ࡱ>   CHl %bjbjVV 7T<<+8MTTދ "###~-~-~-2F~--|~-~-~-## 666~-##f6~-66j oCt#)#.p20ދpJx0^xdCtCtxtD6~-~-~-6~-~-~-ދ~-~-~-~-x~-~-~-~-~-~-~-~-~- :  MACROBUTTON MTEditEquationSection2 Equation Chapter 1 Section 1 SEQ MTEqn \r \h \* MERGEFORMAT  SEQ MTSec \r 1 \h \* MERGEFORMAT  SEQ MTChap \r 1 \h \* MERGEFORMAT  MACROBUTTON MTEditEquationSection2 Equation Section 4 SEQ MTEqn \r \h \* MERGEFORMAT  SEQ MTSec \r 4 \h \* MERGEFORMAT 3. Spatiotemporal field correlations 3.1. Spatiotemporal correlation function. Coherence volume. All optical fields in practice fluctuate randomly in both time and space and are subject to a statistical description  ADDIN EN.CITE Goodman200034295[1]34295342956Goodman, Joseph W.Statistical opticsWiley classics libraryxvii, 550 p.Wiley classics libraryOptics Statistical methods.Mathematical statistics.2000New YorkWiley0471399167 (classics ed.) 047101502412185972Jefferson or Adams Building Reading Rooms GE320.B6; .M86 1999http://www.loc.gov/catdir/description/wiley0310/00708884.htmlhttp://www.loc.gov/catdir/toc/onix06/00708884.html[ HYPERLINK \l "_ENREF_1" \o "Goodman, 2000 #34295" 1]. These fluctuations depend on both the emission process (primary sources) and propagation media (secondary sources). Optical coherence is a manifestation of the field statistical similarities in space and time and coherence theory is the discipline that mathematically describes these similarities  ADDIN EN.CITE Mandel199530115[2]30115301156Mandel, LeonardWolf, EmilOptical coherence and quantum opticsxxvi, 1166 p.Coherence (Optics)Quantum optics.1995Cambridge ; New YorkCambridge University Press0521417112 (hardback)3799509Physics/Astronomy 535.2; M312O[ HYPERLINK \l "_ENREF_2" \o "Mandel, 1995 #30115" 2]. A deterministic field distribution in both time and space is the monochromatic plane wave, which is only a mathematical construct, impossible to obtain in practice due to the uncertainty principle. The formalism presented below for describing the field correlations is mathematically similar to that used for mechanical fluctuations, for example, in the case of vibrating membranes. The analogy between the two different types of fluctuations and their mathematical description in terms of spatiotemporal correlations has been recently emphasized  ADDIN EN.CITE Popescu200716793[3]167931679317Popescu, G.Park, Y. K.Dasari, R. R.Badizadegan, K.Feld, M. S.Coherence properties of red blood cell membrane motionsPhys. Rev. E.Phys. Rev. E.03190276
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2007[ HYPERLINK \l "_ENREF_3" \o "Popescu, 2007 #16793" 3]. A starting point in understanding the physical meaning of a statistical optical field is the question: what is the effective (average) temporal sinusoid,  EMBED Equation.DSMT4 , for a broadband field? What is the average spatial sinusoid,  EMBED Equation.DSMT4 . A monochromatic plane wave is described by  EMBED Equation.DSMT4 . These two averages can be performed using the probability densities associated with the temporal and spatial frequencies, S(() and P(k), which are normalized to satisfy  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Thus, S(()d( is the probability of having frequency component ( in our field, or the fraction of the total power contained in the vicinity of frequency (. Similarly, P(k)d3k is the probability of having component k in the field, or the fraction of the total power contained around spatial frequency k. Up to a normalization factor, S and P are the temporal and spatial power spectra associated with the fields. The two effective sinusoids can be expressed as ensemble averages, using S(() and P(k) as weighting functions,  EMBED Equation.DSMT4  (1a)  EMBED Equation.DSMT4  (1b) Equations 1a-b establish that the average temporal sinusoid for a broadband field equals its temporal autocorrelation, (. The average spatial sinusoid for an inhomogeneous field equals its spatial autocorrelation, denoted by W. Besides describing the statistical properties of optical fields, coherence theory can make predictions of experimental relevance. The general problem can be formulated as follows (Fig. 1):  Figure 1. Spatio-temporal distribution of a real optical