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Abstract
Based on the dual-resonance principle around the dispersion turning point, a scheme of chiral long-period fiber gratings (CLPGs) formed by twisting a high-birefringence (Hi-Bi) fiber is herein proposed to realise ultra-broadband flat-top circular polarizers. The coupling bandwidth is approximately seven times larger than that of traditional CLPGs. In addition, by introducing chirp characteristics in these CLPGs, an ultra-broadband flat-top circular polarizer with ∼200 nm@3 dB was conveniently achieved. Subsequently, by optimising the chirped CLPGs, a circular polarizer with a bandwidth extinction ratio of approximately 30 dB and a high level of ∼100 nm at 1 dB was realised. It was shown that the mode-controlling performances of the CLPGs can be remarkably improved, which has significant applications in light-field regulation. Finally, for the first time, it was proved that the CLPG cannot generate a vortex beam.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, the application of chiral fiber gratings has attracted considerable interest because of their unique optical field control technology, especially in the realms of special sensing, orbital angular momentum (OAM) mode generators, and circular polarisation filtering [1–9]. The study of circularly polarized light has important applications in fiber-optic gyroscopes, fiber-optic current sensors, underwater optical communication, and other areas [10–16]. Chiral long-period fiber gratings (CLPGs) are primarily divided into single- and double-helix structures based on helix symmetry [13,14]. Single-helix CLPGs have been recently shown to be novel all-fiber OAM converters [17–22]. Meanwhile, the double-helix CLPG has polarisation-sensitive characteristics. It can thus be used as an all-fiber circular polarizer which has the characteristics of high optical sensitivity and resolution [12,13,18]. The flexibility and quality of circular polarization controlled by CLPGs are remarkable. Compared with the traditional spatial circular polarizer, the circular polarizer based on CLPG has the advantages of simplicity, compatibility, flexibility, and the use of all fibers [10]. However, like most mode optical field converters, CLPG-based mode converters have a narrow bandwidth of approximately 10 nm at −20 dB, which seriously restricts their applications [13,14]. Thus, broadband generation of all-fiber circular-polarisation converters is expected. For example, Kopp et al. fabricated a non-resonant intermediate-period fiber grating with a broad range of periods to create a broadband circular polarizer [10]. However, to date, there has been no complete theoretical explanation for this phenomenon.
Consequently, Yang et al. proposed an adiabatic coupling method involving slowly varying twist rates of high-birefringence fibers to realise broadband circular polarizers [12]. The CLPG period varies over a large range and length, and the fundamental mode is coupled with multiple cladding modes. Almost all previous reports have focused on circular polarizer mode coupling as an aspect of the work [23–31]. However, the broadband conditions for mode coupling of CLPGs remains to be determined, and their relationship to dispersion turning point coupling is still unclear [32–34]. Moreover, it is uncertain whether the double-helix CLPG can realise OAM mode conversion.
In this study, a method to efficiently generate an ultra-broadband circular polarizer formed by double-helix CLPGs was designed. The dispersion and chirp characteristics were polarized in detail, and their effectiveness was confirmed by simulation. The mechanism of realising the ultra-broadband and flat circular polarizer of the CLPG—which was approximately seven times larger than that of traditional CLPGs—was elucidated. Meanwhile, the phase-matching conditions of mode coupling in the broadband range were examined. A broadband and flatness circular polarizer mode coupling theory for CLPGs was proposed and established. Finally, for the first time, it was proven that the double-helix CLPG cannot generate a vortex beam.
2. Theoretical analysis of CLPGs
2.1. CLPG circular polarization characteristics
In the study of CLPG, the core mode can be coupled with the cladding mode of a higher order by designing the CLPG pitch. Then, the dispersion turning point (DTP) appears in the working wavelength range. Based on the principle of dual-resonance around the DTP, an ultra-broadband circular polarizer formed by the double-helix CLPG is herein proposed for the first time. As shown in Fig. 1, this double-helix CLPG is designed by twisting Hi-Bi fiber with right-hand rotation and a chirped twist rate. The coupling mode equations of the x- and y-polarized core and cladding modes are given directly in local coordinates, where the coupling between the pair of x- and y-polarized core (or cladding) modes is caused by twisting [12]. In this case, the coupled-mode equations of double-helix CLPGs with varietal pitch can be expressed as
In the mode coupling of double-helix CLPG, ${\beta _{co}}$>${\beta _{cl}}$. For right-handed structures, $\tau $>0 (corresponding to the scenario illustrated in Fig. 1). Therefore, the phase-matching condition of the effective coupling can be expressed as follows:
Thus, Eq. (1) becomes
The above formula represents the coupling between the right-hand circularly polarized (RCP) core mode and the left-hand circularly polarized (LCP) cladding mode with right-handed structures, which is consistent with the coupling characterised by the coupled-mode equation of the perturbation theory.
