Interferometric generation of parametrically shaped polarization pulses Stefan M. Weber,* Fabian Weise, Mateusz Plewicki, and Albrecht Lindinger Institut für Experimentalphysik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin *Corresponding author:
[email protected] Received 30 April 2007; revised 19 June 2007; accepted 21 June 2007; posted 22 June 2007 (Doc. ID 82560); published 9 August 2007
We demonstrate the capabilities of the recently introduced interferometric parallel pulse shaper setup and present a method for fully tailoring the three-dimensional electrical field of femtosecond laser pulses. The possibility of producing parametric polarization pulses with arbitrary orientations and ellipticities in time is demonstrated with a selection of example pulses. © 2007 Optical Society of America OCIS codes: 320.5540, 320.7080, 190.7110.
1. Introduction
Shaping other than linearly polarized femtosecond laser pulses [1] is a relatively new endeavor and brings the final missing degree of freedom, namely, the polarization, into play. Polarization pulse shaping has found its first applications in the field of coherent control [2–7]. The first polarization-capable setups [8,9] offered phase and limited polarization control with commercially available liquid crystal modulators. To generate quite arbitrary waveforms in time, interferences between the limited polarization states in the spectral domain could be exploited [10] without reaching full polarization control. Polarization-capable setups were also employed to compensate for polarization mode dispersion and polarization-dependent loss in single-mode fibers using other than the commonplace ⫾45° liquid crystal array orientations to counterbalance polarization and phase distortions [11,12]. Also, improvements to the original setups were made. Two shapers were used sequentially, effectively employing three arrays, to provide more freedom to the polarization transients [13] and a scalar, not yet vectorial, amplitude was incorporated with two round trips [14] through one double array shaper. Full control over the electrical field could still not be reached until, recently, when the modulator was incorporated into a Mach–Zehnder interferometer to shape the 0003-6935/07/235987-04$15.00/0 © 2007 Optical Society of America
field’s x and y components independently. The socalled parallel setup’s qualities in the spectral domain have already been established [15]. We will demonstrate its pulse shaping capabilities in the time domain using tailored, parametric pulses. These can be employed to customize search algorithms [16,17] but can be a useful tool for systematic investigations as well. 2. Experimental Setup
The significant innovation of the parallel setup is a full implementation of amplitude, phase, and polarization using interferometrically overlapped, phase and amplitude shaped pulses. As a linearly polarized electrical field can be fully described by phase and amplitude, a linear combination of two x- and yshaped fields allow for every physically feasible temporal waveform that can be realized within experimental limits. Such pulse forms do not suffer from side effects that arise when the amplitude is not or only partially incorporated, or when there are limitations to the spectral phase. The experimental main idea is that two regions of the used 640 pixel liquid crystal, double array modulator (CRi, Woburn, Massachusetts, SLM-640) shape the x and y components of the electrical field independently in phase and amplitude in a modified 4-f setup [15]. As displayed in Fig. 1, the laser beam, originating from a Coherent Mira (with a wavelength of 783 nm and a FWHM bandwidth of 25 nm) is split into two beams that hit the grating 共600 lines兾mm兲 at differ10 August 2007 兾 Vol. 46, No. 23 兾 APPLIED OPTICS
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reconstruct the component phases is contained. To yield a continuous , has to be added to the principal axis when situated in the second or fourth quadrant. The angle ⫽ 1兾2 arccos关cos  cos共2 arctan Hba兲兴
Fig. 1. (Color online) Interferometric, parallel shaper setup to fully control phase, amplitude, and polarization. The inset labels the ellipse parameters that we used.
ent incident angles at the same spot. After the focusing lens 共f ⫽ 25 cm兲 the spectral components travel through the shaper as two separate, parallel beams. Before recombination inside a polarizing beam splitter, the polarization of one of the beams is rotated by 90°. Its compactness and symmetry has the consequence that vibrations, thermal deformations, and the surrounding air affects both paths similarly and offers interferometric stability for a couple of minutes. Additionally, the shaper is enclosed in a sealed box to minimize air fluctuation and the modulator is triggered nonstop to keep the electronic components under constant load to produce a steady thermal output. 3. Parameterization
For the test pulses, we used a subpulse encoding that consists of the parameters intensity I, position in time, zero-order phase, chirps, and polarization states (major axis angle 兾2, minor to major axis ratio Hba ⫽ Hb兾Ha, and helicity) that are kept constant for the subpulse duration. One could alternatively choose these parameters arbitrarily in time, if fundamental limitations of the temporal derivatives of the respective parameters are observed [18]. We chose a parameter encoding with the x axis serving as an absolute reference, so the phase difference ⫽ y ⫺ x ⫽⫾arctan
tan关2 arctan Hba兴 , sin 
(1)
where the minus sign represents left-hand polarized light and the plus sign represents right-hand polarized light, is well-defined in the laboratory frame (which is not problematic as long as a field component exists in the x direction). Together with the phase sum sum ⫽ x ⫹ y, all the information necessary to 5988
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(2)
determines the bounding box of the resulting ellipse (see the Fig. 1 inset), where a2 ⫽ I cos2 and b2 ⫽ I sin2 are the squares of half of its side lengths. Before assembling the complex field, first, a spectral phase filter N共兲 is constructed for every subpulse N by use of spectral Taylor terms bn⬘. The linear term b1⬘ holds the distance from time zero, the quadratic term b2⬘ the linear chirp, etc., (not to be confused with the y-component’s amplitude b) with equal Taylor components for both x and y directions, except for the y component of the zero-order phase, which has to be set to b0,y⬘ ⫽ b0⬘ ⫹ (whereas b0,x⬘ ⫽ b0), to implement the polarization. To yield the target field Eout, the N subpulses are superposed, still in the frequency domain, as ˜ 共兲 · Eout共兲 ⫽ E in
冉 冊 Ç 兺 N
aNeiN,x共兲 bNeiN,y共兲
,
(3)
H共兲
˜ 共兲 is the linear pulse that enters each part where E in of the modulator. Then, |Hx|2 and |Hy|2 represent the components of the now vectorial, spectral transmission T共兲 to be written on the two modulator regions. The arguments constitute the phase filter 共兲 for the two parallel paths, where energy conservation demands
冕ⱍ
ⱍ
˜ 共兲 2d ⫽ E x,y
冕
ⱍ
ⱍ
˜ 共兲 2d Tx,y共兲 E in
(4)
independently for x and y. 4. Detection
The interferometric nature of the pulse generation most likely will make dual-channel interferometry retrievals, such as polarized light interference versus wavelength of only a glint (POLLIWOG) [19], difficult to implement, as three light fields would have to be interferometrically superposed, independently monitored, and stabilized. To measure our example pulses, we used a simplified version of the time-resolved ellipsometry technique [15,20], which is sufficient for our purposes, as the helicities can be preselected by the pulse form calculation. We detect a multitude of sum-frequency-generated cross correlations by rotating the shaped beam polarization with a 兾2 plate, and then focus and overlap it with the reference pulse in a beta barium borate crystal, which is polarization selective. Interferometric stability while measuring the cross correlations is guaranteed by monitoring destructive interference of a reference spectrum.
