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A rigorous full-vector multi-mode fiber (MMF) model is proposed. It is believed to be the first comparative study of vector and scalar MMF model in terms of differential mode delay (DMD), mode power distribution (MPD), transfer functions as well as eye diagrams. It shows that the vector nature of fiber modes cannot be ignored even though the refractive index difference can be as small as 1%. A standard-compliant methodology for MMF characterization is introduced. The impact of fiber parameters on bandwidth is studied. The statistical transfer function model of OM3 and OM4 fiber is provided. These transfer functions can be applied to the MMF link modeling. Rigorous full-vector MMF model is an essential tool for research and development of MMF link transceivers and standard development of 112Gbps and beyond.
© 2016 Optical Society of America
1. Introduction
Multi-mode fiber (MMF) links are widely used in data communication systems, because of their high density, high reliability, low power and low-cost advantages. The applications of MMF links include: Top of Rack (ToR) switch-to-server connections, storage area networks (SANs), router/switch to transport equipment interconnections, router/switch cluster interconnections, and so on. The advantages of MMF system come from large coupling tolerance and high reliability, because of MMF’s large core diameter and numerical aperture. MMF that employs Vertical-Cavity Surface-Emitting Laser (VCSEL) is one of the most efficient short reach link solutions in terms of power, volume and cost [1].
The standard transmission speed of single MMF link is up to 25Gbps with non-return-to-zero (NRZ) modulation, e.g. 100GBASE-SR4 or 400GBASE-SR16. The next single-lane speed foreseen in the standard would be 56Gbps with pulse amplitude modulation (PAM), e.g. 400GBASE-SR8. The feasibility of 56Gbps PAM-4 transmission over MMF has already been proven [2, 3]. Data rate greater than 56Gbps NRZ signal transmission over 100m MMF using directly modulated 850nm VCSEL has been reported [4–6]. It shows that by using PAM-4, advanced digital signal processing (DSP) algorithm and forward error correction (FEC), it is promising to achieve 112Gbps per lane error free transmission. In this paper, single link/lane corresponds to a transceiver pair.
Improving the speed of single lane is to improve the interconnection density, which is becoming one of the bottlenecks of high throughput systems. MMF link technology is a very promising or even dominant candidate to achieve cost-effective interconnection density improvement [1]. In MMF link analysis, rigorous MMF modeling is very important in the system design process. When data rate grows beyond 10Gbps, the MMF systems apply restricted mode launch (RML) technology [7, 11]. The RML technology is initially adopted in 10GE specification IEEE802.3ae, and is continued in IEEE802.3ba 40GBASE-SR4, 100GBASE-SR10 and 100GBASE-SR4 standards. In RML systems, the transmission characteristic of MMF does not only depend on the MMF itself, but also highly depends on the launch conditions. This makes the analysis and design very complicated and cumbersome. We need to ensure that a specific design of MMF transceiver can guarantee reliable operation over wide range of MMF manufacturing variations and launch condition fluctuations.
Laboratory experimentation is essential for successful MMF system design, however they are time and energy consuming. Only a small portion of available parameter space can be explored. It is difficult to explore the ‘Worst-Cases’ or ‘Marginal Zone’ by experiments alone. Theoretical models have been used to investigate the MMF link performance, to assist in designing, optimizing and testing for MMF systems [7–12]. These models utilize the linearly polarized (LP) or scalar modes, which is also well known as ‘paraxial approximation’ or ‘scalar approximation’. The paraxial approximation assumes that the longitudinal mode field is negligible compared with the transverse mode field, i.e. |Ez/Et|≪1. Such assumption is reasonable for low order modes, but not proper for high order modes. High order modes will be surely excited in offset launched MMFs. The assumption of the scalar mode fields will cause divergences in the coupling coefficients, which are very important to the calculation of MMF transfer functions. The scalar model ignores the vector nature of fiber modes, thus it will provide different estimations both in the MMF bandwidth and the eye opening of signals going through the MMF systems compared with the vector model. The difference will be more severe, when the launch offset value is large, because more high order modes will be excited, and they will be dominant. The essential difference between the vector and scalar models will be discussed in section 4 with some concrete cases.
