**3. State of the art**

and is not available for the 1st zero CD. An overview of the obtainable ZDWs as a function of

Both core size and the radius of holes exhibit significant influence on the location of both ZDWs. Increasing core size tunes the ZDW to longer λ. The increase in air percentage in the cladding could extend the origination of the 1st ZDW at longer wavelengths and the 2nd ZDW at shorter

Some compensating fibres could work at short spectrum of λs. When a compensating fibre works at a certain λ, optical pulses can be transmitted using one or – in general – low amount of WDM channels. As it has already been mentioned above, MOFs are suitable for designing the compensating fibres because they allow huge index contrast and offer many parameters to be optimized (core size, hole radius, lattice pitch, amount of rings of holes) in order to

Currently, flat CD fibres could for example be referred to as near zero CD transmission medium. In Ref. [22], a DCF made of pure (undoped) material is shown. Advanced DCFs with flat CD property over broad spectrum of working λs could be found in Ref. [23]. An interesting fibre with CD parameter close to zero with CD fluctuations less than 0.5 ps in S to L bands is shown in Ref. [24]. Results were obtained by careful optimization of holes in particular rings. To reduce losses, it is often required avoiding doping the MOF's core. Instead, one could consider using octagonal [25] and decagonal [26] lattice, or more

Recent analyses concentrate on the low, flat CD over telecommunication λs for any telecom wavelength compensation. The value of CD and the width of the working range is a compro‐ mise. A DSF could have large negative CD value, but for medium-wide wavelength range, or, acceptable CD parameter, but designed for very broad spectrum of λ. Currently, the strongest demand is to design DCFs mainly for C and L bands, where modern WDM systems can work.

The idea how to obtain flat CD properties is to tune the 1st ZDW to shorter λs and the 2nd ZDW to longer λs, having little negative CD over the whole bandwidth at the same time. Practically, predicted CD diagram is a wide parabola. Considered properties are related to the diameter of the core and the radius of the holes in particular rings. Both were found to have different impact on the origination of each ZDW. There are requirements to locate the zero CD point at the λ of 1.55 μm or, in general, in the C-band [8]. Changing the air filling fraction and lattice pitch is not the only idea that could be used to optimize CD. For example elliptical holes instead of circular holes exhibit potential low and very stable CD properties, i.e. 0.6-1 ps/nm/km in the

Finally, there are a few examples that are worth to be noticed: a DCF in Ref. [27], with CD parameter being –1350 ps/nm/km at 1.55 μm; another one in Ref. [28] has CD of –440 to –480

MOF's structure was shown in Ref. [21].

optimize waveguide dispersion [2].

sophisticated lattice arrangements [2].

range from 1 μm to 1.9 μm [3].

ps/nm/km at the band of 1.5-1.62 μm [3].

λs (i.e. less negative CD is obtained for larger air-filling fraction).

102 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

**2.3. Dispersion flattened fibres and wideband fibres [2][3]**

The employment of the considered fibres is mainly in high-speed transmission systems using wavelength division multiplexing and signal recovery, offering transmission rates of even more than 1 Tb/s, where it is proved that non-optimized CD could destroy the pulse spreading, but as well it could generate some nonlinear effects, such as four wave mixing, among others. Although the properties of the fibres significantly differ from one another, the techniques used for optimized CD could be systematized so that one could design a fibre with desired CD value and slope at a specific λ.

Understanding of the mechanisms governing CD tailoring is necessary not only permits for the fibre design, CD suppression and avoidance, but also to predict the potential manufactur‐ ing tolerances. There are a few techniques used to obtain the expected CD and we describe them in more details in the following sections.

## **3.1. Fibre designs [1]**

One of the techniques applied is to dope the fibre's core. By inserting small doped inclusion in a MOF's core, the index contrast with result in much larger mode confinement. Because the waveguide dispersion is related to it and because the length of propagated light waves is of the range that is comparable to the core size, the inserted region will be responsible for very low waveguide dispersion component.

There is another technique that could be used to optimize dispersion, it assumes to inject some liquids into the microstructured holes. In Ref. [29], the 4th ring of holes is doped by using liquids for this purpose. But liquid's properties are strongly dependent on temperature, which means that constant temperature must be ensured. This means that such a fibre could for example be used for some experiments in laboratory conditions (e.g. sensors), but they can't be used in outdoor installations of telecommunication systems [1].

Germanium dioxide is very often used to raise the effective index. In Ref. [13] and [30], the idea how to use it to dope the microstructured fibre's core is described in details. But this solution is not perfect in terms of effective area of the mode which is rather small in the doped fibres, as well as from the perspective of increased attenuation caused by material absorption, which is high for germanium dioxide [1].

