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Abstract
The developments in polarized light have spawned a multitude of novel applications in optical fiber systems, but the design and fabrication of practical fiber wave plates with high degree of integration still remain a challenging issue. To address this problem, an all-fiber spun wave plate (SWP) for arbitrary state of polarization (SOP) conversion is proposed in this work, and its principle is analyzed with Mueller matrix. Simulations are conducted to exhibit the arbitrary SOP conversion capability of the proposed SWP, and two key parameters, including the maximum spinning rate (ξmax) and linear birefringence (δ), are investigated for efficient conversion of desired SOP. Different functions to increase the spinning rate ξ from 0 to ξmax, computational efficiency and accuracy related to N are discussed in detail. Furthermore, the depolarization effect caused by retardation of SWP is also considered. The results of this research suggest that the proposed SWP exhibits promising performance in arbitrary SOP conversion, and the meticulous analysis of the numerical computation, design, and implementation of SWP presented in this work can provide novel insights for devloping fiber wave plates.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The generation and manipulation of polarized light play essential roles in numerous applications of fiber optic and integrated photonic systems, including advanced optical interconnection and communication networks [1,2], optical vortex generation [3–5], high-performance fiber laser systems [6,7], and practical fiber sensor devices [8–12]. Different from bulk optics, where birefringent crystals are commonly employed to manipulate polarization, the generation and conversion of states of polarization (SOP) in optical fibers confront greater challenges. While it is possible to generate a specific polarization state with a bulk wave plate in free space and subsequently inject it into the optical fiber, this approach not only increases optical losses but also diminishes system stability and practicality. Consequently, the development of all-fiber wave plates has been a persistent pursuit among researchers all around the world.
Figure 1 provides an overview of the existing major fiber-based schemes for SOP conversion from linear polarization (LP) to elliptical polarization (EP). Traditional fiber optic polarization controllers (PC) manipulate polarized light by taking advantage of fiber birefringence, which is induced by bending and lateral pressure. However, these devices are not integration-friendly due to their large size and unstable characteristics. Similarly, fiber polarization devices based on helically wound structures are also not integration-friendly because they require extra ancillary equipment to change the fiber shape [13–15]. More advanced in-line fiber wave plate is made by fusion splicing the polarization maintaining fiber (PMF) of specific length to the previous fiber with their slow axes aligned at particular angle, which includes three fiber sections and is named three-stage wave plate. This wave plate is exactly the fiber optic counterpart of the wave plate in bulk optics with higher degree of integration, but uncontrollable manufacturing errors and long-term instability in the length and the alignment angle remain key barriers in many practical applications [16]. In addition, many research efforts are devoted to new-material-based SOP manipulation. For example, polarization tuning [17] and nonlinear polarization rotation [18] in optical fiber have been demonstrated using graphene. Another material, liquid crystal, has also been introduced into optical fibers to control polarization [19–21]. However, the practicality of these schemes is still not optimistic due to the optical loss and difficulty in fabrication. In the field of optical fiber sensors, scientists studying fiber optics current sensor (FOCS) proposed a method to generate circularly polarized light by rotating the PMF from slow to fast with the fiber heated to almost molten [22,23]. This is the spun quarter wave plate (SQWP). Under the perturbation of PMF rotation, the light wave SOP is re-coupled and forms a novel polarization mode [22], and its efficacy is independent of fiber fusion points and ancillary equipment. The implementation of this technique in FOCS also exemplifies the formidable potential of SQWP. Nevertheless, existing investigations have primarily concentrated on the generation of circularly polarized light, while further development of other applications for spun wave plates remains limited.
Fig. 1. Implementation of polarization conversion in optical fiber, including polarization controller, three-stage wave plate, and spun wave plate proposed in this work. The bottom right corner shows the change in spinning rate $\xi$ as the fiber is rotated.
In this work, inspired by SQWP, an all-fiber spun wave plate (SWP) that can produce an arbitrary polarization state is proposed. Free from solving the complicated differential equations based on coupled mode theory (CMT), all calculations and analysis in this work are conducted by Mueller matrix, which is more conducive to computer numerical calculation. In Section 2, the Mueller matrix model of SWP is discussed in detail. Then, in Section 3, simulations are presented to exhibit the arbitrary SOP conversion based on the SWP, and the relationship between the SOP of interest and the maximum spinning rate is also investigated. Section 4 compares the SOP conversion performance of different functions to increase spinning rate $\xi$, discusses the depolarization effect caused by the retardation of SWP exceeding the coherence time of a broadband light source, and also studies the computational efficiency and accuracy related to the number of tiny retarders that SWP is split into. Compared to traditional PC which requires researchers to continuously adjust and try to get ‘better’ output results, SWP is more suitable for scenarios where the polarization state is determined. This work exhibits a comprehensive study on the numerical calculation, design, and application of SWP, which is useful to the future development of fiber wave plates.
