1. Introduction

The pulse shaping technology of femtosecond laser pulses [1–7] is crucial in controlling ultrafast light pulses and the interaction with matter, such as molecular reaction control [8,9], selective multi-photon excitation of fluorophores [10,11], or order-selective high harmonic generation [12]. Recently, demand has also been greatly increasing for time-dependent polarization shaping of femtosecond laser pulses to control ultrafast light-matter interaction, which is sensitive to the e-field vector [13–16]. Dynamic control of surface plasmon distribution in nanometer structures using vector-shaped femtosecond laser pulses has also been demonstrated [17,18].

In the first experiments on time-dependent polarization shaping of femtosecond laser pulses reported by Brixner and Gerber [19], only the spectral phases of two orthogonally polarized beams were modulated. In principle, to perfectly define an instantaneous elliptical polarization state (ellipticity and angle) and its amplitude, one has to gain control over four parameters, the amplitude and phase of both orthogonal polarization fields, independently. So far, various types of polarization pulse shapers have been reported based on the interference of two separately shaped orthogonally polarized beams [20–22], where the x- and the y-polarization components are first divided into different optical passes by a polarization beam splitter before a spatial light modulator (SLM) and combined again after spectral modulation at the SLM. In the schemes of Ninck et al. [20] and Masihzadeh et al. [22], they designed the separation of two optical paths of the orthogonal polarization beams to be in minimum in a 4-f optical layout. However, fluctuation of optical path difference still causes fluctuation of the polarization pulse shape in an experiment. Weiner et al. [23] developed an ideal monolithic SLM consisting of independently controllable four-liquid-crystal layers. Very recently, the similar experiment was reported by Weise and Lindinger [24]. The cascaded pulse shaping with a perfect common path can generate stable polarization-shaped pulses. Recently, Polachek et al. [25]reported that the Stokes parameter can be controlled by 3-cascaded spectral modulation using two liquid-crystal spatial light modulators (LC-SLMs). The results seem to be very attractive, whereas the system would be space-consuming due to the cascaded 4-f optical configuration.

We demonstrate a new approach for time-dependent polarization shaping by use of a reflective-type two-dimensional (2D) SLM, which can offer a compact and common-path optical configuration with high vector pulse stability. Moreover, since the 2D-SLM can generate spatiotemporally shaped femtosecond laser pulses as demonstrated by Nelson et al. [26], we generate spatially variable vector-shaped femtosecond laser pulses with this pulse shaper.

2. Experimental setup

The optical setup we propose is shown in Fig. 1 along with the diagram of polarization pulse shaping at each reflection at the 2D-SLM (768 x 768 pixels, X8267DB-14, Hamamatsu). The femtosecond laser pulse, which is linearly polarized in the x-direction with a center wavelength at 800 nm and a spectrum width of 40 nm (FWHM), is guided to the first grating (830 lines/mm). A Fourier-plane (FP) is formed on the surface of the 2D-SLM (Line 1 in Fig. 1(a)) by an f = 100 mm cylindrical lens. The spectral resolution of this pulse shaper is defined by the spot size of the beam on the FP, since the spot size of 50 μm is larger than the pixel size of 26 μm x 26 μm. By rotating the incident laser polarization 45 deg., only the e-field parallel to the horizontal axis of the SLM is phase-modulated. The phase modulation at this Line 1 defines the amplitude ratio of the orthogonal polarization components of the shaped laser pulse E_x(ω)/E_y(ω). Then, the polarization is rotated −45 deg. with a second half-wave plate inserted between the 2D-SLM and the cylindrical lens. The FP of the laser pulse is formed again on the 2D-SLM at a vertically shifted line (Line 2), and only φ_x (ω) is modulated. Finally, the polarization is rotated by + 90 deg. with a third half-wave plate, and only φ_y (ω) is modulated at Line 3. The Jones matrix of the 3-cascaded spectral modulation processes is described as follows:

 

Fig. 1 (a) Optical setup for polarization pulse shaping by a 3-cascaded reflective-type pulse shaper with a 2D-SLM. (b) Diagram of cascaded spectral phase modulation at each of three lines.

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Here, φ _i denotes the phase modulation at ω at i-th (i = 1,2,3) reflection at the SLM. From Eq. (1), the amplitude and phase of the orthogonal polarization components are described by φ _i ω) as follows:

Finally, the spectrally shaped laser pulse is transformed to the time domain by the second grating. In this optical setup, the reflected pulse must slantingly travel through the cylindrical lens. Therefore, significant dispersion is added on the transmitted pulses. We measured the dispersion of this 3-reflective pulse shaper by use of spectral interferometry. Then, the dispersion was compensated by applying the reversed dispersion at both Line 2 and 3 of the SLM and attained the zero-dispersion pulse shaper. In principle, such dispersion caused by Fourier lenses could be compensated by adjusting the 4-f optical layout. However, since the three cascaded 4-f layouts do not necessarily exhibit the same dispersion due to the slight difference in the reflection angle at the SLM and the mirror. Therefore, it is not easy to achieve the three cascaded zero-dispersion 4-f layouts just by adjusting the spacing between the common lens and mirror.