Another technique to be described and systematized is the one using air holes located close to the fibre centre. The 1st mode could be confined within the core by the index contrast, and the effective diameter of a mode is then usually small enough to control CD. Stronger confinement could be obtained in a MOF using other type of a lattice, like for example octagonal, decagonal or spiral.

An alternative way, how to control the effective index, is to have core defects. A "defect core" is usually understood as one allowing light coupling to some other regions by some imper‐ fection that is introduced intentionally. This technique uses weak interactions between the core-guided mode and a mode localized in an intentionally introduced defect of the crystal.

Finally, the idea of concentric cores could be considered. What feature in dual core technique is responsible for low CD? It is large value of neff produced by significant asymmetry between the two cores where light is coupled. Light is propagated in both cores concurrently. Dual cores are responsible for the phase matching at a certain λ, and consequently at this wavelength large CD parameter is obtained.

Low value of CD parameter is possible in concentric cores. They could be obtained by removing [7],[10] or reducing [7],[31] the air fraction in some rings of holes [1].

There are many aspects that could influence resultant CD; it could be coupling modes and the phase matching wavelength, at which CD control is possible, could be obtained in many ways, for example by fibre bending. We found these conclusions as interesting, because many works assume that the most important consequence of bending a fibre is increased attenuation. It could be shown that large negative value of CD parameter could be the accompanying process.

#### **3.2. Extension of transmission band towards infrared — Fluoride fibres [32]**

Dealing with the structural parameters of MOFs is a solution to find optimal CD. They could be enhanced by very careful optimizing the material composition. For example fluorides could be used as additives, but they could also be used as a background material instead of silicon dioxide, which could be evaluated as significant progress in the field [32]. The fluorides have low effective index, broad range of λs and low attenuation. One could consider the use of BaF2, CaF2 and more advanced ZBGA (ZrF4-BaF2-GdF3-AlF3) or ZBLAN (ZrF4-BaF2-LaF3-AlF3- NaF) [32]. In addition, because those materials are composite, it is possible to change their composition in a compound in order to obtain neff value or its slope in the investigated area of λ. Last but not least, these materials are transparent in infrared region (as well as in the telecommunication bands) which makes them interesting for some applications in sensors. The properties of fluorides allow using them in MOFs [33]. They would lower the effective index of the cladding or the index at the second, larger concentric core. It is usually combined with doping the main core by using germania. This idea could be considered as a significant progress in fibre optic technology. Then a fibre with a W-type profile or refractive index could be created. For example, the core doped by using germanium dioxide would exhibit the neff of about 1.48 and concurrently the cladding index doped by using F-SiO2 would exhibit the index of about 1.43. Similar solution (in general modified W-type index profile fibre) could be employed in systems with WDM [32],[34].

Material and structural parameters are key features in order to obtain exact CD at each λ, but the accuracy and quality of production could become a critical issue, especially in submicron fibres whose small holes with the diameter being less than one micron is problematic for some manufacturing technologies. All this combined with the application of new materials that should be processed in different way compared to siliceous materials could result in both attenuation and CD properties far from the expectations [32].

At the telecommunication band both ZBLAN and ZBGA showed similar attenuation proper‐ ties like the one of silicon dioxide, but their CD properties significantly differ [35], in addition fluorides have different mechanical properties and they have to be fused at different temper‐ ature than temperature for silica. The theoretical attenuation could even be of 0.01 dB/km, in the range from 0.3 μm to 4.3 μm. The neff in telecom band is 1.47 – 1.52 [32, 36].

## **4. Design approaches**

Finally, the idea of concentric cores could be considered. What feature in dual core technique is responsible for low CD? It is large value of neff produced by significant asymmetry between the two cores where light is coupled. Light is propagated in both cores concurrently. Dual cores are responsible for the phase matching at a certain λ, and consequently at this wavelength

Low value of CD parameter is possible in concentric cores. They could be obtained by removing

There are many aspects that could influence resultant CD; it could be coupling modes and the phase matching wavelength, at which CD control is possible, could be obtained in many ways, for example by fibre bending. We found these conclusions as interesting, because many works assume that the most important consequence of bending a fibre is increased attenuation. It could be shown that large negative value of CD parameter could be the accompanying process.

