Carrier recovery in coherent receiver of optical orthogonal frequency division multiplexing system

Abstract

In this paper, we reviewed our common phase error (CPE) and intercarrier interference (ICI) compensation methods for coherent optical orthogonal frequency division multiplexing (CO-OFDM) system. We first presented a unified CPE estimation framework combining decision-aided (DA), pilot-aided (PA) and decision feedback (DF) algorithms. The DA method is used to estimate the CPE of the current OFDM symbol based on the decision statistics of the previous symbol. DA+ PA helps increase the phase noise tolerance of DA and reduce the overhead of PA, while DA+ DF reduces the overhead to zero, achieving best performance with one more step of estimation, compensation and demodulation. We also described a modified time-domain blind intercarrier interference (BL-ICI) mitigation algorithm over non-constant amplitude formats. The new algorithm is derived from the BL-ICI algorithm over constant amplitude format for wireless networks. A new power estimation scheme was proposed for the BL-ICI algorithm to adapt to non-constant amplitude format. It has the same order of complexity with frequency domain decision-aided ICI (DA-ICI) compensation method and does not suffer from symbol decision errors. The effectiveness of both CPE and ICI compensation algorithms were demonstrated in a simulated 56-Gbit/s CO-OFDM system with various modulation formats.

Keywords

linear phase noise / coherent optical orthogonal frequency division multiplexing (CO-OFDM) / common phase error (CPE) / decision-aided (DA) / intercarrier interference (ICI)

1 Introduction

Today’s telecommunication system strongly depends on data based communication like different online based services, video communication, video sharing, and many more. The rate of increment of this data demand is raising exponentially. Research is going on to enhance the data capacity of optical fiber in the most recent years [1–4]. As a candidate, photonic crystal fiber (PCF) has a great field of attraction to the researchers [5,6]. PCFs are made up of central core of solid silica or air and periodic arrangement of air holes throughout the length. PCFs have many usual optical properties like endless single mode operation [3,7], high birefringence [8–10], dispersion compensation [11,12], large mode area [13–15], and high nonlinearity [16] over the conventional fiber. This is the main reason of using PCFs which is the major topic in the area of optical fiber research.

Dispersion, confinement of light, and birefringence are the most important properties of PCF. Practically, these properties are used in bio-medical and medical diagnostic, telecommunication, and sensing applications [17–19]. Dispersion management of PCFs is very much vital due to significant variation of different methods [20]. PCFs offer higher birefringence than conventional fibers due to large index difference. Asymmetric design or various air hole geometries of PCFs support high birefringence [21–23]. Usually to increase the birefringence, rupture the steadiness of the structures and create refractive index variation involving the two orthogonal polarization states [24]. The elliptical air holes arrangement in PCFs confirm fully complicated fabrication procedures [25–27]. Though it is well established that non-circular air holes like elliptical are difficult to fabricate but it is applied to obtain high birefringence [28,29]. Nonetheless, the shape and size of the elliptical air holes require a conscientious control in the fabrication procedures [30]. Also, stack and draw fabrication methods provide an enormous birefringence to be appreciated on PCFs.

High birefringence is introduced by elliptical air hole PCFs (EHPCFs) which shows birefringence of 1.12 × 10⁻² that is bigger than the traditional fiber (5 × 10⁻⁴), circular PCF (3.7 × 10⁻³), and elliptical shape hollow PCF (EHPCF) (2.35 × 10⁻³) [27]. To increase the birefringence, an imitation imperfections spiral PCF (S-PCF) is proposed which shows birefringence of 0.09 at the excitation wavelength 1.55 µm [31]. An asymmetrical structure is made by using circular air holes at the cladding region and elliptical air holes at the core region that shows birefringence up to 10⁻² [30]. A hybrid PCF (HyPCF) is proposed to achieve birefringence of 1.77 × 10⁻² and confinement loss is less than 10² dB/km at 1.55 µm wavelength [32]. Hasan et al. proposed an equiangular spiral photonic crystal fiber (ES-PCF) which gives a usual negative dispersion of 526.99 ps/(nm·km) from 1.05 to 1.70 µm wavelength and large birefringence of 0.026 at 1.55 µm wavelength [33]. Moreover, modified octagonal PCF (M-PCF) is proposed that demonstrated birefringence of 1.81 × 10⁻² at 1.55 µm wavelength [34]. However, these types of PCFs show highly fabrication complexity especially elliptical air holes at the core region.

