The paradigm of ultrafast pulse shaping in the Fourier plane of a Martinez stretcher [1] was introduced by Heritage and Weiner in the mid-1980’s [2] which allows more complete control over ultrafast pulse dispersion than simple compressors [3, 1, 4, 5, 6]. Implementation of ultrafast pulse shaping technology became widespread with the advent of computer-controlled spatial lightmodulators (SLM) [7]. A number ofmodulator technologies have been introduced over the years besides SLMs [8, 9], including deformable mirrors [10] and acousto-optic dispersive filters [11, 12]. Computer-controlled ultrafast pulse shapers have found numerous applications, such as compensation of dispersion in telecommunications systems [13], pulse compression for amplifiers [14, 15, 16, 17], and coherent control [18, 19, 20, 21].

Shaping the polarization vector of ultrafast laser pulses has recently been addressed. Construction of such a vector pulse shaper requires independent control of the phase and amplitude of two orthogonal polarization states. The first polarization shaper developed by Brixner et al. [22] demonstrated control of the relative phase and polarization of the entire spectrum of an ultrafast laser pulse. Polachek et al. [23] extended control of polarization state of an ultrafast laser pulse by including amplitude control over the spectrum. Thier shaper require three SLM masks and two Martinez 4- f stretchers. Complete, independent amplitude, phase, and polarization control over the spectrum of an ultrafast laser pulses recently demonstrated by Plewicki et al. [24] makes use of a Mach-Zehnder interferometer. The Plewicki shaper suffers from relative phase instabilities due to extensive non-common path propagation of the two orthogonal polarization states that are mixed to form an arbitrary polarization state.

We introduce an approach to complete polarization state control of ultrafast laser pulses that requires only a single linear SLM mask and a single folded Martinez stretcher. Moreover, the two independently-controlled orthogonal states travel a nearly common path throughout the entire pulse shaper, mitigating most phase instability effects. Amplitude shaping of the individual polarizations is accomplished through over-sampling of the spatial mode and subsequent spatial diffraction of individual frequency components by the application of a rapidly-oscillating phase grating. As we have previously reported [25], the large number of pixels and small pixel pitch of the SLM permits a phase grating to control the spectral amplitude independently of the spectral phase imparted on the laser pulse spectrum.

 

Fig. 1. (a) Unfolded shaper schematic: The angular separation between the polarizations due to the Wollaston prism is imaged onto the prism by a 4- f imaging system with a curved mirror (M₁) and a fold mirror (M₂). The remaining focusing optics form a standard foldedMartinez stretcher with the reflective spatial light modulator at the Fourier plane. (b) Mapping of wavelength to pixels for incident s- and p-polarization components.

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In our approach, the input laser pulse is split into two orthogonal polarizations with a Wollaston prism (Karl Lambrecht; Chicago, IL) as sketched in Fig. 1(a). The Wollaston introduces an angle of 2° between two orthogonally-polarized components of the incident ultrafast laser pulse. The two pulses are imaged onto the input plane of a folded zero-dispersion stretcher by a 250-mm focal length mirror folded with a flat mirror [M₁ and M₂, respectively in Fig. 1(a)]. As indicated in Fig. 1(a), a half-wave plate is placed in the s-polarized (vertical polarization) arm to rotate the polarization to p-polarization. This polarization rotation is necessary for efficient transmission through the prism in the Martinez stretcher, and because the SLM only modulates the phase of p-polarized light. The time delay introduced by the wave plate is compensated by a glass window (C) in the p-polarization arm. A silicon prism (apex angle 32°; Argyle International; Princeton, NJ) angularly disperses the two pulses and a 250-mm focal length collimating mirror (M₃) focuses the spectral components onto the SLM(1×12,288 pixels in a 19.7×19.7- mm active area; Boulder Nonlinear Systems; Lafayette, CO). Due to the angular split in the two polarizations, the incident angle for the two pulses on the prism is set near the average minimum deviation. The combination of the angle between the two polarizations and the collimating mirror map the spectrum from each polarization onto a distinct spatial region on the SLM, as shown in Fig. 1(b). The angular separation introduced by the Wollaston prism and angular dispersion of prism are chosen so that a gap is maintained between the two spectra to allow for separate manipulation by the SLM. The reflective SLM is set to a retro condition so that the two polarizations retrace their steps through the pulse shaper. The Wollaston prism recombines the two orthogonal polarizations to a single, shaped field that is subsequently characterized by a tomographic retrieval method for self-referenced measurements of the polarization state of ultrafast laser pulses [26].

