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A Statistical Introduction to Temporal Coherence Effects in Interferometers

Pierre H. Chavel, Jean‐Michel Jonathan

Published Mar 25, 2017• 2 citations
Publisher

Abstract

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.

DOI
10.1117/3.2265058.ch2

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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.
Introduction We will now study the properties of fluctuating electromagnetic fields, paying attention mainly to the optical region of the electromagnetic spectrum. It seems hardly necessary to stress that every electromagnetic field found in nature has some fluctuations associated with it. Even though these fluctuations are, as a rule, much too rapid to be observed directly, one can deduce their existence from suitable experiments that provide information about correlations between the fluctuations at two or more space-time points. The simplest manifestations of correlations in optical fields are the well-known interference effects that arise when two light beams that originate from the same source are superposed. With the availability of modern light detectors and electronic circuitry of very short resolving time, other types of correlations in optical fields began to be studied in more recent times. These investigations, as well as the development of lasers and other novel types of light sources, led to a systematic classification of optical correlation phenomena and the complete statistical description of optical fields. The area of optics concerned with such questions is now generally known as optical coherence theory . The first investigations of coherence phenomena are due to Verdet (1865, 1869) and von Laue (1907a, b). Some early investigations of Stokes (1852) and Michelson (1890, 1891a, b, c, 1892, 1920) although not explicitly mentioning coherence – because this concept is of a much later origin – have also contributed to the clarification and development of this subject.
The interference phenomenon of light is a common but most important effect in physics. In this article, we compare and analyse the interference effects of both coherent and incoherent light; in particular, the interference of spatially incoherent light sources is described. Thermal light and a two-photon entangled source may both be regarded as incoherent sources which can generate similar second-order interference effects, but their underlying physics is quite different. First-order interference can be realized with both spatially coherent and incoherent sources in the same well-designed experimental setup, but exhibit different interference patterns; their different interference mechanisms will also be clarified.
Introduction The degree of first-order temporal coherence, a function denoted by g (1) ( τ ), provides information about the coherence length and the power spectral density of a light source. However, without additional information, g (1) ( τ ) has no bearing on intensity fluctuations and higher-order statistics of the emitted light. A quasi-monochromatic laser beam and the beam of light from an incandescent light bulb, provided that the latter is properly filtered to match the spectral line-shape of the former, will have identical degrees of first-order coherence. Any interferometric experiment involving the splitting and superposition of amplitudes would yield identical results for the laser beam and the (properly filtered) thermal light. Therefore, on the basis of such experiments alone, there is no way to distinguish the two light sources. It turns out, however, that the intensity fluctuations of laser light are fundamentally different from those of thermal light. The two sources can, therefore, be distinguished based on their second-order coherence properties. An ideal photodetector produces an electrical signal proportional to the “cycle-averaged intensity” of the E -field (or B -field) of the light beam at the location of the detector. Assuming that the electrical bandwidth of the detector (including all associated circuitry) is greater than the bandwidth of the incident light wave by at least a factor of 2, the output of the detector should accurately represent the intensity fluctuations of the light beam as a function of time.
The conditions are examined under which interference effects should be observable in the superposition of two light beams derived from independent sources. The quantum-mechanical description of these effects differs from the classical description principally in that it is necessarily based on expectation values of the light intensity at one or more space-time points. It is shown that pure states of the radiation field which are not energy states give rise to interference effects, but in that case the two beams cannot meaningfully be described as statistically independent. For a realistic description mixed states have to be introduced, when the expectation value of the intensity gives no indication of interference effects. On the other hand, the intensity correlation at two space-time points is a periodic function of the separation of the points and indicates the presence of transient interference effects. The effects become readily observable only when the average photon occupation number of each cell of phase space is appreciably greater than 1, and this explains why laser beams were needed for the experimental observations. It is pointed out that, even in these experiments, each photon may be regarded as interfering only with itself.
The physical interaction processes behind the emergence of dark and bright fringes registered by the detectors at the output of optical interferometers is explained. This knowledge should be helpful to interferometrists to make better physical interpretations of their data. The belief in mysterious “interference of single indivisible photon” will disappear once we recognize that the spatial or temporal energy re-distributions are generated by the physical transformation experienced by the detecting molecules drawing energy from all the light beams. The molecules could be photodetectors at the interferometer output, or the beam combining dielectric boundary. The superposition principle (SP), represented by the linear mathematical sum of two or more wave amplitudes, does not represent an observable phenomenon. The superposition effect (SE), represented by the non-linear square modulus of the joint dipolar stimulation of the detectors by all the superposed waves, is observable. We present two laboratory experiments to clarify these points. Both classical beam combiners and quantum detectors are capable of generating superposition fringes of intensity variations. The logic of “quantumness” of light is narrowly relevant only when a quantum detector deciphers the fringes; it is not valid for classical beam combiners. We will also discuss “entanglement” based on these experiments.
The high-order intensity correlation function of NNN-photon interference with thermal light observed in a recent experiment [S. Oppel, T. B\"uttner, P. Kok, and J. von Zanthier, Phys. Rev. Lett. 109, 233603 (2012)] is analyzed. The terms in the expansion of the NthN\mathrm{th}Nth-order correlation function are put into three groups. One group contributes a homogeneous background. Both of the other two contribute $(N\ensuremath{-}1)$-fold super-resolving fringes. In principle they correspond to coherent and incoherent superpositions of classical optical fields, respectively. Therefore similar super-resolving fringes can be obtained without intensity-correlation measurements. We report the experimental results of the coherent and incoherent super-resolving diffraction fringes, which are observed directly in the intensity distribution. The $N\ensuremath{-}1$ sources in both the coherent and incoherent cases are set in certain definite positions. In the coherent case, moreover, the phase difference between two adjacent source fields is $\ensuremath{\pi}$. The fringe visibility is unity in the incoherent case, while it decreases as NNN increases in the incoherent case.