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Abstract
Novel multicolor concentric annular ultrafast vector beams (MUCAU-VB) are firstly generated simply by using cascaded four-wave mixing (CFWM) in a glass plate pumped by two intense vector femtosecond pulses. A proof-of-principle experiment shows that up to 10 frequency up-conversion concentric annular radially polarized sidebands are obtained simultaneously based on CFWM process, where the spectra range of the first 7 order sidebands extending from 545 nm to 725 nm. The results prove the polarization transfer property from the pump beam to the signal beams even in the CFWM, a third-order optical parametric process. The pulse duration of the first order sideband is measured to be 74 fs which is according with those of two input beams. These novel MUCAU-VB, which are manipulated in temporal, spectral, spatial domain and polarization state simultaneously, are expected to apply in wide fields, such as manipulating particles and multicolor pump-probe experiments.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical field modulation in both temporal and spatial domain, such as amplitude, phase and polarization modulation became a hot research topic recently. Polarization modulated vector beams with inhomogeneous polarization distribution have attracted many attention since the first radially polarized laser reported in 1972 [1]. Radially polarized beams and azimuthally polarized beams as two types of vector beams are two solutions of vector Helmholtz equation in the paraxial approximation [2,3]. They can be used in many fields owing to their unique inhomogeneous polarization distribution. The focal spot can have a strong longitudinal electric field component and can be applied to form optical needle when radially polarized beams are tightly focused, which have already been demonstrated by numerical simulations and experiments [4–6]. Azimuthally polarized beams can be tightly focused to get optical bubble, which can be used for trapping particles [7–9]. Vector beams also have important applications in high-resolution imaging [10], plasmon excitation [11], optical storage [12], classical optical communications and quantum optical communications [13–15], science-surface structuring [16], microstructure fabrication [17] and optical field collapse controlling [18].
In the past decades, many methods had been developed to generate useful vector beams, which could be classified as active methods and passive methods [2]. In active methods, typically, mode selected components such as axial birefringent elements were inserted to a laser cavity to drive the laser oscillator in vector mode [1,19,20]. In passive methods, optical modulation device, such as spatial light modulator (SLM) [21–24], metasurfaces [25], subwavelength gratings [26] and liquid crystal polymers (LCP) optical retarders and q-plate [27–29] were used to convert a uniformly polarized beams, such as linearly polarized beams, into vector beams.
With the presented methods, the spectral bandwidth of laser vector beams generated by active methods are usually narrow due to the mode selection in cavity and the limited gain bandwidth of the gain medium. And in passive methods, they only change the polarization state of income beams into a different polarization state with limited bandwidth and can’t create new wavelengths. Usually, it is not easy to generate vector beams in traditional nonlinear process, such as three-wave mixing or second-order nonlinear processes, where an anisotropic nonlinear material, such as BBO or LBO, is used. The process is rigidly constrained by phase matching condition which is extremely sensitive to light polarization. Then it is rather difficult and complex to generate broad bandwidth inhomogeneous distribution in every polarization using one nonlinear crystal [30,31]. How to generate multiple vector beams with different central wavelengths in a broad spectral range simultaneously using compact and economical setup by a nonlinear process, especially third-order nonlinear process, remains to be an unexplored issue and would help further applications of vector beams and multicolor femtosecond pulses.
Cascaded four-wave mixing (CFWM) in transparent bulk media have been studied systematically [32–36] which is promising for multicolored vector beam generation. CFWM should also subject to the energy conservation law and the momentum conservation law, which are ${\omega _{\textrm{ASm}}} = ({m + 1} ){\omega _1} - m{\omega _2}$ and ${k_{\textrm{AS(m - 1)}}} = {k_{\textrm{ASm}}} - ({k_1} - {k_2})$ for wavelength downshift sidebands [33]. Higher order sidebands are generated by the adjacent lower sidebands together with the original pump beams. The spatial distribution of generated sidebands are discrete in one array or multi-arrays. The advantage of CFWM compared to second-order nonlinear process is that isotropic media, such as glass plates, can be used in CFWM which are not sensitive to polarization. Then, it is able to modulate the spatial mode, beam polarization, spectrum and temporal profile of generated sidebands simultaneously.