Dealing with the structural parameters of MOFs is a solution to find optimal CD. They could be enhanced by very careful optimizing the material composition. For example fluorides could be used as additives, but they could also be used as a background material instead of silicon dioxide, which could be evaluated as significant progress in the field [32]. The fluorides have low effective index, broad range of λs and low attenuation. One could consider the use of BaF2, CaF2 and more advanced ZBGA (ZrF4-BaF2-GdF3-AlF3) or ZBLAN (ZrF4-BaF2-LaF3-AlF3- NaF) [32]. In addition, because those materials are composite, it is possible to change their composition in a compound in order to obtain neff value or its slope in the investigated area of λ. Last but not least, these materials are transparent in infrared region (as well as in the telecommunication bands) which makes them interesting for some applications in sensors. The properties of fluorides allow using them in MOFs [33]. They would lower the effective index of the cladding or the index at the second, larger concentric core. It is usually combined with doping the main core by using germania. This idea could be considered as a significant progress in fibre optic technology. Then a fibre with a W-type profile or refractive index could be created. For example, the core doped by using germanium dioxide would exhibit the neff of about 1.48 and concurrently the cladding index doped by using F-SiO2 would exhibit the index of about 1.43. Similar solution (in general modified W-type index profile fibre) could be

Material and structural parameters are key features in order to obtain exact CD at each λ, but the accuracy and quality of production could become a critical issue, especially in submicron fibres whose small holes with the diameter being less than one micron is problematic for some manufacturing technologies. All this combined with the application of new materials that should be processed in different way compared to siliceous materials could result in both

At the telecommunication band both ZBLAN and ZBGA showed similar attenuation proper‐ ties like the one of silicon dioxide, but their CD properties significantly differ [35], in addition fluorides have different mechanical properties and they have to be fused at different temper‐

[7],[10] or reducing [7],[31] the air fraction in some rings of holes [1].

104 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

**3.2. Extension of transmission band towards infrared — Fluoride fibres [32]**

large CD parameter is obtained.

employed in systems with WDM [32],[34].

attenuation and CD properties far from the expectations [32].

To describe propagation of light in a given optical glass, we could consider the constant of propagation of a wave that is related to the phase variation [32]:

$$k = \left(\beta - j\alpha\right) \tag{1}$$

The real part *β* and the imaginary part *ω* could be specified [37]:

$$\beta = \alpha \sqrt{\frac{\mu \varepsilon}{2}} \left[ \sqrt{1 + \left( \frac{\sigma}{\alpha \varepsilon} \right)^2} + 1 \right] \tag{2}$$

$$\alpha = \alpha \sqrt{\frac{\mu \varepsilon}{2}} \left[ \sqrt{1 + \left( \frac{\sigma}{\alpha \varepsilon} \right)^2} - 1 \right] \tag{3}$$

Where *ω* is angular velocity, *σ* is conductivity, a *µ* is permeability and *ε* is permittivity. Because MOFs are dielectric it could be assumed that:

$$\left(\frac{\sigma}{a\kappa}\right) << 1\tag{4}$$

Both real and imaginary part of propagation constant in (3) could be simplified to a form:

$$
\beta = \alpha \sqrt{\mu \varepsilon} \tag{5}
$$

$$\alpha = 1 \tag{6}$$

MOFs could be described by an effective index *neff*, which expresses "the refractive index at the boundary of two entities", which in this particular case are the core and the cladding [32]:

$$m\_{\rm eff} = \frac{\beta}{k\_0} \tag{7}$$

*β* is real part of the constant of propagation of core, *k0* is the constant of propagation in vacuum. Propagation constant depends on *λ*, it could be written that:

$$k = \frac{2\pi}{\lambda} \tag{8}$$

Group velocity that integrates optical waves in an "envelope" could be considered to describe its velocity. It refers to CD and is less than the speed of electromagnetic wave in vacuum [32]:

$$\mathbf{v}\_g = \frac{\hat{\mathcal{C}}\boldsymbol{\alpha}}{\hat{\mathcal{C}}k} = \frac{\mathbf{c}}{\sqrt{1 + \frac{\boldsymbol{\alpha}^2 \hat{\mathcal{X}}^2}{4\pi^2 \boldsymbol{c}^2}}} \tag{9}$$

Then group delay *τg* could be calculated as time necessary for the propagation of wave to certain distance with velocity of *vg* (9):

$$
\pi\_g = \frac{1}{\nu\_g} \tag{10}
$$

In general, chromatic dispersion *CD* is known as the dependence of group delay *τg* on the wavelength *λ*, at which the signal is transmitted [38]:

$$\text{CDD}\left(\lambda\right) = \frac{\partial \tau\_{\text{g}}\left(\lambda\right)}{\partial \left(\lambda\right)}\tag{11}$$

In simulations it is more suitable to use material dispersion equation described by using Sellmeier approximation. CD equation suitable for Sellmeier approximations is following [39]:

$$\text{C}D\left(\mathcal{A}\right) = -\frac{\mathcal{A}}{c} \frac{\partial^2 \text{Re}\left[\ln\_{\text{eff}}\right]}{\partial \mathcal{X}^2} \tag{12}$$

