1. Introduction

Pulsed lasers have become an important tool for nonlinear optical experiments and therefore multiple characterization techniques evolved. The duration of picosecond pulses has first been demonstrated by intensity autocorrelation using a nonlinear medium [1]. The retrieval of phase information is increasingly important with shorter pulse durations. Fringe-resolved autocorrelation (FRAC) measurements which are readily applicable to a microsope setup are sensitive to the pulse phase although with ambiguities [2]. The intensity autocorrelation and FRAC techniques work in the time-domain only. Extension to the time-frequency domain was first demonstrated by Treacy in 1971 [3, 4]. Such spectrograms are still used in e.g. frequency-resolved optical gating (FROG) where the electric field is retrieved [5, 6]. Another method is spectral phase interferometry for direct electric-field reconstruction (SPIDER) [7]. A multitude of different FROG derivatives are described in [8]. Cross-correlation FROG (XFROG) is demonstrated by second-order nonlinear interaction with a well characterized reference pulse [9]. Such schemes are used to characterize complex pulses which are generated in microstructure-fibers [10]. XFROG spectrograms are more intuitively analyzed than FROG spectrograms but require a synchronized reference pulse which is well known. This prerequisite often limits the applicability.

The use of ultrashort laser pulses becomes increasingly important for nonlinear optical microscopy (e.g. multiphoton microscopy [11] and coherent anti-Stokes Raman scattering microscopy [12, 13]). The choice of method is often determined by the geometry in which an ultrashort experiment is implemented. In the field of nonlinear microscopy, typically a collinear geometry is established for high resolution imaging by exploiting the full numerical aperture of the focusing objective. In this paper we present a new method which is compatible to the excitation geometry in typical multiphoton microscopes. It is based on degenerate four-wave mixing (FWM) XFROG between an unknown ultrashort laser pulse and a bandwidth-limited narrowband gate pulse with a flat-top profile of the temporal envelope. Despite the long duration of the gate pulse its shape allows for a high temporal resolution due to the steep leading and trailing edges and a well reproducible structure of the gating pulse between measurements. Excellent phase-matching is realized in a tight microscope focus by degenerate FWM [14]. Weak pulses can sensitively be characterized due to their linear contribution to the FWM signal (see Fig. 1(a)). The sensitivity can be increased by using an intense gate pulse which contributes to the FWM signal with the square of its intensity. Enabled by the flat-top shape of the gate pulse, the spectral phase and the intensity can be extracted in a non-iterative manner from the XFROG spectrogram to reconstruct its temporal envelope.

 

Fig. 1 (a) Frequency conversion diagram of the degenerate FWM process, (b) Schematic spectrograms of the continuum pulse I_c(t,ω), the gate pulse I_g(t,ω) and the FWM signal I_FWM(t,τ,ω), (c) Schematic spectrogram I_XFROG(τ,ω) of the cross-correlation with indicated t_gd(ω) curve.

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2. Theory

For an instantaneously responding medium, the degenerate FWM field E_FWM is expressed in terms of the unknown continuum field E_c(t) and the time τ delayed gate field E_g(t − τ)

The FWM signal is blueshifted and therefore free of interference with the excitation light (see Fig. 1(b)). Direct access to the group delay of the short pulse is possible by analyzing the spectrogram I_XFROG(τ, ω) (indicated by the temporal distortion versus frequency in the spectrogram, see Fig. 1(c)). The expression for the group delay t_gd = dϕ/dω unveils the spectral phase

3. Experiment and analysis

Our light source is based on a two-branch Er:fiber laser at a wavelength of 1550 nm and a repetition rate of 40 MHz [13, 15]. In the first branch the octave spanning continuum (from 900 nm to 1800 nm) is generated in a highly nonlinear fiber whose characterization is demonstrated in this paper. A NSF10 prism compressor is used for chirp control. The second branch delivers the flat-top gate pulse at a wavelength of 775 nm by second-harmonic generation of the fundamental light at a wavelength of 1550 nm (see Fig. 2(a) and 2(b)). The pulse shape is measured by cross-correlation with the ultrashort continuum pulse using sum-frequency generation in a thin LiNbO₃ crystal. A narrow bandwidth of 0.6 nm allows for a good spectral resolution in the spectrogram. The pulse duration of 3 ps yields a time-bandwidth product of 0.9 which corresponds to a bandwidth limited flat-top pulse. The two branches have a low mutual timing jitter of less than 50 as (integrated from 1 Hz to the Nyquist frequency) [16].

