Vectorial sculpturing of spatiotemporal wavepackets

I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. RESULTS AND DISCUSSIO...III. CONCLUSIONREFERENCESPrevious sectionNext sectionIn recent years, ultrafast lasers have become powerful tools for broad science and technological fields. Accompanied by the development of ultrafast lasers, pulse shaping has enabled the generation of programmable ultrafast optical waveforms11. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929 (2000). https://doi.org/10.1063/1.1150614 to meet specific application needs, such as laser machining,22. N. Götte, T. Winkler, T. Meinl, T. Kusserow, B. Zielinski, C. Sarpe, A. Senftleben, H. Hillmer, and T. Baumert, “Temporal Airy pulses for controlled high aspect ratio nanomachining of dielectrics,” Optica 3, 389–395 (2016). https://doi.org/10.1364/optica.3.000389 nonlinear optical microscopy,33. Y. X. Yan, E. B. Gamble, and K. A. Nelson, “Impulsive stimulated scattering: General importance in femtosecond laser pulse interactions with matter, and spectroscopic applications,” J. Chem. Phys. 83, 5391–5399 (1985). https://doi.org/10.1063/1.449708 spectroscopy,44. T. Hornung, J. C. Vaughan, T. Feurer, and K. A. Nelson, “Degenerate four-wave mixing spectroscopy based on two-dimensional femtosecond pulse shaping,” Opt. Lett. 29, 2052–2054 (2004). https://doi.org/10.1364/ol.29.002052 and coherent quantum control of light–matter interactions.55. Y. Silberberg, “Quantum coherent control for nonlinear spectroscopy and microscopy,” Annu. Rev. Phys. Chem. 60, 277 (2009). https://doi.org/10.1146/annurev.physchem.040808.090427 In most pulse shaping techniques, the optical pulse is treated as a scalar field that is normally linearly polarized during the entire process. However, quantum systems are three dimensional in nature; thus, the interactions are inevitably vectorial, which calls for vectorial pulse shaping techniques.The simplest means of polarization control is to use a pair of linearly polarized pulses with twisted polarization directions, which have been utilized to study nuclear motion,66. M. M. Wefers, H. Kawashima, and K. A. Nelson, “Optical control over two-dimensional lattice vibrational trajectories in crystalline quartz,” J. Chem. Phys. 108, 10248–10255 (1998). https://doi.org/10.1063/1.476485 molecular rotation,77. K. Kitano, H. Hasegawa, and Y. Ohshima, “Ultrafast angular momentum orientation by linearly polarized laser fields,” Phys. Rev. Lett. 103, 223002 (2009). https://doi.org/10.1103/physrevlett.103.223002 momentum of magnetization,88. N. Kanda, T. Higuchi, H. Shimizu, K. Konishi, K. Yoshioka, and M. Kuwata-Gonokami, “The vectorial control of magnetization by light,” Nat. Commun. 2, 362 (2011). https://doi.org/10.1038/ncomms1366 etc. The polarization modulation in this case is inter-pulse; hence, the dynamic rate of change in polarization is limited. More elaborate vectorial pulse shaping has been developed to dynamically manipulate the polarization within a single pulse using a one-dimensional (1D) linear array of liquid crystal spatial light modulator (LC-SLM).99. T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. 26, 557–559 (2001). https://doi.org/10.1364/ol.26.000557 Later, techniques that use the common-path Mach–Zehnder interferometer to combine two such polarization shapers for each orthogonal component have been developed for complete vector-field control (spectral phase and amplitude control of both orthogonal components).1010. M. Sato, T. Suzuki, and K. Misawa, “Interferometric polarization pulse shaper stabilized by an external laser diode for arbitrary vector field shaping,” Rev. Sci. Instrum. 80, 123107 (2009). https://doi.org/10.1063/1.3270254 Pulses with twisting-polarization and swept polarization rotation have been generated. However, these methods are, in general, difficult to align, and some of them require active stabilization feedback control. More importantly, these techniques only create alternating polarization states in the temporal domain, leaving the spatial domain completely untouched.Recently, unique phenomena and properties have been discovered for spatiotemporal wavepackets through intentionally engineering the space–time coupling. For examples, precisely engineered spatiotemporal spectral correlations may lead to anomalous refraction that defies expectations from Fermat’s principle;1111. B. Bhaduri, M. Yessenov, and A. F. Abouraddy, “Anomalous refraction of optical spacetime wave packets,” Nat. Photonics 14, 416–421 (2020). https://doi.org/10.1038/s41566-020-0645-6 spatiotemporal optical vortex with purely transverse orbital angular momentum can be generated through spatiotemporal phase control.1212. A. Chong, C. Wan, J. Chen, and Q. Zhan, “Generation of spatiotemporal optical vortices with controllable transverse orbital angular momentum,” Nat. Photonics 14, 350–354 (2020). https://doi.org/10.1038/s41566-020-0587-z Typically, a two-dimensional (2D) LC-SLM is used in a traditional 4-f pulse shaper to introduce spatiotemporally coupled phase modulation. Recently, a linear mapping technique using chirped pulses has been demonstrated,1313. Q. Cao, J. Chen, K. Lu, C. Wan, A. Chong, and Q. Zhan, “Sculpturing spatiotemporal wavepackets with chirped pulses,” Photonics Res. 9, 2261–2264 (2021). https://doi.org/10.1364/prj.439849 where a pure phase modulation displayed on a 2D LC-SLM is linearly mapped to the spatiotemporal domain. Naturally, if we can replace the phase modulation with modulation of other field properties (e.g., amplitude, polarization etc.), the corresponding modulations can be imprinted onto the spatiotemporal domain, which will greatly expand our capability to produce more sophisticated spatiotemporal wavepackets. In this work, we demonstrate spatiotemporal polarization modulation through simply introducing a quarter-wave plate inside the pulse shaper that enables us to generate various wavepackets with sophisticated spatiotemporal vectorial structures.

