1. INTRODUCTION

Shaping of ultrashort laser pulses is a well-established technique [1 –4] and has many applications as, for example, in coherent control [5 –10], ultrafast spectroscopy [11 –15], and microscopy [16,17]. However, most of the conventional setups are limited in their shaping capabilities. Typically either only the phase and amplitude [18] or phase and polarization [19,20] can be modulated. The benefit of polarization shaping was, for instance, demonstrated in coherent control experiments [21 –25]. But the fact that amplitude and polarization cannot be controlled simultaneously with these setups limits the producible polarization states and the generation of multipulse sequences, which are, for example, employed in coherent multidimensional spectroscopy [12,14,15,26,27].

There are two different approaches to achieve complete vector-field control, i.e., independent control over all four degrees of freedom (phase, amplitude, orientation, and ellipticity) of an ultrashort laser pulse. The first method is to shape a single laser pulse directly in amplitude, phase, and polarization. The simplest realization of this is to use a 4-layer spatial light modulator (SLM) with a polarizer sandwiched inside the four layers in a $4f$ setup instead of a 2-layer SLM as necessary for amplitude and phase shaping. One possible implementation would be to control the amplitude with the first two SLM layers and the phase and polarization state with the second pair [28]. However, for the first experimental realizations a 4-layer SLM was not available and thus had to be substituted by other devices. This was done by using up to three SLMs [28,29] or by multipassing a 2-layer SLM [30] or a 2D-SLM [31]. It is worth noting that complete control of the vector field without limitations was so far only possible by using three consecutive SLMs [28]. A custom made 4-layer SLM (Cambridge Research & Instrumentation) was reported [32], however, without a sandwiched polarizer and therefore without the ability to shape the amplitude.

The second approach to vector-field control is to modulate two perpendicularly polarized pulses in two independent arms of an interferometer in phase and amplitude and to recombine these into a single laser pulse. This is, in most cases, achieved by manipulating both pulses simultaneously within one $4f$ setup in the frequency domain and using different pixel groups of the same SLM for the two polarizations [33 –37]. Instead of a pixelated SLM, also a frequency-domain acousto-optical modulator can be used [38]. It is also possible to modulate the two arms with two independent devices. This allows the use of time-domain acousto-optical pulse shapers in an interferometric setup [39]. To spatially separate the two pulses within a single $4f$ setup one can either send the two beams, under different angles of incidence, onto a grating [33,34,37,38] or a prism [35]. Instead of a normal prism, birefringent prisms can also be used [36]. In this case no additional optics are needed to generate the two perpendicularly polarized pulses. The major drawback of all these setups, compared with the 4-layer SLM approach, are the stringent requirements regarding the temporal and spatial stability between the two polarization components. This stability is indispensable since a drift in the relative phase or temporal delay of both components would result in a change in the polarization state of the resulting polarization-shaped pulse. To reduce the phase drift, a common-path setup [34 –36] or an active phase stabilization can be utilized [37 –39].

In this work we present an interferometric setup allowing broadband (740–880 nm) vector-field shaping with a high inherent phase stability. While vector-field shaping has been implemented previously in the literature, as summarized above, in the present work we place particular emphasis on designing the setup such that a number of artifacts are minimized allowing the generation of high-quality predefined polarization pulse sequences. By using a thin-film polarizer for polarization splitting and recombination phase distortions across the beam profile are minimized. We demonstrate a routine to compensate the remaining temporal and phase drift between both arms of the interferometer utilizing the SLM of the pulse shaper. By employing a polarization-quadruple-pulse basis we are able to produce deterministic multipulse sequences with high accuracy.

2. DESIGN OF THE VECTOR-FIELD SHAPER

Since 4-layer SLMs are still not commercially available, we use the concept of shaping two perpendicularly polarized pulses and recombine them in an interferometric common-path setup for improved stability. In a common-path setup the two beams are modulated with different pixel areas of the same SLM, as sketched in Fig. 1(a). The spatial separation of both beams is realized by using two different angles of incidence onto the grating of the $4f$ setup. The grating disperses both beams into their spectral components, which get collimated and focused at the SLM by a cylindrical lens. To improve the shaping capabilities of the setup, the optimization of the frequency distribution across the SLM pixels is crucial. In general, the sampling points in the frequency domain, the shaping window, and the replica pulses for one pulse are given by this distribution [40,41]. Therefore, both polarization components should have a similar frequency distribution and span as many pixels as possible without spatial overlap. (Birefringent) prism-based designs do not offer enough degrees of freedom to achieve this and thus a grating-based setup is preferred.

 

Fig. 1. Design parameters of the setup. (a) Definitions of all relevant quantities for calculating the optimal set of parameters of the setup (see text for detailed explanation). Two beams [beam 1 (orange), beam 2 (blue)] hit the grating of a $4f$ setup under two different angles of incidence ${\mathrm{\Theta }}_{\mathrm{i}}^1$ and ${\mathrm{\Theta }}_{\mathrm{i}}^2$. Their diffracted spectral components are collimated by a cylindrical lens and mapped onto the pixels of the spatial light modulator (SLM). (b) Graphical solutions to Eq. (11a) (blue line), Eq. (11b) (red line), and Eq. (11c) (green line) for two different angles ${\delta }_{\mathrm{in}}$. In the case of ${\delta }_{\mathrm{in}}=10.76°$, all three lines intersect at one point and all three equations are fulfilled for $\beta =-46.57°$ and $g=892.60{\mathrm{mm}}^{-1}$. For a slightly different value of ${\delta }_{\mathrm{in}}=10.26°$, only two of the three equations can be fulfilled.