The ultrafast laser pulses to be shaped are derived from an Er:fiber laser system (Precision Photonics; Boulder, CO) yielding a 37-MHz pulse train, centered at a wavelength of ≈1550 nm. A folded prism compressor before the shaper compensates the chirp of the pulses from the laser, so that the full dynamic range of the SLMcan be optimally utilized for pulse shaping. The prism compressor is adjusted to yield near-transform limited pulses at the output of the shaper, measured to be ~80-fs in duration, and a half-wave plate before the shaper is used to evenly split the s- and p-polarization powers of the 42 mW output of the shaper. We used a telescope to expand and collimate the beam to a 4-mm diameter. Since the imaging between Wollaston and Si prisms introduces no magnification, the spectral foci at the Fourier plane of the shaper are approximately 60 mm in radius. This value is in good agreement with the measured width of a rapidly-oscillating mask width necessary to achieve full attenuation of a single spectral component [25], and corresponds to an over-sampling of the spatial mode by 40×.

Before use, the shaper must be carefully calibrated. This includes the wavelength-to-pixel mapping arising from the prism dispersion; the phase response of each SLM pixel with applied voltage; and optimal parameters and response curve of the amplitude control. The calibration procedures used here are similar to those described in Ref. [25]. The pixel-to-wavelength mapping was calibrated by sweeping a π phase step across the SLM mask, recording the individual s- and p-polarized spectra with an optical spectrum analyzer (OSA, Advantest 8384,Japan). The phase step introduces a spectral discontinuity, from which the we can directly extract the pixels occupied by the wavelengths of each polarization on the SLM. The results of this calibration are displayed in Fig. 1(b).

In order to calibrate the nonlinear phase response of each pixel to applied voltage, we use an intrinsic spectral interferometry (SI) capability of the pulse shaper. Removing the compensating plate leads to a 4.2-ps delay of the s-polarized pulse relative to p, due to the double-passed half-wave plate. A polarizer placed after the shaper at 45° to the polarization axes gives rise to interference fringes, which we use to obtain relative phase information between the two polarizations. We measure a reference interferogram with a flat mask on the shaper, and use a standard Fourier sideband filtering technique [27] as the phases are ramped. First, we record SI fringes while ramping the SLM pixels of the s-polarization through their voltage range, keeping the p-polarization phase fixed, then repeat the measurement, swapping the polarization being calibrated and that being held constant. The measured nonlinear phase responses of ~50 adjacent SLM pixels is averaged to obtain an independent, frequency-dependent, response map for each polarization.

Spectral amplitude shaping is achieved by the application of a high spatial frequency sinusoidal phase grating. Empirically, we found that a periodicity of ~50 pixels yielded optimal attenuation. The transmitted amplitude is expected to follow a Bessel function with modulation depth [25], which was in agreement with the transmitted intensity measured for increasing modulation depths.

To validate our shaper calibration, we inserted a 330-mm long BK7 rod of into the path of the beam. The 150-fs temporally stretched pulses for the s- and p-polarizations were measured with a combination of a polarizer, which selected the relevant pulse polarization, and a SHG frequency-resolved optical gating (FROG) device [28]. The spectral phase obtained from the reconstructed [29] FROG traces for each polarization was inverted and loaded onto the pulse shaper SLM. After a single measure-and-invert phase iteration, sub-80-fs, near-transform limited pulses were again measured.

To further test the fidelity of our calibration, we introduced a slow sinusoidal spectral phase onto the p-polarization. The red curve in Fig. 2(a) indicates the spectral phase posted to the shaper, whereas the spectral phase imparted on the pulse measured with SI as described above is shown in the blue curve. A sinusoidal amplitude modulation was simultaneously applied to the s-polarization, as shown in Fig. 2(b), where the original spectrum, shown by the dashed line, is compared with the shaped spectrum, solid.

 

Fig. 2. (a) Phase shaping: Target (red) and measured (blue) sinusoidal phase applied to the p-polarization. (b) Amplitude shaping: Shaped spectral amplitude of the s-polarization, while maintaining a flat spectral phase.