2.2 Ultra-broadband circular polarization of CLPG at DTP
As in [21], a mode basis set in terms of the circularly polarized mode can be defined as follows:
For the double-helix CLPG, the LP0n cladding mode is coupled to the LP01 core mode. As shown in Eq. (1), these two polarisation states can be combined into two degenerate circular polarisation modes: the right circular polarisation (RCP) mode and the left circular polarisation (LCP) mode. The naming convention for these patterns is $H{E_{1,n}}$, or $\textrm{L}{\textrm{P}_{0,n}}$. In the coupling model, $H{E_{1,1}}$ is the core mode; $H{E_{1,2}}$ is the first $H{E_{1,n}}$ cladding mode; $H{E_{1,4}}$ is the second $H{E_{1,n}}$ cladding mode; $H{E_{1,6}}$ is the third $H{E_{1,n}}$ cladding mode, etc. Moreover, $E{H_{1,3}}$ is the first $E{H_{1,n}}$ cladding mode, $E{H_{1,5}}$ is the second $E{H_{1,n}}$ cladding mode, $E{H_{1,7}}$ is the third $E{H_{1,n}}$ cladding mode, etc. We use the scalar mode $\textrm{L}{\textrm{P}_{0,n}}$ to accurately represent the vector mode, where $H{E_{1,2}}$ is represented as $\textrm{L}{\textrm{P}_{0,2}}$, $H{E_{1,4}}$ is denoted as $\textrm{L}{\textrm{P}_{0,3}}$, $H{E_{1,6}}$ is represented as $\textrm{L}{\textrm{P}_{0,4}}$, etc. The cladding coupling mode of the double-helix CLPG consists of only the $H{E_{1,n}}$ modes.
In the following simulations, the right circularly polarized core modes (RCPCM) in right-handed CLPGs are directly solved by the coupled-mode equations in Eq. (4). The parameters of the double-helix CLPG are the following. The selected refractive indices of the core and cladding are chosen as ${n_{co}}$=1.452 and ${n_{cl}}$=1.444, and the radii of the core and cladding are ${R_{co}}$= 4.15 µm and ${R_{cl}}$= 62.5 µm, respectively. The beat length of the PANDA fiber is 2.5 mm.
According to the phase-matching condition in Eq. (3), the relationship between the resonance wavelength and the grating pitch is calculated and demonstrated, as shown in Fig. 2. For the double-helix CLPG, as shown in Fig. 2(a), the pitch of each mode is positively correlated with the resonant wavelength. In Fig. 2(b), near the chromatic dispersion turning point (the black dot in the figure), one cycle corresponds to two resonant wavelengths, and the DTP is the maximum pitch that can be selected for each mode. It is shown that HE1,20 is in the DTP in the wavelength range of ∼1550 nm. It should be noted that a shorter pitch change results in a change in the resonant wavelength range. To verify the broadband transmission characteristics, we selected the $H{E_{1,20}}$ cladding mode with DTP near the telecommunications band to simulate the transmission spectrum of the double-helix CLPG. Based on these selection principles, the pitch of the corresponding grating is 189 µm at a wavelength of 1550 µm. According to the condition of full power conversion, the $H{E_{1,1}}$ mode is coupled with the $H{E_{1,20}}$ mode with a grating length of 11 mm. For comparison, the coupling between the $H{E_{1,1}}$ mode and the $H{E_{1,2}}$ mode is also shown in Fig. 3. The red line indicates the $H{E_{1,20}}$ mode working at the DTP, and the blue line indicates the $H{E_{1,2}}$ mode not working at the turning point of dispersion.
Fig. 2. Wavelength of cladding-mode resonances versus pitch of the CLPGS. (a) Mode of $H{E_{1,2}}$ to $H{E_{1,14}}$. (b) Mode of $H{E_{1,16}}$ to $H{E_{1,28}}$. (The DTPs for mode $H{E_{1,16}}$ to $H{E_{1,28}}$ are marked by black dots)
Fig. 3. Transmission spectra of the double-helix CLPG working at the DTP ($H{E_{1,20}}$) and not working at the DTP ($H{E_{1,2}}$).
The $H{E_{1,20}}$ mode 3 dB bandwidth of the double-helix CLPG working at the DTP exhibits a broad bandwidth of 190 nm, and the $H{E_{1,2}}$ mode with a 3 dB bandwidth exhibits a bandwidth of only 22 nm. Therefore, an ultra-broadband mode conversion is achieved, the bandwidth increases by nearly seven times with the mode coupling in DTP, and the extinction ratio of the central working wavelength reaches ∼60 dB [13].