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The first parameter of the pulse shaper to be determined is the focal length $f$ of the cylindrical lens within the $4f$ setup. The focal length is independent of all other setup parameters and only depends on the used beam radius within the $4f$ setup. The maximum beam radius $w$ is limited by the pixel height of the SLM (in our case 10 mm) due to possible clipping at the SLM. Considering a Gaussian beam profile, the beam radius should be smaller than $w=2.57\mathrm{mm}$ (defined by $1/e^2$ of the intensity) so that 99.9% of the intensity passes the SLM. The temporal shaping window of the pulse shaper is not only given by the frequency distribution but also influenced by the spot size of the spectral components at the SLM pixels [41]. As a rule of thumb, the spot size should be smaller than one third of the pixel width (in our case $\text{97\hspace{0.17em}}\mathrm{\mu m}$) to avoid narrowing of the shaping window due to the focus size. To reach this value, a cylindrical lens with $f=250\mathrm{mm}$ is used which results in a spot size $w_0<27\mathrm{\mu m}$ along the SLM pixels.

All other parameters of the $4f$ setup depend on each other given that the spectral distribution is a function of the angles of incidence onto the grating of both beams ${\mathrm{\Theta }}_{\mathrm{i}}^1$, ${\mathrm{\Theta }}_{\mathrm{i}}^2$ and of the grating frequency $g$. The optimal set of parameters can be found by numeric calculations. Figure 1(a) shows the relevant parameters for these calculations. The two different angles of incidence ${\mathrm{\Theta }}_{\mathrm{i}}^1$ and ${\mathrm{\Theta }}_{\mathrm{i}}^2$ can be substituted by the parameters $\beta$ and ${\delta }_{\mathrm{in}}$,

The two beams are dispersed by the grating. The angles of diffraction ${\mathrm{\Theta }}_{\mathrm{d}}^{1,2}$ are given by

These angles depend on the wavelength $\lambda$, the grating frequency $g$, the order of diffraction $m$, and the angle of incidence ${\mathrm{\Theta }}_{\mathrm{i}}^{1,2}$. The edges of the frequency distribution (${\mathrm{\Theta }}_{\mathrm{d},\min }^{1,2}$ and ${\mathrm{\Theta }}_{\mathrm{d},\max }^{1,2}$) are given by the minimal and maximal used wavelength:

The angle of aperture $\eta$ of the complete SLM pixel array is given by the width $w_{\mathrm{SLM}}$ of the pixel array and the focal length of the cylindrical lens $f$:

To avoid a spatial overlap of the spectral components of both beams there should be a defined gap $w_{\mathrm{gap}}$ between both spectral distributions. The angle of aperture $\xi$ of this gap is given by

The cylindrical lens and the center of the SLM should be placed in such a way that the angle of the optical axis of the $4f$ setup ${\mathrm{\Theta }}_{4f}^{\text{o.a.}}$ with respect to the grating normal matches the angle bisector of the inner angles of diffraction ${\mathrm{\Theta }}_{\mathrm{d},\max }^1$ and ${\mathrm{\Theta }}_{\mathrm{d},\min }^2$:

The cylindrical lens collimates the dispersed beams and focuses the spectral components. Thus, the position for each spectral component at the SLM can be approximated by the position on the cylindrical lens and can be calculated based on the wavelength-dependent diffraction angle ${\mathrm{\Theta }}_{\mathrm{d}}^{1,2}(\lambda )$ via

The goals of using the SLM pixel array in its full width [Eq. (11a)] without spatial overlap [Eq. (11b)] and with a similar frequency distribution [Eq. (11c)] are described by the following system of equations:

The setup is designed for a spectral range of 740–880 nm. The value $w_{\mathrm{SLM}}=64\mathrm{mm}$ is given by our SLM (Jenoptik SLM-S640d) and a gap of $w_{\mathrm{gap}}=1\mathrm{mm}$ is desired. A focal length of $f=250\mathrm{mm}$ is chosen. The graphical solutions of Eq. (11) for ${\delta }_{\mathrm{in}}=10.76°$ are shown in Fig. 1(b) (top). The three colored lines (blue, red, and green) represent Eqs. (11a )–(11c) and intersect exactly in one point, thus fulfilling all three goals. This results in the optimal parameters of

Since in practice these parameters cannot be achieved exactly with available optics, one has to make trade-offs. Experimentally, ${\delta }_{\mathrm{in}}=10.26°$ was realized and $\beta$ and $g$ were recalculated according to this. In this case, only two of the three equations [Eqs. (11a )–(11c)] can be fulfilled simultaneously [Fig. 1(b), bottom]. In general the consequence is that either the gap size differs from the target size or that some pixels at the SLM edges are not illuminated if both frequency distributions should cover the same amount of pixels. The priority should be: (i) frequency distribution, (ii) sufficient gap size, and (iii) illumination of the pixel array in its full width. The nearest commercially available grating frequency was $g=850{\mathrm{mm}}^{-1}$ and $\beta$ was chosen according to that. The resulting parameters for the presented setup are

