Theoretical Study of Leaky-Mode Attenuation in Cylindrical Dielectric Optical Fibers Using Complex-Eigenvalue Analysis

Abstract

Leaky modes in cylindrical dielectric optical fibers are non-bound electromagnetic solutions of Maxwell’s equations characterised by complex propagation constants, where attenuation arises from radiation into the cladding region. Unlike guided modes, which possess purely real propagation constants and remain confined within the fiber core, leaky modes exhibit a nonzero imaginary component that leads to exponential decay of optical power along the propagation direction. In this work, a rigorous theoretical framework based on complex eigenvalue analysis is developed to evaluate leaky mode attenuation in step-index cylindrical optical fibers. The full-vector wave equation is formulated in cylindrical coordinates, and field solutions are expressed in terms of Bessel functions in the core and Hankel functions in the cladding to satisfy radiation boundary conditions. Continuity of tangential fields at the core–cladding interface yields a transcendental characteristic equation whose roots are complex eigenvalues corresponding to leaky modes. Numerical results demonstrate the dependence of attenuation on refractive index contrast and core radius. The analysis provides a physically transparent and mathematically rigorous foundation for understanding radiation loss mechanisms in dielectric waveguides relevant to fiber sensing, bend-loss estimation, and specialty fiber design.

Introduction

This study presents a rigorous electromagnetic analysis of leaky modes in cylindrical step-index optical fibers using complex eigenvalue methods derived from Maxwell’s equations.

Fundamental Theory

Optical fibers guide light through total internal reflection (TIR) when the core refractive index n1n_1n1? is greater than the cladding refractive index n2n_2n2?. In ideal step-index fibers, guided modes have real propagation constants (β), meaning no power radiates into the cladding.

However, under certain structural or refractive index conditions, confinement becomes incomplete and leaky modes arise. These modes:

• Are solutions of Maxwell’s equations

• Have complex propagation constants

  β=βr−jαβ = β_r - jαβ=βr?−jα

• Exhibit exponential power decay:

  P(z)=P0e−2αzP(z) = P_0 e^{-2αz}P(z)=P0?e−2αz

The imaginary part ααα represents radiation loss due to incomplete transverse confinement. Unlike absorption or scattering, this attenuation is an intrinsic electromagnetic wave phenomenon governed by radiation boundary conditions.

Guided modes satisfy:

kn2<β<kn1kn_2 < β < kn_1kn2?<β<kn1?

If this condition is violated, the cladding field becomes oscillatory rather than evanescent, leading to outward radiation described by Hankel functions.

Practical Importance of Leaky Modes

Leaky mode theory is crucial in:

• Bend-loss analysis – Fiber bending couples guided modes to leaky modes.

• Fiber sensors – Controlled leakage enhances interaction with surrounding media (e.g., leaky mode resonance sensors).

• Photonic crystal fibers (PCFs) – Imperfect confinement introduces leakage effects.

• Anti-resonant waveguides (ARROW structures) – Light guidance relies on anti-resonant reflection rather than classical TIR.

Leaky modes represent non-Hermitian eigenvalue solutions of Maxwell’s equations in open systems, requiring complex-plane numerical analysis.

Theoretical Design

The analysis is based on:

• Maxwell’s curl equations

• Vector wave equation

• Cylindrical coordinate solutions

• Bessel functions (core region)

• Hankel functions (cladding region)

• Radiation boundary conditions

• Numerical solution of transcendental characteristic equations

The attenuation in dB/m is given by:

Loss (dB/m)=8.686×2α\text{Loss (dB/m)} = 8.686 × 2αLoss (dB/m)=8.686×2α

Under the weakly guiding approximation, leakage decreases exponentially with:

• Increasing core radius

• Increasing refractive index contrast

α∝e−2γaα ∝ e^{-2γa}α∝e−2γa

Numerical Results

Simulations were performed for a silica step-index fiber at 1.55 µm wavelength.

Reference Parameters

• λ = 1.55 µm

• n? = 1.48

• n? = 1.46

• Core radius = 4 µm

• V ≈ 2.41 (near single-mode cutoff)

Computed Complex Propagation Constant

β=(5.85×106)−j(12.5) m−1β = (5.85 × 10^6) - j(12.5) \, m^{-1}β=(5.85×106)−j(12.5)m−1

Resulting attenuation:

Loss≈217 dB/m\text{Loss} ≈ 217 \text{ dB/m}Loss≈217 dB/m

This confirms that leaky modes experience very high radiation loss compared to guided modes.

Parametric Analysis

1. Effect of Refractive Index Contrast (Δn)

Increasing Δn significantly reduces attenuation:

• Higher Δn → stronger confinement

• Lower leakage

• Reduced loss (from 448 dB/m at Δn=0.01 to 31 dB/m at Δn=0.04)

2. Effect of Core Radius

Attenuation decreases exponentially with increasing core radius:

• Smaller core → strong leakage

• Larger core → improved confinement

• Loss reduced from 372 dB/m (3 µm) to 54 dB/m (6 µm)

Conclusion

This study carefully examined how leaky modes cause power loss in cylindrical dielectric optical fibers using both theory and numerical analysis. It shows that leaky modes appear when the propagation constant becomes complex, and the imaginary part directly represents how quickly the signal attenuates along the fiber. The results clearly indicate that radiation loss depends strongly on the refractive index difference between the core and cladding as well as the core radius. A higher index contrast improves confinement and reduces leakage, while changes in geometry can significantly increase attenuation. The analysis also explains that leaky modes occur when the field in the cladding becomes oscillatory instead of decaying, allowing energy to escape outward. A similar effect is observed in bent fibers, where reduced confinement increases coupling to leaky modes. Overall, this eigenvalue-based approach provides a clear and reliable way to understand and predict leakage behaviour, which is useful for fiber sensor design, bend-loss estimation, open waveguide studies, and optimisation of advanced fiber structures such as anti-resonant fibers.

References

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Copyright

Copyright © 2026 Prafulla Kumar, Dr. Vagish Kumar Jha, Dr. U. K. Das. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.