Fourier transform along $Ω$ to yield the signal field ${\tilde{E}}_{s i g} (\vec{r}, t, τ)$ , from which the complex pulse at $\vec{r}$ can be retrieved using Eq. (4) directly. In this way by measuring the HcFROG traces at different locations, the spatiotemporal evolution of $E (\vec{r}, t)$ can be probed. We would like to point out that the FROG algorithm is applied only to the trace measured at ${\vec{r}}_{p i v o t}$ , whereas for every other location the corresponding FROG trace is obtained by utilizing the additional phase information contained in the HcFROG trace measured at that location. Thus our retrieval procedure involves minimal computational overhead.
3. Experimental setup
As a proof-of-concept demonstration of our HcFROG technique, we in situ characterize the pulse near the focus of an objective lens, by using a BaTiO₃ micro-cluster air-dried on a cover glass as the nonlinear probe and traversing through the focus of the objective along its axis. As shown in the setup in Fig. 2(a), a Michelson interferometer with a variable delay arm generates two copies of the fundamental pulse which is obtained from a Ti:Sapphire laser (KMLabs, central wavelength ~818nm, average output power ~400mW, repetition rate 88MHz). The two pulses combined at the output of the interferometer first propagated through a short length (~10cm) of photonic crystal fiber before being focused on the BaTiO3 micro-cluster by a 40 $\times$ objective lens to generate the SH cFROG signal. An SEM image of the micro-cluster is shown in Fig. 2(b). We vary the delay in steps of 0.4fs which is enough to satisfy the Nyquist sampling rate for the cFROG signal, although it has been shown that the delay step may be increased to speed up acquisition without affecting the final retrieved pulse [27]. In our setup we have mounted the PCF output end and the collimating and focusing objectives on the same translational stage (as indicated in Fig. 2(a)). This arrangement effectively allows us to probe the focused femtosecond beam at multiple locations along the objective axis without the need to move the micro-cluster itself or the signal acquisition optics, thereby maintaining the optical path of generated cFROG signal. Additionally, a SH reference pulse (~409nm) is generated from the fundamental beam by using a Barium Borate (BBO) crystal, separated by a dichroic filter, and appropriately delayed to interfere with the cFROG signal. The resulting HcFROG trace is filtered by a band-pass filter (D400/70, Chroma Technology) and detected by a spectrograph (PI Acton SP2500i with a liquid nitrogen cooled charge coupled device camera, resolution: 0.03nm at 409nm). Figure 2(c) shows a typical recorded HcFROG trace which exhibits the characteristic fringe structure along both the delay and wavelength dimensions. We acquire such HcFROG traces at five locations as we traverse through the focus: ‘location 2’ at 9.43 μm, ‘location 3′ at 18.11 μm, ‘location 4’ at 27.37 μm and ‘location 5′ at 37.75 μm with respect to the pivot ‘location 1’.
Fig. 2 (a) Schematic of our experimental setup. (b) The inset shows SEM image of the BaTiO3 micro-cluster used for generating FROG signal. (c) Typical recorded HcFROG trace. The fringe structure can be seen along both delay and wavelength dimension.
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4. Results
We implement the procedure described in section 2 to retrieve the complex pulse profile from the recorded HcFROG trace. In our experiment, among the five locations at which HcFROG trace is acquired, the ‘pivot location’ is chosen to be closest to the focusing objective along the axis. As noted in Fig. 2(a), ‘location 4’ is nearest to the focus. First panel in Fig. 3(a) shows the intensity and phase of the pulse at pivot location retrieved by applying the conventional FROG algorithm [22] at that location. The full width at half maximum (FWHM) of the retrieved pulse is 89 fs (at temporal resolution = 4 fs). At other locations, the relative FROG phasor with respect to pivot is computed as prescribed by Eq. (3). Combining the relative FROG phasor with the FROG trace (extracted digitally from the corresponding experimentally measured HcFROG trace at that location), the resulting complex spectrogram can be simply inverse Fourier transformed and processed to obtain pulse amplitude and phase at that location, as described before. The panels in Fig. 3(a) show pulses retrieved at each location. The pulse generally maintains its amplitude profile, displays slightly different phase profile, and is shifted along the time axis in accordance to the separation of the spatial locations as it travels along axis through the focus. To check the validity of the retrieval process, in Fig. 3(b) we compare the measured laser spectra (both fundamental and at SH) with that computed from the retrieved pulse profile at each location. The measured and computed spectra agree reasonably well and do not vary significantly between the five locations. The FWHM bandwidth of the fundamental pulse spectrum is ~12 nm.