3. POLARIZATION SPLITTING AND RECOMBINATION

To achieve a beam separation angle of ${\delta }_{\mathrm{in}}\approx 10°$ one option would be to use a calcite Wollaston prism as in previous designs [34,35]. Figure 2(a) shows a schematic of the polarization separation by a Wollaston prism. The Wollaston prism consists of two orthogonal prisms with perpendicular optical axes. The polarization of a laser beam which is perpendicular to the optical axis is influenced by the ordinary refractive index $n_{\mathrm{o}}$ and the parallel polarization component by the extraordinary refractive index $n_{\mathrm{e}}$. Due to the change in the optical axis at the intersection of the two prisms the beam will be split up into its parallel (p) and perpendicular (s) polarization components: the p component is influenced by $n_{\mathrm{o}}$ in the first half of the Wollaston prism and by $n_{\mathrm{e}}$ in the second half and vice versa for the s component. Wollaston prims feature a high extinction ratio of $>{10}^6:1$ and it is a practical advantage that the separation angle depends only on the angle between both prisms. However, the high birefringence ($\mathrm{\Delta }n\approx 0.166$) and dispersion of calcite has some disadvantages when used in a pulse shaper as described in the following.

 

Fig. 2. Principle of the Wollaston prism and TFP for polarization separation and recombination. (a) The Wollaston prism consists of two birefringent prisms with perpendicular optical axes. The separation of the incoming pulse (red) into a p- and s-polarized pulse (orange/blue) is based on refraction and offers a very high extinction ratio. (b) A TFP is a thin glass substrate (gray) with special coating (green) on both sides. The p-polarized component of the incoming beam (red) is transmitted (orange) and the s polarization is reflected (blue) with the given efficiencies. Multiple reflections inside the TFP lead to minor satellite pulses which are delayed by 3.6 ps. (c), (d) The back-going beams are recombined (red dotted beams) by passing the TFP a second time. The polarization states of the back-going beams determine the intensity which is emitted on both sides of the TFP. If the polarization of both beams is not changed by the SLM in the pulse shaper, the maximum intensity will be emitted on the side of the incident beam (side A) and the minimal intensity on the other side (side B) [(c)]. If the polarizations are rotated by 90°, this behavior is switched [(d)]. Due to this effect amplitude shaping is realized for each beam. Either the beam emitted at side A or B can be used as the shaped beam. The double-pass extinction ratio at side A is $43521:1$ for beam 1 and $5605:1$ for beam 2 [(c)] and $15619:1$ for both beams at side B [(d)].

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The first disadvantage of Wollaston prisms is that the separation is not symmetric for both beams. The exit angle for the s polarization is about 3% larger than the angle for the p polarization. This can be compensated by the alignment of the subsequent optics. However, the wavelength dependency of the refraction at the intersection of both calcite prisms and at the exit plane will lead to an angular dispersion of the exit beams. The difference in the exit angle for the lowest and the highest wavelengths is about 1.5% for 740–880 nm and is also asymmetric. The distance the beam travels as an ordinary or extraordinary beam depends on the beam position in the Wollaston prism. Therefore, there will be an additional inherent varying dispersion along one dimension of the beam profile. The mean value of this dispersion can be compensated with the pulse shaper but not the spatial variation of the dispersion. For instance, the remaining group-delay dispersion across a beam radius of $w=2.57\mathrm{mm}$ is of the order of $\pm 110{\mathrm{fs}}^2$ (note that this value is actually independent of the Wollaston size and depends only on the beam radius, the angle between the two halves of the Wollaston prism, and its material). The Wollaston material ${\mathrm{MgF}}_2$ is for example quasi-achromatic but its small birefringence ($\mathrm{\Delta }n\approx 0.011$) [42] results in a much lower separation angle and therefore is not suited for the presented setup.

Due to these disadvantages, a thin-film polarizer (TFP) is used for beam separation and recombination in this work. The separation angle of a TFP depends on the angle of incidence. Standard small-band TFPs can be used with ${\alpha }_{\mathrm{in}}>40°$ and offer an extinction ratio of about $200:1$. For broadband applications higher angles of incidence are needed resulting in lower extinction ratios. Their performance strongly depends on the used coating. The tailor-made TFP (Laseroptik GmbH) employed here is specified for an angle of incidence of 77° which results in a beam separation angle of $\delta =26°$ [Fig. 2(b)], which will be demagnified to ${\delta }_{\mathrm{in}}=10.26°$ (see Section 4). The glass substrate (wedge angle $<{0.2}^{\prime \prime }$) is coated on both sides to provide an extinction ratio of $210/75:1$ (p/s) for a single pass. As shown in Fig. 2(b), the transmitted beam (beam 1, orange) after the TFP is mainly p, the reflected one (beam 2, blue) primarily s polarized. One drawback of a TFP consists of multiple transmitted and reflected beams due to internal reflections inside the TFP. The intensity of the second transmitted beam is about 0.00/0.42% of the incoming p/s component and for the second reflected pulse about 0.28/0.42%. The consecutive post pulses are mainly s polarized with decreasing intensity. These multiple pulses overlap partially in space due to the finite beam size, but they are separated in time. The delay between consecutive pulses depends on the TFP thickness and is calculated to be about 3.6 ps. The small amplitude and the relative large temporal separation of these satellite pulses therefore do not hinder an application of the TFP in the pulse-shaper setup. The values for the pulse intensities are gained by numerical calculations. For these calculations polarization-dependent measurements by the manufacturer of the total transmission with coating at one and both sides were taken into account. Under the assumption that up to six consecutive pulses contribute to the total measured intensity, the transmission coefficients for the transition air-coating-substrate and substrate-coating-air for both polarizations were calculated for the center wavelength of 810 nm. Since the beam separation is not based on refraction, no angular dispersion occurs when using a TFP. The only chromatic effect is a slight parallel displacement of different wavelengths. However, this chromatic displacement is in the order of $0.3\mathrm{\mu }\mathrm{m}$ for 740–880 nm and is therefore negligible.

