A practicable and economic method for the generation and full characterization of laser pulses ranging down to sub 7 fs duration with energies spanning the full µJ domain is presented. The method utilizes a self-induced and self-guiding filamentation of titanium-sapphire based, amplified pulses in air for spectral broadening, a standard chirp mirror compression scheme and transient grating frequency resolved optical gating for determining the spectral phase over the full visible to near infrared range. In this manner, few-cycle laser pulses with a high quality in the spatial beam profile have been generated in an robust arrangement with a minimal amount of standard optical components for their full characterization. The optical scheme demonstrates an uncomplicated, versatile access to this regime of pulsed laser radiation accompanied by a comprehensive analysis.
© 2008 Optical Society of America
The development of solid-state femtosecond laser technology has provided the basis for the optical nonlinearity fundamental to generating extended coherent bandwidth and ultimately attaining pulsed laser radiation with significantly improved time resolution and enhanced peak power. In recent years, different strategies for improving upon these elementary attributes of ultrashort laser pulses have been pursued. One possible route to these means generally involves propagating pulsed radiation at high field strengths over long optical pathways in a confined volume. This can be realized in several variations, where confinement is generally achieved in micro-structured fibres , hollow core fibres  or via self-guiding filamentation [3, 4, 10, 11]. Active processes in the spectral broadening range form self-phase modulation (SPM) , self-steepening  and pulse splitting , four-wave-mixing processes (FWM)  and coherent Raman cascading  as well as light-plasma interactions [3, 10, 11]. Landmarks in the development of this general principle include initial supercontinuum generation in the solid state and liquids  followed by later work in gaseous media . Building on these developments, interaction lengths were increased through guiding in different forms of nonlinear media. In this framework, solid state micro-structured fibers allow for the generation of pulses with octave spanning spectra in the nano- and recently in the lower mircojoule regime combined with the unique and tunable propagation properties of these materials.[1, 12, 13, 14] Alternatively, the use of gaseous media in hollow core fibres and filamentation allows for high pulse energies to be employed and additionally access light-plasma interactions for spectral broadening and guiding. In this manner, octave spanning bandwidth als well as pulses with compressibility down to a few optical cycles can be obtained with pulse energies ranging to the mJ regime and beyond . This scheme can be extended to the terawatt regime, where white light generation in filaments spanning several hundred meters can be generated with a spectral range extending form approximately 230 nm to 4.5 µm at altitudes beyond 10 km in the framework of atmospheric research with LIDAR. 
Accompanied with their generation, the characterization of extremely short and octave spanning laser pulses is still a issue in current research. Established methods such as interferometric autocorrelation, frequency resolved optical gating (FROG) [17, 18, 19] and spectral phase interferometry for direct electric-field reconstruction (SPIDER) [20, 21] with their respective drawbacks and advantages are summarized in Ref . While nearly single cycle, sub 4 fs pulses have been correlated interferometrically via surface third harmonic generation, access to the spectral phase over the full bandwidth still remains challenging. SPIDER posses the powerful and unique advantage of an analytical approach in obtaining the spectral phase from the respective interferogram, yet the necessity of frequency mixing and pulse shearing can limit the total pulse bandwidth that can be analyzed. FROG techniques substantially vary in the mechanism employed for temporal gating, ranging from second order frequency mixing to third order processes such as self-diffraction, Kerr-based polarization gating and laser induced transient gratings. While FROG intrinsically poses the difficulties associated with an iterative, non-analytical phase retrieval, the variations of this technique that do not rely on bandwidth limited frequency mixing open the noteworthy prospect of a complete phase characterization over octave spanning spectra, as will be demonstrated in this work.
In many cases, the generation, and particularly the characterization, of few cycle laser pulses involve intricate optical schemes that employ non-standard optical components. With the motivation to makes state of the art pulse parameters readily accessible in the framework of commercially available solid-state femtosecond technology, this work describes a means for attaining and fully characterizing octave spanning laser pulses with compressibility in the sub 7 fs regime and energies ranging to upper end of the µJ domain. The strategy described in the following full avoids complex, sensitive optical schemes and relies on an absolute minimal amount of standard optical components. For spectral broadening, self-induced filamentation in air is employed, where the developed setup allows for pulses durations with the highest compression factor attained in a single filamentation process. For the characterization of the obtained pulses, it is demonstrated that transient grating, four-wave mixing based frequency gating permits the full characterization of the spectral phase ranging an octave spanning spectrum, to the best of our knowledge the highest bandwidth so far correlated in a FROG scenario.