In this paper, a rigorous full vector (FV) MMF model is proposed to use. A comparative study of the vector and scalar model is provided in terms of differential mode delay (DMD), mode power distribution (MPD), transfer functions as well as eye diagrams. A standard-compliant methodology for the MMF characterization is used to obtain the statistical models for OM3&OM4 fibers [15]. The remainder of this paper is organized as follows. In Section 2, the detailed formula for the rigorous MMF model is described. In Section 3, the standard-compliant methodology of OM3&OM4 fiber characterization is proposed [15]. In Section 4, a close comparison of the vector and scalar model is provided through specific simulation examples. Some insights of the key difference of vector and scalar model are also given. In Section 5, The influence of refractive index profile perturbation on the bandwidth of MMF is studied; typical parameters for OM3&OM4 fibers are extracted. The combinations of these typical parameters form the ‘Worst Case’ OM3&OM4 fiber samples. In Section 6, the transfer functions of the ‘Worst Case’ OM3&OM4 fiber samples with different light source spot sizes and launch offset values are investigated. These transfer functions can be readily used in the MMF system models, similar to the electrical channel modeling, in which the transfer functions is derived from the measured scattering parameters. Section 7 gives the conclusion.
2. Full vector MMF modeling
2.1. Refractive index profile of MMF
The refractive index profile of the MMF is given by
where When α1 = α2, a single α = α1 = α2 is used to denote the power law parameter throughout this paper. In order to consider the material dispersion, n2(λ) is given by the Sellmeier formula where B1 = 0.696166300, B2 = 0.407942600, B3 = 0.897479400, C1 = 0.00467914826, C2 = 0.0135120631, C3 = 97.9340025, are Sellmeier coefficients for fused silica, and noffset = 0.0223572302 is chosen to make (n1 + n2)/2 matching the effective refractive index of OM3&OM4 MMF specification, i.e. 1.482@850nm and 1.477@1300nm. The slight change of the material dispersion with the dopant (GeO2) level of the core is ignored to simplify the modeling and analysis, because the absolute value of the change is very small i.e. <2ps/nm/km [13]. For short reach applications, the mode dispersion is dominant over the chromatic dispersion. Introduce the core refractive index defect as where , δFWHM is the full width half maximum (FWHM) of core defect profile. The core defect is determined by the width δFWMM and the amplitude A. Further, introduce the core-cladding refractive index defect function as The refractive index profile with core defect and core-cladding defect is given by Equation (6) defines the refractive index profile of MMF similar to [7] and [11]. There are six sets of parameters:- Optical wavelength: λ, which defines n2 through (3);
- Refractive index difference: Δ;
- Refractive index profile parameters: α1 and α2;
- Core radius: a;
- Core defect parameters: A and δFWHM;
- Core cladding defect parameters: r0 and γ.
