The spatio-temporal coupling of shaped laser pulses is measured using scanning SEA TADPOLE as a function of propagation distance through the focal region of a plano-convex lens. A double pulse sequence is measured to have a gradually changing spectral phase across the beam front as a function of propagation distance. When a sinusoidal spectral phase is applied to the shaper a saw-tooth spectral amplitude is measured across the beam front before and after the focal plane of the lens. The measured spatio-spectral phase and amplitude for these two common pulse shapes are consistent with the predictions of a theoretical model.
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Spatio-temporal coupling in focusing ultrashort laser pulses can alter the temporal profile during propagation through the focal plane of a lens. While beam relaying and pixelation effects have been theoretically predicted to change the spatial intensity distribution through the focal volume , experimental measurements of the spectral phase and amplitude of the pulse profile through the focal plane reveals even more exotic features . Spatially-resolved pulse characterization is of interest to the fields of quantum control, multiphoton microscopy, and spectroscopy where atoms and molecules are often probed in the laser focal volume. The outcome of these experiments can be sensitive to inhomogeneous phase profiles caused by pulse shape distortion in the focal plane  and to the spatial intensity as the pulse propagates through the focus . Here we consider spatial spectral and spatial temporal inhomogeneities for shaped laser pulses as a function of propagation distance through a focus.
Pulse shapers commonly used in coherent control have been shown to have space-time profiles that depend on the propagation distance from the last grating of the 4-f compressor . In the first theoretical calculations illustrating space-time coupling  a desired space-time profile was produced using an appropriate spectral filter in the Fourier plane of a 4-f shaper. The filter was calculated using a model based on Martinez’s treatment of the grating , followed by a Fourier optical analysis. Phase-shaping of the spectrally dispersed beam in the Fourier plane in a 4-f pulse shaper results in wavefront modulation and affects the beams spatio-temporal profile. Theory describing space-time coupling resulting from wavefront modulation has been presented previously [7–9] and the resulting space-time coupling has been addressed theoretically [1,4] and experimentally [4,7,10,11]. For instance, the dual spot ablation profile created when a double pulse sequence (created in a 4-f pulse shaper) was focused onto a metal surface reveals evidence for spatio-temporal coupling [10,11]. Such indirect investigations into spatio-temporal coupling motivate direct measurement of the spatio spectral phase and amplitude of the shaped pulse during focusing.
One widely used pulse shape involves the generation of a pump and time-delayed probe pulse to investigate the temporal dynamics of molecules and atoms in laser fields. Pump probe spectroscopy has been used, for example, to study the interaction of ionic wavepackets [12,13], the polarization of high harmonic generation , and the tomographic reconstruction of the highest occupied molecular orbital of nitrogen . A 4-f pulse shaper has been shown by Präkelt  to produce nearly identical experimental pump probe results when compared with those obtained using an optical delay line. Creating an optical delay line with a pulse shaper requires extensive modulation of the spectral phase to create a pi phase step and an cos2 transmission filter applied to the input pulse. We will show that such shaping provides the potential for spatio-temporal-coupling-induced distortion of the pulse shape in the vicinity of and at the focus.
Another widely used pulse shape is sinusoidal spectral phase modulation. Sinusoidal spectral phases have been utilized to control and investigate wavepacket dynamics of polyatomic molecules and atoms [17–20]. The characterization of the spatio-spectral and temporal pulse structure as the laser focuses is motivated by the need for the intended pulse shape to interact with the medium.
The spatio-temporal profile of focusing laser pulses is important in strong and weak field approaches to quantum control and pulse shaping has been employed to manipulate systems ranging from isolated atoms [21–23] to biomolecules in solution . The detection region must be restricted to the Rayleigh range of the focus to ensure that no volume averaging occurs within the focal plane of the ultrashort laser pulse . Volume averaging can overwhelm the nonlinear response of the sample masking the coherent phenomena. A flat intensity profile in the interaction region is achieved by inserting a small slit or aperture transverse to the propagation to the propagation direction to restrict the focal volume observed [3,25]. Pulse shape averaging may be similarly problematic during propagation through the focus and could also serve to mask controllability.
