1 Introduction

One of the main serious limitations in long-haul optical transmission networks and wavelength division multiplexing (WDM) systems is the chromatic dispersion. It widens the optical pulses, limits the information carrying capacity and limits the traveling distance. To mitigate this undesirable impact on the long-haul and high-speed optical transmission systems, the chromatic dispersion must be compensated. Broadband dispersion compensation techniques, such as dispersion-shifted fibers (DSFs) and dispersion compensating modules (DCMs), are two effectively techniques that used to eliminate the chromatic dispersion present in optical transmission systems. DSFs are simply single mode fiber (SMF) with the best 1550 nm performance like G.653. The zero-dispersion point is shifted to the 1550 nm window to match the minimum attenuation window of the fiber. However, theses DSFs can’t be used in DWDM applications because it severe from non-linear effects like four-wave mixing (FWM). Another type similar to the DSF is the non-zero dispersion shift fiber (NZ-DSF) like G.655. The NZ-DSF has a small dispersion in the 1550 nm window. However, when the bit rate increases over 2.5 Gbit’s, even the low dispersion of the NZ-DSFs needs to be compensated using Dispersion Compensation Fiber (DCF) (Wandel et al. 2001).

DCFs with high negative dispersion are used to equalize the positive dispersion caused by the conventional SMFs in the telecommunication bands (Gruner-Nielsen et al. 2005). One of the drawbacks of using DCFs is that total link attenuation is increased. The added attenuation must be compensated with additional gain in amplifiers, which degrades the signal to noise ratio (SNR) and increases the cost of the system. To avoid the excessive losses coming from the confinement losses, DCFs should be as short as possible. This requires the negative dispersion coefficient of the DCF to be as large as possible. Hence, the DCF performance is described by a figure of merit (FOM) which is determined by dividing the fiber dispersion by its loss (dispersion/attenuation) and has a unit of \({\text{ps}}/({\text{dB}}.{\text{nm}}):\)

$${\text{FOM}}=|{\text{D}}|/\mathrm{\alpha }$$
(1)

One of the limitation of the traditional DCF, it has low FOM in the range of 150—200 ps/(dB.nm) (Wandel, et al. 2001). To achieve a very high negative value of D, the core of the compensating fiber has to be doped with relatively high \({{\text{GeO}}}_{2}\) compared with the conventional fibers. Unfortunately, this doping increases the total fiber loss (\(\mathrm{\alpha }\)).

Recently, Photonic-Crystal-Fibers (PCFs) fascinate researchers due to their design flexibility that overcomes the limitations of conventional fibers. In particular, the presence of air holes in the microstructure fibers and the ability to change their sizes, shapes and positions in the cladding provided a great degree of freedom for precise tuning of the optical properties of the PCF. The remarkable properties of PCFs include endlessly single-mode, large mode area, solitons propagation, light guidance in air, higher birefringence, anomalous dispersion properties, and nonlinear effects (Birks et al. 1997; Knight et al. 1998; Knight et al. 2000; Wadsworth et al. 2000; Cregan et al. 1999; Ortigosa-Blanch et al. 2000; Broderick et al. 1999). These special properties have led to the development of several applications in the fields of optical communications, nonlinear optics, ultrafast optics, supercontinuum generation, sensing applications, optical coherence tomography, optical metrology, spectroscopy, high power technology, multi-wavelength generation, etc. (Peucheret et al. 2003; Benabid et al. 2002; Travers et al. 2011; Monro et al. 2001; Diddams et al. 2000; Limpert et al. 2003; Pinto et al. 2011). PCFs are promising candidates as dispersion compensators as they provide relatively higher dispersion values than those of conventional dispersion compensating fibers. Our PCF design here enhances the signal quality and gives much higher FOM than conventional DCF, it can compensate the dispersion of over 100 times its length of standard fiber G.652 with minimum attenuation.

