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Abstract
A kind of circular polarization beam splitter (CPBS) based on dielectrically chiral dual-core photonic crystal fiber (DC-PCF) is proposed and investigated by a chiral plane-wave expansion method. The numerical results demonstrate the achievement of circular polarized beam splitter must introduce the circular asymmetry of structure and chirality of medium, where the former can shorten the coupling lengths for right- and left-handed beams and increase their difference, while the latter can decrease their difference. Then, according to the competing effect of the circular asymmetry of structure and chirality of medium in polarization and the difference of the paired coupling length, a CPBS based on chiral DC-PCF is designed. The simulation demonstrates that it possesses the high extinction ratios and wide bandwidth at the central wavelength of 1.55µm.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization beam splitters (PBS) are one kind of important optical devices that can separate the beams with different optical polarization states and widely applied in the field of optical information processing, communication, routing, sensing and so on. According to the polarization states, they can be divided into linear and circular ones (LPBS and CPBS) based on the different technologies such as metasurfaces [1], nanostructures [2], chiral fiber [3,4] and dual-core fiber [5,6]. With development of optical fiber manufacturing technology, dual-core fiber (DCF), especially dual-core photonic crystal fiber (DC-PCF) possessing tailored coupling length (CL), spectral bandwidth and extinction ratio (ER), that is capably composed of single materials and possesses tailorable structure, is demonstrated important potential applications in the field of optical fiber circuit and novel sensing [5,6,7–18]. Generally, the LPBS used in linearly polarized optical routes can be made up of simple DC-PCF structures [8,11] and elliptical holes [14], holes filled some liquid [9,13], complex microstructures around cores [7], which have been implemented a lot of research and a lot of results on spectral bandwidth and high polarization extinction (ER) are obtained [12,15]. Since the higher connecting loss, it hasn’t been widely used. Research results have been demonstrated that application of all-fiber LPBS can suppress the polarization error or noise in the linearly polarized optical fiber system [19,20]. However, due to the inherent defect such as polarization axis imperfectly aligning during welding and extrusion or tensile force in practical applications, the polarization error and noise would inevitably be introduced [20]. It is luck that research results demonstrate that they can be suppressed, even eliminated by using circularly polarized optical fiber system [21–23], in which the quarter-wave plate and all-fiber LPBS are applied, since the research on all-fiber CPBS has seldom. Therefore, it is very necessary to study all-fiber CPBS, especially the CPBS based on the DC-PCF composed of single materials and possessing tailorable structure, which is focused in this paper.
Generally, the PCF-CPBS can respectively be realized by employing chiral structures [6,24] and chiral medium [25], namely structurally and dielectrically chiral PCF-CPBS, in which the interesting wavelength is respectively far less and much larger than the corresponding chiral structure in size. The former that has the helically twisted or spun structure can be used as filters, sensors, mode converter, circular polarization controller and generating modes with total angular momentum [3,4,26–30], especially application in the sensors with circularly polarized light, such as fiber optic current sensors [30] and fiber optic gyroscope [21]. Most of the research is based on the periodical twisted core or twisted stress around the core, which form a kind of chiral fiber or chiral fiber grating and can split the beam with opposite-handedness by eliminating the beam with one handedness or divide different wavelength by changing the twist angle [3,4,26–29]. It is hardly applied as a connector in ordinary all-fiber system due to only one output end. Recently, theoretical research results demonstrate that the twisted DC-PCF can be viewed as an all-fiber CPBS, whereas there is a challenge of fabricated technology for undergoing symmetric distortions in twisting and drawing [6]. For the latter, theoretical researches demonstrate that they can support the circularly polarized modes with single core [25,31–33], in which left-handedness polarization beam can be leaked by single-polarization single-mode fiber [25]. In addition, the theoretical research shows that the fundamental modes are circularly polarized in chiral multicore PCFs [34]. However, as a kind of CPBS, the dielectrically chiral multicore-PCF has not been explored. Therefore, in this paper, we will focus on the dielectrically chiral DC-PCF to research its coupling characteristics. We believe that our work will be of benefit to further development of circular-polarization related all-fiber PBS, chiral multicore-fiber, sensor and other optical elements.
