In this paper, we propose an expanded Jones complex space model, by discussing the four degenerated modes with arbitrary amplitudes and phases.

Firstly the configuration of spatial modes is introduced. Figure 2A depicts the transformation relation among the CV modes, LP modes and OAM modes and the complete transformation relation can be found in our previous articles [50], [51]. Notice that such a four-dimensional space is not visible. Figure 2A is just the imaginary visible sketch of space beyond three dimensions for demonstration purposes. Any *l*th higher-order spatial state in optical fiber can be decomposed as the superposition of four degenerated modes. If expressed in CV modes, the arbitrary spatial state is $\text{E}={\text{AEH}}_{l-1,m}^{\text{even}}+{\text{BEH}}_{l-1,m}^{\text{odd}}+{\text{CHE}}_{l+1,m}^{\text{even}}+{\text{DHE}}_{l+1,m}^{\text{odd}},$ where the amplitudes and phases of A, B, C, D denote the independent amplitudes and phases of the four degenerated modes. If the values of A, B, C, D are defined, the field is also determined solely. These four independent complex variables can be spanned as a four-dimensional complex space. Each point in the space corresponds to a spatial state. The four degenerated modes work as a group of four bases in the complex space. In LP modes, the mode bases are $\widehat{x}{\text{LP}}_{l,m}^{\text{even}},$ $\widehat{y}{\text{LP}}_{l,m}^{\text{even}},$ $\widehat{x}{\text{LP}}_{l,m}^{\text{odd}}$ and $\widehat{y}{\text{LP}}_{l,m}^{\text{odd}}$ [51], where $\widehat{x}$ represents the *x* polarization or (1, 0)^{T} in Jones calculus, and $\widehat{y}$ represents the *y* polarization or (0, 1)^{T} in Jones calculus. While in OAM modes, the mode bases are $\widehat{x}{\text{OAM}}_{-1},$ $\widehat{y}{\text{OAM}}_{-1},$ $\widehat{x}{\text{OAM}}_{+1}$ and $\widehat{y}{\text{OAM}}_{+1}.$ Similar to CV mode bases, complex vectors (*x*_{e}, *y*_{e}, *x*_{o}, *y*_{o})^{T} and (*x*_{−}, *y*_{−}, *x*_{+}, *y*_{+})^{T} denote the coordinates in respective mode bases [51], where the amplitudes and phases of *x*_{e}, *y*_{e}, *x*_{o} and *y*_{o} represent the amplitudes and phases of $\widehat{x}{\text{LP}}_{l,m}^{\text{even}},$ $\widehat{y}{\text{LP}}_{l,m}^{\text{even}},$ $\widehat{x}{\text{LP}}_{l,m}^{\text{odd}}$ and $\widehat{y}{\text{LP}}_{l,m}^{\text{odd}}.$ So as for *x*_{−}, *y*_{−}, *x*_{+} and *y*_{+}. The spatial state expressed in different mode bases, such as ${\text{HE}}_{l+1,m}^{\text{even}}+{\text{iHE}}_{l+1,m}^{\text{odd}}={\widehat{\text{\sigma}}}^{+}{\text{OAM}}_{+l},$ is equivalent as base transformation in the four-dimensional complex space. And there are base transformation matrices connecting these three mode bases [51].

Figure 2: Sketch of the expanded Jones model.

(A) The transformation among three kinds of spatial states; (B) the mathematical nature and procedure of the expanded Jones model.

In this paper, we show the derivation in LP mode bases. Arbitrary the *l*th order spatial state expressed in LP mode bases is

