Coherence effects from visible light to X-rays

Nuclear Instruments and Methods in Physics Research A 347 (1994) 161-169 North-Holland NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A Coherence effects from visible light to X-rays Eberhard Spiller IBM TJ. Watson Research Center, Yorktown Heights, AT 10598, USA The development of the concept of coherence of light and its application for experiments in the XUV range are reviewed. Coherence was originally defined by the visibility of interference fringes, and the main effort of early work was to produce coherency from non-coherent, chaotic or thermal sources. With the invention of the laser, fully coherent light became available, leading to further clarification of the coherence concept. The laser offered the first possibility to observe easily interference effects between different, independent light sources. By providing many photons in a single mode, non-linear effects, such as the generation of higher harmonics or sum and difference frequencies became possible. The generation of non-classical fields, i.e. fields which are only allowed by quantum theory but have no analogue m classical physics, is a more recent development. Synchrotron sources in the XUV region are now reaching the brightness which visible light had at the beginning of the sixties and the first soft X-ray lasers are available. Brightness is now sufficient to prepare spatially coherent X-ray beam with sufficient intensity for holography or the observation of coherency effects in the fluctuations of the light intensity. Non-linear effects might become observable by mixing synchrotron radiation with laser light. 1. Introduction Practically all light sources have fluctuating amplitudes and phases. The degree of coherence y l,(r, T) defines to what degree fluctuations between two points, separated in space and time by r, r are correlated. We present in this overview a very short description of the coherence functions, their definition, measurement, propagation and their influence on imaging. There are many excellent textbooks, review articles and collections of papers, which give a much more detailed treatment [1-8]. A screen illuminated with incoherent light, such as normal daylight or room light appears uniform and shows no structure. The illumination does not provide any information about the field distribution, origin, direction or phase front of the incident light. No reconstruction of the sources or objects which contribute to the illumination is possible. In contrast, a room illuminated diffusely with the coherent light from a single laser shows high contrast interference pattern (speckle) on any wall, screen or object. This interference pattern contains information about the source and all objects in all possible paths between source and screen. The scenery of the entire room can to some degree be reconstructed from the interference pattern recorded on the screen. Early studies of coherence properties of light defined the conditions to obtain interference patterns with an incoherent light source. The contrast of an interference pattern is used to define the degree of coherence between the beams, and spatial and spectral filters select a coherent sub-beam from the source. This analysis defines a coherence volume or mode, and the number of coherence volumes or modes in a beam defines the information capacity or the number of independent pixels which can be recorded in an image. Interference effects are understood as the superposition of the amplitudes of electromagnetic waves. These amplitudes show random fluctuations in space and time for thermal sources, and the coherence conditions define the volume in space and time in which these fluctuations are correlated. Interference experiments always select first a single coherence volume and then divide and recombine the light with various spatial or temporal delays. The phase differences produced by these delay elements determine the interference fringes ; all fluctuations in the original source are averaged away in the recording. Direct observation of the intensity fluctuations in a light beam and the correlations in the fluctuations between two coherent beams was difficult before the invention of the laser, because detectors were too slow to resolve the fluctuations, and there was very little intensity within a coherence volume (less than one photon). The first observation of correlations in the intensity fluctuations started a decade-long discussion, which included all the paradoxes of the dual wave-particle nature of of light. Because fractions of a photon could not be counted, it appeared difficult to understand how a single photon incident on a beam-splitter could cause correlations in the count-rate at two separate detectors. Quantum theory had produced similar problems for the explanation of the classical interference experiments. Interference fringes remained observable, even 0168-9002/94/$07.00 1994 - Elsevier Science B.V. All rights reserved SSD70168-9002(94)00188-D VI. COHERENCE

