The effects of longitudinal spatial coherence of light in interference experiments

The physical conditions needed to observe the effects of longitudinal spatial coherence of emission of an extended thermal source are considered. Results of experiments on observation of interference pulses of longitudinal spatial coherence, performed using a scanning longitudinal shearing Michelson interferometer, are presented. Distinctions between the effects of temporal and longitudinal spatial coherence of light are established. It is shown that the presence of a noncompensated optical layer in one of the arms of the interferometer makes the interference pulses of the temporal and spatial coherence of light shift relative to each other (“recession”).     

Longitudinal pure spatial coherence of a light field with wide frequency and angular spectra

We have observed the longitudinal pure spatial coherence of a light field in an interference experiment when the length of the temporal coherence is significantly smaller than the length of the longitudinal spatial coherence of the light field. We introduce into consideration new spatial and temporal scales of a light field: the length of the coherent (free) run and the coherent time (the time of life) of a wave train.     

Longitudinal purely spatial coherence of a light field

The effects of longitudinal purely spatial coherence of light and the results of observation of these effects in an interference experiment are considered under the condition that the length of temporal coherence l _c is considerably smaller than the length of longitudinal spatial coherence ρ_‖ of the field. It is shown that, for l _c ? ρ_‖, the longitudinal purely spatial coherence of the light field in fact governs the coherence of the wave train in the process of its propagation. The length and the time of coherent (“free”) path of the wave train are considered as new spatial and temporal scales of a partially coherent light field.     

Spatial information transmission using axial temporal coherence coding

We present an approach that can be used for transmission of information through space-limited systems or for superresolution. The spatial information is coded with different axial temporal coherence by interfering every spatial region in the input with the same region, but with a certain known delay in the longitudinal axis. Every spatial region has different delay. After mixing all of the spatial information, it is transmitted through the space-limited system. At the detection the information is passed through a similar interference setup containing certain axial delay. By temporally scanning along the longitudinal axis, each time a different spatial region that was coded with the corresponding axial delay is reconstructed. To allow coding of different spatial regions with different and small axial delays, we use a thermal light source that has very short coherence length. We include experimental validation of the presented approach.     

Enhancement of spatial coherence during light propagation in bounded media

The spatial coherence of thermal light propagated through passive bounded media is measured. It is found that specular reflection at the boundaries enhances the degree of coherence. This result has considerable implications for the spatial coherence of light from laser and amplified spontaneous emission (A.S.E.) systems.     

What type of coherence of the optical field is observed in the Michelson interferometer

The types and corresponding coherence functions of the optical field are considered depending on the frequency and angular spectra of the field. The main concepts of the theory of coherence effects in the interference experiment with the amplitude splitting of the initial field are discussed. It is shown that, in strict correspondence with the theory of coherence of the random wave fields, the Michelson interferometer reveals manifestations of the transverse and longitudinal spatial (rather than temporal, as it is commonly adopted) coherence of the optical field; the purely temporal coherence of the optical field is revealed only under special conditions of the interference experiment. Beyond these conditions, either spatial or spatiotemporal coherence is revealed.     

Three-dimensional generalization of the Van Cittert-Zernike theorem to wave and particle scattering

Coherence properties of primary partially coherent radiations (light, X-rays and particles) elastically scattered from a 3D object consisting of a collection of electrons and nuclei are analyzed in the Fresnel diffraction region and in the far field. The behaviour of the cross-spectral density of the scattered radiation transverse and along to the local direction of propagation is shown to be described by respectively the 3D Fourier and Fresnel transform of the generalized radiance function of a scattering secondary source associated with the object. A relativistic correct expression is derived for the mutual coherence function of radiation which takes account of the dispersive propagation of particle beams in vacuum. An effect of the spatial coherence of radiation on the temporal one is found; in the Fresnel diffraction region, in distinction to the field, both the longitudinal spatial coherence and the spectral width of radiation affect the longitudinal coherence. A solution of the 3D inverse scattering problem for partially coherent radiation is presented. It is shown that squared modulus of the scattering potential and its 2D projections can be reconstructed from measurements of the modulus and phase of the degree of transverse spatial coherence of the scattered radiation. The results provide a theoretical basis for new methods of image formation and structure analysis in X-ray, electron, ion, and neutron optics.     

