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Introductory remarks

The spatial coherence of optical fields with a narrow spectral bandwidth such as a laser is ordinarily represented by the mutual coherence function in the space-time domain.[1] However, the spectral coherence function is indispensable when we discuss the spatial correlation of the fields with the wide spectral bandwidth. The correlation function of the Fourier spectra of the wavefields is called the cross-spectral density, and the spectral degree of coherence is defined as the normalized formula of the cross-spectral density. It was 1976 that E. Wolf and L. Mandel first proposed the concept of the spectral degree of coherence.[2]

One of the most interesting studies in which the spectral coherence plays a central role may be the correlation-induced spectral changes that are also known as the Wolf effect.[3] In 1986, E. Wolf pointed out that the spectrum of the light, which is propagating from the spatially, partially coherent source with the wide spectral bandwidth, can change depending upon the source correlation. Wolf also suggested that the spectrum is invariant through the propagation in the case where the spectral degree of coherence across the source obeys the Wolf's scaling law. It has long been known that the spectrum of the light changes because of the dispersive diffraction and the Doppler effect. However, the physics of the spectral changes which take place even if the light propagates through free space is indeed different from the dispersive diffraction or the Doppler effect. The age of the universe is estimated at twelve to fifteen billion years according to the latest study, and this value was mainly calculated from observed redshifts of the spectra of stellar objects. Therefore, much attention has been paid from an astrophysical point of view to the correlation-induced spectral changes, which imply the possibility of spectral change through the free propagation. However, it should be noted that the frequency does not shift in the Wolf effect unlike the frequency shift of a line spectrum seen in the Doppler effect.

Although many studies have been published since Wolf first pointed out this phenomenon,[4,5,6,7,8,9] most of the investigations are theoretical [3,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48] and the reports with the experimental verifications or the demonstrations are minor.[49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65] The main reason that makes laboratory experiments difficult may be hardness to separate two causes of the spectral changes, namely, the dispersive diffraction and the correlation-induced spectral changes. Nevertheless, experimental verification for the theoretical properties and establishing reliable techniques for these experiments are strongly required. Such experimental results and the reliable techniques will greatly help not only to justify the existing theoretical analyses but also to prompt the novel discoveries.

In contrast to the correlation-induced spectral changes that come from the problem of the forward propagation of the spatial coherence,[66,67,68,69] the interferometric imaging technique is based on the inverse propagation of the spatial coherence. The interferometric imaging techniques have long been used in practical measurements such as radio astronomy.[70] Although the resolving power of an ideal optical image-forming system is determined by ratio of a wavelength to an aperture size, inhomogeneities of the earth's atmosphere reduce the real resolving power. To overcome the effect of the earth's atmosphere, Fizeau first suggested in 1868 that Young's two-beam interference principle enabled one to measure the angular size of the stellar objects. The experiment for this idea was first conducted by Stephan. With the 1 m telescope of the Marseille Observatory, he succeeded in measuring the visibility of the interference fringes though the fringes were fluctuated owing to the turbulent atmosphere. Michelson also tried and succeeded in determining the angular diameter of a Jupiter's satellite.

One of the recent developments in the interferometric imaging technique was made by James.[71] James' technique is based on the new principle that is called the ``space-frequency equivalence theorem'' which was proposed by himself. The measurement system that uses this theorem enables us to determine the angular size of the source without changing the separation of the double aperture. The experimental demonstrations conducted by using the white light were reported by Kandpal.[72] The experimental results well agreed with the theory, and usefulness of this technique was confirmed. In the late 1990's, a principle of interferometric imaging for three-dimensional (3-D) source distribution was proposed by Rosen.[73,74,75] Their technique is based on the modified van Cittert-Zernike theorem. The experimental demonstration was also reported by themselves. Marks also presented their own principle for 3-D imaging in the latest publication.[76]

As stated above, there are increasing demands for 3-D imaging techniques. However, 3-D imaging techniques are still immature although 2-D imaging techniques have been sophisticated in both the theoretical analyses and the practical measurement systems. In other words, interferometric 3-D imaging is at the dawn. The methods recently proposed for 3-D imaging are based on the complex principles, and they need some fundamental restrictions for the state of the source correlation or locations of the sources because the basis of these new principles is originated from the van Cittert-Zernike theorem which holds for the spatial incoherent planer sources in the paraxial far field. As a result, new principles of interferometric 3-D imaging that can be applied under more general conditions are desired.

