Spectral-dependent electronic-photonic modeling of high-speed VCSEL-MMF links for optimized launch conditions

Shanglin Li; Mohammadreza Sanadgol Nezami; Shubhankar Mishra; Odile Liboiron-Ladouceur

doi:10.1364/OE.411348

1. Introduction

Due to low power consumption [1], high modulation speed [2], and low cost [3], vertical-cavity surface-emitting laser (VCSEL)-multimode fiber (MMF) interconnects represent over 85% of short-reach datacenter links [4]. Further, more than 90% of MMFs are shorter than 100 m [5]. As the demand for greater capacity intensifies, high-speed devices such as VCSELs and photodetectors (PDs) are developed [6,7]. Data rates of VCSEL-MMF transmission systems reach 100 Gbps per lane with equalization [8] and potentially reach 1 Tbps in aggregated capacity through multiplexing techniques [2]. At high-speed transmission, the mode launch condition has significant impact on the performance due to the enhanced fiber sensitivity to mode power distribution. The optimization of the beam launch condition, such as offset launch techniques [9–13] and tilt launch techniques [14], was studied for fiber bandwidth enhancement. As VCSEL-MMF links consist of electrical and optical components, as well as optoelectronic devices, a hybrid electronic-photonic link model plays a significant role in the design and evaluation of high-speed next-generation interconnects.

Device-level modeling previously reported [15,16] do not account for the overall link-level system modeling in the device optimization. At the system level, a 10 GbE system model illustrates modal and chromatic dispersion interference (MCDI) [17]. However, the spectral dependence and other fiber effects are neglected affecting the simulation accuracy of the higher speed link. In addition, existing mode launch models only exploit specific mode-excitation conditions [18,19]. In [20,21], spatial segments of a PD are utilized to detect mode-division-multiplexed signals, where a fairly simple fiber-PD coupling model is given. Nonetheless, they do not further discuss the effects of launch conditions and free-space diffraction phenomenon between the fiber and the PD. Therefore, more accurate models of the VCSEL-MMF and MMF-PD coupling are required to simulate high-speed VCSEL-MMF links for next-generation interconnects. Additionally, signal processing CMOS circuits, such as equalizers, are generally used to extend link capabilities and mitigate the contradiction between increased data rates and limited device bandwidth [22,23]. In such context, hybrid electronic-photonic modeling of a VCSEL-MMF link allows modular extension within an electronic design automation environment.

In this paper, we propose a comprehensive hybrid-electronic-photonic modeling platform sensitive to the coupling conditions for high-speed VCSEL-MMF links. The paper outline is as follows. In Section 2, the spectral-dependent beam coupling from the VCSEL to the free-space is analyzed. Section 3 explains the mode coupling over the fiber input facet. The radial offset and 3-dimensional (3-D) angular tilt are simultaneously included and coupling coefficient matrices are calculated for different coupling conditions. Section 4 introduces the coupling-coefficient-dependent fiber model sensitive to launch conditions. An advanced split-step small-segment (4-S) method calculates the signal evolution over fiber. In Section 5, along with the coupling analysis from the MMF to the PD, a wavelength-sensitive PD model, matching the multi-wavelength characteristic of the VCSEL, realizes the opto-electric conversion. The proposed link model is developed in the commercial Cadence tool suite, a schematic-driven tool for CMOS electronic integrated circuit design. In Section 6, the high accuracy of the electronic-photonic modeling platform is validated by 25 Gbps non-return-to-zero (NRZ) transmission experiments over different coupling conditions and transmission distances. Section 7 presents impact of launching conditions on mode power distribution With spectral-dependent 3-D coupling conditions investigated, it is found that compared to high-order mode groups, further optical power can be distributed to low-order mode groups using a larger radial offset with a tilted beam. In Section 8, we summarize and conclude.

2. VCSEL-to-free-space coupling

The VCSEL dominates short-reach datacenter systems as the laser source, due to low cost, excellent energy efficiency, and high modulation speed. As a part of the VCSEL-MMF link model, an advanced equivalent circuit model of the multiple-quantum-well (MQW) VCSEL has been proposed and detailed in [24] by the authors, which includes the extrinsic parasitic and equivalent circuits. The temperature-dependence and noise effects are accounted in the model as well as carrier and photon dynamisms. With the extracted parameters, the VCSEL model shows a satisfactory ability to reproduce the measurement results in small-signal and large-signal responses.

The VCSEL outputs multiple transverse modes, with typical numbers from 3 to 10 [25]. When each VCSEL transverse mode is coupled into the MMF, the launch condition significantly impacts the link performance. Therefore, it is important to define the corresponding power of each VCSEL transverse mode and model the launch-condition-dependent VCSEL-MMF coupling.

The VCSEL is modelled as a cylindrical weakly-guided step-index waveguide supporting linearly polarized (LP) modes [26]. Each mode exhibits its own specific resonant wavelength meeting the cavity resonance phase condition. The wavelength of the gth VCSEL LP mode, λ_g, is obtained from Eq. (1) using the effective index model [27]:

(1)$${\lambda _g}\textrm{ = }{\lambda _c} - {\lambda _c}\frac{{{\mu _g}^2({{n_a} - {n_c}} )}}{{{n_c}({{\mu_g}^2 + {\omega_g}^2} )}}$$

where λ_c is the central wavelength, n_a and n_c are the refractive indices of the core and the cladding, µ_g and ω_g represent the core and cladding propagation parameter of the gth VCSEL mode. In the simulation, the value of n_a is obtained from the publications [28,29]. Based on the step-index waveguide eigenvalue equation [30], n_c, µ_g and ω_g are derived by fitting the measured VCSEL optical spectrum. It is worth mentioning that the VCSEL wavelength compositions slightly vary between devices due to fabrication process variations. For applications of mass production of VCSEL-MMF modules, statistical mean and worst-case parameters of VCSELs should be included in the model. In this paper, as a proof-of-principle demonstration of the model, a typical multimode VCSEL spectrum is measured with a bias current of 6 mA using a 25 Gbps VCSEL (manufactured by II-VI Incorporated). The measured overall spectral power with an optical spectrum analyzer (Anritsu MS9710C) is shown in Fig. 1(a), fitting a Gaussian distribution. Based on the obtained Gaussian distribution and the wavelengths of VCSEL modes calculated in (1), Fig. 1(b) illustrates the output power of each VCSEL mode normalized to the peak value of the Gaussian distribution curve. The corresponding power of the gth mode (from right to left) is denoted as P_g. As seen from Fig. 1(b), the third and fourth VCSEL transverse modes (i.e., LP₂₁ and LP₀₂) have close wavelengths such that both modes are superposed due to the linewidth of the transverse mode, resulting in a merged peak.

Fig. 1. (a) The measured VCSEL optical output spectrum. The solid blue line is the VCSEL spectrum for a typical bias current of 6 mA; the red dash line is a Gaussian fitting curve. (b) The normalized output power for each VCSEL mode to the peak value of the Gaussian distribution curve.

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The cylindrical coordinate system ($\tilde{r}$, $\tilde{\varphi }$, $\tilde{z}$) is used for modeling the VCSEL mode at its output, as shown in Fig. 2. The radial coordinate axis $\tilde{r}$ and the azimuthal coordinate axis $\tilde{\varphi }$ are at the VCSEL surface with the pole $\tilde{o}$ at the VCSEL aperture center. The coordinate axis $\tilde{z}$ is perpendicular to the VCSEL surface. The normalized transverse field of the gth VCSEL mode, $\widetilde {LP}$_g, is derived using Eq. (2) at $\tilde{z} = 0$ (i.e., at the VCSEL surface).

