Complex Degree Of Spectral Coherence Research Articles

Independent-elementary-field model for three-dimensional spatially partially coherent sources

The independent-elementary-source model recently developed for planar sources [Opt. Express 14, 1376 (2006)] is extended to volume source distributions. It is shown that the far-field radiation pattern is independent of the three-dimensional (3D) distribution of the coherent elementary sources, but the absolute value of the complex degree of spectral coherence is determined by the 3D Fourier transform of the weight function of the elementary fields. Some methods to determine an 'effective' three-dimensional source distribution with partial transverse and longitudinal spatial coherence properties are outlined. Especially in its longitudinal extension, this effective source volume can be very different from the primary emitting volume of the source. The application of the model to efficient numerical propagation of partially coherent fields is discussed.

Two-beam interference experiments in the frequency-domain to measure the complex degree of spectral coherence

Two-beam interference experiments in the frequency-domain to measure the complex degree of spectral coherence

Two-beam interference experiments in the frequencydomain to measure the complex degree of spectral coherence

Correlation-induced spectral changes in two-beam interference experiments in the space-frequency domain are used to determine the amplitude and phase of the complex degree of spectral coherence. The spectral modifications are observed either as a shift in the peak wavelength or as sinusoidal modulations within the bandwidth of the white light spectrum due to the complex degree of spectral coherence of the secondary source and the path difference between the interfering beams. These correlation-induced spectral changes were analysed using a theoretical model to establish the behaviour of the real and imaginary parts of the complex degree of spectral coherence over the entire visible region of the spectrum.

Angular separation and spectra of two point sources determined from measured spectral coherence

The theoretical foundation is described not only of how to determine the angular separation of two point spectral sources, but also on how to recover each spectral profile from the measured complex degree of spectral coherence. It is shown that the so-called space-frequency equivalence law is not always required. A simple simulation experiment is done by incorporating paired superluminescent diodes as the two point sources with different broadband spectra. The experimental results verify the validity of the theoretical prediction.

Determination of the angular separation and the spectra of two-point sources from spectral coherence measurements

We present a novel method not only for determining the angular separation of unknown two-point spectral sources but also for obtaining each of their spectral profiles from measurements of both their complex degree of spectral coherence and their spectra across an observation area in the space-frequency domain. The theoretical predictions are fully proved by an experimental demonstration with a pair of uncorrelated two-point spectral sources.

Complex degree of spectral coherence and spectral changes measured by two-beam interference experiments in the space-frequency domain

A variety of the complex degree of spectral coherence μ is provided for investigating the correlation-induced spectral changes resulting from two-beam interference in the space-frequency domain. Primary two-point sources with variant Gaussian-like spectra illuminate the secondary source plane across which spatially, partially coherent wave fields result to produce μ. The variational aspects of μ are controllable by the primary source parameters. The spectral changes are discussed in detail in association with the μ-dependent interference term. The argument of μ plays a particularly important role in asymmetric changes in spectrum.

Measurements of the complex degree of spectral coherence by use of a wave-front-folded interferometer

A wave-front-folded interferometer composed of a Kösters prism and an image-forming lens provides a good way of measuring the complex degree of spectral coherence. Demonstrative experiments are implemented by incorporation of uncorrelated two-point spectral sources in the primary-source plane; these sources establish a spatially partially coherent wave field over the secondary source plane just in front of the interferometer. The complex degree of spectral coherence of the secondary source is successfully measured after an a priori calibration is made for the interferometer that was constructed.

Correlation-induced spectral changes dependent upon spatiotemporal interference effects

A wave-front folded interferometer consisting of a Kosters prism and an image forming lens provides an excellent way to make a precision measurement of correlation-induced spectral changes. Experiments are successfully made by incorporating a primary spectral source of super-luminescent diodes. The diode emits a Gaussian-like spectrum. Theoretical background for the measurement is given in the framework of geometric optics. It is shown theoretically and experimentally that the spectral changes are induced by two causes: one is the complex degree of spectral coherence of the secondary source, and the other the time delay between the interfering optical waves. No spectral change takes place if the secondary source satisfies a spatially incoherent condition at particular optical frequency, whereas the spectrum changes most clearly if a spatially coherent condition is satisfied.