In this paper, we generated novel multicolor concentric annular ultrafast vector beams (MUCAU-VB) based on the CFWM process simply in a glass plate, which realizes the modulations of a laser beam in spatial mode, polarization, wavelength, and pulse duration simultaneously. Concentric annular beams with radial polarization, multicolor sidebands in a broad spectral range, and femtosecond pulse duration are demonstrated. Up to 10 radially polarized frequency-upconversion concentric annular sidebands with wide bandwidth are observed. The spectral range of the first 7 order extending from 545 nm to 725 nm. And the polarization states of the MUCAU-VB are uniformly polarized distribution analyzed using a linear polarized analyzer. It needs to be noted that different vector beams, such as multicolor azimuthally polarized beams can be generated by carefully change the polarization of the input pump beams. The pulse duration of the first-order sideband is measured to be 74 fs which is shorter than those of the two input pump beams. To the best of our knowledge, this is the first time MUCAU-VB are demonstrated.
2. Experiment
The experimental setup for MUCAU-VB generation is shown in Fig. 1(a). The proof-of-principle experiments are performed with a 1 kHz Ti:sapphire CPA femtosecond laser system (Legend Elite, Coherent) generating 30 fs laser pulse with a central wavelength of 800 nm and pulse energy up to 4 mJ. After a long-pass dichroic mirror, the input laser beam is split into two beams with wavelength longer and shorter than 800 nm for the transmitted beam (beam1) and reflective beam (beam2), respectively. The two beams are further truncated by two band-pass filters to obtain the required spectral range. Then beam1 is focused by a plane-convex fused silica lens with a 200-mm focal length and a fused silica axicon with an apex angle of 178°, while beam2 is focused by another plane-convex fused silica lens with a focal length of 500 mm. Finally, beam1 and beam 2 are combined by the second long-pass dichroic mirror and both focused into a 0.5-mm H-ZLAF90 glass plate which has relatively higher third-order susceptibility than other ordinary glass materials. The optical path length between beam1 and beam2 can be eliminated by carefully tuning the delay line in optical path of beam1. Two liquid crystal polymer vortex half-wave plates (VHWP) with central wavelengths of 780 nm (WPV10L-780, Thorlabs) and 830 nm (WPV10L-830, Thorlabs) are used to change the two linearly polarized input pump beams to radially polarized beams, respectively. Different polarization can be obtained depending on the orientation of the zero-degree fast axis of the VHWP to the polarization direction of the linear polarized input beams. When the zero-degree fast axis of the VHWP is parallel (perpendicular) to the polarization direction of the input beams, radially (azimuthally) polarized beams can be produced. It is convenient to control the polarization state of the pump beams by rotating the VHWP or rotating the polarization of the input beams to form a controlled setup. In this proof-of-principle experiment, the input beams are converted into radially polarized beams which will generate multicolor concentric annular radially polarized beams. Other vector beams such as azimuthally polarized sidebands can also be obtained in the same way. The spectra of the sidebands were measured using a spectrometer (HR4000, Ocean Optics). The maximum input power of beam1 and beam2 before the glass plate are about 820mW and 400mW, respectively.
Fig. 1. (a) Experimental setup for generating MUCAU-VB. (b) Schematic for the generation of multicolor sidebands. VND: variable neutral-density filter. DM1(2): long-pass dichroic mirror. BP1(2): band-pass filter. M: reflective mirror. VHWP1(2): vortex half-wave plate with a central wavelength of 830 nm (780 nm). L1(2): plane-convex fused silica lens with focal length of 500 mm (200 mm). Axicon: fused silica axicon with an apex angle of 178°. G: 0.5-mm H-ZLAF90 glass plate.