In this paper we design suitable broadband PCF structures that can be used as DCF to enhance network performance and capacity in WDM applications. WDM systems are divided into two patterns: coarse-WDM (CWDM) and dense-WDM (DWDM). CWDM is used for the short-range communications (< 100 km) using the wavelengths from 1260 nm up to 1625 nm with a channel spacing of 20 nm (G.694.2). On the other hand, DWDM is used for the long-range communications (> 100 km), where DWDM needs to fully utilize the gain of an erbium-doped fiber amplifier (EDFA) in the C-band (1530–1565 nm). DWDM puts a large number of channels within the C-band with a tight wavelength spacing (0.8 nm) according to the ITU standard (G.694.1). We numerically investigate the dispersion compensation characteristics for three designs of dispersion compensation photonic crystal fiber (DC-PCF) over the six telecommunication optical bands O-, E-, S-, C-, L- and U- (wavelengths ranging from 1260 to 1675 nm) (Paschotta 2022). All the three designs have a large negative dispersion coefficient. The first design has a matched RDS value with standard SMF and can be used in DWDM dispersion compensation applications. The second design has a little modification in the core, and have a higher birefringence, which is needed for sensing and PMD applications. The third design has the exact opposite dispersion curve to the conventional SMF, and has zero dispersion at 1310 nm, which will be more suitable in CWDM applications.

2 Design and theory

2.1 The proposed structure

Figure 1a shows a cross section of the proposed Hexagonal DC-PCF for designs A, B and C. Design A has a circular core while designs B and C have elliptical ones. The structure is based on a pure silica (SiO2) background and a hexagonal arrangement of six air-hole rings with different diameters across the cladding region. The PCF geometrical parameters contribute most significantly to this dispersion compensation capability are as follows: the air hole diameters d, d1, and d2, the pitch \(\Lambda\) for hole-to-hole spacing, and the pitch \({\Lambda }{\prime}\) (2/3 of \(\Lambda\)) for the second air-hole ring. The displacement in the \(\pm {\text{x}}\) directions for the first two rings’ holes is used in designs B and C to make the core elliptical. The fundamental optical field distribution through DC-PCF for X- and Y-polarization modes of design A at the wavelength 1550 nm is shown in Fig. 1b. The suggested designs are kept as simple as possible to simplify the fabrication process and to be cost effective. They can be easily fabricated by employing the stack and draw technique, which is used with the commercially available PCFs (Chillcce et al. 2006).

Fig.1
figure 1

a Cross sectional of the proposed PCF showing pitch Λ, air-hole channels with diameter d, d1, d2 and background solid silica. b The fundamental optical field distribution of different modes in X-polarization and Y-polarization at the operating wavelength 1550 nm

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2.2 The DC-PCF parameters

Silica, used in the PCF, is a dispersive material. So, its refractive index (n) is wavelength dependent. It is calculated as a function of wavelength (λ) through Sellmeier equation (Hossen et al. 2018):

$${n}^{2}\left(\uplambda \right)=1+ \frac{{B}_{1}{\uplambda }^{2}}{{\uplambda }^{2}- {C}_{1}}+ \frac{{B}_{2}{\uplambda }^{2}}{{\uplambda }^{2}- {C}_{2}}+ \frac{{B}_{3}{\uplambda }^{2}}{{\uplambda }^{2}- {C}_{3}}$$
(2)

The Sellmeier coefficients \({B}_{1}, {B}_{2},{B}_{3}\), \({C}_{1}, {C}_{2}\) and \({C}_{3}\) for the silica are as follows: \(0.6961663, 0.4079426, 0.8974794, 4.67914826\times {10}^{-3} {\mathrm{\mu m}}^{2}, 1.35120631 \times {10}^{-2} {\mathrm{\mu m}}^{2}\) and \(97.93400 {\mathrm{\mu m}}^{2},\) respectively (Hossen et al. 2018).

The modal effective refractive index \({n}_{{\text{eff}}}\) of the PCF is used to describe the refractive index of the (air-glass) cladding, assumes a value in between that of glass and air. The \({n}_{eff}\) is a flexible parameter as it can be easily varied by changing the design geometry of the core and cladding structure opening the door to engineer the optical properties of the PCF and tune the design parameters. It used to calculate the PCF parameters such as chromatic dispersion \(D\left(\lambda \right)\), confinement loss, birefringence, effective area and effective V-parameter. The total chromatic dispersion \({D}_{c}\left(\lambda \right)\) is due to the addition of material \({D}_{M}\left(\lambda \right)\) and waveguide dispersion \({D}_{W}\left(\lambda \right)\) components. It can be calculated from the following differential equation, where \(c\) is the speed of light (Agrawal 2012):

$$D_{c} \left( \lambda \right) = - \frac{\lambda }{c} \cdot \frac{{d^{2} {\text{Re}} \left[ {n_{eff} } \right]}}{{d\lambda^{2} }}$$
(3)