In principle, we could set one more reflection path and add fourth spectral modulation at Line 4. Then, we could specify the size of instantaneous polarization ellipse, whereas the present three-reflection scheme can define only the ellipticity and angle of the polaization ellipse with $E_x/E_v$and${\phi }_x-{\phi }_y$.

For evaluating the shaped time-dependent polarization, we utilized a dual-polarization channel, spatially resolved spectral interferometry (SSI) technique and analyzed spectral amplitude and phase of both the x-pol. and the y-pol. components at the same time [27]. In this scheme, since a common reference pulse is used for generating the spectral interference, we can evaluate the relative phase fluctuation between the x-pol. and the y-pol. components.

When setting different spectral modulation in the direction of vertical space at each line, the shaped vector pulse varies in the vertical direction. Therefore, the vector pulse shaping and the 1-D spatial pulse shape can simultaneously be obtained in this setup.

3. Results and discussion

As an example, four different double-pulses consisting of temporally varying polarization generated with the polarization pulse shaper are shown in Fig. 2 . The left figures show the x-pol. and y-pol. temporal waveforms, the azimuthal angle and the ellipticity of the shaped pulse. The right figures show three-dimensional representations of the e-field amplitudes of the corresponding target pulses. An orthogonal double-pulse is generated by adding 200 fs delay on the x-pol. pulse in Fig. 2(a). By producing an x-polarized double-pulse with an alternate 0-π phase mask and adding −200 fs delay on the y-pol. pulse, a double-pulse with major axis angles of 45° and 0° is generated in Fig. 2(b). Figure 2(c) and (d) are generated from the pulse of Fig. 2(b) by adjusting the relative phase or amplitude ratio between the x-pol. and y-pol. pulses. Since more than 10⁵ liquid-crystal pixels are involved in pulse shaping, some errors in phase calibration of each pixel may be accumulated and reflect on the pulse shaping performance. However, fairly good accuracy is obtained as shown in Fig. 2. There is only a minimal deviation from the desired values for the ranges in which there is sufficient intensity. We can generate various types of vector pulses similar to the previously reported polarization pulse shapers, such as a pulse exhibiting temporal polarization change within a single pulse.

 

Fig. 2 Left: experimentally generated double pulses in which polarization state is changed from the first pulse to the second pulse. Right: the targeted polarization pulse shape corresponding to the experimental result.

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The phase stability between the x-pol. and the y-pol. components is plotted in Fig. 3 . The fluctuation is within ± 0.1 rad, which corresponds to the phase resolution of our SSI measurement setup. Therefore, the advantage of the common path layout is experimantally proved. When we vertically divided each of the lines into two individual segments and set a different spectral phase pattern at each segment, the vector-shaped laser output clearly exhibits spatial variation of the vector pulse across one dimension in the beam cross section. An example is shown in Fig. 4 . Figure 4(a) shows the phase pattern applied in the 3-cascaded spectral phase modulation. At Lines 1 and 2, the spectral phase pattern, which is uniform in the vertical direction, was applied. The alternate 0-π phase pattern generates a double-pulse for the x-polarization. The upper and lower spectral phase patterns at Line 3 cause the temporal shift of + 150 fs and −150 fs for the y-polarization components, respectively. The fringe patterns of the SSI measurement (Fig. 4(b)) clearly show a different spatial fringe pattern between the upper and lower beams. The reconstructed time-histories (Figs. 4(c) and (d)) show that the spatiotemporal polarization pulse shaping is accurately performed. The spatial resolution of the spatially variant pulse shaping will be limited by diffraction. By allowing a wider area for Line 3, we will be able to embed more spatial variation of the vector pulse shaping in the beam.

 

Fig. 3 Fluctuation of phase difference between orthogonal polarization beams.

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Fig. 4 (a) Spectral phase modulation patterns at 3-cascaded reflections. Note that Line 3 is divided into upper and lower with different phase patterns. (b) Spectral interference patterns measured for x-polarization and y-polarization. Spatially different fringe patterns are observed in y-polarization beam. (c) Reconstructed polarization pulse shape from spectral interference patterns of (b) for upper half of beam cross section. (d) Reconstructed polarization pulse shape for lower half of beam cross section.

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4. Conclusion

A novel non-interferometric vector pulse-shaping scheme was developed for femtosecond laser pulses using a two-dimensional spatial light modulator (2D-SLM). By employing 3-cascaded spectral phase modulation to form a common optical path for orthogonal polarization components, shaped polarization pulses exhibited no fluctuation. By utilizing spatiotemporal pulse shaping obtainable by the 2D-SLM, we demonstrated spatially variant vector pulse shaping for the first time.