In this paper we analyze and measure space-time coupling from spectral phase and amplitude modulated pulse shapes for two cases, double pulse and a multipulse pulse train. These pulses are generated by applying a pre-calculated phase and amplitude mask to a transform-limited pulse via a spatial light modulator. We investigate a double pulse sequence where the spectral phase is modulated by a pi phase step and a cos2 transmission filter [26,27]. The multipulse train is generated using a sine function in the spectral phase modulation [28–32]. We completely characterize the focus by measuring the spatio-temporal coupling using scanning SEA TADPOLE to determine the two dimensional spatial-spectral content of the pulse propagating through the focus of the lens. Fourier analysis is employed to show full analytical solutions for the electric field after passing through the 4-f zero-dispersion compressor with the SLM in its Fourier plane. Accurate mapping of the dispersed frequencies components in the Fourier plane onto the SLM liquid crystals reduces diffraction effects on the beam profile but still produces unpredictable spatio-spectral patterns through the focus.
2. Experimental setup
The experimental setup is shown in Fig. 1 . A Ti:Sapphire oscillator delivers ~20 nm of phase locked bandwidth centered at 800nm with ~5nJ energy per pulse at a repetition rate of 80MHz. The laser beam is first collimated by a telescope to ~2mm 1/e spot size. The femtosecond pulses traverse a 4-f pulse shaper consisting of a 1200 l/mm gold coated-grating followed by a 210 mm reflective focusing element and a CRI (Cambridge Research Instruments) SLM-2 X 128 modulator in the Fourier plane. Just behind the modulator is a retroreflector that is slightly offset in the vertical direction to return the beam vertically offset by a few mm to clear an input mirror. Following the shaper, a scanning SEA TADPOLE (fiber mode diameter ~5μm) is used to characterize the spatio-temporal features of the generated pulse shapes . The measurements are taken at two positions before the geometric focus, at the geometrical focus and two positions after the geometric focus of the lens. At each position, the measurement consists of scanning the SEA TADPOLE fiber across the beam. The data obtained in this fashion can be used to plot the complex spectrum as a function of spatial coordinates, and the Fourier transform of the data yields the spatio-temporal intensity distribution. A flip mirror mount can be used to redirect the shaped beam into the beam profiler mounted on the rail, as indicated on Fig. 1. The beam profiler is used to directly determine beam waist and any changes in the transverse beam profile throughout the focal region as a function of the applied pulse shape.
3. Experimental results and discussion
3.1 Spatio-temporal coupling in the double pulse experiment
The double pulse temporal structure is the first example we investigate for spatio-temporal coupling. To generate a double pulse sequence a pi step phase modulation and a cos2 transmission filter were programmed onto the liquid crystal arrays to generate two near transform limited pulses delayed in time by 800fs as shown in Fig. 2 . We characterize the pulse as it propagates through the focus with a series of scanning SEA TADPOLE measurements. Figure 3 consists of transverse spatio-temporal and spatio-spectral Wigner distributions (Fourier transform pairs) for 5 consecutive longitudinal positions after propagating through a f = 50mm focal length lens. The focusing lens is placed many focal lengths after the final shaper grating . The geometric focus is determined from a measurement of the spot size using a beam profiler. Due to beam divergence after the shaper the exact geometric focus is between f-2zr and f, where zr is the Rayleigh length of the focused beam. Thus, there are two length scales of importance: the position of the beam waist (which depends on the beam divergence), and the geometric focal length (where spatio-temporal coupling is minimized). The top 5 panels in Fig. 3 show the measured transverse spatio-temporal Wigner distributions as a function of propagation distance. Note that for all positions out of the geometric focus, the two pulses are displaced along the transverse beam coordinate. Before the geometric focus (−4 zr and –2 zr) the pulse at negative 400fs is shifted to positive transverse coordinates and the pulse delayed to 400fs is shifted to the negative transverse coordinates. After the geometric focus, the pulse delayed to −400fs time shifts to positive transverse position and the pulse delayed to positive time shifts to negative transverse position. Shifting spatio-temporal pulse features through the focal plane have been predicted by Sussman et al., and Frei et. al. [1,4] with one measurement showing evidence of spatio-temporal coupling in the focal plane . Here we show the first complete characterization of the spatio-spectral distributions shown in the lower set of panels in Fig. 3 and these reveal the expected spatial focusing with no apparent changes in the interferometric pattern as a function of transverse position. Figure 3 demonstrates that a continuum of different shapes are present in the vicinity of the focus of a shaped laser pulse. The changing pulse shape near the focus provides one more reason for sampling a small section of the focal volume for any experiments employing pulse shaping to avoid averaging the response of atoms and molecules near the focus. Sectioning the focal volume can be implemented with pinholes and slits as are routinely used in the extraction region of a time of flight mass spectrometer .