Designs of MOFs could be calculated by Finite-Difference Frequency Domain (FDFD) method. The distribution of light could be calculated by discretizing electric (13) and magnetic (14) field, it is described in Refs. [32, 40], and [41]:

$$\left(\nabla\_t^2 + k\_0^2 \varepsilon\_r\right) \mathbf{E}\_t + \nabla\_t \left(\varepsilon\_r^{-1} \nabla\_r \varepsilon\_r \cdot \mathbf{E}\_t\right) = \beta^2 \mathbf{E}\_t \tag{13}$$

$$\left(\nabla\_{\iota}^{2} + k\_{0}^{2}\varepsilon\_{r}\right)\mathbf{H}\_{\iota} + \varepsilon\_{r}^{-1}\nabla\_{\iota}\varepsilon\_{r} \times \left(\nabla\_{\iota} \times \mathbf{H}\_{\iota}\right) = \beta^{2}\mathbf{H}\_{\iota} \tag{14}$$

*k0* is wavenumber in free space, *β* is real part of the propagation constant, *εr* is dielectric constant as in (5). The FDFD method uses the discretization scheme shown in Ref. [42][32].

FDFD is used in many commercial simulators of mode distribution and neff calculators. In most of them, it is possible to use user friendly graphical interface, but when a very accurate result is demanded, many iterations ought to be done by creating a loop using an appropriate scripting language. We use a mode solver from the Lumerical Inc. When the fibre's cross section is proposed and the simulation parameters and monitors are set, the simulation is run [3].

Creation of realistic models of sophisticated optical glasses is an interesting feature. To introduce materials to the simulator, we used the Sellmeier approximation [32, 43],[44]:

$$m^2\left(\mathcal{A}\right) = A + \sum\_{i} \frac{B\_i \mathcal{A}^2}{\mathcal{A}^2 - C\_i^2} \tag{15}$$

where *A*, *Bi* , *Ci* are coefficients referring to index *n*. As a result, the wavelength evolution of refractive index could be plot by using the Sellmeier equation. The coefficients for the expand‐ ed version of Sellmeier equation (16) could found in [32][45]:

$$m^2 - 1 = \frac{B\_1 \mathcal{X}^2}{\left(\mathcal{X}^2 - C\_2^2\right)} + \frac{B\_3 \mathcal{X}^2}{\left(\mathcal{X}^2 - C\_4^2\right)} + \frac{B\_5 \mathcal{X}^2}{\left(\mathcal{X}^2 - C\_6^2\right)}\tag{16}$$

where B1,3,5 C2,4,6 are material constants and λ is wavelength.

### **5. Exemplary results**

*β* is real part of the constant of propagation of core, *k0* is the constant of propagation in vacuum.

<sup>2</sup> *<sup>k</sup>* p

l

Group velocity that integrates optical waves in an "envelope" could be considered to describe its velocity. It refers to CD and is less than the speed of electromagnetic wave in vacuum [32]:

> 2 2 2 2 1 4

*c*

*c*

w l

+

Then group delay *τg* could be calculated as time necessary for the propagation of wave to

1

In general, chromatic dispersion *CD* is known as the dependence of group delay *τg* on the

( ) ( )

t l

In simulations it is more suitable to use material dispersion equation described by using Sellmeier approximation. CD equation suitable for Sellmeier approximations is following [39]:

2

Designs of MOFs could be calculated by Finite-Difference Frequency Domain (FDFD) method. The distribution of light could be calculated by discretizing electric (13) and magnetic (14) field,

*c* l

( ) ( ) 2 2 <sup>1</sup> <sup>2</sup> *t r t t r rr t t* <sup>0</sup> *k*

 ee

2 *Re neff*

 b- Ñ + + Ñ Ñ =× *E EE* (13)

l

¶ é ù ë û = - ¶

*<sup>g</sup> CD*

l

( )

l

*CD*

e

( )

l

*g g v*

t

p

<sup>=</sup> (8)

= (10)

¶ <sup>=</sup> ¶ (11)

(9)

(12)

Propagation constant depends on *λ*, it could be written that:

106 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

*g*

*k*

¶ = = ¶

w

*v*

certain distance with velocity of *vg* (9):

it is described in Refs. [32, 40], and [41]:

wavelength *λ*, at which the signal is transmitted [38]:

In this section, we present some selected fibre designs covering the wide area of telecom photonic fibres for CD tailoring, such as DCFs with large negative CD parameter, wideband fibres or slope compensating fibres, or fibre designs suitable for CD prevention, such as near zero CD fibres. We employ different design approaches and optimize them based on para‐ metric investigation.