Following the pulse-shaping procedure via the SLM in the $4f$ setup (see below), the same TFP generates two outgoing beams which can be used as shaped beams. Either the beam is used which is emitted on the same side as the incident one [side A, Figs. 2(c) and 2(d)] or the emitted beam on the opposite side can be used [side B, Figs. 2(c) and 2(d)]. The first variant (using the outgoing beam from side A) has the advantage that the beam path inside the TFP is inverted, which therefore generates a perfect spatial overlap of both beams. By passing the TFP twice, the transmitted beam is slightly more dispersed compared to the reflected beam. This must be compensated with the pulse shaper. The maximum intensity at side A is emitted if the polarization of both beams is not changed [Fig. 2(c)]. The differing efficiencies [Fig. 2(c)] for both beams can also be compensated by the pulse shaper or by adjusting the intensity of the polarization components of the incoming beam. Due to the imperfect extinction it must be considered that the p/s component of the exit beam is not only given by the transmitted/reflected beam but is additionally partly influenced by the respective other beam. This cross talk is, however, only 0.016% (p) and 0.0026% (s) and can thus be neglected. The double-pass extinction ratio is $43521:1$ for the transmitted beam (orange) and $5605:1$ for the reflected beam (blue). Shaping the amplitude of each spectral component is done by changing their polarization state with the SLM. The minimum exit intensity at side A is reached when each polarization is rotated by 90°, respectively [Fig. 2(d)]. The expected minimal outgoing intensity for perfect shaping is about 1.59% for both beams as shown in Fig. 2(d). However, this polarization rotation is not possible in the pixel gaps of the SLM. These gaps cover 3% of the SLM array width and therefore the expected minimum intensity for zero transmission would be in practice around 4.5%. Note that in this configuration the shaped pulse has to be tilted in height to separate it from the incoming pulse.

In the second variant that beam is used which exits the TFP at the opposite side as the incoming beam [side B, Figs. 2(c) and 2(d)]. In this case, the polarization of each arm has to be rotated by 90° to get the maximum output intensity [Fig. 2(d)]. The benefit of this configuration is that not only the efficiencies for beam 1 and beam 2 are automatically equalized, but also their extinction ratio. The dispersion introduced by the TFP is also equal for both beams. The transmission for both beams is 90.7% with a negligible cross talk of 0.0064% and an extinction ratio of $15619:1$. The minimum output intensity is $\approx 2.42\%$ (p) and $\approx 0.86\%$ (s) [Fig. 2(c)] and is not increased by the intensity passing the pixel gaps. Another advantage is that no tilt in the beam height is necessary. One downside of this variant is that the beam paths are not inverted: the initially transmitted/reflected beam is reflected/transmitted on the second pass. This results in a minor lateral offset between both exit polarizations. This displacement depends on the angle of incidence and the thickness of the TFP and is in our case 0.20 mm. For the presented design this last configuration is used. This variant is equivalent to the use of the perpendicular entrance and exit polarizers in a nonfolded amplitude pulse shaper and thus called a crossed-polarizer arrangement.

4. EXPERIMENTAL SETUP

The setup of the vector-field pulse shaper is depicted in Fig. 3(a). The incoming beam (solid red line) is collimated in a $1:1$ telescope. With a pair of glass wedges (GW1) a reference (transparent line) is separated. This has the advantage in comparison to a standard beam splitter that multiple reflections are emitted at different angles and can thus be separated. To transfer the incoming p-polarized pulse to a pulse with equal intensity of the p and s component, a combination of a $\lambda /2$ plate and a 45° periscope is used. The $\lambda /2$ plate can also be used to compensate for different efficiencies between the two arms of the setup. This rotated beam is split up by the TFP. The p polarization is transmitted (beam 1, orange) and the s polarization is reflected (beam 2, blue). Both beams pass a magnifying telescope and are directed onto a volume-phase-holographic grating (Wasatch Photonics) optimized for two different angles of incidence. By using the telescope instead of mirrors to direct the beams onto the grating a (near-)common-path setup is given. The second purpose of the telescope ($f_1=51.6\mathrm{mm}$ and $f_2=125\mathrm{mm}$) is to reduce the separation angle of 26° after the TFP to 10.26° in front of the grating. The beam radius in front of the TFP must be adjusted to result in a radius of $w<2.57\mathrm{mm}$ after the telescope. Due to the non-normal incidence on the lenses the separation angle after the telescope and the beam radius magnification were calculated by ray-tracing simulations, so that the adequate focus length ratio could be chosen. The $4f$ setup is composed of the volume-phase-holographic grating, a cylindrical lens, a 2-layer SLM with 640 pixels, and the folding mirror.

