Fresnel equations
Introduction
The Fresnel equations describe the behavior of light when it transitions between two different media. These equations, named after the French physicist Augustin-Jean Fresnel, are fundamental in the study of optics and are used to predict the reflection and transmission of light, which is incident on the interface of two media with different refractive indices.
Derivation
The Fresnel equations can be derived from Maxwell's equations, which describe how electric and magnetic fields interact. The derivation involves applying the boundary conditions that the tangential components of the electric and magnetic fields are continuous across the interface. The derivation also assumes that the media are isotropic (the same in all directions) and homogeneous (the same at every point).
Reflection and Transmission Coefficients
The Fresnel equations give the reflection and transmission coefficients, which are complex numbers that describe the amplitude and phase shift of the reflected and transmitted waves, respectively. The reflection coefficient for perpendicular (s) polarization is given by:
- r_s = (n_1 cos θ_i - n_2 cos θ_t) / (n_1 cos θ_i + n_2 cos θ_t)
And for parallel (p) polarization:
- r_p = (n_2 cos θ_i - n_1 cos θ_t) / (n_2 cos θ_i + n_1 cos θ_t)
Where n_1 and n_2 are the refractive indices of the two media, and θ_i and θ_t are the angles of incidence and transmission, respectively. The transmission coefficients can be found from the reflection coefficients using the relation:
- t_s = 1 + r_s
- t_p = 1 + r_p
Brewster's Angle
A special case occurs when the light is incident at the Brewster's angle, at which the reflected light is completely polarized parallel to the plane of incidence, and no light is reflected in the perpendicular direction. The Brewster's angle can be found from the Fresnel equations by setting the reflection coefficient for p-polarized light to zero, which gives:
- θ_B = arctan(n_2/n_1)
Total Internal Reflection
Another special case is total internal reflection, which occurs when light traveling in a medium with a higher refractive index hits the boundary with a medium of lower refractive index at an angle greater than the so-called critical angle. In this case, all the light is reflected back into the first medium, and none is transmitted into the second medium. The critical angle can be found from the Fresnel equations by setting the transmission coefficient to zero, which gives:
- θ_c = arcsin(n_2/n_1)
Applications
The Fresnel equations have many applications in optics and engineering. They are used in the design of optical devices such as lenses and mirrors, in the study of optical phenomena such as reflection, refraction, and diffraction, and in the analysis of optical systems such as cameras and telescopes. They are also used in the field of fiber optics, where they help to understand the behavior of light in optical fibers.