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Abstract
We analyzed light confinement in circular step-index cores of tellurite and silica fibers through numerical calculations and also examined crosstalk between the fundamental modes of cores in multicore fibers. Our analysis showed that tellurite fibers have a pixel density about 2.2 times higher and a brightness about 1.4 times brighter than silica fibers. As a result, tellurite multicore image fibers have the potential to provide improved resolution and brightness for near-infrared image transportation compared with silica fibers.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION
In recent years, endoscopic diagnosis and surgery have become increasingly common, particularly for organs and tissues where visible light observation is not obstructed by blood. However, in a blood-filled environment, the use of endoscopes is limited, as visible image fibers produce unclear images. As a result, indirect methods such as x-ray fluoroscopy, magnetic resonance imaging, and ultrasound echography must be used for intravascular diagnosis. Nevertheless, the favorable scattering and absorption characteristics of blood in the near-infrared region between 1.5–1.9 µm make it possible to view structures through blood using near-infrared image fibers [1]. The use of these fibers would enable direct intravascular observation and treatment of pulmonary arterial embolism, heart valve disease, coronary thrombosis, and more. Within the wavelength range of 1.5–1.9 µm, 1.55 µm is considered particularly advantageous because this wavelength band is used in optical communication, allowing for the use of various inexpensive devices such as light sources for optical communication. Thus, by using the wavelength of 1.55 µm, it is expected that practical near-infrared image transmission can be achieved at a low cost.
Currently, available image fibers comprise many high-refractive-index silica-glass cores and a common low-refractive-index silica-glass cladding. The resolution of multicore fibers is limited by the number of cores per cross-sectional area (pixel density) and the degree of coupling between the cores. To increase the pixel density, a dense arrangement of small cores is desirable. However, as the cores are positioned closer to each other, the strength of the crosstalk between them increases. Generally, image fibers tend to have lower resolution with longer wavelengths, which means that infrared image fibers have lower resolution compared with visible image fibers. The strength of light confinement in the core is dependent on the difference in refractive index between the core and cladding. It is thus essential to find a pair of core and cladding materials with a large difference in refractive indices.
Among glasses for optical fibers [2], tellurite glass is characterized by a wide infrared transmission window, high linear refractive index, high nonlinear refractive index, low phonon energy, and high Raman gain coefficient. Tellurite glass optical fibers have been studied for applications such as erbium-doped fiber amplifiers [3,4], rare-earth-doped fiber lasers [3], fiber Raman amplifiers [5], fiber Raman lasers [6], and microstructured fibers for supercontinuum light generation [7]. Recently, the fabrication of photonic bandgap fibers [8] and transversely disordered fibers [9] using tellurite glass pairs with large refractive index differences has been reported. These tellurite glasses are likely to be suitable as materials for infrared image fibers.
This paper addresses the effective refractive index, cross-sectional field profile, effective mode diameter, and penetration length of tellurite and silica fibers with a single circular step-index core. The paper then investigates the coupling lengths and crosstalk parameters between the fundamental modes of different cores of a multicore fiber with many circular step-index cores. Based on these considerations, the potential application of tellurite multicore fibers for image transportation is discussed.
2. METHOD OF MODE CALCULATION OF CIRCULAR STEP-INDEX FIBERS
Our initial investigation focuses on optical fibers that have a single circular step-index core and a transversely infinite cladding, with longitudinal uniformity. The exact solutions of the propagation modes in the step-index fibers are described in [10–12]. Assuming the refractive indices of the core and cladding materials are ${n_{{\rm core}}}$ and ${n_{{\rm clad}}}$, respectively, and the fiber has the transverse index profile
The normalized transverse wavenumbers in the core and cladding are defined as
The $u$ and $w$ are related as
where $v$ is the normalized frequency. When the normalized frequency $v$ is given, and the normalized transverse wavenumbers $u$ and $w$ are determined by solving the set of Eq. (4) and the following dispersion Eq. (5) simultaneously:The propagation constant $\beta$ is obtained by Eq. (2), and the effective refractive index ${n_{{\rm eff}}}$ is given by
The time-averaged Poynting vector component along the longitudinal direction ($z$ axis) per unit area is expressed as
The power-confinement factor in the core is defined as
where ${\rm d}A$ is the area element in the fiber cross-section, and the integration is carried out within the core. The effective mode area ${A_{{\rm eff}}}$ [16] and the effective mode diameter are defined asAll calculations were performed using programs we wrote in the MATLAB programming language.
3. RESULTS AND DISCUSSION
In Sections 3.A–3.F, a comparative analysis of the light confinement properties of tellurite and silica single-core fibers is conducted using analytical solutions. These results review the weak-guidance approximation, as explained in textbooks on optical waveguides [11,12], and aid in understanding subsequent results. Then, Sections 3.G–3.H are devoted to the examination of the coupling coefficients and crosstalk parameters governing the behavior of the fundamental modes of tellurite and silica multicore fibers.
A. Effective Refractive Index
The effective refractive index, which is defined as the ratio of the phase velocity of light in the waveguide to that in a vacuum, serves as a crucial parameter in characterizing a guided mode in an optical fiber. In a step-index fiber, the effective refractive index is uniquely determined for each guided mode and is dependent on several factors, including the refractive index of the core and cladding materials, core diameter, and wavelength of the light. As such, a thorough understanding of the effective refractive index is essential in analyzing and designing optical fibers with specific mode characteristics. Figures 1 and 2 show the effective refractive index ${n_{{\rm eff}}}$ of the fundamental ${{\rm HE}_{11}}$ mode and several higher-order modes in tellurite and silica fibers, respectively. Since the phase velocity of a propagation mode is intermediate between the velocities in bulk materials of which the core and cladding are made, the effective refractive index is within the range of ${n_{{\rm clad}}} \le {n_{{\rm eff}}} \le {n_{{\rm core}}}$. The first higher-order ${{\rm TE}_{01}}$ and ${{\rm TM}_{01}}$ modes appear at the cutoff diameter ${d_{{\rm cutoff}}}$ of 1.92 µm for tellurite fiber and 2.59 µm for silica fiber, respectively. The number of guided modes increases with increasing core diameter. The hybrid ${{\rm HE}_{21}}$ mode also appears in proximity to the cutoff diameter. The group of the ${{\rm TE}_{01}}$, ${{\rm TM}_{01}}$, and ${{\rm HE}_{21}}$ modes correspond to the first higher-order ${{\rm LP}_{11}}$ mode of weakly guiding fibers; thus, the effective refractive index curves of these modes are similar each other. [17] Meanwhile, the fundamental ${{\rm HE}_{11}}$ mode corresponds to the ${{\rm LP}_{01}}$ mode of weakly guiding fibers. [17] In addition to these modes, ${{\rm EH}_{11}}$, ${{\rm HE}_{31}}$, and ${{\rm HE}_{12}}$ modes are supported by tellurite fiber of a core diameter larger than about 3.2 µm. These modes are not found for silica fiber of a core diameter less than 4 µm.
Fig. 1. Effective refractive index of the fundamental mode and several higher-order modes of tellurite fiber as a function of the core diameter at a wavelength of 1.55 µm.
Fig. 2. Effective refractive index of the fundamental mode and several higher-order modes of silica fiber as a function of the core diameter at a wavelength of 1.55 µm.
B. Mode Field Patterns
The Poynting vector provides valuable information about the directional power flow of an electromagnetic field. Figures 3 and 4 show cross-sectional profiles of the longitudinal Poynting vector ${S_z}$ distribution for the fundamental ${{\rm HE}_{11}}$ mode and several higher-order modes in tellurite and silica fibers, respectively, with a core diameter of 4.0 µm. The phase angle $\psi$ in the figures indicates the azimuthal orientation of the mode’s symmetry axis in the fiber cross-section. For hybrid HE and EH modes, $\psi = 0$ and $\pi /2$ correspond to two orthogonal directions in the fiber’s transverse plane. The two orthogonal fundamental ${{\rm HE}_{11}}(\psi = 0,\pi /2)$ modes have an azimuthally symmetric cone pattern and decay in the radial direction. As previously mentioned, the fundamental ${{\rm HE}_{11}}$ mode corresponds to the ${{\rm LP}_{01}}$ mode of weakly guiding fibers. The next four modes, i.e., ${{\rm TE}_{01}}$, ${{\rm TM}_{01}}$, and ${{\rm HE}_{21}}(\psi = 0,\pi /2)$, have a doughnut shape that is similar to each other, and they belong to the first higher-order ${{\rm LP}_{11}}$ mode of weakly guiding fibers. There is no significant difference between tellurite and silica fibers in any of the above mode patterns. In contrast, ${{\rm EH}_{11}}$ and ${{\rm HE}_{31}}$ modes of tellurite fibers have a larger doughnut shape than that of the first higher-order ${{\rm LP}_{11}}$ modes. The ${{\rm HE}_{12}}$ mode has a unique shape, which is a combination of a cone and a doughnut. From these observations, it is clear that the ${{\rm HE}_{11}}$ mode, characterized by a Gaussian-like distribution, is the most suitable for applications involving image transport.
Fig. 3. Cross-sectional profiles of the longitudinal Poynting vector distribution of the fundamental ${{\rm HE}_{11}}$ mode and several higher-order modes in tellurite fiber with a core diameter of 4.0 µm.
Fig. 4. Cross-sectional profiles of the longitudinal Poynting vector distribution of the fundamental ${{\rm HE}_{11}}$ mode and several higher-order modes in silica fiber with a core diameter of 4.0 µm.
C. Power-Confinement Factor in the Core
Figure 5 shows the power-confinement factor in the core ${\eta _{{\rm core}}}$ for the fundamental and several higher-order modes in tellurite and silica fibers. As the core diameter $d$ increases, ${\eta _{{\rm core}}}$ monotonically increases for all modes. For higher-order modes, ${\eta _{{\rm core}}}$ is larger than that for the fundamental mode. A tellurite fiber can confine light in its core more effectively than a silica fiber with the same $d$. For example, a tellurite fiber with $d = {1.00}\;\unicode{x00B5}{\rm m}$ can reach ${\eta _{{\rm core}}} = 0.34$ for the fundamental ${{\rm HE}_{11}}$ mode, while a silica fiber with the same $d$ can only reach ${\eta _{{\rm core}}} = 0.07$.
D. Effective Mode Diameter
Figure 6 shows the effective mode diameter ${d_{{\rm eff}}}$ for the fundamental and several higher-order modes of tellurite and silica fibers. It can be clearly seen that ${d_{{\rm eff}}}$ of the higher-order modes is larger than that of the fundamental mode. The minimum values of ${d_{{\rm eff}}}$ are achieved at different points corresponding to different core diameters. Near the cutoff diameter, ${d_{{\rm eff}}}$ is large due to weak confinement by the core, and ${d_{{\rm eff}}}$ decreases with increasing core diameter $d$. When $d$ is sufficiently larger than the cutoff diameter, ${d_{{\rm eff}}}$ increases monotonically to $d$. The smallest values of ${d_{{\rm eff}}}$ about 1.98 µm is achieved for the fundamental ${{\rm HE}_{11}}$ mode of a tellurite fiber of $d = {1.46}\;\unicode{x00B5}{\rm m}$, which is much smaller than 2.80 µm of a silica fiber of $d = {2.04}\;\unicode{x00B5}{\rm m}$.
E. Penetration Length in the Cladding
The guided modes form evanescent waves in the cladding. The electric and magnetic fields of the guided modes of a circular step-index core are expressed using modified Bessel functions ${K_l}\big(\frac{{wr}}{a}\big) (l = 0,1,2, \ldots)$ in the cladding [12], where $r$ is the distance from the center of the core. The Bessel functions are proportional to $\sqrt {\frac{1}{r}} \exp\! \big(- \frac{{wr}}{a}\big)$ for sufficiently large $r$ [18]. Thus, the degree of light spreading in the cladding is characterized by the penetration length, denoted by $\Lambda = \frac{a}{w}$. Figure 7 shows the penetration length $\Lambda$ for the fundamental ${{\rm HE}_{11}}$ mode and several higher-order modes of tellurite and silica fibers. $\Lambda$ near the cutoffs is large, which means that the field is spread over a wide region in the cladding. When the core diameter $d$ increases, $\Lambda$ decreases to the limiting value ${\Lambda _{{\rm min}}} = \frac{a}{v}$. The higher-order modes have larger $\Lambda$ than the fundamental ${{\rm HE}_{11}}$ mode for the same $d$. $\Lambda$ of tellurite fiber is shorter than that of silica fiber, indicating that light is more concentrated near the core in the cladding for tellurite fiber. Even though the refractive index contrasts between the core and cladding in tellurite and silica fibers are close values, 0.0451 for tellurite fiber and 0.0423 for silica fiber, the large refractive index of tellurite glass would allow tellurite fibers to confine light to a narrow region near the core.
Fig. 5. Normalized power in the core of the longitudinal Poynting vector distribution of the fundamental ${{\rm HE}_{11}}$ mode and several higher-order modes in tellurite fiber (solid curves) and silica fiber (dotted curves).
Fig. 6. Effective mode diameter of the fundamental ${{\rm HE}_{11}}$ mode and several higher-order modes in tellurite fiber (solid curves) and silica fiber (dotted curves).
F. Radial Field Distribution of the Fundamental Mode
The fundamental ${{\rm HE}_{11}}$ mode has a smaller effective mode diameter ${d_{{\rm eff}}}$, shorter penetration length $\Lambda$, and more power concentrated in the core than higher-order modes, making it the most suitable propagation mode for image transportation. Figure 8 shows the radial distribution of the longitudinal Poynting vector ${S_z}$ of the fundamental ${{\rm HE}_{11}}$ mode of tellurite and silica fibers. The field distribution of the fundamental mode is azimuthally symmetric and similar to the Gaussian distribution function, with a discontinuity at the core-cladding boundary. For a fiber with a small $d$, most of the optical power is in the cladding, and the long penetration length $\Lambda$ results in a large spread into the cladding. With the increase in $d$, the power-confinement factor in the core ${\eta _{{\rm core}}}$ increases and $\Lambda$ becomes shorter. As a result, the field intensity is smaller for larger $d$ in the cladding far from the center. In general, since light is more strongly concentrated in the core in tellurite fibers than in silica fibers, it can be seen that the field intensity at a distance from the center in tellurite fibers is smaller than that of silica fibers with the same $d$.
G. Coupling Coefficient of the Fundamental Modes in Two Identical Cores
Multicore structures are used in image transportation and space-division multiplexing optical transmission [19,20]. Crosstalk between modes can occur in such multicore fibers. In image transmission, it is desirable to minimize crosstalk to prevent degradation of the transmitted image. On the other hand, in spatial-domain multiplexing, crosstalk is acceptable in some cases, but it must be controlled to prevent missing signals. In general, crosstalk between modes can effectively occur when cores with identical propagation constants are placed close together.
Fig. 7. Penetration length into the cladding of the fundamental ${{\rm HE}_{11}}$ mode and several higher-order modes in tellurite fiber (solid curves) and silica fiber (dotted curves).
The crosstalk per unit fiber length between the fundamental ${{\rm HE}_{11}}$ modes in two identical circular step-index cores is given by the coupling coefficient [11]
Since the modified Bessel function can be approximated as ${K_0}(x) \cong \sqrt {\frac{\pi}{{2x}}} \exp (- x)$ for large $x$ [18], the coupling coefficient $C$ decreases rapidly with respect to the core-core distance $D$. Light incident on one core travels back and forth between the two cores with a period of $2{L_c}$, where ${L_c} = \pi /2C$ is the coupling length, at which the power of light transferred from one core to another is maximized. Figure 9 shows the coupling length of two identical cores in tellurite and silica fibers as a function of core diameter $d$. As shown in the figure, the coupling length has a maximum with respect to $d$. For small $d$, with increasing $d$, the power-confinement factor in the core ${\eta _{{\rm core}}}$ increases, while the penetration length in the cladding $\Lambda$ and the effective mode diameter ${d_{{\rm eff}}}$ decrease, thus ${L_c}$ increases. For large $d$, ${\eta _{{\rm core}}}$ and $\Lambda$ asymptotically approach a constant value (${\eta _{{\rm core}}} \to 1,\Lambda \to a/v$). On the other hand, ${L_c}$ decreases with increasing $d$ as a result of monotonic increasing in ${d_{{\rm eff}}}$.
Fig. 9. Coupling length of two identical cores in tellurite fiber (solid curves) and silica fiber (dotted curves) as a function of core diameter.
Since the fundamental ${{\rm HE}_{11}}$ mode is isotropic, crosstalk is independent of their core arrangement. Therefore, the discussion here can be generalized to any arrangement such as triangular lattice.
The coupling length shown here is the value when cores of exactly the same size are adjacent. However, in reality, the core diameter of multicore image fibers varies for various reasons, and it is rare for cores of exactly the same size to be adjacent. Therefore, in actual fibers, image transportation with sufficiently high resolution is possible even over distances longer than the coupling length shown here. The discussion of crosstalk considering some variation in core size is conducted in Section 3.H.
H. Crosstalk Parameter of Multicore Imaging Fibers
In commercially available multicore fibers for image transportation, coupling length is not necessarily a good measure of performance due to the structural nonuniformity of the cores. Hosono proposed a crosstalk parameter that can be calculated using the following equation:
where $l$ is the fiber length and is set to 1 m as a typical length for an image fiber. To reduce crosstalk, a small crosstalk parameter $B$ is desirable. However, in practice, there is a trade-off between minimizing crosstalk and maintaining brightness, and this balance needs to be carefully considered in the design of image fibers. Kudo suggested that multicore fibers with a crosstalk parameter $B$ of less than 1000 can be a practical option for transmitting images [21]. Figure 10 shows the crosstalk parameter of the fundamental mode of tellurite and silica fibers as a function of core diameter. On the whole, tellurite fibers have smaller crosstalk parameters than silica fibers. The longer the core-core distance, the smaller the crosstalk parameter. A fiber can be considered practical if the core diameter $d$ and core-core distance $D$ fall below the dotted straight line in the figure. While a diameter of $D = {5}\;\unicode{x00B5}{\rm m}$ is impractical for silica fibers, a diameter of $D = {4}\;\unicode{x00B5}{\rm m}$ can still be practical for tellurite fibers. It can be said that the tellurite fiber with the optimal structure shown here is also practical in terms of being able to be fabricated by adopting the method of fabricating a photonic bandgap fiber reported in [8].Fig. 10. Crosstalk parameter of tellurite fiber (solid curves) and silica fiber (dotted curves) as a function of core diameter.
Figure 11 illustrates the relationship between core diameter and core-core distance of identical cores in tellurite and silica fibers with a crosstalk parameter of $B = 1000$. For commercially available silica multicore image fibers for visible light, $D$ is about 3.3–4.6 µm [22], and the estimated crosstalk parameter $B$ is about 1000. The near-infrared image fibers proposed here is thought to have low crosstalk comparable with commercially available visible image fibers.
Fig. 11. Core-core distance of identical cores in tellurite and silica fibers with a crosstalk parameter $B = 1000$ as a function of core diameter.
The minimum core-core distance ($D$) is approximately 5.74 µm for silica fiber with $d = 2.18\;{\unicode{x00B5}{\rm m}}$ and about 3.87 µm for tellurite fiber with $d = 1.48\;{\unicode{x00B5}{\rm m}}$. The tellurite fiber has a pixel density of ${15.4}\;{\rm pixels}/100\;\unicode{x00B5}{\rm m}^2$, which is more than double that of ${7.0}\;{\rm pixels}/100\;\unicode{x00B5}{\rm m}^2$ the silica fiber for near-infrared light. The pixel density of tellurite fiber is even higher than ${6 {-} 12}\;{\rm pixels/}100\;\unicode{x00B5}{\rm m}^2$ of commercial silica multicore fibers for visible light [22].
Assuming no losses, the brightness of a fiber (${B_f}$) is determined by ${B_f} = ({\rm NA}{)^2} \times {P_f}$, where NA is the numerical aperture, and ${P_f}$ is the packing fraction (ratio of core area to cladding area). The ${B_f}$ values of silica and tellurite fibers with minimum $D$ are 0.11 and 0.16, respectively, indicating that the tellurite fiber can increase the pixel density and brightness as compared with the silica fiber. By changing the composition of tellurite glass, its physical properties can be significantly altered. For example, the refractive index of the glass can be increased by adding more heavy metal-oxides. By exploring combinations of tellurite glass compositions that differ significantly in refractive index, imaging fibers with superior performance can be created.
4. CONCLUSIONS
This paper investigated the optical confinement effect of circular step-index cores in silica and tellurite fibers, using numerical analysis. The fundamental mode is better confined to the core than higher-order modes, making it ideal for image transportation. Tellurite fibers, which have a larger core-cladding refractive index difference, can confine light more tightly to the core than silica fiber. The multicore tellurite fiber, with a core diameter of $d = {1.48}\;\unicode{x00B5}{\rm m}$ and a core-core distance of $D = {3.87}\;\unicode{x00B5}{\rm m}$, is expected to be the most practical option for image transportation at a wavelength of 1.55 µm. In contrast, silica fiber requires a larger core diameter of $d = {1.98}\;\unicode{x00B5}{\rm m}$ and a larger core-core distance of $D = {5.24}\;\unicode{x00B5}{\rm m}$. The tellurite fiber offers approximately 2.2 times the pixel density and 1.4 times the brightness of the silica fiber. Therefore, multicore tellurite image fiber holds great promise for enhancing near-infrared image transportation with superior performance.
In this paper, we have examined step-index multicores. In order to further reduce crosstalk, it will be necessary to conduct further study on core structures such as trench structures that can better confine light to cores.
Funding
Japan Society for the Promotion of Science (19H02203, 23K03964).
Acknowledgment
This work was partly supported by JSPS Kakenhi Grant-in-Aid for Scientific Research.
Disclosures
The authors declare no conflicts of interest.
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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