162 E. Speller/Nucl. Instr. and Meth. in Phys. Res A 347 (1994) 161-169 at such low intensities that only a maximum of one photon existed at any time in the equipment. This had led to the famous statements in Dirac's book [9] that "Each photon interferes only with itself, interference between two different photons never occurs", which seemed to exclude interference effects between independent sources. The invention of the laser provided coherent light of high brightness and demonstrated indeed that interference effects between independent sources were observable. Intensity fluctuations and their correlations could easily be observed, and a consensus on the classical and quantum mechanical description of coherence effects was developed. It also became clear that quantum theory permitted states of the electromagnetic field, which had no classical analogue (the classical harmonic oscillator of all textbooks being one example). The question if and how such states ("squeezed states") could be experimentally obtained was tackled during the last two decades. Practically all experimental work on coherence phenomena has up to now been done at wavelengths in the visible or infrared region. Classical interference experiments with X-rays have been performed ; they require higher precision and stability over longer observation time than those with visible light. The brightness of X-ray sources (number of photons per mode) has up to now been too low to observe correlation in the photon count rate and non-linear effects with X-rays. The X-ray laser and the new advanced X-ray light source will change this scenario. We hope that this tutorial will be a useful preparation of the synchrotron radiation community for future possibilities. 2. Contrast of interference fringes The superposition of two beams (complex amplitude a,, a2) with equal intensity provides the simplest measurement of coherence. The intensity (I)=( Ia,+a2 1 2 ) = ( ai ) + ( az ) +2yt2ata2 cos AO is modulated by the cos-factor, where 0(h is the phase delay (path difference) in the experiment, and the degree of coherence y,2 describes how much the interference term is reduced due to random fluctuation in the phase difference during averaging. For equal intensity of the two beams, the degree of coherence can be directly measured from the contrast in the interference fringes. _Imax -Imm yt2(r' 7) - L max +I mm Fig. 1. Interferometers which determine the degree of coherence y from the fringe visibility. The double slit experiment (a) determines the spatial degree of coherence of the source of size Ox at the points Pt and PZ from the contrast of the fringes on the axis The Michelson interferometer (b) measures temporal coherence from the fringes observed at the detector when the mirror M, is moved. BS is a beamsplitter. The two points are defined by their spatial (r) and temporal separation (7). The Young and Michelson interferometer (Fig. 1) provide a measurement. In the Young double-slit experiment the two slits define the vector r and the center of the fringe pattern corresponds to a time delay 7 = 0. The Michelson interferometer produces fringes for r = 0 for time delays -r defined by the position of the moving mirror. One finds that y,2(0, 0) = 1, that point sources have good spatial coherence, and monochromatic sources allow the observation of fringes for large time delays. The coherence conditions Ax sin Aux < A, Ay sin Au,, < À, AvAt < 1 define the maximum source size Ax, Ay, aperture angles Au x, Au y and the spectral bandwidth Av and time delay At which still produce fringes of good contrast (Fig. 2). The momentum of a photon is defined as p = h/a, therefore sin Au x = Opxlp, and the coherence conditions can also be written as AxAp x < h, AyAp y < h, AzAp, < h. b M1 DETECTOR Fig. 2. Parameters in the >patial coherence conditions M2 (3a) (3b) (3c) (4a) (4b) (4c)

E. Spiller/Nucl. Instr. and Meth. in Phys. Res. A 347 (1994) 161-169 163 >- c D w Z w Z O 0 x a 1000 100 10 An N=2 A2 AvAt. (n>=e" 'kr_ 1 TEMPERATURE ( K) 10 4 10 5 10 6 10 7 10 8 10 2 10 3 0 Zw W a 104 3. 10 5 0.1 1 10 100 1000 TEMPERATURE (ev) Fig 3. Average number of photons (n) per mode for a blackbody radiator. The hatched rectangle represents thermal light sources for the visible. For (n) = 1 the probability for stimulated emission is equal to that for spontaneous emission. Intensity fluctuations caused by interference effects can be observed with good signal/noise for (n) > 1. The coherence conditions define the volume in phase space in which a photon can be localized ; they define the size of a "point source" or the resolution limit of an ideal optical system. A single mode of the radiation field is defined by an equal sign in Eqs. (3) and (4), and the number of modes in a beam emanating from a source of area A into a solid angle.f2 with bandwidth Av, observed for a time At is given by The factor 2 in Eq. (5) represents the two possible polarizations of the field, the second factor gives the number of spatial modes or the maximum number of pixels in an image of area A produced by a beam with solid angle 0, and the third factor represents the number of temporal modes. Planck's formula gives the average number of photons (n) per mode for blackbody