 

Fig. 2 (a) Spectrum of gate pulse with a bandwidth of 0.6 nm, (b) Temporal shape of gate pulse intensity envelope with a duration of 3 ps, measured by cross-correlation with the ultrashort continuum pulse using sum-frequency generation in a thin LiNbO₃ crystal, the time-bandwidth product amounts to 0.9 and indicates a bandwidth-limited flat-top pulse, (c) Experimental setup: D; delay-line, C; beam combiner, T; reflective telescope, O1/O2; Focussing and collecting objective, χ⁽³⁾; susceptibility of microscope slide, F; filter, P; UVFS equilateral prism, C; CCD camera.

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Although the demonstrated FWM-XFROG pulse characterization method is conveniently performed with a multi-branch Er:fiber laser it is not restricted to this type of light source. An ultrashort laser pulse which needs to be characterized can simply be split into two branches. Flat-top pulses can easily be generated in one branch by second-harmonic generation of the ultrashort laser pulse in a long periodically-poled lithium niobate crystal [17]. The flat-top pulse can subsequently be used as a gate for the characterization of the unknown ultrashort laser pulse in the other branch.

The XFROG setup is shown in Fig. 2(c). A mechanical delay line is used for scanning the time-delay τ between the two laser pulses. The two branches are combined by a dichroic mirror, expanded by a reflective telescope and coupled into a home-built microscope. The two laser beams are collinearly focused by a near-infrared corrected water immersion objective (NA 0.85 32×, Carl Zeiss AG, Jena, Germany) into a microscope slide (SuperFrost®). The FWM signal is collected in forward direction by a long working distance objective (NA 0.6, 20×, Edmund Optics, Karlsruhe, Germany) and detected by a spectrometer based on a UVFS prism and a CCD camera (iXon^EM+897 back illuminated, Andor Technology plc., Belfast, Northern Ireland). The XFROG spectrogram is recorded by spectral acquisition of the FWM signal as a function of the time-delay of the gate pulse (see Fig. 3(a)). In order to suppress a background proportional to $E_c(t)E_c^{*}(t)E_g(t-\tau )$ which originates from FWM of frequency component pairs in the spectral wings of the broadband continuum pulse E_c(t) and the narrowband gate pulse E_g(t) two consecutive spectrograms are recorded. Therefore the long wavelength components (> 1160 nm, in the Fourier plane of the prism-compressor) of the continuum are blocked to record a background-free I_FWM which corresponds to the short wavelength part of the continuum (< 1160 nm). For the second measurement the short wavelength components (< 1060 nm) of the broadband continuum pulse are blocked to record the FWM signal which corresponds to the long wavelength part of the continuum (> 1060 nm). In this way the interaction between the wings of the continuum is inhibited. Combination of the two measurements yields the background-free XFROG trace (see Fig. 3). The acquired spectrogram contains the information about the laser pulse relative spectral intensity I_c(t) and group-delay t_gd. With the narrowband gate pulse at 776.7 nm the FWM signal reproduces the laser spectrum. Integrating along the time-axis τ delivers the relative spectral intensity $\propto {\int }_{-\infty }^{\infty }{I}_{\mathit{XFROG}}(\tau ,\omega )d\tau$ from which the laser spectrum E_c(ω) is determined by wavelength conversion (see Fig. 3(c)).

 

Fig. 3 (a) Measured XFROG spectrogram with a CCD camera exposure time of 1 ms and time delay steps of 2 fs, (b) Reference cross-correlation, section at 574 nm indicated by vertical line in (a) (corresponds to 1200 nm in the continuum), (c) Retrieved laser spectrum E_c(ω) by averaging over time delay τ, (d) Retrieved group delay t_gd with a zoomed inset which indicates a temporal error of 2 fs.