A Theoretical Analysis of the Coherence-Induced Spectral Shift Experiments of Kandpal, Vaishya, and Joshi.

The optical system used by Kandpal, Vaishya, and Joshi in their experiments on coherence-induced spectral shifts is analyzed theoretically. An approximate form for the cross-spectral density in the secondary source plane is obtained, and it is shown that, contrary to the assertions of Kandpal, Vaishya, and Joshi, the corresponding complex degree of spectral coherence in this plane is wavelength dependent. After making some assumptions about the behavior of the interference filter used in the system, an approximate form for the spectrum of the light on-axis in the observation plane is obtained. It is shown that the peak wavelengths of this spectrum do not agree with those reported by Kandpal, Vaishya, and Joshi. Possible reasons for this disagreement arc discussed.

Scattering of a light beam from waves at an air–sea interface

The authors report a new theory that describes the scattering of an upward propagating laser beam from the sea through the air-sea interface in the presence of sea waves. The sea is assumed to be a uniform dielectric and conventional scattering theory is employed by using a modification to the first Born approximation that permits treatment of surface refraction phenomena. Methods of statistical radiometry are also used in a new manner by assuming that the surface scattering function for the sea waves can be treated by a quasi-homogeneous source model to calculate the second-order correlation functions for the partially coherent scattered field. These correlation functions yield a simple expression for the radiant intensity of the scattered field as the convolution of the Fourier transform of the complex degree of spectral coherence for the sea waves with the squared modulus of the angular spectrum of plane waves for the incident laser beam. We believe that this theory is a significant improvement over the models that are usually used for modeling this phenomenon.

Simple experimental arrangement for observing spectral shifts due to source correlation.

A filter-lens combination in conjunction with partially coherent light is used to produce a secondary source that violates a certain scaling law. The spectral shifts observed have been found to be dependent on the complex degree of spectral coherence, filter peak transmission wavelength, and half-band-width of the filter.

Field correlations within a Bessel-correlated spherical source

The spatial field correlations within a Bessel-correlated spherical source with large size parameter (radius much greater than the wavelength) are investigated. Exact expressions for the field spectral density, in terms of a partial-wave series, as well as for the magnitude of the complex degree of spectral coherence (correlating the field at an arbitrary internal point with that at the center of the sphere), are obtained. Asymptotic approximations for a large size parameter are found by applying complex angular-momentum techniques. The results are discussed and plotted for a wide range of values of the ratio between the source spatial coherence length and the wavelength. The asymptotic approximations are found to be quite accurate. The field ranges from extreme incoherence to full spatial coherence. The latter is attained only when the source’s coherence length becomes much larger than its Fresnel radius. Results previously obtained in the limit of an unbounded source are verified. Features arising from finite size and boundary effects are discussed.

Coherence properties of some isotropic planar sources

The complex degree of spectral coherence for both the source distribution function and the field amplitude over the source plane are found for isotropic homogeneous planar sources that do not produce evanescent plane waves. This calculation is extended to also include similar sources that are quasi-homogeneous rather than homogeneous. All the calculated coherence functions are expressed in simple mathematical form and are tabulated for easy reference.

Spectral coherence and the concept of cross-spectral purity*

A new measure of correlations in optical fields, introduced in recent investigations on radiometry with partially coherent sources, is studied and applied to the analysis of interference experiments. This measure, which we call the complex degree of spectral coherence, or the spectral correlation coefficient, characterizes the correlations that exist between the spectral components at a given frequency in the light oscillations at two points in a stationary optical field. A relation between this degree of correlation and the usual degree of coherence is obtained and the role that the complex degree of spectral coherence plays in the spectral structure of a two-beam interference pattern is examined. It is also shown that the complex degree of spectral coherence provides a clear insight into the physical significance of cross-spectral purity. When the optical field at two points is cross-spectrally pure, the absolute value of the complex degree of spectral coherence at these points is found to be the same for every frequency component of the light. This fact is reflected in the visibility of the spectral components of the interference fringes formed by light from these points.