In the PCF, the confinement loss \({\alpha }_{loss}\) indicates the light confinement ability of the PCF within its core region and is defined as (Hossen et al. 2018):

$${\alpha }_{loss}=8.686\times \frac{2\pi }{\uplambda }\mathrm{ Im}\left({n}_{eff}\right) \times {10}^{6 }{\text{dB}}/{\text{m}}$$
(4)

The birefringence \(B\) indicates the polarization dependence of the refractive index in a medium. In optical fibers, it is determined by the following equation (Ortigosa-Blanch et al. 2000):

$$B=\left|{n}_{{\text{effx}}}-{n}_{{\text{effy}}}\right|$$
(5)

where \({n}_{{\text{effx}}}\) and \({n}_{{\text{effy}}}\) are the effective refractive index for X and Y polarized fundamental modes, respectively. The effective mode area \({A}_{eff}\) is calculated from the following equation (Arif et al. 2019):

$${A}_{eff}=\frac{{\left(\iint {\left|E\left(x, y\right)\right|}^{2}dxdy\right)}^{2}}{\iint {\left|E\left(x, y\right)\right|}^{4}dxdy}$$
(6)

where \(E\left(x, y\right)\) is the electric field distribution of the fundamental mode in the fiber. Nonlinearity is closely related to the effective area of the fiber. The nonlinear coefficient, \(\gamma\), is calculated by the following equation (Arif et al. 2019):

$$\gamma =\left(\frac{2\pi }{\uplambda }\right)\left(\frac{N}{{A}_{eff}}\right)$$
(7)

where, \(N\) is the nonlinearity index which is about \(2.19 \times {10}^{-20} {m}^{2}/W\) at 1550 nm for SiO2 (Le et al. 2021).

In PCF, the number of guided modes can be determined by the normalized frequency or V-parameter which is defined as follows (Haque et al. 2014):

$${V}_{PCF}=\frac{2\pi }{\lambda }\Lambda {F}^\frac{1}{2}\sqrt{{n}_{eff}^{2}-{n}_{air}^{2}}$$
(8)

where \({n}_{air}\) is the refractive index of the air, and \(F\) is the filling ratio fraction. The higher-order mode cut-off can be associated with a value of \({V}_{PCF}=\pi\) (Mortensen et al. 2003; Nielsen and Mortensen 2003), and the condition of operating a PCF as a single mode is \({V}_{PCF}\le \pi\). If \({V}_{PCF}>\pi\), then the fiber will act as a Multimode fiber (MMF).

For a DCF to be used for multichannel WDM applications, this requires a large negative dispersion coefficient for \(D\left(\lambda \right)\), over a broad range of wavelengths. The dispersion slope of PCF is calculated by differentiating the dispersion \(D\left(\lambda \right)\) with respect to the wavelength \(\lambda\). This is done in order to assess a DCF's capacity to compensate for dispersion over a wide range of wavelengths.

$$DS\left(\lambda \right)=\frac{dD\left(\lambda \right)}{d\lambda }$$
(9)

The relative dispersion slope (RDS), which is the ratio of the dispersion slope \(DS\left(\lambda \right)\) to the dispersion coefficient \(D\left(\lambda \right)\) of the fiber, is another often used factor to assess the capabilities of DCF (Habib et al. 2013a):

$$RDS=\frac{DS\left(\lambda \right)}{D\left(\lambda \right)}$$
(10)

We use the RDS value to judge DC satisfaction over a range of wavelengths. Once the RDS value of the DCF and SMF is matched, the design of the broadband DCF is accomplished. For a PCF, to compensate the accumulated dispersion of the SMFs over a range of wavelengths, this condition must be satisfied:

$${RDS}_{SMF}={RDS}_{PCF}$$
(11)