field Given the optical field distribution  EMBED Equation.DSMT4  that varies randomly in space and time, over what spatiotemporal domain does the field preserve significant correlations? This translates into: combining the field  EMBED Equation.DSMT4  with a replica of itself shifted in both time and space,  EMBED Equation.DSMT4 , on average, how large can  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  be and still observe significant interference? We expect that monochromatic fields exhibit infinitely broad temporal correlations, plane waves are expected to manifest broad spatial correlations. Regardless of how much we shift a monochromatic field in time or a plane wave in space, they remain perfectly correlated with their unshifted replicas. It is difficult to picture temporal correlations decaying over timescales that are shorter than an optical period and spatial correlations that decay over spatial scales smaller than the optical wavelength. In the following we provide a quantitative description of the spatiotemporal correlations. The statistical behavior of optical fields can be mathematically captured generally via a spatiotemporal correlation function  EMBED Equation.DSMT4  2 The average is performed temporally and spatially, indicated by the subscripts r and t. Because common detector arrays capture the spatial intensity distributions in 2D only, we will restrict the discussion to  EMBED Equation.DSMT4 , without losing generality. These averages are defined in the usual sense as  EMBED Equation.DSMT4  3 Often we deal with fields that are both stationary (in time) and statistically homogeneous (in space). If stationary, the statistical properties of the field (e.g. the average, higher order moments) do not depend on the origin of time. Similarly, for statistically homogeneous fields, their properties do not depend on the origin of space. Wide-sense stationarity is less restrictive and defines a random process with only its first and second moments independent of the choice of origin. For the discussion here, the fields are assumed to be stationary at least in the wide-sense. Under these circumstances, the dimensionality of the spatiotemporal correlation function  EMBED Equation.DSMT4  decreases by half,  EMBED Equation.DSMT4  4 The spatiotemporal correlation function becomes  EMBED Equation.DSMT4  5  EMBED Equation.DSMT4  represents the spatially averaged irradiance of the field, which is, of course, a real quantity. In general  EMBED Equation.DSMT4  is complex. Define a normalized version of  EMBED Equation.DSMT4 , referred to as the spatiotemporal complex degree of correlation  EMBED Equation.DSMT4  6 For stationary fields  EMBED Equation.DSMT4  attains its maximum at  EMBED Equation.DSMT4 , thus  EMBED Equation.DSMT4  7 Define an area  EMBED Equation.DSMT4  and length  EMBED Equation.DSMT4 , over which  EMBED Equation.DSMT4  maintains a significant value, say  EMBED Equation.DSMT4 , which defines a coherence volume  EMBED Equation.DSMT4  8 This coherence volume determines the maximum domain size over which the fields can be considered correlated. In general an extended source, such as an incandescent filament, may have spectral properties that vary from point to point. It is convenient to discuss spatial correlations at each frequency  EMBED Equation.DSMT4 , as described below. 3.2. Spatial correlations of monochromatic light 3.2.1. Cross-spectral density Taking the Fourier transform of Eq. 2 with respect to time, we obtain the spatially averaged cross-spectral density  ADDIN EN.CITE Wolf198217785[4]177851778517Wolf, E.Wolf, E Univ Rochester,Dept Phys & Astron,Rochester,Ny 14627New Theory of Partial Coherence in the Space-Frequency Domain .1. Spectra and Cross Spectra of Steady-State SourcesJournal of the Optical Society of AmericaJ Opt Soc Am J Opt Soc AmJournal of the Optical Society of America343-35172319820030-3941ISI:A1982ND83200006<Go to ISI>://A1982ND83200006English[ HYPERLINK \l "_ENREF_4" \o "Wolf, 1982 #17785" 4]  EMBED Equation.DSMT4  9 The cross-spectral density function was used previously by Wolf to describe the second-order statistics of optical fields, the Fourier transform