 

Fig. 3. Setup schematics. (a) Vector-field shaper, see text for a detailed description. The key elements to achieve vector-field control of the incoming pulse (solid red line) are a TFP to generate two perpendicularly polarized pulses and the subsequent telescope to direct them onto the grating of a $4f$ setup in a common-path setup. The transmitted (beam 1, orange) and reflected (beam 2, blue) beams are shaped individually in amplitude and phase and recombined by the TFP to a single beam (red dotted line). The reference beam line (transparent) is used for pulse characterization via dual-channel FTSI. (b) Spectrometer setup. To simultaneously characterize both polarization components of the shaped pulse, two identically configured high-resolution spectrometers are employed. The polarization separation is achieved by a beam-splitter cube and two linear polarizers.

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The shaped pulse (red dotted line) is extracted in the crossed-polarizer arrangement at the TFP, so that for maximum transmission the polarization of each arm has to be rotated at the SLM. This has the additional advantage that also the polarization-dependent efficiencies of the grating and cylindrical lens are automatically compensated. The incoming and outgoing beam can also propagate at the same height parallel to the table. This leads to a huge improvement on the beam profile compared to a tilting of the outgoing beam. By exactly inverting the beam paths, optical aberrations introduced by the telescope due to the small focal lengths and the non-normal incidence can be almost fully compensated.

The shaped pulse and the reference are recombined using a second pair of glass wedges (GW2). The reference is used to characterize the shaped pulse via dual-channel Fourier-transform spectral interferometry (FTSI) [20,43 –45]. It is temporally delayed by a time $\tau$ with respect to the shaped beam using a mechanical stage and its polarization is rotated to 45° and cleaned with a combination of a $\lambda /2$ plate and a linear polarizer. The neutral density (ND) filter wheel allows the tuning of the intensity ratio between the shaped pulse and the reference. The power throughput after all optics shown in Fig. 3(a) is 10%. This could, for example, be improved by coating both pairs of glass wedges and by using a nonstandard antireflective coating for lens 1 and lens 2 customized to the used angles of incidence.

A Ti:Sa oscillator [Coherent Mira Basic, 797 nm, 13.6 nm (FWHM), 61 fs (FWHM)] serves as the light source. However, note that the vector-field shaper itself is designed for a future use with a broadband laser system and therefore is specified for 740–880 nm.

For pulse characterization, the two polarization components are separated with a beam-splitter cube and two orthogonal linear polarizers [Fig. 3(b)] and focused into identical high-resolution spectrometers (Ocean Optics HR4000). The polarization-dependent efficiency of this setup is taken into account by comparing the measured spectra with the separately measured power of both components in front of the spectrometer setup. Using two spectrometers in parallel, rather than performing subsequent measurements for p and s polarization with the same spectrometer, reduces the requirements for interferometric stability between the shaped beam and the reference. If only one spectrometer is used, fluctuations in the time delay $\tau$ between the measurements of the p- and s-polarized components might lead to errors in the characterized total polarization state.

5. PROPERTIES OF THE VECTOR-FIELD SHAPER

Figure 4 compares the measured (orange and blue circles) wavelength distributions as a function of the SLM pixels with the design goal (solid black line, Section 2). Both beams have a similar frequency distribution in the measured wavelength range of 778–817 nm. Extrapolation of the data (red circles) shows that the wavelength range of 740–880 nm covers nearly the whole SLM array without spatial overlap of spectral components. For the simulation the calculated parameters ${\delta }_{\mathrm{in}}$, $\beta$, and $g$ [Eq. (13)] were used. Both data sets agree very well. The differences are either due to simplifications in the simulation, errors in the extrapolation, or unavoidable deviations in the alignment of the setup. We note, however, that for using the pulse shaper always the actual experimentally obtained wavelength calibration is employed.

 

Fig. 4. Comparison of the simulated (black) and measured [orange (beam 1) and blue (beam 2)] wavelength distribution across the SLM pixels. The calibration was measured (orange and blue circles) in the range of 778–817 nm (gray dashed lines) and then extrapolated using a second-order polynomial fit (red circles). In the measured range, both beams span 83 pixels. For 740–880 nm they cover 295/293 (orange/blue) pixels. The simulation for the optimal parameters [Eq. (13)] predicts a range of 85/84 (orange/blue) pixels for 778–817 nm and 301/301 (orange/blue) pixels for 740–880 nm. For better visibility only every fifth data point is plotted.

