1. Introduction

Flexible dispersion control in optical fibers at a desired spectral range has been a major issue in high bit rate long-haul wavelength division multiplexing (WDM) optical communication systems [1]. In recent years, several hundreds of Gbps based dense WDM (DWDM) transmission systems are being successfully introduced in the fields [2]. The intersymbol interference (ISI) between adjacent bits in the channel can occur by the linearly accumulated chromatic dispersion along the transmission fiber, which can significantly deteriorate the communication quality [2]. Chromatic dispersion management in optical fiber design is largely divided into three categories: non-zero dispersion shifted fiber (NZDSF), dispersion flattened fiber (DFF) and dispersion compensating fiber (DCF), which require specific waveguide designs different from standard single mode fiber (SSMF) [3–5].

In recent years, photonic crystal fibers (PCFs) has been intensively investigated to explore their potential in the chromatic dispersion control as an alternative to conventional solid core/cladding optical fibers [6–11]. Due to their unique holey cladding structures, PCFs enabled an exquisite dispersion control by varying structural parameters such as the air hole diameter d, the hole-to-hole pitch Λ, and symmetric arrangements of air holes. In pure silica PCFs, the local refractive index can be effectively controlled by locally varying the air hole diameter such that a layer of air holes with a smaller or larger diameter can serve as a higher refractive index pedestal or a lower refractive index trench, respectively. Flattened near-zero dispersion over a broad spectral range was recently reported in a PCF using a similar technique with five concentric air-hole layers with different diameters in the hexagonal arrangement [6]. Despite notable dispersion characteristics in prior theoretical predictions, such PCFs have not been successfully fabricated due to their strict requirements meticulous control and maintenance of the hole geometry with a very low tolerance.

Recently the authors and other researchers have proposed a new method to control the guiding properties of hexagonal PCF by introducing hollow ring defects (HRDs) [12–14]. These HRDs were adopted from the hollow optical fiber (HOF) structure that consists of a central air hole, a high index ring core, and a silica cladding [15, 16]. In this paper, we further explore the potential of HRDs in a square lattice photonic crystal fiber (SLPCF) to provide a new method to flexibly control the chromatic dispersion characteristics (Fig. 1 ). By adjusting the refractive index of its ring, which would be maintained far more robustly than adjusting the air hole diameter during the high temperature fabrication processes, HRD can significantly relieve the stringent requirements in the prior PCFs with various air hole geometries. Note that HRDs can be routinely fabricated by conventional modified chemical vapor deposition (MCVD, Fig. 1(d)), along with the well-established stack and draw process [17]. Moreover, SLPCFs recently showed significant design advantages: the larger number of air holes per a hole-layer to improve the confinement loss [18] and commercialized square hole silica tube (Fig. 1(c)) to ease the stack-and-draw process during fabrication.

 

Fig. 1 (a) Schematic cross-section of the proposed SLPCF along with its geometrical parameters: air-hole diameter d and its pitch Λ. (b) Enlarged view of GeO₂-doped silica high-index hollow ring defect (HRD) on the top and F-doped silica low-index HRD with the relative index difference Δ_ring and the ring thickness t_ring. (c) Commercially available silica tube with a square hole [17]. (d) Schematic diagram of modified chemical vapor deposition (MCVD) process for GeO₂- or F-doped silica ring tube, which is the preform for HRDs in (b).

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The proposed structure is schematically shown in Fig. 1(a), which is composed of a central silica defect and HRDs within the innermost three layers, and three more air-hole layers outside the HRD structures. In contrast to prior PCFs requiring meticulous hole-diameter control in every layer, the proposed structure has the same hole diameter d and hole-to-hole pitch Λ over the entire lattice as shown in Fig. 1(a), which can significantly enhance fiber fabrication capability. The local refractive index of SLPCF was efficiently controlled by layers of HRDs by individually doping GeO₂ or F in the ring of HRD to increase or decrease its refractive index, respectively as shown in Fig. 1(b). These GeO₂/F-doped HRDs can serve as an efficient index modifying element in the SLPCF, providing a higher tolerance against structural parameter variations.

Through numerical analysis, we successfully achieved two types of dispersion management based upon the SLPCF along with HRDs: 1) ultra-flattened zero-dispersion in wavelength of 1.36~1.89μm (530nm-bandwidth) which covers the entire E, S, C, L, and U bands, 2) efficient dispersion compensation in 1.485~1.70μm covering C, L, and U bands with a high compensation ratio (CR) of 0.96~1.0 in reference to SSMF. Using the full vectorial finite element method (FEM) [19] with the anisotropic perfectly matched layer (PML) [6], we analyzed various optical properties of the proposed SLPCF such as effective modal area mode field distribution, confinement loss, and estimated splice loss as well as chromatic dispersion and its slope. Tolerance of the proposed fibers against the refractive index variation in the HRDs was also discussed.

2. Structural parameters in the proposed SLPCF with GeO₂/F-doped HRDs

In addition to conventional PCF structural parameters such as hole diameter d and hole-to-hole pitch Λ, the proposed fiber has HRD parameters: the relative index difference between the GeO₂/F-doped ring and silica, Δ_ring, and the ring thickness t_ring. Here Δ_ring of the doped ring in reference to the pure silica glass is defined as ${\Delta }_{ring}=\left(n_{doped}-n_{SiO2}\right)/n_{SiO2}\times 100$ in the unit of %, where n_doped and n_SiO2 are the refractive index of the doped ring and pure silica glass respectively. The Sellmeier equation for the refractive index of the pure silica glass [20] was used in the following calculations to take the material dispersion into account. The range of Δ_ring was from −2.0 to 2.0%, which can be readily obtainable by adjusting the mole fraction of dopants, for example, GeO₂-doped silica glass for the positive value and F-doped silica for the negative value [21, 22]. As shown in Fig. 1(a), each HRD layer can have different Δ_ring, and we will use a notation ‘Δ_Li’ for the i^th layer in the following discussions. The proposed SLPCF has six air-hole layers in total, three layers with HRDs and three without HRDs.

3. Numerical analysis and results

3.1 Numerical analysis methodology and investigated optical properties

To analyze optical properties, we use the widely accepted numerical analysis method, the full-vectorial finite element method (FEM) [16] which provides a well-proven reliability. Using FEM with anisotropic perfectly matched layer (PML) assuming the absorbing boundary condition [6], the complex effective indices n_eff are achieved from below eigenvalue equation for magnetic field, $\overrightarrow{H}\left(x,y,z,t\right)=\overrightarrow{H}\left(x,y\right)\exp \left[i\left(\omega t-\beta z\right)\right]$, propagating along z direction,.

Here ω is the angular frequency, β is propagating constant, and $k_0=2\pi /\lambda$is the free-space wave number.

The chromatic dispersion D(λ) is obtained from the real part of n_eff in the unit of ps/nm∙km as below, where λ is the wavelength and c is the speed of light in vacuum [6–9, 23].

The effective mode area A_eff which is calculated from the transverse electric field of the guided mode E_t, in the unit of μm² as in Eq. (3) [6, 8, 9]:

For practical purposes, the designed SLPCFs are supposed to be spliced to SSMF, which causes the splice loss L_splice that can be estimated by evaluating mode field overlap between two spliced fibers as below [24],

3.2 Ultra-flattened dispersion near zero

Firstly, we demonstrate the dispersion controllability of the proposed SLPCFs by realizing the ultra-flattened near-zero dispersion. The effects of the relative index difference Δ of the HRDs over the chromatic dispersion within λ = 1.3~1.9μm are summarized in Fig. 2(a) . Here we fixed other waveguide parameters: the hole-to-hole pitch Λ = 2.0μm and the air hole fraction d/Λ = 0.32, which would have small dispersion value with comparatively small dispersion slope [18], and the HRD thickness t_ring = 0.5μm. In comparison to prior PCFs, we have a uniform hole-diameter and hole-to-hole pitch over the whole cross-section.

 

Fig. 2 (a) Optimization of the relative index differences, Δ_L1, Δ_L2, and Δ_L3 of HRD for the ultra-flattened zero dispersion. (b) Enlarged view of chromatic dispersion (red) and its slope (blue) of the optimized SLPCF. (c) Modal intensity distribution of the fundamental mode at λ = 1.55μm.

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When the HRDs are absent Δ = 0%, the chromatic dispersion decreased monotonically in the spectral range of interest, with the zero dispersion wavelength near 1.4μm. Adopting the high-index HRD that forms a graded-index profile to the first HRD layer (Δ_L1), the chromatic dispersion increased over the spectral range, and especially the amount of increment was significantly larger in the longer wavelength. By enlarging the graded-index region up to the second HRD layer (Δ_L2), we could improve the flatness of the chromatic dispersion because the amount of dispersion decrement in the shorter wavelength was larger than that in the longer wavelength. Lastly, at the third HRD layer (Δ_L3), we applied the lower refractive index ring defects to make the index trench which shifts the overall dispersion value toward zero. It is also noteworthy that the amount of the dispersion change and its spectral dependence reduced with adding HRDs at the outer air-hole layer.

In the optimal HRD condition (Δ_L1 = +0.8%, Δ_L2 = +1.1%, and Δ_L3 = −0.9%) that is alike the graded-index profile with a depressed inner cladding [4], we obtained an ultra-flattened near-zero dispersion with the variation within ± 0.39 ps/nm/km over the 530nm spectral range from 1.36 to 1.89μm covering the entire E, S, C, L, and U communication bands as shown in Fig. 2(b). Its maximum dispersion slope dD/dλ is + 1.57 × 10⁻²ps/nm²/km in this spectral range. Also, the effective normalized frequency, V_eff [25] of the proposed SLPCF is less than 1.29 over the analyzed spectral range, which satisfies the single-mode regime for SLPCFs (V_eff <2.46), so that the chromatic dispersion of the fundamental mode shown in Fig. 2(a) and 2(b) is valid in the given spectral range. The modal intensity of the fundamental mode at λ = 1.55μm is shown in Fig. 2(c).

3.3 Dispersion compensation over standard single mode fiber

As another dispersion management application, we optimized the structural parameters of the proposed SLPCFs to realize wide band dispersion compensation over a SSMF. To compensate the dispersion of an SSMF D_SSMF completely at a wavelength, the dispersion of a PCF D_PCF should satisfy the below condition [7]:

To achieve a normal dispersion in the optical communication bands, the hole-to-hole pitch Λ should be adjusted to be comparable to or smaller than the wavelength of the light so that the waveguide dispersion dominates over the material dispersion [9]. Moreover, it was reported that air hole fraction needs to satisfy d/Λ>0.5 to obtain the negative dispersion slope in the C band [9]. Our SLPCF has Λ = 1.0μm and d/Λ = 0.6 to ensure both the negative dispersion and negative dispersion slope.

As shown in Fig. 3(a) , the SLPCF without HRD (Δ = 0%) had a normal dispersion, −277.8ps/nm/km at λ = 1.55μm, and its CR varied in a broad range from 0.80 to 1.17. In order to achieve more efficient dispersion compensation over a wider spectral range, we needed to further decrease the dispersion value while bring the compensation ratio to near 1. Similar to dispersion flattening, we varied the relative index differences of HRDs in the innermost three layers, Δ_Li, for i = 1, 2, and 3. The thickness of the HRDs, t_ring = 0.18μm was fixed. We summarized successive optimization process for dispersion compensation in Fig. 3(a) and 3(b).

 

Fig. 3 Optimization of the relative index differences, Δ_L1, Δ_L2, and Δ_L3, for efficient wide band dispersion compensation: (a) compensation ratio and (b) chromatic dispersion. (c) The modal intensity distribution of the fundamental mode at λ = 1.55μm.

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In contrast to dispersion flattened SLPCF (Fig. 2), we applied the low-index F-doped HRDs in the first layer (Δ_L1) to form an index trench which decrease the dispersion and let the CR toward the ideal value, 1 over the whole spectral range. Further broadening of the index trench region up to the second HRD layer (Δ_L2) drastically improved CR in λ<1.55μm, but not in λ<1.55μm. This limitation in shorter wavelength could be overcome by adding a high index GeO₂-doped HRD in the third layer (Δ_L3), while it also resulted in the larger negative dispersion. The effective index profile of the proposed SLPCF for the dispersion compensation is consistent with a W-type index profile realized in DCFs with solid core and clad [5].

With the optimal HRD parameters of SLPCF (Δ_L1 = −2.0%, Δ_L2 = −2.0%, and Δ_L3 = +2.0%), we could achieve a large negative dispersion of −352.3ps/nm/km at λ = 1.55μm, and a near unity CR of 0.96 to 1.0 in the spectral range from 1.485 to 1.700μm that includes entire C, L, U band and a part of S band extending over 250nm-bandwidth. Also, its dispersion value at a targeted wavelength, λ = 1.55μm, is −352.3ps/nm/km so that the necessary length of SLPCF is 0.0493 times shorter than that of SSMF, i.e. L_PCF/L_SSMF = 0.0493. For example, the optimal dispersion compensating SLPCF with a length of 4.93km can compensate 100km-long SSMF, and the net dispersion for the SLPCF + SSMF would be within 0 to −0.8ps/nm/km over λ = 1.485~1.700μm. Furthermore, proposed SLPCF shows the single-mode condition with V_eff<1.58 within this targeted spectral range. The third air-hole layer with the GeO₂-doped ring structures (Δ_L3 = + 2.0%) forms the concentric outer core, but its guided mode exists below the effective cladding index.