$$\begin{array}{c}{E}_{in}={x}_{e}\widehat{x}{\mathrm{LP}}_{l,m}^{even}+{y}_{e}\widehat{y}{\mathrm{LP}}_{l,m}^{even}+{x}_{o}\widehat{x}{\mathrm{LP}}_{l,m}^{\mathrm{odd}}+{y}_{o}\widehat{y}{\mathrm{LP}}_{l,m}^{\mathrm{odd}}\\ ={F}_{l,m}\mathrm{(}r\mathrm{)}\left(\mathrm{cos}\text{\hspace{0.17em}}l\xi \left(\begin{array}{l}{x}_{e}\\ {y}_{e}\end{array}\right)+\mathrm{sin}\text{\hspace{0.17em}}l\xi \left(\begin{array}{l}{x}_{o}\\ {y}_{o}\end{array}\right)\right)\\ ={F}_{l,m}\mathrm{(}r\mathrm{)}\left(\begin{array}{l}\left|\text{\hspace{0.05em}}{E}_{1}\text{\hspace{0.05em}}\right|{e}^{i{\alpha}_{1}}\mathrm{cos}\text{\hspace{0.17em}}l\xi +\left|\text{\hspace{0.05em}}{E}_{3}\text{\hspace{0.05em}}\right|{e}^{i{\alpha}_{3}}\mathrm{sin}\text{\hspace{0.17em}}l\xi \\ \left|\text{\hspace{0.05em}}{E}_{2}\text{\hspace{0.05em}}\right|{e}^{i{\alpha}_{2}}\mathrm{cos}\text{\hspace{0.17em}}l\xi +\left|\text{\hspace{0.05em}}{E}_{4}\text{\hspace{0.05em}}\right|{e}^{i{\alpha}_{4}}\mathrm{sin}\text{\hspace{0.17em}}l\xi \end{array}\right)\text{\hspace{0.17em}},\end{array}$$(3)

where ${\text{LP}}_{l,m}^{\text{even}}$ is *F*_{l,m} (*r*) cos *lξ* and ${\text{LP}}_{l,m}^{\text{odd}}$ is *F*_{l,m} (*r*) sin *lξ*, describing the radial and azimuthal amplitude distributions. *F*_{l,m} (*r*) is the cylindrical function to describe the radial distribution of fiber, where *l* is the azimuthal order and *m* is the radial order. Because *F*_{l,m} (0)=0 for higher-order modes (*l*>0), an intensity singularity exists at the center of the intensity patterns. If without the modulation of the azimuthal function, the final pattern only determined by *F*_{l,m} (*r*) is a doughnut shape. The vector (*x*_{e}, *y*_{e}, *x*_{o}, *y*_{o})^{T} is substituted by the equivalent exponential form ${\mathrm{(}\left|\text{\hspace{0.05em}}{E}_{1}\text{\hspace{0.05em}}\right|{e}^{\text{i}{\alpha}_{1}},\text{\hspace{0.17em}|}{E}_{2}\text{|}{e}^{\text{i}{\alpha}_{2}},\text{\hspace{0.17em}|}{E}_{3}\text{|}{e}^{\text{i}{\alpha}_{3}},\text{\hspace{0.17em}|}{E}_{4}\text{|}{e}^{\text{i}{\alpha}_{4}}\mathrm{)}}^{\text{T}}$ in the third row of Eq. (3). |*E*_{1−4}| denote the amplitudes and α_{1−4} denote the phases of *x*_{e}, *y*_{e}, *x*_{o}, *y*_{o}, respectively. To get the information at different polarizations, we assume inserting an imaginary polarizer with the counter-clockwise angle *ψ* related to the defined *x*-axis. After passing the polarizer, the field is

$$\begin{array}{c}{E}_{\text{out}}=\left(\begin{array}{cc}\mathrm{cos}\text{\hspace{0.17em}}\psi & -\mathrm{sin}\text{\hspace{0.17em}}\psi \\ \mathrm{sin}\text{\hspace{0.17em}}\psi & \mathrm{cos}\text{\hspace{0.17em}}\psi \end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\text{\hspace{0.17em}}\psi & \mathrm{sin}\text{\hspace{0.17em}}\psi \\ -\mathrm{sin}\text{\hspace{0.17em}}\psi & \mathrm{cos}\text{\hspace{0.17em}}\psi \end{array}\right){E}_{\text{in}}\\ ={F}_{l,m}\mathrm{(}r\mathrm{)}[{M}_{1}\mathrm{cos}\mathrm{(}l\xi -{\theta}_{1}\mathrm{)}+i{M}_{2}\mathrm{cos}\mathrm{(}l\xi -{\theta}_{2}\mathrm{)}]\left(\begin{array}{c}\mathrm{cos}\text{\hspace{0.17em}}\psi \\ \mathrm{sin}\text{\hspace{0.17em}}\psi \end{array}\right)\\ ={F}_{l,m}\mathrm{(}r\mathrm{)}I{\left(\xi \right)}^{1/2}{e}^{i\gamma \left(\xi \right)}\left(\begin{array}{c}\mathrm{cos}\text{\hspace{0.17em}}\psi \\ \mathrm{sin}\text{\hspace{0.17em}}\psi \end{array}\right),\end{array}$$(4)

where *M*_{1}, *M*_{2}, *θ*_{1} and *θ*_{2} are defined as the CPs. They are changed with the polarizer angle *ψ*, where

$$\begin{array}{l}{M}_{1}=\mathrm{(}\mathrm{(}\left|\text{\hspace{0.05em}}{E}_{1}\text{\hspace{0.05em}}\right|\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\alpha}_{1}+\left|\text{\hspace{0.05em}}{E}_{2}\text{\hspace{0.05em}}\right|\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\alpha}_{2}{\mathrm{)}}^{2}\\ \text{}+\mathrm{(}\left|\text{\hspace{0.05em}}{E}_{3}\text{\hspace{0.05em}}\right|\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\alpha}_{3}+\left|\text{\hspace{0.05em}}{E}_{4}\text{\hspace{0.05em}}\right|\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\alpha}_{4}{\mathrm{)}}^{2}{\mathrm{)}}^{1/2}\\ {M}_{2}=\mathrm{(}\mathrm{(}\left|\text{\hspace{0.05em}}{E}_{1}\text{\hspace{0.05em}}\right|\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\alpha}_{1}+\left|\text{\hspace{0.05em}}{E}_{2}\text{\hspace{0.05em}}\right|\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\alpha}_{2}{\text{\hspace{0.17em}}}^{\mathrm{)}}\\ \text{}+\mathrm{(}\left|\text{\hspace{0.05em}}{E}_{3}\text{\hspace{0.05em}}\right|\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\alpha}_{3}+\left|\text{\hspace{0.05em}}{E}_{4}\text{\hspace{0.05em}}\right|\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\alpha}_{4}{\mathrm{)}}^{2}{\mathrm{)}}^{1/2}\\ {\theta}_{1}=\mathrm{arctan}\mathrm{(}\frac{\left|\text{\hspace{0.05em}}{E}_{3}\text{\hspace{0.05em}}\right|\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\alpha}_{3}+\left|\text{\hspace{0.05em}}{E}_{4}\text{\hspace{0.05em}}\right|\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\alpha}_{4}}{\left|\text{\hspace{0.05em}}{E}_{1}\text{\hspace{0.05em}}\right|\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\alpha}_{1}+\left|\text{\hspace{0.05em}}{E}_{2}\text{\hspace{0.05em}}\right|\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\alpha}_{2}}\mathrm{)}\\ {\theta}_{2}=\mathrm{arctan}\mathrm{(}\frac{\left|\text{\hspace{0.05em}}{E}_{3}\text{\hspace{0.05em}}\right|\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\alpha}_{3}+\left|\text{\hspace{0.05em}}{E}_{4}\text{\hspace{0.05em}}\right|\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\alpha}_{4}}{\left|\text{\hspace{0.05em}}{E}_{1}\text{\hspace{0.05em}}\right|\mathrm{cos}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\alpha}_{1}+\left|\text{\hspace{0.05em}}{E}_{2}\text{\hspace{0.05em}}\right|\mathrm{sin}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\alpha}_{2}}\mathrm{)}.\end{array}$$(5)

Eq. (4) tells us that the field is determined by both the radial function *F*_{l,m} (*r*) and the azimuthal function *I*(*ξ*)^{1/2} e^{iγ}^{(}^{ξ}^{)}, which vary from the polarized angle *ψ*. The last 2×1 vector indicates the field with linearly polarized angle *ψ*.