2.2. Light source model
The electric field of light source (output of probe fiber or VCSEL) is assumed to take Gaussian profile:
where , wFWHM is FWHM of light source, xoffset and yoffset are offset values with respect to MMF center. Note that the light source can also be modeled as superposition of Gaussian beam modes to consider the multimode light source as in [10]; or it can even be measured profile to consider a real light source. In order to keep the concept clear, we use the simple Gaussian profile throughout this paper. The assumption in the light source profile does not hurt the integrality of the vector model that will be discussed in this paper.Define ϕ = tan−1 [(x − xoffset)/(y − yoffset)]. The power normalized vector fields of the light source are given by
where Z = Z0/ng, Z0 is impedance of free space, ng is average group refractive index of fiber modes, 0 ≤ x ≤ 1 defines the polarization. The fields of light source (8) ~ (9) satisfy , where ℜ(x) means the real part of x throughout this paper.2.3. Full vector fiber model
An improved full vector finite difference mode solver is applied to calculate the effective indices neff,i and mode fields of MMF with arbitray refractive index profile, the equations and numerical scheme are exactly the same as in [14]. The MMF refractive index profile is modeled as many cascaded thin homogeneous layers. Zeros boundary condition is applied to consider the guided modes only, because the guided modes will exponentially decay to zero in the cladding. This approach is applicable to MMF solving, we can observed perfect match of DMD curves calulated from this vector model and the scalar model as shown in Fig. 4. The computation window is 62.5µm which is identical to the MMF cladding radius. The vector mode fields are
where 0 ≤ y ≤ 1 defines the mode polarization. The mode fields of (10) ~ (11) are power normalized and satisfy the orthogonality of , where δij = 0 if i ≠ j; δij = 1 if i ≡ j. The first and second order derivative of the propagation constant β over the radian light frequency ω, β1,i and β2,i, and the dispersion coefficient Di are calculated from the effective refractive index neff, i as The power density of the i-th mode is given by where ζ = π/2 when m ≠ 0; ζ = π when m = 0. The mode fields are normalized, such that: .The vector modes of MMF are labeled by two parameters, the azimuthal order m, and radial order Addr, they are mode solver labels. For each azimuthal order m the modes are sorted by the ℜ(neff) in descending order, and the i-th mode is assigned an radial order of Addr = i, where i = 1,2,3,…. The vector model label is related to the mode solver label by Table 1. Further, the vector mode is related to the LP mode by HE1n → LP0n, (TE0n,TM0n,HE2n) → LP1n, (HEm+1,n,EHm−1,n) → LPmn. The LP modes with the same l + 2p − 1 value belong to the same principal model group, where l is the azimuthal and p is the radial order of LP modes. The principal mode group number l +2p−1 can be calculated for a specific vector mode with Table 1 and the relationship of LP mode and vector mode. The mapping is a rule-of-thumb for pure power law profiles, it is not valid for the refractive index profile with defects. Because the defects break down the mapping relationship of Table 1. Fortunately, the exact modes instead of the principal mode groups are used in our approach. We only aggregate the vector modes into mode groups for pure power law fibers throughout this paper, so we can evaluate the impact of neglecting non-degeneracy of vector modes in the same principal mode group, and also compare with the scalar model.
Table 1. Mode Solver Label to Vector Mode Label Mapping.
2.4. Mode power coupling coefficients
When coupling light source into the MMF, the amplitude of the i-th mode is given by
where , , , are given by (8) ~ (9); and Er,i, Eϕ,i, Hr,i, Hφ,i are given by (10) ~ (11). The power coupled into the i-th mode is calculated by all the polarization combinations of the light source and the MMF modes: If we consider the polarization effects, then the coupling coefficient is calculated as where x represents the polarization of the laser source. For a specific fiber length the corresponding mode delay is given by τi = [β1,i − min(β1)] · L. where L is the length of fiber, β1,i is given by (12).If the non-degeneracy of vector modes within a principal mode group is ignored, for the k-th principal mode group, the power coupling coefficients can be aggregated as Pg,k = ∑i∈k-th mode group Pi. Correspondingly, the normalized mode delay β1g,k is calculated by the weighted summation of β1 within each mode group, the weights are the ratio of power coupling coefficients. For a specific fiber length L the mode delay for the k-th mode group is given by τg,k = [β1g,k − min(β1g)] · L.