The schematic for the generation of multicolor sidebands are depicted in Fig. 1(b). The axicon is used to transform the focused beam1 to a divergent beam on the glass plate (G), where the divergent beam1 interacts with the convergent beam2 focused by Lens 2 (L2) with an optimized crossing angle θ. The crossing angle is important in the FWM process, which can be used to control the central wavelengths of generated sidebands.
3. Experimental results and discussion
As we have mentioned, the polarization of the two input pump beams, beam1 and beam2, are controlled using two vortex half-wave plates VHWP1 and VHWP2. Firstly, the polarization properties of beam1 and beam2 are studied based on the method [1] to make sure that the two input beams own radial polarization. When the fast axis of VHWP1 and VHWP2 are parallel to the polarization direction of the linearly polarized input beams, the two-dimensional beam profiles of beam1 and beam2 after DM2 are captured directly by a charge couple device (CCD) (BC106, Thorlabs), as shown in Fig. 2(a) and (b), respectively. An obvious doughnut shape in Fig. 2(b) indicates a polarization singularity in the center of the beam. Detail polarization distribution of beam1 and beam2 after DM2 are analyzed using a nanoparticle linear film polarizer (LP) (LPVIS100, Thorlabs) with an extinction ratio larger than 10000:1 over the bandwidth from 550 nm to 1500 nm. Figure 2(c) to (e) and (f) to (h) are the analyzed results of beam1 and beam2 after the LP when it has been rotated by 0°, 45°, 90°, respectively. The arrows in the upper-left of each figure represent the polarization axis of the LP. The radially polarized distribution of the two pump beams are demonstrated with the beams cutting into two parts in according to the polarization axis of the LP. The results show that the two pump beams are radially polarized beams. The intensity distribution of beam1 in the vertical direction is a little weaker than that of in the horizontal direction due to the polarization dependent effect of DM2 of which the transmission of vertical polarization beam is lower. The dark gaps at 0° and 90° are obvious, while it is less clear for the 45° case, which indicates that the polarization purity of the pump beams at 0° and 90° are higher, but lower around 45°, this polarization purity degradation mainly due to DM2 isn’t a perfect polarization-independent optical element and shows depolarization effect at 45°. Moreover, the VHWP is a vortex half-wave plate, which is a wavelength sensitive optical element, of which the optimized wavelength does not match the broadband input beam completely.
Fig. 2. Beam profiles of beam1 and beam2 after DM2. (a) and (b) Beam profiles of beam1 and beam2 captured using a CCD, respectively. (c) to (e) and (f) to (h) Beam profiles of beam1 and beam2 after pass through the LP rotated by 0°, 45°, 90°, respectively. The arrows in the upper-left of each figure represent the polarization axes of the LP.
When the two radially polarized pump beams are overlapped both in spatial and temporal domain in the H-ZLAF90 glass plate, which has a very high refractive index and then high third-order nonlinearity, beautiful multicolor concentric annular sidebands are generated and shown in Fig. 3(a). It should be noted that only few and weak sidebands generated with ordinary glass plates such as fused silica and BK7. As a simple explanation, it is the interaction between every overlapped part for both beam1 and beam2 with ring profile that produce the sidebands. Only if the polarization of the two overlapped parts are consistent, the energy transfer efficiency in CFWM process is maximum. Since it is FWM process, which is not sensitive to the polarization of the two input beams if both of them own the same polarization in the glass plate, the whole beams with different polarization directions will generate CFWM signals. Owing to the ring shape of the two interacting pump beams, beautiful multicolor concentric annular vector beams are generated. Multicolor annular vector sidebands up to 10 orders appeared simultaneously which are well separated in space and concentric with beam1 and beam2. The photographs in Fig. 3. show multicolor sidebands on a sheet of white paper placed about 35 cm after the glass plate at a maximum pump power of about 820 mW and 400 mW for beam1 and beam2, respectively. We mark the sideband rings from the inner smallest ring to the outer largest ring as AS1 to AS10, respectively. Figure 3(a) and (b) are the photographs of the multicolor sidebands taken without and with a pump beam filter placed after the glass plate, respectively. The intensity of the sidebands in the horizontal direction are a little bit stronger than that of in the vertical direction, this is due to the pump power distribution of beam1 that is a little bit stronger in the horizontal direction, which had been analyzed in the previous discussion.