For example, according to the recommendations of the International Telecommunication Union ( ITU), the dispersion coefficient \({D}_{SMF}\left(\lambda \right)\) and the dispersion slope \({DS}_{SMF}\left(\lambda \right)\) for the G.652 SMF at the wavelength 1550 nm is \(17\mathrm{ ps}/({\text{nm}}.{\text{km}})\) and \(0.06\mathrm{ ps}/({{\text{nm}}}^{2}.{\text{km}})\) respectively (G.652). So, we can calculate the \({RDS}_{SMF}\) that exhibits \(3.529 \times {10}^{-3 }{{\text{nm}}}^{-1}\) at \(\uplambda =1550\mathrm{ \mu m}\). Once the RDS value of the DC-PCF is close to this value, an acceptable design of the DCF is achieved. The effective dispersion after compensation, \({D}_{e}\left(\lambda \right)\), on the fiber link in series can be written as (Kaijage et al. 2009):

$${D}_{e}\left(\lambda \right)=\frac{{D}_{SMF}\left(\lambda \right){L}_{SMF}+{D}_{PCF}\left(\lambda \right){L}_{PCF}}{{L}_{SMF}+{L}_{PCF}}$$
(12)

where, the SMF length is \({L}_{SMF}\), the SMF dispersion coefficient is \({D}_{SMF}\left(\lambda \right)\), the PCF length is \({L}_{PCF}\), and the PCF dispersion coefficient is \({D}_{PCF}\left(\lambda \right)\). For full compensation, the value of \({D}_{e}\left(\lambda \right)\) should be zero.

Splice loss, which occurs during splicing between two fibers, is an important parameter in fiber design. We can calculate the splice loss between PCF and SMF using the following equation (Limpert et al. 2003):

$${L}_{S}=-20{{\text{log}}}_{10}\frac{2{W}_{SMF}{W}_{PCF}}{{W}_{SMF}^{2}+{W}_{PCF}^{2}}$$
(13)

where \({W}_{SMF}\) is the MFD of the SMF and \({W}_{PCF}\) is the MFD of the PCF. The small effective area causes potential difficulties in the input and output coupling of light causes a very high splice loss. However, it has been reported that we can interface tapered intermediate PCF between proposed C-PCF and SMF successfully to minimize the splice loss (Pinto et al. 2011). So that, the proposed PCF can be interfaced to SMF using existing technology without any complications.

3 Simulation results and discussion for design A

The practical design for a broadband PCF with a large negative dispersion coefficient can be achieved by selecting appropriate design parameters. As shown in Fig. 1a, there are four design parameters (d, d1, d2 and Λ). These parameters are adjusted separately and their influences on the dispersion curve, effective area, RDS and confinement loss are investigated until reaching a satisfying performance for our suggested structure.

We start with the air-hole diameter of the second ring, d1, and test the PCF properties by increasing d1/Λ from 0.39 to 0.45 while the other parameters are kept constant as Λ = 1 μm, and d/Λ = d2/Λ = 0.96. At 1550 nm wavelength, the dispersion, \({D}_{M}\) and the confinement loss, \({\alpha }_{loss}\), are illustrated in Fig. 2a. The dispersion, \({D}_{M}\) varies from − 1600 to − 2009, then − 1608 ps/(nm.km) again. We have the best value for dispersion at d1/Λ between 0.41 and 0.42, which is − 2010 ps/(nm.km). Also, as d1/Λ varies from 0.39 to 0.45, the confinement loss, \({\alpha }_{loss}\), at 1550 nm monotonically decreases from 0.01727 dB/m to 0.006618 dB/m. The air-hole diameter d1 can be used to tune the desirable RDS as shown in Fig. 2b. The RDS at 1550 nm matches the SMF of \(0.0036 {{\text{nm}}}^{-1}\) at about d1/Λ = 0.443. Thus, for the wide band DC-PCF we select d1/Λ value to be 0.443.