Fig. 3 Pulse characterization at multiple locations using FROG holography. (a) Retrieved pulse profile (amplitude and phase, FWHM ~89 fs) at each position; (b) Comparison of both fundamental and SH power spectrum of the retrieved pulse at each position with measured power spectra (black squares) (FWHM ~12 nm) (c) Comparison between retrieved and experimentally recorded spatial separation between each location with respect to pivot. The table in inset lists actual values in $μ m$ .
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In general, if we express the total spectral phase of the pulse using Taylor series $ϕ (\vec{r}, ω) = ϕ (\vec{r}, ω_{0}) + ϕ^{'} (\vec{r}, ω_{0}) Δ ω + 0.5 ϕ^{″} (\vec{r}, ω_{0}) Δ ω^{2} + ...$ then $H (\vec{r}, Ω, τ)$ coherently ‘links’ the complex pulse profiles by effectively measuring the carrier phase and the group delay in addition to the O(2) terms that are measured by conventional FROG. Thus the pulse phase is uniquely determined at each location $\vec{r}$ , aside from a common overall undetermined constant for all locations. We find the group delay by computing the relative pulse spectral phase i.e. $Δ ϕ (i, j; ω) = ϕ ({\vec{r}}_{i}, ω) - ϕ ({\vec{r}}_{j}, ω)$ , use linear polynomial fit to approximate its gradient, i.e. $d Δ ϕ (i, j; ω) / d ω$ and finally multiply this value by speed of light in vacuum to numerically obtain the separation between each pair of locations. Figure 3(c) and the table within show the comparison between numerically obtained and experimentally measured distances from the pivot to every other location. The agreement not only validates our retrieval procedure but also serves as proof-of-concept demonstration of the potential of FROG holography to fully characterize the spatiotemporal evolution of an ultrashort optical field.
5. Conclusion
In summary, our technique of FROG holography utilizes a reference beam to record spectral holograms of conventional cFROG traces at multiple spatial locations. In contrast to other existing techniques, our method does not require the reference beam to be characterized, as long as it stays stable during the entire measurement. We have shown that it contains not only the standard non-collinear FROG term but also the relative FROG phasor and thereby coherently ‘links’ the measurements at different spatial locations. We have discussed a method to retrieve the complex pulse profile from the recorded HcFROG traces, and demonstrated the measurement of group delay of the pulse in the vicinity of the focal point of an objective lens. In short, FROG holography combines high sensitivity of spectral holography with FROG to characterize the detailed temporal phase structure of the pulse as well as provide key information about group and carrier velocities of pulse propagation. The improved sensitivity of homodyne detection is useful for probing “weak” spots or for increasing the scanning speed by reducing the integration time of the spectrometer. In principle, FROG holography makes it possible to measure the spatio-temporal evolution of any complicated ultrashort pulse that can be characterized by the general FROG technique. In combination with nonlinear nanoprobes [25], FROG holography has the potential to image the spatiotemporal evolution of ultrashort optical fields in femtosecond scale and 3D nanometer scale, which could benefit emerging applications in biomedical imaging, plasmonics, and metamaterials.
Acknowledgments
The authors acknowledge the support from the National Science Foundation (Award # ECCS 0925591, ECCS 1128587).
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Characterization of the spatiotemporal evolution of ultrashort optical pulses using FROG holography

Abstract

1. Introduction

2. Theory and pulse retrieval using FROG holography

3. Experimental setup

4. Results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (4)

Optics Express