We analyze the azimuthal function *I*(*ξ*)^{1/2} e^{iγ}^{(}^{ξ}^{)} at the beginning. When the imaginary polarizer rotates, different values of *M*_{1}, *M*_{2}, *θ*_{1} and *θ*_{2} are obtained. Using Eq. (4), the azimuthal intensity and phase after passing through the polarizer is

$$I\mathrm{(}\xi \mathrm{)}={M}_{1}^{2}{\mathrm{cos}}^{2}\mathrm{(}l\xi -{\theta}_{1}\mathrm{)}+{M}_{2}^{2}{\mathrm{cos}}^{2}\mathrm{(}l\xi -{\theta}_{2}\mathrm{)},$$(6)

$$\gamma \mathrm{(}\xi \mathrm{)}=\mathrm{arctan}\left(\frac{{M}_{2}\mathrm{cos}\mathrm{(}l\xi -{\theta}_{2}\mathrm{)}}{{M}_{1}\mathrm{cos}\mathrm{(}l\xi -{\theta}_{1}\mathrm{)}}\right).$$(7)

The intensity and phase of a spatial state connects with both the azimuth *ξ* and the polarized angle *ψ*, which should be a two-variables function. The intensity expression Eq. (6) is able to be converted as

$$I\mathrm{(}\xi \mathrm{)}={I}_{D}+{I}_{A}{\mathrm{cos}}^{2}\mathrm{(}l\xi -\delta \mathrm{)},$$(8)

where

$$\{\begin{array}{l}{I}_{\text{A}}=[\mathrm{(}{M}_{1}^{2}\mathrm{cos}\text{\hspace{0.17em}}2{\theta}_{1}+{M}_{2}^{2}\mathrm{cos}\text{\hspace{0.17em}}2{\theta}_{2}{\mathrm{)}}^{2}\\ \text{}+\text{\hspace{0.17em}}\mathrm{(}{M}_{1}^{2}\mathrm{sin}\text{\hspace{0.17em}}2{\theta}_{1}+{M}_{2}^{2}\mathrm{sin}\text{\hspace{0.17em}}2{\theta}_{2}{\mathrm{)}}^{2}{]}^{\frac{1}{2}}\\ {I}_{\text{D}}=\frac{{M}_{1}^{2}+{M}_{2}^{2}-{I}_{\text{A}}}{2}\\ \delta =\frac{1}{2}\mathrm{arctan}\mathrm{(}\frac{{M}_{1}^{2}\mathrm{sin}\text{\hspace{0.17em}}2{\theta}_{1}+{M}_{2}^{2}\mathrm{sin}\text{\hspace{0.17em}}2{\theta}_{2}}{{M}_{1}^{2}\mathrm{cos}\text{\hspace{0.17em}}2{\theta}_{1}+{M}_{2}^{2}\mathrm{cos}\text{\hspace{0.17em}}2{\theta}_{2}}\mathrm{)}.\end{array}$$(9)

Here, *I*_{D} and *I*_{A} indicate the direct current (DC) term and the alternative current (AC) term, respectively, which is the function of polarized angle *ψ*.

First, considering the term cos^{2} (*lξ*–*δ*) in Eq. (8), which changes with the spatial angle in the sinusoidal trend, it reaches the maximum when *lξ*–*δ*=k*π*(k∈Z) and the number of the maximum is 2*l* in the range *ξ*∈[−*π*, *π*). If the radial field distribution *F*_{l,m}(*r*) is also considered, that is the pattern of a 2*l* lobes pattern (or the pattern like an LP mode) with amplitude axes at $\xi =\frac{\delta +\text{k}\pi}{l}.$ If the intensity is invariant with *ξ* [left term in Eq. (8)], the pattern should be a doughnut shape.

Eq. (8) tells us that, for *l*th order spatial states, the intensity consists of a DC term *I*_{D}, and an *l*th order AC term *I*_{A} cos^{2}(*lξ*–*δ*). The initial phase *δ* describes the angle of the maximum $\mathrm{(}\xi =\frac{\delta +\text{k}\pi}{l}\mathrm{)},$ or named AC angle. In physics, an arbitrary higher-order intensity pattern should be the superposition of a ring pattern with intensity *I*_{D} and a 2*l* lobes pattern with maximum intensity *I*_{A} and amplitude axes $\xi =\frac{\delta +\text{k}\pi}{l},$ as shown in the last three graphs in Figure 3C. Notice that, the radial intensity in Figure 3C has been normalized such that the maximum of *F*_{l}_{,}_{m}(*r*) is 1. Thus, the variables *I*_{D} and *I*_{A} in Figure 3C indicate the maximum intensity directly.