3.4 Effective mode area and optical loss analysis

To evaluate the practicality of the optimized SLPCFs, we analyzed the optical losses and the effective mode area. The effective mode area A_eff (Eq. (3)) is mainly affected by the pitch Λ and related with the splice loss L_splice between a SSMF and a SLPCF (Eq. (4)). For the proposed SLPCFs, A_eff was calculated in the spectral range and the results are summarized in Fig. 4(a) and 4(b), respectively. At λ = 1.55μm, the effective mode area of dispersion flattened SLPCF was 10.19μm², and that of dispersion compensating SLPCF was 5.14μm².

 

Fig. 4 Estimated splice loss (red) and effective mode area (blue) for (a) ultra-flattened zero dispersion SLPCF, (b) dispersion compensating SLPCF. (c) Comparison of hole-collapsed splicing condition between the conventional SLPCF and proposed SLPCF with GeO₂/F-doped HRDs. (d) Comparison of intensity distribution at the hole-collapsed region for each case.

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As shown in Fig. 4(a) and 4(b), we have calculated two types of splice loss: 1) direct butt-coupling splice between SLPCF with air holes intact and SSMF, 2) splice between SLPCF with adiabatically collapsed air holes and SSM. For the direct butt-coupling splice, we assumed the holes of SLPCF were maintained as in Fig. 1(a). The dispersion flattened SLPCF assumed to have a direct butt coupling splice loss of 0.67dB at λ = 1.55μm, which is significantly lower than the previously reported structure [9]. The dispersion compensating SLPCF would have a rather high splice loss of 4.78dB at λ = 1.55μm because of the mode field diameter mismatch.

One of salient and differentiating features of the proposed SLPCF is that we can reduce the splice loss furthermore by adiabatically collapsing GeO₂/F-doped HRDs, which still sustain a good optical guidance even in the case of complete air hole closure. The numerically achieved splice loss for SLPCFs with adiabatically collapsed air holes is plotted in a dotted line in Fig. 4(a) and 4(b). In these cases, the estimated splice loss substantially reduced to 0.15dB and 1.34dB at λ = 1.55μm for the ultra-flattened dispersion and dispersion compensation respectively. When the air-holes are completely collapsed in prior SLPCFs, the collapsed cross-section becomes plain pure silica as shown on the top of Fig. 4(c), where the core guidance is no longer present. In contrast, the proposed SLPCF with GeO₂/F-doped HRDs can provide the core-guiding capability even at the fully collapsed region as well as in the adiabatically collapsed region. As a result, we could obtain splice losses at least an order of magnitude lower than 12.8dB-loss in the prior PCF [12].

In a PCF with either a small air hole fraction d/Λ or a sub-wavelength pitch Λ, it is known that the confinement loss inevitably increases [27]. The proposed ultra-flattened dispersion near-zero SLPCF having d/Λ corresponds to the former case, and dispersion compensation SLPCF having Λ = 1.0μm corresponds to the latter case so that they all suffer from a relatively high confinement loss for the structure with six air hole layers as shown in Fig. 1(a). However, this obstacle can be overcome by either increasing the number of air-hole layers or inserting few air-hole layers with larger diameter just outside the proposed structure [6, 24, 27]. Adding a layer of large air hole can be realized by selectively controlling the capillary pressure as in prior reports [28, 29]. For example, by inserting a larger air-hole layer (d/Λ = 0.95) outside the 6th air-hole layers of the ultra-flattened zero dispersion SLPCF, the confinement loss was estimated (Eq. (3)) to be 8.97 × 10⁻⁷dB/m at λ = 1.55μm. For dispersion compensating SLPCF, we could achieve the confinement loss of 3.14 × 10⁻⁴dB/m at λ = 1.55μm by inserting two larger air-hole layers, which is similar to the level of Rayleigh scattering loss (~0.15dB/km at λ = 1.5μm [30]). In both cases, the insertion of few larger air-hole layers hardly affects the dispersion value: as a reference, 0.015ps/nm/km-variation in dispersion for the ultra-flattened zero dispersion SLPCF and 0.0025-variation in compensation ratio for the dispersion compensating SLPCF. Moreover, the first cladding mode of both cases has more than 500 times larger loss level than the fundamental mode.

