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Design of Silica Multimode Optical Fibers with Extremely Enlarged Core Diameter for Laser-Based Multi-Gigabit Short-Range Optical Networks^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Related Works

_{i}for the preform refractive index profile approximated by a simple power function and some optimal α-parameter that is also the best for a multimode regime. Here, local profile-grade parameter correction is performed by reproducing the optimal for the total mode staff DMD diagram—the DMD value distribution on the corresponding mode orders. It is formed by the corresponding modification of the initial DMD diagram of the basic model of optical fibers under overfilled launching conditions, which is the “worst case” for bandwidth, while centralized launching is well known to be the simplest method of implementation under field conditions for bandwidth improvement of a fiber optic link with MMFs operating in a few-mode regime [1,5,10].

#### 2.2. Design of an LDMDF-Graded Refractive Index Profile

_{REF}. The designed LDMDF structure weakly satisfies the guiding optical waveguide approximation. It contains a fused silica core doped by germanium and fluorine, bounded by a pure fused silica outer solid cladding. Here, unlike the known solutions, we utilize a stratification method [17] approach to describe the desired refractive index profile. As a result, the designed weakly guiding optical fiber with an arbitrary axially symmetric refractive index profile is represented in the form of a multilayered optical fiber in the core region. It is considered as a finite set on N layers, where the refractive index value stays constant:

_{k}is the refractive index of k layers (k = 0…N); n

_{max}is the maximal core refractive index; n

_{N}is the outer cladding refractive index; $\Delta =\left({n}_{max}^{2}-{n}_{N}^{2}\right)/2{n}_{max}^{2}$ is the profile height parameter; R

_{k}= r

_{k}/a is the normalized radial coordinate of the k-th layer; r

_{k}is the radial coordinate of the k-th layer; a is the designed LDMDF core radius.

_{d(j)}is the desired value of delay for the j-th guided mode LP

_{lm(j)}computed at the corresponding synthesis iteration; t

_{REF}is some reference value of mode delay that is applied for DMD diagram equalization; M is the total number of the mode components transferring a laser source excited few-mode optical signal with a normalized amplitude that is not less than A

_{j}> 0.1, and with a core power (known also as an optical confinement factor) that is not less then ${P}_{co}^{\left(j\right)}\ge 0.5$. Here, the total number of modes M taken into account depends on the following factors: (1) the designed fiber basic geometry parameters (e.g., core diameter and profile height parameter), (2) the launching conditions, (3) the emission of the initial transverse mode staff at the laser output defined by the type of source—vertical surface-emitting laser (VCSEL) or single-mode laser, e.g., Fabry–Perot laser diode (LD)/distributed feedback laser (DFB-laser), and (4) the prediction of a new guided mode of excitation, with A

_{j}> 0.1 during the following optical signal propagation over MMF, due to mode mixing and power diffusion effects provided by real fiber irregularities and its micro-/macro-bends and tensions/stress occurring under fiber optic cable manufacturing, installation, and maintenance. Unlike the known solutions, we propose setting the reference delay t

_{REF}from the range of values containing the DMD diagram formed for only M selected guided modes, with particular orders propagating in the new generation of LOMFs of Cat. OM2 + –OM4. The objective function F in Equation (4) is minimized by the Nelder–Mead simplex method, whose efficiency was demonstrated in [18,19] concerned with the design of optical fibers.

#### 2.3. Extension of Modified Gaussian Approximation

_{k}that completely describe the optical fiber refractive index profile. Therefore, a fast and simple method for the computation of both the fundamental and higher-order mode parameters propagating in MMF is required. We propose utilizing earlier on the developed modification of a Gaussian approximation [20] that is generalized and extended [21] for the evaluation of arbitrary order-guided mode dispersion parameters propagating in a weakly guiding optical fiber, with an arbitrary axially symmetric refractive index profile in the core region, bounded by a single solid outer cladding. This extended modified Gaussian approximation (EMGA) is based on a combination of “classical” Gaussian approximation [22] and a stratification method [17] for the researched MMF complicated graded refractive index profile representation. The proposed approach permits the derivation of an analytical formula for the square core mode parameter U

^{2}in the form of finite nested sums [21] from a well-known integral variational expression [22]:

