A Hanbury Brown and Twiss renascence: measurement of photon correlations yields spatio-temporal coherence

Investigations of photon correlations similar to the Hanbury Brown and Twiss experiment are since the advent of quantum optics in the center of both fundamental and application-oriented research. We derive a factorial moment-generating function of the probability distribution of electromagnetic radiation emitted by a source and absorbed by a photodetector from which we calculate an analytical expression for the second-order correlation coefficient g(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g^{(2)}$$\end{document} which reflects spatial coherence properties of the emitted light. The measurement of g(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g^{(2)}$$\end{document} thus allows to retrieve the spatial coherence parameter of the light source in terms of the combination of source area, source-detector distance, wavelength and detector area. The validity of the concept is proven by investigating g(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g^{(2)}$$\end{document} of true thermal light from a Xenon arc lamp and from the sun for various detector areas, in excellent agreement with the theory. Finally, we suggest a novel scheme for determining light source diameters by exploiting the spatially dependent statistics and confirm its validity by exemplary calculations. These results based on radiation thermodynamics and photon statistics give fresh insight into quantum optical properties of classical light sources, photon correlations and photodetection with promising applications perspectives, more than 68 years after the original Hanbury Brown and Twiss experiment.


Introduction
The existing theory of photodetection goes back to Glauber [1], where he formulated in this pioneering work the quantum theory of photodetection and optical coherence.This theory is central to quantum optics and has played a key role in understanding light-matter interactions [2,3].This work has been followed by Mandel's semiclassical counting formula [4] relating the photo-clicks of the detector via a so-called Poisson Transform to the probability distribution of the impinging absorbed field or the intensity statistics [5].However, even before this advent of quantum optics the Hanbury Brown and Twiss (HBT) experiment [6], now 67 years ago, set the foundation of nowadays understanding of light statistics in terms of correlations [7,8].The fundamental observation of photon bunching of a thermal light source, as also acknowledged in the Nobel award lecture of Glauber [9] led to the classification of light in terms of its second-order or intensity correlation coefficient.HBT had been astronomers from their profession driven by the search for improving stellar interferometry for determining the star diameter [10].After the lab experiment with the observation of photon bunching of a thermal light source they returned back to the stellar interferometer in applying this newly proven and realized intensity interferometer to this original task [11].The success story was huge and finds even today a perpetual continuation with new detector approaches and signal processing technologies investigating photon bunching [12,13] and resulting already in a sort of HBT revival in astronomy [14][15][16][17] and with new aspects of light correlations [18,19].The original HBT intensity interferometer always exploits two intensity-detectors with the baseline separation L in 162 Page 2 of 14 between them for determining the second order intensity cross correlation G (2) (L) = ⟨I 1 (0)I 2 (L)⟩ between the two intensities I 1 and I 2 , respectively.From the characteris- tic decay of G (2) as a function of the baseline separation L finally the star diameter is derived.Our approach is a based on the autocorrelation determination G (2) (A) = ⟨I(A)I(A)⟩ of the intensity I of a single detector as a function of the detector aperture A. Already in 1966, Martienssen and Spiller showed that the photon counting statistics of a so-called pseudo-thermal (PT) light source (laser light scattered at a rotating diffuser ( [20,21]) measured with a single photomultiplier shows a characteristic transition from Bose-Einstein statistics to Poisson statistics if either the separation between detector and source is decreased or, equivalently the diameter of the source is increased [22], thus revealing spatial coherence effects [23].It has been also recognized that the measured statistics is effectively a Negative Binomial or Polya distribution [5] depending on the mode number N, being the effective mode number at the emitter surface or being N = 1 at infinity separation between source and detector, respectively.Bures et.al. [24,25] performed comprehensive studies of the photon electron counting statistics of various types of this PT source, already using one single photon multiplier and analyzing the photon count statistics in terms of an ansatz for the moment-generating function and considering the degree of coherence of an extended light source in the framework of the Van-Zittert Zernike theorem [26,27].This g (2) approach has also been applied to investi- gations of scattered light [28].