Fig. 3. Photographs of the generated multicolor sidebands on a sheet of white paper placed about 35 cm after the glass plate. (a) and (b) Photographs of the multicolor sidebands directly taken, without and with a filter after the glass plate, respectively. L, R, T, D in (a) represent four symmetric positions for spectra measuring. (c) to (o) Photographs of the multicolor sidebands captured with a LP rotated from 0° to 180° with a 15 degrees step after the glass plate, respectively.
Other photographs from (c) to (o) are taken when the multicolor sidebands pass through a LP at different rotated angles from 0 degree to 180 degree with a 15 degrees step. It is very clear that the rings are divided into two components with a dark gap perpendicular to the polarization axis of the LP in every photograph, which indicates the generated MUCAU-VB are radially polarized beams [1]. These also means all the colorful sidebands own the same polarization state. The photographs show that the dark gaps at 0° and 90° are rather obvious and clearer than those of near 45°, which indicates the polarization purity of the generation sidebands are higher around 0° and 90° and a little bit degrading for the directions near 45°. The polarization purity degradation is due to the relative lower polarization purity of the two pump beams at 45° which has been discussed previously. These low polarization purity of two pump beams may generate lower polarization purity of sidebands at the same positions. A polarization independent DM is hoped to improve the polarization purity of the two pump beams at all directions and result in an improving purity of the generation sidebands. The weak spots in the center of MUCAU-VB in each photograph is a weak leakage of the pump beam. The experimental results and their analysis demonstrate that the polarization distribution of the multicolor sidebands are in consistent with the two pump beams very well. The radially polarization of the two pump beams are transferred to the generated MUCAU-VB.
The spectra of the generated multicolor sidebands are measured using a spectrometer. The spectral profiles of seven order concentric annular sidebands are shown in Fig. 4. To demonstrate the spectral uniformity in spatial domain on every concentric annular sideband, the spectra at four symmetric positions which are marked with L (left), R (right), T (top), D (down) in Fig. 3(a) are measured under the condition of maximum pump power. The obtained spectra are shown in Fig. 4(a). The wavelengths at four symmetric positions are overlapped to each other very well for every different sidebands, which indicates the spectral uniformity of every generated multicolor sideband in spatial domain. Furthermore, we also measured the spectra at position L under three different pump powers of beam2, which are 260 mW, 348 mW, 400 mW, respectively. The results are shown in Fig. 4(b). The spectra of the same sideband under different pump powers are consistent. The result indicates the nice spectral stability in according to the pump power. The spectral range of the seven generated multicolor concentric annular sidebands extends from green (545 nm) to red (725 nm), where the spectral bandwidth of low order sidebands have more than ten nanometers.
Fig. 4. Spectra of the sidebands. (a) Spectra at four symmetric positions of every sideband, which are marked with L (left), R (right), T (top), D (down) in Fig. 3(a). (b) Spectra at position L under three different pump powers. 3, 5, 7 order sidebands are obtained when pump powers of beam2 are 260 mW, 348 mW, 400 mW, respectively. The first 5 order sidebands are show when pump power of 400 mW. The spectra of the same sideband are consistent under different pump powers.
To further prove the process is a CFWM process, simple simulation is done to obtain the central wavelength of every generated sideband based on the CFWM formulas depicted previously [33]. The simulated results are shown in Fig. 5 together with the measured values. The experimental curve match well with the simulated curve. In the simulation process, to obtain the central wavelength of each sideband, the spectral range of both pump beams are scanned to find a pair of pump wavelengths which results in minimum phase mismatch in the glass plate.
