Polarization is one of the fundamental characteristics of light [1]. A fundamental Poincaré sphere (PS) can be used to describe fundamental polarization states [2, 3], which include linear, circular and elliptical polarizations. However, fundamental polarization states always have a homogeneous spatial distribution, while a vector beam may have a spatially inhomogeneous state of polarization. So the conventional fundamental Poincaré sphere is not enough to describe the polarization characteristic of a vector beam. Therefore, the concepts of a high-order Poincaré sphere (HOPS) [4-6] and a hybrid-order Poincaré sphere [7] are put forward to describe the evolution of phase and polarization state of cylindrical vector beams [8, 9].

Full Poincaré beam (FPB) is a class of beams with inhomogeneous polarization distribution which covers all possible polarization states over a fundamental PS in its transverse plane. It can be generated by coaxial super-position of two Laguerre-Gauss (LG) beams with different azimuthal vortex indices. Researchers have already developed some different experimental methods to generate them [10-12]. In this work, we propose a straightforward system which can adjust arbitrary parameters without changing the experimental devices and is easy to operate. Moreover, the range of parameters can also be increased. By controlling the intensity and phase patterns of the incident light and tuning the phase difference between them, we can get a FPB over a HOPS of any order.

For a fundamental PS, the north and south poles correspond to the right- and left-handed circular polarization states. They are the two eigenstates of a PS. The points located on the equator represent linear polarization states which means an equal-weight superposition of the two eigenstates. Other points on the surface represent elliptical polarization states. Notice that points inside a PS correspond to partial polarization states where the degree of polarization (DOP) P < 1 [13].

On this basis, a HOPS or a hybrid PS can be established by superposing two LG beams with different vortex topological values as two orthogonal circular polarization eigenstates. The schematic of a HOPS or a hybrid PS is shown in Fig. 1.

A family of FPB is defined as:

where the two eigenstates are:

Here ê_x and ê_y are unit vectors in x and y directions, respectively. The fundamental LG mode can be expressed as:

And LG_0,n can be expressed as:

where A₀ is a constant, k is the free-space wave number, ω₀ is the beam waist size. R(z) is the beam wavefront curvature radius. ξ(z) = tan^-1(z / z_R) is the Gouy phase and is the Rayleigh length, (r, φ) represent the polar coordinates.

Now the Stokes parameters can be represented as:

where .

is the radius of the HOPS and are the sphere’s coordinates.

On this basis, we can establish a HOPS or a hybrid order PS by superposing two LG beams with vortex topological charge l and m playing the role of the two orthogonal circular polarization eigenstates. When l = m we get a HOPS while when l ≠ m we get a hybrid order PS. With this in mind we assume HOPS for both cases for simplicity. Notice that when l = m = 0 the HOPS reduces to a fundamental PS.

With Stokes parameters, the complex Stokes field is defined as:

The phase pattern can be written as:

Then the winding number of the singularities can be written as:

Here, Δφ₁₂ = [φ₁₂]_final − [φ₁₂]_initial is the winding angle of φ₁₂.

In an inhomogeneous field, several kinds of singularities may appear [14]. Here we focus on two of them:

(1) Vector-point singularities (V-points). They are characterized by the Poincaré-Hopf index η.

(2) Elliptic point singularities which include points of linear polarization (L-points) and points of circular polarization (C-points). They are characterized by the singularity index I_C.

The relationship between I_C , η and σ₁₂ is [15]:

A Mach-Zehnder interferometer method is set to generate full Poincaré beams. The experiment scheme is shown in Fig. 2.

In our experiment, a solid-state laser with center wave-length at 532 nm is used as a source. It emits a linearly polarized coherent fundamental Gaussian mode. A half-wave plate (HWP) is put in front of source to adjust the intensity of the incident light by changing its polarization direction. A combination of one polarizer (P1) and one quarter-wave plate (QWP1) are used to convert linear polarization state to any homogeneous spatial distribution.

Then the light propagates into a polarization beam splitter (PBS) and splits into two beams. One is horizontal polarized and the other is vertical polarized. The horizontal polarized one propagates through PBS while the vertical one gets reflected. Two phase-only spatial light modulators (SLM1 and SLM2, produced by Thorlabs, EXULUS-HD1) are used to modulate them separately. Thus they can acquire two helical vortex phases with any desired topological orders.

These two modulated Gaussian beams are good approximations of vortex-bearing LG beams LG_0,l and LG_0,m . So they can be regarded as eigenstates of a HOPS. Notice that the phase-only SLMs used in the experiment take the clockwise direction as positive, so we will follow this through this work.

A Soleil-Babinet compensator (SBC, produced by Thorlabs, SBC-VIS) is put in the horizontal polarized light path. This device can be used as a phase retarder. In our experiment it is very hard to ensure that the two beams both pass through the same propagation distance z. However, due to Gouy phase [16], the phase of LG beams will change as z changes. That means different z will bring unknown phase difference between the two beams. The SBC is used for bringing in an additional phase difference to offset it.

Then the two modulated beams will get combined again. The exit light is an FPB after it propagates through a quarter-wave plate (QWP2) whose fast-axis is at an angle of 45 degrees to the x-axis. Here we define the horizontal direction as the x-axis.