2.3. Flat-top characteristics of chirped CLPG
To obtain flat and ultra-broadband filtering characteristics, the chirp effect was introduced to the spectrum of the given mode. A CLPG with length L can generally be divided into M∼100 sections of equal length, and the pitch of each section is as follows:
where ${\Lambda _0}$ is the pitch during which the mode operates at the chromatic dispersion turning point, z changes from 0 to L on the axis of the fiber length, and c is the chirp coefficient. The formula used is the following: where $\Delta \Lambda $ is the total cycle change of chirped double-helix CLPG. Next, we simulated a chirped double-helix CLPG with a length of L. From the phase-matching condition of Formula (5), it can be observed that different pitches correspond to different resonant wavelengths. For a double-helix CLPG that is divided into 100 segments and has a pitch of gradual change, all the resonant wavelengths work together to achieve broadband transmission. Note that the entire chirp process requires the grating to achieve a long range of periodic variation, and the mode of working at the DTP can effectively solve this problem.For the $H{E_{1,2}}$ mode, to obtain a total bandwidth of 290 nm from 1.39 µm to 1.68 µm, the pitch must be changed from 437 µm to 661 µm by 224 µm, as shown in Fig. 4(a). And the resonance wavelength ranges of the two cladding modes will overlap each other. Therefore, to achieve a broadband flat response, an extremely long grating is required. For the $H{E_{1,20}}$ mode, which works at the turning point of dispersion, a pitch change of 1 µm can be achieved to obtain the same resonant wavelength range.
Fig. 4. (a) Phase-matching curves for $H{E_{1,4}}$ (blue line) and $H{E_{1,2}}$ modes (red line). A pitch change of 224 µm corresponds to the 290 nm resonance wavelength range. (b) Phase-matching curves for $H{E_{1,18}}$, $H{E_{1,20}}$ and $H{E_{1,22}}$ modes. The resonance wavelength with a bandwidth of 290 nm requires only 1 µm pitch change near the DTP of the $H{E_{1,20}}$ mode.
In the simulation, we chose a grating length of 4.97 mm. The grating was divided into 100 segments. Figure 5(a) compares the transmission spectra between the chirped and non-chirped double-helix CLPG operating around the DTP. The chirped double-helix CLPG with 1 µm grating pitch change ranging from 188.5 µm to 189.5 µm is adopted in this simulation. After chirping the double-helix CLPG, a broad 20-dB bandwidth of ∼223 nm has been successfully obtained, which is slightly larger than that of the non-chirped double-helix CLPG. The $H{E_{1,20}}$ mode is shown with chirp added, the bandwidth is wider after optimisation, and the coupling depth is reduced. This is because, when chirp is introduced, the pitch of the grating is changed, resulting in effective coupling of the grating during the coupling process. As the length decreases, the coupling depth also decreases. The broader spectrum range is due to the combination of all the resonance modes, and a flatter response is obtained by losing part of the extinction ratio.
Fig. 5. (a) Transmission spectra of double-helix CLPG working at DTP of $H{E_{1,20}}$ mode with a constant pitch of 189 µm (black line) and linearly chirped double-helix CLPG with a pitch change from 188.5 µm to 189.5 µm. (red line) (b) Transmission spectra under raised-cosine section length apodization of the chirped double-helix CLPG operating near the DTP of the $H{E_{1,20}}$ mode.
In Fig. 5(a), there is a small concavity in the middle of the flat-top band. For the circular polarizer formed by double-helix CLPG, the flatness of the resulted spectrum directly affects its performance. To obtain a flatter transmission spectra, we shape the spectrum by apodizing the section lengths of the chirped double-helix CLPG with raised-cosine function near the DTP of $H{E_{1,20}}$. In the simulation, the grating was divided into 100 segments and the grating coupling length was 4.97 mm. For the linear function, it is adopted for apodizing the section lengths, which can be expressed as:
where L and N correspond to the total grating length and the number of grating sections to be divided, respectively. n denotes the n-th order grating section ranging from 1 to N, and Nsum indicate the sum of the arithmetic progression from 1 to N with a common difference of one. And for shaping the spectrum with the raised-cosine function, the length of the n-th section can be expressed as: where F(n) is the raised-cosine function and is given by: where ${z_n} = L\frac{n}{N}$ is the longitudinal position of the n-th section of the chirped double-helix CLPG ranging from 0 to L. As shown in Fig. 5(b), the unevenness of the bottom is improved, and its extinction ratio reaches 30 dB. Therefore, by establishing a dispersion transition model in a chirped CLPG, the parameters of the ultra-broadband flat-top circular polarizer through the double-helix CLPG are obtained.3. Angular momentum conversion of CLPGs
The study of OAM modes generated by CLPGs has attracted considerable attention in recent days. However, the debate continues about the double-helix CLPGs that can generate the OAM mode. Therefore, in this section, the OAM mode conversion of double-helix CLPG is demonstrated. A general linear combination of two l = ±1 OAM modes has a topological charge with an integer total OAM. With the mode definition [21], the relationship between the angular order and charges of the spin angular momentum (SAM), OAM, and the total angular momentum (AM) of the modes, denoted s, l, j, respectively, can be clearly observed, as summarised in Table 1.