radiation as Fig. 3 is a plot of (n) for various blackbody temperatures and photon energies. We see that the hottest sources for visible light have only a value of ( n ) = 10-3, i.e. most of the modes or wavetrains contain not even a single photon. The number of photons per mode becomes larger than one in the infrared and is huge for the radio-frequencies. For soft X-rays (hp = 1 kev) and a blackbody temperature of 0.3 ev we have (n) = 10-145 1 Even a temperature of 10 6 C provides less than 10-1 photons per mode. The brightest synchrotron sources now under construction will have a brightness which corresponds to (n) = 10-3 for photon energies around 1 kv and (n) = 1 around 100 ev or A = 100 Â. 3. Intensity fluctuations A classical thermal source is thought to consist of a large number of oscillators (atoms) which emit wavetrains with an average duration At, randomly and independent from each other. All these wavetrains interfere with each other and produce a complicated interference pattern. The most monochromatic thermal sources (spectral lamps) have coherence times around 10-9 s. Therefore the interference pattern is visible only for 10-9 s, after that time it changes to one, which has the same statistics but differs in the position of the maxima and minima. A screen, illuminated by the thermal source, appears uniform to our eye, because we cannot resolve the fast fluctuations. A fast detector could resolve the fluctuations, and the correlations in the fluctuations at different points in space-time are defined by the coherence function y12 (r, r) in a similar way as from the visibility of interference fringes. Let us consider a screen, illuminated by a wellmonochromatized extended thermal source and observed at each point for a short time period. Each point on the screen is illuminated by many source points simultaneously, and the complex amplitudes (intensities and phases) from all source points are added at random. The superposition of all these vectors is equivalent to a random walk in two dimensions, and the probability p(a) of observing a certain amplitude a is a Gaussian around the origin : p( a) = 1 e -a 2 /(2o z ) > 23To, 2 (~) where a2 = ax + ay2, ; a., ay are the x - y components of the amplitude in the complex plane. The variance 0-2 can be expressed by the expectation value of the intensity (I) = (a2 ) : 2o, 2 = (1). (8) The probability of observing an intensity I is given by 1 I p(i) = (I)expwith a variance (10) VI. COHERENCE

164 E Seller/Nucl. Instr. and Meth. in Phys Res. A 347 (1994) 161-169 Fig. 4. Short time exposure of the far field interference pattern from an extended round (top) and rectangular (bottom) light source. The pattern changes within a coherence time and the correlation in the intensity fluctuations is the Fourier transform of the intensity distribution of the source. The average size of an interference maximum (speckle) corresponds to one coherence volume or mode. This simulation of a thermal source is obtained by illuminating a random diffuser (white paper or ground glass screen) with a He-Ne laser. The average number of photons per mode (speckle gram) has to be larger than one, to observe the pattern from a thermal source. The "frozen" interference pattern from an extended source is a speckle pattern. The most probable intensity is zero and the contrast in the intensity fluctuations is one. The smallest distance between interference maxima Ax or the average size of the elements is determined by the angular size of the source by Eqs. (3a) and (3b), while Eq. (3c) determines how fast the pattern changes in time. Fig. 4 shows two examples of such interference patterns. The thermal source is simu- lated by a ground glass screen illuminated with a He-Ne laser. The ground glass screen randomizes the phases at different positions on the screen as in a thermal source. However, in contrast to a thermal sources, the amplitudes do not change in time and the interference pattern can be seen by eye or be photographed. For Fig. 4 the source was a round, nearly Gaussian spot, while the width of the source was reduced by a slit in Fig. 4 (bottom). The coherence area corresponds to the average size of one speckle and is larger in the y-direction in the bottom figure due to the smaller source size in the y-direction. The average size of the patterns in Fig. 4 defines the distances over which the temporal fluctuations of a thermal source would be correlated. The patterns in Fig. 4 are closely related to the diffraction pattern of the sources, i.e. the diffraction pattern produced by a plane wave with the same intensity distribution. We can interpret the diffuser as a device which splits the incoming plane wave into numerous plane waves of different directions. The speckle pattern is the superposition of the diffraction patterns produced by all these waves. The average size of a speckle, or more precise, the width of the speckle autocorrelation function, is proportional to the inverse of the source size. Let us consider a scan through a pattern of Fig. 4. At an intensity maximum a large number of source points add up nearly in phase. Moving side-ways