(2)$${\widetilde {LP}_g}(\tilde{r},\tilde{\varphi }) = {N_g} \cdot \left\{ {\frac{{{J_l}({{{\mu_g} \cdot \tilde{r}} / {{a_v}}})}}{{{J_l}({\mu_g})}} \cdot [{1 - H(\tilde{r} - {a_v})} ]\textrm{ + }\frac{{{K_l}({{{\omega_g} \cdot \tilde{r}} / {{a_v}}})}}{{{K_l}({\omega_g})}} \cdot H(\tilde{r} - {a_v})} \right\} \cdot {e^{ - il\tilde{\varphi }}}$$

where N_g is a normalization constant, a_v is the VCSEL oxide aperture radius, l is the azimuthal index of the gth LP mode, i is the imaginary unit. H(·) is the Heaviside Step Function, J_l(·) denotes the lth-order Bessel function of the first kind, and K_l(·) represents the lth-order modified Bessel function of the second kind.

Fig. 2. Launching system between the VCSEL and the MMF using cylindrical coordinates ($\tilde{r}$, $\tilde{\varphi }$, $\tilde{z}$) referenced to the VCSEL and ($\hat{r}$, $\hat{\varphi }$, $\hat{z}$) referenced to the fiber. R is the fiber core radius. The optical system between the VCSEL and the MMF is defined by a 2×2 matrix.

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As the VCSEL beam propagates in free space after emission, the diffraction will change its amplitude and phase. As a result of the symmetric circular structure of the VCSEL, modes are expanded over the Laguerre-Gaussian (LG) mode basis, accounting for the diffraction phenomenon. Equation (3) expresses the normalized spectral-dependent mode profile of the hth LG mode, ${\widetilde {{LG}}_h}$, in the cylindrical coordinate ($\tilde{r}$, $\tilde{\varphi }$, $\tilde{z}$), with f₁ and f₂ representing the amplitude and phase functions of the LG mode beam.

(3)$${\widetilde {LG}_h}(\tilde{r},\tilde{\varphi },\tilde{z},\lambda ) = {f_1}({\tilde{r},\tilde{z},\lambda } )\cdot {e^{ - i \cdot {f_2}({\tilde{\varphi },\tilde{z},\lambda } )}}$$

The functions f₁ and f₂ are defined in Eq. (4) and Eq. (5),

(4)$${f_1}({\tilde{r},\tilde{z},\lambda } )= \sqrt {\frac{{2n!}}{{\pi (n + |m |)!}}} \cdot \frac{1}{{\omega ({\tilde{z},\lambda } )}} \cdot {\left( {\frac{{\sqrt 2 \tilde{r}}}{{\omega ({\tilde{z},\lambda } )}}} \right)^{|m |}} \cdot L_n^{|m |}\left( {\frac{{2{{\tilde{r}}^2}}}{{\omega {{({\tilde{z},\lambda } )}^2}}}} \right)$$

(5)$${f_2}({\tilde{\varphi },\tilde{z},\lambda } )= \frac{{\pi {{\tilde{r}}^2}}}{{\lambda q({\tilde{z},\lambda } )}} - f({\tilde{z},\lambda } )\textrm{ + }\frac{{2\pi \tilde{z}}}{\lambda }\textrm{ + }m\tilde{\varphi }$$

where n and m are the radial and azimuthal indices of the LG mode, λ stands for the beam wavelength, ${L}_n^m\textrm{(}\cdot\textrm{)}$ is the generalized Laguerre polynomial, ω(${\tilde{z}}$,λ) denotes the spot size, ϕ(${\tilde{z}}$,λ) expresses the Gouy phase shift, and q(${\tilde{z}}$,λ) is a complex beam parameter. The expressions for q(${\tilde{z}}$,λ), ω(${\tilde{z}}$,λ), and ϕ(${\tilde{z}}$,λ) are defined by Eqs. (6) to (8):

(6)$$q({\tilde{z},\lambda } )= \frac{{\pi {\omega _0}^2\textrm{A}({\tilde{z}} )- i\lambda \textrm{B}({\tilde{z}} )}}{{\pi {\omega _0}^2\textrm{C}({\tilde{z}} )- i\lambda \textrm{D}({\tilde{z}} )}}$$

(7)$$\omega ({\tilde{z},\lambda } )={-} \frac{\lambda }{{\pi {\mathop{\rm Im}\nolimits} ({q({\tilde{z},\lambda } )} )}}$$

(8)$$\phi ({\tilde{z},\lambda } )= ({2n + |m |+ 1} )\textrm{arctan}\left( {\frac{{\textrm{B}({\tilde{z}} )\cdot \lambda }}{{\textrm{A}({\tilde{z}} )\cdot \pi {\omega_0}^2}}} \right)$$

where ω₀ is the beam waist. A($\tilde{z}$), B($\tilde{z}$), C($\tilde{z}$) and D($\tilde{z}$) are the elements of a 2×2 ABCD matrix, describing the optical system between the VCSEL and the MMF (see in Fig. 2) [31,32]. The optical system represented by the transfer matrix will depend on the type of optical module and may include a collimating lens or a 45° reflective metal, or both.

With Eqs. (2) and (3), the coupling coefficient c_gh between the gth VCSEL mode, $\widetilde {{LP}}$_g, and the hth LG mode, ${\widetilde {{LG}}_h}$, is obtained by the overlap integral as follows:

(9)$${c_{gh}}({\lambda _g}) = {K_f}\int\!\!\!\int_{{A_\textrm{v}}} {{{\widetilde {LP}}_g}({\tilde{r},\tilde{\varphi }} )\cdot {{[{{{\widetilde {LG}}_h}({\tilde{r},\tilde{\varphi },0,{\lambda_g}} )} ]}^ \ast }\tilde{r}d\tilde{r}d\tilde{\varphi }}$$

where K_f is the Fresnel coefficient for the fiber–air interface, A_v is the VCSEL aperture area, and the asterisk denotes the complex conjugate. K_f is approximately 0.98 because of the small refractive index difference between the air and the glass [28].

3. Free-space-to-MMF coupling

MMF is considered as a weakly-guided graded-index (GI) waveguide with the refractive index (RI) profile parameter α. Due to the cylindrical structure of the fiber, it is common to use the cylindrical coordinate system. As such, ($\hat{r}$, $\hat{\varphi }$, $\hat{z}$) describes the MMF LP mode field distribution. The radial coordinate axis $\hat{r}$ and the azimuthal coordinate axis $\hat{\varphi }$ are at the MMF input facet with the pole $\hat{o}\; $at the center, and the $\hat{z}$-axis is along the fiber axis, as shown in Figs. 2 and 3. The analytical expression in Eq. (10) elucidates the uth MMF LP mode within the profile parameter range (1.8<α<2.2) of manufactured optical fibers [18] at $\hat{z} = 0$ (i.e., at the fiber input facet).

