Feynman's optics
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Feynman's optics |
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Richard Feynman popularized an interesting approach to optics in his book QED, the Strange Theory of Light and Matter. His ultimate interest is the study of quantum mechanics using his 'sum over paths' method, which is an alternative to the standard description using a wave equation (Schroedinger), or matrix mechanics (Heisenberg). The aim in optics is to obtain an intuitive understanding of interference phenomena, diffraction, etc., without taking everything as a consequence of Maxwell's equations (which lead to the wave equation in electromagnetism).
The technique of phasor analysis that was used to analyze the double- and multi-slit aperture interference patterns made certain assumptions (that parallel wave fronts are superimposed at infinity, which in the laboratory can be accomplished by focussing them on a screen using a converging lens), and used stationary waves whose phase difference gave rise to the interference effects. The same idea was used for Fraunhofer diffraction by a single slit, except that amplitudes were added up for waves which were assumed to originate at different spots across the aperture. Feynman's approach also adds amplitudes using arrows which are added according to the vector addition rules. The meaning of the arrows is different, however. The idea is to follow straight-line paths between source and screen. The paths can bend sharply at some common distance (e.g., the location of an aperture). Along these paths one accumulates the temporal phase of the light wave: it travels with velocity c, and its frequency f can be obtained from the wavelength lambda using lambda*f = c. Different paths will have different lengths and the wave will oscillate by a different amount. The oscillation of the wave can be represented by a handle on a clock, which we will allow to rotate counterclockwise (to have the clock's advancement correspond to positive phase angles). The objective is to consider the various paths, and to add the amplitudes (arrows that correspond to final clock handle orientations) to form an overall amplitude. The claim is that the square of the length of the final amplitude is a measure of the light intensity observed at that location. Before proceeding with an interesting application, such as Fraunhofer scattering, we will consider simply a source of photons, and a screen separated by some distance X, which can be measured in multiples of the wavelength of the light (this way we do not have to redo the problem as a function of two parameters, the source-screen distance, and the wavelength of the light separately, but we just vary a dimensionless parameter). At the midway distance between source and screen the photon is allowed to pass through any vertical height, i.e., not only the straight-line connection between source and screen is allowed. 
A number of questions can be asked: a) why should we consider anything but the straight connection? b) why restrict the photon paths to have a kink only half-way between source and detector? c) how do we deal with the infinite number of possibilities of chosing paths that cross at arbitrary height y? The first question is easily answered: if we are interested in exploring the wave nature of light, we have to do more than ray optics (which would consider only the straight-line connection). The second and third questions are harder to answer. Let us take the attitude that we would like to explore what happens when we give the photons some freedom as to how they move, and then add the contributions coherently (add amplitudes) to form a total amplitude. We use a finite number of paths. This limits us to a finite range of heights y at the midpoint of each photon's path. The magnitude of the final amplitude is somewhat arbitrary as a result of the chosen values of y that contribute to our sum over the paths. One learns from the resulting diagram (called a Cornu spiral) that the biggest contribution to the final amplitude comes from paths near the straight connection. The sum over the arrows starts with negative heights y, resulting in the addition of arrows that nearly cancel out (spiral near the origin in the diagram below), then as one approaches y=0 the contributions do add up, and finally as y becomes large and positive it spirals to make the tip of the total amplitude arrow move in a circular way. Explore the applet by changing the source-detector separation (measured in multiples of the wavelength) and observe what happens as the separation approaches values less than the wavelength. The orientation of the final amplitude arrow is arbitrary. What happens as the source-detector distance exceeds the wavelength substantially? Apparently any paths but the ones close to the straight-line connection become irrelevant. The range of y values covered by the 20 paths chosen corresponds to the source-detector distance. What can we learn from this exercise? The wave nature of light will be most noticeable when the distances travelled are not too large compared to the wavelength of the light. It remains to be seen whether the method is capable to explain Fraunhofer diffraction and other phenomena related to the wave nature of light. Those wave phenomena are important even if the lengths involved are much larger than the wavelength of the light. |
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