In this paper, as a kind of CPBS, the dielectrically chiral DC-PCF composed of single chiral material is considered. Using the chiral plane-wave expansion (PWE) method [31], the effects of circular asymmetry of structure, chirality of medium and other structural parameters are analyzed. It is found that the chirality can tailor the CL, ER, and the corresponding polarization states, which change completely from linear to elliptical, even circular ones due to the competition between circular asymmetry of structure and chirality of medium in polarization. Finally, a PCF-CPBS with 205.5 mm is designed by optimizing the structural parameters and chirality strength.
2. Model and theory
The typical schematic of the dual-core chiral PCF is considered, as shown in Fig. 1, where the two defects suggest the two cores (A and B), the darker part and white circles denote dielectrically chiral background and air holes, respectively. Λ, d and Dth respectively represent the lattice constant, the diameters of air holes and tailored holes (yellow color), which is applied to regulate the CL.
The Drude-Born-Fedorov’s constitutive relations are chosen with D = ε0εr(E + ξ∇×E) and B = μ0μr(H + ξ∇×H) to describe the chiral background [31], where ξ indicates chirality strength related to the specific rotatory power δ of the chiral medium through δ=-k2n2, in which k and n respectively represent the wavenumber in vacuum and the average refractive index of chiral medium. The chirality strength ξ is characterized by the specific rotatory power δ, and hence the air can be viewed as chiral medium with δ=0 (achiral medium).
The coupling characteristics of chiral DC-PCF are analyzed by the chiral PWE method [31]
In the simulation, polymethyl methacrylate (PMMA) doped griseofulvin is selected as the chiral background of the DC-PCF. Since more dilute dopant hardly affects the material dispersion of PMMA, the material dispersion is described by using the formula of the pure PMMA
where A1 = 0.4963, l1 = 71.8 nm, A2 = 0.6965, l2 = 117.4 nm and A3 = 0.3223, l3 = 9237 nm [35]. Chirality can be introduced by griseofulvin with solution doping technique [36], and the corresponding optical rotatory dispersion could be expressed by the empirical Boltzmann formula δ0= B1/λ2 + B2/λ4, where B1 = 1.46 × 104 °•nm4/mm, B2 = 1.82 × 1010 °•nm4/mm are employed as in the Ref. [37]. In this paper, the interesting wavelength is fixed as 1.55 µm and the corresponding specific rotatory powerδ0 = 0.0092 °/mm.Generally, the guided modes are elliptically polarized in chiral DC-PCF due to the competitive effect between the circular asymmetry of structure and chirality of medium in polarization. Thus, the normalized third Stokes parameter S3 with weighted intensity is selected like the chiral single-core PCF [25,32,33].
Like the structurally chiral DC-PCF, the CL (Lci) with i-handedness of dielectrically chiral DC-PCF can be defined as [6]
According to the coupling equation, when a fundamental mode power PAi_in with i-handedness is inputted into core A, after propagating a distance L, the output powers PAi_out in the output port A are calculated as
Generally, the transmission loss can be ignored since the length of the coupler is short. Then, the normalized powers are as follows
In order to successfully separate the RHP and LHP light, the total physical length L of the device are required to satisfy the sufficient condition L = mLCr = nLCl, where LCr and LCl respectively represent the CLs for RHP and LHP light, m and n are positive integers with opposite parity. Therefore, the coupling length ratio (CLR) is define as [6]
From the above analysis, we can know that the key of designing a DC-PCF-CPBS is to obtain an appropriate value of CLR by tailoring the structural parameters and chirality strength.
The extinction ratio (ER) is one of the important parameter to characterize the split ability of light for opposite handedness at the identical output port [6,12]. In other words, when r- and l-polarized light beams with a certain central wavelength (λ0) are incident at the input port of core A and propagate after a distance (the length of PBS), the output power at core A for r- and l-polarized beams are respectively PA-out_r and PA-out_l. The ER can be defined as
The larger the value for |ER| is, the more thorough the split ability for different polarized light and the better performance the PBS. The minimum value of |ER| equals zero that means the output powers at core A for the two polarized light are same, which indicates they cannot be separated at all.