The output powers of the generated multicolor rings are measured at three different pump powers by tuning the VND. The power of beam1 is about 2 times that of beam2. The dependence of the output powers of the sidebands on pump power are shown in Fig. 6(a). The maximum output powers of AS1 are about 9.85 mW, 7.73 mW, 4.35 mW, when total pump powers are 1.232 W, 1.073 W, 0.811 W, respectively, and the output powers of the sidebands decreased rapidly with order number increasing. The power conversion efficiency from total pump power to sidebands are about 1%, 0.9%, 0.6%, respectively. Conversion efficiency increased as the total pump power increasing. The crossing angles between sidebands and beam2 are recorded by measuring the ring radius corresponding to the distance of the white screen which was placed about 654 mm after the glass plate. After passing through a focus lens and an axicon, beam1 forms a small ring in the focal plate and an annular beam after propagating certain distance. The divergence angles of beam1 are also recorded by measuring it’s inner and outer radius in the screen. The corresponding divergence angles are 1.3° and 2.6°, respectively. The dependence of the crossing angles on sideband order number are shown in Fig. 6(b). The distances between two adjacent rings decreased with the order number increasing which is the results of phase matching process.
Fig. 6. (a) Dependence of the output powers of the generated multicolor sidebands on order number when total pump powers are 1232 mW, 1073 mW, 811 mW, respectively. (b) Dependence of the crossing angles between multicolor sidebands and beam2 according to the order numbers.
The pulse durations of the two pump beams and AS1 are measured using a home-made SHG-FROG. The retrieved pulse profiles and temporal phase of two incident pump beams are shown in Fig. 7(a). The measured SHG-FROG trace of AS1 is shown in Fig. 7(b). Figure 7(c) and (d) show the retrieved pulse profiles, the measured and retrieved spectra, the corresponding temporal and spectral phase of AS1, respectively. The FWHM pulse width of beam1, beam2, and AS1 are 150 fs, 152 fs, and 74 fs, with small retrieved error of 0.0023, 0.0125, and 0.0059, respectively. Due to the spectra truncated effect of dichroic mirror and band-pass filter, the input pump beams are stretched. The pulse width of AS1 is compressed and narrower than those of the two pump beams. The retrieved and measured spectra agree well with each other. The temporal and spectral phase of AS1 show that there are some chirp in the pulse, which may be induced by the optical elements, such as the focus lens and axicon.
Fig. 7. (a) The retrieved pulse duration and phase of beam1 and beam2. (b) The measured FROG trace of AS1. (c) The retrieved pulse duration and phase of AS1. (d) The measured and retrieved spectra of AS1 with recovered phase.
4. Conclusion
In conclusion, we have proposed a simple method to generate novel MUCAU-VB for the first time. Up to ten multicolor sideband rings with identical radial polarization are obtained simultaneously. So far, the maximum energy conversion efficiency of the sidebands from input pump beams is 1%. The spectra, polarization distribution, spatial shape, output power, exiting angles and pulse profile of the multicolor sidebands are studied. This method can be used to generate other polarization state vector beams, such as azimuthally polarized beams, conveniently, by just using variable vortex half-wave plates or adjustable wave plates before vortex half-wave plates to change the polarization of the pump beams. Owing to the CFWM process can be operated in any transparent medium, MUCAU-VB in a broad spectral range, from deep ultraviolet to infrared, can be obtained by shifting the central wavelengths of two input pump beams. For our MUCAU-VB, the difference of focal length of different color wavelength have promising applications in forming multi-focus which would benefit the material process. And it may find great potential application in pump-probe experiments [37], in single-cycle pulse synthesis [38], in trapping microparticles where a perfect vortex beam is used [39]. Furthermore, multi-channel ultrafast optical switch can be obtained due to the fast responding of CFWM using the multicolor annular beams. Another potential application is to produce ultralong optical needle and ultralong optical bubble by focusing multicolor radially polarized and azimuthally polarized beams, respectively.
Funding
National Natural Science Foundation of China (61527821, 61905257, U1930115); Chinese Academy of Sciences (Instrument Developing Project (YZ201538), Strategic Priority Research Program (XDB16)).
Disclosures
The authors declare no conflicts of interest.
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