Fig. 2
figure 2

a Dispersion and confinement loss with the effect of d1 variation while other parameters are kept constant (Λ = 1 μm, and d/Λ = d2/Λ = 0.96). b Effect of varying d1 diameter on RDS. c Dispersion and Confinement Loss with the effect of d variation. (All results are taken at λ = 1550 nm). d Dispersion and Confinement Loss with the effect of ± 1% to ± 2% variation in pitch size Λ. e Dispersion curve and effects of changing Λ with fabrication tolerance of ± 1% and ± 2% (f) The effective refractive index, \({n}_{eff}\), as a function of wavelength

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Figure 2c shows the effect of the hole diameter variation d/Λ on \({D}_{M}\) and \({\alpha }_{loss}\) at 1550 nm, while the other parameters are kept constant as d2 = d, Λ = 1 μm, and d1/Λ = 0.443. It can be seen that with the increase of the air-hole diameter d, the value of the negative dispersion coefficient increases and the confinement loss decrease. We could obtain a higher negative dispersion value and a much lower confinement loss if the air-hole diameter d was further increased, but this would have had a negative effect on the mode field diameter (MFD) as well as the nonlinearity coefficient, and make the fabrication process more difficult. The variation in RDS with varying d is not significant and can be neglected. We choose d/Λ value to be 0.96.

From Fig. 2d, the value of the negative dispersion coefficient decreases appreciably as Λ increases due to the greater index difference and vice versa. So, we need to keep the pitch as small as possible. However, there is a trade-off between the dispersion coefficient and the confinement loss: as Λ decreases dispersion coefficient becomes much higher with a noticeable increase in the confinement loss. So, we choose the pitch size to be in the middle (Λ = 1 μm). From these results, the selected parameters for design A are Λ = 1 μm, d/Λ = d2/Λ = 0.96 and d1/Λ = 0.443. The characteristics of design A such as dispersion curve, effective area, RDS, and confinement loss will be discussed in the following subsections.

3.1 Dispersion in design A

The variation of the chromatic dispersion, \({D}_{M}\left(\lambda \right)\) against wavelength of design A is depicted in Fig. 2e. The negative dispersion coefficient decreases appreciably as the wavelength increases roughly till the wavelength 1600 nm. Also, it is found that it is possible to have a large negative dispersion coefficient of about − 100 to − 1870 ps/(nm.km) in the wavelength range 1260–1650 nm, and about − 1716 ps/(nm.km) at the operating wavelength 1550 nm. Thus, the proposed PCF can compensate the accumulated dispersion of the standard fiber which has dispersion coefficient of about 17 ps/nm.km (G.652) at 1550 nm of about 100 times its length. For fabrication purposes, we need to evaluate the impact of changing the pitch size in dispersion accuracy. It is known from the experimental results that in a standard fiber draw and during the fabrication process, tolerances in the structural parameters can occur up to 2% (Kaijage et al. 2009; Agrawal 2013). Figure 3a also shows the effect of ± 1% to ± 2% variation in the pitch Λ. It can be concluded that we can tolerate ± 2% variation in the global diameter with no considerable effect on the performance of DC-PCF, which is good for fabrication.

Fig. 3
figure 3

a Variation of RDS of the Design A as a function of wavelength at (Λ = 1 μm, d/Λ = 0.96, d2/Λ = 0.9, d1/Λ = 0.443). b Design B dispersion properties of PCF for varying the x-displacement while keep other parameters constant when keeping the other geometrical parameters (d and Λ) constant

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3.2 Confinement loss in design A

The effective refractive index \({n}_{eff}\) curve against wavelength for design A is presented in Fig. 2f. It is shown that as the wavelength increases, \({n}_{eff}\) decreases. This indicates that at higher wavelengths the confined power inside the core decreases and that the confinement loss increases. For design A, the confinement loss equals 0.007692 dB/m at 1550 nm and remains less than 0.01 dB/m for all the wavelengths below 1562 nm. The confinement loss values for this design are little high, but it is in the acceptable level for the transmission systems (Le et al. 2021; Xu et al. 2009). However, we can provide stronger confinement ability and achieve much lower confinement loss by changing the d2 value. For example, from Table 1, the confinement loss at 1550 nm can reach \(4.144\times 1{0}^{-5}\) dB/m for d2/Λ = 0.75. On the other hand, there is a trade-off between the confinement loss and the dispersion; the dispersion coefficient increases quickly when the air hole diameter d2 increases, while the confinement loss decreases. We can use new design parameters (Λ = 1 μm, d/Λ = 0.96, d2/Λ = 0.9, d1/Λ = 0.443) as design A′, which gives us much better enhancement in confinement loss for about 0.001108 dB/m (or just 1.1 dB/km) with a lower dispersion coefficient of about − 1070 ps/(nm.km).

