1. Introduction

An axisymmetrically polarized (AP) mode, a kind of the cylindrically polarized modes or the vector vortex modes, has the symmetry of C_∞ in both of the polarization and the intensity distributions [1]. The high symmetry of the AP modes has attracted attention for the various research topics such as super-resolution microscopy [2], particle acceleration [3–6], laser processing [7–14], and optical tractor [15].

Recently, some applications using ultrashort or broadband AP pulses have been reported such as quantum communication [16, 17], generation of extreme ultraviolet AP beams [18], and nonlinear spectroscopy [19–22]. In order to fully explore the potential of the AP modes, it is important to develop a generation system of AP pulses with an ultrashort duration or a broadband spectrum, while there are fewer researches on the generation of AP pulses in comparison with the generation of AP continuous wave beams. We have reported on the generation of AP broadband pulses by using the combination of a spatial light modulator (SLM) in the 4-f configuration and a coherent combining system, and showed that the arbitrary AP states can be generated by superposing the left and right circularly polarized (LCP and RCP, respectively) optical vortex (OV) broadband pulses [23]. Thanks to the 4-f configuration SLM (4-f SLM) [24, 25], this method enables us to modulate the complex amplitude distribution in the radial axis of AP broadband pulse beams. However, a separate-path interferometer in the coherent combining system, being sensitive to disturbance (e.g. vibration and air turbulence), causes a practical issue.

One way to solve this issue is the use of common-path interferometers, which are robust over disturbance compared with our previous method. The generation of arbitrary AP beams using common-path interferometers has been demonstrated by utilizing a space variant wave plate (SVWP) or a q-plate [26], SLMs [27–29] or anisotropic crystal beam displacers [30]. In all cases, the generation methods need a SLM [27–29] including a digital micromirror device (DMD) [30] for complex amplitude modulation, preventing from generation of a point vortex [26] or a hypergeometric-Gaussian state [31]. However, the configurations of such devices do not compensate for the spatial chromatic dispersion caused by grating hologram patterns displayed on the DMD or the SLM. These optical systems are, therefore, unsuited for broadband pulses.

In the present paper, we generate arbitrary AP pulses with spatial complex amplitude modulation by the system composed of a 4-f SLM and a SVWP. The benefit of using the SVWP is robust against perturbations, but the complex amplitude modulation in the radial direction cannot be done with the SVWP. The use of 4-f SLM, which is suitable only for linear polarized states, can perform the complex amplitude modulation with compensation of chromatic dispersions. In contrast, the combination of the 4-f SLM and SVWP gives the complex amplitude modulation in all the directions without chromatic dispersions for not only linear polarization states but AP states. Moreover, we evaluate the properties of the AP pulses qualitatively and quantitatively through interference fringe pattern and extended Stokes parameters (ESPs) [23, 32–34], respectively. The present paper is organized as follows. We propose the system capable for the generation of arbitrary AP broadband pulses with spatial complex amplitude modulation. After that, we show the experimental results of the generation of p = 0 and p = 1 Laguerre-Gaussian (LG) AP pulses. Here, p denotes the radial index of LG modes [35]. The AP state and the spatial phase distribution are characterized by using the Stokes polarimetry and capturing the interference fringe pattern, respectively. We discuss the experimental results and comment prospects of this research. Finally, we summarize the conclusion.

2. Experiments

In this section, after proposing a generation system of arbitrary AP pulses possessing a broadband spectrum with spatial complex amplitude modulation, we show an experimental setup for evaluation of generated AP pulses. Finally, we describe experimental results of the generation of p = 0 and p = 1 LG AP pulses.

2.1. Proposal of system capable for generation of arbitrary AP pulses with a broadband spectrum

Our schematic generation system is shown in Fig. 1(a). This generation system has four principle components: a 4-f SLM system, a relay lens system, a polarization converter system and a SVWP.

 

Fig. 1 (a) System setup capable for generation of arbitrary AP broadband pulses. SLM1,2, a liquid crystal on silicon spatial light modulators; L1-L4, convex lenses (f_1,2,3 = 500 mm, f₄ = 1000 mm); I1,2, irises; AHWP1,2, achromatic half-wave plates; AQWP1, an achromatic quarter-wave plate; SVWP, a space variant wave plate (Photonic Lattice SWP-808). |s, l〉 (= |s〉 |l〉) represents the state of spin and orbit angular momentum of light in the ħ units [23, 32]. (b) an AP state of final generated pulses plotted on the l =1 normalized extended Poincaré sphere (EPS). The final generated state is manipulated by the angles of AHWP1 (θ_H1) and AHWP2 (θ_H2) in the polarization converter. (c) (ϕ_l=1, θ_l=1)-plane of the l =1 normalized EPS. Here, ϕ_l=1 and θ_l=1 are, respectively, azimuthal angle and polar angle. The rotations of AHWP1 and AHWP2 make the moves of the final generated AP state parallel to the θ_l=1 and ϕ_l=1 axes, respectively.