(10)$${\widehat {MLP}_u}(\hat{r},\hat{\varphi },\lambda ) = \frac{1}{R} \cdot \sqrt {\frac{{{V^{|{l^{\prime}} |\textrm{ + }1}}\Gamma (p^{\prime})}}{{\pi \,\Gamma (p^{\prime} + |{l^{\prime}} |)}} \cdot } {\left( {\frac{{\hat{r}}}{R}} \right)^{|{l^{\prime}} |}} \cdot L_{p^{\prime} - 1}^{|{l^{\prime}} |}\left( {\frac{{{{\hat{r}}^2}}}{{{R^2}}} \cdot \textrm{V}(\lambda )} \right) \cdot {e^{ - \frac{{{{\hat{r}}^2}}}{{2{R^2}}} \cdot \textrm{V}(\lambda )- il^{\prime}\hat{\varphi }}}$$

where l’ and p’ are the azimuthal and radial indices of the MMF LP mode, R is the fiber core radius, $\textrm{V}(\mathrm{\lambda } )\textrm{ = 2}\mathrm{\pi }\textrm{R}\sqrt {{\textrm{n}_\textrm{1}}{{(\mathrm{\lambda } )}^\textrm{2}} - {\textrm{n}_\textrm{2}}{{(\mathrm{\lambda } )}^\textrm{2}}} /\mathrm{\lambda }$ is the wavelength-dependent normalized frequency, and Γ(m) is the gamma function equal to the factorial of m−1. The modeling of the wavelength dependence of the core peak RI, n₁, and the cladding RI, n₂, uses the third order Sellmeier equation [33]. It is worth noticing that Eqs. (3) and (10) are both dependent on the optical wavelength λ, which implies that the field distribution of LG modes and MMF LP modes are different for diverse wavelengths.

Fig. 3. The geometrical schematic diagram of beam launch. Point P is the center of the VCSEL, and Point A is the projection of P onto the fiber input facet. ψ is the incidence tilt angle between the $\hat{z}$-axis and the ž-axis, and θ is the azimuthal tilt angle between the $\hat{y}$-axis and the $\mathop y\limits^\vee $-axis.

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Figure 3 illustrates an arbitrary launch of LG beams with a radial offset and a 3-D angular tilt. The reference point P is at the center of the VCSEL, z_d is the distance between the VCSEL center and the light spot center at the fiber input facet. Three right-handed Cartesian coordinate systems are used for modeling. The plane $\hat{x}$-$\hat{y}$ of the coordinate system $\hat{o}$-$\hat{x}$-$\hat{y}$-$\hat{z}$ is located at the input facet of fiber, where the origin $\hat{o}$ is the center of the fiber input facet, $\hat{x}$-axis coincides with the direction of the beam radial offset and $\hat{z}$-axis coincides with the fiber axis. $\bar{o}$-$\bar{x}$-$\bar{y}$-$\bar{z}$ has a lateral translation of a_off from $\hat{o}$-$\hat{x}$-$\hat{y}$-$\hat{z}$ along the $\hat{x}$-axis. Different from the first two coordinate systems, $\mathop o\limits^\vee $-$\mathop x\limits^\vee $-$\mathop y\limits^\vee $-ž is at the beam wave front plane, where $\mathop y\limits^\vee $-axis corresponds to the intersection line between the MMF input facet and the beam contour across $\mathop o\limits^\vee $, and ž-axis is along the beam optical axis. At the input facet of MMF, the free-space LG beam excites MMF LP modes, with the radial offset a_off, the incidence tilt ψ between the $\hat{z}$-axis and the ž-axis, and an azimuthal tilt θ between the $\hat{y}$-axis and the $\mathop y\limits^\vee $-axis. The combination of a_off, ψ and θ ensures an arbitrary launch condition in three-dimensional space, overcoming the restriction of launch conditions in reported literatures [18,19]. ψ has a value ranging from 0 to 90°. θ is between −180° and 180°, relying on the coordinates of the point P, ($\overline {{\textrm{P}_\textrm{X}}} $, $\overline {{\textrm{P}_\textrm{Y}}} $, ), with respect to the coordinate system $\bar{o}$-$\bar{x}$-$\bar{y}$-$\bar{z}$. When $\overline {{\textrm{P}_\textrm{Y}}} $> 0, θ is less than 0; otherwise, θ ≥ 0.

Equation (3) in the coordinate system ($\tilde{r}$, $\tilde{\varphi }$, $\tilde{z}$) is rewritten as Eq. (11) in the cylindrical coordinate system ($\hat{r}$, $\hat{\varphi },\hat{z}$),

(11)$${\widehat {LG}_h}({\hat{r},\hat{\varphi },\hat{z},\lambda } )= {\widetilde {LG}_h}({\tilde{r},\tilde{\varphi },\tilde{z},\lambda } )$$

where

\tilde{r} = \sqrt {{{\tilde{x}}^2} + {{\tilde{y}}^2}},

\tilde{x} = \hat{r}\sin \hat{\varphi }\cos \psi \sin \theta - ({{a_{off}} - \hat{r}\cos \hat{\varphi }} )\cos \psi \cos \theta - \hat{z}\sin \psi,

\tilde{y} = ({{a_{off}} - \hat{r}\cos \hat{\varphi }} )\sin \theta + \hat{r}\sin \hat{\varphi }\cos \theta,

\tilde{z} = \hat{z}\cos \psi - ({{a_{off}} - \hat{r}\cos \hat{\varphi }} )\sin \psi \cos \theta + \hat{r}\sin \hat{\varphi }\sin \psi \sin \theta + {z_d},

\tilde{\varphi }\textrm{ = }\left\{ {\begin{array}{c} {\pi \textrm{ + }\arctan \left( {\frac{{\tilde{y}}}{{\tilde{x}}}} \right),\;\,when\;\tilde{x} < 0}\\ {\quad \quad \frac{\pi }{2},\;\;\quad \quad \,\,\,when\;\tilde{x} = 0}\\ {\arctan \left( {\frac{{\tilde{y}}}{{\tilde{x}}}} \right),\;\quad \,when\;\tilde{x} > 0} \end{array}} \right. .

Based on Eqs. (10) and (11), the coupling coefficient for the wavelength λ between the hth decentered and tilted LG mode and the uth MMF LP mode, d_hu(λ), is expressed as Eq. (12).

(12)$${d_{hu}}(\lambda )= \int_0^{\textrm{ + }\infty } {\int_0^{2\pi } {{{\widehat {LG}}_h}(\hat{r},\hat{\varphi },0,\lambda ) \cdot {{[{{{\widehat {MLP}}_u}(\hat{r},\hat{\varphi },\lambda )} ]}^ \ast }\hat{r}d\hat{r}d\hat{\varphi }} }$$

The expansion of LG modes over the orthogonal MMF LP modes is obtained in Eq. (13),

(13)$${\widehat {LG}_h}(\hat{r},\hat{\varphi },0,\lambda ) = \sum\nolimits_{u = 1}^{{N_{MLP}}} {{d_{hu}}(\lambda ){{\widehat {MLP}}_u}(\hat{r},\hat{\varphi },\lambda )} + \sum\nolimits_{k = 1 + {N_{MLP}}}^\infty {{d_{hu}}(\lambda ){{\widehat {MLP}}_u}(\hat{r},\hat{\varphi },\lambda )}$$

where N_MLP denotes the number of guided LP modes propagating along MMF. The first term on the right-hand side of Eq. (13) indicates the guided LP modes, while the second term represents the lossy LP modes which powers radiate. It is worth mentioning that mode power distribution is independent of the coordinate system. Therefore, coupling coefficients c_gh are the same in the coordinate systems ($\tilde{r}$, $\tilde{\varphi }$, $\tilde{z}$) and ($\hat{r}$, $\hat{\varphi }$, $\hat{z}$). At the fiber facet ($\hat{r}$, $\hat{\varphi }$,0), the expansion of the gth VCSEL LP mode over the MMF LP modes is expressed in Eq. (14).