of the temporal cross-correlation between two distinct points,  EMBED Equation.DSMT4   ADDIN EN.CITE Wolf198217785[2, 4]177851778517Wolf, E.Wolf, E Univ Rochester,Dept Phys & Astron,Rochester,Ny 14627New Theory of Partial Coherence in the Space-Frequency Domain .1. Spectra and Cross Spectra of Steady-State SourcesJournal of the Optical Society of AmericaJ Opt Soc Am J Opt Soc AmJournal of the Optical Society of America343-35172319820030-3941ISI:A1982ND83200006<Go to ISI>://A1982ND83200006EnglishMandel19953011530115301156Mandel, LeonardWolf, EmilOptical coherence and quantum opticsxxvi, 1166 p.Coherence (Optics)Quantum optics.1995Cambridge ; New YorkCambridge University Press0521417112 (hardback)3799509Physics/Astronomy 535.2; M312O[ HYPERLINK \l "_ENREF_2" \o "Mandel, 1995 #30115" 2,  HYPERLINK \l "_ENREF_4" \o "Wolf, 1982 #17785" 4]. This function describes the similarity in the field fluctuations of two points, for example in the two-slit Young interferometer. Two points are always fully correlated if the light is monochromatic, because, at most, the field at the two points can differ by a constant phase shift. Across an entire plane, the phase distribution is a random variable. To capture the spatial correlations in a ensemble-averaged sense, most relevant to imaging, we use the spatially averaged version of  EMBED Equation.DSMT4 , defined in Eq. 9.  Figure 2. Mach-Zender interferometry with spatially extended fields. One configuration that allows measurement of W is illustrated in Fig. 2 via an imaging Mach-Zehnder interferometer. Here the monochromatic field  EMBED Equation.DSMT4  is split in two replicas that are further re-imaged at the CCD plane via two 4f lens systems, which induce a relative spatial shift  EMBED Equation.DSMT4 . The question of practical interest is: to what extent do we observe fringes, or, more quantitatively, what is the spatially averaged fringe contrast as we vary  EMBED Equation.DSMT4 ? For each value of  EMBED Equation.DSMT4 , the CCD records a spatially resolved intensity distribution, or an interferogram. We compute the spatial average of this quantity as  EMBED Equation.DSMT4  10 Assuming that the interferometer splits the light equally on the two arms. Once the average intensity of each beam,  EMBED Equation.DSMT4 , is measured separately (e.g. by blocking one beam and measuring the other), the real part of  EMBED Equation.DSMT4 , as defined in Eq. 9, can be measured experimentally. Clearly, multiple CCD exposures are necessary corresponding to each  EMBED Equation.DSMT4 . The complex degree of spatial correlation at frequency  EMBED Equation.DSMT4  is defined as  EMBED Equation.DSMT4  11  EMBED Equation.DSMT4  is nothing more than the average optical spectrum of the field,  EMBED Equation.DSMT4  12  EMBED Equation.DSMT4 , where the extremum values of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  correspond to complete lack of spatial correlation EMBED Equation.DSMT4  and full correlation, respectively. The area over which  EMBED Equation.DSMT4  maintains a significant value defines the correlation area at frequency  EMBED Equation.DSMT4 , e.g.  EMBED Equation.DSMT4 . 13 Often, we refer to the coherence area of a certain field, without referring to a particular optical frequency. In this case, what is understood is the frequency-averaged correlation area,  EMBED Equation.DSMT4 . In practice we deal many times with fields that are characterized by a mean frequency,  EMBED Equation.DSMT4 . In this case the spatial coherence is fully described by the behavior at this particular frequency. A broad band field is fully spatially coherent if  EMBED Equation.DSMT4 , for any ( in the domain  ADDIN EN.CITE Wolf200931210[5, 6]312103121017Wolf, E.Solution of the Phase Problem in the Theory of Structure Determination of Crystals from X-Ray Diffraction ExperimentsPhys. Rev. Lett.Phys. Rev. Lett.2009Mandel198117793177931779317Mandel, L.Wolf, E.Mandel, L Univ Rochester,Dept Phys & Astron,Rochester,Ny 14627 Univ Rochester,Inst Opt,Rochester,Ny 14627Complete Coherence in the Space-Frequency DomainOptics CommunicationsOpt Commun Opt CommunOptics CommunicationsOpt Commun247-24936419810030-4018ISI:A1981LJ57200001<Go to ISI>://A1981LJ57200001English[ HYPERLINK \l "_ENREF_5" \o "Wolf, 2009 #31210" 5,  HYPERLINK \l "_ENREF_6" \o "Mandel, 1981 #17793" 6].  