#### **5.1. Dual core dispersion compensating fibre with large negative dispersion parameter [1]**

A concentric core DCF based on microstructured optical fibre is proposed [1]. There is an assumption for the proposed structure to keep the 1st mode in a central core over the wide working spectrum of λ and in order to obtain low CD at 1.55 μm, and with low theoretical losses. The refractive index is optimized by making smaller the hole diameter in the 2nd ring. The design is specified in Table 1, the cross section is shown in Figure 1 [1].


**Table 1.** Parameters of a dual core MOF for CD suppression [1].

**Figure 1.** Structural parameters and the cross section of a dual core MOF for CD suppression purposes [1].

CD of the considered fibre is shown in Figure 2. Obtained CD parameter was -1460 ps/nm/km. The designed fibre work in the C-band, and the lowest CD is at 1.55 μm. Theoretical losses are 3.8 10-4 dB/cm. It could be concerned as one of the advantages of the MOF, when we have a look at losses presented for some other fibres [1].

In order to obtain very accurate results, a number of iterations were performed to find the most suitable structural parameters of a fibre. One could conclude that smaller holes in the 3rd ring, and concurrently larger radius of holes in the 2nd ring could result in improved CD properties (Figure 3). Optimized solution could be found in Table 2 [1].

**Figure 2.** Chromatic dispersion in a dual core MOF for CD suppression at 1.55 μm [1].

**Quantity [unit] Value Value Value Value Value** Lattice pitch Λ [μm] 1.55 1.52 1.50 1.48 1.45 Minimum D at 1.55 μm [ps/nm/km] -1010 -1180 -1300 -1460 -1690 Core index [-] 1.44 1.44 1.44 1.44 1.44 Theoretical loss [dB/cm] 9·10-10 7·10-10 3.8·10-7 3.8·10-4 7·10-2 Full width at half maximum [nm] 142 132 129 125 122 Hole diameter d1 [μm] 1.33 1.34 1.314 1.27 1.23 Hole diameter d2 [μm] 0.55 0.54 0.574 0.63 0.67 Hole diameter d3 [μm] 1.35 1.35 1.2 1.04 0.92 Number of rings [-] 6 6 6 6 6

108 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

**Figure 1.** Structural parameters and the cross section of a dual core MOF for CD suppression purposes [1].

CD of the considered fibre is shown in Figure 2. Obtained CD parameter was -1460 ps/nm/km. The designed fibre work in the C-band, and the lowest CD is at 1.55 μm. Theoretical losses are 3.8 10-4 dB/cm. It could be concerned as one of the advantages of the MOF, when we have a

In order to obtain very accurate results, a number of iterations were performed to find the most suitable structural parameters of a fibre. One could conclude that smaller holes in the 3rd ring, and concurrently larger radius of holes in the 2nd ring could result in improved CD properties

**Table 1.** Parameters of a dual core MOF for CD suppression [1].

look at losses presented for some other fibres [1].

(Figure 3). Optimized solution could be found in Table 2 [1].


**Table 2.** Specification of parameters for dual core MOF for CD suppression [1].

**Figure 3.** Dispersion in proposed dual core microstructured DCF [1].

In Figure 3, it could be observed that λ at which there is minimum CD is function of normalized hole diameter d/Λ of the holes; CD is mostly sensitive to the holes located in the 2nd ring. To move the operating λ towards 1.5 μm, one has to increase the size of the holes in this ring. Concurrently, the value of lowest CD is less. Summarizing, a one km long section of the designed DCF should be sufficient to tailor CD in a network that is created by using 75 km long conventional SMF with CD parameter being 17 ps/nm/km, (this value is in agreement with the recommendation of ITU-T for SMFs). The insertion losses of such a CD compensator are about 0.04 dB [1].

#### **5.2. DCF with optimized dispersion slope [3]**

Another design is done with the scope on fibres that could compensate CD in each channel of a system using wavelength multiplexing. It means that the fibre should be wideband, and its CD mustn't be flat, but it should have exactly opposite CD to CD of a fibre that is used in a WDM system. The proposal is shown in Figure 4 and in Table 3 [3].