_{0}= ρ

_{0}/a is the equivalent (as a result of Gaussian approximation) normalized mode field radius (MFR); ρ

_{0}is the equivalent MFR; a is the MMF core radius; l is the azimuthal mode; m is the radial mode of LP

_{lm}orders. $V={k}_{0}a{n}_{max}\sqrt{2\Delta}$ is the normalized frequency; ${k}_{0}=2\pi /\lambda $ is the wavenumber; λ is the operating wavelength; ${b}_{p}^{\left(l,m\right)}$ is the expansion factor of the Laguerre polynomial representation in the form of a finite power series [23,24]:

_{0}is the result of the numerical solution of the characteristic Equation (7) after substitution of the researched optical fiber geometrical parameters and the particular azimuthal and radial orders of the analyzed mode. In EMGA (as well as in the “classical” Gaussian approximation), R

_{0}is a basic single variational parameter, which completely describes the mode dispersion characteristics. Following the substitution of R

_{0}to the variational expression expressed in Equation (5), this permits the evaluation of the core mode parameter U, which relates with the propagation constant β by the following well-known ratio [17,20]:

_{core}as the second criterion for the identification of the “ghost” solutions:

#### 2.4. Mode Delay

#### 2.5. Mode of Chromatic Dispersion Parameter

^{2}and the normalized equivalent mode field radius R

_{0}will be obtained:

#### 2.6. Material Dispersion and Refractive Index Profile Parameters

_{i}and B

_{i}are Sellmeier’s coefficients (B

_{i}is also denoted as the resonance wavelength), which has been empirically measured for GeO

_{2}–SiO

_{2}glass [26,27] under several particular dopant concentrations. Here, we shall apply the method described in detail in the published work [28], to estimate the Sellmeier coefficients at the graded-index profile points.

_{k}lead to the following expressions:

#### 2.7. Model of a Piecewise Regular Multimode Fiber Optic Link Operating in a Few-Mode Regime under Laser-Excited Optical Pulse Propagation

_{lm}over a regular multimode fiber with length z [1]:

_{Tx}(t) is the initial pulse at the transmitter end; ${A}_{p}^{\left(0\right)}$ and α

_{p}are the starting amplitude and the mode attenuation of the p-th guided mode LP

_{lm}(p = 1...M); ${\beta}_{1}^{\left(p\right)}$ and ${\beta}_{2}^{\left(p\right)}$ are first- and second-order dispersion parameters. These dispersion parameters are elements of the well-known Taylor series expansion approximation of the propagation constant frequency dependence β(ω) [1,17,29]:

_{lm}.

_{z}= E(z/∆z); E(x) is the integer part of the real number x.

^{−1}is the inverse Fourier transform; [x]* is the complex conjugate of x.

_{0}(λ) is the attenuation of lower-order modes (it is supposed to be equal to the attenuation at the correspondence wavelength mentioned in fiber specification); M

_{0}is total number of modes satisfying the cutoff condition for the analyzed fiber:

#### 2.8. Mode Coupling

_{m}and ρ

_{n}are the injected LP

_{lm}and excited LP

_{ln}mode field radiuses, respectively.

_{1}F

_{1}is the confluent hypergeometric function of the first kind [23,24]:

_{θ}is the refractive index of the launching medium (air gap, core of adjusting/exiting fiber, etc.).

## 3. Results

#### 3.1. Low Differential Mode Delay Fiber with a Large 100 µm Core Diameter

_{REF}range. The reference graded refractive index profile was set by a data protocol of measurements, performed by a certified optical fiber analyzer for a sample of a commercially available LOMF of ISO/IEC Cat. OM2+/OM3 [34], and scaled up to a 100 µm core. This is represented in Figure 2.

_{lm}(l = 0, …, 24; m = 1, …, 11) satisfying the cutoff conditions with an optical confinement factor P

_{co}> 0.5, whose distribution over the total mode staff at wavelength λ = 1310 nm is shown in Figure 3.