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The 4-f SLM system, which is composed of two SLMs (SLM1,2), two lenses (L1,2) and an iris (I1) (I1 selects the first diffraction ray), can modulate the complex amplitude distribution of the input horizontally-polarized Gaussian pulses without the spatial chromatic dispersion [24,25]. Normally, a 4-f SLM system is used to convert Gaussian broadband pulses into OV broadband pulses by applying both amplitude and phase modulations. In our system, we apply just amplitude modulation to the input Gaussian broadband pulses by giving the depth modulated grating pattern [36–39] on SLM2.

The pulse after SLM2 is no longer an eigenmode of the paraxial Helmholtz wave equation. The relay lens system (L3 and L4) provides the amplitude-modulated image on SLM2 to SVWP. Here, I2 in the relay lens system selects only the first diffraction ray from SLM2. The complex amplitude on the input facet of SVWP can be described by $f_{\text{SLM}2}^{1\text{st}}(r{f}_4/f_3)$, where $f_{\text{SLM}2}^{1\text{st}}(r)$ is the complex amplitude of the first diffraction beam in the beam cross section on SLM2,(f)_3,4 are, respectively, the focal lengths of L3 and L4.

A polarization converter system is located between L4 and SVWP. The system is composed of a sequence of achromatic wave plates in the order of an achromatic half-wave plate (AHWP1), an achromatic quarter-wave plate (AQWP1) and another achromatic half-wave plate (AHWP2). The Jones vector after the combination of AHWP1 and AQWP1 with a horizontally-polarized light input can be expressed by

In our proposing system, SVWP is a q = 1/2 half-wave plate (or q-plate [40]), which transforms the uniformly-distributed polarization state into an AP state, i.e., l = 1 cylindrically polarized (CP) state. Here, q and l describe the fast-axis distribution and the rotational symmetry of CP state [40, 41], respectively. In terms of the spin-to-orbit angular momentum conversion, the space variant wave plate can be regarded as a converter that flips spin angular momentum of light s and gives orbital angular momentum of light l [Fig. 1(a)] [32, 40, 42–44] because the Jones matrix of a SVWP is described by

The complex amplitude state f (r, ϕ) of the final generated pulse in the beam cross section can be described as

The normalized ESPs ${\tilde{\mathbf{S}}}_{l=1}^{\mathrm{E}}={\left(\begin{array}{ccc}{\tilde{S}}_{1,l=1}^{\mathrm{E}}& {\tilde{S}}_{2,l=1}^{\mathrm{E}}& {\tilde{S}}_{3,l=1}^{\mathrm{E}}\end{array}\right)}^{\mathrm{T}}$ [23, 32–34], which is an extension of conventional Stokes parameters into spatial dependent polarization states, describe the AP state of the final generated pulse as following:

Thereby, this generation system can generate arbitrary AP broadband pulses with spatial complex amplitude modulation.