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The group delay versus wavelength is directly observable in the contour of the spectrogram (see Fig. 3(a)). For group delay extraction a reference gate in the spectrogram at an arbitrary wavelength (here λ_FWM =574 nm, which corresponds to 1200 nm in the continuum) is chosen (see vertical line in Fig. 3(a)). The section is shown in Fig. 3(b) and corresponds to the squared intensity envelope of the gate pulse (compare Fig. 2). This temporal profile is reproduced at all wavelengths throughout the spectrogram but with a temporal shift which corresponds to the group delay of the continuum pulse. Tracking this reference shape along all wavelength positions of the FWM-XFROG spectrogram unveils the corresponding group delay. This result is achieved by calculating the cross-correlation between the reference gate at 574 nm and the sections through the spectrogram along all wavelengths. Since the respective profiles have an overall rectangular shape, the cross-correlations are triangular and therefore their maximum is precisely defined. The temporal position of the maximum of the cross-correlations directly yields the group delay t_gd. Due to the steep slopes at the leading and trailing edge of the gate pulse, a good temporal resolution is obtained. In the inset of Fig. 3(d) a zoomed graph is shown which indicates a temporal error of 2 fs. One should note that a small error in the determination of t_gd has a relatively small effect on the determination of the pulse duration. Plugging t_gd in Eq. (3) leads to the spectral phase ϕ(ω). In this way the electric field can be completely determined, except for a constant phase offset which is not relevant for the determination of the envelope. This finding shows that the characterization of complicated laser pulses is possible with our technique.

An analysis of the short wavelength part of the continuum is demonstrated by blocking the wavelength components above 1500 nm of the supercontinuum in the Fourier plane of the prism compressor (see Fig. 4(a)). Characterization of the compressed spectrum unveils a pulse duration of τ_FWHM=11.2 fs (bandwidth limit: 8.4 fs) in the focus of a microscope objective as well as the spectral phase information. Any linear phase function can be subtracted from the experimentally determined trace which affects the absolute time delay of the pulse envelope but not its shape. Therefore a linear phase is subtracted from the experimental phase for best visualization of the phase curvature (see Fig. 4(a)). Fourier transform yields the pulse intensity envelope as well at the phase in the time domain (see Fig. 4(b)). The validity of the XFROG scheme is verified by comparison of a calculated second-harmonic FRAC trace based on the XFROG retrieved electric field with an independent second-harmonic FRAC measurement. Although the FRAC technique shows some phase ambiguities it is never the less a frequently used pulse characterization technique and therefore we think suitable for our validation. A good agreement between the two traces is observable in Fig. 5. This result demonstrates the applicability of the technique for ultrashort laser pulses in the few cycle regime.

 

Fig. 4 (a) Retrieved intensity and phase spectra as well as the intensity spectrum measured by a linear spectrometer, (b) The retrieved temporal intensity envelope and phase show a pulse duration of 11.2 fs.

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Fig. 5 Line: Calculated second-harmonic fringe-resolved autocorrelation based on the XFROG retrieved spectrum and phase, dots: Measured second-harmonic fringe-resolved autocorrelation.

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4. Conclusion

The presented method allows for direct extraction of the group delay as a function of frequency of an octave spanning supercontinuum output of the highly nonlinear fiber enabled by the broad phase-matching bandwidth of the FWM process in the microscope focus. High temporal and spectral resolution is exploited by the flat-top shape of the gate pulse. Direct reconstruction of the electric field envelope is demonstrated by analyzing the XFROG spectrogram without relying on iterative calculations. A broad range of shapes can be analyzed from highly chirped pulses to short transients close to the bandwidth limit. The collinear excitation geometry is well suited for pulse analysis in multiphoton microscopes which are applied to biomedical imaging. The method may also find application in the characterization of single-cycle laser pulses [18]. The need for a gate pulse seems to be a drawback at first sight but pulse replicas from ultrashort laser sources which are typically present in multiphoton microscope laboratories can be used for its generation.