3.2.2. Spatial power spectrum Since  EMBED Equation.DSMT4  is a spatial correlation function, it can be expressed via a Fourier transform in terms of a spatial power spectrum,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  14 The spatial correlations of fields at two different frequencies vanish,  EMBED Equation.DSMT4 , 15 The meaning of  EMBED Equation.DSMT4  is: performing the spatial correlation measurements in Fig. 2 with each field of the interferometer now having two different optical frequencies,  EMBED Equation.DSMT4  for each spatial shift  EMBED Equation.DSMT4 , the spatial integration averages to zero the effect of the cross term. Maximum contrast fringes in a Mach-Zehnder interferometer like in Fig. 2 are obtained by having the same spectral content on both arms of the interferometer.  Figure 3. Measuring the spatial power spectrum of the field from source S via a lens (a) and Fraunhofer propagation in free space (b). The spatial correlation function  EMBED Equation.DSMT4  can also be experimentally determined from measurements of the spatial power spectrum, as shown in Fig. 3. Both the far field propagation in free space and propagation through a lens can generate the Fourier transform of the source field, as illustrated in Fig. 3,  EMBED Equation.DSMT4  16 The CCD is sensitive to power and detects the spatial power spectrum,  EMBED Equation.DSMT4 . In Eq. 16, the frequency component  EMBED Equation.DSMT4  depends either on the focal distance, for the lens transformation (Fig. 3a), or on the propagation distance z, for the Fraunhofer propagation (Fig. 3b),  EMBED Equation.DSMT4 . 17 In the Fraunhofer regime, the ratios x/f and x/z describe the diffraction angle; therefore sometimes  EMBED Equation.DSMT4  is called angular power spectrum. For extended sources that are far away from the detection plane, as in Fig. 3b, the size of the source may have a significant effect on the Fourier transform in Eq. 16. This effect becomes obvious if we replace the source field U with its spatially truncated version, U, to indicate the finite size of the source  EMBED Equation.DSMT4  18  EMBED Equation.DSMT4  is the 2D rectangular function, a square of side a. The far field becomes  EMBED Equation.DSMT4  19 * denotes convolution and sinc is sin(x)/x. The field across detection plane (x, y),  EMBED Equation.DSMT4 , is smooth over scales given by the width of the sinc function. This smoothness indicates that the field is spatially correlated over this spatial scale. Along x, this correlation distance,  EMBED Equation.DSMT4 , is obtained by writing explicitly the spatial frequency argument of the sinc function,  EMBED Equation.DSMT4  20 We can conclude that the correlation area of the field generated by the source in the far zone is of the order of  EMBED Equation.DSMT4  21  EMBED Equation.DSMT4  is the solid angle subtended by the source. This relationship allowed Michelson to measure interferometricaly the angle subtended by stars. For example, the Sun subtends an angle  EMBED Equation.DSMT4 , i.e.  EMBED Equation.DSMT4 . Thus, for the green radiation that is the mean of the visible spectrum,  EMBED Equation.DSMT4 , the coherence area at the surface of the Earth is of the order of  EMBED Equation.DSMT4 . Measuring this area over which the sun light shows correlations (or generates fringes) provides information about its angular size. For angularly smaller sources, far field spatial coherence is correspondingly higher. This is the essence of the Van Cittert-Zernike theorem, which states that the field generated by spatially incoherent sources gains coherence upon propagation. This is the result of free-space propagation acting as a spatial low-pass filter  ADDIN EN.CITE Goodman199629370[7]29370293706Goodman, Joseph W.Introduction to Fourier opticsMcGraw-Hill series in electrical and computer engineeringxviii, 441 p.2ndFourier transform optics.1996New YorkMcGraw-Hill00702425424455392Jefferson or Adams Bldg General or Area Studies Reading Rms QC355; .G65 1996http://www.loc.gov/catdir/description/mh022/95082033.htmlhttp://www.loc.gov/catdir/toc/mh022/95082033.html[ HYPERLINK \l "_ENREF_7" \o "Goodman, 1996 #29370" 7]. Zernike employed the spatial filtering concept to develop phase contrast microscopy  ADDIN EN.CITE Zernike194266[8, 9]666617Zernike, FPhase contrast, a new method for the microscopic observation of transparent objects, Part 1PhysicaPhysica686-6989