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The interferometric stability of the setup over almost 24 h is demonstrated in Fig. 5(a). As a benchmark for the stability we use the phase difference between the p and s component of the shaped pulse at the center frequency of the compressed pulse. Since the p component is generated with beam 2 and the s component with beam 1 this is a direct measure for the relative phase stability between both beams. The pulses were characterized via dual-channel FTSI. The pulse was initially compressed by applying the inverted phase of the uncompressed pulse to the SLM. For the blue curve [Fig. 5(a)] only this initial phase offset was applied to the SLM. The phase difference at $t=0$ is larger than zero because the relative phase has already changed with respect to the original pulse compression. The blue curve shows a significant long-term drift of the relative phase but also a very high stability along shorter time scales (see the inset). The standard deviation of the relative phase is $\sigma =28.3\mathrm{mrad}$ $(\approx \lambda /222)$ in this period of 60 min. Similar values are reproduced at different temporal sections of this measurement. Due to the long-term drift, we implemented an on-the-fly phase reduction and stabilization (OPRAS) utilizing the SLM. Using this routine, the remaining phase of the pulse is automatically calculated after each point of the blue curve via a dual-channel FTSI measurement and added to the previous offset phase. The pulses measured with this new offset phase show a much lower phase difference and an improved stability (green curve). By a second iteration of OPRAS (red curve) the relative phase was stabilized to $\sigma =31.9\mathrm{mrad}$ $(\approx \lambda /197)$ with a remaining mean value of 28.5 mrad over almost 24 h [Fig. 5(b)]. After this process, up to four different pulse shapes were generated and measured (Section 6.B). This explains the relative long time between subsequent data points of the blue curve ($\approx 94\mathrm{s}$). The needed time for OPRAS is mainly given by the integration time of the spectrometers and not by the data evaluation. By reducing this time and measuring only the essential spectra, one could significantly improve the speed of the stabilization. An improvement of the automatic pulse evaluation could also make the second iteration redundant. Since the setup shows a very good short-time stability, it would also be sufficient for time-critical measurements to repeat OPRAS at longer time intervals and not for every data point. The SLM temperature [Fig. 5(c)] basically shows the same trend as the curve for the unstabilized phase [blue, Fig. 5(a)] but slightly delayed. Hence, it can be assumed that the drift in the relative phase is mainly due to temporal fluctuations in the laboratory which result in delayed temperature variations of the SLM. The advantages of OPRAS are that no additional stabilization hardware and no moving mechanical parts are required. OPRAS cannot only compensate a temporal drift, but in general phase fluctuations of any order, due to the complete characterization of the pulse. Employing dual-channel FTSI, information about the relative phase between both polarization components is directly gained, since the whole vector field is simultaneously characterized. This also eliminates the necessity to characterize or compress the pulse in advance and makes this routine highly suitable to stabilize the output of an interferometric vector-field shaper.

 

Fig. 5. Phase stability of the setup. (a) Stability over nearly 24 h. As a quantity for the phase stability, the spectral phase difference at the center frequency between both polarizations is used. In blue, the unstabilized phase difference is shown. In green and red, the first and second iterations of an OPRAS (see text) using the pulse shaper itself are plotted. The unstabilized curve (blue) reveals a significant long-term phase drift, but good short-term stability. The inset shows a period of 60 min with a standard deviation of $\sigma =28.3\mathrm{mrad}$ $(\approx \lambda /222)$. (b) By employing two iterations of OPRAS, a long-term phase stability of $\sigma =31.9\mathrm{mrad}$ $(\approx \lambda /197)$ over nearly 24 h (iteration 2, red) is achieved. The phase difference is reduced to a mean value of 28.5 mrad (gray line). (c) SLM temperature over the course of the measurements. Note that the small gaps in the data set arise from loading times of new measurement parameters.

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6. PULSE MEASUREMENTS

A. Verification of OPRAS

The polarization state of a pulse is given by the time-dependent phase difference and intensity ratio between the p- and s-polarized components. The actual polarization state can be described as an ellipse (ellipticity $ϵ$) with a certain orientation $\theta$ of its major axis [46]. The compressed pulse generated by OPRAS should not only have a flat spectral phase but also a relative phase of zero between both polarization components. The resulting pulse in the time domain should then be short and linearly polarized (ellipticity $ϵ(t)=0\mathrm{rad}$). The actual orientation $\theta$ of the polarization ellipse depends on the intensity ratio $I_{\mathrm{p}}(t)/I_{\mathrm{s}}(t)$. This ratio was adjusted to be $\approx 1$ and therefore the orientation should be $\theta (t)=0.25\pi \mathrm{rad}=45°$.

The stabilized pulse via OPRAS was characterized via second-harmonic-generation frequency-resolved optical gating (SHG FROG) [47]. The trace for the p polarization [Fig. 6(a)] of the stabilized pulse as well as the trace for the s polarization [Fig. 6(b)] indicate that the pulse compression was successful. The reconstructed phase for the p [Fig. 6(c)] and s polarization [Fig. 6(d)] agree very well with the phases retrieved by the FTSI characterization. Also, the reconstructed and measured spectra match in their overall shape. However, both measured spectra exhibit slight oscillations. These oscillations are lower in amplitude for the p polarization [Fig. 6(c)] than for the s polarization [Fig. 6(d)]. These oscillations arise from multiple reflections inside the TFP (Section 3). Fourier transformation of the measured spectra and the FTSI phase show that the first subpulse appears 4.05 ps after the main pulse. The difference between the calculated delay of 3.6 ps and the measured value can be a result of an aberration of the desired angle of incidence onto the TFP and the negligence of the coating for the calculations. The intensity ratio between the first subpulse and the main pulse is $I_{\mathrm{p}}^{\mathrm{sub}1}/I_{\mathrm{p}}^{\mathrm{main}}=0.010\%$ for the p and $I_{\mathrm{s}}^{\mathrm{sub}1}/I_{\mathrm{s}}^{\mathrm{main}}=0.18\%$ for the s polarization.

