Comprehensive quantitative analysis of vector beam states based on vector field reconstruction

We demonstrate a comprehensive quantitative analysis of vector beam states (VBSs) by using a vector field reconstruction (VFR) technique integrating interferometry and imaging polarimetry, where the analysis is given by a cylindrically polarized Laguerre–Gaussian (LG) mode expansion of VBSs. From test examples of cylindrically polarized LG mode beams, we obtain the complex amplitude distributions of VBSs and perform their quantitative evaluations both in radial and azimuthal directions. The results show that we generated (l, p) = (1, 0) LG radially polarized state with a high purity of 98%. We also argue that the cylindrically polarized LG modal decomposition is meaningful for the detail discussion of experimental results, such as analyses of mode purities and mode contaminations. Thus the VFR technique is significant for analyses of polarization structured beams generated by lasers and converters.

symmetry of a VBS, respectively 38,40,42,43 . However, these parameters, being calculated from just intensity distributions of several polarization components, do not provide a phase distribution. Moreover, information of VBSs with respect to the radial and azimuthal coordinates is lost because these parameters are spatially averaged. In order to make full fine evaluations of VBSs, we need to acquire complex amplitude vector distributions of VBSs.
The acquisition of complex amplitude vector distributions of VBSs has been reported by using a phase-shifting interferometry method [44][45][46] . This method requires taking multiple (over five) interference patterns. If an interferometer is not stable or arms are too long to keep the relative global phase fluctuate between the object and the reference beams, it is hard to correct the relative global phase of acquired interference patterns. The authors' group has developed a field reconstruction technique for a uniformly polarized optical vortex (OV) state 47,48 . This field reconstruction technique gives a complex amplitude distribution of a certain uniform polarization component in the beam cross section. Since VBSs are expressed by the superposition of two orthogonal polarization components 49 , we can acquire the complex amplitude vector distribution of VBSs by taking the complex amplitude distributions in each orthogonal polarization through the field reconstruction technique ( Fig. 1(b)). Figure 1(d) shows the system that integrates the imaging polarimetry and the field reconstruction, which is basically an extension of the polarization analyzer system introduced in Fig. 1(c). We call hereafter this technique reconstructing two orthogonal polarization states "the vector field reconstruction (VFR) technique".
Moreover, VBSs can be decomposed in orthonormal basis such as cylindrically polarized Laguerre-Gaussian (LG) modes 41,50 . The cylindrically polarized LG modes, having two indices; the azimuth index l and the radial index p, are solutions to the paraxial Helmholtz wave equation. A single cylindrically polarized LG mode propagates with scaling its intensity and polarization distributions. As the LG modal expansion of a uniformly polarized OV state 48,51,52 , the mode expansion coefficients φ c lp r , can be calculated from the complex amplitude vector distribution of VBSs. Here r, φ are radial and azimuthal coordinates, respectively. In the present paper, we expand VBSs into cylindrically polarized LG modes through an interferometric method.
We demonstrate cylindrically polarized LG modal expansion of VBSs by using a VFR technique based on interferometric field reconstruction, for the first time. The present paper is organized as follows. First, we describe a procedure of the VFR technique ( Fig. 1(b)) in the section of Reconstruction of Complex Amplitude Vector Distribution. We experimentally demonstrate reconstructions of complex amplitude vector distributions of VBSs. The polarization distributions of the reconstruction results are compared with ones obtained by imaging polarimetry (IP) depicted in Fig. 1(a). Here, the IP method is regarded as VFR method without interferometry. Thus, we measured the object beams through both the IP method (blocking the reference beam) and the VFR technique by using the same optical setup. Second, we introduce a cylindrically polarized LG modal decomposition of VBS states in the section of Decomposition of Vector Beam State in Cylindrically Polarized Laguerre-Gaussian Modes, following which we show and discuss the decomposition results of the complex amplitude vector distributions. Finally, we summarize the conclusion in the section of Conclusion. We describe specific methods and definitions in the section of Methods.  are the polarization basis for RCP and LCP states. As mentioned in introduction, the RCP and LCP components can be individually reconstructed by using the field reconstruction (FR) technique 47 , which was originally invented by Takeda et al. 53 . The FR technique is reviewed in the subsection of Field reconstruction of uniformly polarized optical vortex states, in Methods. Experimental setup for acquisition of interference pattern is shown in Fig. 1(c). We call the system a polarization analyzer system. The polarization analyzer system was based on a rotating-retarder type imaging polarimetor [39][40][41] , which was composed of a quarter-wave plate (QWP), a polarizer (POL1) and a charge coupled device (CCD) camera. A non-polarizing beam splitter (BS) was inserted between POL1 and the CCD camera in order to make interference between the VBS beam and a reference beam. When the reference beam is blocked, the polarization analyzer system is just a general imaging polarimeter. Thus, we can say that we integrated the imaging polarimeter with an interferometer into one system in order to reconstruct vector fields.