Table 1. Charges of the SAM, OAM, and AM of CLPGs
In the double-helix CLPG, coupling between the core and cladding modes is considered. Table 2 summarises all possible coupled cladding modes with SAM, OAM, and AM charges when excited by the core mode HE11 and when the phase-matching condition is not considered.
Table 2. All Possible Mode Coupling Rules in Double-helix CLPGs for Co-propagating Mode Coupling with OAM Conversion
In this CLPG, we can obtain the interaction of the transmission light based on the conversion conditions of the linear momentum and AM, which can be expressed as the following AM matching conditions:
where M1,2 respectively represent the propagation constants and AM charges of the core and cladding modes that propagate independently in the SMF. The upper and lower signs – and + denote the left- and right-handed CLPGs, respectively. The case of the right- and left-handed double-helix CLPGs follows from the AM and mode coupling selection roles, as shown in Eq. (11).Furthermore, according to the phase matching and AM matching conditions in Eq. (4) and Eq. (11), Table 2 can be simplified to Table 3. Hence, for right-handed double-helix CLPGs, the phase-matching condition can be realized only when the RCP core mode ($HE_{11,co}^ -$) is coupled to the LCP cladding mode ($HE_{1n,cl}^ +$). Due to the SAM of cladding mode $EH_{1n,cl}^ +$ is -1, which corresponds to the RCP cladding mode, it cannot be effectively coupled to the RCP core mode. Similarly, left-handed double-helix CLPGs have corresponding results. The latter table shows the relationship between the core and cladding modes that can be effectively coupled in the double-helix CLPG. Under the interaction between the OAM and SAM with double-helix CLPGs, the AM matching condition in Eq. (11) represents the conversion of AM in mode coupling.
Table 3. Effectively Coupled Cladding Modes in Core and Cladding Mode Couplings in Double-helix CLPGs
Table 3 summarises the effectively coupled cladding modes, SAM, OAM, and AM charges of CLPGs when excited by the core modes $HE_{11,co}^ \pm$. For the right-hand double-helix CLPG, only the RCP core mode ($HE_{11,co}^ -$) is coupled to the LCP cladding mode ($HE_{1n,cl}^ +$) via the AM selective rule. Moreover, only the LCP core mode ($HE_{11,co}^ +$) is coupled to the RCP cladding mode ($HE_{1n,cl}^ -$) using a left-handed double-helix CLPG. As shown in Table 3, the OAM of the double-helix CLPG does not change during the mode coupling process. Therefore, for the first time, it was proven that double-helix CLPGs cannot generate a vortex beam.
4. Conclusion
By analyzing the mode coupling of the double-helix CLPG near the dispersion turning point, we demonstrated that an ultra-broadband flat-top circular polarizer can be generated by the excitation of the dual-resonance fundamental core mode in the respective CLPGs. A circular polarizer with ∼200 nm@3 dB was readily achieved. It was approximately seven times larger than that of the CLPG without operating at the DTP. Moreover, to optimise the flatness of the circular polarizer, the grating length was adjusted according to the variation in the coupling coefficient. Consequently, a circular polarizer with a bandwidth extinction ratio of 30 dB and a high level of ∼100 nm at 1 dB was readily achieved. Finally, the interactions of SAM and OAM accompanied by mode couplings in the CLPGs were investigated. It was demonstrated that the OAM mode could not be generated in the double-helix CLPGs. Therefore, the OAM mode has significant potential for beam OAM and circular-polarisation control.
Funding
State Key Laboratory of Transient Optics and Photonics (SKLST202107); Natural Science Foundation of Shaanxi Provincial Department of Education (20JK0928); Natural Science Foundation of Shaanxi Province (2021JQ-720, 2022JM-357); National Natural Science Foundation of China (12104368).
Acknowledgments
The authors would like to thank Nan Liu in Northwest University for the help.
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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