2.5. Frequency and impulse response of MMF
The frequency response of MMF is calculated by
where σλ is RMS spectral width of light source.If non-degeneracy of vector modes within a mode group is ignored, MMF transfer function can also be written as
where D is group dispersion coefficient of MMF. If we set all the power coupling coefficients Pk or Pg,k equal we can get the transfer function under over fill launch (OFL) condition. Then OFL bandwidth can be estimated from these transfer functions. OFL bandwidth is not applicable for RML systems, thus they are beyond the scope of this paper.Let H(f, r) to be the MMF transfer function with launch offset r, The corresponding impulse response is given by its inverse Fourier transform as:
where is Fourier transform operator, and means inverse Fourier tranformation.The mode mixing in the connectors of multi-segment links can also be considered in a recursive manner. For a link of “source → fiber 1 → connector 1 → fiber 2 → connector 2 → fiber 3 → detector”, we can solver the output field of “connector 1” by solving the segment of “source →fiber 1→connector 1”. And the output of “connector 1” is then considered as a new “source”. Such kind of procedure can be proceeded until the signal reaches the detector. This procedure is exactly the same as in [9, 10], the only difference is that vector fiber modes are applied in our approach. In real short reach applications, the length of fiber 1 and fiber 3 are usually much shorter than the length of fiber 2, and the connector mismatch can be well controlled. Thus, the offset launch effect is dominant over the mode mixing in the connectors, the “source→fiber → detector” model can cover most of the scenarios.
3. Multi-mode fiber characterization
The differential mode delay (DMD) for a specific mode is defined by its relative mode latency:
where β1,i is given by (12). It means the walk off with respect to the fastest fiber mode. And the maximum value of DMDi defines the DMD of the fiber.The output beam of probe fiber for DMD measurement can be well modeled as a Gaussian beam described by Eqs.(8) ~ (9). The mode field diameter of the probe fiber shall be , i.e 5µm@850nm and 9µm@1310nm [15]. The mode field diameter of a Gaussian beam is , Thus, for a given wavelength λ, we choose .
Choose an optical pulse waveform R(t) as reference, let ΔTPULSE to be the 25% width of R(t). Assume the pulse has a Gaussian shape
The temporal response U(r,t) for a series of radial offsets r, is calculated by U(r,t) = R(t)∗h(t,r). where “*” is convolution calculation and h(t,r) is given by (19).If the power coupling coefficient for i-th mode at a specific launch offset r is Pi, the near field intensity I(r) is given by
where Γi(r) is the radial power distribution of the i-th mode given by (13). The encircled flux (EF) is calculated as where rmax is typically 1.15 times of the core radius. An acceptable light source offset launch should satisfy the EF constraint: It is initially specified by 10GE specification IEEE802.3ae, and is also used in IEEE802.3ba 40GBASE-SR4, 100GBASE-SR10 and 100GBASE-SR4 standards. Define the radial intensity as The resultant output temporal response Po(t) is calculated by W(r) in (26) can also be weighting values provided in [15] Annex D. The effective modal bandwidth (EMB) is evaluated from the MMF transfer function The EMB is given by where minEMBc is the minimum value of calculated modal bandwidth (EMBc) of (27) under various offset launch conditions. These launch conditions should satisfy the EF constrain given in (24), which means we rule out the EMBc values that violate the EF constraint. The EMBc of (27) is calculated by the 1.5dB bandwidth and Gaussian extrapolation, i.e. the 1.5dB bandwidth is scaled up by a factor of 1.414 to get the 3dB bandwidth. 1.13 in (28) is an adjusted scaling factor. A fiber is qualified as OM3 if the EMB at 850nm is over 2000MHz · km, or OM4 if EMB at 850nm is over 4700MHz · km.There are four methods to calculate the EMB,
Method 1) and 2) are simulated experiment methods; Method 3) and 4) are theoretical methods. 1) is a recurrence of standard MMF measurement process by using the measured DMD waveform U(r,t) and weight function W(t) calculated from the measured near field intensity I(r); 2) is a recurrence of standard VCSEL launched MMF measurement, by using the measured DMD waveform U(r,t) and weight function W(t) in Annex D of [15]. Note that 4) is only valid for pure power law MMFs and it ignores non-degeneracy of modes in the same principal mode group. We list it here as a ‘method’ is to investigate the impact of neglecting non-degeneracy of vector modes in the same principal mode group.Figure 2 (left) gives the simulated DMD measurement waveform sets U(r,t) for different α. The 25% pulse width ΔPULSE is 100ps. Figure 2 (right) gives the DMD values versus the principal mode group number under different α. It can be seen that there exists an optimized profile parameter αOPT ≈ 1.9812. The results in Fig. 2 are consistent with the scalar model. An one-to-one DMD curve comparison between vector and scalar model can be found in Section 4.