II. RESULTS AND DISCUSSION

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ChooseTop of pageABSTRACTI. INTRODUCTIONII. RESULTS AND DISCUSSIO... <<III. CONCLUSIONREFERENCESPrevious sectionNext sectionThe experimental setup is illustrated in Fig. 1, along with the measurement system for characterizing the generated spatiotemporal vectorial fields. The generation system constitutes a standard reflection type 4-f pulse shaper with one quarter-wave-plate (QWP) oriented at 45° inserted between the cylindrical lens and the 2D LC-SLM. The combination of QWP and LC-SLM forms a polarization rotator,1414. Q. Zhan, “Cylindrical vector beams: From mathematical concepts to applications,” Adv. Opt. Photonics 1, 1–57 (2009). https://doi.org/10.1364/aop.1.000001 whose Jones matrix can be written asMPR=−i⋅RϕSLM2⋅ei⋅ϕSLM2,(1)where ϕSLM is the applied LC-SLM phase and the rotation matrix is defined as Rϕ=cos⁡ϕsin⁡ϕ−sin⁡ϕcos⁡ϕ. Therefore, the polarization of each spatio-spectral component reflected off the device will be rotated from the incident polarization by ϕSLM/2 and gain an extra geometrical phase of ϕSLM/2,1414. Q. Zhan, “Cylindrical vector beams: From mathematical concepts to applications,” Adv. Opt. Photonics 1, 1–57 (2009). https://doi.org/10.1364/aop.1.000001 hence forming a spatiotemporal polarization pulse shaper. In essence, such a combination represents a common-path interferometer for the two orthogonal components, eliminating the requirements for alignments and stabilization observed in the previous designs. More importantly, the use of a 2D LC-SLM allows us to design spatiotemporally coupled vectorial properties for the first time.After the polarization pulse shaper, the generated spatiotemporal vectorial wavepacket is recombined with a transform-limited probe wavepacket on a CCD camera. The polarization and relative time delay of the probe are controlled by a half-wave-plate (HWP) and an optical delay line, respectively. Through setting the polarization of the probe to the x-direction and y-direction and scanning the relative time delay, we record the interference patterns between the object and the probe pulses. The images are later used to reconstruct the 3D intensity profile of the object wavepacket via the algorithm described in Ref. 1515. J. Chen, K. Lu, Q. Cao, C. Wan, H. Hu, and Q. Zhan, “Automated close-loop system for three-dimensional characterization of spatiotemporal optical vortex,” Front. Phys. 9, 31 (2021). https://doi.org/10.3389/fphy.2021.633922.To demonstrate the polarization modulation capability, we first generate a pulse train with polarization that rapidly alternates between x-polarization and y-polarization on a hundred-femtosecond scale within each pulse. Such a pulse train is generated by applying a periodical binary phase mask along the spectral direction of the LC-SLM. Figure 2(a) plots the measured 3D intensity profile of the generated vectorial field. In the plot, the red iso-surface corresponds to the field polarized along the x-direction and the blue iso-surface corresponds to the field polarized along the y-direction. The applied LC-SLM phase mask is shown in Fig. 2(b). The binary phase mask changes between 0 and π, which correspond to a 0° and 90° polarization rotation, respectively. The phase mask can be analytically written asϕΩ=π/2+π/2⋅sgn(sin(Ω/ΔΩ)),(2)where ΔΩ stands for the spectral period and sgn(x) is the sign function. The period of the phase mask shown in Fig. 2(b) is 10 THz. For a linearly chirped input pulse, the applied polarization modulation in the spectral domain is mapped to a polarization modulation in the temporal domain. Figure 2(c) plots the integrated intensity profile. The polarization state of the field switches every 275 fs. With the same chirping condition, further reducing ΔΩ can generate vectorial optical fields with even faster polarization switching. Figures 2(d) and 2(e) show the measurement results when the modulation period is 5 THz. The polarization state switches between x-polarization and y-polarization every 140 fs.It should be noted that this vectorial wavepacket can be easily converted to have its polarization rapidly switching between left-handed circular polarization (LCP) and right-handed circular polarization (RCP) following a QWP with its fast axis aligned at 45° with respect to the x-polarization direction. The photons within the wavepacket will have spin angular momentum (SAM) rapidly switching between +ℏ and −ℏ, which is of great interest in applications such as selective production of enantiomers1616. S. Rozen, A. Comby, E. Bloch, S. Beauvarlet, D. Descamps, B. Fabre, S. Petit, V. Blanchet, B. Pons, N. Dudovich, and Y. Mairesse, “Controlling subcycle optical chirality in the photoionization of chiral molecule,” Phys. Rev. X. 9, 031004 (2019). https://doi.org/10.1103/physrevx.9.031004 and magnetization.88. N. Kanda, T. Higuchi, H. Shimizu, K. Konishi, K. Yoshioka, and M. Kuwata-Gonokami, “The vectorial control of magnetization by light,” Nat. Commun. 