The Jones vector of the beam after QWP2 can be written as [14]:

where γ is the angle between the transmission axis of P1 and the x-axis, ψ is the angle between the fast axis of QWP1 and the x-axis and δ is the final phase difference between the two eigenstates.

The parameters of a HOPS can be represented as:

It is clear that SBC will only change Φ but will not affect θ . Here Φ is the latitude angle and θ is the longitude angle of the HOPS. So by tuning SBC the light will travel on a certain latitude on the Poincaré sphere. Experimental details will be introduced in the next chapter.

To examine the polarization state of FPB, a combination of a quarter-wave plate (QWP3) and a polarizer (P2) are set to calculate the Stokes parameters of the field. Normalized Stokes parameters are given by [17]:

where represents the light intensity recorded by CCD. i and j represents the optical axis directions of QWP3 and P2 respect to the x-axis.

It is clear that S₃ = ±1 represents a circular-polarized beam, while S₃ = 0 represents a linearly-polarized beam. For a completely polarized beam, there should be .

When l = −m, the FPB located on the equator is linearly polarized. In this condition we can acquire two special beams: a radially polarized one and an azimuthally polarized one.

To acquire a radially polarized beam, we set the parameters as l = −1, m = 1, γ = 0, ψ =π / 4 and δ = π . Thus θ = π / 2. The beam is located on the equator. The final field can be written as:

Theoretical and experimental results are shown in Fig. 3. The green lines in S₀ represent polarization directions of the elliptic field. The Poincaré-Hopf index of the central point can be measured by calculating Δφ₁₂. Here it is easy to see η_r = 1.

Let δ = 0 and the other parameters remain unchanged, and we can get an azimuthally polarized beam. The final light field can be written as:

Theoretical and experimental results are shown in Fig. 4. The Poincaré-Hopf index of the central point here η_a = 1.

The FPB will travel on a certain latitude on HOPS by changing δ . In other words, phase difference between the two eigenstates will affect Φ directly. Some theoretical examples are shown in Fig. 5. This kind of ‘travel’ can be achieved simply by tuning the SBC since it can bring controllable and continuous phase differences.

When l + m ≠ 0, we will find inhomogeneous polarization distributions on HOPS. According to XiaoHui Ling et al. [10], when l = ±1, m = 0, a lemon-type or a star-type C-point exists in the light field on the equator of HOPS. Furthermore, the value of m is an arbitrary integer instead of a stationary zero in our setup so some kinds of new points can be found in this work.

To acquire a star-type polarization distribution, the parameters are set as l = −1, m = −2, γ = 0 , ψ=π / 4 and δ = 0. It represents an FPB on the equator. Theoretical and experimental results are shown in Fig. 6.

The polarization distribution is star-type with . Note that the central point is no longer a conventional C-point but a disclination because the intensity value is 0 there.

Note that there is an error in the center of S₃ which cannot be neglected. This is because high-order vortices are not stable during propagation. A high-order optical vortex will divide into two or more independent first-order vortices according to its topological charge [17, 18]. Figure 7 shows an example. The distance between the vortices will grow as z increases.

Similarly, to acquire a lemon-type polarization distribution, the parameters are set as l = 1, m = 2, γ = 0, ψ =π / 4 and δ = 0. Theoretical and experimental results are shown in Fig. 8.

From Fig. 8 it can be seen that the winding number . The central point is also a disclination.

We come to this result that for a point which is located on the equator, the polarization pattern has a topological charge of:

When l = −m and l,m ≠ 0, the beam located on the equator is linearly polarized and the central point is a dark V-point. When l ≠ −m, if l = 0 or m = 0, the central point is bright which means it is a C-point. If l,m ≠ 0, the central point is dark which means it is a disclination.

In our setup, it is quite easy and convenient to adjust all the parameters. In this way we can get arbitrary points on a HOPS of arbitrary order Poincaré sphere without replacing any devices. Thus, the capacity and stability of the system are greatly improved.

Figure 9 shows an example of generating a high-order singularity point. In this case, we set l = −1, m = 2, γ = 0, ψ =π / 4 and δ = 0. The topological charge of the central point is [14].

In this work, a convenient experimental method to generate full Poincaré beams over an arbitrary order Poincaré sphere by using two phase-only SLM and a Soleil-Babinet compensator is proposed. With the help of PBS a beam will split into two orthogonal polarized components and get modulated separately. The Soleil-Babinet compensator is used to compensate Gouy phase and provide controllable phase difference. Then the two beams get combined and pass through a quarter-wave plate to form the desired FPB. A combination of one polarizer and one quarter-wave plate together with CCD are set to measure the Stokes parameters and determine polarization distribution of the light field. The experimental results are in good coincidence with theory. By simply controlling the optical axis directions of the P1 and QWP1 and controlling the two SLMs, any desired beam over HOPS of any order can be obtained. This work may be helpful to study the polarization characteristics of light such as polarization singularities [19], anisotropy polarization [20], transverse spin angular momentum [21], scattering characteristics of Poincaré beams [22-24], etc.