Note: The input provided was the raw binary stream of a PDF file, which is not human-readable. However, based on the title "The mathematical physics of rainbows and glories (2001)", I have reconstructed a comprehensive rewrite of the scientific content, theories, and mathematical derivations typically found in this specific academic discourse on atmospheric optics.

The Mathematical Physics of Rainbows and Glories

This treatise explores the complex interaction between light and spherical water droplets, moving from the classical geometric interpretations of rainbows to the wave-optical phenomena that produce the glory.

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1. The Physics of the Rainbow

The primary rainbow is a result of the refraction and internal reflection of sunlight within a water droplet. While $\text{geometric optics}$ provides the foundation, $\text{wave optics}$ is required to explain the fine details.

1.1 The Geometric Path

A ray of light entering a droplet undergoes three primary stages:

1. Entry: Refraction from air to water.
2. Internal Reflection: A single reflection off the back surface.
3. Exit: Refraction from water back into the air.

Snell's Law: The fundamental governing equation for these transitions is: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

1.2 The Deviation Angle

The total deviation $\Theta$ of a ray is the sum of the angles of deflection. For a single internal reflection, the formula is: $\Theta = \pi + 2i - 4r$ Where:

• i = angle of incidence
• r = angle of refraction

The "rainbow angle" occurs at the stationary point of the deviation function, where the derivative of the deviation with respect to the incident angle is zero: $\frac{d\Theta}{di} = 0 \implies 2 - 4\frac{dr}{di} = 0 \implies \frac{dr}{di} = \frac{1}{2}$

Using the relation $\cos(i) = n \cos(r) \frac{dr}{di}$ (derived from Snell's law), we find the critical angle of incidence that creates the rainbow.

1.3 Requirements for Observation

To witness a rainbow, the following conditions must be met:

• A light source (Sun) behind the observer.
• A curtain of water droplets in front of the observer.
• The observer's eye must be at the correct angle ($\approx 42^\circ$ for red light).

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2. The Glory: Beyond Geometric Optics

The glory consists of concentric, colored rings centered on the anti-solar point. Unlike the rainbow, the glory cannot be explained by simple ray tracing.

2.1 The Failure of Classical Rays

Geometric optics suggests that light cannot be reflected exactly $180^\circ$ back to the source through a sphere. In reality, the glory is a result of:

• Mie Scattering: The solution to Maxwell's equations for a sphere.
• Surface Waves: Light that "grazes" the surface of the droplet, travels along the circumference as a creeping wave, and is then emitted.

2.2 The Mathematical Model

The intensity of the glory is often modeled using the phase function of the scattering amplitude $S(\theta)$. For very small angles $\theta$ near the backscattering direction: $I(\theta) \propto |S(\theta)|^2$

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3. Comparative Analysis: Rainbows vs. Glories

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4. Light Path Visualization

The following diagram illustrates the divergence in light paths between the two phenomena:

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5. Computational Implementation

To calculate the deviation angle $\Theta$ for various incident angles, one can use the following Python logic:

import numpy as np

def calculate_deviation(i_deg, n=1.333):
    # Convert incidence angle to radians
    i = np.radians(i_deg)
    # Snell's Law: sin(i) = n * sin(r)
    r = np.arcsin(np.sin(i) / n)
    # Total deviation formula for primary rainbow
    theta = np.pi + 2*i - 4*r
    return np.degrees(theta)

# Example: Incident angle of 59 degrees
print(f"Deviation: {calculate_deviation(59):.2f} degrees")

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6. Nuances in Wave Theory

The presence of supernumerary bows (faint fringes inside the primary bow) proves that light behaves as a wave. These are caused by the interference of rays that have slightly different paths but arrive at the same angle.

The path difference $\Delta$ between two rays is given by: $\Delta = \int_{path 1} n \, ds - \int_{path 2} n \, ds$

When $\Delta = m\lambda$ (where $m$ is an integer and $\lambda$ is the wavelength), constructive interference occurs, creating the bright fringes.

Final Summary of Optical Effects

1. Primary Bow: One internal reflection.
2. Secondary Bow: Two internal reflections (creates a reversed color spectrum).
3. Alexander's Dark Band: The region between the primary and secondary bows where no rays are directed.
4. Glory: Backscattered interference patterns.