At the beginning of quantum optics [29] measuring photon statistics has been a useful approach for understanding photon statistics of various light sources [23,[30][31][32], but even nowadays, g (2) measurement via HBT based correla- tion investigations or equivalently of the characteristic statistics of light is THE tool for the understanding of electromagnetic radiation [33] and the search and understanding of light emitters [18,[34][35][36], and even applicable to fermions [37] or photons in biological systems or photosynthesis [38][39][40].Even super-bunched light with a g (2) value exceeding two had been achieved by doubly scattering of laser light [41][42][43].Another very recent approach to generate new statistics focused on nonlinear optical processes [44,45] or superbunching in twin photon generation [46][47][48].These developments and the realization of novel photon source schemes have been accompanied by a comparable advancement in the detection and in the analysis of photon events [49][50][51].This development demonstrates that today it is still a challenging field of research with impressive progress giving complete insight into the photon properties by novel detector signal analysis strategies and reconstruction techniques [52][53][54][55].This research may incorporate inherent novel light sources or strategies to modify and tailor the light statistics according to the needs of applications, as e.g. in ghost modalities [56][57][58] as ghost imaging [59][60][61] or ghost spectroscopy [62] where tailored photon bunching directly determines the metrology advantages [63].
The aim of the present publication is to investigate the statistical properties of photo clicks or absorption events originating from the photon statistics.The method is traditional stochastic theory based on probability distributions and a factorial moment generating function finally being condensed in the second order intensity correlation coefficient.We start with a characteristic factorial moment generating functions consisting of a composite of a binomial function involving the photo-detection and a generalized Poisson distribution originating from the light source properties.We derive an analytical expression for the central second order correlation coefficient g (2) in terms of spatial and temporal coherence Z s and Z t , respectively [27].Its dependencies are comprehensively discussed in terms of spatial coherence and thermodynamics of the source.Finally, we conceive and realize an experimental set-up which investigates g (2) of four different true thermal light source -detector configurations, a Xe arc lamp with three different detector dimensions and sun light.The good agreement between theory and experimental results verifies that this novel modified single-detector HBT concept allows to determine e.g. the source area if source distance is known, and this with a single-detector concept without any baseline separation, thus giving rise to a renascence of HBT.

Schematical emission and detection scheme
The schematic of the emission and detection configuration for the basis of the calculations and the experiments is depicted in Fig. 1.Radiation with an emission wavelength is emitted by a light source with an emission area of A source and is detected by an absorption process in a photodetector at a distance R and with detector area A detector resulting in a number n of click events during a time interval Δt with a normalized probability distribution P(n).We define the coherence parameter Z, which factorizes into the temporal coherence Z t and the spatial coherence Z s : Z s may or may not again factorize into two parameters Z x and Z y in an obvious way, depending on the geometric shape of the light source and detector.

The factorial moment generating function
To calculate photon correlations we make use of the factorial moment-generating function (fmgf) G(s), which is defined as and together with the normalization of the probability distribution ∑ ∞ n=0 P(n) = 1 leads to G(s = 1) = 1.We start with a factorial moment-generating function G(s) of the probability distribution of radiation absorbed by a photodetector yielding a distribution of clicks events as a consequence of the absorption events which has the type of a generalized binomial distribution.The derivation of this factorial moment-generating function G(s) is performed in appendix C and D and we obtain with the average number of photons in a unit coherence volume n Ph , the quantum efficiency of the detector , and an a-priori probability p Z which will be discussed below and in the appendix D.