(14)$$\begin{array}{l} {\widehat {LP}_g}(\hat{r},\hat{\varphi },{\lambda _g}) = \sum\nolimits_{u = 1} {\sum\nolimits_{h = 1} {({{c_{gh}}({\lambda_g}) \cdot {d_{hu}}({\lambda_g}){{\widehat { \cdot MLP}}_u}(\hat{r},\hat{\varphi },{\lambda_g})} )} } \\ \quad \quad \quad \quad \quad \;\, = \sum\nolimits_{u = 1} {{e_{gu}}({\lambda _g}) \cdot {{\widehat {MLP}}_u}(\hat{r},\hat{\varphi },{\lambda _g})} \end{array}$$

where $\mathop \sum \nolimits_{{h = 1}} {c}_{gh}d_{hu}$ can be defined as e_gu. As illustrated in Fig. 4, (a) power coupling coefficient (PCC) matrix, Η’, whose element is η_gu, is obtained, where η_gu is equal to |e_gu|². In addition, a corresponding wavelength matrix Ω’ is generated, where the element, Λ_gu, is equal to λ_g. In view of the large value of N_MLP (over 200 for 850 nm wavelength), the number of elements in Η’ and Ω’ is on the order of 10³. As a result, tremendous computation cycles are needed to iterate each mode propagation. We further degenerate MMF modes into mode groups, which are indexed by the principle mode number (PMN) v=2p'+|l'|−1. Figure 5 depicts the MMF modes included in the first ten mode groups. Within the same mode group, modes share approximately the same propagation constant [34,35].

Fig. 4. The conversions of matrices Η’ to H and Ω’ to Ω. l and p are the azimuthal and radial indices of the VCSEL mode, while l’ and p’ are the azimuthal and radial indices of the MMF LP mode. v is the principle mode number of the MMF mode group. The element ξ of H is the summation of elements, η, circled by the same color in Η’; The elements in the gth row of Ω’ and Ω are equal to λ_g.

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Fig. 5. The azimuthal and radial index distribution in the first ten MMF LP mode groups. The first and second element in the parentheses are the azimuthal and the radial indices, respectively.

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A PCC matrix H with the element ξ_gv and a wavelength matrix Ω with the element ζ_gv are built for each mode group. Η’ and Ω’ are converted into H and Ω to express the coupling relation between VCSEL LP modes and MMF mode groups, as shown in Fig. 4. The number of guided mode groups N_MG is typically around 20 to 30. Therefore, the size of H and Ω is only about 10% of that of H’ and Ω’. The use of mode groups improves calculation efficiency of the proposed coupling model by approximately 90%. Due to nearly the same propagation characteristics of the degenerated modes, the use of mode groups maintains the computation accuracy very well, which is verified by works [36,37].

4. MMF model

Attenuation, dispersion, fiber mode mixing and mode partition noise (MPN) play significant roles in the signal distortion, especially in the VCSEL-based MMF link. Due to the MMF multimode excitation and the VCSEL multi-transverse-mode output, the model should include mode-dependence and wavelength-dependence of these effects. With a coherent source and mode-selective loss devices, it is worth pointing out that modal noise may occurs when the link suffers from source frequency changes or environmental vibrations such as mechanical stress. Modal noise results in speckle pattern variations at the fiber output translating into signal-to-noise ratio fluctuations [38]. The model assumes no source frequency fluctuations nor environment vibrations; thus, modal noise is not studied in this paper.

Attenuation and fiber mode mixing lead to mode power fluctuation. As a result of intrinsic waveguide properties and material defects [39], LP modes experience differential mode attenuation, where higher-order modes suffer more loss. Equation (15) defines the spectral-dependent modal attenuation for the νth MMF LP mode group excited by the gth VCSEL mode, γ_gv:

(15)$${\gamma _{gv}} = {\gamma _0}\textrm{ + }{\gamma _0} \cdot {\textrm{I}_\rho }\left( {\eta \cdot {{\left( {\frac{{{\zeta_{gv}}v}}{{\pi R}}} \right)}^{{{2\alpha } / {(\alpha + 2)}}}} \cdot {{\left[ {\frac{{\alpha + 2}}{{2\alpha }} \cdot \frac{1}{{{n_1}{{({\zeta_{gv}})}^2} - {n_2}{{({\zeta_{gv}})}^2}}}} \right]}^{{\alpha / {(\alpha + 2)}}}}} \right)$$

where γ₀ is the attenuation per unit length of the fundamental mode, η is a weighting constant, ζ_gv is the wavelength element in the matrix Ω, n₁(ζ_gv) and n₂(ζ_gv) is the core peak RI and the cladding RI of the MMF, and I_ρ(·) is the ρth-order modified Bessel function of the first kind. Thus, the attenuated output power P_out(gv), as a function of the transmission length z, is obtained as follows:

(16)$${P_{out(gv)}}(z )= {P_g} \cdot {\xi _{gv}} \cdot {e^{ - {\gamma _{gv}}z}}$$

In addition to differential mode attenuation, fiber mode mixing phenomenon contributes to power transfers between modes, inducing mode power fluctuation as well as inter-symbol interference (ISI). Based on coupled power equations, fiber mode mixing is expressed by Eq. (17),

(17)$$\frac{{\partial P_{gu}^M}}{{\partial z}} + \tau \frac{{\partial P_{gu}^M}}{{\partial t}} = \sum\nolimits_{\hat{u} \ne u} {d_{u\hat{u}}^M({P_{g\hat{u}}^M - P_{gu}^M} )}$$

where ${P}_{{gu}}^{M}$ is the power of the uth MMF LP mode induced by the gth VCSEL LP mode, ${d}_{{u\hat{u}}}^{M}$ is the coupling coefficient between the uth and ${\hat{u}}$th MMF LP modes [40,41], and τ is the time delay of the uth MMF LP modes per unit fiber length. The superscript of ${P}_{{gu}}^{M}$ and ${d}_{{u\hat{u}}}^{M}$, M, denotes the MMF mode.