Table 1 Effects of varying d2/Λ on DC-PCF properties while other parameters (d/Λ = 0.96 and Λ = 1 μm) are kept constant at λ = 1550 nm
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3.3 Birefringence in design A

Due to the symmetric core of the proposed PCF, the value of the birefringence is very little which is about \(5.11\times {10}^{-8}\) at 1550 nm. This low value is acceptable in telecommunication transmission and in chromatic dispersion compensation. For example, the standard SMF is basically circularly symmetric with a little Polarization Mode Dispersion (PMD) value. It doesn’t maintain the polarization state of the guided mode along their length, so there is a small birefringence value. However, in some situations we cannot predict the birefringence properties of the fiber because of the uncontrolled variation in its shape due to small twists, stresses and bends (Agrawal 2013).

3.4 Effective area in design A

Figure 4a depicts the effective area and the nonlinearity for design A as a function of wavelength, which is calculated from Eqs. 5 and 6. The effective Area tends to increase with increasing the wavelength and results in lowering the nonlinearity coefficient. The optical effective area is almost \(1.48 {\mathrm{\mu m}}^{2}\) and the corresponding nonlinear coefficient is about \(74 {W}^{-1} {km}^{-1}\) at λ = 1550 nm. The small effective area causes potential difficulties in the input and output coupling of light causing a very high splice loss (Xu et al. 2009). However, it has been reported that we can interface tapered intermediate PCF between proposed PCF and SMF successfully to minimize the splice loss (Xiao et al. 2007). So, the proposed PCF can be interfaced to SMF using existing technology without any complications.

Fig. 4
figure 4

a Effective area and nonlinear curve as a function of wavelength for design A (b) V-parameter curve as a function of wavelength for design A (c) Effective dispersion curve as a function of wavelength for design A (Λ = 1 μm, d/Λ = d2/Λ = 0.96 and d1/Λ = 0.443). d Chromatic dispersion curve against wavelength with different values of the x-displacement for design B (e) Effect of varying the x-displacement on Birefringence for design B (f) RDS curve with wavelength for design B (Λ = 1 μm, d/Λ = 0.96, d2/Λ = 0.9, d1/Λ = 0.4 and x ± 0.15 μm)

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3.5 V parameter

The V parameter curve is shown in Fig. 4b, we see that the proposed design A behaves as a SMF over the entire telecommunication optical bands as the cut-off wavelength is at about 1232 nm. Also, the effective dispersion after compensating the positive dispersion of 100 km of conventional SMF by 1 km of the proposed DC-PCF for Design A is shown in Fig. 4c. The maximum value of the effective dispersion in the C-band after the compensation is about ± 0.15 ps/(nm.km), and at 1550 nm the effective dispersion is about zero. We would like to note that the effective dispersion should be lower than ± 0.8 ps/(nm.km) to compensate for a 40 Gbps signal (Belahlou et al. 2002). Also, we have a maximum variation of ± 2 ps/(nm.km) in the effective dispersion over the E-, S-, C- and L- bands.

3.6 DC-PCF design A and the broadband dispersion compensation

Figure 3a shows the variation of RDS of the proposed PCF as a function of wavelength. The RDS value of the proposed DC-PCF Design A for the optimum design parameters: Λ = 1 μm, d/Λ = 0.96, d1/Λ = 0.443 at 1550 nm wavelength is 0.0036 nm − 1, which is perfectly match to SMF of 0.0036 \({{\text{nm}}}^{-1}\). As a result, the proposed PCF is capable enough to compensate broadband dispersion especially over the C-band which refers to the wavelength range 1530–1565 nm, which corresponds to DWDM applications and the amplification range of erbium doped fiber amplifiers (EDFAs) (Paschotta 2022). Our design improves the signal quality. With the dispersion value of -1716 ps/(nm.km) and the silica attenuation value by only 0.4 dB/km at 1550 nm, we can calculate the FOM of this PCF to a high value of 4290 ps/(dB.nm). This FOM value is much higher than conventional DCFs that have an FOM in the range of 150–200 ps/(dB.nm). (Wandel, et al. 2001).