3. Simulation and discussion
In general, both the parameters of structure and materials can influence the coupling characteristics. Figure 2 exhibits the effect of chirality on CLs (a), CLRs (b) and S3 (c), where Λ and d are respectively employed 2 µm and 0.4Λ, and Dth are respectively chosen as 0.6d, 0.8d, and d. The same color of lines indicates fiber with identical parameters, and the solid and dotted lines respectively denote the characteristics of RHP and LHP modes.
It is clearly that the CLs for RHP and LHP modes respectively shorten and lengthen with increase of the chirality strength, and the difference of the paired CLs lessens. The paired CLs shorten whether chiral or achiral case when Dth approaches to d, and their difference gradually reduces, even almost vanishes at Dth =d, where the CLRs trend to unity, even equal to unity, as shown in Fig. 2(b). Meanwhile, because of the tailoring of chirality, the polarization states change from LP to EP, even CP with increase of the chirality, as shown in Fig. 2(c), where the |S3| respectively change from zero to 0.78, 0.95 and 1for the case of Dth = 0.6d, Dth = 0.8d and Dth =d, especially Dth =d, |S3| can reach 0.96 at δ=230δ0 and 1 at δ=660δ0. This indicates that both of the chirality of medium and circular asymmetry of structure can modulate the CLs, CLRs and the corresponding polarization states.
This is because there are competing effects in polarization and distribution of mode field between chirality of medium and circular asymmetry of structure [32,33]. The circular asymmetry of structure decreases when Dth approaching to d, which results in that the more light is localized in the core area and the coupling effect for paired modes strengthens, thus the paired CLs shorten, the CLR trends to unity, and the corresponding polarization states change trends to circular polarization, where the chirality of medium plays an increasingly important role in polarization, even it is dominated.
The difference of the paired CLs is larger at δ=0, with introduction of the chirality, it shortens slowly, the polarization states of the modes are LP, and then EP, even CP with gradually increasing chirality, as shown in Fig. 2(c). This means that the circular asymmetry of structure is still dominated in CL and polarization when the chirality of medium is relatively weaker.
For the same chirality strength, with Dth approaching to d, the polarization states trend to circular polarization in virtue of the decrease of the circular asymmetry. This demonstrates that the smaller the circular asymmetry of structure, the more circular the polarization ellipses. For the case of Dth =d, the CP states can be easily realized, but the corresponding CLRs almost equal to unity, which indicates this kind of splitter is inapplicable since it needs a very long physical length though it can separate the CP light with opposite handedness. Therefore, the reality of DC-PCF for CPBS must introduce the circular asymmetry of structure and chirality of medium to respectively modulate CLs and the degree of circular polarization retention.
Figure 3 shows the relation of the CLs, CLRs and S3 with Dth, in which d and δ are respectively employed as 0.4Λ and 5000δ0, and Λ are fixed as 1.8 µm, 2 µm and 2.2 µm, respectively.
In Fig. 3(a), all the variation trends of CLs are the same and shorten whether the chiral or achiral cases. For the achiral cases, the CLs of x-polarized modes are always longer than that of the y-polarized modes until Dth =0.61Λ, Dth =0.67Λ and Dth =0.73Λ for Λ=1.8 µm, Λ=2 µm and Λ=2.2 µm. As exhibited in Fig. 3(b), the corresponding CLRs (solid lines) respectively increase until Dth =0.38Λ and Dth =0.36Λ for Λ=2 µm and Λ=2.2 µm, then reverse till Dth = 0.52Λ and Dth =0.56Λ, and then reverse again, whereas for Λ=1.8 µm, the CLRs (solid lines) always increases. Thus it can be explained as a reversal of the difference of effective refractive indices of the x- and y-polarized modes at Dth =0.4Λ due to the reverse of the structural asymmetry between x- and y-direction.