Wigner functions are calculated to model the space-frequency and space-time coupling in the pulse after propagation through a pulse shaper. Phase or wavefront modulation paradigms can be used to describe the mode of operation of a pulse shaper and each has a different effect on spatio-temporal coupling. Pure phase modulation is only feasible mathematically and is not possible in practice due to space-frequency mapping in the Fourier plane of the pulse shaper. However, we model the spatio-spectral pulse distributions through the focus with Fourier optics to compare the (abstract) effect of pure phase modulation to wavefront modulation [8,35]. Analytical solutions are obtained for a shaped pulse propagating through the focus of a lens. For these simulations, pixelization of the SLM is neglected and smooth functions are used to obtain analytic solutions. The methods used to obtain the electric-field description after each optical component have been described previously [1,4,8,9,35] and will be only briefly summarized here.
The electric field incident on the first grating of the 4f shaper setup is described by Eq. (1),9], the grating is included in the form in Eq. (2):Eq. (4):Eq. (3).Eq. (5) consists of three terms. The first term is the spectral envelope, the second term describes the frequency mapping onto the transverse spatial coordinate x, and the last term is the phase accumulated by propagating a distance of 2f.
Wavefront modulation shaping is described by A6x(x,ω), whereas pure spectral phase modulation is shown by A6ω(x,ω) in Eq. (6).9] is included by Eq. (7):
The outcome for spectral phase shaping is described by Eq. (8):Fig. 3 are present in the shaper output. The expression for wavefront modulation is shown in Eq. (9).1]. The quadratic phase factor βx appears in both the numerator and the denominator of the expression. The term in the denominator introduces spatial beam broadening along x transverse coordinate. βx is coupled with angular frequency, which, for sufficiently high βx values, will lead to spatial chirp in an out going beam. The last term, with the imaginary arguments of the exponential function, describes the phase as a function of both spatial position and angular frequency. The first two arguments describe the result of spatial shaping in Fourier plane on the spectral phase. Since δx and βx act like linear and quadratic spectral phases, respectively; we will refer to them as linear phase and quadratic phase through the rest of the paper. The next argument of the exponent is a function of both position and frequency, which will lead to spatio-temporal coupling of the spectral and spatial phases. Furthermore, the remaining two terms are functions of position that can be regarded as a wavefront description. The last three arguments of the exponent are constant with respect to position and frequency and we can therefore omit them in any further analysis. The impact of each of the arguments will scale with their k coefficients.
Equation (9) describes the spatio-spectral properties of the field directly after the final pulse shaper grating. The equation is useful for demonstrating the origin of spatio-spectral coupling but, practically speaking, experiments are usually carried using a focusing geometry. The electric-field for a focusing geometry is obtained by propagating the field to and through a lens, including the lens-induced wavefront curvature, and then propagating to the focus. The Fourier transform of the spatio-spectral electric-field yields spatio-temporal distribution. An analytical solution for the electric-field in both the spectral and temporal domains with first- and second-order phase parameters allow for a theoretical examination of spatio-spectral and spatio-temporal coupling effects. We also analyzed the impact of the position of the focusing element relative to the position of the last shaper element. A lens placed at the focal distance (f) from the grating directly images the pulse at the output grating of the shaper onto the focal plane of the lens .