4 Simulation results and discussion for designs B and C

Design A gives a good dispersion compensation and an acceptable confinement loss but with low birefringence. For some applications we need to attain a higher birefringence value and a lower confinement loss. We can use the birefringence value to eliminate the effect of PMD in SMF in the same way the DCF is used to equalize the positive dispersion caused by the conventional SMFs. These birefringent fibers control the polarization state of the propagating light and eliminate the PMD. We can achieve this by adding some asymmetric design to the DC-PCF core. Furthermore, highly birefringent with a high sensitivity PCFs have been investigated for various sensing applications (Arif et al. 2019; Cameron and Anumula 2006). The high birefringence value is essentially required for sensing application and PMD effect elimination. To improve these properties, designs B and C are introduced. They are identical to design A except that we make the core elliptical by shifting the first two rings’ holes in the right and left sides with a displacement in the ± x direction, as shown by the red dashed holes in Fig. 1a. The shift in design B is 0.15 μm and for design C is 0.2 μm. Simulation results show that, the elliptical core enhances the birefringence, and provide strong confinement ability. So, designs B and C may have lower dispersion coefficient but better birefringence properties.

Figure 3b also shows the effect of x displacement variation on the dispersion and Relative Dispersion Slope (RDS) of the DC-PCF at λ = 1550 nm. It is shown that with different ellipticities (due to displacement); the dispersion decreases with the increase in the displacement. In addition, the displacement has an effect on RDS. From the results, it can be seen that displacement ellipticity can be used to tune the desirable RDS so that it can match that of the SMF.

Figure 4d shows the effect of the displacement x on the chromatic dispersion. It shows that with different ellipticities (due to displacement), the magnitude of the dispersion decreases with the increase in the displacement. It is possible to obtain large negative dispersion coefficient of − 49 to − 1007 ps/(nm.km) for wavelengths ranging from 1250–1650 nm. In particular, the negative dispersion at 1550 nm is − 807.8 ps/(nm.km) for x ± 0.15 μm. From Fig. 4e the birefringence increases with increasing x. The elliptical design of the core causes an increase in the linear birefringence properties, which is suitable for Polarization Mode Dispersion applications. The proposed design exhibits a high birefringence value of about \(1\times {10}^{-2}\) at 1550 nm and x displacement =  ± 0.15 μm. The displacement ellipticity can be used to tune the desirable RDS, so that it can match that of the SMF value of 0.0036 nm − 1 at 1550 nm. At this point, we would like to list the selected parameters of design B: Λ = 1 μm, d/Λ = 0.96, d2/Λ = 0.9, d1/Λ = 0.4 and x displacement ± 0.15 μm. Figure 4f shows the effect of the wavelength variation on the RDS for design B. The RDS at 1550 nm matches the SMF of \(0.0036 {{\text{nm}}}^{-1}\) for broadband dispersion compensation.

With the help of the V parameter, as shown in Fig. 5a, design B will support only a single mode rather than multimode within the entire telecommunication optical bands. Figure 5b shows the variation of the effective dispersion with wavelengths after compensating the positive dispersion of 47.5 km of conventional SMF by 1 km of the proposed Design B. The maximum value of the effective dispersion after the compensation is about ± 0.15 ps/(nm.km) in C-band and about zero at 1550 nm. As mentioned before, the effective dispersion should be lower than ± 0.8 ps/(nm.km) to compensate for a 40 Gbps signal (Belahlou et al. 2002). According to dispersion value of design B, the length required to compensate the accumulated dispersion of the SMF is about 2 times the Design A length but with much lower confinement loss. Figure 5c shows the variation of the confinement loss of design B as a function of the displacement x. It shows clearly that the confinement loss decreases smoothly with the increase in displacement. It reaches about \(3.94\times {10}^{-4}\) dB/m for x =  ± 0.15 μm at wavelength 1550 nm.