The fundamental concept through this work is the propagation of the spatial coherence. Although statistical optics which deals with the nature of the wave propagation in free space or scattering media has been considerably completed, there are much potentialities in the phenomena and the applications originated from or resting on these basic principles. From this background, the correlation-induced spectral changes and the interferometric imaging technique as the phenomenon and the application that are derived from the forward and the inverse propagation of the spatial coherence are studied theoretically and experimentally in this work. Studies on the correlation-induced spectral changes are devoted mainly on the experimental verification and the establishment of the reliable measurement method. Studies on the interferometric imaging techniques are, on the other hand, mainly on the proposal of new principles and concepts such as the 3-D imaging or the detection of the incoherent sources. Contents of chapters are listed with brief summaries in the followings.

In Chapter 2, the fundamental concepts of the second-order spatial coherence are reviewed. Definitions of the spatial coherence functions that are described in the space-time domain and the space-frequency domain, and their propagation law are introduced. The phenomenon of the correlation-induced spectral changes is also brought up.

In Chapter 3, the correlation-induced spectral changes are studied theoretically and experimentally. First, an experimental analysis of the spectral changes caused by the source correlation and the dispersive diffraction is presented. The spectral changes in a Gaussian-like spectrum that depend on ratio of the coherence area to the source area are investigated. Next, the spectral changes which are caused by only the source correlation are investigated. It is proven that the wavefront folding image-forming system is considerably reliable for observing the spectral changes released from the dispersive diffraction. Redshifts and blueshifts of the spectrum are experimentally observed depending on the source correlation and the observing location.

In Chapter 4, new principles for determining the two-dimensional source image are presented. One is a technique to determine not only the angular separation of two point sources but also their spectral profiles. The principle is based on measuring both the spectral degree of coherence and the uniform spectra across an observation area. The theoretical predictions are proven by an experimental demonstration with uncorrelated two point sources. The other one is for retrieving the cross-spectral density propagating in free space. The principle of the technique is based on the propagation law of the angular spectrum. Since this law holds without the paraxial approximation, the cross-spectral density across the off-axis reference plane can be retrieved. Results of an experiment demonstrating the retrieval of the cross-spectral density across any reference plane are also presented.

In Chapter 5, a novel interferometric 3-D imaging technique based on retrieving the sequential cross-spectral densities is presented. It is shown that the cross-spectral density propagating from the source enables us to retrieve the information of the second-order spatial coherence conveyed through the 3-D space. In addition to the 3-D imaging principle, a new concept to detect the spatially incoherent sources is presented. These principles make it possible to know the state of the spatial coherence across an arbitrary transverse plane and to find the incoherent sources even in the high background intensity. Full mathematical description and the analysis of the point spread function are also given.

In Chapter 6, an interferometric 3-D imaging technique that is based on retrieving the spatial distribution of the generalized radiance function is proposed. Although the technique that is presented in Chapter 5 requires the four-dimensional Fourier transform in the 3-D image retrieval, this imaging principle with the generalized radiance function enables us to retrieve the 3-D information by the two-dimensional Fourier transform. Consequently, the data processing time is much reduced. The point spread function of the system is derived and the spatial resolution is discussed.

In Chapter 7, noise-limitations of the interferometric imaging system described in Chapter 4 are studied in two limiting cases, namely the photon-noise-limit and the detector-noise-limit cases. The signal-to-noise ratio is theoretically derived and an experiments are conducted. The noise statistics obtained from the experimental results agree with the theoretical expectations.

In Chapter 8, concluding remarks about the entire dissertation are given.


next up previous contents
Next: Introduction to second-order spatial Up: No Title Previous: Contents
Hidenobu ARIMOTO
2000-02-17