Three-dimensional coherenceimaging in the Fresnel domainDaniel L. Marks, Ronald A. Stack, and David J. BradyWe show that three-dimensional incoherent primary sources can be reconstructed from finite-apertureFresnel-zone mutual intensity measurements by means of coordinate and Fourier transformation. Thespatial bandpass and impulse response for three-dimensional imaging that result from use of thisapproach are derived. The transverse and longitudinal resolutions are evaluated as functions of aper-ture size and source distance. The longitudinal resolution of three-dimensional coherence imaging fallsinversely with the square of the source distance in both the Fresnel and Fraunhofer zones. We exper-imentally measure the three-dimensional point-spread function by using a rotational shear interferom-eter. © 1999 Optical Society of AmericaOCIS codes: 030.1640, 110.1650, 110.4850, 100.3010, 100.6890, 070.4550.1. IntroductionImprovements in electronic sensors, automated po-sitioning systems, and data processing equipmentrender optical coherence imaging of complex three-dimensional ~3D! objects increasingly practical.Two-dimensional ~2D! imaging based on the far-fieldvan Cittert–Zernike theorem has been used in radioastronomy for more than two decades.1Recently,coherence imaging techniques have begun to shiftback to the optical domain,2,3and a number of opticalsystems have been implemented or are under devel-opment.4Several researchers have generalized the vanCittert–Zernike theorem to 3D source distributionsand have shown that 3D inversion is possible in thefar field.5,6LaHaie7describes modal 3D reconstruc-tion techniques that also work in the near andFresnel zones. Zarubin notes that the 3D general-ized van Cittert–Zernike theorem applies in theFresnel zone under certain coherence assumptionsand that the theorem can also be applied to x-ray andparticle scattering.8More recently, 3D source re-construction from a finite far-field aperture by use ofthe generalized 3D theorem was analyzed and exper-imentally demonstrated.9–11Unlike pseudo-3Dtechniques such as holography and stereo imaging,coherence imaging provides a true 3D model of objectsources.In this paper we show that Fourier reconstructiontechniques can be applied to Fresnel-zone reconstruc-tion by application of a coordinate transformation tothe generalized van Cittert–Zernike theorem. Thisextension is important because the object distancemay be much less for a given aperture and wave-length in the Fresnel zone than in the Fraunhoferzone. Because longitudinal resolution falls as thesquare of object distance, longitudinal resolution inthe Fresnel zone may exceed longitudinal resolutionin the Fraunhofer zone by several orders of magnitude.In Section 2 of this paper we review the Fourier-transform relationship between the source intensitydistribution and the far-field mutual intensity anddescribe the coordinate transformation by which asimilar relationship is obtained between the sourcedistribution and the Fresnel-zone mutual intensity.In Section 3 we explore the bandpass and resolutionlimits of 3D coherence imaging. Resolution con-straints are easily visualized by use of the 3D spatialbandpass, or band volume, because in limited-aperture systems the band volume has preciseboundaries.12,13The resolution along any given di-rection is inversely proportional to the extent of theband volume along that direction. In Section 3 weanalyze the band volume and the impulse responsefor two particular coherence measurement systems,The authors are with the Beckman Institute for Advanced Sci-ence and Technology and Department of Electrical and ComputerEngineering, University of Illinois at Urbana—Champaign, Ur-bana, Illinois 61801. D. J. Brady’s e-mail address [email protected] 5 January 1998; revised manuscript received 28 Octo-ber 1998.0003-6935y99y081332-11$15.00y0© 1999 Optical Society of America1332 APPLIED OPTICS y Vol. 38, No. 8 y 10 March 1999the Michelson stellar interferometer and the rota-tional shear interferometer. In Section 4 we showexperimental reconstructions obtained from an im-plementation of the rotational shear interferometer.2. Fourier Inversion of the Generalized VanCittert–Zernike TheoremThe mutual intensity for a quasi-monochromatic 3Dincoherent primary source can be expressed in termsof the source radiant power density by use of theHopkins integral:J~r1, r2! 5Sk02pD2*sI~rs!3exp@jk0~ur12 rsu 2 ur22 rsu!#ur12 rsir22 rsud3rs, (1)where s is the source volume, I~rs! is the 3D sourceradiant power density, J~r1, r2! is the mutual inten-sity at field sample positions r1and r2, and k052pyl0is the wave number of the optical field at wave-length l0.14The geometry of the radiation and mea-surement space is illustrated in Fig. 1. rsis theposition vector in the source volume. J~r1, r2! ismeasured between pairs of points drawn from anaperture labeled the correlation plane. The correla-tion plane lies a distance R along the z axis from thecenter of the correlation volume. As is shown in thefigure, r1and r2are vectors from the origin of thesource volume to field sampling points on the corre-lation plane. J~r1, r2! is the zero-delay mutual co-herence function between the field at r1and that atr2. Systems for measuring J~r1, r2! are described insection 3.The goal of coherence imaging is to invert Eq. ~1!and reconstruct I~rs! from measurements of J~r1, r2!.Inversion was previously shown to be straightfor-ward in the Fraunhofer zone, where Eq. ~1! reduces tothe generalized van Cittert–Zernike theoremJ~r1sˆ1, r2sˆ2! 5 I˜F~sˆ12 sˆ2!l0Gexp@ jk0~r12 r2!#l02r1r2, (2)where sˆ1and sˆ2are unit vectors in the r1and r2directions, r15 ur1u, r25 ur2u, and˜I~u! is the 3DFourier transform of the source intensity distribu-tion.5,6u is the position vector of I~rs! in 3D Fourierspace. According to Eq. ~2!, measurement of J~r1,r2! over a range in r15 r1sˆ1and r25 r2sˆ2yieldssamples of˜I~u! for u over the range ~sˆ12 sˆ2!yl0.Inasmuch as variations in r1and r2do not affect therange sampled in u, it is sufficient to measure J~r1,r2! for r1and r2drawn from a surface surroundingthe object rather than a volume. Doing so reducesthe six-dimensional measurement space of J~r1, r2! tofour dimensions. Even for r1and r2drawn from asurface, redundant values of ~sˆ12 sˆ2!yl0will be ob-tained. To characterize I~rs! to wavelength-limitedresolution it is necessary only to sample J~r1, r2! overa 3D subspace of r1R r2that fully samples