Visibility in differential phase-contrast imaging with partial coherence source

This paper gives theoretical analysis of visibility of fringes, which is influenced by distances, temporal and spatial coherence of source, in hard x-ray differential phase-contrast imaging with microfocus x-ray source. According to the character of longitudinal periodicity of the interferogram, the setup is insensitive to mechanical drift and vibrations. The effect of temporal coherence of x-ray source is investigated and its related bandwidth is derived. Based on the theory of partially coherent light, it shows that the requirement for the spatial coherence of x-ray source is not strict and can be met by the general microfocus x-ray tube for x-ray differential phase-contrast imaging.     

A contrast variation of image speckle intensity under illumination of partially coherent light

Speckle patterns are formed by completely and partially coherent light at the image plane of a diffusing object. The general formula is derived for the contrast variation of the image speckle intensity distribution as a function of the spatial coherence of the illuminating light, the amplitude point-spread function of the optical imaging system and the statistical characteristic of the diffusing object. The effect of spatial coherence of the illuminating light on the contrast variations of the image speckle intensity distribution is theoretically evaluated under various statistical conditions of the diffusing object. The theoretical results are confirmed experimentally.     

Spatial and temporal coherence effects in interference microscopy and full‐field optical coherence tomography

Low‐coherence optical microscopy or optical coherence microscopy uses light with short coherence length. The well‐known case is: “white‐light interferometry”, which became recently more known as: “optical coherence tomography”. However, when lenses and microscope objectives are used to create interferometric images, in what is known classically as “interference microscopy” or today as “full‐field optical coherence tomography” the spatial coherence starts to play a critical role. In this article the coherence effects in low‐coherence optical microscopy are reviewed. As this technology is becoming increasingly publicized due to its importance in three‐dimensional imaging, particularly of scattering biological media and optical metrology, the understanding of the fundamental physics behind it is essential. The interplay between longitudinal spatial coherence and temporal coherence and the effects associated with them are discussed in detail particularly when high numerical apertures are used. An important conclusion of this study is that a high‐contrast, high‐resolution system for imaging of multilayered samples is the one that uses narrowband illumination and high‐NA objectives with an index‐matching fluid. Such a system, when combined with frequency‐domain operation, can reveal nearly real‐time three‐dimensional images, and is thus competitive with confocal microscopy.     

大功率单色LED的空间相干特性

基于van Cittert-Zernike定理,从理论上计算出大功率单色发光二极管(LED)辐出光,在空间中传输后的空间相干性分布。计算结果表明,自发辐射的LED光源其辐射光在空间中传输后,由非相干光变为部分相干光,且其空间相干性与LED芯片的结构有关。实验上采用双缝干涉对LED辐射光的空间相干性进行测量,由干涉条纹的可见度与两点间的相干度之间的关系得出,非相干LED光源的辐射光在传输后为部分相干光。而采用间隔可调的双缝干涉测量两点间的相干度后发现,LED的芯片发光区域决定了其辐射光在传输中的空间相干性分布。理论计算的空间相干性分布与实验测量结果基本吻合。     

Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the mutual coherence function of quasi-monochromatic electromagnetic Gaussian Schell-model (EGSM) beams propagating through turbulent atmosphere is derived analytically. By employing the lateral and the longitudinal coherence length of EGSM beams to characterize the spatial and the temporal coherence of the beams, the behavior of changes in the spatial and the temporal coherence of those beams is studied. The results show that with a fixed set of beam parameters and under particular atmospheric turbulence model, the lateral coherence of an EGSM beam reaches its maximum value as the beam propagates a certain distance in the turbulent atmosphere, then it begins degrading and keeps decreasing along with the further distance. However, the longitudinal coherence length of an EGSM beam keeps unchanging in this propagation. Lastly, a qualitative explanation is given to these results.     

Factorial moment of the photon counting distribution of gaussian-lorentzian light under various spatial coherence

The generating function of the photon counting distribution is studied under consideration of the spatial coherence. The high order factorial moments are induced from the generating function, and are found to be experimental proof for the theory. The behavior of the second factorial moment is investigated for the purpose of spectroscopy of Rayleigh scattered light. The factorial moment measurements are made by using a digitalized electric system. The spectral line shape is determined quantitatively under various spatial coherence conditions.     