The general scheme for a simple and robust access to high-energy, few-cycle laser pulses in the framework of standard Ti:Sa amplifier technology is depicted in Fig. 1. Beginning with the output of a commercial femtosecond laser system,1 continuum generation is attained through filamentation in air at atmospheric pressure by focusing the laser pulses with a spherical mirror SM1 (f=2 m), as shown in upper part of Fig. 1. Through the spatial confinement and temporal modulation resulting primarily from optical Kerr effect in competition with weak, multi-photon gas ionization and thin plasma formation, this self-initiating and self-guiding process circumvents problems of pointing stability, as well as aperture matching and the optimization of propagation length involved in fiber guiding. Direct filamentation in air make vacuum components as well as entrance and exit windows obsolete and unwanted spectral as well as spatial phase distortions are avoided, vide infra. Assuming 1.0 mJ pulse energy, 7.2×1013 W/cm2 at 1/e2 intensity is achieved assuming a geometric focus of the transversal mode with the corresponding pulse power of 25 GW. This value constitutes approximately eight times the critical power for self-focussing of approximately 3 GW in air . Regardless of hereinafter presented input energies, the transmission after filamentation maintained high values of approximately 95%. An important setting is the adjustment of the beam diameter through iris I1 at 1.5 m ahead of SM1. Reduction of the initial diameter (9.3 mm) from 10% to 30% prevents fluctuation in the filamentation process as well as multi-filamentation in air at atmospheric pressure. In this configuration, the onset of filamentation can be observed through a weak fluorescence roughly 8 cm before the geometrical focus, extending approximately 25 cm. Subsequent to spontaneous termination of the filamentation, the spectrally broadened pulse propagates with a reduced beam diameter and divergence of 2.5 mrad, which is determined by the filamentation process rather than the initial focusing optic SM1. An analysis of spatial, spectral and temporal behavior as well as transmission and conversion efficiency of the initial output of the filament will be given in the following section.
Variably, the filament output can be directed to a FROG for immediate analysis or to a set of chirped mirrors for compression with subsequent correlation, as shown in the lower part of Fig. 1. Dispersion management is achieved by four bounces on commercial available chirped mirror pair CM1 & CM2, (Layertec) with GVD oscillation compensation and average negative GVD of approximately -60 fs2 per bounce in the wavelength range from 700–900 nm. The total reflectivity is > 99.8% from 650–1000 nm at incident angle of 0°. A distance of approximately 4 m is spanned between iris I2 and I3, in order to provide for the beam diameter of 6 mm on I3, necessary for the correct beam geometry for correlation. Divergence of the filament output sketched behind iris I2 is overstated for better visualization. For characterization of the spectral phase over the complete bandwidth of the filament output, a transient grating based frequency resolved optical gating (TG-FROG) is employed that is achieved with only eight standard optical components. The working principle of this TG-FROG is illustrated schematically in the inlet on the right bottom of Fig. 1, where the highly economic design is inspired by previous optical configurations from autocorrelation scenarios of ultrashort laser pulses and transferred here to a three beam four-wave mixing (FWM) scheme. Entrance iris I3 consists of three holes of 2 mm diameter for spatial beam separation. Considering material dispersion and the availability of octave exceeding beam splitters necessary for the case of amplitude splitting, the advantages of this geometrical separation are evident and have been recently demonstrated [23, 24]. The illumination on iris I3 is achieved by closing I2 down to 1mm and free travel of the beam for 4 m. The holes for this geometric beam separation are drilled in the corners of a square with 4 mm separation, as shown in the schematic view in Fig. 1. By passing iris I3, three phase-locked, parallel running beams with the respective wave vectors (k1, k2, k3) are obtained within 90% of 1/e2 intensity of the transversal beam profile. This arrangement constituting a forward folded-box configuration for a FWM scenario. Two of these subpulses (k1 and k2), drawn as dashed line in Fig. 1 are separated vertically and reflected from a single flat mirror FM onto the focusing mirror SM2 with f=250 mm. The third subpulse, k3 represented by the upper straight line in Fig. 1 is directed via a separate, movable mirror TM toward SM2, where a high resolution translation stage (MICOS, Model PLS-85) allows for variable time delay. From SM2, the three parallel beams are focused to spot size of approx. 