**Table 3.** Structural parameters of the HNPCF with dispersion evolution opposite to one in conventional MOF [3].

**Figure 4.** Cross section of a MOF with negative CD parameter [3].

**Figure 5.** Reversed CD slope of the proposed DCF [3].

In Figure 3, it could be observed that λ at which there is minimum CD is function of normalized hole diameter d/Λ of the holes; CD is mostly sensitive to the holes located in the 2nd ring. To move the operating λ towards 1.5 μm, one has to increase the size of the holes in this ring. Concurrently, the value of lowest CD is less. Summarizing, a one km long section of the designed DCF should be sufficient to tailor CD in a network that is created by using 75 km long conventional SMF with CD parameter being 17 ps/nm/km, (this value is in agreement with the recommendation of ITU-T for SMFs). The insertion losses of such a CD compensator

Another design is done with the scope on fibres that could compensate CD in each channel of a system using wavelength multiplexing. It means that the fibre should be wideband, and its CD mustn't be flat, but it should have exactly opposite CD to CD of a fibre that is used in a

**Table 3.** Structural parameters of the HNPCF with dispersion evolution opposite to one in conventional MOF [3].

are about 0.04 dB [1].

**5.2. DCF with optimized dispersion slope [3]**

**Figure 4.** Cross section of a MOF with negative CD parameter [3].

WDM system. The proposal is shown in Figure 4 and in Table 3 [3].

110 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

neff [-] 1.47 Hole diameter [μm] 3.64 Core index [-] 1.48 Normalized hole diameter d/Λ [-] 0.58 Core size [μm] 4.46 Number of rings [-] 5

From Figure 5 it could be concluded that CD is reversed CD of conventional MOFs. The fibre is wideband. It could be optimized in terms of larger values of CD parameters, for example to exactly match CD evolution of conventional fibres. One shall also pay attention to the fact that larger d/Λ would result in multimode operation [3].

#### **5.3. Dispersion flattened fibres and wideband dispersion compensating fibres [1][3]**

A MOF with flat CD is demanded in design of transmission fibres (not suppression fibres) where it is d to have identical CD at each λ channel. Such a fibre could be doped in the core by using germania. The proposed core diameter could be 7.4 μm. In Figure 6, the 1st mode is kept in the core and the fibre is single-mode. Manufacturing of the MOF's core could be challenging, because of huge doping area; the fibre has large mode area. The proposed geometry is described in Table 4 [3].


**Table 4.** Structure parameters for HNPCF flattened CD curve [3].

Obtained CD property could be evaluated as flat and oscillating around the value of -0.025 ps/ nm/km. In Figure 7, comparison of CD in conventional MOF and the designed fibre is shown [3].

**Figure 6.** Wideband large mode area MOF with near zero flattened CD [3].

**Figure 7.** Dispersion in conventional microstructured fibre and proposed highly nonlinear MOF [3].

Wideband fibre could exhibit negative CD parameter, too. We propose a MOF for suppression in the band of 1.25-1.7 μm. We do not use any additives. Mode confinement is done by optimizing the geometry of the core and the normalized hole diameter d/Λ, the accepted view is that the core size should be small. In the 1st ring, there is d1/Λ=0.9. The proposal could be found in Figure 8. Doping the core could additionally improve CD properties, but it would surely worsen attenuation. One should pay attention to the fact that optimization of CD shall be done in the context of attenuation properties. Optimizing one parameter and ignoring another is unacceptable in high-speed transmission system fibres [1].

**Figure 8.** Cross section of designed microstructured fibre for wideband suppression of CD [1].

**Figure 6.** Wideband large mode area MOF with near zero flattened CD [3].

112 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

**Figure 7.** Dispersion in conventional microstructured fibre and proposed highly nonlinear MOF [3].

Wideband fibre could exhibit negative CD parameter, too. We propose a MOF for suppression in the band of 1.25-1.7 μm. We do not use any additives. Mode confinement is done by Having larger holes in the 1st t ring, d1, is responsible for lower CD. On the contrary, by making smaller all the other holes (d) results in increasing dispersion. In Table 5, optimized structure is shown and its cross section is in Figure 9 [1].


**Table 5.** Specification of parameters for wideband CD suppressing MOF [1].

**Figure 9.** Resultant flattened negative value of CD over broad band of λs 1.25-1.7 μm (lattice pitch is a parameter) [1].

Concerning the values of CD, the best design is the one for Λ=0.62 μm, for which CD is -2259 ps/nm/km, obtained at 1.55 μm. Theoretical losses are 4.8 dB/cm. The optimized fibre is the one with Λ=0.70 μm, where CD parameter is -1580 ps/nm/km at 1.55 μm. Theoretical loss is lowered to 9.1 10-6 dB/cm. In this case it is possible to compensate dispersion of about 90 km of standard SMF.

#### **5.4. Fluoride doped dispersion compensating fibres [32]**

The considered fibre is a MOF employing the idea of a W-profile fibre with the core doped by using BaF2 (nBaF2 = 1.468 to raise its refractive index) and containing three holes in the 1st ring doped by using CaF2, nCaF2 = 1.426 to reduce effective cladding index. Increased index contrast is responsible for enhanced CD, as in eq. (12). As for example, models optical glasses could be expressed using coefficients shown in Table 6.