Moment calculations and second order correlations coefficient for the fmgf
From the fmgf G(s) (Eq.2) we can directly obtain the factorial moments ⟨n(n − 1)(n − 2)...(n − 1 + k)⟩ according to [64] The second order correlation coefficient g (2) (We remind that throughout the manuscript g (2) means the central sec- ond order correlation coefficient g (2) ( = 0) for delay time = 0 [64]) is defined as the ratio between the second order factorial moment ⟨n(n − 1)⟩ and the square of the first order factorial moment ⟨n⟩ resulting finally in [64] (1) For our fmgf in Eq. ( 3) we get: This looks relatively simple, but calculating p Z is quite com- plicated and will therefore be performed in the appendix D.
Here it shall be sufficient to note that p Z → 1 for Z >> 1 , so in this case g (2) ≈ 1 and p Z ≈ Z for Z << 1 and g (2) ≈ 2 .For un-modulated thermal light the photon correlation is thus restricted to values between 1 and 2.
We also note here that the result above is independent of the temperature of the emitter.Still, for real experiments high temperatures are strongly favored, as one then has more photons per single measurement on average and a small enough statistical error is reached faster.
3 The key result: the second order correlation coefficient g (2)   Motivated by a realistic star-like experimental configuration of a circular source and a rectangular detector geometry, we concentrate in the following on the g (2) function obtained in the appendix E for this geometry (Eq.E.3 and Eq. 8).
In addition and as also discussed in the appendix F, we have to consider partial coherence effects due to dark counts and the extended filter transmission curves of the Fabry-Perot interferometers by a factor 2 in front of the four-fold Bessel integral which leads finally to the key result: the second order correlation coefficient g (2) for a circular source -rectangular detector geometry with integration variables r 1 and r 2 denoting points within the detector area S: With x i and y i being the components of r i , the argument of the Bessel function amounts to , with the source radius d/2 and the source-detector-distance R. The Papoulis simplification (see appendix E) reduces this to a double integral for a single vector (x, y) over the detector width and height dimensions Δx and Δy , and we finally obtain the key expression for g (2) for a circular source -rectangular detec- tor geometry: (5) g (2) = 4 Discussions of g (2)   In the framework of the calculations we always consider "good" temporal coherence Z t << 1 (see also section experi- mental set-up).According to Eq. D.1 and Eq. 8 the second order correlation coefficient g (2) depends on the spatial coherence parameter Z s for good temporal coherence Z t .g (2)  thus depends on the source geometry and its dimensions, the detector geometry and its dimensions, and the distance between source and detector R. In the following, we concentrate on modelling these dependencies for a circular source geometry and a rectangular detector geometry and with parameters which will be also realized in the experimental section.
In order to compare and understand the g (2) results, we numerically integrate Eq. 8 for discrete parameter values and visualize the modelling results in terms of 3D-Plots and cuts through them.First, we depict graphically in Figs. 2 and 3 the results in terms of a corresponding 3D-Plot of g (2)  as function of source diameter d and detector dimensions Δx and Δy (being here chosen equal for simplicity) for two source-detector distances R of 180 m and 1000 m .We then discuss the calculated results for g (2) in terms of cuts through these 3-D plots as shown in Figs. 2 and 3 in dependence on the characteristic parameters of g (2) and we explore what we can learn from the statistics regarding the properties of the emitting light source in terms of a real-world ( 8) application.Finally, at the end after the discussions of the modelling results, we show experimental results for four true thermal light source configurations and compare and discuss them with the developed model and derive a novel scheme in the spirit of the Hanbury Brown and Twiss experiment.
The general overall tendency of the modelling results as both found in the 3-D plots (Figs. 2, 3) and even more clearly visible in the cuts thereof, Figs. 4, 5, 6, and 7 can be best summarized as: • The g (2) results as a function of the abscissa exhibit always the same kind of shape (in the 3-D plots) having the highest value of approximately 1 + 2 at an abscissa value of zero.• With increasing abscissa values (d, Δx or Δy ) g (2) is monotonously and asymptotically decreasing to a g (2)  value of 1.0.• With increasing d there is a decrease of g (2) , and this decrease occurs faster for larger Δx = Δy values.Fig. 4 Second order correlation coefficient g (2) as a function of rectangular detector width Δx for five rectangular detector heights Δy for a circular source diameter d = 85 m and a source-detector distance R = 180 mm • For larger detector source distances R this overall decrease of g (2) both in the d and Δx = Δy direction occurs slower.
• This behaviour is also reflected by the cuts through the 3D plots as depicted in Figs. 4, 5, 6, and 7. • The comparison between Fig. 4 and Fig. 6 shows that for larger source diameters d also the decay of g (2) with increasing Δx is faster.• The maximum value of g (2) both for nearly zero Δx and d values decrease with increasing Δy.• This decrease is slower for larger source -detector distance R and is faster for larger source diameter d. • If we realize a quadratic detector ( Δx = Δy ) and investi- gate the g (2) behaviour as a function of the source diam- eter d the general behaviour is reproduced but becomes more and more steep with increasing Δx = Δy.
All these results, however most importantly the uniqueness of g (2) as a function of the spatial coherence parameter Z s = , which is determined by the source properties and by the experimental setup allow now reversely to determine the spatial coherence parameter from a measurement of g (2) .This opens a completely novel possibility for a method for the determination of an unknown parameter of the spatial coherence Z s (i.e.e.g.source area or source-detec- tor distance) when the other three parameters are known by measuring g (2) .The essence of this here derived new HBT measurement scheme based on the fmgf and the g (2) function will subsequently be considered in the second part of the discussions after the depiction of the experimental set-up and the discussion of the experimental results.

Experimental set-up
Therefore, motivated by the modelling results for g (2) in dependence of the spatial coherence as depicted in Figs. 2, 3, 4, 5, 6, and 7, we have conceived and realized an experimental set-up (depicted in Fig. 8) to investigate this relation experimentally [65,66].
As light source we have chosen two different thermal black-body sources, a Xe arc lamp and sun light.The light  8 Experimental set-up for the investigations of g (2) of the sun and the Xe arc lamp as a function of their spatial coherence; L lenses, IF interference filter, LP Linear Polarizer, E1, E2, and E3 Etalons, PH pin hole determining the effective source area A source and at a distance R from it a second aperture A determining the effective detector area A detector (c.f.Fig. 1), g (2) photon counting system for the determination of g (2) by a coincidence set-up 162 Page 6 of 14 emitted by both sources is collected and in the collimated beam we perform a highly selective spectral filtering in order to meet the requirement to have a sufficiently good temporal coherence corresponding to a small temporal coherence parameter Z t = Δ ⋅ Δt .A similar type of exper- imental configuration has been used by Kurtsiefer's group to comprehensively study temporal photon bunching of various terrestrial and astronomical sources [67,68].We used three etalons E 1 , E 2 and E 3 in combination with an interference filter (IF) with free spectral range and spectral resolution data as depicted in Table 1.With our multiple Fabry-Perot filter or etalon configuration at a wavelength of 775 nm we realized a Δ = 150 MHz which together with the time resolution of the photon counting set-up of 1.05 ns results in a temporal coherence Z t = Δt ⋅ Δ = 0.26 [65].
The optical beam emitted by the different sources is then focused onto a pinhole (PH, diameter = 85 m) which serves as the effective light source with an area A source = 5.67 ⋅ 10 −9 m 2 .The beam then incoherently illuminates an aperture (A) with variable width Δx and fixed height Δy which defines the effective detector area A detector = Δx ⋅ Δy .The distance R between these two apertures has been kept constant.In the case of the Xe arc lamp we used three different apertures with fixed height of Δy = 0.6 mm, 1.4 mm, and 2.5 mm yielding the data Xe06, Xe14, and Xe25, respectively.For the sun light investigations Δy amounts to 0.7 mm and R has been cho- sen to 0.195 m (data sun).Here, the light had been guided via a sun-tracker to the set-up.