from a maximum modifies the phases by adding a changing geometrical phase difference. The angular extent of the source determines how fast the geometrical phases change. The geometrical phase differences are exactly the same as in a diffraction experiment, where the source pattern is illuminated by a plane wave. These arguments led to the Van-Cittert- Zernike theorem r(ar) = (I(r)r(r+ or)) =FT[I(x, y)], (11) where x, y are the source coordinates, r are the coordinates in the far field speckle pattern and FT represents a Fourier transform. The autocorrelation function of the intensity fluctuations in the far field is the Fourier transform of the source intensity. T(Ar) becomes the degree of coherence y, if we normalize it such that T(0) = l. For the measurement of the autocorrelation function we could move two detectors, separated by Or, over the frozen speckle pattern in Fig. 4, or we could observe the time average product of the intensity fluctuations at any position for each detector spacing, if the speckle pattern changes in time. For a thermal (ergodic) source both methods give the same result. Knowledge of F(r) allows to reconstruct the power spectrum of the source intensity. In contrast, recording of the complete (amplitude and phase) speckle pattern

E. Spiller/Nucl. Instr and Meth. in Phys. Res. A 347 (1994) 161-169 165 allows a complete reconstruction of the complex amplitude in the source. 4. Photon statistics The count rate n of a photon detector will fluctuate, first because photons arrive at random, even if there are no fluctuations in the classical intensity, and second because the classical intensity fluctuates. A constant intensity produces a Poisson distribution for the counting statistics, and for a field with intensity fluctuations the counting statistics can be obtained by a Poisson transform (Mandel's formula): P(n) =~-(aw)ne`p(w) dw, (12) n! where W(t) = fôi(t) dt is the average intensity in the time interval T and a is the quantum efficiency of the detector. We obtain for a perfectly coherent beam without any fluctuations the Poisson distribution (n )n p(n) = e_ ~n) (13) n! with a variance Q z = ((n - (n))2) = (n). (14) For a speckle pattern described by Eq. (9) we obtain the Bose-Einstein statistics (n )n p(n) _ ' ((n) + 1)n+1 (15) 0,2 = (n)2 + (n). ( 16 ) For an interference pattern which changes in space and time, (n) is the number of photons per coherence volume as defined by Eqs. (3) or (4) and given by Eq. (6) for a blackbody radiator. The first term in Eq. (16) represents the fluctuations caused by the wave nature of light, due to the superpositions of waves with random phases, while the second term is the shot effect for photon noise. We see that first term becomes dominant only for (n) > 1. For thermal sources and wavelengths of visible light and shorter, we observe only the shot effect. Observation of the fluctuations with a detector of finite spatial and temporal resolution will reduce the fluctuations. We characterize the resolution of the detector by the number of modes N which are seen within a resolution element and can describe the reduction in the fluctuations by (n +N- 1)! (n) N ( N n p(n)=- I1+ ( N- 1)!n! 11+ N` ) ( n) ) (17) 0,12 c 0- >- 0.08 J m am0 c â r m a m 0 0.04 0 10-3 10-6 10-9 10-12 0 2 4 6 8 PHOTON NUMBER n Fig. 5. Probability to observe n photons with an ideal detector which intercepts one coherence volume for a thermal source (Bose-Einstein statistics B-E) and for an ideal laser without any fluctuations (Poisson) for a source with (n) =10 (a), and distributions for a detector which averages over N coherence volumes either to time or space. N = 2 represents also unpolarized light with a high resolution detector. For (n) < 1 all curves are very similar, the differ mainly for the very rare multi-photon events (b) This equation approaches the Poisson distribution for N- oo and the Bose-Einstein distribution for N= 1. For unpolarized light and a perfect point detector we have N = 2. The plots in Fig. 5 demonstrate the transition from the Bose-Einstein to the Poisson distribution for large N. For a thermal source with (n ) «1 all distributions are very similar and differ only in the probability for the very rare events that several photons arrive in the same mode (Fig. 5b). One has to collect data for a long time to observe the excess fluctuations of the Bose-Einstein statistics. An individual speckle pattern produced by a thermal source cannot be observed, because the number of photons per mode is much smaller than one. The pattern changes so rapidly that not even a single photon can be recorded during the lifetime of the pattern for each interference peak. N : 1 2 5 20 VI. COHERENCE

