Point radiance calculation of GI-MMF and its application in a fiber connection

Xuesong Mao; Xuesong Mao; Manabu Kagami

doi:10.1364/OE.476494

1. Introduction

Growing demands for high-speed data communication in short-reach applications have led to considerable research interest pertaining to multimode fibers (MMFs) in recent years [1], [2]. In the field of data center communications, fiber to the home/node/curb/business (FTTx), and in-vehicle networks, the data rate requirements are rapidly increasing with the adoption of the internet of things. Thus, data speeds exceeding the bandwidth capacity of MMF are required, which puts forward the proposition of various modulation formats and multiplexing technologies [3]–[5]. In addition, the magnitude of transmission loss and modal dispersion still constitute the bottlenecks in MMF transmission, especially the latter, as it is significantly affected by the MMF installation conditions and the insertion conditions of passive optical components such as filters and connectors, which causes fluctuations in the transmission characteristics.

In comparison to step-index MMFs (SI-MMF), graded-index MMF (GI-MMF) holds more potential in achieving the Gigabit transmission speed owing to its lower modal dispersion [6]-[8]. Although the graded refractive index profile significantly improves the transmission capacity, the attenuation and MPD variation along the transmission line still limit the bandwidth and distance of the transmission link [9], [10]. The launch condition of the optical signal, including the power transfer angle and power spot illuminated on the fiber end-face, considerably affects the transmission property of the GI-MMF [11]. Consequently, several methods have been developed to define [12], [13], theoretically predict [14], and measure [15] the MPD inside MMFs, and to calculate the coupling coefficients in GI-MMFs [16]. Unfortunately, no rigorous mathematical model could predict the variation of MPD at fiber discontinuous points, such as the insertion points of fiber connectors, couplers, and splitters. Therefore, link-level channel evaluation has not been realized to date, and network designers are forced to employ the data measured in worst-case scenarios such as various fiber lengths and misalignment at connectors.

Since the advent of MMF in the 1960s, researchers have continuously studied the loss introduced by the fiber connector and its influence on signal transmission. Early studies individually focused on calculating the losses caused by lateral [17], longitudinal [18], and angular [19], [20] misalignments in fiber connectors. Recently, the combined effects of the former two misalignments [21] have been theoretically researched, and the complete effects occurring in the fiber connector have been experimentally determined [22]. However, the loss is not the only factor that should be considered in the fiber connectors. In case we cannot predict the MPD variation from the emitting fiber to receiving fiber connected by the fiber connector, the simulation of the subsequent fiber will be terminated owing to the lack of inputs. For the losses or MPD, the theories were developed based on geometric optics because the radius of the MMF is much larger than the wavelength of the light propagating inside it. The basic principle involves the division of the emitting fiber end-face into small units and calculates the power along a certain angle that arrives at the receiving fiber. In this case, the power of a small unit on the emitting fiber end-face transferring along a certain angle is defined as the point radiance, which is unquantifiable via experimental measurements. The physical instruments, such as charge-coupled device (CCD) cameras, record the near-field pattern (NFP) and far-field pattern (FFP) of the fiber, which correspond to the integration of point radiance over power transfer angle and fiber core radius, respectively. Prior research assumed the use of a Lambertian source at the emitting side [19], indicating that the point radiance is a constant. Recent studies assumed that the point radiance is equal to the fiber FFP under the condition that the points on the emitting fiber end-face are identical [21]. In our previous study, for SI-MMF, we theoretically established that point radiance equals the multiplication of fiber NFP and FFP [23]. However, for GI-MMF, the point radiance has not been discussed yet because of the complicated refractive index profile. Notably, the numerical aperture (NA) of points on the GI-MMF end-face decreases as the range of points to the fiber core center increases [18]. Therefore, the form of the point radiance of GI-MMF alters from that of the fiber FFP. Thus, research should focus on establishing theories to accurately model the radiance of points on the GI-MMF end-face for correctly evaluating the loss and MPD variation introduced by the GI-MMF connector.

In this study, we derived a set of equations to compute the point radiance of GI-MMF according to the fiber NFP and FFP, which were represented using analytical functions. Similar to the treatment of SI-MMF in [23], we assumed that the point radiance of GI-MMF exhibits the same form as the fiber FFP, but the aperture of the point radiance varies with the distance from the point to the fiber core center. Thereafter, to ensure that the fiber NFP and FFP determined from the point radiance are similar to the NFP and FFP analytical functions, correction terms were added to the previously assumed point radiance function. As the analytical solution to the obtained integration equation is yet to be determined, we used a polynomial with unknown coefficients and an unknown function to correct the shape of the point radiance function relating to the distance of the point to the fiber core center and the power transfer angle of the point, respectively. Moreover, we identified the analytical solution to the unknown function that corrects the power transfer angle. Thereafter, we defined a cost function considering the relation of the NFP obtained according to the point radiance and fiber NFP, which yielded the optimal polynomial coefficients. After obtaining the point radiances of GI-MMF, we substituted them into the matrix equations established for modeling the MMF connectors to appropriately investigate the influences of the connector misalignments on the GI-MMF connection.

The point radiance we obtained is a general solution to GI-MMF, as long as the FFP and NFP of the emitting fiber are symmetric patterns, and NA of the core is determined by Eq. (26) in [18]. Currently, in the market, there are two main types of MMFs: glass optical fiber (GOF) and plastic optical (POF) depending on the core material. The solution to the point radiance is effective to both types of fibers because the core NA is determined by the core refractive index distribution, while not the core material. Meanwhile, NFP and FFP change during propagation in the fiber because of mode coupling and achieve symmetric after certain length propagation even if they are not launched by a symmetric pattern.

2. Point radiance of GI-MMF

The point radiance, denoted as g_u(r, θ), is defined as the power transferred along a certain angle θ w.r.t. the fiber axis from a point on the fiber end-face, which is dependent on the distance r from the point to the fiber core center, assuming fiber symmetry. As illustrated in Fig. 1, to calculate the power coupled into the receiving fiber, the power distribution on the plane of the receiving fiber end face, which is the superposition of the point radiance from points on the emitting fiber end face, should be known in advance. However, no instrument can measure the point radiance function at present. Therefore, we must derive it according to the data that are obtainable in experiments, typically the fiber FFP, NFP, and the fiber characteristic parameters. The relation of the point radiance with the fiber FFP and NFP can be expressed as

(1)$$g\left( \theta \right) = 2\pi \int_0^a{g_u\left( {r,\theta } \right)rdr},$$

and

(2)$$N\left( r \right) = \int_0^{\theta r,\textrm {max}}{g_u\left( {r,\theta } \right)d\theta },$$

where g(θ) and N(r) represent the fiber FFP and NFP, respectively; a denotes the fiber core radius; θ_r_,max indicates the maximum radiance angle of the point at a distance r to the fiber core center. Here, the power propagation angle θ with respect to the fiber axis can be considered a continuous value because, for communication channels, the number of principal modes can be several thousand for typical GI-MMFs.

Fig. 1. Point radiance g_u(r, θ) of GI-MMF.

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Suppose that the refractive index profile of the emitting GI- MMF has the form

(3)$${n^2}(r )= \left\{ \begin{array}{l} n_0^2[{1 - 2\varDelta {{({{r / a}} )}^\eta }} ]\quad \quad \;r < a\\ n_1^2\quad \quad \quad \quad \quad \quad \quad \quad \;r \ge a \end{array} \right., $$

where n₀ and n₁ denote the refractive indices of the fiber core center and cladding, respectively; η denotes the profile exponent; Δ represents the refractive index difference defined as

(4)$$\varDelta = \frac{{n_0^2 - n_1^2}}{{2n_0^2}}. $$

Therefore, the maximum radiance angle of a point on the GI-MMF end-face can be referred from [18] (Eq. (26), assuming sinθ ≈ θ) as

(5)$${\theta _{r,\max }} = {\theta _{0,\max }}\sqrt {1 - {{\left( {\frac{r}{a}} \right)}^\eta }}. $$

Similar to the treatments used for SI-MMF [23], if the FFP of the fiber g(θ) is obtained in experiments, then the point radiance of GI-MMF would have the form

