# A Brief History of Optics and Wave Theory

In 1655, a gentleman named Grimaldi observed curious diffraction fringes produced by a narrow slit.  Surprised by the observation, he concluded that light had to consist of a very fine fluid of some sort in a state of constant vibration.  As it turns out, quite an erratic conclusion, but one that would spark the beginning of a search for the theory to describe  light.  It was not until 1678, however, that wave theory come to the forefront.  In this year, Huygens proposed his wave propagation theory which would get the ball rolling for future research.   All the while, Sir Isaac Newton had also gotten his feet wet with diffraction.  His studies culminated in the publishing of his book, Optiks, in 1704.  The book grappled with the concepts of refraction, dispersion, and the discovery of the spectrum as well as analyzing Grimaldi's experiences with diffraction.   The book essentially proposed two modes of propagation, a corpuscular and wave theory.  His corpuscular theory, as it turns out, was a fairly decent method for measuring wavelength.  Newton's work remained at the forefront for the majority of the 1700's until Thomas Young performed his famous Double Slit Experiment in 1802.   From his results and conclusions, it was determined that light propagates in wave form and therefore possesses the ability to interfere with itself.  Young also is responsible for the Principle of Coherence and Principle of Superposition, two of the most fundamental principles in the fields of Interferometry and Optics.   Thus, the Wave Theory of Light came to be thanks to the refinement efforts of several distinguished and important people.

# The Basic Principles of Wave Theory

Before jumping in and talking about light and its specific properties, one must understand how waves propagate and behave in general.  Electromagnetic waves are very complex entities.  The mathematics behind their descriptions, Maxwell's equations, are very belaboring and strenuous.  Therefore, one must look at the very basics first.

The properties of electromagnetic waves can be deduced using Maxwell's four equations of electricity and magnetism.  A lot about the way we think of light in a wave form, however, comes from Huygen's Principle of 1678.  His principle states, according to Serway, that "all points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate outward with speeds characteristic of waves in that medium" (1036).  Huygen arrived at this principle by assuming that a source of light emits a spherical wave.   This principle becomes very important when discussing topics such as diffraction.   A visual description of Huygen's Principle can be seen below:

The planar wave front (I discuss planar waves further below as they are important to the wave nature of light) on the left above is denoted by A'A.  The three arbitrary points in this plane are what Huygen would consider "point sources" for the subsidiary wavelets which are seen in red on the plane B'B.  The same construction of wavelets can be seen for a spherical wave on the right.  Now that they have been mentioned, what exactly constitutes a planar wave?  To be planar means that there exists large enough distance, in comparison to the wavelength of the light, from its source to consider the waves at that displacement to be parallel to each other.  This situation is diagramed below to better help you visualize the definition:

Near Source:

From the picture on the right, it becomes clear that at large distances from a point source, the wavefronts are nearly parallel planes and the rays are parallel lines perpendicular to these planes.  Thus, we are able to make the assumption that waves can be planar.  Next, the motion of these waves must be taken into account.  If there exists a scalar wave propagating in the x direction, then at time t=0 its y position can be described by the equation .   The A term represents the amplitude of the wave, and lambda describes the wavelength.  For any time t, however, the vertical displacement will not be the same.   This change in vertical will repeat itself along the x-axis as the distance changes.  Thus, for a planar wave moving with a phase velocity v in the x-direction, its motion can be described by the equation Now, if the equation is written in terms of the angular wave number , and the angular frequency the more familiar description of a sinusoidal wave can be obtained.  The Phi term represents the phase of the wave determined by the initial conditions that one may impose upon it.  This equation represents the general picture of a plane wave travelling in the x-direction at time t, however, it represents a scalar entity.  It becomes more useful to us when we make the jump to electromagnetic waves to write this expression in terms of a vector quantity.  For a vector quantity alpha, which would include both the electric (E) and magnetic (B) field vectors for light, the wave equation becomes where c is the speed of light.  This expression comes from looking at the second derivatives with respect to position and time of the equation for a sinusoidal wave, and then using the chain rule to rearrange some terms.

Since we now have a description of how our wave moves, lets examine some of the properties specific to a light wave.  Imagine a planar light wave travelling in the x-direction, the electric field vector will be on the y-axis and the magnetic field vector the z-axis as seen below:

The electric and magnetic field at any point on the x-axis are dependant solely on position and time , not y or z positions.  The case here is that of a linearly polarized/plane polarized ray of light.  The electric and magnetic fields of this wave can be described by:

and

Again, there is no y or z dependence, only position in terms of direction of travel, and time.  Notice further their resemblance to the wave equation!  Also, at any instant, the ratio of the electric field to the magnetic field equals c, the speed of light.

The Principle of Superposition also helps our understanding of wave forms.  Picture two coherent light waves of the same wavelength emitted from the same source arriving at point S on different paths.  The Principle of Superposition states that the resultant amplitude can be described as the sum of the individual wave amplitudes at that point.  Mathematically, this can be showed as follows:

To get to the last step, a trigonometric identity for Sin(a) + Sin(b) was used.  The resultant wave, as seen from above, has the same frequency and wavelength as the individual waves comprising it and the amplitude is twice the initial amplitude times the cosine of half the phase constant.  This means that if the two waves arrive at the spot S in phase (i.e. Phi=0), then the cosine term equals one and the waves interfere constructively with new amplitude twice that of their initial ones.  On the other hand, if the phase difference is somewhere between 0 and pi, then the resultant wave will have amplitude somewhere between zero and twice the initial amplitude of the two waves.   Now, we have all the basic knowledge of wave behavior to go ahead and apply it to actual physical situations.  Therefore, use the table of contents below to choose an experiment that interests you and see how it exemplifies the Wave Nature of Light.

### References:

1. Corson, Dale R. and Paul Laurrain.  Electromagnetism: Principles and Applications.  W.H. Freeman:  New York.  1997.
2. Davis, Christopher C.  Lasers and Electro-Optics.   Cambridge University Press:    Great Brittain. 1996.
3. Serway, Raymond A.  Physics For Scientists and Engineers with Modern Physics.  Saunders College Publishing:   Philadelphia.  1996.
4. Tolansky, S.  An Introduction to Interferometry. William Clowes and Sons Ltd.:  London.  1966.
5.