_{01}and LP

_{11}, at the mentioned wavelength. According to the model described above for the simulation of a piece-wise regular multimode fiber optic link operating in a few-mode regime, mode mixing and power diffusion due to micro- and macro-bends are simulated as connections of the researched MMF regular spans performed with weak angular misalignment ϕ. This supposition was confirmed by experimental verification of the model [21], while it was found that for MMF coiled on a typical fiber spool, this angular misalignment can vary as much as ϕ = 2.0–3.5°. Therefore, we performed the computation of normalized amplitude distribution dynamics over a guided mode staff under centralized mode LP

_{01}and LP

_{11}launching conditions, and further mode power redistribution due to bends for 11 regular spans of piece-wise regular representation. The results of the mode staff normalized amplitude dynamics calculation are shown in Figure 4. Here, we noticed that only 21 guided modes LP

_{lm}(l = 0, …, 7; m = 1, …, 5) transfer the majority of the optical signal during its propagation over the reference 100 µm core MMF, and that their delays vary from 4.922 up to 4.924 ns/km, defining the range for the reference value t

_{REF}setting, while normalized amplitudes of other components are less than 0.1.

_{BASE}and α-parameters of a first iteration graded profile. The result of 100 µm core LDMDF refractive index profile optimization under t

_{REF}= 4923.08 ns/km and α = 1.900 for centralized launching conditions is shown in Figure 5, while a comparison between the DMD diagrams for the reference MMF and a sample of LDMDF is shown in Figure 6.

_{LP01}(λ) computed over the “O”-band for both the reference 100 µm core MMF and LDMDF samples are represented in Figure 7: (a) corresponds to the total “O” band, while (b) shows the LDMDF and DMD curve fragment over a λ = 1310 nm wavelength region. For the reference MMF, DMD varies from 1589.7 ps/km up to 1950.5 ps/km over the “O”-band, while the optimized refractive index profile provides a decrease of DMD from 5.0–6.0 times at the boundaries of the “O”-band, and up to 9.0 and more times at a λ = 1310 nm region in comparison with the reference MMF. Here, the total value of DMD

_{LP01}is not more than 300 ps/km over all of the “O”-band; the lowest DMD

_{LP01}is 176.2 ps/km, and it corresponds to 1285 nm; for λ = 1310 nm, DMD

_{LP01}= 255.7 ps/km.

#### 3.2. Simulation of 10GBase-LX Fiber Optic System Optical Pulse Propagation

_{01}and the higher-order mode LP

_{11}. Here, the “worst case” of launching conditions was considered, which corresponds to a connection via a conventional fiber optic adapter simulated as a connection with an angular misalignment θ = 4.20° [36] between a standard single-mode optical fiber (ITU-T Recommendations G.652) of a laser source pigtail and a link 100 µm core MMF.

## 4. Discussion

_{REF}= 7.04, which corresponds to the “worst case” BER = 10

^{−12}required by a 10GBase-LX specification without a forward error correction (FEC) technique [40]. As a result, a maximal 100 µm core MMF length L = 0.286 km was localized: here, the total pulse dispersion is D = 62.329 ps (Figure 12a), which provides a desired Q-factor of 8.236 (Figure 12b).

## 5. Conclusions

_{LP01}= 176.2 ps/km, and it corresponds to 1285 nm, and DMD

_{LP01}= 255.7 ps/km at λ = 1310 nm. We utilized a previously developed alternative method for the simulation of piecewise regular multimode fiber optic links operating in a few-mode regime for the computation of laser-excited 10GBase-LX optical pulse dynamics during its propagation over irregular 100 µm core optical fibers—the reference MMF and the optimized LDMDF. While most of the commercially available MMFs with a 100 µm core diameter are targeted to only multimode regimes, and the utilized transceivers based on multimode LEDs provide low bit rates of 10–100 Mbps of data transmission, our simulation results demonstrate the implementation of 10GBase-LX channels over both short-length MMFs with a LOMF-graded refractive index profile, and scaled up to a 100 µm core and extended distance links provided by LDMDFs with an optimized graded refractive index profile. Here, in spite of uncontrolled launching conditions corresponding to connection via a conventional fiber optic adapter simulated as a connection with an angular misalignment θ = 4.20° between a standard single-mode optical fiber (ITU-T Recommendations G.652) of a laser source pigtail and a link optical fiber, even the 100 µm core MMF keeps a pulse envelope for a distance of up to 0.5 km, with a total dispersion of about 85 ps, while LDMDF blocks up the DMD over the entire 2 km length with a total pulse dispersion of not more than D = 32 ps under a length L = 0.5 km, and D = 92.50 ps under a length L = 2.0 km. According to “the worst case link model” developed by the IEEE 802.3z Task Force, we estimate the Q-factor for the 10GBase-LX channel based on 9.2 dB budget IEEE 802.3ae LX-transceivers and a 100 µm core reference MMF and LDMDF without any special launching conditions under a particular fiber length. As a result, the maximal lengths of both MMF and LDMDF were localized to provide the least reference Q = 7.04 for BER = 10