While for the chiral cases, the corresponding CLs emerge great change, in which all of the CLs of RHP modes are always longer than that of the LHP modes until Dth =d, where the corresponding CLR≈1 (dotted lines), then reverse till Dth =0.61Λ, Dth =0.67Λ and Dth =0.73Λ for Λ=1.8 µm, Λ=2 µm and Λ=2.2 µm, and then reverse again, as shown in Fig. 3(b). This can be deduced that the RHP mode for right-handed chirality of medium corresponds to the ones with larger effective index for achiral case, and vice versa.
In addition, the larger lattice constant the longer the CLs, since the larger lattice constant can enlarge the distance between the two cores, which would decrease the coupling effect. The CLR for chiral case is closer to unity than the corresponding achiral case due to the modulation of chirality, which leads to better circular symmetry of mode field and decreases/increases the difference of effective indices for the paired LHP/RHP modes.
In Fig. 3(c), the |S3| approaches to unity when Dth approach to d, which indicates the polarization states trend to CP because of the decrease of the circular asymmetry of structure. With increase of lattice constant, the |S3| also trend to unity due to the more light distributed in the chiral medium. When Dth =d, the polarization states are always CP, but this kind of CPBS cannot be manufactured since the corresponding CLR≈1. Meanwhile, the |S3| at the left of Dth =d is larger than that at the right of Dth =d when | Dth -d| is equal, which suggests the smaller tailored holes may decrease the dependence on the chirality of medium. Therefore, the reality of CPBS need that Dth is close to size of d and a relatively larger lattice constant to reduce the dependence of chirality of medium.
Figure 4 exhibits the relations of CLs, CLRs and S3s with the lattice constant, in which Dth and δ are respectively employed as 0.7d and 103δ0, and d is respectively chosen as 0.35Λ, 0.4Λ and 0.45Λ. The solid and dotted lines indicate the CLs for the RHP and LHP modes, respectively.
In Fig. 4(a), it is clearly that all of the CLs lengthen with increase of the lattice constant and the differences of the paired CLs are always smaller. The CLs first increase faster for case of the smaller d than that for the case of the larger d. Since the smaller d reduces the bound of the mode field, which would spread into cladding more. This means the coupling effect increase, and with increase of the lattice constant, the paired mode fields are localized in the core area, the CLRs decrease and then trend to unity, as shown in Fig. 4 (b)
From Fig. 4(c), one can see the |S3| trends to unity with increasing lattice constant, which demonstrates the role of the chirality is gradually enhanced, even dominant in the case of larger lattice constant. Further, the smaller the diameter of the air holes the more circular the polarized ellipse. This is because the smaller lattice constant can shorten the distance of the paired cores to strengthen the coupling effect and larger lattice constant lengthen the distance of the dual cores to strengthen the chirality effect, which is incompatible. Therefore, it is necessary to reasonably optimize structural parameter to decrease the dependence on chirality of medium or CL.
Figure 5 shows the effect of air holes on the CLs, CLRs and S3, where λ, δ and Λ are respectively chosen as 1.55 µm, 5000δ0 and 2 µm, Dth are respectively 0.1d, 0.2d and 0.4d, and the dotted and solid lines denote coupling characteristics of the LHP and RHP modes, respectively.
It is clearly that the paired CLs and their differences increase at first, and then decrease, where all the peak values slightly move to the right with decrease of Dth. As shown in Fig. 5(b), all the peak values of CLRs respectively move to the right and emerge at d = 0.41Λ, d = 0.43Λ, d = 0.435Λ for Dth =0.4d, Dth =0.2d, Dth =0.1d with 1.185, 1.393 and 1.498. This can be interpreted that the smaller air holes can weaken the bound of the mode field and increase the difference of the paired mode fields due to the circular asymmetry. For the bigger d, the paired mode field is bounded in the core areas and their difference reduces, and with further increase of d, the more light is squeezed in the core area near the tailored holes and the difference enlarges. Although the smaller tailored holes means the larger circular asymmetry, the CL is longer with decrease of the tailored holes because of Dth < d, which reduce the bound of the mode field.
