Full Quantitative Analysis of Arbitrary Cylindrically Polarized Pulses by Using Extended Stokes Parameters

Cylindrically polarized (CP) modes are laser beam modes which have rotational symmetry of the polarization distribution around the beam axis. Considerable attention has been paid to CP modes for their various applications. In this paper, by using the extended Stokes parameters and the degree of polarization defined for the spatial distribution (DOP-SD), we fully-quantitatively characterize the spectrally-resolved polarization states of arbitrary CP (axisymmetrically polarized and higher-order cylindrically polarized) broadband pulses generated by coherent beam combining. All the generated pulse states were fully-quantitatively analyzed for the first time and proved to have high symmetry (DOP-SD ≳ 0.95) and low spectral dependence of polarization states. Moreover, we show the DOP-SD, which cannot be defined by the conventional higher-order and hybrid Stokes parameters, enables us to make a quantitative evaluation of small degradation of rotational symmetry of polarization distribution. This quantitative characterization with high precision is significant for applications of precise material processing, quantum information processing, magneto-optical storage and nonlinear spectroscopic polarimetry.

Scientific RepoRts | 5:17797 | DOI: 10.1038/srep17797 spatial distribution (DOP-SD; modified DOP) 37,38 , and have already shown their availability of quantitative characterization of l = 1 CP narrowband pulses 38 (the pulses having C ∞ symmetry of their transverse electric fields; the definition of l is described in our report 37 ). In the present paper, to demonstrate the importance of the fully-quantitative characterization of CP beams, we generate l = 1 and l = 2 CP broadband pulses and make fully-quantitative spectrally-resolved characterization by using the ESPs and their DOP-SD. To our knowledge, the fully-quantitative characterization of various CP broadband pulse states is conducted for the first time. CP pulses recently began to be used in some applications such as material processing 39 and nonlinear spectroscopic polarimetry 40 , where broadband or ultrashort CP pulses give us more information in the frequency or temporal domain. In this sense, the fully-quantitative characterization of broadband or ultrashort CP pulses here is significant.