Fig. 2 Output pulse waveforms for DMD measurement (left) and DMD values for different principal mode groups (right). α values for curves of 0 ~ 6 are 1.930, 1.955, 1.974, 1.983, 1.994, 2.015, 2.040.
Figure 3 shows the relationship of EMBc and launch offset of two examples with these four different methods. It shows that methods 1~3 are self-consistent in EMBc calculation. However, the simulated experimental methods 1) and 2) have the best EMBc stability with respect to the launch offset. The EMBc evaluated from the theoretical methods 3) and 4) fits well with each other when launch offset values are small; the results diverge for large launch offsets, because method 4) ignores the non-degeneracy of vector modes in the same principal mode group. The divergence of method 3) and 4) at larger launch offset values indicates that the vector nature of fiber modes cannot be ignored. Through out this paper, 1) is used to characterize the performance of MMF, because it is closest to the experiments.
Note that bandwidth calculation is not the final goal of MMF modeling, the EMB bandwidth reflects the frequency response of the low frequency band (with insertion loss around 1.5dB). It cannot well characterize the ‘notches’ of the frequency response curve in the high frequency band. These ‘notches’ can be dominative in the performance of high speed MMF link systems. Actually, we need to extract the transfer function over a wide frequency range to fully cover the signal bandwidth.
4. Comparison of vector and scalar model
In order to make a fair comparison of vector and scalar model, we choose three fiber samples from [8] and calculate the differential mode delay (DMD), mode power distribution (MPD), transfer functions as well as eye diagrams. The results of vector model are compared with the results of scalar model given by [8]. We choose the number 3, 51, 99 fiber samples, these fibers have pure power law profile and the refractive index profile α of them are 1.89, 1.97 and 2.05, respectively. The fiber samples in [8] are 62.5µm OM1 fibers simulated at wavelength of 1300nm; the launch beam is a 7µm FWHM Gaussian beam with offset values of 17, 20 and 23µm. We choose these simulation parameters because the simulation results of the scalar model are independently calculated and they can be readily obtained from [8]. However the conclusions of the comparative study are also applicable to the 50µm OM2, OM3, OM4 fibers at wavelength of 850nm. Theoretically, OM1, OM2, OM3, OM4 fibers follow the same governing equations, they have no difference in the sense of modeling.
Figure 4 gives the DMD and MPD calculated by the vector and scalar models. We can see perfect match of the DMD curves. This is meaningful because DMD is obtained from the propagation constant which is stable with respect to the mode field perturbation. The assumption of the scalar model is on the mode field (i.e. scalar model ignore the z-components of the mode fields). However, the assumption of scalar model make its MPD curves quite different from the results of vector model. The peaks of MPD curves shift to high order principal mode groups, and the MPD curves become more asymmetric. The difference in the MPD curves directly comes from the mode field assumption of scalar model.
The inter-group DMD cannot be ignored either as shown in Fig. 5. In Fig. 5 the circles connected by the line belong to the same principal mode group. The chain with more circles corresponds to higher order principal mode group. Figure 5 shows that the degeneracy of vector modes in the same group is broken down for high order mode groups, the DMD values of the vector modes belong to the same group do not equal anymore. The inter-group DMD can be large and cannot be ignored. The non-degeneracy of the vector modes within the same principal mode group explains the inconsistency of the EMBc calculated from method 3) and 4) as shown in Fig. 3. It also explains the inconsistency of transfer function estimated from method 3) and 4) in the high frequency range as shown in Fig. 6.