We can acquire the interference patterns for RCP and LCP components by setting the rotation angle θ π = /4 and 3π/4, respectively 42 , where θ is the angle of the fast axis of QWP from the horizontal direction (See the subsection of Field reconstruction of uniformly polarized optical vortex states, in Methods). When measurement is made at = z z 0 , the complex amplitude vector distribution on the measurement plane is described as follows: FR are the reconstructed RCP and LCP complex amplitude components, respectively. We note that the relative phase δ between RCP and LCP components cannot be generally neglected because it is not always true that the measurements of the interference patterns of the RCP and LCP components are made at the same time. Since we perform the measurements of the interference fringes in succession with a single CCD camera ( Fig. 1(c)), which enables us to simplify the superposition of the RCP and LCP components without geometry transformation, we need to extract the relative phase δ. An earlier study 54 acquired the interference patterns of x-and y-polarized components simultaneously by displacing them using a beam displacer. This manner can make reconstructing the complex amplitude vector distribution imprecise because there should remain the issue of spatial registration between x− and y− polarized components. In order to extract the relative phase δ, it is sufficient to show the consistency in intensity distributions of a certain polarization component between IP and VFR methods. We estimated experimentally the relative phase δ through searching the minimum of the mean-squared error G δ of intensity profile of the x-polarized component: where R is a cutoff radius, and φ E r z ( , , ) x IP 0 2 is the intensity profile of the x-polarized component calculated through IP (Fig. 1(b)). The detail is described in the subsection of Evaluation of relative phase δ, in Methods. Finally, we acquire the complex amplitude vector distribution.

Results and discussions.
In this subsection, we report the analysis results of the cylindrically polarized LG pulses generated by using the combination of a 4-f spatial light modulator and a common-path optical systems in ref. 42 . Figure 2 shows an essence of the whole system. The detail of the generation system is shown in Fig. 8 of ref. 42 . The light source that we used was a Ti:Sapphire oscillator (central wavelength, 800 nm; bandwidth, 60 nm). A bandpass filter (central wavelength, 800 nm; bandwidth, 3 nm) narrowed the bandwidth so that we can apply VFR technique. A Mach-Zehnder interferometer was installed in the setup for VFR. One path was for the reference beam of the polarization analyzer system, the other path was for the generation of cylindrically polarized LG pulses. We generated the cylindrically polarized pulses with a complex amplitude modulation in the radial axis by using a spatial light modulator in the 4-f configuration 55,56 and a space variant wave plate 57,58 .
In the present paper, we analyze two examples of the cylindrically polarized LG modes. The first one is = l 1 radially polarized state with spatial modulation of a p = 0 LG mode (We call it p = 0 radially polarized LG pulses). The second one is = l 1 radially polarized state with spatial modulation of a p = 1 LG mode (We call it p = 1 radially polarized LG pulses). Figure 3 shows a comparison of the analysis results of p = 0 radially polarized LG pulses between the IP and FR methods. The intensity distributions of the RCP and LCP components measured by the IP method (i.e. a measurement without the reference beam in Fig. 1(c)) and the FR method are displayed in Fig. 3(a) and the upper row in Fig. 3(b), respectively. The intensity distributions were well reconstructed through the FR method. The fine noisy structure shown in the intensity distribution taken by the IP method ( Fig. 3(a)) was not reproduced in the reconstructed ones thanks to the spatial frequency filtering in the reconstruction process. The mean-squared error of the RCP and the LCP intensity distributions were evaluated as 3 × 10 −4 and 6 × 10 −4 , respectively. The low mean-squared errors (~10 −4 ), which are the same orders as the previous study 47 , means that the reconstruction of the RCP and LCP components was well succeeded in. Unlike the IP method, the FR method outputs phase distributions (lower row in Fig. 3(b)) as well as the intensity distributions. From the phase profiles, the RCP and the LCP components were mainly l = −1 and = l 1 OV states, respectively. That indicates that the VBS was mainly a = l 1 cylindrically polarized state, that is, an axisymmetrically polarized (AxP) state 49 .