**Table 6.** Sellmeier coefficients for the fluoride additives used in investigated MOF [32].

The structural and material properties are in Table 7. Large radius of CaF2 holes and the low core size are necessary for flat CD (Figure 10). CD is -413 ~ -415 ps/nm/km at 1.4 - 1.65 μm. Theoretical loss is 1.75 10-4 dB/cm. CaF2 doped holes affect the CD (Figure 11). The larger are the doped holes, the lower is resultant CD [32].

**Figure 10.** Fluoride-doped W-profile fibre's cross section for infrared and telecom band [32].

**Figure 9.** Resultant flattened negative value of CD over broad band of λs 1.25-1.7 μm (lattice pitch is a parameter) [1].

Concerning the values of CD, the best design is the one for Λ=0.62 μm, for which CD is -2259 ps/nm/km, obtained at 1.55 μm. Theoretical losses are 4.8 dB/cm. The optimized fibre is the one with Λ=0.70 μm, where CD parameter is -1580 ps/nm/km at 1.55 μm. Theoretical loss is lowered to 9.1 10-6 dB/cm. In this case it is possible to compensate dispersion of about 90 km

The considered fibre is a MOF employing the idea of a W-profile fibre with the core doped by using BaF2 (nBaF2 = 1.468 to raise its refractive index) and containing three holes in the 1st ring doped by using CaF2, nCaF2 = 1.426 to reduce effective cladding index. Increased index contrast is responsible for enhanced CD, as in eq. (12). As for example, models optical glasses could be

The structural and material properties are in Table 7. Large radius of CaF2 holes and the low core size are necessary for flat CD (Figure 10). CD is -413 ~ -415 ps/nm/km at 1.4 - 1.65 μm. Theoretical loss is 1.75 10-4 dB/cm. CaF2 doped holes affect the CD (Figure 11). The larger are

**Material B1 C2 B3 C4 B5 C6** Silica 0.6961663 0.0684043 0.4079426 0.11624 0.897479 9.89616 BaF2 0.6433560 0.0577890 0.5067620 0.109680 3.82610 46.3864 CaF<sup>2</sup> 0.5675888 0.0502636 0.4710914 0.10039 3.848472 34.6490

**Table 6.** Sellmeier coefficients for the fluoride additives used in investigated MOF [32].

of standard SMF.

**5.4. Fluoride doped dispersion compensating fibres [32]**

114 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

expressed using coefficients shown in Table 6.

the doped holes, the lower is resultant CD [32].


**Table 7.** Design specification of investigated W-profile MOF with doped holes in the 1st ring [32]

The air holes influence the dispersion slope. Larger d/Λ would limit the confinement losses. An interesting feature is that we use fluorides, ZBGA or ZBLAN, not just to dope the cladding, but we propose to use it as a background material instead of silicon dioxide (Figure 12). As a result, optimization of CD is more flexible. Last but not least, temperature and mechanical properties of such a fibre wouldn't be worse [32].

Right combination and the composition of additives is responsible for optimized properties of a compound. Let us consider zirconium (Zr), barium (Ba), gadolinium (Gd) or aluminium (Al) that are the compounds of ZBGA material: ZrF4 – BaF2 – GdF3 – AlF3 [32].The neff of ZBGA material could be represented by modified Sellmeier equation (17) [46], using the coefficients in Table 8 [45].

**Figure 11.** Wavelength evolution of CD with hole diameter of CaF2 as a parameter [32].

$$m\left(\mathcal{X}\right) = \frac{A}{\mathcal{X}^4} + \frac{B}{\mathcal{X}^2} + C + D\mathcal{X}^2 + E\mathcal{X}^4 \tag{17}$$


**Table 8.** Sellmeier coefficients for the fluoride-background ZBGA fibre [45].

ZBGA has larger neff comparing with silicon dioxide. Tailored CD is expected [32].

Parametric sweep performed for the hole diameter showed that flatten CD is possible, which is shown in Figure 6. For hole diameter being equal to 0.44 μm, the evolution of CD is similar to one obtained for silicon dioxide. Larger neff of the ZBGA fibre requires larger radius of airholes in the cladding to tailor CD [32].

BaF2 could be used in the core and CaF2 in the 1st ring, ZBGA as a background material (Figure 13). CD parameter is then possible within the range of -435 ~ -438 ps/nm/km and over the entire telecommunication band. At the same time, attenuation properties are not worse than those obtainable in fibres with silica as a background material [32].