Regardless of any preexisting spatio-temporal coupling in the laser beam, propagating the beam through a lens modifies the spatio-temporal profile of a shaped and unshaped pulse. When a linear phase is applied in the shaper Fourier plane the beam is relayed in the transverse coordinate after the pulse shaper. Translation of the beam on the face of a lens, after the shaper, will not affect the transverse spatio-spectral profile at the focus, but the beam outside the focal plane will be shifted off the center line of the lens. Moreover, beam relaying has unexpected consequences for the generating of multiple pulses, as they can be described as a superposition of pulses with different linear spectral phases . Each pulse will be vertically shifted in proportion to their relative delay outside the focal plane and only in the focal plane will the intended spatial pulse structure be present.
The comparison of the consecutive Wigner functions along the beam focus for a focusing geometry where the lens is placed a distance of 10f is shown in Fig. 4 . The simulation and experimental parameters match and are summarized in Table 1 . The focusing geometry affects only the beam divergence compared with a lens placed one focal length from the shaper grating (not shown). The spectral plots show the characteristic amplitude modulation of a double-pulse. The space-time plots show the measured shift from Fig. 3 of the pulses in transverse coordinate before and after the focus. The magnitude of the shift is proportional to the delay and changes sign after the focal point.
To gain a better understanding of the evolution of the spectral phase in the presence of spatio-temporal coupling, Fig. 5a displays the measured spatio-spectral distribution of the beam at 2z before the focus. The insets in Fig. 5 are slices of the spectral phase and amplitude at different transverse positions indicated by the solid red line. The spectral fringes in Fig. 5 do not tilt. The spectral phase is not constant across the transverse coordinate of the beam cross section and reflects the shift in position of the temporal sub pulses as seen before and after the focus in Fig. 3.
To model the measurements displayed in Fig. 5 the Wigner distributions (Fig. 4) are calculated and displayed in Fig. 6 for comparison. The spectral fringes of Fig. 6 do not shift along the transverse beam position in agreement with measured cross section in Fig. 5. The sign of the spectral phase is shifted from the top half of the beam relative to the bottom half. Figure 6 is in good agreement with the measured cross section of Fig. 5 for both phase and amplitude.
3.2 Spatio-temporal coupling for multi-pulse trains, sine phase modulation
The measurements of the spatio-spectral and temporal amplitude of a sinusoidal phase modulation (spectral phase amplitude of +/−π) are shown in Fig. 7 as a function of propagation distance through the focus. The upper panels display the measured space-time profiles and reveal a transverse shift in the spatial position of the pulse train across the beam front. The shift decreases as the beam approaches the geometric focus and at the focus, the maxima in the pulse train are aligned through the zero spatial position. On the positive side of the focus transverse shifting occurs in the opposite direction in comparison to the negative side.
The spatio-spectral features shown in the lower panels of Fig. 7 reveal a tilt with respect to the transverse coordinate before and after the focus. The tilt of the spectral features can be attributed to the addition of the parabolic wavefront from the lens with the maxima and minima of the sinusoidal spectral phase. In the spectral domain the tilt angle in the transverse beam coordinate of each feature decreases as the beam approaches the focus. As the beam defocuses, the features in the spectrum again acquire a tilt in the transverse position however, and are inverted with respect to the tilt before the focus.
