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:

huygensplane.gif (3072 bytes)                                         huygens.gif (3902 bytes)

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:                                                                                                      

planar.gif (3493 bytes)                                                             planar2.gif (3091 bytes)

 

    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:

efield.gif (2643 bytes)

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. 

    Finally, there are several principles that one needs to be familiar with to be able to understand the experiments included in thisproject.  The first of which is the Principle of Coherence.  This principle depends upon the phase of the light in question.  If light beams from two independent sources reach a point S, there is no way to tell about phase history of the two light sources, and thus they are not able to combine to form stationary waves known as fringes. Actually, the light may produce interference effects, but the vibration frequency of light is so fast, as would be the phase differences from the separate sources, that the interference would occur to rapidly to be detected. These types of waves are referred to as incoherent and only their intensities combine locally at the point S where they interact.  Coherent light, however, is of particular interest to us.  An example of coherent beams would be light emitted from a single source that comes back together at a later time and position.   These waves, since they have the same phase history, are able to superpose and produce interference patterns because their amplitudes can combine.  This kind of  light, coherent light,  represents the foundation for the Principle of Coherence. The Principle of Coherence relies upon, and describes, the phase of the light in question. If two beams of light from separate sources reach a point S in space, there is no definite relation between their phase histories. They are therefore unable to superpose to form stationary waves. In reality, these light beams may actually produce interference effects, but because the changes in phase between the waves are so fast they would occur to rapidly to be detected. These types of waves are called incoherent and only their intensities combine at the point S where they interact. Coherent light, however, is of particular interest to the field of Optics. Light emitted from a single source that diverges and then comes back together at a later time and position would be an example of coherent waves. These light waves, because they both have the exact same phase history, are able to combine locally and produce interference effects. This kind of light represents the foundation for the Principle of Coherence and plays a major role in the study of wave theory. For example, lasers have a property known as coherence length that pertains to how long its emitted light maintains phase unity. Analogous to temporal coherence, it relates to how constant the phase of the light, or laser light, remains at any given point over a period of time. If throughout a specific time interval the phase of light in question remains constant, then the light is said to exhibit perfect temporal coherence. If the time period over which the light exhibits perfect temporal coherence is limited, then the light manifests partial temporal coherence. The time period over which the light maintains perfect temporal coherence is called the coherence time. This time, when multiplied by the speed of light (3x10^8 m/s), gives the coherence length. Coherence length is very important due to the fact that laser light is mostly monochromatic, but not wholly comprised of one frequency. Over time, these different frequencies of emitted light will spread apart because they all propagate at different velocities. Therefore, the interference effects characteristic to this light disappear for a displacement greater than the coherence length. Imagine, for example, laser light radiating from a single source. If the light is split up, then the difference in displacement of the two paths must not be longer than the coherence length to be able to see interference patterns. This phenomenon occurs because if the two paths differ by a distance larger than the coherence length, then the phases of the beams will be too different for their amplitudes to combine locally. Remember that coherence length dictates how far the light can travel apart before the phase histories of the two beams will become non-uniform. This problem concerns experiments using equipment like the interferometer that splits monochromatic light up into two separate beams for the purpose of recombination. For light with a small linewidth (frequency bandwidth of 1kHz), particularly that of a laser, it might require a path length difference of up to 300km to make the interference patterns non-visible. With white light, however, which has a large linewidth (includes wavelengths from 400-700 nm), it is simple to make the interference effects disappear because it exhibits a coherence length on the order of 10^-3 mm. This means that if the two arms of the interferometer differ by that tiny amount, then no interference patterns will be visible. This requirement causes difficulty in viewing the interference phenomena of white light because it takes such precision to align the interferometer correctly enough to view them.

   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.  

Table of Contents:

  1. Abstract
  2. Stationary Michelson Interferometer
  3. Travelling Mirror Michelson Interferometer
  4. Fraunhofer Diffraction
  5. Fabry-Perot Interferometer

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