Study on the spatial coherence of vortex Gaussian beams passed atmospheric turbulence

In this paper, the spectral degree of coherence formula was derived by using the generalized Huygens–Fresnel principle and the method of Rytov phase structure function quadratic approximation. The spatial coherence of vortex Gaussian beams passed atmospheric turbulence was studied. It showed that the spatial coherence was mainly affected by the coherence of light source, the number of topological charges and the transmission distance. The distance of transmission was farther, the spatial coherence of beam was better. The number of topological charges were more, the spatial coherence of beam was better. In a certain transmission distance, the coherence of light source was better, the spatial coherence of beam was worse. In addition, there were coherent vortexes which spectral degree of coherence was zero after the partially coherent vortex beams getting through atmospheric turbulence transmission.     

用非近似方法分析双光栅干涉仪

利用简单直观的非近似方法分析双光栅干涉仪,证明了该干涉仪的干涉条纹反差不受照明光源相干性限制时,光源的空间位置不影响条纹形状,仍是等间距直条纹.给出了消除光源相干性限制的一般条件,并讨论了条件失配对干涉仪的影响.     

Spatially periodic wave fields: an experimental demonstration of the relationship between the lateral and the longitudinal spatial frequencies

The relationship between the lateral and the longitudinal periodicities of the coherence function of a propagating quasi-periodic field, which has such well known manifestations as the Talbot and the Lau effects, is verified experimentally in the more general case of partial coherence. The 3-D nature of spatial periodicity of such a field might find applications in some areas of image processing or in 3-D spatial encoding of optical information.     

热光源产生贝塞尔光束的理论与实验

利用热光源获得了近似贝塞尔光束。对多波长光波同时入射轴棱锥的情形进行分析和数值模拟,结果显示轴棱锥后会形成近似贝塞尔光束,但受光场非相干叠加的影响截面光强对比度降低。实验采用卤素灯杯作为产生贝塞尔光束的热光源,针对热光源相干性差的特点设计出一套光学系统,提高光波的空间相干性,再使光波平行透过轴棱锥得到了近似贝塞尔光束。将实验结果与数值计算对比,发现实验所得贝塞尔光束的特性与理论基本吻合,但最大无衍射距离较理论计算所得的短,并对此进行了分析。     

An analysis of the second order spatial coherence of stationary laser oscillators with a Fabry-Perot resonator

We investigate in this paper the results yielded by a rigorous application of second order spatial coherence theory to a Gaussian TEM-multimode laser beam. Formulae for a qualitative discussion and a quantitative evaluation of the second order spatial coherence properties of a laser oscillator with an arbitrary number of longitudinal and/or transverse Gaussian TEM-modes of oscillation are derived and considered for some practically important cases.     

Mutual effects of temporal and spatial coherence of laser radiation during wind refraction

Conclusions Generalizing our results, one can identify the basic mechanism responsible for the mutual influence of temporal and spatial fluctuations in the amplitude of a light field in a moving medium with thermal nonlinearity.A necessary condition for this type of mutual interaction is a nonlinear incoherent effect connected nonlinearly with the induced fluctuations in the refractive index n_n. The mutual coupling of temporal and spatial variations is detectable only when the fluctuating part of n_n is comparable in magnitude to the normal refractive index, and when the distribution is nonuniform over the beam cross section and unstable during the pulse.The fundamental relationships of mutual interaction are that the slow fluctuations, which peel off the nonlinear medium over time, lead to a degradation of the spatial coherence of the light beam while it is being nonlinearly refracted. In all cases the spatial coherence of the laser radiation is reduced. The reverse effect of spatial fluctuations on the degradation of temporal coherence of a light pulse that is spatially-incoherent at the entrance to the medium is similar in its characteristics. This mutual interaction is substantially reduced when the nonstationary self-interaction regime promotes smoothing of fluctuations in the induced optical channel.Finally, we note that if the noise component of the pulse is a stable, uniform process, the transformations of the temporal and spatial coherence of a light field will occur independently in a nonlinear medium.Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 32, No. 7, pp. 816–822, July, 1989.     

Resolution and noise in ghost imaging with classical thermal light

The resolution and classical noise in ghost imaging with a classical thermal light are investigated theoretically. For ghost imaging with a Gaussian Schell model source, the dependences of the resolution and noise on the spatial coherence of the source and the aperture in the imaging system are discussed and demonstrated by using numerical simulations. The results show that an incoherent source and a large aperture will lead to a good image quality and small noise.