Figure 6 plots the transfer functions of L=200m, α = 1.89 and α = 2.05 fibers with offsets of 17, 20, 23µm. Obvious difference between the results of vector and scalar model can be observed in the 5~10GHz band, the maximum insertion loss difference can be as large as 5dB. There is an obvious ‘notch’ in the 5~10GHz band that fails to be predicted by the scalar model. That ‘notch’ will dominate the link performance (as shown in Fig. 7). To compare the transfer function obtained from the vector model, i.e. method 3) i.e. equation (17) and method 4) i.e. equation (18), we can also see obvious difference, the difference will become larger if the offset values increase from 17 to 23µm. It also indicates that the non-degeneracy of vector modes in the same principal mode group cannot be ignored.
Figure 7 and 8 give the eye diagrams of 10Gbps NRZ signal going through 300m, α = 1.89 fiber; and 112Gbps PAM-4 signal going through 150m, α = 1.97 fiber, respectively. The eye diagrams are simulated in order to evaluate the difference of vector and scalar model on the link simulation. We can see that the eye diagram qualities simulated with vector and scalar model are quite different, even though the simulation setups are identical. The scalar model always gives open eyes for offset values of 17, 20, 23µm. However the eye diagram simulated with the vector model closes gradually when the offset value increases from 17µm to 23µm. Under the same offset value, the scalar model always gives optimistic evaluation on the eye quality compared with the vector model. The simulation results shows obvious divergence of scalar and vector model. And the divergence is directly induced by neglecting the vector nature of high order fiber modes in scalar model.
5. Statistical modeling of OM3&OM4 fibers
In order to establish a statistical model for OM3 and OM4 fibers, the impacts of Δ, α, a, A, γ, on the EMB are investigated at the operating wavelength of 850nm. The core defect width and the core-cladding defect radius are chosen to be δFWHM = 2.4µm and r0 = 22µm, respectively. The results in Fig. 9 show that EMB is less sensitive to a and Δ, but more sensitive to α, A and γ. The refractive index profiles for OM3 and OM4 fibers are constructed by the combinations of parameters shown in Table 2. For a and Δ, the typical values of a = 25µm and Δ = 1% are chosen. Three power-law values are used corresponding to over-compensated, near optimum and under-compensated refractive index profile. Power-law value of 1.9812 represents the near optimum value. The other two power-law values are chosen since pure power-law profiles with these parameters result in an EMB of 2000MHz·km for OM3 and 4700MH·km for OM4 at operating wavelength of 850nm. For A and γ, besides zero, which means defect-free, another two values are chosen since the MMFs with these two values and near optimum power-law value degrade to 2000MHz·km for OM3 and 4700MH·km for OM4 at operating wavelength of 850nm. There are 81 different profiles for OM3 and OM4 fibers, respectively. The 81 different profiles compose of three core defects, three core cladding defects, three power-law values for the inner core and three power-law values for the outer core.
Fig. 9 Impact of Δ, α1 = α2 = α, a, A, γ on the EMB. The α values for curves of (a) ~ (e) are 1.9551, 1.9737, 1.9812, 1.9936, 2.0149.
Table 2. OM3 & OM4 Fiber Parameters
Figure 10(left) gives the EMB of simulated OM3 and OM4 fiber samples at 850nm and 1300nm. The fiber samples are sorted in ascending order of EMB at 850nm. It shows that there are abundant samples around 2000MHz·km and 4700MHz·km for OM3 and OM4 fibers. The EMB at 1300nm fluctuates around the EMB at 850nm. Figure 10(right) show that there are more fibers with larger EMB at 850nm than 1300nm. For the OM3/OM4 samples, there are 58/61 out of 81 fiber samples that 850nm EMB is larger than 1300nm EMB. It means these simulated OM3&OM4 samples are 850nm optimized. In order to obtain the ‘Worst Case’ fibers, the length of the fibers whose EMB larger than the OM3 and OM4 requirement is scaled up, i.e. ×EMB(850nm)/2000 and ×EMB(850nm)/4700, for OM3 and OM4 fibers respectively. Such that these samples satisfy the minimum requirement of OM3&OM4 fiber.