(6)$${\hat{g}_u}({r,\theta } )= g\left( {{\theta / {\sqrt {1 - {{\left( {\frac{r}{a}} \right)}^\eta }} }}} \right). $$

Evidently, if Eq. (6) is substituted into the right-hand side of Eq. (1), the obtained integration result deviates from the FFP of the fiber g(θ). Therefore, a correction term should be added to Eq. (6) to obtain

(7)$${g_u}({r,\theta } )= f(\theta ){\hat{g}_u}({r,\theta } ), $$

where f(θ) represents an unknown function related only to the power transfer angle θ. Subsequently, Eq. (7) was substituted into Eq. (1) to obtain

(8)$$g\left( \theta \right) = \int_0^a{2\pi rf\left( \theta \right)g\left( {{\theta / {\sqrt {1-{\left( {\displaystyle{r \over a}} \right)}^\eta } }}} \right)dr} = 2\pi f\left( \theta \right)\int_0^a{rg\left( {{\theta / {\sqrt {1-{\left( {\displaystyle{r \over a}} \right)}^\eta } }}} \right)dr},$$

which was divided by the integration term on both sides to derive

(9)$$f\left( \theta \right) = {{g\left( \theta \right)} / {2\pi \int_0^a{rg\left( {{\theta / {\sqrt {1-{\left( {\displaystyle{r \over a}} \right)}^\eta } }}} \right)dr} }}$$

and further substituted into Eq. (7) to obtain

(10)$$g_u\left( {r,\theta } \right) = g\left( \theta \right)\displaystyle{{g\left( {{\theta / {\sqrt {1-{\left( {\displaystyle{r \over a}} \right)}^\eta } }}} \right)} \over {2\pi \int_0^a{rg\left( {{\theta / {\sqrt {1-{\left( {\displaystyle{r \over a}} \right)}^\eta } }}} \right)dr} }},$$

As observed, Eq. (1) is satisfied if Eq. (10) is substituted into its right-hand side. Therefore, Eq. (10) presents a perfect solution of the point radiance of the GI-MMF if the NFP of the fiber is neglected. However, for GI-MMF, even at equilibrium mode distribution (EMD), the NFP can manifest a different shape owing to a distinct refractive index profile. Thus, we introduced an additional correction term into Eq. (7) to obtain

(11)$${g_u}({r,\theta } )= N^{\prime}(r )f(\theta )g\left( {{\theta / {\sqrt {1 - {{\left( {\frac{r}{a}} \right)}^\eta }} }}} \right). $$

Upon substituting Eq. (11) into Eq. (1), we obtain

(12)$$G\left( \theta \right) = f\left( \theta \right)\int_0^a{2\pi r{N}^{\prime}\left( r \right)g\left( {{\theta / {\sqrt {1-{\left( {\displaystyle{r \over a}} \right)}^\eta } }}} \right)dr}.$$

Thereafter, we substituted Eq. (7) into Eq. (2) to derive

(13)$$N\left( r \right) = {N}^{\prime}\left( r \right)\int_0^{\theta \max}{f\left( \theta \right)g\left( {{\theta / {\sqrt {1-{\left( {\displaystyle{r \over a}} \right)}^\eta } }}} \right)d\theta }.$$

As the integrand function g(·) equals zero for θ > θ_r_,max, the integration upper limit can be simplified as θ_max, which depicts the maximum radiance angle of the fiber core center.

In Eqs. (12) and (13), there are two unknown functions, f(θ) and N′(r). Currently, the analytical solutions are yet to be determined, and therefore, we propose to use a polynomial to approximate N′(r). Suppose that we use a fifth-order polynomial,

(14)$$N^{\prime}(r )= 1 + {b_1}r + {b_2}{r^2} + {b_3}{r^3} + {b_4}{r^4} + {b_5}{r^5}. $$

Then, Eq. (11) can be substituted into the right-hand side of Eqs. (1) and (2) to derive

(15)$$f\left( \theta \right)\; = \displaystyle{{g\left( \theta \right)} \over {\int_0^a{2\pi r{N}^{\prime}\left( r \right)g\left( {{\theta / {\sqrt {1-{\left( {\displaystyle{r \over a}} \right)}^\eta } }}} \right)dr} }}.$$

and

(16)$$\tilde{N}\left( r \right) = {N}^{\prime}\left( r \right)\int_0^{\theta \textrm {max}}{f\left( \theta \right)g\left( {{\theta / {\sqrt {1-{\left( {\displaystyle{r \over a}} \right)}^\eta } }}} \right)d\theta }.$$

In particular, Eqs. (15) and (16) can be determined if the coefficients of the polynomial N′(r) are known. Note that $\tilde{N}(r )$ is distinct from the fiber NFP N(r) because the coefficients in Eq. (14) are arbitrary at the current stage.

Upon substituting Eq. (11) into the right-hand side of Eq. (1), the integration of the point radiance defined in Eq. (1) is equal to the FFP of the fiber if f(θ) is expressed using Eq. (15) for any selection of polynomial coefficients in Eq. (14). Therefore, we only need to assure that Eq. (16) approaches fiber NFP by appropriately selecting the polynomial coefficients. Moreover, a cost function is defined as the measure of the approximation error and expressed as

(17)$$error = \int_0^a{\left[ {N\left( r \right)-\displaystyle{{\tilde{N}\left( r \right)\max \left( {N\left( r \right)} \right)} \over {\max \left( {\tilde{N}\left( r \right)} \right)}}} \right]dr}.$$

Herein, the objective is to determine a group of b₁, b₂, …, b₅ such that the error function approaches the minimum (optimum) value.

As the typical value of the refractive index profile exponent of the commercial GI-MMF is η = 2, we selected three values of η approximately equal to 2, and we employed Eqs. (14)–(17) to determine the optimal polynomial coefficients. In simulations, the fiber NFP N(r) and FFP g(θ) are pre-assumed, which are generally obtained by physical measurements. Here, we assumed the fiber NFP as

(18)$$N(r )= 1 - {\left( {\frac{r}{a}} \right)^\eta },$$

depicting the same form as that obtained in [24]; Eq. (38), in case the MPD approaches the EMD and fiber FFP, following

(19)$$g(\theta )= {{\exp \left( { - c \cdot \frac{{{\theta^2}}}{{\theta_{0,\max }^2}}} \right)} / {({\pi {a^2}} )}}$$

as a Gaussian function, where c denotes a constant incorporating the expected percentage of optical power inside the fiber NA. Although NFP and FFP change during propagation in the fiber owing to mode coupling and fiber loss, we used the NFP and FFP represented using Eqs. (18) and (19) to calculate the point radiance; the calculation method given by Eqs. (11) and (14)–(17) is effective for estimating the point radiance when the NFP and FFP have symmetric patterns.

The polynomial coefficients and the corresponding errors obtained for a fifth-order polynomial are listed in Table 1. In the simulations, θ_0,max and c were assumed as 0.2 rad and 2.5², respectively. The fiber core radius equals 25 µm, which is a typical value defined in IEC standards (A1 MMF in IEC60 793-2-10). If η ≈ 2, the fifth-order polynomial can adequately approximate the point radiance because the residual error of fiber NFP calculated from point radiance is relatively low. Therefore, we observe a prominent trend in which the error increases for large values of η, implying that a higher order polynomial is required for N′(r) to correct the distance-related component of the point radiance function defined in Eq. (11) if higher accuracy is required.

Table 1. Polynomial coefficients defined in Eq. (14) and corresponding error defined in Eq. (17)

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The fiber NFPs obtained based on the point radiance for the polynomial coefficients listed in Table 1 (approximated NFP) and the pre-assumed fiber NFPs (theoretical NFP) are illustrated in Fig. 2. Notably, although the coefficients listed in Table 1 are largely different from b₁ to b₅, it has a minimal negative effect on the numerical stability because fiber core radius r is on the level of 10⁻⁶ m. As observed, even for large values of η, the error was still under the negligible level for an estimated MPD at the receiving fiber side. As described earlier, the fiber FFP calculated from the point radiance coincided with the pre-assumed fiber FFP in theory, and thus, the comparison was not displayed in a figure to save space.

Fig. 2. Comparison of approximated NFP and theoretical NFP if refractive index profile exponents are proximate to η = 2. A specific situation for η = 10 is illustrated to demonstrate the residual errors existing in the obtained point radiance.

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The point radiance g_u(r, θ) obtained for fiber refractive index profile exponent η = 2 and r = 5, 10, 15, 20, and 24.9 µm is illustrated in Fig. 3. The propagation angle is wider at small values of r, which accords with the theoretical values of Eq. (5). For η deviation from parabolic, the method developed here can be used for obtaining the point radiance, which can be seen from Fig. 2. The FFP of the emitting fiber is a Gaussian function, which corresponds to Eq. (32) in [23]. The NA of the point at a distance r to the core center (expressed in Eq. (5)) was plotted using circles in corresponding colors. Evidently, the NA of the point radiance is in accordance with the theoretical value. Another observable phenomenon included the deviation of the angle with the maximum point radiance from 0 to a higher value as the point approaches the fiber core center, which was not observed in the SI-MMF.

Fig. 3. Power radiance g_u(r, θ) at points on GI-MMF end-face. Circles of various colors indicate theoretical NA expressed in Eq. (5).

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3. GI-MMF connector simulation

Similar to the prior study on SI-MMF connector [23], we performed three case studies for GI-MMF after evaluating the point radiance using the method described in the previous section: calculate the connector insertion loss, effective FFP at the input of receiving fiber, and effective NFP illuminating the receiving fiber end-face in case the connector alignment parameters involve random combinations of lateral, longitudinal, and angular offsets. Herein, we use the term “effective” to indicate that we considered only the power transferred along an angle less than the acceptance angle of the incident point w.r.t. the receiving fiber axis. Note that the acceptance angle of the points on the GI-MMF end-face varies according to Eq. (5), which is distinct from that of the SI-MMF. As such, the matrix method developed in [23] to model the fiber connector is universal upon obtaining the point radiance of the emitting fiber. Therefore, it can be used to investigate the MPD variation introduced by the GI-MMF connector.

The positions of the two connected fibers in 3D coordinates x′y′z′ established on the emitting fiber end-face are depicted in Fig. 4. The two planes on which the emitting fiber end-face and receiving fiber end-face are placed are termed as the emitting and receiving planes, respectively. Moreover, the origin is set at the emitting fiber core center. The x′-axis is parallel to the intersection line of the emitting and receiving planes, whereas the z′-axis is parallel to the emitting fiber axis. Accordingly, the y′-axis was selected such that the coordinate satisfied the right-hand rule. The z′-axis crossed the receiving plane at A. The receiving fiber was placed in the polar coordinate established on the receiving plane with the origin at A, and the x-axis is parallel to the x′-axis. Therefore, the position of the receiving fiber is denoted as the polar coordinate (r_o, γ_o) of the fiber core center O.

Fig. 4. 3D coordinate for determining positions of receiving and emitting fibers.

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In the simulations, the maximum connector tilt angle α_max = 5°; refractive index profile exponent η = 2; step-size for dividing receiving plane: Δr = 0.125 µm, Δγ = π/100; step-size for dividing emitting ellipse: Δφ = π/100; and step-size for dividing power transfer angle: Δθ = θ_max/100.

Case 1: loss in GI-MMF connection

To demonstrate the effects of the longitudinal, lateral, and angular offsets on the power loss of the connector, we calculated the power entering the receiving fiber relative to the power transferred by the emitting fiber under the condition that the lateral offset was in the range of 0–3 times that of the fiber core radius and the longitudinal offset was in the range of 0–100 times that of the fiber core radius, whereas the angular offset was fixed at 0°, 2.5°, and 5°. The obtained relative losses are illustrated in Fig. 5.

Fig. 5. Relative transferred power of GI-MMF from emitting fiber to receiving fiber; (a), (c), and (e) show relative transferred power vs. absolute lateral and longitudinal offsets under 0°, 2.5°, and 5° connector tilt angle; (b), (d) and (f) show a comparison of relative transferred powers obtained for GI-MMF and those obtained for SI-MMF in [23].

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Compared to the results of SI-MMF obtained in [23], we determined that GI-MMF is more sensitive to the connector absolute angular offsets. This is evident because the radius of the GI-MMF is much smaller than that of the SI-MMF. Thus, a slight fiber tilt will transmit a large amount of emitting power outside the end-face of the receiving fiber. Normalizing the connector lateral offset using fiber core radius reveals that the lateral offset has nearly the same impact on SI-MMF and GI-MMF when the tilt angle is 0, as shown in Fig. 5(b). When the connector tilt angle increases, the difference between the relative transferred power of GI-MMF and SI-MMF is more pronounced, as shown in Fig. 5(d) and (f), because the absolute tilt angles were used in the comparison. In Fig. 5(d) and (f), longitudinal offsets are set as core radius multiplied by the tangent of the tilt angle because when the fiber is tilted, the longitudinal offset cannot be zero. For a valid comparison of the effect of connector tilt and longitudinal offsets on GI-MMF and SI-MMF, the tilt angles and longitudinal offsets were normalized using the maximum core NA and core radius of the corresponding fiber, respectively. Relative transferred power is plotted against tilt angles and longitudinal offsets in Fig. 6(a) and (b), respectively. Similar to the cases in Fig. 5(d) and (f), longitudinal offsets in Fig. 6(a) are set as core radius multiplied by the tangent of the tilt angle. When the normalized tilt angle is small, the relative transferred power of GI-MMF is nearly the same as that of SI-MMF. Prior research has reported that the longitudinal offset causes the slightest effects on the connector loss for SI-MMF [18]. We determined that although the connector loss rapidly increased with the absolute longitudinal offsets in the small offset region for GI-MMF, it is less sensitive to the normalized longitudinal offset, as shown in Fig. 6(b).

Fig. 6. Comparison of relative transferred powers of GI-MMF and SI-MMF studied in [23].

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Case 2: calculate FFP incident on receiving fiber

FFP inside the fiber considerably affects the modal bandwidth of the fiber, which should be calculated in the high-speed transmission link design. If the point radiance of the GI-MMF expressed by Eq. (11) is known, we can apply the method developed in [23] to calculate the FFP at the input of the receiving fiber.

As depicted in Fig. 7, we illustrate the cases for connector angular offset α = 3°. The FFPs incident on the receiving fiber end-face at various lateral offsets for longitudinal offset z₀ = 100, 300, and 500 µm are illustrated in Fig. 7(a), (b), and (c), respectively. Evidently, the FFPs at the receiving fiber does not correspond to the Gaussian shape or that of the emitting fiber. Similar to their counterparts of SI-MMF, the lateral and longitudinal connector offsets rapidly reduce the power of the lower-order mode. In case the offsets are enlarged, the peak power displaces from the lower- to middle-order mode, as depicted in Fig. 7(c), which was observed in the SI-MMF connector loss studies when the connector tilt was neglected [21]. Unlike the SI-MMF, we observed that more power is received at the lower-order mode for the GI-MMF for any combination of the misalignment parameters. According to the simulation results, we can conclude that the power transferred along the large angles w.r.t. the receiving fiber axis suffers a relatively high loss because a larger component of power transferred from the inner component of the emitting fiber cannot enter the receiving fiber if the incident point is at the outer side of the receiving fiber, as the acceptance angle of the GI-MMF gradually decreases as the incident point on the receiving fiber is displaced from the fiber core center.

Fig. 7. Angle-dependent transferred power of GI-MMF under combined effects of connector offsets; α denotes connector offset; z₀ represents longitudinal offset.

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Case 3: calculate effective NFP on receiving fiber

Effective NFP is defined as the power spot on the receiving fiber end-face, excluding the power outside the fiber acceptance angle. Although it does not affect the modal bandwidth of the SI-MMF, it influences the transmission properties of the GI-MMF. Along with the power transfer angle, it constitutes the launch condition of GI-MMF. With the knowledge of the point radiance of emitting fiber, the effective NFP of the receiving fiber can be easily evaluated using the matrix developed for the modeling fiber connector [23].

Moreover, four examples of the effective NFP on the receiving fiber for the core center placed at four positions equidistant to point A are illustrated in Fig. 8(a). The angular coordinates of the four positions are γ₀ = 0°, 90°, 180°, and 270°, respectively. The tilt angle was α = 3°, and the longitudinal offset z₀ = 50 µm. The red circles in the figures indicate the positions of the core of receiving fiber on the receiving plane. Similar to the SI-MMF, the spots on the end-face of the GI-MMF displayed a centralized pattern, although the radiance patterns of the points on the emitting fiber were unequal. Note that the power beyond the acceptance angle of the receiving fiber was excluded because this component of the energy does not propagate in the followed transmission lines.

Fig. 8. NFP of receiving fiber and angle-dependent transferred power of GI-MMF.

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The angle-dependent transferred powers of the receiving fiber at eight positions on the receiving plane are illustrated in Fig. 8(b). In particular, the power entering the receiving fiber was the smallest for the receiving fiber placed at γ0 = 90°, whereas the highest power was observed for γ0 = 270°. This happens because a large amount of power is transferred into the high-order mode at γ0 = 90° owing to the fiber tilt. We observed that the angle-dependent power transfer functions were the same when the receiving fiber was placed at symmetric positions relative to the line γ0 = 90° on the receiving plane. Evidently, at two symmetric positions, the power transfer angle conversions caused by the fiber angular offset are identical. This explains the angle-dependent transferred power as that depicted in Fig. 8(b), wherein the receiving fiber was placed at γ0 = 45° and 135°, γ0 = 0° and 180°, and γ0 = 225° and 315°.

4. Conclusions

In this paper, we report a calculation method for the point radiance of GI-MMF based on the emitting fiber NFP and FFP, which can be experimentally measured using CCD cameras. Subsequently, we use the obtained point radiance to study the power transfer properties in GI-MMF connectors in case of a random combination of connector misalignments. In conventional methods, the point radiance is considered either as a constant within the range of fiber NA or as the same distribution of the fiber FFP. Evidently, these assumptions yield large errors in general cases because the integrations of the assumed point radiance according to Eqs. (1) and (2) cannot be ensured to converge to the emitting fiber FFP and NFP. Unlike the conventional assumptions, the proposed method derived the point radiance from the data available in the experiments. Therefore, the result is more approximate to the real quantities than the conventional assumptions. The proposed method, along with the matrix for modeling the fiber connector [23], can contribute toward the evaluation of the channel properties of the fiber link cascaded by the GI-MMF. In GI-MMF connection simulations, we observed that lateral, longitudinal, and angular offsets significantly influence the NFP and FFP that enter the receiving fiber. Similar study [10] that examined launching GI-MMF by a single-mode fiber revealed the variation in launching modes with different lateral and angular offsets. Future research will be focused on a more accurate evaluation of the proposed method by conducting GI-MMF connection experiments and the development of the point radiance theory of optical components such as composite lens, resulting in a study on mode array in optics (MAO) to aid short-reach network design.

Funding

National Institute of Information and Communication Technology (NICT) (2020-003); the Ministry of Economy, Trade and Industry (METI) of Japan.

Acknowledgment

The authors would like to thank the National Institute of Information and Communication Technology (NICT), Japan, and the Ministry of Economy, Trade and Industry (METI) of Japan for financial support in this research.

Disclosures

The authors declare that we have no conflicts of interest to disclose.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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η	b₁	b₂	b₃	b₄	b₅	error
1.5	−350	3e7	−8.4e13	7.1e18	−2.2e23	3.5e−4
2	−355	2.5e7	−8.7e13	7.05e18	−2.144e23	5.2e−4
3	−230	−0.115e7	−7.9e13	6.58e18	−2.14e23	1.4e−3
10	−2226	−17.9e7	40e12	0.39e18	1.06e23	2.8e−2

Point radiance calculation of GI-MMF and its application in a fiber connection

Abstract

1. Introduction

2. Point radiance of GI-MMF

3. GI-MMF connector simulation

4. Conclusions

Funding

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (1)

Equations (19)

Optics Express