Equation (17) aims at fiber modes, such that the simulation process calls this equation tens of thousands of times (in the order of 10⁴). With today’s computing capabilities, the computation load is important resulting in simulation time up to several hours. As modes within the same mode group exhibit the same transmission characteristics, after some mathematical operations, a modified steady-state coupled-power Eq. (18) is derived for the mode groups instead of Eq. (17),

(18)$$\frac{{\partial P_{gv}^G}}{{\partial z}} = \sum\limits_{\hat{v} \ne v} {d_{v\hat{v}}^G\left( {\frac{{P_{g\hat{v}}^G}}{{{N_{g,\hat{v}}} - {N_{g,\hat{v} - 1}}}} - \frac{{P_{gv}^G}}{{{N_{g,v}} - {N_{g,v - 1}}}}} \right)}$$

where ${P}_{{gu}}^{G}$ represents the power of the νth mode group excited by the gth VCSEL LP mode, and ${Ng,v}$ denotes the number of modes within the first ν mode groups. The superscript, G, represents the MMF mode group. ${d}_{{u\hat{u}}}^{G}$ expresses the coupling coefficient between the νth mode group and the ${\hat{v}}$th mode group, which is derived from ${d}_{{u\hat{u}}}^{M}$ following Eq. (19):

(19)$$d_{v\hat{v}}^G\textrm{ = }\sum\limits_{u = ({N_{g,v - 1}} + 1)}^{{N_{g,v}}} {\sum\limits_{\hat{u} = ({N_{g,\hat{v} - 1}} + 1)}^{{N_{g,\hat{v}}}} {d_{u\hat{u}}^M} }$$

While differential modal attenuation and fiber mode mixing influence the amplitude, modal dispersion and chromatic dispersion result in time delay difference, leading to ISI and MPN penalties. Using the well-known WKB (Wentzel-Kramers-Brillouin) analysis of the Helmholtz equation [42], the modal propagation constant, β_gv, is obtained using Eq. (20).

(20)$${\beta _{gv}}({{\zeta_{gv}}} )\textrm{ = }\frac{{\textrm{2}\pi {n_1}}}{{{\zeta _{gv}}}}\sqrt {1 - \frac{{{n_1}^2 - {n_2}^2}}{{{n_1}^2}}{{\left[ {\frac{{({\alpha + 2} )\cdot {\zeta_{gv}}^2 \cdot {v^2}}}{{\alpha \cdot 2{\pi^2}{R^2} \cdot ({{n_1}^2 - {n_2}^2} )}}} \right]}^{\frac{\alpha }{{\alpha + 2}}}}}$$

It is worth noting that n₁ and n₂ are wavelength-dependent as well based on the Sellmeier equation. The modal delay per unit length, ${t}_{{gv}}^{m}$ is proportional to the first-order derivative of β_ν,

(21)$$t_{gv}^m = {\left. { - \frac{{{\lambda^2}}}{{2\pi c}}\frac{{d{\beta_{gv}}(\lambda )}}{{d\lambda }}} \right|_{\lambda \textrm{ = }{\zeta _{11}}}}$$

where c is the speed of light in vacuum.

Due to the multi-transverse-mode nature of the VCSEL, its output shows a relatively wide spectrum as seen in Fig. 1. The chromatic dispersion is usually neglected for the low-speed data rate transmission because the chromatic delay is far less than the bit period; however, it is of vital significance for the performance evaluation of the next-generation high-speed (≥ 10 Gbps) VCSEL-MMF interconnects [36,43]. The chromatic delay per length, $ {t}_{{gv}}^{c}$, is expressed in (22),

(22)$$t_{gv}^c = \frac{{{S_0}}}{4}\left[ {\frac{{{\zeta_{gv}}^4 - {\lambda_0}^4}}{{{\zeta_{gv}}^3}}} \right]({{\zeta_{gv}} - {\zeta_{11}}} )$$

where λ₀ is the zero-dispersion wavelength, and S₀ is the zero-dispersion slope. With the interaction of the modal and chromatic dispersion, the total delay, t_gv, is equal to ${t}_{{gv}}^{m}$+${t}_{{gv}}^{c}$.

Mode competition among the lasing transverse modes of a multi-mode VCSEL contributes to random correlated fluctuations in the mode powers. In the dispersive fiber link, the time delay induced by dispersion effects de-correlates VCSEL modes, resulting in MPN. The standard deviation of MPN, σ_MPN, is expressed below [29],

(23)$${\sigma _{MPN}}^2 = {\kappa _{MPN}}^2 \cdot \left\{ {\sum\nolimits_g {{a_g}{{[{{F_g}(t )} ]}^2}} - {{\left[ {\sum\nolimits_g {{a_g}{F_g}(t )} } \right]}^2}} \right\}$$

where κ_PMN is the mode partition parameter. F_g is the signal waveform carried by the gth VCSEL mode after propagation over the fiber, and it is given as

(24)$${F_g}(t )= \sum\nolimits_{v = 1}^{{N_{MG}}} {{\xi _{gv}}{f_g}({t - {\tau_{gv}}} )}$$

where ξ_gv is the power coupling coefficient from matrix H, τ_gv is the temporal delay of the modulated signal induced by the MMF dispersion after fiber transmission, and f_g is the waveform before coupling to the MMF fiber for a given sequence.

A highly-efficient and highly-accurate fiber effect simulation approach, named split-step small-segment (4-S) method, is proposed to emulate the signal evolution along the MMF, as illustrated in Fig. 6(a). Using the 4-S method, the MMF fiber is equally divided into Ns segments with an identical length of Δz. The algorithm steps, outlined in the flowchart of Fig. 6(b), are implemented as follows:

Fig. 6. (a) Illustration of the split-step small-segment (4-S) method. The fiber length L is divided into several small segments with the length of Δz. (b) The flowchart describes the 4-S method, where N_V is the number of VCSEL modes, and N_MG is the number of guided MMF mode groups.

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Step 1: At the fiber input facet, the transmission distance, z, is 0, where the power of the vth MMF mode group excited by the gth VCSEL mode is a_gξ_gv;

Step 2: the first MMF mode group excited by the first VCSEL mode is selected;

Step 3: the selected mode group propagates over the first half of the small segment with the length of Δz/2, while only the mode mixing effect is numerically calculated;

Step 4: at the mid-point of the segment, attenuation and dispersion effects over the whole segment are imposed;

Step 5: the mode group propagates over the second half of the segment, while only the mode mixing is considered;

Step 6: the steps 3 to 5 are repeated until all MMF mode groups for all wavelengths are iterated over;

Step 7: the steps 2 to 6 are repeated over the next segments until the light reaches the fiber output facet;

Step 8: MPN is determined by Eqs. (23) and (24).

The error of the 4-S method is positively correlated with Δz; however, the computational cost is inversely proportional to Δz. Therefore, a shorter segment results in a more accurate result, while the simulation time increases. In our model, the small segment Δz is chosen as one-twentieth of the fiber length for the tradeoff between high accuracy and low computation burden. With this small segment, the simulation can reproduce the measurement satisfactorily as we will see in Section 6.