30 µm into a centrosymmetric dielectric medium (BK7 glass) and four-wave mixing in this beam geometry leads to, among others, a signal beam in the direction ks under the phase-matching condition kS=k1-k2+k3. The two stationary beams (k1, k2) act as the gating mechanism and interact simultaneously in the medium, while the third (k3) is swept temporally via TM for correlation. Important to note is the phase-conjugation of k1 and k2 for the detected signal direction. This provides for a temporal gating mechanism via spatial phase interference that is void of temporal phase information. This translates to pure amplitude gating resulting in an unambiguous phase determination. The two dotted lines after broadband FWM indicate a small angular dispersion of kS, denoted with kSb and kSr, which occurs when octave spanning white light with an unsymmetric spectral profile is correlated. In order to prevent angular effects in the acquisition of the spectrum when recorded via a fiber spectrometer (Ocean optics, Model USB2000), the signal beam is focused onto a scatterer providing for diffuse reflection that is independent of angular dispersion of kS.
At this point, two geometrical aspects concerning the distortion of the temporal measurement should be discussed. First, movement of mirror TM causes a parallel spatial shift of k3 on surface of SM2. In order to minimize this shift, the incident angle on delay mirror is only 1° which cause a parallel shift of 2 µm for a scan range of 200 fs. Since parallel shift does not change focus position in first approximation, the shift of 2 µm is negligible. More important is an analysis of the temporal resolution for a non-collinear interaction of three beams. The intersection angle θ inside the medium between two k-vectors introduces a geometrical time smearing Δτ. Therefore, angle θ is kept small (0.6°) by use of relatively long focal length f combined with a small beam separation, di. Assuming Gaussian beams, the temporal smear can thus be calculated according to ref.  by Δτ=(fθλ)/(diπc) where c is speed of light and λ denotes laser wavelength. For the present TG-FROG setup, Δτ possess a maximum value of 0.63 fs when long wavelength of 900 nm is assumed as upper limit. Δτ blurs the physical value τ 0 and gives a measured value with a scaling constant x which depends on beam geometry. For three intersecting beams, ξ ≈ 5/3 in a forward box configuration. Based on above equations, the blurring of a 6 fs pulse results in pulse lengthening of less than 6% through the measurement in this beam geometry. The determination of pulse duration was therefore carried out by evaluating the raw data obtained directly from FROG trace. The Wigner representation was projected onto the horizontal time axis and fit assuming a Gaussian temporal profile. Circles in Fig. 4 correspond to measured data points whereas unbroken lines display the Gaussian fit. Different than the case of two beam correlations, the deconvolution factor for Gaussian functions is for a three beam  geometry in a χ (3) process. Retrieval of spectral phase from the FROG data was accomplished utilizing the generalized projection method [18, 26]. As a further advantage of employing FWM based correlation, it should be noted that in standard correlation techniques involving frequency conversion, the common method for increasing phase matched bandwidth involves reducing the propagation length through the nonlinear medium, currently down to a crystal thickness of 5-10 µm. Thus, χ (2)nonlinear processes, i.e. three wave mixing, reach physical limitation in their dimensions as well as transparency when octave spanning spectra are gated. In contrast to this, FWM can easily be carried out in a spectral range extending into the deep UV as will as into the IR domain, with the only requirement being the transparency of the medium. As shown here, phase-matching of broadband spectra extending over an octave is automatically realized in this process through the dispersion of the signal direction. From this, FWM mixing can be viewed as one of the most flexible nonlinear processes with respect to the spectral range and bandwidth in the correlation of pulsed laser radiation. For the experiments carried out here, standard BK7 microscope objectives where employed and provided that the focus was placed at front face of the medium, a crosscheck for varying thickness and different composition of the medium showed no significant change of the pulse duration when correlating few cycle laser pulses.
