1.

INTRODUCTION

The conversion between SOP of polarized light has attracted great interest, and the manipulation of different polarization states in optical fibers has become a very popular research topic at present [1-3]. Meanwhile, it is essential to utilize only fiber optic devices for circular SOP conversion [4-6], which presents a multitude of advantages over traditional methods. The creation of circular SOP is finished by applying bending and stress to birefringent fibers through fiber polarization controllers, but this method is easily affected by environmental temperature and stress, resulting in instability and difficulties in integration. Another approach involves using a fusion-spliced QWP to transform linearly polarized light into circularly polarized light. This technique faces two major hurdles during manufacturing: the inability to accurately adjust the axis angle of the polarization-maintaining fiber (PMF) to 45 degrees, and the difficulty in precisely achieving a quarter beat length due to the PMF’s typically millimeter-scale beat length, leading to incorrect phase delay [7]. Additionally, significant losses at the fusion splice junction are not tolerable for certain applications.

Decades ago, H. Huang and his colleagues proposed that the variably spun birefringent fiber can replace the fiber QWP for SOP conversion [8]. The structure, which changes gradually due to spinning, has been demonstrated to be effective in generating circular SOP. This approach significantly minimizes the phase delay errors associated with non-ideal fiber QWP, thereby ensuring highly accurate state of polarization conversion without the necessity for fusion splicing, and it exhibits low losses and excellent stability, being regarded as a more practical solution for fiber circular SOP generation. The fabrication process of the spun QWP is as follows: Initially, the PMF with linear birefringence (δ) is heated until it is nearly at its melting point; after that, the entire PMF is rotated by a customized machine at a rate that increases from slow to fast (the spinning retarder section, from z₀ to z₁), such that one end of the fiber has no spinning rate while the other end is supposed to possess a spinning rate value of ξ_max, as shown in Figure 1. It can be observed from the fabrication process of the SQWP that the function representing the change in spinning rate from 0 to ξ_max, which is traditionally chosen to be cosine and linear functions, plays a pivotal role in the SOP conversion of the SQWP. To achieve a more circular SOP, a larger ratio of ξ_max to δ is usually required. However, this will also increase the unexpected ellipticity fluctuation due to the large slope and step in the traditional cosine and linear spinning functions, as pointed out by the published scientific paper [9]. This work also predicts that the performance of the SQWP in terms of polarization conversion should be closely related to the higher order continuity of the spinning rate functions. Nevertheless, the theoretical details and solutions for this phenomenon have not been thoroughly investigated yet.

Figure 1.

The spinning rate distribution along the fiber according to cosine function.

Herein, we study the impact of the continuity of the spinning rate function on the performance of circular SOP conversion inside SQWP. To address the issue of ellipticity fluctuation, we introduce a novel spinning rate function for SQWP. As the spinning rate ξ approaches its maximum value ξmax, the proposed function shows higher order continuity. We conduct a theoretical examination of the newly proposed spinning rate function for the generation of circular SOP. Our findings reveal that, given the same maximum spinning rate ξ_max, the ellipticity of the circular SOP produced using the proposed spinning rate functions exhibits less fluctuation compared to the traditional linear and cosine functions, leading to a more stable generation of circular SOP. The design of this novel high order continuous spinning rate functions and the study on generating circularly polarized light according to the new spinning rate function hold significant reference value for future development of SQWP.

2.

THEORY

The simulation section of this work follows the matrix model introduced in references [9-10], which is an approach that does not require solving differential equations and can improve computation efficiency. Referring to references [9-10], the number of small units divided by SQWP (N) is set to 50000 and the length of SQWP (l) is set to 100 mm in this simulation. The input light is set to vertically linear polarization, which is

The output can also be characterized by a Jones matrix. Considering that the Jones matrix include both amplitude and phase of the output light polarization, which causes difficulty for us to compare different SOP, we here use ellipticity in Equation (2) to evaluate the output features, with α and ϕ referring to the ellipse auxiliary angle and the phase difference between the output lights in x and y axis.

Then, two factors are proposed to judge the SOP conversion performance of SQWP: 1) Higher ellipticity. Similarity between the ellipticity of the circularly polarized state and that of the emerging light is directly proportional to the performance of the system. In other words, a smaller difference in ellipticity values indicates a more desirable outcome, as it suggests that the emerging light closely resembles the circularly polarized state.2) Smaller fluctuation. During SOP conversion, it is preferable to maintain minimal fluctuations in ellipticity. Such fluctuations can introduce uncertainty into the SOP of the output light, which may adversely affect the optimal functioning of the SQWP. We also use ΔΕ to characterize the ellipticity fluctuation, as shown in Equation (3).

In Equation (3), EP is the true ellipticity calculated from Equation (2), EP_i is the smooth ideal ellipticity curve obtained by seventh-order Savitzky-Golay filter [11]. The Equation (3) shows that smaller value of ΔΕ results in smaller ellipticity fluctuation. The real ellipticity curve and the smooth ideal ellipticity cure obtained by Savitzky-Golay filter involved in the Equation (3) are shown in Figure 2.

Figure 2.