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1942Zernike194288098809880917Zernike, FPhase contrast, a new method for the microscopic observation of transparent objects, Part 2PhysicaPhysica974-9869
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1942[ HYPERLINK \l "_ENREF_8" \o "Zernike, 1942 #66" 8,  HYPERLINK \l "_ENREF_9" \o "Zernike, 1942 #8809" 9]. It had been known since Abbe that an image can be described as an interference phenomenon  ADDIN EN.CITE Abbe187368[10]686817Abbe, E.Beitrge zur Theorie des Mikroskops und der mikroskopischen WahrnehmungArch. Mikrosk. Anat. Arch. Mikrosk. Anat.43191873[ HYPERLINK \l "_ENREF_10" \o "Abbe, 1873 #68" 10]. Image formation is the result of simultaneous interference processes that take place at each point in the image.  Figure 4. Spatial filtering via a 4f system. To turn transparent specimens visible, Zernike employed spatial filtering (in a way that is similar to Fig. 4) and extended the coherence area of the illuminating field to exceed the field of view of the microscope. This type of illumination is called coherent. An average field over the entire image could be defined, because the phase relationship among different points was stable. The simultaneous interferences at all points that generate the image have a common phase reference, which is the phase associated with the mean field. Controlling the phase delay of the mean field adjusts the contrast of the entire image, like in typical interferometry experiments. Phase contrast microscopy is a major breakthrough in microscopy and an important precursor to QPI. Another major method for intrinsic contrast microscopy, Differential Interference Contrast (DIC, or Nomarski)  ADDIN EN.CITE Smith195558[11, 12]585817Smith, F. H.Microscopic interferometryResearch (London) Research (London)38581955Pluta19883204032040320406Pluta, MaksymilianAdvanced light microscopy3Microscopy Technique.1988Warszawa Amsterdam ; New YorkPWN ; Elsevier : Distribution for the USA and Canada, Elsevier Science Publishing Co.0444989390 (v. 1) 0444989188 (v. 2) 044498819X (v. 3)1005903Jefferson or Adams Building Reading Rooms QH207; .P54 1988http://www.loc.gov/catdir/enhancements/fy0601/87024605-d.html[ HYPERLINK \l "_ENREF_11" \o "Smith, 1955 #58" 11,  HYPERLINK \l "_ENREF_12" \o "Pluta, 1988 #32040" 12], which renders maps of phase gradients across a transparent sample, is categorized under incoherent methods. DIC does use incoherent illumination (no spatial filtering), yet the high contrast images are generated by interfering an image field with a replica of itself that is slightly shifted transversally. The numerical aperture of the microscope is finite, i.e. the imaging system itself performs spatial filtering. According to Eq. 21, the spatial coherence is of the order of  EMBED Equation.DSMT4 , NA is the numerical aperture of the microscope. This equation states that the image field is fully correlated across a region of the order of the diffraction spot. In DIC, shifting the two replicas of the image field by less than a diffraction spot generates high contrast fringes. 3.2.3. Spatial filtering From the properties of Fourier transforms, we infer that higher spatial coherence at frequency  EMBED Equation.DSMT4 , i.e. a broader  EMBED Equation.DSMT4 , can be obtained by narrower  EMBED Equation.DSMT4 . When dealing with extended sources, it is common practice in the laboratory to perform low pass filtering on  EMBED Equation.DSMT4 , such that the coherence area extends over the desired field of view. This procedure, commonly encountered in QPI experiments, is called spatial filtering, and is illustrated in Fig. 4. In Fig. 4, the extended, source S emits light at a multitude of frequencies  EMBED Equation.DSMT4 , and spatial frequencies k. At a given frequency  EMBED Equation.DSMT4 , lens L1 performs the spatial Fourier transform. If an aperture is placed at this Fourier plane to block the high spatial frequencies, the field reconstructed by lens L2 at plane S (conjugate to S) approximates a plane wave of wavevector  EMBED Equation.DSMT4 . With this procedure, from an extended source, we obtain a highly spatially coherent field. This procedure is lossy, as the energy carried by the high spatial frequency is lost. Asymptotically, closing down the aperture generates a field that approaches a plane wave at plane S. Conversely, all sources exhibit spatial coherence at least at the scale of the wavelength. This is easily understood by noting that a  EMBED Equation.DSMT4 -correlated source,  EMBED Equation.DSMT4  requires  EMBED Equation.DSMT4  infinitely broad, i.e.  EMBED Equation.DSMT4 . This is impossible, because a planar source can only emit in a  EMBED Equation.DSMT4  solid angle. Thus the minimum coherence area for an arbitrary source is of the order of  EMBED Equation.DSMT4  22 Spatial coherence of a field over a given plane describes how close the field is to a plane wave. Alternatively, spatial coherence describes how well the field can be focused to a point (this point corresponds to a delta-function in the frequency domain). Spatial coherence plays an important role in microscopy and is used to differentiate between two classes of methods: (spatially) coherent vs. incoherent. Quantitative phase imaging requires that the illumination field is spatially coherent, such that a phase shift can be properly defined over the entire field of view of interest. 3.3. Temporal correlations of plane waves 3.3.1. Temporal autocorrelation function We now investigate the temporal correlations of fields at a particular spatial frequency k (or a certain direction of propagation). Taking the spatial Fourier transform of ( in Eq. 2, we obtain the temporal correlation function  EMBED Equation.DSMT4  22  Figure 5. Michelson interferometry. The autocorrelation function  EMBED Equation.DSMT4  is relevant in interferometric experiments of the type illustrated in Fig. 5. In a Michelson interferometer, a plane wave from the source is split in two by the beam splitter and subsequently recombined via reflections on mirrors M1,and M2. The intensity at the detector has the form (we assume 50/50 beam splitter)  EMBED Equation.DSMT4  23 The real part of  EMBED Equation.DSMT4  is obtained by varying the time delay between the two fields. This delay can be controlled by translating one of the mirrors. The complex degree of temporal correlation at spatial frequency k is defined as  EMBED Equation.DSMT4  24  EMBED Equation.DSMT4  represents the intensity of the field, i.e.  EMBED Equation.DSMT4  25 The complex degree of temporal correlation has the similar property with its spatial counterpart  EMBED Equation.DSMT4 , i.e.  EMBED Equation.DSMT4  26 The coherence time is defined as the maximum time delay between the fields for which  EMBED Equation.DSMT4  maintains a significant value, say . If we cross-correlate temporally two plane waves of different wave vectors (directions of propagation), the result vanishes unless  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  27 At each moment t, the two plane waves generate fringes parallel to  EMBED Equation.DSMT4 . If the detector (e.g. a CCD) averages the signal over scales larger than the fringe period, the temporal correlation information is lost. As ( changes, the fringes run across the plane such that the contrast averages to 0. For this reason, for example, the two beams in a typical Michelson interferometer are carefully aligned to be parallel. 3.3.2. Optical power spectrum The temporal correlation  EMBED Equation.DSMT4  is the Fourier transform of the power spectrum,  EMBED Equation.DSMT4  28  EMBED Equation.DSMT4  can be determined via spectroscopic measurements, as exemplified in Fig. 6.  