166 E. Spdler /Nuel. Instr and Meth. in Phys Res. A 347 (1994) 161-169 The interferometers in Fig. 1 and all classical interferometers can record stationary fringes at path-differences around zero, because the interference pattern is not influenced by the phase fluctuations of the source. Only geometrical, unchanging path differences determine the interference pattern and photons can be accumulated for arbitrarily long time and fringes can be recorded for arbitrarily weak sources. Dirac's statement that "each photon interferes only with itself" is certainly correct for this case. When we observe a changing interference pattern with poor spatial or time resolution, as in the case of a white screen in sunlight, we average over many coherence volumes and observe no intensity fluctuations. The photon statistics in this case is described by the Poisson distribution (limit N- - in Eq. (17)). A thermal, completely chaotic incoherent source, a perfectly stabilized fully coherent source, and a weak source ((n) << 1) all produce the same counting statistics, but the reason is quite different for the three cases! 5. Intensity interferometers The Hanbury-Brown-Twiss intensity interferometer (Fig. 6) derives the degree of coherence from correlations of the intensity fluctuations at two points in space-time. Sideways translation of one detector produces a spatial separation analogue to the double slit experiment of Young, and longitudinal translation produces a time delay. Time delays can also be introduced in the electronics ; in that case only one detector is required. The output of the correlator for a time delay T gives (01,(t +T)OI2 (t)) = (I X,2) I Y12( r)1 2. ( 1S) An analogous formula describes spatial separations. The interpretation of the result poses no problem, Fig. 6. Intensity interferometer of Hanbury-Brown-Twiss to measure correlation in the intensity fluctuations of the two detectors P, and P2. COR is a correlator or coincidence counter and its output is recorded at REC. One detector can be translated sideways to measure spatial correlations. Tem poral correlations are measured by a delay between the two detector signals. when one considers only classical intensities I ; the fluctuations of the intensity will be the same at both detectors for T = 0 and produce a strong correlation signal. There were controversies in the fifties, when the experiment was interpreted as a photon counting experiment. (A single photon can only be observed in one channel or the other, but not in both simultaneously.) The correlations in the photon counting are due to the photon bunching in the Bose-Einstein statistics described by the first term in Eq. (16). This term is due to the wave nature of light caused by the fluctuations produced by interference effects. When one photon is observed in one channel, there is an enhanced probability that the classical intensity or the probability to detect a photon is higher than average, and therefore we have an enhanced probability to observe a photon in the other channel. For a small number of photons in a coherence volume the correlations are small compared to the shot-noise term (second term in Eq. (16)). The Bose- Einstein and the Poisson distribution are very similar (see Fig. 5b), and differences between the two distributions can be seen only by accumulating a large number of the rare occurences of several photons per mode. In the first experiments of Hanbury-Brown-Twiss, the average number of photons per mode was around (n) = 10-3 and a long integration of the correlator signal was required to demonstrate the correlation between the two signals. There was also some loss of signal due to the fact that the time resolution of the electronic circuits was longer than the coherence time, reducing the signal further. The brightest synchrotron radiation source can now provide intensities in the soft X-ray region which correspond to (n) = 10-3 and experiments to observe intensity correlations are being prepared [101. 6. Non-classical fields Many of the difficulties in the interpretation of the intensity interferometer were due to the fact that quantum theory allows states which do not have a corresponding classical field. The treatment of the harmonic oscillator in nearly all textbooks is such an example. Fig. 7 is a picture of the standard solution in phase space with coordinates p, q. The ground state is described by a Gaussian around the origin (black area), its area is determined by Heisenberg's uncertainty condition and its energy is the zero-point energy E = zhv. Each subsequent circle in Fig. 7 represents the states which are occupied by 1, 2, 3... photons. The area between subsequent states is constant and equals the minimum area defined by the uncertainty condition. Therefore the uncertainty condition requires that a state with a fixed energy or photon number is spread

