Wave Optics – Introduction

Wave optics describes light with a second order differential equation known as the wave equation. Using the wave equation, optical phenomena that fall outside of the scope of ray optics can be described, such as interference and diffraction. One limitation of wave optics is in it’s inability to describe polarization effects and other phenomena that require a vector formulation.

The wave equation used in wave optics is as follows:

wavefunction

Optical intensity I(r,t) is proportional to the squared wavefunction and is defined as watts/cm^2. This is also referred to as irradiance.

intensity

A monochromatic wave does not vary in intensity over time:monoc

The time-independent equation is known as the complex amplitude of the wave: monoccomplex

And the above formula is the solution to the Helmholtz equation, another important formula used in wave optics:

helmholtz

wavenumber

Where k is the wavenumber.

 

Plane Waves, Spherical Waves, Paraboloidal Waves are the main wave formations.  A plane wave is seen as a wave that continues infinitely along a continuous plane with a constant intensity. The vector along which the plane exists is known as the wavevector. The spherical wave consists of spheres centered about a single point. It may also originate from the centered point or travel inwardly towards the center. A paraboloidal wave is an approximation to the spherical wave using the Fresnel approximation. A paraboloidal approximation is useful for simplifying the interaction of a spherical wave in calculations for diffraction and other situations. Paraxial Waves are a ideal approximation of the interaction of these waves that simplifies many calculations. The approximation leading to the paraboloidal wave is done by considering the spherical wave at a sufficiently great distance, at which only curvatures in the wave are detected. For points very far from the center of the sphere, waves may be treated as plane waves.

paraboloidal

Paraxial optics is an idealization of the directionality of optical rays that allows for approximated results of a number of optical systems. A wave is a paraxial wave under the condition that the wavefront normals are paraxial rays. Under the paraxial approximation of waves, optical rays are perpendicular to optical waves. wavefronts

 

The paraxial Helmholtz equation is as follows:

paraxhelmholtz

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