The arm length of the interferometer was over 6 m. Disturbance (e.g. vibration and air turbulence) easily led to a fluctuation of the arm length with respect to time, and therefore we needed to estimate the relative phase δ (see the subsection of Evaluation of relative phase δ, in Methods for the detail). From Eq. (4), we evaluated δ was 2.92 rad, thereby reconstructing the complex amplitude vector distribution.
Since the relative phase δ is a known parameter, we are able to change the basis from the cylindrically polarized ones to AxP ones:  Figure 2. Experimental Setup. Ti:Sa Osc., a Ti:Sapphire oscillator; BPF, a bandpass filter; BS, a beam splitter; 4-f SLM system, a system of a spatial light modulator in the 4-f configuration; SVWP, a space vatiant wave plate; M1,2, mirrors; Polatization Analyzer System, a polarization analyzer system for the VFR method depicted in Fig. 1(d). Here, e r ≡ (cosφ, sinφ) T and e φ ≡ (−sinφ, cosφ) T are, respectively, the basis for radially polarized and azimuthally polarized states.
are, respectively, the radially polarized and the azimuthally polarized components. Figure 4(a) shows intensity and phase distributions of the radially polarized and the azimuthally polarized components (| . Since the intensity of the azimuthally polarized component was small enough compared to the radially polarized one, the beam under test was regarded as a radially polarized state. We note that the phase distribution of the radially polarized component had no phase ramp along the azimuthal direction, which indicates the object beam under test was an AxP state. We discuss quantitatively it in the section of Decomposition of Vector Beam State in Cylindrically Polarized Laguerre-Gaussian Modes. We thereby accomplished the VFR of the object beam.
Since we know the complex amplitude vector distribution φ  r z E( , , ) 0 , we can obtain Stokes parameter on the measurement plane through the VFR method. Figure 4(b) gives polarization distributions of the beam obtained from the IP method (left) and the VFR method. The both polarization distributions represented the radially polarized state. It was hard to find the difference between them qualitatively. In order to make a quantitative comparison of the results between VFR and IP methods, we calculated the ESPs of the polarization distributions given by the IP method and the VFR method (S l 1 , respectively). Definitions of ESPs are described in the subsection of Extended Stokes parameters, in Methods and Refs. 38,40,41,43 . Table 1  In the same way, we investigated p = 1 radially polarized LG pulses generated by the system in ref. 42 . Figure 5(a) shows intensity and phase distribution for  www.nature.com/scientificreports www.nature.com/scientificreports/ beam under the test was regarded as a radially polarized state. The intensity distribution of  E r FR had two rings, which indicated a p = 1 LG mode. It is well known that there is a π phase shift between the inner and the outer rings of p = 1 LG mode 59 . The phase distribution of  E r FR in Fig. 5(a) showed that the phase jump of ~π was located on the boundary of the inner and outer rings. The quantitative mode decomposition results is described in the section of Decomposition of Vector Beam State in Cylindrically Polarized Laguerre-Gaussian Modes. The polarization distribution derived from the IP method and the VFR method are displayed in Fig. 5(b). These polarization states were radially polarized ones. Table 2 gives the corresponding ESPs for the polarization distributions. The angle formed by the two ESPs was evaluated to be ,VFR rad = 0.059 deg, which shows that we excellently reconstructed the polarization distribution through the VFR method.
Several research groups have reported the fiber mode expansion after a propagation in multimode fiber. Shapira et al. demonstrated the decomposition of eigenmodes of a photonic-band gap fiber through a noninterferometic approach 60 . This approach needs taking into account both of far-field and near-field intensity profiles. The issue of spatial registration 61 (e.g. the uncertainty of beam centers) can easily decrease the accuracy of the decomposition results, and it is inevitable for this approach. Fatemi et al. reported LP mode decompositions of vector beams via an interferometric approach 54 . However, it is hard to restore the original complex amplitude vector distribution in the beam cross section because each polarization component is measured after spatial separation of two orthogonal polarization states and there also can remain the issue of spatial registration. In contrast to them, we can reconstruct complex amplitude vector distributions without the issue of spatial registration since our proposed method does not, in principle, need to capture images at different propagation positions or spatially separate the optical path of the two orthogonal polarization states.