2.2. Experimental setup

Figure 2(a) shows the schematic experimental setup (See Appendix for the complete experimental setup). A horizontally-polarized femtosecond pulse (central wavelength, 800 nm; bandwidth, 60 nm) from a Ti:Sa oscillator passed through a bandpass filter (BPF; central wavelength, 800 nm; bandwidth, 3 nm) and its pulse duration was evaluated to be ∼ 4 ps. Although the generation system we proposed supports broadband pulses, we used here a narrowband pulse as an input in order to perform fundamental demonstration experiments. The pulse was divided into two by a broadband non-polarizing beam splitters (BS1); they were guided into the main and the reference branches, respectively. The reference pulse was necessary for evaluating the spatial phase distribution of the finally generated pulse but unnecessary just for generation of AP pulses. The pulse in the main branch passed through the generation system and finally the generated pulse was evaluated in terms of two aspects: polarization and phase distributions. The former and the latter were, respectively, evaluated by using a rotating retarder-type imaging polarimeter [23, 45] and by capturing a spatial interference with the reference pulse beam. A rotating retarder-type imaging polarimeter was composed of an achromatic quarter-wave plate (AQWP2), a polarizer (POL1) and a charge coupled device camera (CCD). When we measured polarization distributions of the final generated pulses, we blocked the reference pulse beam. For the evaluation of phase distributions, we acquired the interference patterns of |s = 1〉 (|s = −1〉) component of the final generated pulse with the reference pulse beam, setting the angle of AQWP2 θ_Q2 = 135 deg (45 deg) [Fig. 2(b)]. When θ_Q2 = 135 deg (45 deg), the polarization analyzer composed of AQWP2 and POL1 outputs the |s = 1〉 (|s = −1〉) component as a vertically polarized state, thus the interference pattern with the vertically polarized reference pulses was observed. The crossing angle in the interference was 0.7 deg. The reference pulse was enlarged enough in the beam cross section by a beam expander, composed of L7 and L8, to cover the whole beam pattern of the final generated pulses. Since a pinhole (P2) was inserted into the beam expander for the sake of beam cleaning, the intensity distribution of the reference pulse was almost Gaussian shape. Its beam size was estimated to be 3 mm. An achromatic half-wave plate (AHWP3) and a polarizer (POL2) purified vertically-polarized reference pulses and adjusted their intensity in order to maximize visibility of the interference patterns.

 

Fig. 2 (a) Outline drawing of whole experimental setup. Ti:Sa Osc., a Ti:Sapphire oscillator (central wavelength, 800 nm; bandwidth, 60 nm); BPF, a bandpass filter (central wavelength, 800 nm; bandwidth, 3 nm); BS1,2, broadband non-polarizing beam splitters; L5-8, convex lenses (f_5,8 = 200 mm, f_6,7 = 100 mm); P1,2, pinholes; AQWP2,3, achromatic quarter-wave plates; POL1,2, polarizers; a delay stage is put in the reference pulse branch (see the Appendix). (b) Method for measurement of interference pattern of the |s = ±1〉 component of the final generated pulses. When θ_Q2 = 45 [deg], the polarization analyzer outputs the |s = −1〉 component as a vertically polarized state. By combining vertically polarized reference pulses, we observe an interference fringe pattern of the |s = −1〉 component of the final generated pulses with the reference pulse beam. When θ_Q2 = 135 [deg], we observe that of the |s = 1〉 component of the final generated pulses.

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A pinhole (P1) purified the spatial rotational symmetry of the final generated pulse because SVWP that we used was a 12-segmented half-wave plate and deteriorates somewhat its spatial rotational symmetry into C₁₂. If we use a smoothly fast-axis distributed SVWP, P1 is essentially not necessary.

In the experiment, we applied two patterns of amplitude modulations on SLM2; the one [Fig. 3(a)] and the other [Fig. 3(b)] were to generate p = 0 and p = 1 LG AP pulses, respectively. Here, the patterns were calculated by using the Davis’s method [37, 39]. In particular, the p = 1 LG mode has phase difference of π between the inner ring and the outer ring, which reflected the half-cycle gap of grating pattern between these rings [Fig. 3(b)]. As we mentioned before, we omitted the spiral phase term e^imϕ of LG modes from these fringe patterns. If we apply the spiral phase e^imϕ to the pattern on SLM2, the final generated pulse is not an eigenmode of LG AP mode any longer; in other words, the vector vortex state which is the superposition of |s = 1, l = m − 1〉 and |s = 1, l = m + 1〉 OVs is generated in that case.

 

Fig. 3 Grating patterns on SLM2 in order to generate (a) p = 0 and (b) p = 1 LG AP pulses.

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2.3. Experimental results

2.3.1. Generation of radially polarized pulses with spatial complex amplitude modulation

First, we generated the p = 0 and p = 1 LG radially polarized (RP) pulses in order to see how precisely the amplitude and phase modulation operated. The polarization and intensity distribution for p = 0 and p = 1 LG RP pulses are shown in Figs. 4(a) and 4(e), respectively.

 

Fig. 4 Generation results of (a–d) p = 0 and (e–h) p = 1 LG RP pulses. (a,e) Intensity and polarization distributions of (a) p = 0 and (e) p = 1 LG RP pulses. (b,f) Intensity on the radial axis r of (b) p = 0 and (f) p = 1 LG RP pulses. Red and cyan curves represent experimental and fitting ones, respectively. (c,d) Interference fringe patterns of (c) |s = 1〉 and (d) |s = 1〉 components in p = 0 LG RP pulses with the reference pulse beam. (g,h) Interference fringe patterns of (g) |s = −1〉 and (h) |s = 1〉 components in p = 1 LG RP pulses with the reference pulse beam. Note that the images in (e), (g) and (h) are displayed on a 75% scale of the images in (a), (c) and (d).

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As shown in Table 1, the polarization distributions were quantitatively evaluated by using the normalized l = 1 first ESP (strictly speaking, all the values of ESPs in this article were normalized by the energy of the temporally- and spatially-perfect-polarized states [23]) and the degree of polarization for the spatial distribution (DOP-SD) [23, 33]. Since the normalized l = 1 first ESP $S_{1,l=1}^{\mathrm{E}}$ gives the energy ratio of the RP state to the azimuthally polarized state as $\left(1+{\tilde{S}}_{1,l=1}^{\mathrm{E}}\right)/\left(2{\mathcal{P}}_{l=1}^{\text{space}}\right):\left(1-{\tilde{S}}_{1,l=1}^{\mathrm{E}}\right)/\left(2{\mathcal{P}}_{l=1}^{\text{space}}\right)$ [23], over 99.5% power of both the generated p = 0 and p = 1 pulses were RP states. Moreover, the DOP-SD ${\mathcal{P}}_{l=1}^{\text{space}}$ represents the energy purity in the symmetry of C_|l−1| (when l = 1, C_∞) polarization distribution [23]. The energy proportions of the states having C_∞ polarization distributions were ∼99% in both the generated pulses. These facts indicate that we generated RP pulses with high purity in terms of polarization distribution.

Table 1. The values of the first ESP ${\tilde{S}}_{1,l=1}$ and the DOP-SD ${\mathcal{P}}_{l=1}^{\text{space}}$ of generated p = 0 LG RP pulses [Fig. 4(a)] and p = 1 LG RP pulses [Fig. 4(e)].

View Table

Figures 4(b) and 4(f) show the intensity distributions (red curves) on the radial axis r of p = 0 and p = 1 LG RP pulses, respectively. The intensity on the radial axis I(r) were obtained by using the following equation:

Figure 4(c) [(d)] shows the interference pattern of |s = −1〉 (RCP) [|s = 1〉 (LCP)] component of p = 0 LG RP pulses with the reference pulses. Figure 4(g) [(h)] shows the interference pattern of |s = −1〉 [|s = 1〉] component of p = 1 LG RP pulses with the reference pulses. All the patterns had two-pronged fork patterns, which indicated that the absolute values of the dominant topological charge of these components were 1 [47, 48]. By considering the direction of the folk patterns, both the p = 0 and 1 LG RP pulses possessed |s = −1, l = +1〉 and |s = 1, l = −1〉 components. We surely generated the true LG RP pulses. Moreover, in-phase and out-of-phase positions of the inner and the outer rings of p = 1 LG RP pulses [Figs. 4(g) and 4(h)] flipped. The flip reflected the π phase jump between inner and outer rings of p = 1 LG modes. Thus, the spatial phase distribution as well as intensity distribution was successfully modulated by SLM2.

2.3.2. Generation of arbitral axisymmetrically polarized pulses with spatial complex amplitude modulation

In the next place, we demonstrated the generation of various AP pulses with p = 0 or p = 1 LG modes in order to show the arbitral AP pulses can be generated.

By using the polarization analyzer, we observed the transition of AP states when AHWP1 was rotated with the angle of AHWP2 fixed to θ_H2 = −22.5 deg. Figures 5(a) and 5(c) are, respectively, the transition of polarization distribution for p = 0 and p = 1 LG AP pulses with a 10 deg increase in θ_H1. We observed generation of elliptic AP states by rotating AHWP1, and hence the move of the AP state on the meridian of the l = 1 EPS [Fig. 1(b)] or the move of that parallel to θ_l=1 axis of (ϕ_l=1, θ_l=1)-plane [Fig. 1(c)] was realized. In contrast, Figs. 5(b) and 5(d) show the transition of AP states when AHWP2 was rotated with the angle of AHWP1 fixed to θ_H1 = +22.5 deg. The results show that the rotation of AHWP2 made the move of the AP state on the latitude line of the l = 1 EPS.

 

Fig. 5 Transition of polarization distribution when (a,c) AHWP1 was rotated under the condition that θ_H2 = −22.5 deg, or when (b,d) AHWP2 was rotated under the condition that θ_H1 = 22.5 deg. The grating pattern on SLM2 was (a,b) the pattern in Fig. 3(a) (p = 0 LG AP mode) or (c,d) the pattern in Fig. 3(b) (p = 1 LG AP mode). These polarization distributions are colored under the following rule: black, linear polarization; red, left-handed elliptical polarization; blue, right-handed elliptical polarization. The images in (c) and (d) are displayed on a 75% scale of the images in (a) and (b).

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Polarization distributions in Fig. 5 are visually apparent but give only qualitative information, hence we fully quantitatively evaluated these polarization distribution by using the normalized l = 1 ESPs and the l = 1 DOP-SD [23, 33]. Figures 6(a)–6(d), respectively, show the quantitatively-evaluated values of the polarization distributions depicted in Figs. 5(a)–5(d). No significant differences of ESPs and DOP-SD were found between the modulations, indicating that the manipulation of AP states by the polarization converter works well with independent of the spatial beam patterns. Basically, the values of ESPs followed the curves of Eq. (6). However, while ${\tilde{S}}_{2,l=1}^{\mathrm{E}}$ in Figs. 6(a) and 6(c), and ${\tilde{S}}_{3,l=1}^{\mathrm{E}}$ in Figs. 6(b) and 6(d) should be constant functions with respect to θ_H1 and θ_H2 respectively, they slightly deviated from the constant functions. This was ascribed to retardation errors of waveplates. We plotted the experimental data in Fig. 6 on the (ϕ_l=1, θ_l=1)-plane (Fig. 7), which showed that the shifts of the AP state parallel to θ_l=1 and ϕ_l=1 axes on the (ϕ_l=1, θ_l=1)-plane were achieved by rotating AHWP1 and AHWP2, respectively. Thus, this generation system can generate arbitrary AP pulse states. The values of DOP-SD ${\mathcal{P}}_{l=1}^{\text{space}}$ were over 0.988 in all the evaluated results, which meant that all the generated AP pulses had as high as over 98.8% purity of C_∞ polarization distribution.

 

Fig. 6 (a–d) The value of normalized l = 1 ESPs ${\tilde{S}}_{i,l=1}^{\mathrm{E}}(i=1-3)$ and l = 1 DOP-SD corresponding to the polarization distributions displayed in Fig. 5(a–d). Solid curves represent ideal ones described by Eq. (6).

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Fig. 7 Experimental results for (a) p = 0 and (b) p = 1 LG AP states plotted on the (ϕ_l=1, θ_l=1)-planes. Dashed lines are ideal trajectories [Eqs. (7)–(8)]

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3. Discussions and prospects

The experimental demonstration in Sec. 2.3.1 yielded high purity (≳ 93%) of p = 0 and p = 1 LG RP pulse states. Some researches have been shown that the p = 0 LG mode with high purity can be easily generated from a point vortex state by using a low-spatial-frequency-pass filter such as a pinhole. In comparison, the spatial quality of our generated beam was superior to that in an earlier research [49]. This improvement in our method was done by as follows: first, we pre-shaped the spatial beam pattern to the target pattern (except the phase profiles), which is imaged on the SVWP, by using a 4-f SLM so that the higher-order mode components are reduced. Second, by use of the low-spatial-frequency-pass filter, the residual higher-order modes (p > 0 LG modes) were fully eliminated. Thereby, in the present study, we essentially modulated the spatial complex amplitude through SLM and generated p = 0 and p = 1 LG RP pulse states. In addition to that, the improvement of the purity can be achieved by means of a feedback mechanism [50]. Some researches [51, 52] have reported on generations of single mode LG RP CW beams with high spatial quality. Their method utilized a Fabry-Perot interferometer as a mode cleaner after a SVWP. They, however, cannot be applied to broadband pulses. Unlike systems using Fabry-Perot interferometers, our system is suitable for broadband pulses, being able to generate superposition states of AP LG modes with different p indices.

If we replace the q = 1/2 half-wave plate with a q = Q half-wave plate, the final output pulse is l = 2q (= 2Q) CP pulse. Moreover, when we make complex amplitude modulation of l = 2q LG mode, the spatial mode of the final generated pulse will be l = 2q LG CP mode, which is an eigenmode of the paraxial Helmholtz wave equation. The system we proposed has thereby scalability of any order of LG CP modes.

As we mentioned in the Sec. 2.2, when the hologram grating pattern with e^imϕ is applied on SLM2, the finally generated pulses are the superposition of |s = 1, l = m − 1〉 and |s = −1, l = m + 1〉 OVs. Thus, they are no longer AP state pulses (since AP state is the superposition of |s = 1, l = −1〉 and |s = −1, l = 1〉 OVs), but vector vortex pulses. Since it is difficult to distinguish AP states and vector vortex state solely through polarization distributions, we observed the interference patterns of |s = 1〉 and |s = −1〉 OV of the final generated pulses with the reference pulses (Fig. 4(c,d) and (g,h)), respectively. Thereby, we conclude that we generated AP pulses. In a precise sense, the evaluation of mode decomposition is needed, but it is beyond the area of this paper and we will report on it in another article.

The system we proposed in Sec. 2.1, which supports the generation of broadband AP pulses, can generate ultrashort pulses when we just apply a frequency chirp compensation to the linear polarized broadband pulses before SLM1. Moreover, when a femtosecond polarization pulse shaper [53] is used in place of the polarization converter, we can realize the transformation of AP state in a single pulse, which may enable us to precisely manipulate light-matter interactions in nonlinear spectroscopy.

Recently, space division multiplexing transmission experiments by using LG modes with the higher radial index p in the free space have been demonstrated [54, 55]. In addition to that, the radial index of LG modes is proved to be a quantum number [56–58] and the fundamental experiments of Hong-Ou-Mandel interference was performed [59]. Our technique of generation of LG AP pulses can be applied in the fields of classical and quantum communication.

4. Conclusion

We have proposed a system to realize broadband generation of arbitral AP pulses with spatial complex amplitude modulation. In this system, a 4-f SLM and a space variant wave plate, respectively, compensates spatial chromatic dispersion and gives stability to perturbation. On the basis of our proposal, we constructed the demonstration system and generated various AP pulses by applying p = 0 and 1 LG mode modulations. The polarization purity evaluated by ESPs was high in each modulation (DOP-SD ${\mathcal{P}}_{l=1}^{\text{space}}>0.988$), and there was no significant difference between the modulations. We confirmed that the complex amplitude modulation worked well through intensity patterns and interference fringes with not so broadband pulses. The evaluation for broadband pulses is a future study. Our proposed system, which is easily extended to generation of ultrashort LG CP pulses, can be utilized in classical and quantum communication with higher-order spatial modes.

Appendix

In this section, we describe the detail of experimental setup. Figure 8 shows the actual experimental setup. Single mode fiber (SMF) performed a function as a mode cleaner. A pair of an achromatic half-wave plate (AHWP0) and a polarized beam splitter (PBS) modulated pulse energy. The 4-f SLM system was realized with the folded path configuration as depicted in Fig. 9. The grating patterns for SLM1 and SLM2 in Fig. 1(a) were shown on the dimidiate areas of a single spatial light modulator (SLM). The sequence of relay lenses in the reference pulse beam path (Fig. 8, the dashed line) formed an image on CCD and stabilized the pointing of the reference pulse beam. P2 cleaned the spatial beam profile of reference pulses.

 

Fig. 8 Whole experimental setup. Ti:Sa Osc., a Ti:Sapphire oscillator (central wavelength, 800 nm; bandwidth, 60 nm); OL1,2, objective lenses; SMF, single mode fiber (∼10 m); M1-15, mirrors; L1-L14, convex lenses (L1-3,9,10,12,14, f = 500 mm; L4,11, f = 1000 mm; L5,7,13, f = 200 mm; L6,8, f = 100 mm); PBS, a polarizing beam splitter; BS1,2, broadband non-polarizing beam splitters; SLM, a liquid crystal on silicon spatial light modulator; MZ, a zero-incidence mirror; I1,2, irises; GL1,2, Glan-Laser Calcite polarizers; AHWP0-2, achromatic half-wave plates; AQWP1,2, achromatic quarter-wave plates; SVWP, a space variant wave plate (Photonic Lattice SWP-808); Delay, a delay stage; POL2, a polarizer; BPF, a bandpass filter (central wavelength, 800 nm; bandwidth, 3 nm).

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Fig. 9 A three-dimensional configuration of optical components in a 4-f SLM system.

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Funding

Japan Society for the Promotion of Science (JSPS) (JP26286056, JP15J00038, JP16H06506); Japan Science and Technology agency (JST) (CREST).

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