 

Fig. 6. Characterization of the stabilized pulse via SHG FROG. (a) Measured FROG trace of the p and (b) s polarization. (c), (d) Reconstructed (red) spectrum (solid lines) and phase (circles) compared with the spectrum and phase measured via FTSI (blue) of p [(c)] and s polarization [(d)]. The FROG error is 0.26%/0.64% [(a/b)].

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The information about the relative phase between both polarizations, their relative intensity, and thus the ellipticity and orientation of the pulse, cannot be gained by these FROG measurements due to the ambiguity of the SHG FROG technique regarding the absolute phase of laser pulses [47]. But this information can be extracted for a compressed pulse [Fig. 7(a)] by measuring the intensity of the pulse behind a linear polarizer as a function of the polarizer angle [Fig. 7(b)]. The minimum normalized intensity after the linear polarizer is 0.023 for an angle of 134.43°. This results in $|ϵ|=0.152\mathrm{rad}$ and $\theta =0.773\mathrm{rad}=44.43°$. The pulse in the time domain was calculated by Fourier transformation of the spectral data gained via FTSI. Both measurements show a pulse which is nearly linearly polarized with $\theta \approx 45°$, but with a difference of $\mathrm{\Delta }ϵ=0.13\mathrm{rad}$ for the ellipticity [Fig. 7(c)]. This difference of the ellipticity corresponds for this pulse to a change in the relative phase of $\mathrm{\Delta }{\varphi }_0\approx 0.26\mathrm{rad}$ [46], which is probably due to a slight temporal phase drift between OPRAS and the polarizer measurement, caused by the imperfect stability of the setup [see Fig. 5(a)]. Also, the imperfect extinction ratio of the polarizer and the mainly s-polarized subpulses emitted from the TFP contribute to a remaining intensity behind the polarizer and hence to the discrepancy in $ϵ$. We conclude from the FROG and the polarizer measurements that the OPRAS routine is correct and an efficient tool to characterize, compress, and stabilize polarization-shaped laser pulses. The coherent superposition of the s and p components [from beams 1 and 2 in Fig. 3(a)] leads to a single linearly polarized and compressed pulse.

 

Fig. 7. Characterization of the stabilized pulse by rotating a linear polarizer. (a) Intensity (solid line) and phase (circles) of the p- (blue) and s-polarized (red) components in the time domain, gained by FTSI analysis. (b) Measured intensity of the pulse after a linear polarizer as a function of the polarizer angle (black crosses). The orientation of $\theta =0.77\mathrm{rad}=44.43°$ and the ellipticity of $|ϵ|=0.15\mathrm{rad}$ is extracted by fitting a squared sinusoidal (red). (c) Comparison of the FTSI analysis (red dots), the mean of the FTSI analysis (black dot), and the polarizer measurement (black cross) in a Poincaré plot [46].

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B. Polarization-Shaped Multipulse Sequences

An example of a pulse shaped in amplitude, phase, and polarization is given in Figs. 8(a)–8(d). The parameterization of the pulse sequence is based on the quadruple-pulse basis described in [15] and extended for polarization control based on [46,48]. Quadruple-pulse sequences are particularly relevant for coherent two-dimensional spectroscopy. The relative temporal delay and the relative total phase between four subpulses can be defined. In addition, for each subpulse the total intensity, ellipticity, and orientation in the time domain can be specified.

 

Fig. 8. Double pulse with different orientations and ellipticities. (a), (b) Intensity (solid) and phase (circles) of the p (blue) and s polarization (red) in frequency [(a)] and time domain [(b)]. (c) Poincaré plot of the temporal ellipticity $ϵ$ and orientation $\theta$. The measured subpulses are shown in red circles. The color saturation is proportional to the instantaneous subpulse intensity. The target polarization states are marked with black crosses. (d) Pseudo 3D representation of the pulse sequence in the time domain. The instantaneous frequency $\omega (t)$ is color-coded. A scan of the orientation of the second subpulse is shown in Media 1. (e) In another example, the temporal delay between two nearly linearly polarized subpulses with an orientation of 45° is scanned (Media 2). In this frame, the delay of the first subpulse is $-800\mathrm{fs}$. (f) The relative spectral phase offset between two subpulses can also be manipulated. The p- and an s-polarized pulses are delayed by 80 fs with respect to each other and overlap therefore partially in time. Their relative phase difference in the frequency domain is $\mathrm{\Delta }\varphi ({\omega }_0)=-0.49\mathrm{rad}$. This has the effect that the orientation and ellipticity varies in the region of temporal overlap. A scan of their phase difference is presented in Media 3.