Decomposition of Vector Beam state in Cylindrically polarized Laguerre-Gaussian Modes
Description of mode decomposition. We here describe a cylindrically polarized LG mode decomposition. Since the cylindrically polarized LG modes u l p , CPLG are one of the orthonormal basis for paraxial beams, a complex amplitude vector distribution at the measurement plane can be decomposed as follows: l p l p l p r l r l p l 0 0 , , sin cos , l r l are the lth cylindrically polarized basis 43 of radially polarized and azimuthally polarized states, respectively. U(x) is a step function that if x ≥ 0, U(x) = 1, otherwise U(x) = 0. Here, R is the cutoff radius introduced in Eq. (4). U(R − r) means that we put an imaginary aperture whose radius is R at the measurement plane. E 0 represents an amplitude of the complex amplitude vector distribution, defined by  Fig. 6(a,b), respectively. A definition of the cylindrically polarized LG modes u l p , CPLG is described in the subsection of Cylindrically polarized Laguerre-Gaussian modes, in Methods.
From Eq. (6), the mode coefficients are . Thus, we can discuss proportions of modes in the complex amplitude vector distribution through the mode coefficients c l p r , and φ c l p , . Strictly speaking, the decomposition result is equivalent to the mode distribution at z = z 0 when an iris whose diameter is 2R is placed at z = z 0 . We note that the mode coefficients depend on the beam waist radius w 0 and the distance from the beam waist z 0 − z w , which are described in the subsection of Cylindrically polarized Laguerre-Gaussian modes, in Methods (z w is a position of the beam waist on the z axis). We set z 0 − z w = 0 because the object beam was almost collimated. We describe how we decided the beam waist radius in the next subsection.

Results and discussions.
In this subsection, we describe decomposition results of the complex amplitude distribution into cylindrically polarized LG modes. Figure 7 shows the cylindrically polarized LG mode decomposition results of the complex amplitude distribution of p = 0 radially polarized LG pulses [ Fig. 4(a)]. Since we intended to generate (l, p) = (1, 0) radially polarized LG state, we searched the best w 0 maximizing the intensity of the (l, p) = (1, 0) mode coefficients | | . The other mode intensities were not zero but less than 0.005. We intended to generate an (l, p) = (1, 0) radially polarized state, and thus we succeeded in the modulation to the target cylindrically polarized beam with high purity. Thereby, we can quantitatively analyze the quality of cylindrically polarized beams.
We decomposed the complex amplitude distribution of p = 1 radially polarized LG pulses shown in Fig. 5 into cylindrically polarized LG modes with the same beam waist radius as the (l, p) = (1, 0) radially polarized beam. We  Fig. 8(a,c), the (l, p) = (1, 1) radially polarized LG mode was the dominant one (| | = .  www.nature.com/scientificreports www.nature.com/scientificreports/ radially polarized LG mode) whereas there were somewhat ignorable other unwanted modes. The unwanted modes mainly appeared on = l 1 radially polarized modes [ Fig. 8(a)], which indicated that the polarization distribution of the object beam was an = l 1 radially polarized state but the complex amplitude distribution on the radial axis was deviated from that of a (l, p) = (1, 1) radially polarized LG mode. The contamination of other modes was ascribed to that a spatial filter (PH in Fig. 8 of ref. 42 ) in front of a CCD camera to improve the beam rotational symmetry had a slightly small hole and excited = l 1, p ≠ 1 radially polarized modes. The cylindrically polarized LG mode decomposition thus offers us information in order to discuss experimental results in detail.
In general, the VBSs generated by spatial light modulators, spiral phase plates, or space variant wave plates are Hypergeometric Gaussian modes 62,63 . In contrast, our generation system modulates the intensity pattern on the radial axis by using SLMs so that we can generate LG modes but not Hypergeometric Gaussian modes 42 . Thus, we choose cylindrically polarized LG modes as the base functions.

Conclusion
We demonstrated the reconstruction of complex amplitude vector distributions of VBSs through the VFR technique and the cylindrically polarized LG mode decomposition of them. The reduction of the number of interferometric image in comparison with the earlier studies 44-46 enabled us the reconstruction by using an unstable interferometer even with arm length over 6 m. We evaluated accuracy of VFR technique through comparing the ESPs calculated from the polarization distributions to the ones acquired by the IP method. The difference between them was less than 1 degree on the extended Poincaré sphere. Since the VFR method is the integration of interferometry and the IP method, the VFR gives not only spatial polarization profiles of vector beam states but also their spatial phase profiles with high accuracy. We showed that the cylindrically polarized LG modal decomposition of VBSs is meaningful for the detail discussion of experimental results, such as analyses of mode purities and mode contaminations.
The radial index p, which gives a node number of a cylindrically polarized LG mode on the radial axis, had been nearly forgotten or regarded as a trivial feature 64 . However, it is theoretically and experimentally proved that the radial index p as well as the azimuth index l of LG modes is attributed to a quantum number of photon [64][65][66][67] . Thus, we note that the full quantitative characterization of VBSs is significant for not only classical optics but also quantum optics.