Results and Discussions
Arbitrary manipulation of cylindrically polarized pulse states. We here describe the basic concept of generating arbitrary CP broadband pulses ( Fig. 1(a)). The detail of the experimental setup is shown in Supplementary Fig. S1. First, x-polarized = l 0 broadband (or ultrashort) pulses are converted into x-polarized = + l m optical vortex (OV) by the spatial light modulator in the 4-f configuration (4-f SLM). Here, l is referred to as the azimuthal index of LG modes 41 . A super-achromatic half-wave plate (HWP1) based on the design by Pancharatnam 42 and a coherent combining system coherently superpose x-polarized = + l m and y-polarized = − l m OV broadband pulses, whose energy ratio is controlled by HWP1; θ θ ( ) ( ) cos 2 : sin 2 2 H1 2 H1 . After that, the x-and y-polarized components of E 3 are converted into = − s 1 and = + s 1 circularly polarized states by a super-achromatic quarter-wave plate (QWP1), respectively. Here, s is the spin angular momentum of photon in units of ħ 38 . The pulse passes through a super-achromatic half-wave plate (HWP2), following which the sign of spin angular momentum of light is flipped 36 and the relative phase between = + s 1 and = − s 1 states can be adjusted by the rotation angle of HWP2 θ H2 :  which is represented by the point (θ, φ) = (4θ H1 , π/2 + 4θ H2 ) on the extended Poincaré sphere ( Fig. 1(b)). Hence arbitrary manipulation of CP broadband pulse state can be achieved by adjusting the rotation angles of HWP1 and HWP2. In the present paper, we characterize generated pulse states and spatial symmetry by using parameters of the normalized extended Stokes vectors and the lth DOP-SD P l space , respectively. The definition of P l space is in Supplementary materials. Full quantitative analysis of cylindrically polarized states. We respectively generated seven states for l = 1 and l = 2 CP broadband pulses: (θ, φ) = (0, 0), (π/4, 0), (π/4, π/4), (π/4, π/2), (π/2, 0), (π/2, π/4), (π/2, π/2). For simplicity, (θ, φ) is omitted hereafter. The light source used is a Ti:sapphire laser amplifier (center wavelength 800 nm, bandwidth of ~40 nm, pulse duration ~25 fs, and repetition rate 1 kHz). Figure 2 shows characterization results for l = 1 (π/2, 0) and l = 2 (π/2, 0) CP pulses as typical examples. Spectrally-resolved polarization distributions are shown in Fig. 2(a,d); (a) is for l = 1 (π/2, 0) CP pulses (or radially polarized pulses) and (d) is for l = 2 (π/2, 0) CP pulses. From the polarization distributions in Fig. 2(a,d), the values of ,  S 1 1 E ( Fig. 2(b)) and P 1 space (Fig. 2(c)), and ,  S 2 1 E ( Fig. 2(e)) and The characterization results for all states are described in Fig. 3; (a) and (b) are for l = 1 CP pulse states and (c) and (d) are for l = 2 CP pulse states. Figure 3(a,c) respectively represent the l = 1 and l = 2 extended Poincaré sphere, on which the spectrally-resolved values of normalized ESPs ( ,  ,  ) , , , in l = 1 and l = 2 CP states are plotted. The spectrally-resolved values of DOP-SD corresponding to the CP states in Fig. 3(a,c) are shown in Fig. 3(b,d), respectively.
All polarization distributions of l = 1 (π/2, 0) CP pulses at measured wavelengths (780, 790, 800, 810 and 820 nm) in Fig. 2(a) are almost purely radially polarized. This fact is well indicated by the obtained results that ,  S 1 1 E and P 1 space were respectively over 0.99 and 0.98 in all spectral regions ( Fig. 2(b,c)). Since ,  S 1 1 E is associated with the energy ratio between (π/2, 0) (radially polarized) state and (π/2, π) (azimuthally polarized) state 38 , which is given by ( , over 99% energy of the temporally-and spatially-perfect-polarized 37 (TSPP) state was radially polarized. Moreover, DOP-SD P 1 space enables us to evaluate the over 98% of the temporally-perfect-polarized state of the generated pulses were TSPP state. Consequently, the pulses generated from a coherent combining system had high purity of l = 1 (π/2, 0) CP state and high symmetry of polarization distribution.
The contamination comes from two factors. One is the deformation of incident OV pulses into the coherent combining system; the other is the degradation of extinction ratio of the polarizing beam splitter in the coherent combining system because of inclining incident angle. Figure 4(a-d) respectively depict the intensity and polarization distributions of (l, θ H1 ) = (1, 0), (1, π/4), (2, 0) and (2, π/4) cases. The measurements of Fig. 4(a,c) and (b,d) are respectively conducted under blocking the blue blanch and the magenta branch in Supplementary Fig. S1, which means E 5 should be proportional to = + = − s l m 1 and = − = + s l m 1 . However, these intensity distributions are of twofold symmetry rather than axisymmetry. This result is attributed to the slight superimposition of = ± l m 2 component on = l m OV pulses because of deformation passing through optic elements. Though the polarization distribution should be circularly polarized, the polarization states are elliptic. The fact can be ascribed to the degradation of extinction ratio of the polarizing beam splitter in the coherent combining system because of inclining incident angle (in other words, the contamination of s-and p-polarized components at the polarizing beam splitter). The actual electric field of E 5 is approximately described as leads to degradation of C |m−1| rotational symmetry. The value of DOP-SD of l = 2 CP pulses are thus smaller than that of l = 1 pulses. Figure 3(a,c) respectively indicate the spectral dependence of polarization states of l = 1 and l = 2 CP pulses. All the pulse states have quite low spectral dependences thanks to optics for broadband pulses such as super-achromatic wave plates and a low-group-velocity-dispersion polarizing beam splitter. All the values of DOP-SD for l = 1 and l = 2 CP pulses have low spectral dependence (∆ = , P l 1 2  0.01), while the DOP-SD values for l = 2 CP pulses are somewhat less than those for l = 1 CP pulses by 0.02 to 0.03 ( Fig. 3(b,d)) because of the previously described reasons. These results clearly show that our system employing coherent beam combining is able to generate arbitrary CP broadband pulse states with high symmetry and low spectral dependence, which is fully-quantitatively investigated by ESPs and DOP-SD with high precision.

Comparison with simulation.
In this section, we mention the comparison between the experimental and the simulation results. We conducted simulation for l = 1 (π/2, 0) and l = 2 (π/2, 0) CP states. The simulation results are shown in Fig. 5(a,b) and Table 1. Both intensity and polarization distributions in Fig. 5(a,b) well agree with that of the experimental results for l = 1 (π/2, 0) ( Fig. 2(a)) and l = 2 (π/2, 0) ( Fig. 2(d)) states, respectively. The values of ,  S l 1 E and DOP-SD P l space in Table 1 are also in good agreement  Supplementary Fig. S1, respectively. The white and black arrows are placed to emphasize the twofold symmetry of the intensity patterns. All the polarization distributions are colored under the rule in Fig. 2.
with the experimental results in Fig. 2(b,e) and (c,f), respectively. In particular, there is a small (~0.02) difference between l = 1 and l = 2 cases in the simulation results for DOP-SD, which also appears in the experimental results. Therefore, it should be stressed that our measurement method is able to detect such small asymmetricity.