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Figure 8(a) shows the modulated spectrum and phase for both polarization components as obtained with dual-channel FTSI. The modulation generates a double-pulse sequence in the time domain with the target delay of 300 fs [Fig. 8(b)]. The temporal phase is still flat and both subpulses remain compressed. The different intensity ratios of the p and s components and their respective relative phase define the polarization state of each subpulse. The target polarization states are marked with black crosses in Fig. 8(c). The measured polarization states (red dots) agree very well with the targeted ones. The fact that red dots concentrate in the target areas and do not extend further out shows that the polarization state is constant throughout each subpulse as desired. The pulse sequence in the time domain is visualized using a pseudo 3D representation in Fig. 8(d). The elliptical and the linear polarizations are clearly visible. To further prove the degree of polarization control, a scan of the orientation angle $\theta$ of a linearly polarized subpulse within a double-pulse sequence is provided in Media 1. Also, changing the time delay between two subpulses is challenging because both beams have to be amplitude modulated, precisely delayed in time, and kept at a fixed relative phase to avoid a change in their polarization state. A scan of the delay between two under 45° linearly polarized subpulses was successfully performed where no significant change in their polarization states was observed (Media 2). The “edge-on” view of the two subpulses for one particular delay [Fig. 8(e)] demonstrates the excellent fidelity as the 3D representation has vanishing “thickness” in the undesired polarization direction. The relative phase between the subpulses can also be controlled. The change in the relative phase of two temporally overlapping subpulses is visualized in Media 3, and one particular example is shown in Fig. 8(f). In the range of temporal overlap the relative phase change results in a changing polarization state, which is not the case in the region of no temporal overlap.

These examples demonstrate deterministic control of a double pulse in all degrees of freedom, but many spectroscopy applications require sequences of three or four subpulses. This demands a more complex SLM transfer function. In one example [Figs. 9(a) and 9(b)], a four-pulse sequence is generated. Every subpulse exhibits a different polarization state. The temporal delay between subpulse 1 & 2 and subpulse 3 & 4 is 200 fs, while the temporal delay between 2 & 3 is only 80 fs. Due to the temporal overlap, the two middle subpulses form a complicated polarization profile that varies in time. The measured values of $ϵ$ and $\theta$ (red dots) show good agreement with the target polarization states (black crosses), see Fig. 9(a). The small deviations are due to unavoidable pulse-shaping artifacts which are common for complicated modulation functions.

 

Fig. 9. Four-pulse sequences. (a), (b) Relative time delays are 200 fs, 80 fs, and again 200 fs. The overlap between subpulse 2 and 3 leads to varying ellipticity $ϵ$ and orientation $\theta$. The target polarization states for all four subpulses are marked with black crosses in the Poincaré plot [(a)]. The temporal evolution is visualized in a 3D representation [(b)]. (c), (d) Sequence with four subpulses with linearly s-polarized subpulses 1 and 4, while the target orientation angle of subpulses 2 and 3 is $-45°$ and $+45°$. The delay between each subsequent subpulse is 300 fs. The measured polarization states [(c), red dots] agree very well with their target states [(c), black crosses]. A clean polarization multipulse sequence is obtained [(d)].

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The second sequence [Figs. 9(c) and 9(d)] consists of four linearly polarized subpulses ($\theta =-90°$, $-45°$, 45°, 90°) with a fixed time delay of 300 fs between each of them. In Fig. 9(c), the target (black crosses) and measured (red dots) polarization states are compared. Overall, the measured pulse sequences agree very well with the defined polarization profiles which proves that multipulse sequences with highly complex and deterministic polarization states are feasible with the presented polarization pulse-shaper design.

7. CONCLUSION

Our vector-field shaper is based on the concept to shape two perpendicularly polarized laser pulses in phase and amplitude with just a single SLM in two separate arms of a common-path interferometer. By recombining these pulses to a single pulse, arbitrary control of the amplitude, phase, and polarization state as a function of frequency and time is achieved. The advantage of such a setup is that in principle every common liquid-crystal-based pulse shaper can be extended to a vector-field shaper without the necessity of a second or third SLM. We discussed in depth the advantages and disadvantages of a Wollaston prism and a TFP for separating the laser beam into two perpendicularly polarized beams and for recombining them. In our case, the benefits of the TFP outbalanced the advantages of a Wollaston prism. The TFP does not lead to angular dispersion and phase variations across the beam profile. The extinction ratio of the TFP is improved by double passing it for the polarization recombination and by choosing the crossed-polarizer arrangement. In this way, an extinction ratio of the order of $15000:1$ is achieved. The shaping capabilities of the setup were optimized by finding optimal parameters for the setup through numerical calculations and by optimizing the frequency distribution for a broad spectrum of 740–880 nm across the SLM. A high phase stability of $\sigma =28.3\mathrm{mrad}$ $(\approx \lambda /222)$ over 60 min is achieved by employing a common-path setup. To further improve this stability over very long measurement times we implemented on-the-fly phase reduction and stabilization utilizing the pulse shaper itself. With this method we were able to reach a phase stability of $\sigma =31.9\mathrm{mrad}$ $(\approx \lambda /197)$ over nearly 24 h. We demonstrated that this method leads to a compressed pulse with minimized relative phase between the two polarization components. Our experimental results prove that shaped pulses with arbitrary phase, amplitude, and polarization states can be generated. In all cases, very high agreement between the target parameters and the experimental data was achieved. Employing two spectrometers simultaneously for dual-channel spectral interferometry, vector-shaped pulses can be analyzed reliably without stringent stability requirements for the delay between the shaped pulse and the reference. By employing an advanced parameterization, further properties like the temporal chirp of each subpulse could also be controlled [48]. Possible future applications are the generation of specific multipulse sequences for various experiments such as ultrafast (multidimensional) spectroscopy [11 –15,26,27,38], quantum control [5 –10,19,20,49,50], (nonlinear) microscopy [16,17,51], or endoscopy [52,53].