3.4 Tolerance to the variation in the index differences of HRDs

Even if the MCVD process has been already matured, it would be useful in an experimental point of view to consider impacts of deviation in the index differences from the optimum values over the optical properties of the proposed SLPCF. It is known that the numerical apertures (NA) of commercialized optical fibers made by the MCVD process can have a maximum tolerance of 10% [31], which results in 4~5% of tolerance in Δ. Here, we discuss impacts of deviation in the HRD’s relative index differences of the ring δΔ_Li up to ± 5% (Fig. 5(a) and 5(b)) and deviation in the air-hole fraction δ(d/Λ) up to ± 2% (Fig. 5(c) and 5(d)). Also, we further considered the combined effect of deviation in the air-hole fraction, δ(d/Λ) = ± 1% along with δΔ_Li = ± 2% (Fig. 5(e) and 5(f)).

 

Fig. 5 Impacts of deviation in (a), (b) the relative index differences δΔ_Li of HRD, (c), (d) in the air hole fraction δ(d/Λ), and (e), (f) the combined deviation in δΔ_Li and δ(d/Λ) on chromatic dispersion of ultra-flattened zero dispersion SLPCF and dispersion compensation ratio of the dispersion compensating SLPCF respectively.

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As in Fig. 5(a), the chromatic dispersion of ultra-flattened SLPCF maintained its value and flatness within 0 ± 0.92ps/nm/km over the same flattened spectral range (λ = 1.36~1.89μm) even with δΔ_Li = ± 5%. Especially in S and C band, the standard deviation in dispersion was less than 0.04ps/nm/km. When applying variation in structural parameter δ(d/Λ) = ± 2% (Fig. 5(c)), the optimized SLPCF kept its dispersion within 0 ± 1.69ps/nm/km and its standard deviation over the analyzed spectral range was 0.715ps/nm/km. By comparison, the recent report on dispersion flattened SLPCF [8] has the average standard deviation of 0.865 over the same spectral range only with ± 1% structural deviation. Furthermore, even if we consider the effect of δ(d/Λ) = ± 1% and δΔ_Li = ± 2% simultaneously (Fig. 5(e)), the average standard deviation is 0.408ps/nm/km, which is again much less than the previous study.

In the case of dispersion compensation SLPCF, the compensation ratio was maintained within a range of 0.952<CR<1.002 over the spectral range from 1.485 to 1.700μm against δΔ_Li (Fig. 5(b)). The relative length for a compensating module (L_PCF/L_SSMF) did not exceed 0.502. As described in Fig. 5(d) and 5(f), the optimized dispersion compensating SLPCF successfully maintained CR within 0.951<CR<1.003 even if assuming the deviation in structural parameter δ(d/Λ) = ± 2% or the combined effect of δ(d/Λ) = ± 1% and δΔ_Li = ± 2%, which proved the robustness of the proposed SLPCFs against structural parameters deviations caused by fabrication imperfection.

4. Conclusion

A new hollow ring defect doped with either GeO₂ or F was introduced in a uniform square lattice photonic crystal in order to provide a flexible and efficient design platform to manage the chromatic dispersion over a wide spectral range. Due to unique ring structure, the proposed defect could vary the effective index locally flexibly while maintaining the uniform hole parameters over the whole cross-section. The proposed structure provided an effective solution alternative to prior techniques requiring meticulous control of the hole size and pitch. Near zero chromatic dispersion of 0 ± 0.39ps/nm/km was achieved over a 530nm-wide spectral range, 1.36 to 1.89μm, covering the entire E, S, C, L, and U communication bands. Highly efficient dispersion compensation with respect to standard single mode fiber was also obtained with a compensation ratio close to the ideal value of 1, 0.96 to 1.00 over a 220nm-wide spectral range covering the C, L, and U band. The proposed SLPCFs provided intrinsically low splice loss since the GeO₂ or F doped-ring structures maintained very good beam guidance along the core region even in the case when all the holes were completely collapsed during splicing. Low splice loss of 0.15dB and 1.34dB at λ = 1.55μm were predicted for the ultra-flattened dispersion and dispersion compensation respectively. The proposed structure also showed an excellent robustness in the chromatic dispersion characteristics against variations in the structural parameters, which can enhance actual fiber fabrication feasibility.