This VFR and modal decomposition techniques are utilized in mode distribution analysis of lasers emitting VBSs 51,68-72 and VBS converters [73][74][75][76][77][78][79] . Moreover, in the context that some researches [80][81][82] claim that multimode waveguides and fibers should be treated with exact modes, that is VBSs, instead of LP modes. Our VFR methods can be also suitable for characterization of spatial mode properties of multimode waveguides and fibers.
We built a rotating-retarder type imaging polarimeter consisting of a quarter-wave plate, a polarizer and a CCD camera (Fig. 1(c)). The polarization axis of the polarizer is parallel to the y axis. The complex amplitude distribution at the imaging plane is written by We here note that = (the RCP component). Therefore, the intensity distribution recorded by the CCD camera I(x, y, z 0 ) is described as Since the Stokes parameters have different cycles against the rotational angle θ each other, we can acquire each Stokes parameter distribution by measuring images of over four different θ. In the experiments, we captured four intensity distributions of θ = π/8, π/4, 3π/4 and 7π/8. The Stokes vector is described as follows: We, thereby, obtain the Stokes parameter distributions on the beam cross section at the measurement position. Here we choose θ π π π = /8, /4, 3 /4 and 7π/8 because S 1−3 are described by the linear combinations of the differences of two intensity distributions at θ π π π = /8, /4, 3 /4 and 7π/8. Field reconstruction of uniformly polarized optical vortex states. We briefly review the method of field reconstruction of uniformly optical vortex states 47 . We assume that a y-polarized object beam propagating (2019) 9:9979 | https://doi.org/10.1038/s41598-019-46390-7 www.nature.com/scientificreports www.nature.com/scientificreports/ on the z axis interferes with a y-polarized reference beam propagating on the k 0 (ω) = k(ω)(sinθ in x + cosθ in z) direction, where k 0 is a wave vector of the reference beam and x and z represent the unit vectors of x and z axes in the Cartesian coordinate. When the bandwidths of the object beam and the reference beam are narrow enough, the electric fields of the object beam E obj and the reference beam E ref at the measurement plane z = z 0 can be approximately written by where  E obj and  E ref are, respectively, the complex amplitude of the object and the reference beams, ω 0 is the center wavelength of the object and the reference beams, and α is a constant phase between the object and the reference beams. The two dimentional Fourier transform (2D-FT) of the interference pattern |E obj + E ref | 2 consists of four terms: where ⋅ ( )  denotes the 2D-FT in the (x, y) plane. The first and the second terms are, respectively, the 2D-FTs of the intensity distributions of the object and the reference beams, whose intensity peaks appears on the origin of the wavenumber space (k x , k y ) = (0, 0). The third and the fourth ones are described as The peaks of the intensity of |I AC± | 2 in the wavenumber space are at (k x , k y ) = (0, ±ksinθ in ), respectively. Filtering around the (k x , k y ) = (0, +ksinθ in ), we can extract the third term having information of the complex amplitude component of the object beam. The size of the reference beam we used in experiments was estimated to be 3 mm, which is large (over ~5 times larger than the size of the object beam) enough to consider that  E ref was constant. Hence, a 2D-FT of the complex amplitude of the object beam x y obj where β = (k − k 0z )z 0 − α is a constant phase. Thereby, the complex amplitude of the object beam  E x y z ( , , ) obj FR 0 is reconstructed by using the two dimentional inverse Fourier transform  where A is an area of interest, and l is the azimuthal index of the ESPs. In the present paper, we regarded the area as a circle, whose center was (x, y) = (c x , c y ). Modified coordinates   x y ( , ) are (x − c x , y − c y ). The azimuthal angle φ is described by   y x arctan( / ). Here, the component of temporally-perfect-polarized state in S l 0, E are described by which gives a measure of symmetry in the lth cylindrically polarized state. The normalized ESPs (or temporallyand spatially-perfect-polarized Stokes vector) are described by   evaluation of relative phase δ. We here propose a method to evaluate the relative phase δ. First, we observe four interference patterns when θ π π π = /8, /4, 3 /4, and 7π/8 by using the polarization analyzer system shown in Fig. 1(c), following which we reconstruct their complex amplitude distributions θ π  By using Eq. (4), we search δ which minimizes G δ . The center position (r = 0) was selected to be the position maximizing the DOP-SD 38,40,41,43