Figure 11 gives the relationship of the EMBc with respect to the launch offset for OM3 and OM4 fiber samples at 850nm and 1300nm. In Fig. 11 the length of all the samples are scaled according to the OM3&OM4 requirement. It shows that the minEMBc occurs at the encircled flux boundaries, i.e. 3µm or 15µm. For the launch offset values of 3~15µm the bandwidth is usually larger than 2000MHz · km for OM3 fibers and 4700MHz · km for OM4 fibers at 850nm. The circle lines in Fig. 11 represent the mean value of bandwidth. The EMBc variation is also flattened in the offset range of 3~15µm.
Fig. 11 Normalized EMBc versus launch offset values. The circle lines represent the mean value of bandwidth.
6. Transmission properties of OM3&OM4 fibers
The transfer function of MMF is actually composed by many individual ‘notch’ in the frequency spectrum. Each of the ‘notch’ is induced by two dominant modes beating at a specific frequency. For two modes with amplitudes of κ and 1−κ and relative delay of τ, the frequency response induced by their resonance is given by H(f) = (1−κ) + κ · exp(−j ·2π f · τ). The notch peak locates at 1/(2·τ); the notch depth is governed by the relative amplitude of these two modes 0 ≤ κ ≤ 0.5. The deepest notch happens when these two modes have the same amplitude i.e., κ = 0.5. Figure 12 gives an example of two modes beating of different κ. These two modes have a relative latency of τ = 20ps. The resonant frequency is 1/(2·τ) = 25GHz. There are two ways to suppress the notches: (A) reducing the relative latency of these two modes, i.e. reducing τ, this can move the ‘notch’ outside the critical frequency window; (B) enhance the amplitude difference between these two modes i.e. reduce α. The relative latency τ relates to β1 (DMD) and the fiber length L by τ = β1 · L. So method (A) corresponds to “using better fibers with smaller DMD values” or “using shorter fibers”. Method (B) corresponds to “using offset launch” or “using light source of different FWHM” to adjust the modal power distribution. As shown in Fig. 13 and 14, the “launch offset” and “light source FWHM” exhibits non-correlation with the depth of the “notches”, i.e. the “notches” may become shallower or shaper with the increase/decrease of the “launch offset” and “light source FWHM”.
The transfer functions of 81 OM3 and OM4 “Worst Case” fibers are plotted in Fig. 15. Three offset values 6, 9, 12µm and three FWHM values of the incident light are used, i.e. 5, 6 and 7µm for λ = 0.85µm and 6, 7 and 8µm for λ = 1.3µm. The OM3 fiber length is 100m and OM4 fiber length is 235m. There are totally 81 × 3 × 3 = 729 transfer functions. The circle lines of each sub-figure represent the average value of the 729 transfer function curves. The insertion loss mask of OIF-CEI-56G-VSR-PAM and OIF-CEI-56G-MR-PAM channels are also plotted, it shows that the 100m OM3 and 235m OM4 fibers satisfy the OIF-CEI-56G-VSR-PAM template. The transfer functions can be used directly in the MMF system models, similar to the electrical channel modeling, in which the transfer function is derived from the measured scattering parameters.
Fig. 15 Frequency responses of 81 OM3&OM4 fibers. The circle lines represent the mean value of the amplitude.
7. Conclusion
A rigorous full vector MMF model has been proposed, full vector fiber modes instead of LP modes are applied in this model. Comparative study of vector and scalar MMF model in terms of differential mode delay (DMD), mode power distribution (MPD), transfer functions as well as eye diagrams showed that the vector nature cannot be ignored in the modeling of offset launched MMFs. Ignoring the vector nature of fiber modes led to divergent results, which highly impacts accuracy, especially when the launch offset value is large. A standard-compliant methodology has been used to evaluate the MMF bandwidth. The impact of the profile parameters on the MMF performance has been investigated. Statistical model of OM3 and OM4 fibers has been provided, which is believed to be essential for the research and development of MMF link transceivers and standard development of 112Gbps and beyond.