^{−12}: for a 100 µm core MMF, this distance was L = 0.286 km under a total pulse dispersion D = 62.329 ps and Q = 8.236; for the 100 µm core LDMDF with a designed special refractive index profile, even under the length L = 1.020 km, the Q-factor is a desirable reference value (Q = 8.660) under a total pulse dispersion of D = 43.669 ps.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Piecewise regular representation of 200 m of irregular optical fiber with varying core diameters.

**Figure 2.**Reference graded refractive index profile reconstructed by a report of real laser-optimized multimode fiber (LOMF) sample profile measurements and scaled up to a 100 µm core.

**Figure 3.**Optical confinement factor distribution over the mode staff of a reference 100 µm core multimode optical fiber (MMF).

**Figure 4.**Normalized amplitude distribution dynamics over the guided mode staff of the reference MMF under a centralized mode LP

_{01}and LP

_{11}launching, and following mode mixing and power redistribution due to the fiber bends being represented as a connection of regular spans with weak angular misalignment, which is empirically defined as a range of ϕ = 2.0° … 3.5°.

**Figure 5.**Optimized graded refractive index profile for a 100 µm core low differential mode delay fiber (LDMDF).

**Figure 6.**DMD diagrams computed for selection under a centralized launching condition mode staff for the reference MMF and LDMDF.

**Figure 7.**Spectral DMD (λ) curves over the “O”-band: (

**a**) the reference MMF and the LDMDF samples; (

**b**) the λ = 1310 nm wavelength region.

**Figure 9.**Variation of the equivalent angular misalignment at the splice of regular spans along the fiber length simulating fiber micro- and macro-bends.

**Figure 10.**10GBase-LX optical pulse dynamics during propagation over a 100 µm core MMF with a total length of 2 km without special launching conditions: (

**a**) pulse dynamics; (

**b**) diagram of pulse propagation.

**Figure 11.**10GBase-LX optical pulse dynamics during propagation over a 100 µm core LDMDF with a total length of 2 km without special launching conditions: (

**a**) pulse dynamics; (

**b**) diagram of pulse propagation.

**Figure 12.**10GBase-LX data transmission over a 100 µm core MMF without special launching conditions: (

**a**) pulse response; (

**b**) eye diagram envelope.

**Figure 13.**10GBase-LX data transmission over a 100 µm core LDMDF without special launching conditions: (

**a**) pulse response; (

**b**) eye diagram envelope.

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**MDPI and ACS Style**

Bourdine, A.V.; Burdin, V.A.; Janyani, V.; Ghunawat, A.K.; Singh, G.; Zhukov, A.E.
Design of Silica Multimode Optical Fibers with Extremely Enlarged Core Diameter for Laser-Based Multi-Gigabit Short-Range Optical Networks. *Photonics* **2018**, *5*, 37.
https://doi.org/10.3390/photonics5040037

**AMA Style**

Bourdine AV, Burdin VA, Janyani V, Ghunawat AK, Singh G, Zhukov AE.
Design of Silica Multimode Optical Fibers with Extremely Enlarged Core Diameter for Laser-Based Multi-Gigabit Short-Range Optical Networks. *Photonics*. 2018; 5(4):37.
https://doi.org/10.3390/photonics5040037

**Chicago/Turabian Style**

Bourdine, Anton V., Vladimir A. Burdin, Vijay Janyani, Ashish Kumar Ghunawat, Ghanshyam Singh, and Alexander E. Zhukov.
2018. "Design of Silica Multimode Optical Fibers with Extremely Enlarged Core Diameter for Laser-Based Multi-Gigabit Short-Range Optical Networks" *Photonics* 5, no. 4: 37.
https://doi.org/10.3390/photonics5040037