Perspective
It has been pointed out that precise measurement of polarization state is important in quantum information 43 . Applications using polarized pulses such as material processing 44 , magneto-optical storage 45 and nonlinear spectroscopic polarimetry 40 also need to know their polarization states precisely. Using CP pulses instead of the conventional uniform polarized pulses is a manner to extend the degree of freedom in these applications, which have been already demonstrated in quantum information science 5-7 , material processing 16 and nonlinear spectroscopic polarimetry 40 . Our fully-quantitative measurement method for CP pulses hence can improve the sophistication of these applications. We think that frequency chirp compensation can be easily achieved because of optics components for broadband pulses in our system. Characterization results in Fig. 3(a,c) show that the dispersions of spectrally-resolved polarization states in individual CP pulse states are small (�0.05 in propagation distance on the extended Poincaré sphere). CP ultrashort pulses with steady polarization state in the pulse duration, which is especially important for applications for magneto-optical storage and nonlinear spectroscopic polarimetry, can be therefore generated with our system. Our experimental setup, where the accessible spectral range covers the region from 690 nm to 1080 nm (limited by the polarizing beam splitter and half-wave plates), offers us the capability of generating ultrashort CP pulses below 10 fs without polarization distribution dispersion. Moreover, by insertion of a femtosecond polarization pulse shaper 46 after the 4-f SLM system, in place of HWP1 and HWP2, our experimental setup will be able to generate the CP pulses with arbitrary control of temporal CP states on one extended Poincaré sphere. Although the issue of fully-spatiotemporal characterization method for ultrashort pulses with nonuniform polarization distribution still remains, our measurement method is quite useful for precise characterization of ultrashort pulses.
The good agreement between the experimental and simulation results indicates that the degradation in DOP-SD is ascribed to the deformation of incident OV pulses and the contamination of perpendicular polarized components at the polarizing beam splitter, and ensures that we can quantitatively investigate even the small differences of rotational symmetry of polarization distributions or the small contamination of unwanted modes by using DOP-SD. At least ∆ .  P 0 02 space is significant and detectable in our measurement system. Though the earlier studies have not taken account of DOP-SD, DOP-SD as well as ESPs is an important parameter for full-quantitative characterization of CP states.

Methods
Generation of broadband optical vortex pulses. The generated pulses from a Ti:sapphire laser amplifier are attenuated by ND filters, following which the 4-f SLM converts into x-polarized l = 1, p = 0 or l = 2, p = 0 OV pulses. Here, p denotes the radial index of LG modes 41 . The 4-f configuration in the (a) ( b) Figure 5. The simulation results for (a) l = 1 (π/2, 0) and (b) l = 2 (π/2, 0) pulse states. All the polarization distributions are colored under the rule in Fig. 2. SLM system enables us to compensate for angular dispersion 47,48 . We furthermore utilize a complex-amplitude modulation technique with a phase-only SLM 49,50 as means to convert to broadband arbitrary single LG mode OV pulses.
Finding the zero delay in the coherent combining system. Using a polarizer (POL2 in Supplementary Fig. S1) and a spectrometer, we find the zero delay with the aid of the spectrum interference method. A charge-coupled-devise (CCD1 in Supplementary Fig. S1) monitors the intensity profile of the x-polarized component of E 4 in order to ensure the delay time is unchanged within the polarization measurement.
Measuring polarization distributions. In the polarization measurement system, the pulses are spectrally-resolved by bandpass filters (BPF in Supplementary Fig. S1; center wavelengths, 780, 790, 800, 810, 820 nm; bandwidths, 10 nm), then their polarization distributions are acquired by using a rotating-retarder type imaging polarimeter 51 , which is composed of an achromatic quarter-wave plate (QWP2 in Supplementary Fig. S1), a Glan-Laser polarizer (GLP in Supplementary Fig. S1) and a charge-coupled-devise camera (CCD2 in Supplementary Fig. S1). Simulation. We respectively evaluated δ 1,2,4,5 and δ 3,6 from the intensity and the polarization distributions in Fig. 4(a-d) (the values are in Table 1). The intensity distributions were plotted by using the following equation based on equation 3: