2.1. Related Works
Laser-excited optical signal transfer over MMF by a limited number of mode components primarily differs from a multimode regime by the appearance of the DMD effect [
1,
2,
3,
4,
5,
6]. DMD is a main issue for short-range multi-gigabit optical networks based on MMFs. Generated by a coherent source, optical emission is launched to an MMF core with a much greater diameter then the laser spot size. This great difference between the fiber core diameter and the laser spot size leads to the excitation of a limited number of particular order modes from the total MMF guided mode staff. Moreover, during propagation in MMF, the amplitudes of these selected mode components transferring the optical signal are non-equal. The description of the amplitude difference is mainly defined by its launching conditions: the typical fiber optic adaptor, radial or angular misalignments, a special matching device, etc. Therefore, the optical pulse at the transmitter end of the fiber optic link contains several modes with non-equal amplitudes and differing mode velocities (e.g., mode delays). As a result, these modes come to the receiver end with the spread of both amplitudes and delays relative each other, which becomes critical at multi-gigabit rates. DMD strongly distorts the pulse form; it splits at the receiver, and this generates a “glove” effect. Moreover, DMD for each particular “laser–MMF” pair may strong differ, due to the launching uncertainty provided by the variations of sources, connections, and fiber parameters.
A detailed overview of the known solutions concerned with the development of MMF with improved bandwidth is represented in published monographs [
10]. Some of them were designed as LOMFs, so that they are targeted not only for modal dispersion reduction, but also for DMD suppression. Most of the known approaches for LOMFs are based on DMD monitoring during fiber preform manufacturing. As proposed in [
11,
12,
13], the objective function estimates the desired refractive index profile proximity to a profile, which is optimal for the smallest deviation of delays between the total MMF mode staff that satisfies the cut-off condition that corresponds to a multimode regime. In [
14,
15,
16], the object function is the difference between the integral and the local value of the profile grade parameter α
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].
Therefore, we suppose that the following outlined factors should be taken into account during the design of MMFs with reduced DMD, or so-called low differential mode delay fibers (LDMDFs) that are optimized for operation under a few-mode regime: (1) Real and approved commercially available LOMFs of Cat. OM2 + –OM5 structure, geometry, refractive index profile, and parameters of data should be used initially/approximated, but not the first generation MMFs of Cat. OM1 and OM2, as well as the model of optical fibers with the ideal graded refractive index profiles as represented by one or a set of simple power-law α-functions. (2) The development or modification of a fiber structure/refractive index profile has to be based on comparison with the DMD diagram. (3) The laser source type and its parameters, including the generated emission initial transverse mode staff and the launching conditions, should be taken into account. (4) Reduced DMD should be provided, not only at particular wavelengths but over the spectral range (for example, conventional telecommunication “O”-band with a reference wavelength λ = 1300–1310 nm).
2.2. Design of an LDMDF-Graded Refractive Index Profile
The proposed solution is based on the design of a specialized form of refractive index profile providing a selected guided mode staff delay equalization in relation to some reference delay value,
tREF. 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:
and the profile function
f(R) describing the refractive index profile by a known relation:
is written as follows:
where
is the local profile parameter;
nk is the refractive index of
k layers
(k = 0…N);
nmax is the maximal core refractive index;
nN is the outer cladding refractive index;
is the profile height parameter;
Rk = rk/a is the normalized radial coordinate of the
k-th layer;
rk is the radial coordinate of the
k-th layer;
a is the designed LDMDF core radius.