2. Mueller matrix of SWP
For better understanding of the polarization conversion inside SWP, the matrix-based interpretation of SWP is required. Here, the Mueller matrix is chosen because it facilitates the observation of SOP movement on the Poincare sphere. The utilization of Mueller matrix also facilitates investigations into the polarization characteristics of SWP, including the SOP evolution and the extent of depolarization. As introduced in our previous work [24–26], spun structure can be analyzed with differential element method and can be regarded as a cascade of many thin retarders with corresponding alignment angles. Given that SWP is fabricated by spinning the PMF, these thin retarders are naturally linear birefringence (LB) retarders, with the basic concept shown in Figure 2. Then, several LB retarders can be classified into a mini unit, with $\xi$ (rad/mm) between two adjacent LB retarders being a constant. Therefore, the Mueller matrix of the mini unit is
withIn Eq. (1), $R(\theta _i)$ and $M_{LBi}$ respectively denote the coordinate rotation matrix (in Mueller matrix form) and the Mueller matrix of LB retarder in the $i^{\rm th}$ section, where $\theta _i=\xi _i\frac {l}{QN}$ and $\phi _i=\delta \frac {l}{QN}$. $l$ is the length of SWP, $Q$ is the number of LB retarders in each mini unit, and $N$ is the number of mini units that SWP is divided into [24]. $\xi _i$ is the spinning rate for the $i^{\rm th}$ section as shown in Figure 2, and $\delta$ (rad/mm) is the LB in the original PMF. The sign of $\xi$ only refers to rotation direction, while the absolute value of $\xi$ represents the extent of rotation. After diagonalization of $R(\theta _i)M_{LBi}$ on the right-hand side of Eq. (1) by solving the eigenvector, it can be seen that $M_i$ shares the same expression as the elliptical birefringence (EB) retarders [25–27], which is
This indicates that SWP can be regarded as a cascade of many EB retarders, and each EB retarder shares the same $\delta$ but different $\gamma _i$, and here $\gamma _i = \sqrt {\delta ^2/4+\xi _i^2}$. In Eq. (1), the perspective of the observer is rotated together with PMF along $L$ axis [25]; therefore, after the light wave passes through the $i^{\rm th}$ retarder, the output Stokes vector calculated by Eq. (1) is based on local coordinates, while the Stokes vector in global coordinates should be written as
The rotation matrix $R(-\Theta _i)$ can be used to replace the first matrix on the right-hand side of Eq. (5). Additionally, when $i=N$, which means the last Stokes vector is being observed, the rotation angle becomes the number of turns that the entire SWP is rotated. This Stokes vector $S_{\rm out}$ is also the final output, which is
The value of $N$ is expected to be as large as possible, and the Mueller matrix of the entire SWP $M_{\rm SWP}$ can be obtained, which is
Fig. 2. On a microscale, SWP consists of many EB retarders, and each EB retarder includes many LB retarders.
Equations (1)–(8) thoroughly describe how to obtain Mueller matrix of SWP on a microscale, and more importantly, the scheme of cascading retarders is more conducive to the iterative operation in computer numerical calculation. It can be seen that the Mueller matrix of SWP is closely related to both $\xi$ (or say $\xi _{\rm max}$) and $\delta$, and different combinations of these two parameters are also the key to generating arbitrary SOP, which is demonstrated in the next section. In addition, the value of $N$ cannot be expanded indefinitely as this will hinder the computational efficiency. The computational efficiency and accuracy are also discussed in the following sections.
3. Simulation results
Both $\xi _{\rm max}$ and $\delta$ are crucial parameters that can produce varying Mueller matrices of SWP, allowing SWP to generate any desired SOP. By introducing a horizontally linearly polarized light into the SWP, we record the changes in the Stokes vector along the length of the SWP, as shown in Figure 3. Besides, Table 1 summarizes all the simulation parameters in each figure for researchers to facilitate their understanding and replication of this study, including $l$ (the length of the SWP), $\delta$ (the LB in the original PMF), $\xi _{\rm max}/\delta$ (the ratio of the maximum spinning rate to LB), function (how spinning rate changes), and $N$ (number of mini units).