The true ellipticity evolution and the smooth ideal ellipticity evolution obtained by Savitzky-Golay filter in SQWP when the spinning rate ξ satisfies cosine function when ξ_max is 5 rad/mm.

3.

RESULTS AND DISCUSSION

Figure 3(a) shows three different spinning rate scenarios, which are: case 1: the spinning rate has slight jumps at the start and end points; case 2: the spinning rate is a linear function; case 3: the spinning rate obeys a modified cosine function. Then, Figure 3(b) shows the ellipticity transformation performance of the above-mentioned situations. The spinning rate functions for cases 1 are discontinuous, spinning rate function for case 2 is first-order continuous, and the spinning rate function for case 3 is second-order continuous. As can be seen from Figure 3(b), the ellipticity fluctuation is relative to the continuity of the spinning rate functions when l and ξ_max are unchanged. Higher-order- continuity can yield smaller fluctuation.

Figure 3.

(a) The evolution of the spinning rate ξ from 0 at position z₀ to the maximum spinning rate ξ_max at position z₁ according to four different spinning rate functions with ξ_max changing from 5, 10, and 20 rad/mm. (b) The polarization evolution in SQWP when the spinning rate ξ satisfies four different spinning rate functions distribution.

From the cases 1, it is evident that the discontinuity of the spinning rate function will directly disable the circular SOP conversion functionality of SQWP. Only when the spinning rate function is continuous can the linear SOP be transformed into a SOP that approximates a circular polarization. However, from the first-order derivatives of cases 2 and 3, it is obvious that the discontinuity in the first-order derivative induces variations in ellipticity and consequently, the oscillations in the linear function are markedly more pronounced than those in the cosine function. Moreover, even when the spinning rate adheres to the cosine function, it is unable to fully mitigate the issue of ellipticity variation. As observed in the third case, with an increase in the maximum value of ξ_max, the ellipticity approaches 1, but it also exhibits increased fluctuations.

This is because the second-order derivative of the cosine function still has discontinuities. The larger ξ_max causes more significant step in the first or second derivative function, then leading to severer fluctuation. As a result, we believe that the discontinuity of the derivative is the main cause of the ellipticity fluctuation.

The existing spun fiber structure include cosine, linear, constant [12], and tangent [13]. It is obvious that the continuity of these functions does not satisfy the requirement. Therefore, we propose a novel high-order continuous spinning rate function, predicated on the property that sine and cosine functions can be interconverted through differential operations. The conventional cosine function exhibits continuity in its first derivative, however, its second derivative presents a discontinuity, rendering the third derivative non-existent. Consequently, by considering trigonometric functions as the second derivative function, we can design a continuous function. The original spinning rate function can then be derived through the application of calculus principles. This approach provides a robust and effective solution for addressing the discontinuity issue inherent in traditional functions. In this design, we use Equation (4) as the second derivative function, as shown below:

and then its primitive function is

In Equation (4) and (5), z denotes any position of the SQWP.

With the proposed spinning function, Equation (5) can gradually change from 0 to ξ_max and have a higher order continuity than the traditional cosine function. The curves of the first, second, and third derivative of Equation (5) are shown in Figure 4.

Figure 4.

(a) The evolution of the spinning rate ξ from 0 to ξ_max according to the new function with ξ_max changing from 5, 10, and 20 rad/mm. (b) The first derivative of the new function. (c) The second derivative of the new function. (d) The third derivative of the new function. (e) The polarization evolution in SQWP when the spinning rate ξ satisfies the newly proposed function.

Additionally, we use ΔΕ in Equation (3) to quantitatively compare the ellipticity obtained from different spinning rate functions. As shown in Figure 5, the ΔΕ of the new function is two and three orders of magnitude smaller than linear and cosine function. ΔΕ obtained from the proposed spinning rate function demonstrates almost 2 order smaller than that from cosine function, and 3 order smaller than linear function. It is witnessed that the higher-order-continuity indeed enhanced the SOP conversion performance, and the new spinning rate function can significantly improve robustness of circular SOP conversion in SQWP.

Figure 5.

The ellipticity fluctuation in SQWP when the spinning rate ξ satisfies (a) linear, (b) cosine and (c) the new functions.

4.

CONCLUSION

In this paper, we study the influence of the continuity of the spinning rate function on the performance of circular SOP conversion in SQWP. Our analysis encompasses four unique spinning rate scenarios, employing both theoretical exploration and numerical computations. Our findings suggest that the issue of substantial ellipticity fluctuation, instigated by an increase in ξ_max, can be effectively mitigated through the higher order continuity of the spinning rate function. Moreover, a new spinning rate functions, which are higher order continuous compared to traditional linear and cosine functions, are introduced. We propose using ΔΕ to measure ellipticity fluctuation to assess the robustness of SQWP. The results demonstrate that the values of ΔΕ for the proposed rate function are two and three orders of magnitude smaller than those for the traditional linear and cosine functions. This discovery substantiates the assertion that the proposed rate function surpasses the existing linear and cosine functions in terms of efficiency and robustness for circular SOP conversion. The outcomes of this research bear significant referential value and provide strategic guidance for the future evolution and design of SQWP and this underscores the potential of our approach to contribute meaningfully to advancements in this research field.