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The final required calibration step for the polarization pulse shaper is to ensure mutual temporal overlap of the orthogonally polarized, shaped pulses. SI can only be used to adjust the delay compensator to about the transform-limited pulse duration, or about 100 fs, since the fringe spacing observed in SI is inversely proportional to the relative delay of the two interfering pulses. A more accurate delay calibration, as well as verification of spatial overlap, was performed using a type-II collinear cross-FROG (XFROG) measurement as shown in Fig. 3(a). A 50-mm focal length spherical lens focused the output of the shaper into a 2.5-mm long BBO crystal (EKSPLA; Vilnius, Lithuania) cut at 23.2°. The crystal was rotated to achieve a θ=28.7° angle relative to the optic axis for type-II sum frequency generation (SFG) at 1550 nm [30]. The SFG signal was separated from the fundamental beam with a dichroic mirror and focused into a spectrometer (OceanOptics; Dunedin, FL). Since the type-II SFG signal is generated by orthogonally-polarized fundamental pulses, a correlation measurement produces a signal that is sensitive to the mutual overlap of two orthogonally polarized pulses. We used the shaper to introduce calibrated time delays between the s and p-polarized pulses by applying linear spectral phases of opposing slopes to each polarization component. A resulting XFROG trace is shown in Fig. 3(b). From this measurement, we can deduce a residual delay of τ _d ≈50 fs that is imperfectly balanced by the delay compensator. Rotation of the compensation plate and repeated polarization XFROG measurements can correct for this residual delay; alternatively, the measured relative time delay can be compensated by the shaper.

 

Fig. 3. Type-II SFG for XFROG calibration measurement. (a) Experimental arrangement; OA, optic axis, θ is angle with respect to the optic axis. (b)Measured XFROG signal as the relative delay between the s- and p-polarized pulses is adjusted with the SLM. The residual delay τ _d is due to imperfect delay compensation.

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Complex polarization states such as those generated by polarization shapers have traditionally been characterized using dual-channel spectral interferometry [31] with a well-characterized reference pulse. Due to the long path length in the polarization shaper, the requisite interferometric stability between shaped and references pulses was not expected to be good. We thus developed a self-referenced technique, the tomographic ultrafast retrieval of transverse light E-fields (TURTLE, [26]), to characterize the pulse resulting from the shaping shown in Fig. 2, with ultrafast polarization state retrieved results shown in Fig. 4. We measure FROG traces after an analyzing polarizer to measure projections at angles 0,90°, ±45° relative to the p-polarization axis. Measurements of the p and s polarization components are reconstructed using the standard FROG algorithm [29] to obtain the complex spectral field envelopes, denoted Ẽ_p(Ω) and Ẽ_s(Ω), shown in Figs. 4(a) and (b), respectively. We adopt a “phantom” trace display, where the left half-plane displays the (time-symmetric) measured FROG, and the right half-plane the (also time-symmetric) reconstructed trace.

The combined field is the vector sum,

where r, τ and θ denote the relative amplitude, delay, and phase between the polarizations. Suitable normalization of the fields Ẽ_s,p(Ω) allows the amplitude ratio r to be determined experimentally from power measurements [26]. To find the remaining unknowns, the TURTLE algorithm searches over τ and θ space to minimize an error calculated from the difference between measured and retrieved FROGs at ±45° and the measured r. The best fit to the data are shown in Figs. 4(c) and (d) using phantom traces of the measured and retrieved fields at these angles. The resulting field determined from Eq. (1) is shown in panel (e). For comparison, panel (f) shows the expected field for a 90-fs, transform limited Gaussian pulse, applying the equivalent phase and amplitude shaping to the different polarizations.

In summary, we have shown simultaneous phase, amplitude, and polarization control of ultrafast laser pulses using a single linear LC-SLM. The independent polarization shaping is done by splitting the pulse into separate polarization, then shaping the spectrum of each polarization. The amplitude shaping is achieved with over-sampling each spectral focus and by writing high-frequency phase grating, and spectral phase shaping is imparted by applying a slower spectral phase mask across the spectrum. The common path, common optics geometry of the setup guarantees a stable relative phase and delay between the two polarizations. We note that the 45°-polarized FROG traces, Figs. 4(c), (d), are devoid of noise fluctuations over the ~15- minute duration acquisition of the FROG measurement, indicating an excellent relative phase stability between the s- and p-polarized pulses.

 

Fig. 4. Self-referenced TURTLE characterization of phase, amplitude, and polarization-shaped pulse. (a), (b) FROG reconstructions of p and s polarizations, respectively, including the reconstructed pulse shape (horizontal), spectrum (vertical), and spectral phase (dashed). (c), (d) Retrieved FROG traces for ±45° polarization projections. In each case, the “phantom” trace shows the measured data in the left half-plane and the reconstructed or retrieved trace in the right half-plane. (e) The retrieved field evolution, compared to (f) simulation for a Gaussian transform limited initial pulse.

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