As a result, the desired refractive index profile form in the LDMDF core region is selected in such a way that it should satisfy the minimization of some objective function
F described by the proposed simple formula:
where
td(j) is the desired value of delay for the
j-th guided mode
LPlm(j) computed at the corresponding synthesis iteration;
tREF 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
Aj > 0.1, and with a core power (known also as an optical confinement factor) that is not less then
. 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
Aj > 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
tREF 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
During objective function (4) minimization, a direct problem of the optical fiber with the current refractive index profile should be solved during each iteration. Here, objective function arguments are an array of local parameters
hk 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
U2 in the form of finite nested sums [
21] from a well-known integral variational expression [
22]:
where
where
R0 =
ρ0/
a is the equivalent (as a result of Gaussian approximation) normalized mode field radius (MFR);
ρ0 is the equivalent MFR;
a is the MMF core radius;
l is the azimuthal mode;
m is the radial mode of
LPlm orders.
is the normalized frequency;
is the wavenumber; λ is the operating wavelength;
is the expansion factor of the Laguerre polynomial representation in the form of a finite power series [
23,
24]:
The characteristic equation of the equivalent MFR that has the generalized form according to the Gaussian approximation [
22] is
and it also leads to the following analytical expression:
where
Here, the normalized equivalent MFR
R0 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),
R0 is a basic single variational parameter, which completely describes the mode dispersion characteristics. Following the substitution of
R0 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]:
The solution of the characteristic Equation (7) by taking into account the further substitution to Equation (5) and then to Equation (8) should satisfy the guided mode cut-off condition [
1,
17,
20]:
We also propose considering the optical confinement factor
Pcore as the second criterion for the identification of the “ghost” solutions:
The last one is also described by the following analytical expression earlier on, which is derived within the Gaussian approximation approach [
20,
21]:
The proposed EMGA provides low degrees of error, with a reduction of total computational time, especially for the calculations of higher-order mode parameters under low error, even during the analysis of large core optical fibers with complicated forms refractive index profiles, including real commercially available MMF graded-index profiles with refractive index fluctuations and technological defects. The results of EMGA verification by the rigorous mixed finite element method are represented by details in earlier published studies [
25].
2.6. Material Dispersion and Refractive Index Profile Parameters
The well-known Sellmeier equation [
1,
17] is utilized to take into account the material dispersion:
where
Ai and
Bi are Sellmeier’s coefficients (
Bi 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.
A passage from the differentiation operator to its square is utilized analogously to the propagation of constant derivatives. This passage will greatly simplify the expressions for the first and second derivatives of the refractive index written via the Sellmeier equation:
Therefore, the first and second derivatives of the profile height parameter Δ can be expressed as follows:
while derivatives of the local profile parameter
hk lead to the following expressions:
Finally, by applying Equations (22) and (23), the first and second derivatives of the normalized frequency
V can be expressed by the following formulas:
2.7. Model of a Piecewise Regular Multimode Fiber Optic Link Operating in a Few-Mode Regime under Laser-Excited Optical Pulse Propagation
We propose utilizing a previously developed model of a piecewise regular multimode fiber optic link operating in a few-mode regime to simulate laser-excited optical pulse propagation, and to estimate the potentiality of a 100 µm core MMF, as well as an optimized 100 µm core LDMDF for laser-based multi-gigabit data transmission. This model is described in detail in the published work [
21], where its experimental verification is also demonstrated. The proposed solution is based on piecewise regular representation, combined with the general approach of the split-step method [
29] to simulate the processes of mode mixing and power diffusion occurring due to mode coupling (
Figure 1). Here, single silica weakly guiding circular MMF with an arbitrary axially symmetric refractive index profile with single continuous outer cladding is considered. According to the piecewise regular representation, the fiber is divided into regular spans with length ∆
z. Inside of the span, the fiber geometry parameters are considered as a constant, and the modes propagate independently without interaction and mixing. It is supposed that each excited guided mode with a propagation constant, varying from one regular span to another span, satisfies a cut-off condition for whole regular spans composing the fiber. Additionally, it is assumed that at the link transmitter end, each excited guided mode begins transferring a single optical pulse with a particular form that is identical to the input signal (for example, Gaussian). During pulse propagation over a regular span, its amplitude decreases, due to mode attenuation. The signal is mainly distorted due to the difference between the group velocities and the amplitudes of modes, i.e., the DMD effect. Additionally, the transfer by the pulse spreading of each mode due to chromatic dispersion for a particular mode is taken into account.