5. MMF-PD coupling model and PD model

A photodiode (PD) detects and demodulates the optical signal after fiber transmission. The beam coupling into the PD after the MMF influences the photocurrent magnitude, inducing the signal-to-noise ratio variation. In addition, the coupling between the MMF and the PD significantly affects the channel crosstalk in mode-group-division-multiplexing (MGDM) systems [21]. Due to the multi-mode VCSEL output and its wide spectrum, the PD model used in the VCSEL-MMF link should be wavelength-sensitive for high simulation accuracy. Figure 7(a) shows the beam propagation from the MMF output facet to the active area of PD. The cylindrical coordinate system ($\widehat {\hat{r}}$, $\widehat {\hat{\varphi }}$, $\widehat {\hat{z}}$) is built at the fiber output facet. Its origin $\widehat {\hat{o}}$ is at the facet center, and $\widehat {\hat{z}}$-axis directs towards the outside of the fiber along the fiber axis. The coordinate systems ($\widehat {\hat{r}}$, $\widehat {\hat{\varphi }}$, $\widehat {\hat{z}}$) and ($\hat{r}$, $\hat{\varphi },\hat{z}$) meet the following relations: $\hat{r} = \widehat {\hat{r}}$, $\hat{\varphi } = \widehat {\hat{\varphi }}$, and $\hat{z} = \widehat {\hat{z}} + \textrm{L}$. The MMF LP mode profile is described by Eq. (10), and Eq. (3) expresses the free-space LG mode after fiber to characterize beam diffraction. Using Eqs. (3) and (10), at the fiber output facet, the coupling coefficient between the uth guided MMF LP mode excited by the gth VCSEL mode and the wth free-space LG mode, denoted as f_guw, is given by the overlap integral as follows:

(25)$${f_{guw}} = \int_0^{\textrm{ + }\infty } {\int_0^{2\pi } {{{\widehat {MLP}}_u}({\hat{\hat{r}},\hat{\hat{\varphi }},{\Lambda _{gu}}} )\cdot {{[{{{\widetilde {LG}}_w}({\hat{\hat{r}},\hat{\hat{\varphi }},0,{\Lambda _{gu}}} )} ]}^ \ast }\hat{\hat{r}}d\hat{\hat{r}}d\hat{\hat{\varphi }}} }$$

where Λ_gu is the element from the mode wavelength matrix Ω’, due to the wavelength stability during fiber propagation. Based on the orthogonality of the generalized Laguerre polynomials [44, Eq. 7.414], the LG modes after the MMF, ${\widetilde {{LG}}_{w}}$, have a waist of ω₀=$\textrm{R}\sqrt {\textrm{2/V}({{\lambda }{g}} )} $, where R is the fiber core radius. This simplifies the expansion of MMF LP mode profiles over LG modes. Consequently, the MMF LP mode with the azimuthal index l’ and the radial index p’ is fully coupled onto the free-space LG mode with the radial and azimuthal indices n and m if the relations m = l’ and n = p'−1 are satisfied.

Fig. 7. (a) Beam launch with misalignment between the MMF and the PD. ($\widehat {\hat{r}},\widehat {\hat{\varphi }}$, $\widehat {\hat{z}}$)and ($\widetilde {\tilde{r}}$, $ \widetilde {\tilde{\varphi }}$, $ \widetilde {\tilde{z}}$) are cylindrical coordinate systems. The purple region represents the PD active area where photons are converted into electrons. (b) The equivalent circuit model of a PIN photodiode.

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A new cylindrical coordinate system ($\widetilde {\tilde{r}}$, $ \widetilde {\tilde{\varphi }}$, $ \widetilde {\tilde{z}}$) is built at the PD, where the axes $\widetilde {\tilde{r}}$ and $\widetilde {\tilde{\varphi }}$ are located at the surface of the PD active area, and the pole $\tilde{o}$ is at the center. Under this system, Eq. (26) expresses the electrical field for the wavelength λ_g at the PD surface,

(26)$${E_{PD(g)}}({\tilde{\tilde{r}},\tilde{\tilde{\varphi }}} )= \sum\nolimits_u {\sum\nolimits_w {[{{E_{gu}}({t - {\tau_{gu}}} )\cdot {e^{ - i \cdot {\beta_{gu}} \cdot L}} \cdot {f_{guw}} \cdot {{\widehat {LG}}_w}({\tilde{\tilde{r}},\tilde{\tilde{\varphi }},{d_{pd}},{\Lambda _{gu}}} )} ]} }$$

where E_gu is the waveform carried by the uth guided MMF LP mode, τgu is the relative time delay of the uth mode induced by modal and chromatic dispersion after fiber propagation, d_pd is the beam transmission distance between the MMF and the PD surface, and β_gu is the propagation coefficient.

The wavelength-dependent responsivity of a PIN photodiode, R_pd, is modeled as follows,

(27)$${R_{pd}}\textrm{(}\lambda \textrm{) = }\frac{{{\eta _{pd}}q\lambda }}{{2\pi \hbar c}}$$

where η_pd is the quantum efficiency, q is the electron charge, ħ is reduced Planck constant. Therefore, the photocurrent is

(28)$${I_p} = \sum\nolimits_{g\textrm{ = }1}^{{N_V}} {\left[ {{R_{pd}}({{\lambda_g}} )\cdot \int_0^{{r_{pd}}} {\int_0^{2\pi } {{{|{{E_{PD(g)}}({\tilde{\tilde{r}},\tilde{\tilde{\varphi }}} )} |}^2}\tilde{\tilde{r}}d\tilde{\tilde{r}}d\tilde{\tilde{\varphi }}} } } \right]}$$

where N_V is the number of VCSEL modes, and r_pd is the radius of the PD active area. A spectral-dependent equivalent circuit model of the photodiode with noise effects describes the behavior of a PIN photodiode, as shown in Fig. 7(b). R_sh and C_sh denote the shunt junction resistance and capacitance, R_S models the contact resistance, L_b and C_b are the parasitic inductance and capacitance due to the bond wires. The noise currents consist of four sources: the shunt-resistance thermal noise I_t__sh, the series-resistance thermal noise I_t__s, the shot noise I_s and the flicker noise I_f, which variances are expressed as (29)–(32), respectively:

(29)$$\left\langle {{I_{t\_sh}}^2} \right\rangle = \frac{{4kT\Delta f}}{{{R_{sh}}}}$$

(30)$$\left\langle {{I_{t\_s}}^2} \right\rangle = \frac{{4kT\Delta f}}{{{R_s}}}$$

(31)$$\left\langle {{I_s}^2} \right\rangle = 2q({I_d}\textrm{ + }{I_p})\Delta f$$

(32)$$\left\langle {{I_f}^2} \right\rangle = \frac{{{K_f}{I_d}^{{\varepsilon _f}}\Delta f}}{{{f_e}^{{\gamma _f}}}}$$

where k is the Boltzmann constant, T is the device temperature, K_f is the flicker noise coefficient, I_d is the dark current, f_e is the frequency at which the noise is measured, ɛ_f is the flicker noise exponent, γ_f is the flicker noise frequency coefficient and Δf is the bandwidth of the photodiode.

6. Experimental validation

A comprehensive spectral-dependent link model is presented, where the offset condition and 3-dimension tilt conditions are simultaneously considered. The model is developed within a commercial Cadence tool suite, so that the modular extension can be realized easily by designers to verify the performance of the targeted device. The simulation results and experimental validation will be given in this section.

An experimental setup is built on a vibration-isolated optical table to validate the model, as depicted in Fig. 8. First, a bias-tee is employed to combine the bias current and the high-speed signal generated by the pulse pattern generator (PPG). After the bias-tee, a wire-bonded 850 nm VCSEL chip (manufactured by II-VI Incorporated) is driven by the biased signal. The VCSEL beam is butt-coupled into the OM4 MMF, using a 6-axis micro-positioning stage to tune the relative position between the VCSEL and the MMF. After transmission over the MMF, the beam profiles and eye diagrams are measured, respectively. For beam intensity profile measurement, a charge-coupled device (CCD) camera beam profiler (Thorlabs BC106-VIS), located after the MMF, is used to detect the beam intensity profiles. To minimize mode-dependent fiber effects and maximize the influence of the launch condition, a back-to-back (B2B) configuration is employed with a short MMF of 1 meter. For eye diagram measurements, the light is fully coupled to the PD active region without misalignment. The oscilloscope (OSC) is synchronized in time with the PPG. The data signal is then analyzed on an oscilloscope.