2, 362 (2011). https://doi.org/10.1038/ncomms1366 Thus, we can view this type of wavepacket as a spatiotemporal spin grating.So far, we have only demonstrated the capability to dynamically modulate polarization in the temporal domain. We then apply a checkerboard phase mask to generate a spatiotemporal vectorial optical field whose polarization switches between x-polarization and y-polarization both spatially and temporally. Figure 3(a) plots the measured 3D intensity profile. Besides fast polarization switching in the time domain, the light field also switches its polarization in the spatial direction (y-direction). The phase mask for generating the field is shown in Fig. 3(b), which can be expressed asϕΩ,y=π/2+π/2⋅sgn(sin(Ω/ΔΩ))⋅sgn(sin(y/Δy)),(3)where Δy is the spatial period along the y-direction. Here, ΔΩ is 10 THz and Δy is 374 µm. Figure 3(c) shows the integrated intensity profile of x-polarized light in the spatiotemporal domain. The spatiotemporal vectorial optical field switches its polarization every 300 fs and 360 µm. Again, if this wavepacket passes through a QWP oriented at 45°, a spatiotemporal checkboard pattern with alternative spin would be obtained; thus, this type of wavepacket can be termed spatiotemporal spin lattice.This simple setup is capable of generating much more complicated spatiotemporal vectorial waveforms. For example, we can apply a spiral phase with a topological charge of l on the LC-SLM,ϕΩ,x=l⋅(θy−Ω−θ0),(4)where θy-Ω is the y-Ω plane polar angle and θ0 is the polar angle offset. We set the topological charge l = +2 so that there is a total phase variation of 4π along the azimuthal direction. According to Eq. (1), the phase modulation will be converted into polarization rotation as well as a geometrical phase, which is then linearly mapped to the spatiotemporal domain asE⃗y,t=ei⋅(ϕsty,t−θ0)e⃗y⁡cos(ϕsty,t−θ0),(5)where ϕsty,t=tan−1(y/ct) is the azimuthal angle defined in the normalized spatiotemporal coordinates that moves with the wavepacket. Thus, the produced vectorial wavepacket is a spatiotemporally twisting polarization modulated with a spatiotemporal optical vortex phase.By setting θ0 to be 0 and π/2, we generate two vectorial wavepackets with spatiotemporally twisting polarization. The iso-surface plots in Figs. 4(a) and 4(d) correspond to the field that is measured when the probe is x-polarized (red iso-surface) and y-polarized (blue iso-surface), respectively. However, it is noteworthy that the polarization of the light field rotates continuously in the spatiotemporal domain. To better visualize the vectorial behavior, we draw two arrowed lines in Figs. 4(a) and 4(d). Figures 4(b) and 4(e) show the theoretical polarization evolution along the indicated lines. Take Fig. 4(b) as an example, the upper half of the wavepacket has its polarization continuously rotating clockwise from y-polarization to x-polarization and then back to y-polarization while the lower half rotates counter-clockwise. Figures 4(c) and 4(f) show the measured polarization evolution. The results are in good agreements with theoretical anticipations, showing that the light field has a polarization twisting spatiotemporally while different spatiotemporal locations have different twisting features.Finally, it is also worthy pointing out that this setup can be easily adapted for amplitude modulation of spatiotemporal wavepackets by using a linear polarizer as a filter in combination with the polarization modulation. This is illustrated in Fig. 5 by producing spatiotemporal pulses that spell the letters “USST.” Here, binary phase masks, as shown in the insets of Fig. 5, are used to generate the prescribed “USST” wavepackets.

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