E. Spiller/Nucl. Instr. and Meth.m Phys. Res. A 347 (1994) 161-169 167 other, are very interesting and could even have technical application. A source with a sub-poissonian (antibunched) photon statisticwould have less intrinsic noise than a classical source and allow information transfer with lower energy. Such sources have been demonstrated using various tricks such as a single-atom source, sub-poisson excitation by electrons, correlated twophoton emission using one photon to act on the other and negative feedback [8]. These states have not yet found technical application ; any interaction with such a state tends to modify it towards a coherent state. Fig. 7. Eigenstates of the harmonic oscillator in phase space. The circle at the center is the ground state, with its area determined by the uncertainty relation. Subsequent concentric circles represents the modes occupied by 1, 2, 3... photons. The area between these circles is constant and equal to that defined by the uncertainty relation. Phases can not been defined for these fixed energy states. The off-axis circle represents a coherent state, where the energy is unsharp and a phase can be defined. This state becomes a classical wave with amplitude and phase for (n ) -> -. over a 360 -ring in phase space. We cannot assign a phase to such a state and observe no interference effects between such a photon and another similar one from another source. We can perform interference experiments with such a source only by splitting and recombining light beams, i.e. by letting a "photon interfere with itself' in defining a geometrical phase difference between two arms of the interferometer, which does not depend on the absolute phase of the source. However, the field of a laser or a blackbody radiator or practically any light source is not described by the fixed photon number states of the harmonic oscillator, but by states where the photon number is not sharp. This permits the definition of an area in phase space which has phase and photon number connected by the uncertainty condition AnAO = 21r. (19) Such a state is illustrated in Fig. 7. It consists of a superposition of fixed-energy harmonic oscillator states, weighted with a Poisson distribution with An = v6n_. This state converges to a classical, coherent electromagnetic wave with well-defined amplitude and phase for the limit of large photon numbers. Coherent states give the proper description of practically all light sources. It turned out to be difficult to produce light which is not described by the coherent states. Such squeezed states, where the shape of the uncertainty region is compressed in one coordinate and elongated in an- 7. Interference between independent sources The interference pattern produced by two independent sources fluctuates in time and space, and we have to have an observation time shorter than the coherence time to see it. Furthermore, at least one photon per interference maximum has to be available. This leads to the condition (n) > 1. Quantum mechanically a phase can be defined if An > 1 (Fig. 7), the condition i1 n = (n) for coherent states permits the definition of a phase for the case that (n) > 1. The value (n) = 1 corresponds to the brightness, where the probability for induced emission is equal to that for spontaneous emission and is used for a definition of the lasing threshold in laser physics. Even for a blackbody radiator the induced emissions became dominant for long wavelength, and interference between different sources can be observed. A simple demonstration experiment can be performed by feeding the thermal noise (blackbody radiation) from two resistors in the audio-frequency range through a narrow bandwidth amplifier (bandwidth = 0.1 Hz) to an oscilloscope which displays the sum of the two signals in addition to each signal. The sum-trace will show interference effects between two signals which are stable for about 10 seconds. It is expected that such interference effects would not be observable for a non-classical source which is represented by a fixed photon number state. 8. Imaging The propagation of light in space and time preserves the information content, i.e. the number of modes defined by Eq. (5). The number of modes is equal to the number of resolvable pixels in one plane. If the object is illuminated coherently (y = 1) the imaging system is linear in amplitudes. Every plane in the radiation field contains the same information and the field in any plane can be reconstructed from the information in any other plane. Vl. COHERENCE

168 E. Spiller/Nucl. Instr and Meth. to Phys. Res. A 347 (1994) 161-169 For an incoherent system (y = 0 in Eq. (1)) all interference terms are averaged away during the observation time and the system is linear in intensities. The time-averaged intensity in each plane in the far-field is constant and does not contain any information about the structure of the object. A detector with high spatial and temporal resolution might be able to detect fluctuations of the intensity ((n) > 1 is required). In that case the correlation function of the intensity fluctuations contains information about the geometry of the source. The autocorrelation function in the far field is the Fourier transform of the intensity distribution of the source (Van-Cittert-Zernike theorem). A perfect lens can transfer the intensity distribution from an object plane to an image plane without loss of information. For an incoherent system, information about other planes is washed out, in an image of a 3-dimensional scene, the in-focus plane appears sharp on a background of unsharp intensity distributions from other planes. A 3-D scene can be recorded by taking multiple images by focussing on different planes. In a coherently illuminated object each plane carries the full information of the 3-D radiation field, and each