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Journal of Optics

Waves, amplitude waves, intensity waves

Tiberiu S. Tudor

Published Nov 1, 1991• 8 citations
Publisher

Abstract

The interference of two or more waves give rise, at intensity level, to another wave phenomenon, a second-order wave structure. The interference of different frequency waves is the frame in which this fact becomes most evident. A theoretical treatment of the interference of two homogeneous plane waves of different frequencies is given. The notion of amplitude wave is introduced. Ordinary interference and beating are discussed as special cases, in the frame of this general theory. Some experimental illustrations are given.

DOI
10.1088/0150-536x/22/6/005

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The interference of two or more waves give rise, at intensity level, to another wave phenomenon, a second-order wave structure. The interference of different frequency waves is the frame in which this fact becomes most evident. A theoretical treatment of the interference of two homogeneous plane waves of different frequencies is given. The notion of amplitude wave is introduced. Ordinary interference and beating are discussed as special cases, in the frame of this general theory. Some experimental illustrations are given.
Optical interference is regarded as a classical phenomenon and is usually completely described in classical terms, in which optical fields are represented by classical waves. Does it mean that optical interference, which is fundamentally ascribable to the phenomenon of a superposition of wave amplitudes, cannot be applied to test quantum phenomena? The superposition principle is at the heart of quantum physics and one could expect that it should distinguish the quantum nature of light from thewave nature. Our experience based on extensive theoretical analysis and experimental observations shows that classical and quantum theories of optical interference readily explain the presence of an interference pattern resulting from the first-order coherence. However, there are interference effects involving the higher order (second-order) coherence that distinguish the quantum (photon) nature of light from the wave nature. Quantum interference has recently returned to prominence because of its utility in manipulating spontaneous emission and other radiative properties of atomic systems. In this chapter, we present elementary concepts and definitions of both the classical and quantum theory of optical interference and discuss similarities and differences between these two approaches [80].
In paying tribute to the many contributions of Adolf Lohmann to optical science, we must highlight his inspiring talent as a lecturer and teacher as well as his ability to introduce original concepts and ideas.We have chosen to devote our contribution to one aspect of the teaching of physical optics. When teaching interferences, the lecturer always encounters the question:Why do two different frequencies never interfere? A two-beam interferometer illuminated by light of a different frequency on each arm does not show fringes. Or to be more precise, in the quite common case of thermal light sources,1 common experience shows that interferences between two different frequencies of the source spectrum cannot be observed. Interferences with light that is not strictly monochromatic are observed but merely result from the incoherent superposition of all component monochromatic interference intensities. Providing a convincing explanation of this fact in an elementary course is not easy because statistical tools are required that are only available to the students at a more advanced level—typically, senior undergraduate or graduate. However, a full account of the phenomenon based on solid statistical grounds is not even commonly found in textbooks designed for such advanced students. Such is the aim of the present chapter. Indeed, the superposition of two different frequencies should be expected to show a beat phenomenon in time. Nevertheless, common experience shows that interference effects are stationary in time and show no beat effect. If we start from the definition of interferences as the phenomenon whereby the intensity resulting from the superposition of two light beams differs from the sum of the two intensities, we are led to clarifying the temporal aspects in the detection of light beams. First, the common situation of describing the effect at a less advanced level is examined, together with the difficulties that automatically arise. After a review of the necessary statistical concepts such as, in particular, the Wiener-Khintchin theorem, the issue of what exactly is meant by average intensity and the associated “classical detection noise” is addressed, leading to a complete model of temporal coherence in thermal light.
IT is well known from experiment, as well as from classical theory, that strictly monochromatic light waves originating from different sources do interfere.' An example in which the waves are nearly monochromatic is found in the case of radio waves, and it is known that radio waves from two stations that have the same frequency show an interference effect. We may describe the phenomenon quantum mechanically if we represent the light quanta emitted by both sources by waves in Schroedinger configuration space. So long as we consider the quanta emitted by one source as different from those emitted by the other, the total wave function will be the product of the wave functions corresponding to the quanta emanating from both sources, and there will be no interference effect. In the present paper, we will see how things are changed when we symmetrize or antisymmetrize the wave function, i.e. assume either Bose or Fermi statistics for the particles. In the simple quantum mechanical picture that we will use here, we will assume a definite number of photons coming out from each source, and this implies that the phase relations will be undetermined.2 As a result, even the symmetrized wave functions must be expected to give, at any point in space, the same density of quanta that classical theory determines, when the intensity is averaged over the phase differences between the two sources. This implies that no interference is observed between the waves from the two sources. If we want to look for an interference effect when dealing with a quantum picture in which we have a definite number of photons, we must look for a phenomenon whose classical description should be independent of the phase. We shall make use of the fact that, in classical theory,