## Light

We all know what ‘light’ is… Light, or ‘visible light’ is a form of energy, an electromagnetic radiation (waves) visible to the human eye. It allows us to see. Visible light has an **approximate** wavelength range of **400** nano-meters to **700** nano-meters with **infra-red** having longer wavelengths and **ultra-violet **having shorter wavelengths. It is a well known fact to young scholars that light travels on a straight line. That’s what I was told when I was really young (I cant remember exactly how young). One interesting thing I have learnt about light, other than it allows us to see, is that it suffers some physical distortions or deviations when it encounters objects in its path. This fact makes imaging an interesting exercise and a hobby that some people master, and others fail to master. When an imaging system takes an image of an object, two things can happen that makes the image less perfect or less ideal. these two things are aberration and diffraction effects. Aberration, in the simplest of terms, is a deviation in light through optical elements like lenses, causing images of objects to be blurred. I know what you are thinking, no, its not necessarily caused by defects in the lens. Let’s leave it at that. Aberration is not that interesting, or is it? On the other hand, diffraction is the bending of light as it passes through media of different optical densities, or encounters small object(s) or small opening(s) of a size of the order of the wavelength of the light. I am sure I got that right 🙂 . The reference to the order of a wavelength is because diffraction effects are then most pronounced and hence clearly visible. Here is the interesting bit: diffraction of light causes interference patterns. So, “what is the difference between diffraction and interference?”, you might ask. Well, **Richard Feynman** said, “No-one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them”. Who am I to argue with that guy? His reasoning was, when there are two sources, we refer to the effect as interference between the sources, and when they are many, we refer to the phenomena in a generalized manner as diffraction. I know someone else will argue that diffraction is the underlying process of light interacting with the object or the opening, and interference is the way in which the resulting phenomena presents itself, the patterns. Somehow, both these guys might just be right, each under a different point of view. When the object provides multiple closely spaced points or there are multiple closely spaced openings, diffraction results in a complex pattern of varying intensity, mainly due to the superposition principle. The superposition principle states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. Then there is the Huygens-Fresnel principle, which covers the idea that every point in the object that light reaches becomes a secondary source of a spherical wave-front, the sum of which determine the future form of the wave, following the superposition principle. What I am leading to is, diffraction can be used to formally describe the way that light waves propagate in free space.

## Light Propagation Modeling

To put it into perspective, or in mathematical form, we will need to get into the theory of electrodynamics a little bit, if not a lot. In electrodynamics, we usually start with the four Maxwell’s equations:

[latexpage]

begin{equation} label{eq:one}

boldmath nabla . vec{D} = rho

end{equation}

begin{equation} label{eq:two}

boldmath nabla . vec{B} = 0

end{equation}

begin{equation} label{eq:three}

boldmath nabla times vec{E} = frac{- partial}{partial t} vec{B}

end{equation}

begin{equation} label{eq:four}

boldmath nabla times vec{B} = frac{- partial}{partial t} vec{E}

end{equation}

I am not going to derive these equations. A good reference book for that is the Classical Electrodynamics, 3rd Edition by John David Jackson.

Equation (3) contains vectors for both magnetic and electric fields. These fields can be decoupled, say for example, when an imaging sensor capture an image, it samples the electric charge due to the electric field only. Mathematically, we can decouple the electric field from the magnetic field in equation (3) by taking the curl of that equation and using the vector identity

We end up with a homogeneous equation:

[latexpage]

begin{equation} label{eq:five}

boldmath nabla ^2 vec{E} – mu epsilon frac{partial^2}{partial t^2} vec{E} = 0

end{equation}

Similarly, for magnetic field, we end up with the homogeneous equation:

[latexpage]

begin{equation} label{eq:six}

boldmath nabla ^2 vec{H} – mu epsilon frac{partial^2}{partial t^2} vec{H} = 0

end{equation}

where is the permeability and the permitivity, of the medium. For a vacuum, and

These values, when we compare equations (5) and (6) to that of a mechanical system, reveal that Corresponds to the inverse of the speed of light.

This value matches very closely with experimental results. This allows light to be modeled well within the theory of electromagnetism. It also shows that light, and in general electromagnetic waves do not require matter to propagate, but rather the speed of such waves in propagation is determined by the material they may be travelling through. A general equation can be written of the form:

[latexpage]

begin{equation} label{eq:seven}

boldmath frac{partial ^2}{partial t^{2}} vec{psi} – c^{2} nabla ^2 psi = 0

end{equation}

where could stand for , the electric field or , the magnetic field. Equation (7) is a linear second order differential equation. The fact that this equation is linear leads to the superposition principle which means that if is a solution to the wave equation, then is also a solution of the wave equation where a_{j} is a real or complex arbitrary constant. The wave equation can therefore accept or take a variety of solutions. This leads us to the conclusion that all light fields in a homogeneous medium must then be solutions to the wave equation. Some solutions are easy or possibly takes much less effort to describe in theory than in practice. It is through the control of physical conditions in a set up that the light waves can be guaranteed to fit into a particular solution of the wave equation. One example is; by using coherent light (laser or light from a pinhole), we can approximate the plane wave solution to the wave equation at a far field. At near field, the wave can be approximately viewed as a spherical wave.

The plane wave solution can be written as:

Where is the amplitude and is the phase phase of the wave, which can also be written as .

For a plane wave, is a constant perpendicular to the direction of propagation of the wave. A plane or surface of constant phase is known as a wavefront. The wavefronts of a plane wave are perpendicular to the direction of wave propagation.

When equation (7) is written in Cartesian coordinates:

[latexpage]

begin{equation} label{eq:eight}

frac{1}{boldmath c^2} frac{partial ^2}{partial t^{2}} psi = frac{partial ^2}{partial x^{2}}psi plus frac{partial ^2}{partial y^{2}}psi plus frac{partial ^2}{partial z^{2}}psi

end{equation}

Assuming that the the wave is made up of a complex amplitude riding over a carrier frequency , then :

[latexpage]

begin{equation} label{eq:eight}

boldmath psi(x,y,z,t) = real { psy (x,y,x) e^{jomega_o t}}

end{equation}