plane can be reconstructed from the information in any other plane (holography). However, fidelity of the reconstruction of each plane is not equivalent to that obtained in an incoherent system. The interference structures produced by details in any plane of the object appear as a high contrast pattern in each other plane ; each plane is reconstructed on the background of the speckle pattern caused by all other planes. Most parts of a complex object are not even illuminated, because the intensity zero has the highest probability in Eq. (9). Multiple recordings with different illuminations are required to reconstruct a 3-D scene. An efficient algorithm which can reconstruct a general 3-D scene with M image planes from M hologram recordings has up to now not been found. 9. Experiments in the X-ray region Coherent addition of amplitudes scattered from the lattice planes of crystals occurs in every diffraction experiment, and X-ray diffraction has always been used to increase the coherence of an X-ray beam in crystal collimators and monochromators. Multilayer mirrors and zone plates are more recent devices which use the coherent addition of amplitudes. These devices can concentrate soft X-rays into a diffraction-limited size focal spot, if they are illuminated by the single mode of a spatially coherent source. A scanning microscope [11,12] uses this focal spot to produce images which correspond to the case of completely incoherent illumination. All interference effects between different parts of the object are eliminated by recording the different pixels at different times. Present synchrotron sources provide between 10' and 10 11 photon/s in a spatial coherence volume and permit the recording of diffraction patterns from stationary or slowly varying objects. Examples are holography, holography or interferometry with time shift, and the observation of speckle patterns and their changes due to phase transition in a specimen [13,14]. Experiments which require (n) > 1 are more difficult ; present synchrotron radiation sources provide only a brightness around (n) = 10-3. An X-ray intensity interferometer is under construction and is expected to measure the coherence function with an integration time of 1000 s [10]. The main advantage of an intensity interferometer over an amplitude interferometer is the relaxed requirement on the stability. An intensity interferometer requires only stability in the path difference of the order of the coherence time, while stability in the wavelength region is required for the amplitude interferometer. Experiments in non-linear optics with synchrotron radiation alone are probably still in the far future. Mixing of synchrotron light with other sources like X-ray or visible light lasers appears much more promising. 10. Summary All light sources show fluctuations in amplitude and phase. Coherence conditions define the volume in phase space over which these fluctuations are correlated. The definition of a coherence volume is equivalent to the volume of a single mode ; the information content of a light beam and the number of resolvable pixels are represented by the number of modes. The propagation of the correlation function in space-time is described by the wave equation and the normalized coherence of a source is very closely related to its diffraction pattern. The autocorrelation function of the intensity fluctuation of a source in the far field is the Fourier transform of the intensity of the source. Interference effects between independent sources require a minimum source brightness of 1 photon/mode. Above this brightness level stimulated emission becomes dominant and fluctuations due to the wave nature of light can be easily detected. References.. [1] L. Mandel and E Wolf, eds., Coherence and Fluctuations in Light (Dover, 1970), reprinted in SPIE Milestone Series, Vol. MS. 19 (1990). [2] J.C. Dainty (ed), Laser Speckle and Related Phenomena, Top. Appl. Phys. Vol. 9, 2nd ed. (Springer, 1984). [3] J.W. Goodman, Statistical Optics (Wiley, 1985). [4] A.S. Marathay, Elements of Optical Coherence Theory (Wiley, 1982)

E. Spiller /Nucl. Instr. and Meth. in Phys. Res. A 347 (1994) 161-169 169 [5] R. Loudon, The Quantum Theory of Light (Clarendon, [11] C. Jacobsen, S. Williams, E. Anderson, M.T. Browne, 1973). C.J. Buckley, D. Kern, J. Kirz, M. Rivers and X. Zhang, [6] J. Perina, Coherence of Light (Van Norstrand Reinhold, Opt. Commun. 86 (1991) 351. 1971). [12] C. Jacobsen, J. Kirz and S. Williams, Ultramicroscopy 47 [7] M. Bertolotti, Masers and Lasers, an Historical Ap- (1992) 55. proach (Hilger, 1983). [13] C. Jacobsen, M. Howells, J. Kirz and S. Rothman, JOS A [8] M.C. Teich and B.E.A. Saleh, Photon Bunching and 7 (1990) 1847. Antibunching, in : Progr. in Optics, Vol. 26, ed. E. Wolf [14] M. Sutton, S.G.J. Mochrie, T. Greytak, S.E. Nagler, L.E. (North Holland, 1988). Berman, G.A. Held and G.B. Stephenson, Nature 352 [9] P.A.M. Dirac, The Principles of Quantum Mechanics, (1991) 608. 2nd ed. (Clarendon, Oxford, 1958) section 1.3. [10] E. Gluskin, I.M. McNulty, P.J. Vicaro and M.R. Howells, Nucl. Instr. and Meth. A 319 (1992) 213. VI. COHERENCE