Fig. 8. Experiment schematic. DC: direct current; PPG: pulse pattern generator; CCD: charge-coupled device; PC: personal computer; OSC: oscilloscope.

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The beam profiles in the top row of Fig. 9 are obtained from experiments, while those in the bottom row of Fig. 9 are the simulation results. The measurement and simulation show visually similar spatial beam intensity distribution to prove that the model is able to evaluate the guided fiber modes for different beam launch conditions. Their discrepancies are attributed to VCSEL polarization instability [45], imperfect parallelism between the VCSEL aperture and the CCD sensing surface, the CCD camera resolution, and the stray light. The donut profile in Fig. 9(a) has a dispersed intensity distribution over the CCD camera, thus it is more sensitive to these adverse impacts.

Fig. 9. (a): The beam intensity profile at the fiber input facet. (b)-(e): The intensity profiles of guided modes excited by different launch conditions at the fiber output facet. The launch conditions are: (b) a_off is 0 µm, ψ is 20° and θ is 0°; (c) a_off is 10 µm, ψ is 10° and θ is 0°; (d) a_off is 10 µm, ψ is 20° and θ is 45°; (e) a_off is 10 µm, ψ is 10° and θ is 90°. The top figures in (a)-(e) are the experimental results; the bottom ones present the simulation results.

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Utilizing the Cadence Tool Suite (Virtuoso Environment 5.10.41), we simulate the VCSEL-MMF link based on the proposed model. The parameters used in the simulation are summarized in Table 1, which values are extracted from the devices’ datasheets, publications [28,29,35,46–48], experimental measurement, and parameter fitting.

Table 1. Parameters Used in the Simulation

View Table

In the experimental setup, multi-mode fibers manufactured by Corning Inc., with the length 100 m and 200 m, are used for the transmission. A 25 Gbps PRBS7 NRZ signal is generated by the pulse pattern generator (Anritsu MU181020A) combined with a multiplexer (Anritsu MU1813A). Eye diagrams for center launch (no tilt nor offset) in experiments and simulations are shown in Fig. 10(a). Figure 10(b) plots peak-to-peak jitters, bit error rates (BERs) and eye-opening ratios versus fiber lengths. In the experiment, the peak-to-peak jitters are 13.9 ps, 16.5 ps and 20.0 ps for the B2B, 100 m and 200 m transmission, respectively, while the corresponding jitters are 14.1 ps, 16.6 ps and 19.8 ps in the simulation. The experimental BERs are 3.3${ \times }$10⁻⁷, 5.3${ \times }$10⁻⁶, and 1.3${ \times }$10⁻⁴ for the B2B, 100 m and 200 m transmission, while BERs in the simulation are 3.5${ \times }$10⁻⁷, 5.4${ \times }$10⁻⁶, and 1.3${ \times }$10⁻⁴. The BER values are limited by the parasitic bandwidth of the wired-bonded VCSEL [24] and expect to have lower values using the large-bandwidth VCSELs [49,50]. The eye-opening ratio is an important metric to show the eye diagram quality [51,52]. The vertical eye-opening ratio (a ratio of the eye height to the amplitude difference between one and zero logical levels) measured in the experiment are 0.379, 0.287 and 0.170 for the B2B, 100 m and 200 m transmission, while the simulation results are 0.382, 0.289 and 0.167. The horizontal eye-opening ratios normalized to the unit interval are 0.653, 0.588 and 0.500 for the B2B, 100 m and 200 m transmission, while the simulation results are 0.648, 0.585 and 0.505. The experiment and simulation have satisfactory match, which validates that the proposed model have strong ability to predict the transmission performance.

Fig. 10. (a) Eye diagrams for B2B, 100 m, and 200 m transmission. a_off is 0 µm, ψ is 0° and θ is 0°. (b) The fiber length versus jitters, BERs, and eye-opening ratios in the experiment (in orange) and simulation (in blue).

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To validate the launch condition sensitivity of the proposed model, a coupling condition is realized with a_off = 10 µm, ψ = 20° and θ = 45°. Figure 11 shows the eye diagrams for the center launch and the given launch. Due to misalignment, the amplitude of the PD output signal is approximately 25 mV, less than 110 mV relative to the center launch. However, fewer modes are excited so that mode-dependent signal distortion is effectively suppressed. With the fiber length 200 m, the experimental BER is 2.3${ \times }$10⁻⁶, while the simulation result is 2.5${ \times }$10⁻⁶. The experimental vertical eye-opening ratio is 0.337 and the horizontal eye-opening ratio normalized by the unit interval is 0.631. The values of 0.331 for the vertical eye-opening ratio and 0.636 for the horizontal eye-opening ratio are obtained in the simulation. The peak-to-peak timing jitter values in the measurement and simulation are 14.8 ps and 14.5 ps, respectively. The simulation results match the experiment and validate the ability of the model to simulate the transmission performance with arbitrary launch conditions. The mode power distributions under the center launch and the given misalignment launch are shown in Fig. 11(c). Compared to the center launch, the misalignment launch improves the transmission performance, but the additional attenuation resulting from coupling loss needs to be compensated.

Fig. 11. (a) Eye diagrams for 200 m transmission where a_off is 0 µm, ψ is 0° and θ is 0°. (b) Eye diagrams for 200 m transmission where a_off is 10 µm, ψ is 20° and θ is 45°. (c) Mode power distribution at the fiber input facet. The top plot refers to the center launch, while the bottom one corresponds to the misalignment launch with a_off = 10 µm, ψ = 20° and θ = 45°.

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7. Impact of launching conditions on mode power distribution

Mode power distribution (MPD) is of vital significance due to mode-dependent propagation characteristics over the MMF. When different launch conditions are applied, the model can evaluate the corresponding MPD and determine the optimum coupling scheme of VCSEL-MMF links. In Fig. 12, mode power distributions induced by a multi-mode VCSEL are shown, with only the radial offset a_off included. With an increase in radial offset, higher-order mode groups obtain more power, which matches our expectation for the offset excitation. Figure 12(e) shows the effective modal bandwidths for different radial offsets without including fiber mode mixing and MPN. The horizontal axis is normalized to the bandwidth-distance product of the center launch. Compared to the center launch, the bandwidth-distance product of the MMF link has a 1.1-fold, 1.4-fold and 1.8-fold improvement for radial offsets of 10 µm, 15 µm and 20 µm, respectively.

Fig. 12. (a)-(d) The mode power distribution only with the radial offset: (a) The offset is 0 µm (center launch); (b) the offset is 10 µm; (c) the offset is 15 µm; (d) the offset is 20 µm. (e) The frequency response for different radial offsets.

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MPD patterns of purely angular launch are shown in Fig. 13(a)-(d) for angular tilts of 0°, 5°, 10° and 15°. When ψ is 15°, low-order mode groups vanish, and the power concentrates on high-order guided mode groups. As a result, the bandwidth-distance product improves by 3.5-fold as seen in Fig. 13(e). Based on Fig. 12 and Fig. 13, the tilt tuning plays a better role for the control of selective mode excitation, due to the limited impact of radial offsets.