From Fig. 5(c), one can see that all the paired S3 decrease with increase of the d or decrease of Dth. This means that the smaller the air holes the shorter CL and the more circular the polarization ellipse. Thus it can be deduced that the CPBS may be realized only a stronger chirality. In order to achieve the CPBS, the air holes should be as small as possible, whereas the CLR trends to unity, which is conflicting. Therefore, to reduce the dependence on chirality, the CLRs must be considered approach to 10/9, 11/10 and so on, where the CLs would be longer than the corresponding achiral case. From Fig. 5 (b) and (c), one can see that the CLRs and |S3| respectively equal to 1.125, 1.119, 1.19, and 0.73, 0.75, 0.76 at d = 0.34Λ, d = 0.335Λ, d = 0.355Λ for the case of Dth =0.1d, Dth =0.2d, Dth =0.4d, which indicates that the CPBS may be realized only enhancing the chirality of medium based on these parameters.
Because the CLRs trend to unity with increase of the chirality, the larger CLRs must be considered for the achiral case. Figure 6 shows the variation of the CLs (a) CLRs (b) and S3 (c) with chirality strength for different lattice constants. To reduce the dependence of chirality of medium, the larger lattice constants are chosen as 2.2 µm, 2.5 µm and 2.8 µm. d and Dth are respectively 0.37Λ and 0.01d. The dotted and solid lines respectively denote the characteristics of RHP and LHP modes.
In Fig. 6(a), the paired CLs shorten with decrease of lattice constant, with increase of the chirality, shorten and lengthen for the RHP and LHP modes, and the difference of the paired CLs decrease, which is consistent with Fig. 2(a) and Fig. 4(a). Compare to the Fig. 6 (b) and (c), the |S3| approaches to 0.932 for the case of Λ=2.2 µm at δ=12000δ0, while for the case of Λ=2.5 µm and 2.8 µm, it can respectively achieve 0.96 at δ=11800δ0 and δ=9000δ0, where the corresponding CLRs are respectively 1.127 and 1.138. This suggests further optimizing the structure or strengthening the chirality of medium can obtain the integer ratio of paired CLs, the shorter CL and the better circular-polarization containing characteristics.
Figure 7 gives the optimized relation of the CLs, CLRs and the S3 with chirality strength, where λ, Λ, d and Dth are respectively chosen as 1.55 µm, 2.6 µm, 0.36Λ and 0.2d. In Fig. 7 (a), one can know the paired CLs are respectively 20.64 mm, 18.60 mm, and 20.55 mm 18.68 mm for CLR = 10/9 and CLR = 11/10 at δ=9500δ0 and δ=10500δ0, where the specific rotatory power are respectively δ=87.4°/mm and δ=96.6 °/mm. Comparing to the Fig. 7(b), the corresponding paired S3 are respectively 0.956, -0.954 and 0.965, -0.963, which indicates that a nearly CPBS and a purely CPBS can be realized with 186.8 mm and 205.5 mm
Since the CLs of RHP and LHP modes are different, the periodic output power that can be obtained from Eq. (6) is different. Assuming that the incident light port is core A at λ= 1.55µm, the normalized power with propagation distance can be obtained from Eq. (7). Figure 8(a) shows the normalized transmission power variation for RHP and LHP modes along the propagation distance in core A. Obviously, when the light for RHP and LHP modes is respectively incident in core A, the light will propagate along the fiber in two cores alternately. After a propagation distance of 205.5 mm, the RHP mode will remain in core A, whereas the LHP ones can be completely coupled into core B in the optimized PCF-CPBS with L = 10LCr = 11LCl = 205.5 mm, whose |S3| is larger than 0.96 from Fig. 7(b).
Fig. 8. (a) Normalized power in port A along propagation distance, (b) Dependence of the ER on wavelength in chiral PCF-CPBS.
Figure 8(b) shows the dependence of the ER on wavelength in the optimized CPBS with the length of L = 205.5 mm. The ER can reach 120 dB at λ= 1.55 µm. Furthermore, the wavelength range is from 1.533 µm to 1.565 µm when |ER|>20 dB and 1.523 µm to 1.573 µm |ER|>10 dB. Therefore, a PCF-CPBS can be obtained with the ER remain above 20 dB over a 32 nm and 10 dB over a 50 nm range at the central wavelength of 1.55 µm.