Funding
Huwei Technologies.
References and links
1. D. Mahgerefteh and C. Thompson, “Techno-economic Comparison of Silicon Photonics and Multimode VCSELs,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper M3B.2.
2. K. Szczerba, P. Westbergh, M. Karlsson, P. A. Andrekson, and A. Larsson, “60 Gbits error-free 4-PAM operation with 850 nm VCSEL,” Electron. Lett. 49(15), 953–955 (2013). [CrossRef]
3. K. Szczerba, P. Westbergh, M. Karlsson, P. A. Andrekson, and A. Larsson, “70 Gbps 4-PAM and 56 Gbps 8-PAM using an 850nm VCSEL,” J. Lightwave Technol. 33(7), 1395–1401 (2015). [CrossRef]
4. P. Westbergh, E. P. Haglund, E. Haglund, R. Safaisini, J. S. Gustavsson, and A. Larsson, “High-speed 850 nm VCSELs operating error free up to 57 Gbit/s,” Electron. Lett. 49(16), 1021–1023 (2013). [CrossRef]
5. D. Kuchta, A. V. Rylyakov, C. L. Schow, J. Proesel, C. Baks, P. Westbergh, J. S. Gustavsson, and A. Larsson, “64Gb/s Transmission over 57m MMF using an NRZ Modulated 850nm VCSEL,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th3C.2.
6. D. M. Kuchta, A. V. Rylyakov, F. E. Doany, C. L. Schow, J. E. Proesel, C. W. Baks, P. Westbergh, J. S. Gustavsson, and A. Larsson, “A 71-Gb/s NRZ modulated 850-nm VCSEL-based optical link,” IEEE Photon. Technol. Lett. 27(6), 577–580 (2015). [CrossRef]
7. M. Webster, L. Raddatz, I. H. White, and D. G. Cunningham, “A statistical analysis of conditioned launch for gigabit ethernet links using multimode fiber,” J. Lightwave Technol. 17(9), 1532–1541 (1999). [CrossRef]
8. J. Ingham, R. Penty, and I. White, “University of Cambridge Multimode-Fiber Model Results,” http://ieee802.org/3/aq/public/tools/108fiberModel/
9. P. Pepeljugoski, M. J. Hackert, J. S. Abbott, S. E. Swanson, S. E. Golowich, A. J. Ritger, P. Kolesar, Y. C. Chen, and P. Pleunis, “Development of system specification for laser-optimized 50µ m multimode fiber for multigigabit short-wavelength LANs,” J. Lightwave Technol. 21(5), 1256–1275 (2003). [CrossRef]
10. P. Pepeljugoski, S. E. Golowich, A. J. Ritger, P. Kolesar, and A. Risteski, “Modeling and simulation of next-generation multimode fiber links,” J. Lightwave Technol. 21(5), 1242–1255 (2003). [CrossRef]
11. R. E. Freund, C. Bunge, N. N. Ledentsov, D. Molin, and C. Caspar, “High-speed transmission in multimode fibers,” J. Lightwave Technol. 28(4), 569–586 (2010). [CrossRef]
12. M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol. 27, 1248–1261 (2009) [CrossRef]
13. M. J. Li and D. A. Nolan, “Optical transmission fiber design evolution,” J. Lightwave Technol. 26(9), 1079–1092 (2008). [CrossRef]
14. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry,” J. Lightwave Technol. 26(13), 1868–1876 (2008). http://www.mathworks.com/matlabcentral/fileexchange/45809-fiber-grating-solver [CrossRef]
15. FOTP-220: Differential Mode Delay Measurement of Multimode Fiber in the Time Domain, TIA-455-220-A, (2003).