Fig. 3. The evolution of the Stokes vector in SWP and the SOP moving trajectory on the Poincare sphere with (a) $\xi _{\rm max} = -0.08\pi$ rad/mm and (b) $\xi _{\rm max} = 0.32\pi$ rad/mm.
Table 1. The summary of the parameters we use in the simulation
Figure 3(a) and Figure 3(b) show that $s_1$ and $s_2$ oscillate persistently with $L$ and this oscillation becomes more violent as $\left |\xi \right |$ increases, while $s_3$ shows an overall upward or downward trend. Therefore, the SOP starts from a linear polarization, then revolves around the Poincare sphere as the SWP rotates, and finally reaches another elliptical polarization as it emits from the SWP. Specifically, SWP with positive $\xi$ (which means the optical fiber is twisted counterclockwise when viewed parallel to the direction of forward propagation of light) can generate left elliptical polarization while that with negative $\xi$ can bring right elliptically polarized light. Moreover, when $\delta$ remains constant, the output SOP approaches ‘more circular’ polarization (getting closer to the pole) as $\left |\xi _{\rm max}\right |$ increases. In addition, as depicted in Figure 4, SWP fabricated with varying $\left |\xi _{\rm max}/\delta \right |$ enables the SOP of the emitted light to be positioned at any desired location on the Poincare sphere. Similarly, the elliptical SOP maintains the same rotational direction as the direction in which the fiber is twisted. The reason for choosing these values of $\xi _{\rm max}/\delta$ is to graphically display the SOP evolution. In fact, the output SOP can traverse every point of the Poincare sphere, making it possible for arbitrary SOP conversion.
Fig. 4. Arbitrary SOP conversion using SWP, with (a) $\xi _{\rm max}/\delta = -0.7226$, (b) $\xi _{\rm max}/\delta = -0.2701$, (c) $\xi _{\rm max}/\delta = 0.1883$, (d) $\xi _{\rm max}/\delta = 25$. Red: input, Blue: output.
To gain more insight into the relationship between the ellipticity of output SOP and the maximum spinning rate, simulations about the ellipticity of emerging SOP are conducted, with parameters shown in Table 1. Simulation results in Figure 5 show that the ellipticity of output SOP is closely related to $\left |\xi _{\rm max}/\delta \right |$ which refers to the ratio of the maximum spinning rate to LB in PMF. Larger $\left |\xi _{\rm max}/\delta \right |$ can generate the ellipticity closer to 1. Additionally, it can be seen from Figure 5 that all of $s_1$, $s_2$ and $s_3$ can reach any value between $-1$ and $1$, indicating that arbitrary SOP conversion is achievable. In terms of practical SWP fabrication, different combinations of $\xi _{\rm max}$ and $\delta$ can be chosen according to experimental conditions and equipment performance so that SWP can produce any desired SOP. For instance, it is difficult to rotate the fiber many turns in a short distance (large $\left |\xi _{\rm max}\right |$), and therefore, in applications where circular SOP is required, PMF with weak LB can be utilized to obtain a very large $\left |\xi _{\rm max}/\delta \right |$.
4. Discussion
4.1 Cosine function versus linear function
Notably, the way $\xi$ changes from 0 to $\xi _{\rm max}$ generally includes cosine function and linear function [23,28], as shown in the lower right corner of Figure 1. The impact of these two functions on the output SOP is discussed using Mueller matrix and Stokes vector. Assuming that the length of SWP is $l$ and the maximum spinning rate is $\xi _{\rm max}$, the expressions for cosine function and linear function are shown in Eq. (9) and Eq. (10).
