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Aluminum has become the material of choice for overhead transmission lines, surpassing copper due to its abundance and cost-effectiveness. The most prevalent type is the aluminum conductor, steel-reinforced (ACSR), which combines aluminum strands around a steel core. Other variants include all-aluminum conductors (AAC), all-aluminum alloy conductors (AAAC), aluminum conductor alloy-reinforced (ACAR), and aluminum-clad steel conductors. Advanced designs, such as aluminum conductors with steel...
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A household microwave and lasers are examples of standing electromagnetic waves in a cavity. When two conducting metal plates are placed parallel at the nodal planes, it creates a cavity where standing waves are formed. The cavity between the two planes is analogous to a stretched string held at the points x = 0 and x = L. Here, the distance 'L' between the two planes must be an integer multiple of half of the wavelength. The wavelengths that satisfy this condition are given by:
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Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a period T. The wave on the string is sinusoidal and is translated in the positive x-direction as time progresses. Sine is a function of the angle θ, oscillating between +A and −A and repeating every 2π radians. To construct a wave model, the ratio of the angle θ and the position x is considered.
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Related Experiment Video

Updated: Feb 19, 2026

Fabrication And Characterization Of Photonic Crystal Slow Light Waveguides And Cavities
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Optical waveguide coupling and the golden number.

Olivier Parriaux

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    |November 2, 2017
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    Summary
    This summary is machine-generated.

    Numerical modeling of a four-waveguide directional coupler reveals field extrema ratios near the golden ratio (ϕ). Analytical and coupled wave analyses confirm this finding, especially in weakly coupled systems.

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    Area of Science:

    • Photonics and Wave Phenomena
    • Integrated Optics
    • Electromagnetism

    Background:

    • Directional couplers are fundamental components in integrated optics.
    • Understanding mode behavior in multi-waveguide systems is crucial for device design.
    • The golden ratio (ϕ) appears in various natural and mathematical phenomena.

    Purpose of the Study:

    • To investigate the ratio of field extrema for propagating modes in a four-waveguide directional coupler.
    • To confirm numerical findings through analytical derivations.
    • To explore the relationship between coupling coefficients and the field extrema ratio.

    Main Methods:

    • Numerical modeling of a four-identical-slab-waveguide directional coupler.
    • Analytical derivation of the dispersion equation for eigenmodes.
    • Coupled wave analysis to determine propagation constants and field extrema ratios.

    Main Results:

    • The ratio of field extrema for the four propagating modes is consistently close to the golden ratio (ϕ).
    • Analytical derivation confirms the numerical results and provides an exact expression for the ratio.
    • Coupled wave analysis shows the ratio strictly equals ϕ in weakly coupled structures.

    Conclusions:

    • The golden ratio (ϕ) is a significant characteristic of mode field distribution in four-waveguide directional couplers.
    • Weakly coupled structures exhibit a precise golden ratio relationship in their mode field extrema.
    • This finding offers insights into the fundamental physics of multi-waveguide optical devices.