To enhance the birefringence, another report proposed increasing the diameter of the center air holes which provides 1.4 × 10⁻³ birefringence at 1.55 µm wavelength [35]. At 1.55 µm wavelength, the birefringence of 8.7 × 10⁻³ is observed for index-guiding PCF applying the complex unit cells in cladding [24]. A hybrid PCF is presented which shows high dispersion of -242.22 to -762.6 ps/(nm·km) and 0.0264 birefringence at 1.55 µm wavelength [36]. Another report proposed a dispersion compensation PCF (DC-PCF) design with a negative dispersion from -388.72 to -723.1 ps/(nm·km) in the telecommunication bands for broadband dispersion compensation with a birefringence and nonlinear coefficient of 3.79 × 10⁻² and 40.1 W⁻¹·km⁻¹ at 1.55 µm [37]. One more report demonstrated high birefringence of 5.501 × 10⁻³ with low confinement loss 7.30 × 10⁻⁵ dB/km at 1.55 µm wavelength supported by single rectangular midpoint ring of minor rounded air holes in the fiber core [38]. Moreover, at 1550 nm wavelength the maximum birefringence of 0.87 × 10⁻² with confinement loss less than 0.01 dB/km is proposed by Ref. [39]. However, the guiding properties of these models especially the dispersion values are comparatively small at 1.55 µm wavelength. Also, the birefringence did not follow a constant value over a broad wavelength range. As a result, the fiber length and production cost increase. Moreover, different broadband DC-PCFs have been considered but none of them has provided birefringence value more than 2.1 × 10⁻² [34,40,41]. Recently, a highly birefringence solid core PCF through asymmetric cladding is proposed (elliptical air hole) which presents birefringence of 8.56 × 10⁻³ at 1.55 µm wavelength but the dispersion value is not satisfactory [42].

In the proposed design, a high birefringence and low confinement loss PCFs with asymmetric design from 1.2 to 1.8 µm wavelength are analyzed. By using circular air holes in the first two rings and elliptical air holes for the remaining rings through different rotation angle show significant asymmetry in the cladding region. Moreover, two circular air holes are omitted from the core region to increase the light confinement. The proposed models present higher birefringence due to introduce the asymmetry in the cladding region. In addition, high negative dispersion, small effective mode area, and high nonlinear coefficient are presented.

2 Materials and methods

2.1 Designing methodology

Figure 1 shows the cross section of the proposed PCFs which is designed using the combination of circular and elliptical air holes in the cladding. The proposed PCFs contain five air hole rings where first two circular air hole rings are hexagonal with diameter d₁ and other three air holes rings are elliptical shaped. The core of the PCF is formed by eliminating two air holes of the first hexagonal air hole ring in the horizontal direction where the mode field is well confined in the core region. The total number of circular air holes on the 1st and 2nd rings are respectively 4 and 12. Usually circular air holes rings are rotated to obtain high birefringence [37]. In this report, the rotation angle of elliptical air holes on the 3rd, 4th and 5th rings are 22.5°, 10° and 15° for model-1 and 36°, 16.36° and 25.71° for model-2, respectively. Moreover, to identify the effect of angle rotation on PCF guiding properties, the third ring rotation angle is varied. Mainly rotation angle determine the number of air holes in the cladding area. As rotation angle= 360°/number of air holes in the cladding region, then taking rotation angle as 26° in the 3rd elliptical air hole ring the number of air holes in the cladding is 13.84 which is a frictional number and usually air holes value cannot be fractional. As elliptical air holes are taken 15, 16 and 17 which is the reason behind to take rotation angle 21.18°, 22.5° and 24°.