The simulation results in Figure 6 and Figure 7 demonstrate that the cosine and linear functions have varying effects for different values of $\left |\xi _{\rm max}/\delta \right |$. For example, when $\xi _{\rm max}/\delta =0.1$ (as shown in Figure 6), the Stokes vector obtained from both functions reaches nearly identical values after oscillation. This suggests that when the value of $\left |\xi _{\rm max}/\delta \right |$ is relatively small, there is little difference between cosine and linear functions. However, when $\left |\xi _{\rm max}/\delta \right |$ further increases, the trend that the ellipticity should approach 1 still holds true for the cosine function (as shown in Figure 7) while the linear function in this case will produce larger errors. This discrepancy arises because even though both functions allow $\xi$ to vary from 0 to $\xi _{\rm max}$, their changing rates differ. The cosine function is continuously derivable, resulting in a much smoother changing rate of $\xi$. In contrast, the changing rate of the linear function changes abruptly from 0 to a constant, making the SOP conversion uneventful. This effect is more pronounced when $\left |\xi _{\rm max}\right |$ is large, thus introducing errors.
Fig. 6. Comparison between two different types of SWP with $\xi$ belongs to cosine function and linear function. When $\xi _{\rm max}/\delta$ is small ($\xi _{\rm max}/\delta = 0.1$), the difference between these two functions is trivial.
Fig. 7. Comparison between two different types of SWP with $\xi$ belongs to cosine function and linear function. When $\xi _{\rm max}/\delta$ is large ($\xi _{\rm max}/\delta = 20$), significant error occurs in the output Stokes vector.
For practical manufacturing, it is much easier to control the motor by linearly increasing the rotation speed according to a linear function, which can reduce lots of errors in terms of machine faults. Therefore, in application scenarios with small ellipticity, a linear function can be used, while in scenarios that require a ‘more circular’ SOP (larger ellipticity), the cosine function is more recommended. In terms of the diattenuation effect, both cosine and linear functions can provide a relatively low loss scheme because SWP is fabricated under the high temperatures, where the spinning process is thus considered to be smooth enough and will not generate extra stress and attenuation [23,29].
4.2 Depolarization in SWP
In a real-world fiber system, the linewidth broadening of the light source increases the possibility for the retardation of SWP to surpass the coherence time of the light source. This phenomenon results in the depolarization effect [30–32], which can generate partially polarized light and can be described by the degree of polarization (DOP). The LB of the original PMF in this case will become $\delta (\lambda )=\delta \frac {\lambda _0}{\lambda }$. Suppose the power distribution of the light source is a Gaussian function of wavelength $\lambda$, which is
with $\sigma =\Delta \lambda /2\sqrt {2\ln {2}}$. Here, $\lambda _0$ and $\Delta \lambda$ are the central wavelength and the linewidth, respectively. Based on Eq. (5), we can obtain the emerging Stokes parameters $S'_{mi}$ ($m=0,1,2,3$) that describe both completely polarized light and non-polarized light, which can be written as [30]Consequently, the DOP after light wave passes through the $i^{\rm th}$ retarder is [30]
Through Eq. (13), the trend of DOP versus $L$ can be easily obtained. Figure 8 shows the simulation results when $l=200$ mm, $\delta =0.8\pi$ rad/mm, $\lambda _0=1550$ nm. The DOP evolution is investigated under different linewidth of the light source, different $\left |\xi _{\rm max}\right /\delta |$, and different spinning function.
Fig. 8. Investigation of DOP inside the proposed SWP under different linewidth of the light source ($\Delta \lambda$), different $\left |\xi _{\rm max}\right /\delta |$, and different spinning function (cosine or linear). The sub-figures show the details of the DOP.
As depicted in Figure 8, the DOP perturbations within SWP are intrinsically linked to the linewidth of the light source, namely $\Delta \lambda$. This connection is fundamental to the depolarization mechanism. A narrower linewidth closely correlates with a DOP that approaches 1.
In terms of cosine function, the DOP decreases at the beginning of SWP for all the cases. But the final output DOP increases as the the parameter $\left |\xi _{\rm max}\right /\delta |$ rises. It can be explained in the following way. When $\left |\xi _{\rm max}\right /\delta |$ is large, spinning is strong and it generates significant perturbation to SOP. Upon entering the SWP, linearly polarized light experiences strong spinning, leading to a certain degree of depolarization. As the light wave moves forward, the SOP conversion ability of SWP gradually transforms the linearly polarized light into the elliptically polarized light. This elliptically polarized SOP increasingly approximates the eigenmode of the optical fiber under spinning [33]. Hence, even undergoing strong perturbation (large $\left |\xi _{\rm max}\right /\delta |$), the depolarization reduces and DOP eventually approaches 1 at the SWP output port. On the contrary, when $\left |\xi _{\rm max}\right /\delta |$ is small, the spinning still causes depolarization to input linearly polarized light. However, a small $\left |\xi _{\rm max}\right /\delta |$ is insufficient to convert input light into a ‘more elliptical’ SOP. A mismatch occurs between the SOP and the eigenmode of the fiber, resulting in a DOP less than 1.
