DISPOSICIONES ADICIONALES Primera

The relatively short history of long baseline optical interferometry is borne upon the much longer history of discoveries in optics leading to acceptance of a wave theory of light. The first astronomical interferometers would not come until late in the 19th century and the development of large-scale high precision instrumentation.

1.3.1 Newton’s Rings

The first description of the phenomenon of interference comes from Robert Boyle and Robert Hooke (Born & Wolf 1999), working independently. Boyle described the colors apparent in thin films and noted that the color was dependent upon the thickness of the film. In his book

MicrographiaHooke described the colors present in thin sheets of Muscovy-glass (mica) and was able to reproduce the effect using lenses (Hooke 1665). Newton analyzed these effects by pressing a convex lens above a flat glass plate (Newton, Isaac 1704). This produces a pattern of bright and dark concentric rings. These rings occur in both transmission and reflection and, when generated with white light, display a pattern of colors. Newton was able to derive a formula that predicts the radius of the ring for a given wavelength:

rn=

r

Rλ(N ₋ 1

2), (1.1)

where N is the ring number,λis the wavelength, and R is the radius of curvature of the lens.

1.3.2 The Wave Nature of Light

The contributions of Boyle, Hooke, and Newton to the understanding of the colors of thin films as well as those of Grimaldi (Grimaldi 1665) on diffraction lend support to a theory that

presents light as some sort of wave. This concept of a wave nature of light was first published by Christian Huygens in 1690 (Huygens 1690). Despite this evidence, the predominant view held by scientists of the time was one put forth by Newton of a corpuscular nature of light. As a result of conflicting personal accounts and rivalries, it is essentially lost to history and rivalry just how much each individual contributed to the explanation of the colors of thin films; but, it is this work that first began to describe, if not explain, the wave nature of light and the

principle of interference.

Young in 1801. Young devised the now famous double-slit experiment (Figure 1.1) in which a narrow-band light source far behind two narrow but closely-placed slits projects light through the two slits and onto a screen. A portion of the light that passes through one slit overlaps with part of the light from the other slit. If light is composed of particles one would expect a distribution of light cast on the screen behind each slit. If corpuscular theory is correct and light is a particle, then where the light from the two slits overlapped, Young should have seen a region of increased brightness behind each slit. In 1803, Young performed his experiment and observed instead a striped pattern of constructive and destructive interference. This effect can only be satisfactorily explained by a wave theory of light.

Not only was corpuscular theory overturned but Young was able to calculate the wavelength of light from the spacing between the light and dark interference bands. In 1807, Young published this work inA Course of Lectures on Natural Philosophy and the Mechanical Arts

(Young 1807). From this work,tanθ = _Dy relates the position of the fringes with angular spacing,θ _≈ λ_d, wheredis the separation of the two slits,Dis the distance between the slits and the screen,yis the distance from the center of the screen to the interference band, andθ

is the angle from the center of the slits to the interference band maximum. For different orders it can be shown that: dsinθ=mλ. The equation,

I(x) =I0cos2 πxd λD , (1.2)

gives the intensity of light, whereI0 is the maximum intensity on the screen. Figure 1.2shows monochromatic fringes of three different wavelengths and the polychromatic fringe resulting

from their superposition.

Figure 1.1: A figure from Young’s 1807 “A course of lectures on natural philosophy and the mechanical arts” showing his double slit experiment.

Work on interference and polarization by Arago and Fresnel (Arago & Fresnel 1819) in 1817 advanced the wave theory of light, demonstrating that the wave must be transverse as opposed to longitudinal like sound. The wave theory of light would remain dominant until the quantum revolution led by Max Planck in 1900 and by Einstein’s explanation of the

photoelectric effect in 1905, for which he won the Nobel prize, refined the theory of light into the more enigmatic wave-particle duality we hold today.

−20 −10 0 10 20 a 0.0 0.2 0.4 0.6 0.8 1.0 I(a)

Figure 1.2: Example monochromatic fringes and their superposition resulting in a polychro- matic fringe packet.

1.3.3 Fourier

In developing his work “The Analytical Theory of Heat” in 1822, Joseph Fourier proposed that any function could be represented as the weighted sum of a series of sines (Fourier 1822). Fourier analysis is now an integral part of the modern world, and the Fourier transform freely allows the processing of information between the frequency or spatial domains. The Fourier transform and its inverse are used extensively in the analysis of interferometric data:

X(f) = Z ∞ −∞ x(t)e−i2πf tdt (1.3) x(t) = Z ∞ −∞ X(f)ei2πf tdf. (1.4)