The course of intensity, major axis angle, and minor to major axis ratio in time can then be calculated by fitting the angular profile of each point in time to the function that describes the projection intensity of an ellipse toward a certain angle [15]. The temporal intensities displayed would still have to be deconvoluted as they originate from cross correlations; for the separated, unstructured subpulse presented in this contribution, the cross correlation causes only subpulse broadening. 5. Example Pulses
Figure 2 depicts three interferometrically generated double pulses with distances of 400 fs, calculated by the above-stated formalism, showing the time-dependent temporal intensity, major axis angle, and minor to major axis ratio. An orthogonal double pulse is generated in Fig. 2(a) with major axis angles of 0° and 90°. There is only a minimal deviation from the desired values for the ranges in which
there is sufficient intensity to provide a reliable pulse reconstruction. In three-dimensional representation of the field amplitudes, the noise originates from the employed method of detection and is due to the ratio of major to minor axis involved; it is also responsible for the frayed edges of the vertical subpulse. As can be seen, it is not distorted by side pulses that are a common annoyance for angular planes that deviate from the orientation of the setup’s liquid crystal arrays (which are ⫾45° for our case). The next example pulse in Fig. 2(b) shows an intended double linear and circular combination whereby a minor to major axis ratio of 0.89 could be achieved for the circular subpulse. The last pulse form, depicted in Fig. 2(c), consists of an elliptical subpulse (at 30°, with a minor to major axis ratio of 0.27; 0.33 was intended) and a consecutive linearly polarized pulse. The example pulses offer good agreement with the expected values. Creating temporally expanded pulse forms does not seem to impair the interferometric
Fig. 2. (Color online) Experimental parametric test pulses produced by the parallel setup: (a) orthogonal 0° and 90° linearly polarized double pulse, (b) linearly兾circularly polarized double pulse, (c) elliptical (ellipticity of 0.3, 30° major axis angle) and linear (major axis angle of 90°) pulse form. The dashed and dotted curves represent ellipticity Hba and the major axis angle 兾2 from Fig. 1, respectively. (d)–(f) Three-dimensional representations of the electrical field amplitudes of the corresponding pulses (having the same time scale and arbitrary colors). The shadows are the field projections to the respective planes. 10 August 2007 兾 Vol. 46, No. 23 兾 APPLIED OPTICS
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overlap quality compared with the single pulse examples given in Ref. [15], which is apparent from the low ellipticities of the quasi-linear pulses and the small intensities between the subpulses.
7.
6. Conclusion and Outlook
To the best of our knowledge, we have presented the first interferometerically generated, parametrically tailored polarization pulse forms using the recently introduced parallel setup. The experimental example pulses lack the typical shortcomings of restricted setups such as limited control over the amplitude or restrictions to the polarization transients. Such pulses could be beneficially employed to systematically investigate, for example, complex molecular systems with nonorthogonal dipole moments, highlighting the significance of parametric tailoring without inherent restrictions. The parametric encoding allows univocal parameter scans and could be put to good use in fields that start employing polarization shaping such as femtochemistry, polarization compensation in fibers, for supercontinuum generation, filamentation in air, inducing lattice vibrations in solids, in nano-optical manipulation efforts, or periodic electron circulation in molecules. The authors acknowledge Ludger Wöste and the Deutsche Forschungsgemeinschaft (SFB 450) for financial support. References 1. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929 –1960 (2000). 2. T. Suzuki, S. Minemoto, T. Kanai, and H. Sakai, “Optimal control of multiphoton ionization processes in aligned I2 molecules with time-dependent polarization pulses,” Phys. Rev. Lett. 92, 133005 (2004). 3. T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004). 4. N. Dudovich, D. Oron, and Y. Silberberg, “Quantum control of the angular momentum distribution in multiphoton absorption processes,” Phys. Rev. Lett. 92, 103003 (2004). 5. T. Brixner, F. J. Garc´ia de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic ultrafast space-time-resolved spectroscopy,” Phys. Rev. Lett. 95, 093901 (2005). 6. F. Weise, S. M. Weber, M. Plewicki, and A. Lindinger, “Appli-
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