Fig.1 Cross-section of the proposed PCFs. (a) Model-1; (b) model-2

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In the proposed design, the corresponding PCF as model-1 for Fig. 1(a) and model-2 for Fig. 1(b) is shown. In model-1 and model-2, the value of d₁ and d₂ is 0.83 and 0.807 µm and distance between two air holes is 0.85 µm. In the same manner x₃ and y₃, x₄ and y₄, and x₅ and y₅ denote the two axes of elliptical air holes along the x and the y-direction. The values of x₃ and y₃, x₄ and y₄, and x₅ and y₅ are taken respectively as 0.75 and 0.5 µm, 0.5 and 0.25 µm, and 1 and 0.75 µm for both the models. The rotation angle of 3rd elliptical air hole ring is varied as 21.18°, 22.5°, and 24° for model-1 and 32.72°, 36°, and 40° for model-2, respectively. Moreover, the difference between model-1 and model-2 is the position of elliptical air hole. In model-1, elliptical air holes are organized in vertical direction whereas in model-2 the air holes are horizontal direction. The modal analysis is done by applying finite element method (FEM-Comsol Multiphysics. version 5.2) from 1200 to 1800 nm with 10 nm wavelength increments united with perfectly matched layer (PML) boundary condition.

2.2 Theoretical analysis

The modal properties of the proposed fiber are simulated and analyzed by using FEM [43]. Energy loss can be minimized by using PML and a scattering boundary condition [44]. Refractive index of the back-ground material depends on wavelength and can be obtained by using Sellmeier formula [45,46]

(1)$n({\lambda }^{\mathrm{2}})=\mathrm{1}+{\displaystyle \frac{{{\displaystyle B}}_{\mathrm{1}}{\lambda }^{\mathrm{2}}}{{\lambda }^{\mathrm{2}}-{{\displaystyle l}}_{\mathrm{1}}}}+{\displaystyle \frac{{{\displaystyle B}}_{\mathrm{2}}{\lambda }^{\mathrm{2}}}{{\lambda }^{\mathrm{2}}-{{\displaystyle l}}_{\mathrm{2}}}}+{\displaystyle \frac{{{\displaystyle B}}_{\mathrm{3}}{\lambda }^{\mathrm{2}}}{{\lambda }^{\mathrm{2}}-{{\displaystyle l}}_{\mathrm{3}}}},$

and the generalized equation can be written as

(2)$n({\lambda }^{\mathrm{2}}-\mathrm{1})={\displaystyle \sum }_{i=\mathrm{1}}^j{\displaystyle \frac{{{\displaystyle B}}_i{\lambda }^{\mathrm{2}}}{{\lambda }^{\mathrm{2}}-{{\displaystyle l}}_i^{\mathrm{2}}}},$

where j = 3, B₁ = 0.6961663, B₂ = 0.4079426, B₃ = 0.8974794, l₁ = 0.0684043 µm, l₂ = 0.1162414 µm, and l₃ = 9.896161 µm, λ is the operating wavelength, and n($\lambda$) is the wavelength-dependent refractive index. From Maxwells curl equation, the vectorial equation can be written as follows [47]

(3)$\nabla \times ({[s]}^{-\mathrm{1}}\nabla \times E)-{{\displaystyle k}}_{\mathrm{0}}^{\mathrm{2}}{n}^{\mathrm{2}}[s]E.$

Here, E is the electric field vector, ${{\displaystyle k}}_{\mathrm{0}}={\displaystyle \frac{\mathrm{2}\mathrm{\pi }}{\lambda }}$ is the wave number in the vacuum, [s]⁻¹ is an inverse matrix of [s], and n_eff is effective refractive index given as ${{\displaystyle n}}_{\mathrm{eff}}={\displaystyle \frac{\beta }{{{\displaystyle k}}_{\mathrm{0}}}}$, where $\beta$ is the propagation constant. The effective refractive index difference of the two fundamental modes is defined as birefringence which can be calculated through the following relation [48]

(4)$B=\left|({{\displaystyle n}}_x-{{\displaystyle n}}_y)\right|,$

where n_x and n_y are the effective refractive indices of the two-orthogonal polarization fundamental modes. All power do not propagate through the core region and some power emits through the air holes that arises confinement loss. Calculation of confinement loss requires imaginary part of effective refractive index of modes and can be calculated by using the following equation [49,50].

(5) $L=\mathrm{8.686}\times {\displaystyle \frac{\mathrm{2}\mathrm{\pi }}{\lambda }}\mathrm{Im}({{\displaystyle n}}_{\mathrm{ef}\mathrm{f}})\times {\mathrm{10}}^{\mathrm{3}}\hspace{0.17em}\mathrm{dB}/\mathrm{km},$

where Im(n_eff) is the imaginary part of the refractive index of the propagating mode. Through the following relation, dispersion can be calculated [51].

(6)$D(\lambda )=-{\displaystyle \frac{\mathrm{2}\mathrm{\pi }c}{{\lambda }^{\mathrm{2}}}}\beta \simeq -{\displaystyle \frac{\lambda }{c}}{\displaystyle \frac{d^{\mathrm{2}}{{\displaystyle n}}_{\mathrm{eff}}}{d{\lambda }^{\mathrm{2}}}},$

where c is the speed of light in vacuum. The value of the effective area and nonlinear coefficient can be obtained by the following formula [52]

(7)${{\displaystyle A}}_{\mathrm{eff}}={{\displaystyle \left(\iint {\left|E\right|}^{\mathrm{2}}\mathrm{d}x\mathrm{d}y\right)}}^{\mathrm{2}}/\iint {\left|E\right|}^{\mathrm{4}}\mathrm{d}x\mathrm{d}y,$

(8)$\gamma =\left({\displaystyle \frac{\mathrm{2}\mathrm{\pi }}{\lambda }}\right){\displaystyle \frac{{{\displaystyle n}}_{\mathrm{2}}}{{{\displaystyle A}}_{\mathrm{eff}}}},$

where n₂ is nonlinear refractive index coefficient of silica.

3 Results and discussion

Figure 2 demonstrates the optical field distribution of the fundamental mode of the proposed PCFs. The rotation angle of elliptical air hole on 3rd ring is 22.5° for model-1 and 36° for model-2 at $\lambda =\mathrm{1.4}\hspace{0.17em}\mathrm{\mu m}$ wavelength. Figure 2 also illustrates the mode field in model-1, is strongly confined than the other one. For the reason of small rotational angle of model-1 which increases the number of elliptical air holes on third ring. Therefore, the light is strongly confined in the core region. The dark red color indicates the strong confinement of light in the core region. Moreover, high color density represents the high intensity of absorption. The figure also illustrates the optical field that extends broadly in the direction of x-axis than y-axis due to adequate space given in the x-direction by omitting the air holes.

Fig.2 Optical field distribution of the proposed PCFs. (a) x-polarization, (b) y-polarization for model-1; (c) x-polarization, (d) y-polarization for model-2

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Fig.3 Refractive index (real part) with respect to wavelength for proposed PCF model-1

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Fig.4 Refractive index (real part) with respect to wavelength for proposed PCF model-2

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Figure 3 represents the variation of effective index (real) for the proposed PCF model-1 with respect to wavelength for both x- and y-polarization. For both the polarizations, with the increase of the wavelength, the effective index is decreased. Higher effective index is observed at y-polarization with all the three rotational angle. At 1.55 µm wavelength, the effective index difference is 0.0275, 0.0145, and 0.0075 for 21.18°, 22.50°, and 24.00°, correspondingly. As small rotational angle 21.18° shows higher index difference then it gets the high birefringence.

Figure 4 shows the variation of effective refractive index (real) for the proposed PCF model-2 with respect to wavelength for both x- and y-polarization. For both the polarizations increasing wavelength the effective index is decreased, as usual. Moreover, higher effective index is observed for y-polarization of all the three rotational angles. At 1.55 µm wavelength, the effective index difference is observed 0.0115, 0.0116, and 0.0101 for 32.72°, 36.00°, and 40.00°, correspondingly. Model-2 represents slightly different result than model-1 at 1.55 µm. At this specific wavelength, the effective index difference is higher for 36.00° rotational angle not 32.72°. Increasing wavelength from 1.2 to 1.8 µm, the effective index reduction rate is shown 4.60%, 4.65%, 3.76% for x-polarization, and 5.01%, 5.13%, and 4.33% for y-polarization with the three rotational angles, respectively. However, this reduction rate is observed 3.42%, 5.90%, 3.21% for x-polarization, and 4.03%, 4.75%, and 3.88% for y-polarization, respectively of model-1.

Fig.5 Refractive index (imaginary part) with respect to wavelength for proposed PCF model-1 and model-2

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Figure 5 shows the variation of effective refractive index (imaginary part) for the proposed PCF model-1 and model-2 with respect to wavelength for x-polarization only. Generally, with the increase of the wavelength, the effective imaginary index raises. From 1.2 to 1.5 µm wavelength range, the imaginary index shows very small. After 1.5 µm wavelength, the imaginary part is increased upward. Moreover, the lowest imaginary index is obtained at 36° with model-2 at 1.55 µm wavelength. Also, at 1.55 µm wavelength, model-2 exhibits the highest imaginary index at 40° rotational angle.

The birefringence is calculated through the effective refractive index difference of the fundamental guided modes. Figure 6 exhibits the birefringence with respect to wavelength for PCF model-1. The maximum birefringence of 2.75 × 10⁻² is obtained at 21.18° rotational angle. However, from 1.4 to 1.55 µm wavelength range, this high constant birefringence is observed. However, with the increase of the rotational angle, the birefringence reduces significantly. This is due to decrease the effective index difference of x- and y-polarizations as shown in Fig. 3. Moreover, the birefringence is gradually decreased with all the three rotational angle for 1.55 to 1.8 µm wavelength range. Birefringence actually depends on structural asymmetry. In the proposed design, the structural asymmetry is achieved by introducing elliptical air holes in the cladding region as well as by changing rotation angle of air holes in the third ring. Changing rotation angle or number of air holes, then the structural asymmetry will change which will lead to enhance birefringence.

Figure 7 illustrates the results of birefringence with wavelength for model-2. The maximum birefringence of 1.29 × 10⁻² is obtained at 32.72° rotational angle at 1.45 µm wavelength range. However, with the increase of the rotational angle, the birefringence reduces similarly to model-1. Interestingly, at 1.55 µm wavelength, 36.00° rotational angle gives the highest birefringence. Increasing wavelength, birefringence is decreased for all the rotational angle due to reduce the effective index difference of x- and y-polarizations as shown in Fig. 4. Thus, model-1 gives the higher birefringence than model-2 with smaller rotational angle.

Fig.6 Birefringence with respect to wavelength for proposed PCF model-1

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Fig.7 Birefringence with respect to wavelength for proposed PCF model-2

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Figures 8 and 9 exhibit the effect of rotation angle of elliptical air hole on 3rd ring to the dispersion behavior when the other parameters are kept constant for model-1 and model-2, respectively. Figure 8 shows the rotational angle of 21.18°, 22.50°, and 24.00°, the dispersion at 1.55 µm are –300.074, –399.185 and –536.296 ps/(nm·km), respectively. However, the maximum dispersion –955.25, –650.98, and –625.56 ps/(nm·km) are obtained in 1.8, 1.7, and 1.45 µm wavelength for the three rotational angle. Though, it is observed that 24.00° rotational angle shows maximum dispersion from 1.4 to 1.55 µm wavelength range, it gives the lowest birefringence at the specific wavelength range. However, changing the structure parameters, these guiding properties can be enhanced.