3. Results and discussion
In the following, an analysis of the initial filament output will be given for varying input pulse parameters with regards to the spatial beam profile as well as the efficiency and mechanism in spectral broadening. Following this, the capability to characterize the spectral phase of femtosecond laser pulses with bandwidths of up to 500 nm, spanning the visible to NIR range will be discussed. From this, the optimal filamentation conditions for obtaining few cycle laser pulses will be analyzed in view of the spectral phase retrieved form the supercontinuum pulse. Within this analysis, different filamentation conditions will be presented that allow for obtaining an output with a spectral phase suitable for compression of selected spectral regions to the sub 7 fs regime. Furthermore, different pulse energies in the filament output, obtained from the radial selection of the transversal mode, are compared with respect to their compressibility.
3.1. General attributes of the filament output
The transversal mode obtain from the filamentation of standard amplified pulses (1.3 mJ, 40 fs @ 807 nm) is illustrated by means of a CCD camera image of the supercontinuum output in Fig. 2b. The high radial symmetry of the intensity profile in the spatial mode generally obtained from plasma filamentation in gaseous media can clearly be seen from the horizontal and vertical cross section in Fig. 2a. Important to consider in the framework of filamentation is the transversal homogeneity of spectral broadening reflected in the output mode. In order to characterize this quality of the filamentation, the spectral behavior is recorded at different positions by scanning a small diaphragm transversely across the mode, as shown in Fig. 2c. Within a radius of 80% of peak intensity, all general characteristic spectral signatures remain virtually constant except the high frequency cut off. Particulary noteworthy in Fig. 2c is the conformity of the amplitude modulations in the spectral signature form approximately 500 to 950 nm within 25% of peak intensity of the transversal mode. An estimate of conversion efficiency toward visible wavelengths region ranging from 450–720 nm was carried out by numerically integrating the corresponding area in plot of Fig. 2c. Under the given conditions, typically 15%-20% of overall intensity are transferred to the visible range. Contrary to the center of the mode, the peripheral region at 10% of maximum intensity clearly exhibits narrower bandwidth with a different spectral signatures and a slight red-shift.
3.2. Temporal gating of octave spanning white light
Figure 3 shows the acquired FROG trace as well as the spectral phase retrieval when the full bandwidth obtained form filamentation is directly coupled into the TG-FROG described above. Settings of the input pulse for filamentation are chosen to demonstrate the capability of generating octave exceeding spectra within single filamentation in atmospheric air as well as capability to characterize the spectral phase spanning this spectrum via TG-FROG. This demonstrates a simple means for the full characterization of pulsed radiation in this regime and establishes the tools for gaining new experimental insight to the fundamental mechanisms underlying filamen- tation via spectral phase analysis. Widest spectra were generated when iris I1 was nearly fully open, providing a pulse energy of 1.3 mJ and a prechirp of roughly 500 fs2. These settings denote limiting values for air at atmospheric pressure before intensity fluctuations and spatial distortion of beam profile occur. When aiming for broadest spectra, two different sections in the FROG trace become distinguishable. The leading part of the pulse is temporally and spectrally significantly modulated, whereas the spectrum from 650 down to 380 nm shows a predominately smooth spectral profile as well as a smooth phase profile down to 600 nm. For a simple and qualitative interpretation of these to distinct pulse regions, a briefly summary of the current understanding of spectral broadening through filamentation is necessary. In gas phase filamentation, it is primarily the competition between Kerr effect and plasma generation that cause the pulse to experience spatio-temporal modulations with the corresponding spectral modifications. Propagation is influenced by self-focusing  and multi photon ionization , where the latter acts to arrest the collapse of beam. The balancing of both effects results in extended self-guiding . On the spectral side, Kerr induced SPM  accompanied by a minor red shift  leads initially to symmetric broadening accompanied by the respective phase distortions. Onset of plasma generation and thus lowering of the refractive index forces asymmetry of the temporal phase and spectral envelop [8, 6], respectively. This general plasma blue-shift together with the shock dynamics  resulting form the trailing pulse flank, give rise to asymmetric blue shift of the spectrum. Further spectral asymmetry may also be attributed to non-resonant FWM . Providing sufficient intensity in nonlinear evolution, the pulse undergoes temporal splitting [3, 31, 7].
The acquired FROG trace of the filament output in Fig. 3 together with the retrieved spectral phase distinctly reflect the different mechanism involved in the spectral broadening. The strong modulations in the FROG trace around the laser fundamental arise when nonlinear propagation through the entire filament extend beyond a symmetric SPM to complex phase distortions manifesting themselves in the apparent temporal splitting. The corresponding spectral phase in the inlet of Fig. 3b retrieved form the FROG trace for this region further reflects phase distortions beyond SPM and its highly irregular contour clearly shows that under these filamentation conditions, compressibility with conventional methods is unpracticable. A very different picture is seen in the visible part of the obtained FROG trace and the spectral phase in Fig. 3. Spectral phase retrieval for this the unmodulated part of continuum primarily obeys a smooth quadratic behavior. Here, truncation of the spectral phase function at 560 nm is caused by numerical unwrapping when dealing with very steep slopes as well as termination of the retrieved spectral amplitude. However, the total hight stroking over 300 radian was obtained by the retrieval. This phase function corresponds to a GVD of approximately 900 fs2 assuming a purely linear chirp. Considering the propagation in air amounting to 150 fs2, the remaining positive chirp is attributed to the spectral broadening mechanism in the filament. This is supported by the observation that the phase function remains constant for a different prechirp of the input pulse. The distinct and manageable phase function in this region clearly reveals straight forward compressibility. This is in contrast to the band ranging the red to NIR spectral domain under filamentation conditions set for maximal spectral broadening.
3.3. Compression to few cycle pulses
While the compression of the visible region in the filament output has been discussed in the literature, different filamentation conditions for compression of the high amplitude NIR component of the continuum output will be explored in the following. The analysis includes different energies and radial confinement of the input for filamentation as well as radial mode selection for different energies of the filament output. An overview of the compression und these varying parameters is given in the series of Fig.4 a–c. Attaining the shortest pulse durations was highly sensitive to small changes of I1, achieved when closing down to 7 mm, reducing the input energy from 1.5 mJ to 1.0 mJ. Fine adjustment of grating separation in the amplifier compressor was necessary for finding the shortest pulses. In general, best compressibility was achieved with a slightly positive prechirp of 400 fs2 of the input pulse for filamentation, which agrees with other observations by Hauri etal. . Noteworthy, the total amount of positive dispersion accounts to 550 fs2 when including the optical path in air but the implemented negative GVD of four bounces on chirped mirrors is only -240 fs2. It is again assumed that this is an indication of self-induced temporal changes taking place during nonlinear propagation in the filament. Furthermore, I2 is set to select the inner core of the filament output, reducing the filament output energy form 950 approximately 30 µJ/pulse, which corresponds to a mode radius of 500 µm in Fig. 2a. In this manner, pulse durations as short as 6.3 fs were generated with a single filament. Figure 4a shows a representative FROG trace which displays a long term average measurement. Repeatability was verified by taking 41 FROG traces during a period of 10 days. The mean FWHM value for 41 measurements during this time period is 6.3±0.3 fs. Important to note for this case are the satellite structures clearly evident in the FROG trace and the projection on the time axis. The appearance of these structures can partially be understood from the reconstructed spectral phase for this pulse as shown in Fig. 5a. The spectral phase function is inhomogeneous with two distinct spectral regions. While the wavelength region from 650–820 nm shows phase jumps of π or 2π, respectively, the spectral domain above 820 nm exhibits an irregular spectral phase profile. From this, the satellite structures around the primary pulse can be attributed to the compromise made when trying to balance these two distinct spectral regions with chirped mirror compression having a nearly quadratic phase correction over the full bandwidth. In this context, it is worth mentioning that a reduced amplifier output of 1mJ and the opening of iris I1 increases the pulse width and the strength of satellite structures, i. e. not only the absolute amount of input energy but also the spatial beam characteristics and diffraction effects influence the filament output. However, a fully opened iris I1 and reduced input energy of 650 µJ for filamentation enables the generation of clean temporal profiles with reduced spectral modulations as depicted in Fig. 4b. Again, I2 is set to select the inner core of the filament output, reducing