#### **5.5. Submicron dispersion compensating fibres with modified geometry [2]**

One of the trends is to use a lattice with the pitch being less than one micrometre (so-called submicron lattice). Then, by careful adjustments of the diameter of holes (d1-d3 in Figure 14),

**Figure 12.** CD in SiO2 and ZBGA used as a background material [32].

( ) 2 4

ZBGA at 25°C 2.98316e-6 3.39740e-3 6.81447e-3 -1.20276e-3 -5.48085e-6

Parametric sweep performed for the hole diameter showed that flatten CD is possible, which is shown in Figure 6. For hole diameter being equal to 0.44 μm, the evolution of CD is similar to one obtained for silicon dioxide. Larger neff of the ZBGA fibre requires larger radius of air-

BaF2 could be used in the core and CaF2 in the 1st ring, ZBGA as a background material (Figure 13). CD parameter is then possible within the range of -435 ~ -438 ps/nm/km and over the entire telecommunication band. At the same time, attenuation properties are not worse than those

One of the trends is to use a lattice with the pitch being less than one micrometre (so-called submicron lattice). Then, by careful adjustments of the diameter of holes (d1-d3 in Figure 14),

l

 l

= + ++ + (17)

*CD E*

4 2 *A B*

**Material A B C D E**

ZBGA has larger neff comparing with silicon dioxide. Tailored CD is expected [32].

**5.5. Submicron dispersion compensating fibres with modified geometry [2]**

l l

**Figure 11.** Wavelength evolution of CD with hole diameter of CaF2 as a parameter [32].

116 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

*n* l

**Table 8.** Sellmeier coefficients for the fluoride-background ZBGA fibre [45].

obtainable in fibres with silica as a background material [32].

holes in the cladding to tailor CD [32].

**Figure 13.** Proposed MOF with ZBGA material used as a background compared to a fibre with SiO2 as a background [32].

where d1 is hole diameter in the 1st ring of holes, one could obtain nearly zero CD for the λs from the window 1.25-1.7 μm. Hole diameter and lattice pitch are submicron. Then, high confinement of light waves and strong waveguide dispersion is possible. The geometrical parameters of the fibre are in Table 9 and in Figure 14 [2]. CD and loss are simulated for the 1st light mode. Confinement losses are 6.10-6 dB/km (Figure 15). Wavelength evolution of CD is shown in Figure 15 [2].

**Figure 14.** Cross section of a near zero CD flattened MOF with modified rings [2].

**Figure 15.** Near zero flattened CD for wideband utilization in telecommunications [2].

Obtained CD parameter is from -7.7 ps/nm/km to 3.1 ps/nm/km at the band of (1.25, 1.7 μm) and its average value is around 0.51 ps/nm/km. In the C-band, it is 1.35±0.46 ps/nm/km. For both C and L bands CD is 0.12±1.32ps/nm/km [2]. Results for parametric iterations for air fraction changed in three most internal rings are shown in Figures 15 and 16. From the results summarized in Table 10 it could be concluded that the fibre has CD slope of -0.09 ps.nm-2.km-1 is obtained for little variation of d1 from other holes and is suited to the slope of conventional ITU-T G.657 fibres. The slope is less than 0.09 ps.nm-2.km-1 [2].


**Table 9.** Geometrical parameters of a fibre with nearly zero flattened CD [2].

**Figure 16.** Optimized CD slope by varying the hole diameter in the 1st ring [2].


**Table 10.** Optimized dispersion slope by varying diameter of holes in the 1st ring [2].

**Figure 15.** Near zero flattened CD for wideband utilization in telecommunications [2].

**Figure 14.** Cross section of a near zero CD flattened MOF with modified rings [2].

118 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

conventional ITU-T G.657 fibres. The slope is less than 0.09 ps.nm-2.km-1 [2].

Obtained CD parameter is from -7.7 ps/nm/km to 3.1 ps/nm/km at the band of (1.25, 1.7 μm) and its average value is around 0.51 ps/nm/km. In the C-band, it is 1.35±0.46 ps/nm/km. For both C and L bands CD is 0.12±1.32ps/nm/km [2]. Results for parametric iterations for air fraction changed in three most internal rings are shown in Figures 15 and 16. From the results summarized in Table 10 it could be concluded that the fibre has CD slope of -0.09 ps.nm-2.km-1 is obtained for little variation of d1 from other holes and is suited to the slope of

Larger CD slope is here obtained by adjusting the value of d3 (see Table 11, Fig. 16). At d1 being fixed, linear wavelength evolution of CD is possible. It becomes nonlinear when the slope exceeds -1 ps.nm-2.km-1. The obtained property could be applied in CD suppression performed on a fibre with the slope of 0.5 ps.nm-2.km-1 [2].


**Table 11.** Large CD slope obtained by tuning the hole size in the 3rd ring [2].