Fig.8 Dispersion with respect to wavelength for proposed PCF model-1, x-polarization

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Figure 9 demonstrates the dispersion variation with wavelength for proposed model-2. The dispersion variation is observed –110 to –575 ps/(nm·km). The maximum dispersion is achieved from 1.4 to 1.55 µm wavelength range at 40.00° rotational angle which usually shows the minimum birefringence. Increasing wavelength negative dispersion is usually increased, though 40.00° rotational angle shows the lowest dispersion variation comparing to other rotational angles. Therefore, it can be concluded that rotational angle has great impact on dispersion and birefringence.

Fig.9 Dispersion with respect to wavelength for proposed PCF model-2, x-polarization

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Fig.10 Confinement loss with respect to wavelength for proposed PCF model-1

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Figures 10 and 11 depict the difference of confinement loss with wavelength for model-1 and model-2, respectively. Simulation results imply that with the increase of wavelength confinement loss is increased and this increment is more severe at the higher wavelength range. Lowest confinement loss is achieved at 22.50° and 36.00° rotational angle for both the models. However, model-2 shows comparatively small loss than model-1 at higher wavelength. In addition, both the models show higher loss at higher rotational angle. Because at higher rotational angle, light is not strongly confined in the core. However, the proposed models represent smaller confinement loss compared to the exiting work [32,53].

Fig.11 Confinement loss with respect to wavelength for proposed PCF model-2

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Fig.12 Effective area with respect to wavelength for proposed PCF model-1

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The effective area is shown in Figs. 12 and 13 for model-1 and model-2, respectively against wavelength. Effective area increases with wavelength in both the models. For both the PCFs minimum effective area is observed at 22.50° or 36.00° rotational angle. For model-1, the effective area variation is almost constant from 1.6 to 1.8 µm wavelength range. However, increasing wavelength after 1.4 µm there is a large variation of effective area depending on rotational angle for model-1. The similar trend is observed for model-2 though 40.00° rotational angle which shows the maximum effective area at higher wavelength. However, the effective area is almost unvaried (model-2) in the lower wavelength range for the others two rotational angles.

Figures 14 and 15 show the variation of nonlinear coefficient with respect to wavelength. In both the models, with the increase of the wavelength nonlinear coefficient decreases. At 1.55 µm wavelength, the maximum nonlinear coefficient 43.7 and 41.98 W⁻¹·km⁻¹ was observed at 22.50° and 36.00° rotational angle, respectively for both the models. Noted that model-1 shows maximum nonlinear coefficient at low wavelength whereas model-2 represents maximum nonlinear coefficient at high wavelength. Therefore, rotational angle shows great impact on different guiding properties.

Fabrication possibility is a leading apprehension in designing PCFs. The fabrication progression of elliptical shape PCFs can be vulnerable to fall down and to alter into circular shape for the surface tension. Bunches of PCFs fabrication methods are offered at the present time resembling drilling, die cast, extrusion, stack and draw, sol-gel. Newly, Liu et al. have been made-up mixed air hole diameters at the cladding area by using the usual stack and draw method [54]. Moreover, multi-step method of creating perform is presented by Falkenstein et al., where elliptical PCFs model were tested in practically [55].

Fig.13 Effective area with respect to wavelength for proposed PCF model-2

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Fig.14 Nonlinear coefficient with respect to wavelength for proposed PCF model-1

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Fig.15 Nonlinear coefficient with respect to wavelength for proposed PCF model-2

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4 Conclusion

An ultra-high birefringence with moderate negative dispersion and negligible confinement loss PCFs have been proposed. Also, effective area and nonlinear coefficient have been investigated. Numeric analysis shows highly birefringence of 2.75 × 10⁻² with moderate dispersion at 1.55 µm wavelength. Rotating the angle on the third ring of the elliptical air holes has significant impact on the various guiding properties. Results also imply that model-1 shows maximum birefringence whereas model-2 represents low confinement loss. This reported results can be widely applicable in sensing applications, polarization, and long distance data transmission.

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Acknowledgements

The authors would like to thank the support of AcRF Tier 2 Grant MOE2013-T2-2-145 from MOE Singapore.

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg