Polarization Eigenmodes and Mode Coupling in Single-Mode Fibers

Elliptical Core Fiber Analysis

As illustrated in Figure 4-1, the major and minor axes represent the intrinsic birefringence axes. Here, a₁ and a₂ are the major and minor semi-axes respectively, and the ellipticity e is defined as e = [1 - (a₂/a₁)²]¹/², with q being the semi-focal length. The scalar wave equation for an elliptical core fiber in elliptical coordinates (ξ, η) takes the form of Mathieu equations, whose field solutions are Mathieu functions.

Using the boundary conditions of continuous tangential fields, we can derive the eigen equations for the two polarization eigenmodes oHE₁₁ and eHE₁₁, polarized along the x and y directions respectively. When the ellipticity is very small (e → 0), the Mathieu functions can be expanded in terms of Bessel functions, neglecting higher-order terms of the refractive index difference Δ and e². This approximation allows us to obtain simplified eigen equations for the propagation constants βₒ and βₑ of the oHE₁₁ and eHE₁₁ modes.

After omitting the arguments of the Bessel functions, the functions F and G are expressed as follows. These relationships are particularly important in the design of high-performance optical systems, including those using armored fiber optic cable where environmental stability is paramount.

Intrinsic Birefringence Calculation

Under the condition e → 0, equation (4-1) can be expanded in a Taylor series around the eigen equation uJ₁/uJ₀ = wK₁/K₀ of the HE₁₁ mode in weakly guiding circular fibers, resulting in the intrinsic birefringence δβ of the elliptical fiber:

Furthermore, we can obtain the differential group delay between the polarization eigenmodes:

In the above equations, u and w are the transverse propagation constants in the fiber core and cladding, respectively; v is the normalized frequency; J and K are Bessel functions and modified Bessel functions, respectively. These parameters are critical in determining the performance characteristics of both standard and specialized fibers, including armored fiber optic cable designed for specific polarization properties.

The parameter f(v) in equation (4-2) is given by:

Figure 4-2: Birefringence characteristics in elliptical core fibers as a function of normalized frequency v. The graph illustrates how intrinsic birefringence varies with fiber design parameters, a critical consideration in armored fiber optic cable applications where environmental stability is required.

Mode Coupling Theory

When polarization eigenmodes propagate along the fiber, their modes couple with each other, transferring signal energy from one polarization mode to the other orthogonal polarization mode and vice versa. In uniform fibers, this coupling follows a periodic pattern. There are different ways to describe mode coupling in fibers. Here, we expand the electromagnetic field of an elliptical cross-section fiber in terms of the modes of an ideal waveguide (weakly guiding circular fiber), where the expansion coefficients are functions of the transmission direction z, thus forming coupled mode equations.

In practical applications such as armored fiber optic cable systems, understanding mode coupling is essential for predicting signal degradation and designing appropriate compensation techniques. The robust construction of armored fiber optic cable can help minimize external perturbations that would otherwise enhance mode coupling effects.

Coupled Mode Equations

The general form of the coupled mode equations is:

where a_μ and a_ν are the mode amplitudes of the coupled modes. The physical meaning of this equation is:

The change in amplitude a_μ (and thus power) of the μ mode over a propagation distance dz consists of two parts. The first part is the change occurring during propagation itself, determined by the propagation constant β_μ. The second part is the change in the μ mode due to power coupling from the ν mode within the distance dz. The sum of these two terms represents the total change in the μ mode.

The coupling coefficient in equation (4-5) is:

Here, n and n₀ are the refractive index distributions of the deformed fiber (elliptical fiber in this case) and the ideal circular fiber, respectively; E_t and E_z are the transverse and longitudinal components of the electric field; and P is the mode power.

From equation (4-6), we see that the mode coupling coefficient includes contributions from both the coupling between transverse fields (E_μt·E_νt) and the coupling between longitudinal fields (E_μz·E_νz). In specialized fiber systems, including those using armored fiber optic cable, controlling these coupling mechanisms is crucial for maintaining signal integrity over long distances.

Coupling in Weakly Guiding Fibers

As a result of our analysis, we know that linearly polarized modes (LP_lm) in weakly guiding fibers (Δ << 1) are quasi-transverse modes, meaning their transverse fields are much larger than their longitudinal fields. Therefore, when discussing mode coupling in multimode fibers, it is usually sufficient to consider coupling between transverse fields.

However, in uniform elliptical core single-mode fibers, the transverse electric fields of the two polarization modes of HE₁₁ (HE_11x and HE_11y) are always orthogonal, so their scalar product is zero, i.e., E_xt·E_yt = 0. Therefore, from equation (4-6), we can see that coupling between the two orthogonal polarization modes in single-mode fibers can only be achieved through coupling between their longitudinal fields, so equation (4-6) becomes:

This finding has significant implications for the design of polarization-maintaining fibers, which are often implemented as armored fiber optic cable to protect against external influences that could disrupt polarization states. The armored fiber optic cable construction provides mechanical stability, reducing external stresses that would otherwise introduce additional birefringence and mode coupling.

Orthogonal Electric Field Components

A pair of orthogonal electric field components of HE₁₁ in single-mode fibers (within the core) are:

Power Transformation Due to Mode Coupling

Next, we discuss the power transformation caused by coupling between the two polarization modes in uniform elliptical fibers. The general form of the fiber interface deformation function can be given by equation (4-9):

The m = 2 component in the Fourier series expansion of equation (4-9) corresponds to the elliptical deformation interface:

where a is the core radius of the ideal circular fiber; for uniform fibers, f(z) is a constant representing the ellipticity: e = [1 - (a₂/a₁)²]¹/².

The coupling coefficient between the two orthogonal polarization modes in elliptical core single-mode fibers is:

Figure 4-3: Power coupling between orthogonal polarization modes in elliptical core fibers as a function of propagation distance. The periodic nature of power transfer is clearly visible, which is an important consideration in armored fiber optic cable systems where stable signal transmission is required.

Substituting the expressions for E_xz and E_yz of the ideal fiber from equation (4-8) into equation (4-11), we obtain:

where n₁ is the average refractive index of the fiber core and cladding; k₀ is the vacuum wave number; δ is the propagation constant. The fiber length required for complete power conversion from one polarization mode to the other orthogonal polarization mode in single-mode fibers is:

This result shows that the periodic variation of polarization state along the length of the elliptical fiber (beat length) is a consequence of power conversion between the two polarization modes due to coupling. In practical applications, especially those utilizing armored fiber optic cable in challenging environments, engineers must account for these periodic variations when designing systems that rely on maintaining specific polarization states.

The armored fiber optic cable provides a protective layer that helps maintain the geometric integrity of the fiber core, preserving the predictable nature of mode coupling and polarization behavior. This is particularly important in applications such as fiber optic sensors, coherent communication systems, and other precision optical systems where polarization stability is critical.

Understanding the fundamental principles of polarization eigenmodes and mode coupling allows for the development of advanced fiber optic systems with enhanced performance characteristics. Whether implementing standard communication links or specialized armored fiber optic cable solutions, this knowledge forms the foundation for optimizing system design and ensuring reliable operation under various conditions.

As fiber optic technology continues to evolve, the principles discussed here remain relevant, guiding the development of next-generation optical systems. From high-speed data transmission to sensitive sensing applications, the behavior of polarization eigenmodes in single-mode fibers, including those deployed as armored fiber optic cable, will continue to be a key area of study and innovation.