In contrast to the cosine function scenario, a linear function is less effective in SOP conversion as discussed in Figure 6 and Figure 7. Therefore, a large $\left |\xi _{\rm max}\right /\delta |$ does not adequately convert the light wave to match the corresponding eigenmode in the fiber under spinning; instead, it induces violent perturbation and continuously decreases DOP.
In any case, the extent of depolarization induced by the SWP is extremely low, and the value of DOP is close to 1. This is because, in our design, the linearly polarized light enters the wave plate along the axis of the PMF. The physical structure of SWP changes smoothly so that SOP conversion of the light wave is continuous.
4.3 Computational efficiency and accuracy related to $N$
When designing the SWP, it is important to properly select the value of $N$ to ensure that the calculation is performed efficiently and that the simulation remains unaffected by numerical artifacts. The following simulations explore the relationship between the convergence of Stokes vector and the value of $N$ in order to determine the appropriate $N$ to maintain computational efficiency and accuracy.
Figure 9 illustrates the convergence tests for Stokes vector calculations as a function of $N$ with parameters shown in Table 1. As evidenced by Figure 9(a) and Figure 9(b), the values of $s_1$, $s_2$ and $s_3$ exhibit significant fluctuations when $N$ is insufficiently large. As the value of $N$ increases, $s_1$, $s_2$ and $s_3$ converge towards a stable value. The absolute values of the first-order differences of the Stokes vector are calculated to determine the discrepancy between larger and smaller values of $N$ as shown in Figure 9(b), which further describes the decline of numerical errors as $N$ grows. Moreover, in the case of larger $\left |\xi _{\rm max}/\delta \right |$, a larger $N$ should be employed for results convergence as shown in Figure 9(c). This is because large value of $\left |\xi _{\rm max}/\delta \right |$ means SWP is rapidly rotated in a small range of $L$, and then the length of each mini unit should accordingly be shortened to ensure that the difference between $\xi _i$ and $\xi _{i+1}$ is small enough to meet the demand of differential element method. Thus, larger $N$ is required. The results depicted in Figure 9(c) indicate that for each $\xi _{\rm max}/\delta$, the $\left |{\rm 1}^{st} diff.\right |$ of $s_3$ exhibits a downward trend as $N$ increases, with the $\left |{\rm 1}^{st} diff.\right |$ falling below $10^{-8}$ when $N$ approaches $10^4$. Consequently, we choose $N = 2\times 10^4$ for all simulations presented in this work in order to ensure both efficiency and accuracy.
Fig. 9. (a) Convergence tests for Stokes vector calculations. (b) The absolute values of the first-order differences ($\left |{\rm 1}^{st} diff.\right |$) of Stokes vector are utilized to estimate the proper value of $N$. (c) $\left |{\rm 1}^{st} diff.\right |$ of $s_3$ versus $N$ with different $\xi _{\rm max}/\delta$.
5. Conclusion
Overall, an all-fiber SWP for arbitrary SOP conversion is proposed in this work. Based on Mueller matrix which is conducive to numerical calculations, the principles and performance of SWP are thoroughly studied. The ability of SWP to transform arbitrary polarization states is verified by simulation, and the results also show that the ellipticity of the output SOP increases with $\left |\xi _{\rm max}/\delta \right |$. The subjects of diattenuation and depolarization are also discussed, with the results indicating that neither diattenuation nor depolarization will impede the effective operation of SWP. Furthermore, different functions to increase spinning rate $\xi$ and computational efficiency and accuracy related to $N$ are discussed in detail. For high ellipticity SOP, it is more recommended to increase $\xi$ with cosine function, which can be a guideline for practical SWP fabrication. The findings of this research suggest that the proposed SWP exhibits auspicious performance in arbitrary SOP conversion. This work demonstrates a comprehensive and in-depth analysis of the numerical computation, design, and application of SWP, offering novel perspectives for the development of fiber wave plates. Furthermore, the introduction of SWP furnishes a new methodology for SOP manipulation within optical fibers, and its compact and stable scheme affords greater potential for exceptional performance in practical scenarios.
Funding
China Postdoctoral Science Foundation (2022M711243).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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