Equilibrium Mode Distribution Research Articles

New method for the investigation of mode coupling in graded-index polymer photonic crystal fibers using the Langevin stochastic differential equation

The mode coupling in a graded-index polymer photonic crystal fiber (GI PPCF) with a solid core has been investigated using the Langevin equation. Based on the computer-simulated Langevin force, the Langevin equation is numerically integrated. The numerical solutions of the Langevin equation align with those of the time-independent power flow equation (TI PFE). We showed that by solving the Langevin equation, which is a stochastic differential equation, one can successfully treat a mode coupling in GI PPCFs, which is an intrinsically stochastic process. We demonstrated that, in terms of effectiveness, the Langevin equation is preferable compared to the TI PFE. The GI PPCF achieves the equilibrium mode distribution (EMD) at a coupling length that is even shorter than the conventional GI plastic optical fiber (POF). The application of multimode GI PCFs in communications and optical fiber sensor systems will benefit from these findings.

Wavelength dependent transmission in multimode graded-index microstructured polymer optical fibers

Up to now, there have been no commercial simulation tools accessible for researching the transmission properties of multimode microstructured optical fibers (MOFs). In order to avoid this problem, this study uses the time-independent power flow equation (TI PFE) numerical solution to examine the wavelength dependency of the equilibrium mode distribution (EMD) and steady state distribution (SSD) in multimode graded-index microstructured polymer optical fibers (GI mPOF) with a solid core. We showed that the lengths zs at which an SSD is obtained in GI mPOF and the coupling length Lc necessary to create an EMD are shorter at λ = 568 nm than they are found to be at λ = 633 nm. The lengths Lc and zs stay constant when the wavelength decreases further from λ = 568 to 522 and then to 476 nm. As a result, it is anticipated that a faster bandwidth enhancement in the tested GI mPOF will take place at wavelengths around λ = 568 nm as opposed to λ = 633 nm. Such a bandwidth improvement is not brought about by additional wavelength reduction. The study’s findings can be used in communication and sensory systems that use multimode GI mPOFs at different wavelengths.

Investigation of mode coupling in strained and unstrained multimode step-index POFs using the Langevin equation

The Langevin equation (LE) is used to evaluate mode coupling in multimode step-index polymer optical fiber (SI POF) that is both unstrained and strained. The numerical solution of the LE matches the numerical solution of the power flow equation (PFE). Strain-induced mode coupling is noticeably stronger in strained fiber than in unstrained fiber of the same types. Therefore, compared to similar lengths for unstrained fibers, the coupling length of the equilibrium mode distribution (EMD) is attained and the length of fiber required to produce a steady-state distribution (SSD) are both much shorter for strained fibers. We have demonstrated that the mode coupling in strained and unstrained multimode SI POFs that comes from the random perturbations (RPs) of the fiber can be successfully treated by the LE. The study's findings can be used to improve communication and sensory systems that use multimode SI POFs under different bending circumstances. Additionally, it is crucial to be able to compute the modal distribution of the SI POFs used in the optical fiber sensory system at a specific length and under various bending scenarios.

High bandwidth performance of multimode graded-index microstructured polymer optical fibers

The investigation of the bandwidth in multimode graded-index microstructured polymer optical fiber (GI mPOF) with a solid core is proposed using a modal diffusion approach. For a variety of launch radial offsets of multimode GI mPOF, bandwidth is reported by numerically solving the time-dependent power flow equation (TD PFE) using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). The decline in bandwidth with fiber length becomes slower at fiber lengths close to the coupling length Lc at which an equilibrium mode distribution (EMD) is attained, showing that mode coupling enhances bandwidth at longer fiber lengths. As fiber length is increased, bandwidth approaches complete independence from radial offset, suggesting the steady-state distribution (SSD) has been reached. We compare multimode GI mPOF performance in terms of bandwidth with that of traditional multimode GI POFs made of the same material. Higher bandwidth performance and quicker bandwidth improvement are displayed by the GI mPOF. To enhance fiber performance in GI mPOF links, such a fiber characterization can be used.

Power Flow in Multimode Graded-Index Microstructured Polymer Optical Fibers.

We investigate mode coupling in a multimode graded-index microstructured polymer optical fiber (GI mPOF) with a solid core by solving the time-independent power flow equation (TI PFE). Using launch beams with various radial offsets, it is possible to calculate for such an optical fiber the transients of the modal power distribution, the length Lc at which an equilibrium mode distribution (EMD) is reached, and the length zs for establishing a steady-state distribution (SSD). In contrast to the conventional GI POF, the GI mPOF explored in this study achieves the EMD at a shorter length Lc. The earlier shift to the phase of slower bandwidth decrease would result from the shorter Lc. These results are helpful for the implementation of multimode GI mPOFs as a part of communications and optical fiber sensory systems.

Wavelength dependent equilibrium mode distribution and steady-state distribution in double-clad W-type microstructured polymer optical fibers

This paper investigates wavelength dependence of equilibrium mode distribution and steady-state distribution in W-type (double-clad) microstructured polymer optical fibers (mPOFs) with a solid core for parametrically varied refractive index and width of the intermediate layer (IL) (inner cladding) by solving the time-independent power flow equation (TI PFE). In the case of wider IL, independent of wavelength, the lengths for establishing the equilibrium mode distribution and steady-state distribution are larger. We have demonstrated that the wavelength has no effect on these lengths for IL’s width that is larger. These lengths drop in a wavelength-dependent manner as the IL's width decreases. Equilibrium mode distribution and steady-state distribution occur at shorter optical fiber lengths as the depth of the IL diminishes, which is due to the similarly declining number of leaky modes. The smaller the depth of the IL, the shorter the fiber length is required for completion of the coupling process. These programmable characteristics allow double-clad W-type mPOFs to be easier customized for a particular use in optical fiber sensors and communications at various wavelengths.

Influence of the Width of Launch Beam Distribution on the Transmission Performance of Seven-Core Polymer-Clad Silica Fibers

We propose a space division multiplexing (SDM) in a newly constructed multicore polymer-clad silica fiber (PCSF) with seven cores arrayed in a hexagonal array, each carrying a centrally launched beam. This enables a higher SDM capacity at longer fiber lengths in the proposed seven-core PCSF if compared with previously proposed angular division multiplexing (ADM) in single-core (SC) PCSF. As a result, the SDM is not limited to short fiber lengths in the proposed seven-core PCSF, as it is in the case of the ADM channels due to mode coupling in the SC PCSF. In addition, the time-independent power flow equation (TI PFE) is used to analyze the effect of the width of the launch beam distribution on the equilibrium mode distribution (EMD) and steady state distribution (SSD) in each of the seven cores of the investigated PCSF. The width of the launch beam distribution has a considerable impact on the fiber length at which the EMD and SSD are attained, according to our numerical results. Thus, by decreasing the full width at half maximum (FWHM) of the launch beam distribution from 20 to 2°, the length at which EMD is established increases from Lc = 1020 to 1250 m, and the length at which SSD is attained increases from zs = 2650 to 3250 m. A narrow launch beam distribution leads to higher bandwidth at small and intermediate fiber lengths. On the other hand, at shorter fiber lengths, a wider launch beam distribution induces a bandwidth change from 1/z proportional to 1/z1/2 proportional curve, e.g., a slower bandwidth reduction. When building a multicore optical fiber transmission system for SDM, such characterization of multicore PCSFs under various launch conditions should be taken into account.

Influence of launch beam distribution on power flow and angular division multiplexing in seven-core silica optical fibers

We analyze the effect of launch beam distribution on space-division multiplexing (SDM) performance in multimode multicore silica optical fibers (MM MC SOF) with seven cores. The time-independent power flow equation (TI PFE) is used to explore the effect of the width of the distribution of the Gaussian launch beam on power flow in each of the seven cores. We show that the optical fiber length at which the equilibrium mode distribution (EMD) and steady-state distribution (SSD) are obtained is greatly influenced by the width of the Gaussian launch beam distribution. We further show that when the width of the Gaussian launch beam distribution widens, the optical fiber length at which angular division multiplexing (ADM) in each of the seven cores can be realized with minimal crosstalk between neighboring angular optical channels decreases. We demonstrate that, for increasing the capacity of an optical fiber transmission system, an SDM system with two- and three-channel ADM and multicore optical fiber multiplexing can be implemented with the proposed seven-core MM MC SOF at optical fiber lengths up to ≈1 km (2 ADM channels × 7 cores) and ≈200 m (3 ADM channels × 7 cores), respectively. Such characterization of MM MC SOFs under various launch conditions is important for building a multicore optical fiber SDM transmission system.

Transmission performance of multimode W-type microstructured polymer optical fibers.

By solving the time-independent power flow equation (TI PFE), we study mode coupling in a multimode W-type microstructured polymer optical fiber (mPOF) with a solid-core. The multimode W-type mPOF is created by modifying the cladding layer and reducing the core of a multimode singly clad (SC) mPOF. For such optical fiber, the angular power distributions, the length Lc at which an equilibrium mode distribution (EMD) is achieved, and the length zs for establishing a steady state distribution (SSD) are determined for various arrangements of the inner cladding's air-holes and different launch excitations. This information is useful for the implement of multimode W-type mPOFs in telecommunications and optical fiber sensors.

Calculation of the Coupling Coefficient in Step-Index Multimode Polymer Optical Fibers Based on the Far-Field Measurements

Using the power flow equation (PFE), this article investigates mode coupling in step-index (SI) multimode (MM) polymer optical fiber (POF). This equation’s coupling coefficient was initially fine-tuned so that it could appropriately reconstruct previously recorded far-field (FF) power distributions. The equilibrium mode distribution (EMD) and steady-state distribution (SSD) in the SI MM POF were found to be obtained at lengths Lc = 15 m and zs = 41 m, respectively. These lengths are substantially shorter than their glass optical fiber counterparts. Such characterization of the investigated POF can be used in its employment as a part of the communication or sensory system. Namely, the POF’s bandwidth is inverse linear function of fiber length (z−1) below the coupling length Lc. However, it has a z−1/2 dependence beyond this equilibrium length. Thus, the shorter the coupling length Lc, the sooner transition to the regime of slower bandwidth decrease occurs. It is also important to be able to determine a modal distribution at a certain length of the POF employed as a part of optical fiber sensory system.

Investigation of mode coupling in step-index air-clad silica optical fibers

This paper analyzes the state of mode coupling in a high numerical aperture (NA) step-index (SI) air-clad silica optical fiber using the power flow equation. The coupling coefficient of this equation was fine-tuned at first to allow it to correctly reconstruct previously reported experimental intensity distributions of output far-field (FF) patterns. At a length of 2100 m, the equilibrium mode distribution in the investigated fiber was obtained to be established. In our best knowledge for the first time, this paper reports on the strength of mode coupling and output beam quality of SI air-clad silica optical fiber. The results obtained can be exploited to employ SI air-clad fibers in broadband high-NA astro-photonics systems.

Investigation of bandwidth in multimode graded-index plastic optical fibers.

A new method is proposed for investigating the bandwidth in multimode graded-index plastic optical fibers (GI POFs). By numerically solving the time-dependent power flow equation, bandwidth is reported for a varied launch conditions (radial offsets) of multimode GI POF. Our theoretical results are supported by the experimental results which show that bandwidth decreases with increasing radial offset. This decrease is more pronounced at short fiber lengths. At fiber length close to the coupling length Lc at which an equilibrium mode distribution (EMD) is achieved, this decrease becomes slower, indicating that mode coupling improves bandwidth at larger fiber lengths. With further increase of fiber length, bandwidth becomes nearly independent of the radial offset, indicating that a steady-state distribution (SSD) is achieved. Such a fiber characterization can be applied to optimize fiber performance in POF links.

Power flow in multimode step-index plastic photonic crystal fibers

We investigate the state of mode coupling in a multimode step-index plastic photonic crystal fiber (SI PPCF) with a solid-core by solving the time-independent power flow equation. For various arrangements of air-holes, and therefore a different numerical apertures (NAs), as well as a different widths of launch beam distribution, the length Lc for achieving equilibrium mode distribution (EMD) and length zs at which a steady state distribution (SSD) is established are determined for such fiber. We show that the larger the air holes in the cladding (higher NA), the longer length of the fiber it takes for the modal distribution-transients to reach their equilibrium and steady state. This is as a consequence of the greater participation rate of higher-order modes in higher-aperture photonic-crystal fibers. In contrary, in the case of a wide launch that excites more guiding modes, these lengths shorten. This is because the energy of a wide launch beam is more uniformly distributed among guided modes in the fiber, thus the EMD and SSD are reached at shorter distances than for a narrow launch beam. Such information is of interest for application of multimode photonic crystal fibers in telecommunication and fiber optic sensors.

Wavelength dependent transmission in W-type plastic optical fibers with graded index core distribution

This paper investigates the wavelength dependence of the equilibrium mode distribution (EMD) and steady state distribution (SSD) in W-type plastic optical fibers (POFs) with graded index (GI) core distribution for parametrically varied width of the fiber's intermediate optical layer and refractive index of the outer cladding. The numerical solution of the time-independent power flow equation is used to determine the transmission characteristics of the W-type GI POF. We demonstrated that the coupling length Lc required to achieve an EMD in W-type GI POF is shorter than the length determined experimentally for the original SC GI POF at 633 nm. We also demonstrated that as the wavelength increases, the EMD and steady-state distribution (SSD) are achieved at shorter W-type GI POF lengths. This is explained by the increase in leaky mode losses as wavelength increases. This makes it easier to tailor W-type GI POFs to a specific application at different wavelengths.

Method for investigation of mode coupling in multimode step-index silica photonic crystal fibers

We propose a new method for investigation the state of mode coupling in a multimode step-index silica photonic crystal fiber (SI SPCF) with a solid-core by solving the time-independent power flow equation. For various arrangements of air-holes (different numerical apertures (NAs)), as well as a different widths of launch beam distribution, the length Lc for achieving equilibrium mode distribution (EMD) and length zs at which a steady state distribution (SSD) is established are determined for such fiber. We obtained that the larger the air holes in the cladding (higher NA), the longer length of the fiber it takes for the modal distribution-transients to reach their equilibrium and steady state. In the case of a wide launch that excites more guiding modes, these lengths shorten. Such information is of interest for application of multimode SI SPCFs in telecommunication and fiber optic sensors.

Influence of width of launch beam distribution and mode coupling on transmission characteristics of W-type plastic optical fibers

We show strong effects of width of the launch beam’s angular distribution on bandwidth and mode coupling in W-type plastic optical fibers (POFs). In general, the broader this width, the stronger the coupling and the lower the bandwidth—but the bandwidth drop due to this effect tapers off with distance from the input end and ceases at lengths over 25 m as the steady state distribution is reached. With increasing the strength of mode coupling, the equilibrium mode distribution is reached at shorter fiber lengths, which results in slower bandwidth decrease and finally in higher bandwidth at longer fiber lengths for fibers with larger coupling coefficients D. These results are aimed at aiding designers and application integrators of W-type POFs, particularly in employing laser, LED or vertical cavity surface emitting laser (VCSEL) light source.

Numerical and experimental study of a multimode optical fiber sensor based on Fresnel reflection at the fiber tip for refractive index measurement

This paper presents the numerical and experimental study of a multimode optical fiber (MMF) sensor based on Fresnel reflection at the fiber tip. Considering different modal power distributions (uniform mode distribution, gaussian and equilibrium mode distribution), the effective fraction of the incident light reflected at the interface between the end of the fiber and the surrounding medium is first calculated numerically with a ray model. Even when considering modal power distribution and non-normal incidence influence, simplifying assumptions such as normal incidence hypothesis can be made to facilitate experimental data exploitation, with a maximum error on the deduced medium refractive index of the 10-4 order. A statistical analysis of noise is then made, to study the influence of a random noise between 0.1% and 5% noise level: results show that the approximation error made by using normal incidence formula for a Fresnel sensor based on a MMF is significantly smaller than the repeatability error. These assumptions are then verified experimentally with two experimental setups by measuring the refractive index at 1550 nm of reference solutions (distilled water-isopropanol and distilled water-sodium chloride mixtures) with a Fresnel based on a multimode fiber, and a Fresnel based on single-mode fiber for comparison. Good agreement is found with refractive index values from the literature with a maximum relative difference of only 0.1%, and the maximum experimental uncertainties in this work is 1.1×10-4. Differences between refractive index measured with a Fresnel based on single-mode and multimode fiber are found to be in accordance with values from the statistical study.

Power flow in multimode W-type plastic optical fibers with graded index core distribution

A method is proposed for investigation of transmission along a multimode W-type (doubly clad) plastic optical fiber (POF) with graded index (GI) distribution of the core. The multimode W-type GI POF is designed from a multimode singly clad (SC) GI POF fiber upon modification of the cladding layer of the latter. The transmission characteristics of such proposed W-type GI POF are determined from the numerical solution of the power flow equation. We have shown that coupling length Lc at which an equilibrium mode distribution (EMD) is achieved in W-type GI POF is shorter than this length experimentally determined for the original SC GI POF. This is a consequence of leaky mode losses which reduce the number of higher guided modes involved in the coupling process, thus reducing the length Lc. We have shown that the shorter the length Lc, the earlier the bandwidth switches from the functional dependence of 1/z to 1/z1/2. We found that bandwidth improves with thinner and shallower intermediate layers. Finally, the bandwidth of the W-type GI POF analyzed in this work is greater if compared to the bandwidth of the original SC GI POF.

Method for calculating coupling coefficient in W-type multimode optical fibers

A simple method is proposed for determining the mode coupling coefficient D in W-type (double clad) multimode optical fibers. It only requires the observation of the far-field output pattern for one fiber length with the input light launched centrally along the fiber axis (θ 0= 0). For illustration, the coupling coefficient determined by this simple method for a W-type multimode glass optical fiber was used to calculate the coupling length L c at which the equilibrium mode distribution is achieved.

Observation of strong mode coupling in multimode step index silica optical fibers with fluorinated thermoplastic polymer cladding

Using the power flow equation, the state of mode coupling in step index multimode silica optical fiber with fluorinated thermoplastic polymer cladding is investigated in this work. Coupling coefficient is determined by employing our previously reported calculation method. The silica optical fiber with fluorinated thermoplastic polymer cladding shows much stronger mode coupling than the all plastic-clad silica fibers investigated in earlier works. Consequently, it takes much shorter fiber length for the equilibrium mode distribution or steady state distribution to develop (faster launch beam narrowing), which is of great interest in employing silica optical fiber with fluorinated thermoplastic polymer as a part of telecommunication, medical and sensory system.