the full filament output energy from 620 to 20 µJ/pulse with a radial selection of 500 µm in Fig. 2a. This is noteworthy, since generation of sub 10 fs with a single filament has previously been presented with distinctly higher energies  than 1 mJ. The absence of satellites in the FROG trace of Fig. 4b is accompanied by smoother NIR spectrum and a higher wavelength cut off around 550 nm. Furthermore, the corresponding spectral phase is mostly linear as shown in Fig. 5b for this low energy case. In a last variation, the compressibility of the full filament output is tested without radial selection of the output mode. Here, the input pulse is reduced from 1.4 to 0.76 mJ/pulse by closing I1. The filament output energy of 720 µJ/pulse is obtained when I2 is set to pass the full radius of the output, which is then guided to the TG-FROG via a 4 % surface reflection. Important to not is the discrimination of the outer, more instable region of the mode through I3 as well as the inner core corresponding to the dimensions of I3 in the methodology section, see Fig. 2a. The FROG trace for these conditions is shown in 4c, where the shortest pulse required different dispersion compensation achieved by adjusting the optical pathway through air. It should also be noted that, in comparison to the previous measurements, correlation of the inner mode yields a pulse duration of approximately 7.6 fs for this filament input conditions. From the FROG trace, representative pulse duration obtained for these conditions are at 8.5 fs. This slightly higher pulse durations for the full transversal mode correspond to theoretical predictions of radial dependency in the spectral broadening in the filamentation process . With the three measurements given above, different input and output conditions for pulse compression are shown for the highly practicable scenario of filamentation in air at atmospheric pressure and the insight gained by a full spectral phase analysis is demonstrated with the economical TG-FROG setup.
The strategy employed for the generation as well as full characterization of few cycle pulses and octave exceeding supercontinua described above is distinguished in both cases by the ease of realization and the high economy in optical components. The white light generation via filamentation in atmospheric air under conventional laboratory laser conditions (0.6-1.5 mJ) and a characterization by a FWM based FROG scheme can be easily realized. The noteworthy capability to frequency gate the full coherent bandwidth emitted from the filament allows for a general insight to the mechanism of filamentation as well as the optimal conditions for pulse compression of different spectral regions with varying input energies and radial output selection. The work presented above opens a variety of aspects for future investigations. The utilization of a more complex phase correction provided by liquid crystal base pulse shapers opens the prospect of compressing the full bandwidth of the filament output as well as the generation of complex pulse forms for spectroscopic applications . Furthermore, the high quality in the spatial beam profile allows for a second filament to be ignited with the given output, bringing a higher conversion into the visible components of the spectrum and eventually compression to shorter pulse durations. While the qualitative spectral phase retrieval presented here has provided insight to the mechanism of spectral broadening and compressibility of filament output, a truly quantitative retrieval remains challenging for high bandwidths with intricate phase structure. This will be approached by employing slicing techniques as well as incorporating genetic algorithms in the iterative optimization of a quantitative spectral phase assignment. Further work will also focus on a better understanding of spectral broadening via filamentation through spectral phase analysis under varying filament conditions. The presented achievements and outlook given here aspire to establish few cycle laser pulses as a readily achievable standard for implementation in a wide spectrum of spectroscopic applications.
The financial support provided by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich 450 “Analysis and Control of Ultrafast Photoinduced Reactions” and the Sonderforschungsbereich 546 “Structure, Dynamics and Reactivity of Aggregates of Transition Metal Oxides” as well as the Leibniz Graduate School for Molecular Biophysics is gratefully acknowledged by the authors. The authors greatly appreciate the support form Dr. Daniel J. Kane for generously providing the algorithm used for the spectral phase retrieval. fruitful discussion with Dr. Kamil Stelmaszczyk, Philipp Rohwetter and Dr. Estelle Salmon are gratefully acknowledged.
|1The Laser system consists of cw (Verdi V5, Coherent) pumped oscillator (Femtosource, Femtolasers; 80 MHz, 6 nJ/pulse) which is coupled into multi-pass amplifier system (Odin C, Quantronix; 1 kHz, 0.6-1.5 mJ/pulse, 40 fs (FWHM) centered at 807 nm, Δλ=35 nm (FWHM).|
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