Obviously, the length of chiral CPBS is far longer than that (millimeter, micron scale) of the existing achiral ones and the corresponding bandwidth is narrower [1,2,5,7–18]. This is because the competing effect between the circular asymmetry of structure and chirality of medium in polarization, CL and CLR. When the former is larger, the difference of the effective mode indices for the identical polarized direction or handedness modes is also large, and the corresponding polarization states trends to LP. While for the latter with right-handedness, it would increase the difference of the effective indices for the RHP modes and decrease the ones for LHP modes, and the corresponding polarization states trends to CP. Thus, in order to obtain the circular-polarization-maintaining chiral DC-PCF as a kind of CPBS, the circular asymmetry of structure should not too large, and the length of CPBS would be longer than the existing achiral ones.
In addition, the length (205.5 mm) of this CPBS is far longer than that (24.76 mm) of the structurally chiral PCF-CPBS [6]. This is because the twist rate in Ref. [6] is 15.7 rad/mm (≈900 °/mm), which is far larger than 96.6 °/mm. This indicates that the length of the dielectrically chiral PCF-CPBS can shorten to equal and even shorter than that of the structurally chiral PCF-CPBS as long as the specific rotatory power of medium is comparative to the twist rate of structural chirality or larger and further strengthen the circular asymmetry of the cross-section of fiber. Further, the spectral bandwidth (32 nm) of dielectrically chiral PCF-CPBS is wider than that (11.43 nm) of the structurally chiral PCF-CPBS when |ER|>20dB. This can be understood that the introduction of the microstructure can tailor the coupling characteristics including CL, polarization states and spectral bandwidth.
4. Conclusions
A CPBS based on the dielectrically chiral DC-PCF was proposed in this paper. The influence of geometric parameters and chirality strength on the CL and the corresponding polarization were investigated by modified PWE method. It is found that the chiral CPBS must introduce the circular asymmetry of structure because there is competition between chirality of medium and circular asymmetry of structure in CL, CLR and |S3|, where the former and latter respectively decrease and increase the difference of the paired CLs, the CLR is respectively closed to and away from unity, and the |S3| respectively trends to unity and zero. In detail, the paired CLs increase with increase of the lattice constant, and both of the CLR and |S3| trend to unity. The CLs respectively shorten and lengthens for the RHP and LHP mode with increase of the chirality of medium for the right-handed case. And all of the CLRs and |S3| trends to unity, which is contrary to the case of increase of the circular asymmetry of structure. The paired CLs and CLR respectively increase first and then decrease with increase of the air holes, and the |S3| trends to zero.
Based the on competition between chirality of medium and circular asymmetry of structure in polarization, a CPBS with the length of 205.5mm based on the chiral DC-PCF is obtained by optimizing the geometric parameters and chirality strength (δ=96.6°/mm). When |ER|>10dB, the bandwidth is about 50 nm range from 1.523 µm to 1.573 µm, and when |ER|>20dB, the bandwidth is about 32 nm range from 1.533 µm to 1.565 µm. This technique provides a new method to realize PCF-CPBS with high ER. Due to introduction of the microstructure, the corresponding bandwidth is respectively narrower and wider than that of the achiral case and structurally chiral case, respectively.
Finally, in order to obtain the proposed high-performance polarization splitter, it is very important to obtain medium with stronger chirality. Besides solution doping technique [35], the chiral medium can also be realized by organic chemical synthesis [39,40]. Furthermore, the chiral DC-PCF can be fabricated like the ordinary achiral ones, such as stack-and-draw technique, performs drilling method and so on. Therefore, with the development of relevant techniques, it is feasible to fabricate the proposed chiral DC-PCF as CPBS, whose length and spectral bandwidth can be comparative, even respectively shorter and wider than that of the structurally chiral DC-PCF-CPBS as long as the chiral medium have larger specific rotation power or more optimized optical fiber structure.
Funding
Fundamental Research Funds for the Central Universities (3072022CF0806).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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