OAM Modes in Fiber - Science and Technology

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OAM Modes in Fiber : Optical fibers support different types of modes depending on their refractive index profile. In step-index fibers, the fundamental modes are linearly polarized (LP), but in ring-core fibers (RCFs), the natural modes are hybrid modes, specifically HE (hybrid electric) and EH (hybrid magnetic) modes. These modes arise due to the vector nature of Maxwell’s equations in cylindrical coordinates.

Orbital angular momentum (OAM) modes in fibers are constructed by taking superpositions of these hybrid modes, leading to a well-defined azimuthal phase dependence of the form $e^{i l θ}$ , where $l$ is the topological charge of the mode.

2. Expressing OAM Modes in Terms of HE and EH Modes

The hybrid modes in cylindrical waveguides are solutions of Maxwell’s equations that exhibit both electric and magnetic field components in all three directions $(E_{r}, E_{θ}, E_{z})$ and $(H_{r}, H_{θ}, H_{z})$ . The azimuthal structure of these solutions is given by:

$E, H \propto e^{i (l θ + β z)}$

where $l$ is the azimuthal mode index, and $β$ is the propagation constant. The electric field components for HE modes satisfy:

$H E_{l, m}^{even} = \cos (l θ) J_{l} (k_{t} r) e^{i β z}$
$H E_{l, m}^{odd} = \sin (l θ) J_{l} (k_{t} r) e^{i β z}$

For OAM modes, we construct circularly polarized superpositions of these HE modes:

$O A M_{\pm l, m}^{\pm} = H E_{l + 1, m}^{even} \pm j H E_{l + 1, m}^{odd}$

Similarly, for negative helicities:

$O A M_{\pm l, m}^{\mp} = H E_{l - 1, m}^{even} \pm j H E_{l - 1, m}^{odd}$

These equations define the OAM modes for $l > 1$ . The complex weighting factor $j$ ensures that the mode carries a defined helical wavefront with angular momentum properties.

3. Special Case for $l = 0$

For the fundamental case when $l = 0$ , the OAM mode composition changes, as there are no higher-order HE modes to form the standard superposition. Instead, the lowest-order modes in a ring-core fiber are the TE and TM modes:

$O A M_{\pm l, m}^{\mp} = T M_{0, m} \pm j T E_{0, m}$

This equation states that for $l = 0$ , the OAM mode is a combination of transverse electric (TE) and transverse magnetic (TM) modes, rather than HE modes.

4. Interpretation of These Equations

The derived expressions indicate that OAM modes are formed by combining even and odd HE modes with a phase shift of $\pm j$ . This phase shift introduces the required azimuthal dependence, ensuring that the resulting mode carries orbital angular momentum with a well-defined topological charge.

When $l > 1$ , OAM modes are obtained from HE modes of order $l + 1$ and $l - 1$ .
For $l = 0$ , the modes originate from TE and TM modes, as HE modes do not exist in this case.
The term $\pm j$ ensures a $\pm π / 2$ phase shift between the even and odd HE modes, creating circular polarization.

5. Conclusion

The derivation of OAM modes in ring-core fibers highlights the fundamental role of hybrid HE and EH modes in supporting stable OAM transmission. The key takeaway is that OAM modes can be constructed as superpositions of hybrid modes with appropriate phase relationships. The special case for $l = 0$ involving TE and TM modes ensures completeness in describing the full mode spectrum in RCFs.

These findings are crucial for applications in optical communication, where OAM multiplexing can significantly enhance data transmission capacity. By leveraging RCFs and their unique mode structure, next-generation fiber-optic networks can achieve unprecedented efficiency and bandwidth.

Ref:

Ultra-low loss polymer-based photonic crystal fiber

Design and optimization of photonic crystal fiber

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2. Expressing OAM Modes in Terms of HE and EH Modes

3. Special Case for l=0

4. Interpretation of These Equations

5. Conclusion

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3. Special Case for $l = 0$