Figure 3: Corresponding relation between the expanded Jones model and mode patterns of a random *l*th order spatial state, where **|E|=**(2.4, 2.7, 3.1, 1.5) and *α*=(2.1, 0.2, 0.9, 3.1) or $1.72{\text{e}}^{2.47\text{i}}{\text{EH}}_{l-1,m}^{\text{even}}+1.01{\text{e}}^{-1.21\text{i}}{\text{EH}}_{l-1,m}^{\text{odd}}+1.01{\text{e}}^{1.43\text{i}}{\text{HE}}_{l+1,m}^{\text{even}}+2.73{\text{e}}^{0.58\text{i}}{\text{HE}}_{l+1,m}^{\text{odd}}$ expressed in CV mode bases.

(A) The trend of four CPs, *M*_{1}, *M*_{2}, *θ*_{1} and *θ*_{2}; (B) the trend of *I*_{D}, *I*_{A} and *δ* along with the polarized angle *ψ* and the patterns at several polarized angles when (D) *l*=1, *m*=1; (E) *l*=2, *m*=1. (C) Diagram to depict the decompositions of the hybrid state when *ψ*=0 and *l*=1. (F) Diagram to demonstrate the decomposition of patterns for arbitrary higher-order modes. Inten., intensity pattern; Infer., interference pattern.

It should be highlighted that Eq. (6) and Eq. (8) are just the different forms of the intensity distribution. Eq. (6) indicates that the final intensity pattern is the superposition of two 2*l* lobes patterns with different weights and amplitude axes, while Eq. (8) indicates that it consists of a ring pattern and a 2*l* lobes patterns. Figure 3C depicts the difference. If using Eq. (6) to describe the intensity pattern, even if we know the two composed 2*l* lobes patterns, it is still hard to intuitively imagine the final results. However, if using Eq. (8), the physical meaning is clear enough without further calculation. In the next section, we just use Eq. (8).

Also, Eq. (7) can be deformed as an equivalent form,

$$\gamma \mathrm{(}\xi \mathrm{)}=\mathrm{arctan}\left[\frac{{M}_{2}}{{M}_{1}}\mathrm{cos}\mathrm{(}{\theta}_{2}-{\theta}_{1}\mathrm{)}+\frac{{M}_{2}}{{M}_{1}}\mathrm{sin}\mathrm{(}{\theta}_{2}-{\theta}_{1}\mathrm{)}\mathrm{tan}\mathrm{(}l\xi -{\theta}_{1}\mathrm{)}\right],$$(10)

The right term contributes a helical phase component and the left term can be regarded as a constant. When sin(*θ*_{2}–*θ*_{1})>0, the right term contributes an *lξ* helical phase. Thus, the final interference pattern should be a counter-clockwise vortex shape. A clockwise vortex shape interference pattern happens when sin(*θ*_{2}–*θ*_{1})<0. Thus, it is able to predict the vortex direction of the interference pattern by the value of sin(*θ*_{2}–*θ*_{1}). When sin(*θ*_{2}–*θ*_{1})=±1, if *M*_{2}=*M*_{1} is also satisfied, *γ*=±*lξ*∓*θ*_{1} are just the phases of the pure OAM modes. Supplement 1 elaborates the complete derivation from Eq. (3) to Eq. (10).

Figure 2B concludes the analytical procedure and mathematical nature of the proposed expanded Jones complex space model. First, the spatial state should be expressed in a specific group of mode bases (LP mode bases in this paper). Then, the four mode bases are converted as another four parameters to make the physical meanings more intuitive, namely, *M*_{1}, *M*_{2}, *θ*_{1} and *θ*_{2}, which are the function of polarized angle *ψ*. As the coordinates change along with *ψ*, there should be a trace *f*(*ψ*) in the four-dimensional space for a specific spatial state. By drawing the trend of four CPs (or their equivalent variables *I*_{D}, *I*_{A}, *δ*, *γ*) along with *ψ*, we are able to tell the corresponding spatial state at any polarization. Thus, the information of the spatial state is described totally. In the following section, we list some examples to verify the model and introduce how the proposed model works.

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