Figure 6. Spectroscopic measurement using a grating: G grating, D detector, diffraction angle. The dashed line indicates the undiffracted order (zeroth order) By using a grating (a prism, or any other dispersive element), we can disperse different colors at different angles, such that a rotating detector can measure  EMBED Equation.DSMT4 directly. To estimate the coherence time for a broad band field, let us assume a Gaussian spectrum centered at frequency  EMBED Equation.DSMT4 , and having the r.m.s. width  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  29 S0 is a constant. The autocorrelation function is also a Gaussian, modulated by a sinusoidal function, as a result of the Fourier shift theorem  EMBED Equation.DSMT4  30 If we define the width of  EMBED Equation.DSMT4  as the coherence time, we obtain  EMBED Equation.DSMT4  31 and the coherence length  EMBED Equation.DSMT4  32 The coherence length depends on the spectral bandwidth in an analog fashion to the coherence area dependence on solid angle (Eq. 21). This is not surprising as both types of correlations depend on their respective frequency bandwidth. 3.3.3. Spectral filtering The coherence length values can vary broadly, from kilometers for a narrow band laser, to microns for LEDs and white light. Figure 7 shows qualitatively the relationship between  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 .  Figure 7. a) Broad (marker) and narrow (solid line) power spectrum. b) Temporal autocorrelation functions associated with the power spectra in a. Of course, using narrow band filters has the effect of enlarging the coherence length of the field. The short coherence length of a broad band source is the starting point in low-coherence interferometry and optical coherence tomography  ADDIN EN.CITE Huang19911833[13]1833183317Huang, D.Swanson, E. A.Lin, C. P.Schuman, J. S.Stinson, W. G.Chang, W.Hee, M. R.Flotte, T.Gregory, K.Puliafito, C. A.Fujimoto, J. G.Mit,Dept Elect Engn & Comp Sci,Cambridge,Ma 02139 Mit,Electr Res Lab,Cambridge,Ma 02139 Mit,Lincoln Lab,Lexington,Ma 02173 Harvard Univ,Sch Med,Dept Ophthalmol,Boston,Ma 02115 Massachusetts Eye & Ear Hosp,Laser Res Lab,Boston,Ma 02114 Massachusetts Gen Hosp,Wellman Labs,Boston,Ma 02114Optical Coherence TomographyScienceScience ScienceScience1178-11812545035nerve-fiber layermicroscopereflectancethicknessglaucomasystemhead1991Nov 22ISI:A1991GQ83400038<Go to ISI>://A1991GQ83400038[ HYPERLINK \l "_ENREF_13" \o "Huang, 1991 #1833" 13], as discussed in Chapter 7. Phase can only be defined via correlation functions; there is no absolute origin for measuring a phase shift in time or space. To define a quantitative phase image, the phase shift itself across the image must be well defined. The illumination field must be spatially coherent over the field of view. This does not mean that the illumination has to be monochromatic, as long as, on average, each frequency component ( has a correlation area larger than the field of view. Phase contrast microscopy  ADDIN EN.CITE Zernike194266[8, 9]666617Zernike, FPhase contrast, a new method for the microscopic observation of transparent objects, Part 1PhysicaPhysica686-6989
686
1942Zernike194288098809880917Zernike, FPhase contrast, a new method for the microscopic observation of transparent objects, Part 2PhysicaPhysica974-9869
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1942[ HYPERLINK \l "_ENREF_8" \o "Zernike, 1942 #66" 8,  HYPERLINK \l "_ENREF_9" \o "Zernike, 1942 #8809" 9] is a white light method where the phase shift is meaningful across the entire field of view  ADDIN EN.CITE Born199929201[14]29201292016Born, MaxWolf, EmilPrinciples of optics : electromagnetic theory of propagation, interference and diffraction of lightxxxiii, 952 p.7th expandedOptics.1999Cambridge ; New YorkCambridge University Press0521642221 0521639212 (pbk.)2144840Reference - Science Reading Room (Adams, 5th Floor) QC355.2; .B67 1999 Jefferson or Adams Bldg General or Area Studies Reading Rms QC355.2; .B67 1999http://www.loc.gov/catdir/samples/cam041/98049429.htmlhttp://www.loc.gov/catdir/description/cam029/98049429.htmlhttp://www.loc.gov/catdir/toc/cam021/98049429.html[ HYPERLINK \l "_ENREF_14" \o "Born, 1999 #29201" 14]. QPI with white light illumination provides certain advantages over laser illumination.      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