The boundaries of regular spans can be represented generally as ideal axial alignment splices of two almost similar optical fibers with mismatching parameters. However, it is correct only for “straight” fibers. Therefore, we propose simulating fiber bends by the representation of boundaries in the form of splices of two mismatched fibers with some low angular misalignment [
21]. The mode power redistribution between the amplitudes of signal components as a result of mode interaction is estimated by mode coupling of coefficient matrix computing at the joints of regular spans. Here, only guided modes are considered as the main issue under pulse dynamics research during the propagation over a multimode fiber link in a few-mode regime. However, power loss due to component transformation from guided to leaky mode and reflections are also indirectly taken into account.
At the receiver end, the resulting pulse envelope is considered as a superposition of all mode components of the signal. Here, it is proposed that a well-known expression is applied for the frequency response of the signal transferred by the
M mode components
LPlm over a regular multimode fiber with length
z [
1]:
where
F is the direct Fourier transform;
hTx(t) is the initial pulse at the transmitter end;
and
αp are the starting amplitude and the mode attenuation of the
p-th guided mode
LPlm (
p = 1...M);
and
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]:
where
Here, is the mode delay and is the group velocity dispersion associated with the chromatic dispersion of the p-th guided mode, LPlm.
Therefore, according to introduced piecewise regular representation of the irregular multimode fiber link, the frequency response of a few-mode optical signal, transferred by
M guided modes over an irregular MMF with length
z under a given particular length of regular span ∆
z by taking Equation (30) into account, can be written in the following form [
21]:
where
Nz =
E(
z/∆
z);
E(x) is the integer part of the real number
x.
The resulting pulse response at the receiver end of the irregular multimode link is computed by the following simple expression:
where
F−1 is the inverse Fourier transform; [
x]
* is the complex conjugate of
x.
Therefore, EMGA is applied for computing the dispersion parameters of the desired selected order guided modes at each regular span of the researched irregular MMF, including mode delays (or group velocity) and chromatic dispersion parameters. Differential mode attenuation is estimated by the known empirical expression proposed by Yabre in [
30], which is based on experimental data represented by the same author in [
31]:
where
is principal mode number;
α0(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);
M0 is total number of modes satisfying the cutoff condition for the analyzed fiber:
where
g is the gradient factor of the smoothed
α-profile.
The model presented passed experimental approbation: a good agreement between its simulation and direct experimental measurements of pulse response was obtained, which was described in detail in the published work [
21].
2.8. Mode Coupling
Research on the irregularity of MMF is concerned with core diameter variations. Therefore, we propose setting it directly via an array from the reports of optical fiber diameter measurements that were produced during fiber drawing. Micro- and macro-bends are simulated by random equivalent low (
θ = 2.0°…4.0°) angular misalignments [
21] at the boundaries of regular spans, while mode coupling coefficient redistribution at the span boundaries as well as at the optical interconnections of the transmitter and receiver ends may be estimated by the well-known mode field overlap integral method [
1,
17,
22], taking into account the particular angular misalignment inserted. Here, we utilized the overlap integral method in combination with the introduced EMGA, which takes into account the local features of the real silica optical fiber refractive index profile and decreases the computational error. A passage from the generalized form of the overlap, integral to the analytical expression for the arbitrary order mode coupling coefficient estimation at the centralized splice of the optical fibers with mismatched parameters without any misalignments, was proposed in [
32]. The analytical expression derivation is described in detail in [
33]. Finally, the formula for the arbitrary order mode coupling coefficient at the central splice is written as follows:
where mode coupling occurs only for modes with the same azimuthal order
l; Γ is the gamma function;
ρm and
ρn are the injected
LPlm and excited
LPln mode field radiuses, respectively.
The analytical expression for the arbitrary order mode coupling coefficient at the optical fiber splice under a low angle misalignment
θ < 10° derived and represented in [
33] has the following form:
where
1F1 is the confluent hypergeometric function of the first kind [
23,
24]:
where
and
are Laguerre polynomial expansion factors of Equation (6);
nθ is the refractive index of the launching medium (air gap, core of adjusting/exiting fiber, etc.).