To simulate the propagation of the sinusoidal spectral phase through the focus Fourier optical analysis is used to calculate an analytical description of the field with the Jacobi-Anger identity in Eq. (10) :
A sinusoidal spectral phase modulation can be simulated by summation of pulses with linear phase ramps defined as and amplitudes given by the Bessel function. Because decays very rapidly as it is reasonable to limit summation to The results from the simulation are presented in Fig. 8 . The upper panel of Fig. 8 displays the calculated space-time Wigner distributions before, at, and after the geometric focus. The superposition of phase ramps described by Eq. (9) leads to displacement of the sub-pulses across the transverse beam coordinate similar to the double-pulse experiment. The space-frequency Wigner plots presented in Fig. 8 also contain the saw-tooth like spectral features which evolve through the focal plane. The saw tooth distribution can be derived from the interference of series of pulses equally spaced in time, shifted in transverse coordinate and with the intensity governed by a Bessel function. The space-time Wigner plot at z = f + zr can be obtained by reflection of the distribution at z = f-zr through the t = 0 plane. The spatio-spectral features on the –zr side of the geometric focus are the reflection around ω0 frequency of the features at the zr side of the focal plane, which is in qualitatively agreement with the experimental results in Fig. 7.
The measured spectral amplitude as a function of transverse position for the sinusoidal spectral phase shaped pulse at a position before the geometric focus is shown in Fig. 9a . Figure 9b displays spectral phase and amplitude lineouts for three different transverse beam positions highlighted by the red lines in 9a. Each transverse position contains a different spectral amplitude distribution and the spectrum changes as the beam propagates through the focus. The spectral phase lineouts remain constant as a function of transverse beam coordinate. Using the Wigner distributions of Fig. 8, a cross section of the sinusoidal phase distribution at one focal position is shown in Fig. 10 to qualitatively compare with experimental results in Fig. 9.
The simulations reveal that the spectral amplitude is not constant as a function of transverse coordinate, as shown in Fig. 10b. The spectral features in 10b and d, increase in modulation depth, as the distance from the center of the transverse position in the beam increases. In addition the spectral features also gradually shift spectral position as a function of the spatial position in the transverse direction across the beam. Each half of the transverse direction across the beam is shifted by an equal amount in the spectral domain; however the shift is in the opposite direction in frequency space. Despite the non-constant spectral amplitude across the beam front the spectral phase remains unchanged regardless of position in the beam showing good agreement between the simulation and measurement.
The tilted spectral features are an unintuitive effect and are qualitatively the result of the addition of the parabolic wave front from the lens with the sinusoidal spectral phase. Each spectral phase peak and trough roughly approximates a parabola, and may add with the parabolic wave front of the lens to produce a local lensing effect. Addition of the spectral phase to that of the lens may lead to the saw-tooth like pattern in the spatio-spectral cross sections. Our measurements and calculations demonstrate that the spectral phase of the pulse and wave front of the lens are tightly coupled.
We have theoretically and experimentally investigated focusing shaped pulses. The experimental measurements were performed using scanning SEA TADPOLE to determine the temporal electric-field intensity distribution as a function of transverse and longitudinal position. We have shown that shaped pulses have significant spatio- temporal and spatio-spectral coupling through the focus. For example, spatio-temporal coupling leads to a gradually changing spectral phase through the focus resulting in translation in the transverse position as a function of propagation length. Consequently, the desired double pulse structure is present in the immediate region of the focus and a continum of pulse shapes dominate the regions outside the focus. We observe for the first time that pulses with sinusoidally-modulated spectral phases result in a saw tooth spatio-spectral pattern. The saw tooth pattern gradually corrects as the beam approaches the focus and the spectral features are vertical in the vicinity of the focus. After the focus the features again tilt in the spatio-spectral domain. The spatio-spectral saw-tooth pattern is inverted through the zero transverse position in the beam. The spectral phase, however, does not change in the transverse beam coordinate.
The spatio-temporal coupling of focusing of shaped laser pulses is important to characterize for extracting mechanisms from coherent control experiments. The spatial profile of the pulse shape should be taken into consideration when considering mechanisms for coherent control experiments. Care should be taken to actively select the most constant portion of the focusing beam with respect to the changing spatio-spectral beam profile for example, by using an aperture.
Authors would like to thank Professor Rick Trebino and Dr. Pamela Bowlan for their help with SEA TADPOLE and Dr. S. M. Weber for the parametric shaping algorithm. This work was supported by grants from the National Science Foundation CHE No. 331390111 and the Army Research Office No. 311390121.
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