Fig. 13. (a)-(d) The mode power distribution only with tilt ψ: (a) ψ is 0° (center launch); (b) ψ is 5°; (c) ψ is 10°; (d) ψ is 15°. (e) The frequency response for different angular tilts.

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When the radial offset a_off and the tilt ψ are simultaneously changed, the MPD can be optimally controlled. MPD figures are plotted in Fig. 14(a)-(c) and (e)-(g). In (a)-(c), a_off is fixed at 10 µm, and the values of ψ are 5°, 10° and 15°, respectively. With the same radial offset, higher-order mode groups are excited with a larger ψ, which matches the conclusion of purely angular launch. This result is explained by the fact that the input beam has a better overlap with the high-order MMF modes for large angular tilts. However, the improvement of the bandwidth-distance product for 15° is only 1.5-fold over the center launch. In (e)-(g), the tilt ψ is fixed to 10°, and a_off is varied to 5 µm, 10 µm, and 15 µm, respectively. The lower-order mode groups obtain even more portion of power for a larger radial offset. Therefore, the bandwidth-distance product decreases with the radial offset increasing, as shown in Fig. 13(h). The reason is that the radial offset is counteracted by the angular tilt misalignment, which implies the combination of a radial offset and an angular tilt does not necessarily result in expected high-order-mode excitation.

Fig. 14. (a)-(c) The mode power distribution with a fixed radial offset of 10 µm and varied tilts ψ: (a) ψ is 5°; (b) ψ is 10°; (c) ψ is 15°. (d) The corresponding frequency response for different angular tilts. (e)-(g) The mode power distribution with a fixed angular tilt (10°) and changed radial offsets: (e) a_off is 5 µm; (f) a_off is 10 µm; (g) a_off is 15 µm. (h) The corresponding frequency response for different radial offsets.

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Finally, Fig. 15(a)-(d) show MPD patterns, with a radial offset a_off, tilt ψ, and tilt θ concurrently analyzed. a_off is fixed at 10 µm; ψ is fixed at 10°; the values of θ are 45°, 90°, 135° or 180°. Higher-order mode groups are excited by a larger tilt θ. The modal bandwidths under these launch conditions are shown in Fig. 15(e). The bandwidth-distance product increases with the increasing value of θ. When a_off is 10 µm, ψ is 10° and θ is 180°, the bandwidth-distance product has a 3.8-fold improvement over the center launch. The additional coupling loss is 4.5 dB of due to lower coupling efficiency from the laser to the fiber. Figure 15(f) displays simulated eye diagrams for 25 Gbps NRZ transmission over 200 m fiber length, with θ of 0° (corresponding to the magenta curve in Fig. 15(e)) and 180° (corresponding to the aqua curve). The vertical and horizontal eye-opening ratios of the top eye diagram are 0.17 and 0.56, respectively. The bottom one is more widely open by virtue of better selective mode excitation, with the vertical eye-opening ratio of 0.32 and the horizontal eye-opening ratio of 0.62. The results are based on simulation only and provide a sense of the possible improvement in the frequency-distance curves with respect to launch conditions.

Fig. 15. (a)-(d) The mode power distribution with a_off, ψ and θ concurrently analyzed: (a) a_off is 10 µm, ψ is 10° and θ is 45°; (b) a_off is 10 µm, ψ is 10°, and θ is 90°; (c) a_off is 10 µm, ψ is 10° and θ is 135°; (d) a_off is 10 µm, ψ is 10° and θ is 180°. (e) The frequency response for different θ values. (f) Eye diagrams of 25 Gbps NRZ after 200 m transmission. The launch conditions are: (top) a_off = 10 µm, ψ=10°, θ = 0°; (bottom) a_off = 10 µm, ψ=10°, θ = 180°.

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8. Conclusion

In this paper, we propose a spectra-dependent VCSEL-MMF link model for the next-generation high-speed interconnects. Spectral-dependent coupling analysis is realized with the offset and tilt angles included. The coupling model is sensitive to mechanical misalignments (radial offset, axial offset, and 3-dimensional tilt) and optical spectra. With mode-dependent fiber effects explained, the signal evolution over the MMF is simulated by an advanced split-step small-segment method. In addition, the equivalent circuit model of a wavelength-sensitive PD, matching the multi-wavelength characteristic of the VCSEL, realizes opto-electric conversion. The link model is validated using a 25 Gbps NRZ transmission experiment. Measured eye diagrams and bit error rates show good agreement with the simulation. While small-signal measurements have not been done, it is considered in future work. A CMOS-compatible electronic-photonic simulation platform is built in Cadence tool suits to help users design their devices. The model verifies that the misalignment launch can improve the effective modal bandwidth greatly compared to the conventional center launch (e.g. 3.8-fold improvement with a_off of 10 µm, ψ is 10° and θ is 180°), although the concomitant additional attenuation needs compensation. As one of the applications, it can be used to explore the optimum 3-D coupling solution of VCSEL-MMF links. With the transfer matrix representing the optical system between the VCSEL and the MMF, our model also provides a strong tool for optimizing components in the practically produced multimode optical module, such as the focal length of a lens inserted between the VCSEL and the MMF. While a typical VCSEL was used in this work, the statistical variations in the VCSEL parameters can be included in the model to investigate the variances in the performance obtained with various launch conditions.

Funding

Mitacs Elevate program (IT12564) ; Natural Sciences and Engineering Research Council of Canada Strategic Partnership Grant Program (494385-2016); Fonds de recherche du Québec – Nature et technologies (282433).

Disclosures

The authors declare no conflicts of interest.

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Symbol	Description	Value	Symbol	Description	Value
a_v	VCSEL oxide aperture radius	3 µm	λc	Central wavelength	852 nm
n_a	VCSEL core RI	3.6	r_pd	PD active area radius	40 µm
n_c	VCSEL cladding RI	3.59	η_pd	Quantum efficiency	0.87
R	GI-MMF core radius	25 µm	R_S	PD contact resistance	0.72 mΩ
n₁	GI-MMF core peak RI	1.47 (at 850 nm)	R_sh	PD shunt junction resistance	430 kΩ
n₂	GI-MMF cladding RI	1.45 (at 850 nm)	C_sh	PD shunt junction capacitance	0.18 pF
α	RI profile parameter	2	C_b	PD parasitic capacitance	98 fF
γ₀	Attenuation coefficient	0.53 $\times$ 10⁻³ /m (at 850 nm)	L_b	PD parasitic inductance	0.44 nH
ρ	Order of Bessel function	9	I_d	Dark current	0.1 nA
η	Attenuation weighting constant	7.35	K_f	Flicker noise coefficient	$1 \times$ 10⁻¹²
S₀	Zero-dispersion slope	0.101 ps/(nm²·km)	ɛ_f	Flicker noise exponent	1
λ₀	Zero-dispersion wavelength	1310 nm	γ_f	Flicker noise frequency coefficient	1
κ_PMN	Mode partition parameter	0.03

Spectral-dependent electronic-photonic modeling of high-speed VCSEL-MMF links for optimized launch conditions

Abstract

1. Introduction

2. VCSEL-to-free-space coupling

3. Free-space-to-MMF coupling

4. MMF model

5. MMF-PD coupling model and PD model

6. Experimental validation

7. Impact of launching conditions on mode power distribution

8. Conclusion

Funding

Disclosures

References

Cited By

Figures (15)

Tables (1)

Equations (37)

Optics Express