Fig. 5
figure 5

a V parameter curve with wavelength for design B (b) The effective dispersion curve after compensating 47.5 km of SMF by 1 km for design B (c) The variation of the confinement loss of the proposed DC-PCF at 1550 nm for Design B as a function of the displacement x (d) The variation of chromatic dispersion against wavelength for design C parameters (Λ = 1 μm, d/Λ = 0.96, d2/Λ = 0.9, d1/Λ = 0.65 and x displacement ± 0.2 μm) (e) V parameter curve with wavelength for design C (f) The effective dispersion curve after compensating 13.3 km of SMF by 1 km for design C

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Design B can be used to compensate the dispersion of SMF for both DWDM and CWDM systems. However, we should note that most of CWDM systems use the standard SMF G.652 with a total chromatic dispersion that passes through zero around 1310 nm (G.652). To fully nullify the positive and negative dispersion caused by the standard SMFs, the chromatic dispersion of the DC-PCF should be engineered to be the exact opposite of the dispersion in the fiber link. To achieve this, we demonstrate design C which is similar to design B with the following parameters: (Λ = 1 μm, d/Λ = 0.96, d2/Λ = 0.96, d1/Λ = 0.65 and x displacement ± 0.2 μm). Figure 5d shows the chromatic dispersion for this design. It is found that the proposed design C shows the exact opposite of the SMF dispersion curve. It has positive dispersion before 1310 nm, a negative dispersion after 1310, and a zero dispersion in the 1310 nm. Figure 5e depicts the V parameter for design C. Design C will support only a single mode within the five-telecommunication optical bands for CWDM systems. The negative dispersion at 1550 nm is − 228.9 ps/(nm.km), which is smaller than preceding values, but very sufficient to CWDM systems for short-range communications. The effective dispersion after compensating the positive dispersion of 13.3 km of conventional SMF by 1 km of the proposed Design C is shown in Fig. 5f. It has two zero values at 1310 nm and 1550 nm, with a variation of about ± 1.7 ps/(nm.km) through the CWDM, O- to L- optical bands.

5 Comparison between different designs

Table 2 shows the design parameters for the three proposed designs A, B and C. Table 3compares between these proposed designs and other published designs with respect to the dispersion coefficient, birefringence, nonlinearity, RDS, and confinement loss at operating wavelength of 1550 nm. From Table 3 it is clearly noticed that our proposed PCF designs \({\text{A}}\), \({{\text{A}}}{\prime}\) and B have better dispersion coefficient than other designs and can compensate the dispersion from (50 times for B) up to (100 times for A) of the conventional SMF G.652D length. Respectively, it is clearly indicating that our structures are better for dispersion compensation applications and makes them a promising cost-effective candidate for commercial use.

Table 2 The best parameters values of design A, B and C
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Table 3 Comparison between properties of the proposed DC-PCFs and other DC-PCFs at 1550 nm wavelength
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Also, the designs have a matched RDS values with an acceptable confinement loss for DWDM applications, and have a higher non-linear co-efficient for nonlinear applications. Moreover, Design B have a higher birefringence property, which is widely used in sensing and PMD applications. So, the proposed designs can be used for fiber optic transmission system as well as sensing. Design C has a new feature as it is engineered for CWDM applications by having the exact opposite dispersion curve of the conventional SMF with zero dispersion at the wavelength 1310 nm. All three designs cover the six-telecommunication optical bands O-, E-, S-, C-, L- and U- bands (wavelengths ranging from 1260 to 1675 nm).

6 Conclusion

In this paper, we present three designs for a hexagonal dispersion compensation photonic crystal fiber (DC-PCF). All three designs cover the six-telecommunication optical bands O-, E-, S-, C-, L- and U- bands (wavelengths ranging from 1260 to 1675 nm). Design A demonstrates a chromatic dispersion coefficient of − 1716 ps/(nm-km) at the wavelength 1550 nm, which allows a section of Design A of a length L to compensate the dispersion of a G.652D standard fiber of a length 100L. The RDS of the suggested fiber is closely matched with standard SMF. We also successfully achieved an effective dispersion of ± 0.15 ps/(nm.km) over the C-band with a small confinement loss. Also, we can provide confinement loss of about 0.001108 dB/m with a lower dispersion coefficient of about − 1070 ps/(nm.km) by a slight change in one of the holes’ diameters d2. Thus, the proposed Design A is a suitable choice for broadband dispersion compensation in DWDM high bit rates optical transmission systems. The elliptical core of Design B causes an increase in the birefringence properties, and exhibits a high birefringence value of about \(1\times {10}^{-2}\) at 1550 nm, which is suitable for sensing and PMD applications. With a little modification in the hole diameter of the second ring, Design C can be engineered to have the exact opposite dispersion curve of the conventional SMF with zero dispersion at the wavelength 1310 nm, which will be more suitable in CWDM applications.