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Mathematics

Mathematical Physics

Optics Principles

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Reflection

The principle that states the angle of incidence is equal to the angle of reflection. Expressed as

\theta_{incidence} = \theta_{reflection}

Snell's Law

The relationship between the angles of incidence and refraction for a wave impinging on an interface between two media with different refractive indices is given by

n_1 \sin(\theta_1) = n_2 \sin(\theta_2)

where

n_1

and

n_2

are the refractive indices of the two media, and

\theta_1

and

\theta_2

are the angles to the normal of the interface.

Fermat's Principle

The principle that states the path taken between two points by a ray of light is the path that can be traversed in the least time.

Refraction

The change of direction of a wave passing from one medium to another caused by its change in speed. For light, this is typically from air to another transparent substance (like water or glass).

Lambert's Law

Describes the intensity observed from a diffusely reflecting surface, which is directly proportional to the cosine of the angle $\theta$ between the direction of the incident light and the surface normal, represented as

I = I_0 \cos(\theta)

, where

I_0

is the intensity of the incident light.

Fresnel Equations

These equations describe the reflection and transmission (refraction) of light when incident on an interface between two media. They are derived using the boundary conditions of Maxwell's equations.

Rayleigh Criterion

A criterion for deciding when two light sources are resolved in an optical system; two sources are considered resolved if the principal diffraction maximum of one image coincides with the first minimum of the other, given by the equation

\theta = 1.22 \frac{\lambda}{D}

where

\lambda

is the wavelength and

D

is the diameter of the aperture.

Diffraction

The bending of waves around obstacles and the spreading out of waves past small openings. It is described by the Huygens-Fresnel principle.

Critical Angle

The minimum angle of incidence above which total internal reflection occurs when light travels from a medium with higher refractive index to a medium with lower refractive index, given by

\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)

where

n_1

is the refractive index of the denser medium and

n_2

is the refractive index of the less dense medium

Malus's Law

Describes the intensity of polarized light after passing through a polarizer, given by the equation

I = I_0 \cos^2(\theta)

where

I_0

is the initial intensity and

\theta

is the angle between the light's initial polarization direction and the axis of the polarizer.

Huygens-Fresnel Principle

Every point on a wavefront is considered a source of wavelets, which spread forward at the same speed. These wavelets interfere with each other to form a new wavefront. This is the basis for understanding wave propagation and diffraction.

Brewster's Angle

The angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. For unpolarized light, it predicts the angle where reflected light will be polarized and is given by

\tan(\theta_B) = \frac{n_2}{n_1}

where

n_1

and

n_2

are the refractive indices of the incident and transmitting mediums, and

\theta_B

is Brewster's angle.

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