The second order correlation coefficient g (2) could have been measured with a single photon-counting photomultiplier (PMT) with subsequently following counting electronics where mean and variance are derived from the photon counting probability distribution [69].We have chosen a coincidence method instead by using a beamsplitter and two PMTs, thus reducing dead-time effects [70].Typical count rates for the sun as light source to be investigated have been from 30.500 cps to approx.80.000 cps, depending on the realized spatial coherence yielding in measurement times from 20 min up to 40 min and for the Xe lamp typical count rates have been from 40.000 cps to approx.90.000 cps with measurement times between 40 min up to 14 h, respectively.The geometrical parameters of the experimental set-up represent all ingredients of the spatial coherence parameter Z s = , entering the subsequently following modelling of the experimental g (2) values which both are jointly plotted in Figs. 9 and 10.
Figures 9 and 10 show the experimental results for g (2) as a function of the detector width Δx for fixed detector height Δy for the four true thermal light source configurations, 3 Xenon arc lamp results (Fig. 9) and one for the sun light (Fig. 10), Fig. 9 Experimental results for g (2) measured as a function of the rectangular detector width x for three detector heights y = 0.6 mm (Datax06), 1.4 mm (Dataxe14), and 2.5 mm (Dataxe25) with a xenon arc lamp source together with modelling results according to Eq. 8 for g (2) considering a source diameter d = 85 m and a source detector distance R = 180 mm (c.f.Fig. 8).Temporal coherence with Z t << 1 has always been realized Fig. 10 Experimental results for g (2) measured as a function of the rectangular detector width Δx for a detector height Δy = 0.7 mm with the sun as source together with modelling results according to Eq. 8 for g (2) with the corresponding parameters.Temporal coherence with Z t << 1 has been realized Page 7 of 14 162 respectively together with the model values already shown in Fig. 4. The experimental results clearly exhibit the S-shaped curve discussed above.Besides this qualitative agreement, we find excellent quantitative agreement with the modelling results if the real-world parameters for R, d, Δx and Δy are inserted.The coherence reduction factor as discussed in subsection F of the appendix has been calculated to be = 0.63 resulting in 2 = 0.397.We find that with increasing Δx values the second order correlation coefficient g (2) is decreas- ing from approximately 1.4 towards an asymptotic value of 1.0 for Δx values beyond 12 mm.
We remind here as outlined before that the experimentally observed maximum value g (2) (0) is far from the theoretically expected value of g (2) (0) of 2.0 of ideal thermal sources and with ideal detection.We have to consider that in the experiment we do not only have the "coherent" light, but also a large number of spectrally distributed low intensity side lobes in the arrangement of the etalons as discussed in the appendix F. The light in these spectral side lobes sums up to quite a substantial incoherent contribution to the measurement, which effectively reduces g (2) .With this modification as considered by the factor , our the experimental data are in very good agreement with the theoretical g (2) values.

Renascence of Hanbury Brown and Twiss experiment towards star diameter determination
Finally, this first single-detector g (2) measurement of a true thermal light source as a function of the spatial coherence parameter together with the analytical expression for g (2)  derived from the fmgf suggests a novel measurement technique for determining one of the ingredient parameters of Z S , specifically the source area A source , because usually the detector area A detector , the wavelength and the distance R between source and detector are known.An hypothetical example for such a measurement procedure is illustrated by Fig. 11 and Table 2.
• Let us assume that we have an experimental configuration as listed in Table 2 with a wavelength = 775 nm, a detector source distance of R = 180 mm, and a detec- tor height of Δy = 1.0 mm and a source diameter d = 0.63 mm, where the source diameter could not be measured directly in the experiment.Within our elaborated approach, we therefore perform a full hypothetical series of measurements of g (2) for 3 detector widths Δx = 0.5 mm, 1.5 mm, and 10.0 mm with the results as depicted in Table 2.
• These experimental results for g (2) for the three detec- tor widths Δx are then plotted in the framework of the modelling results for g (2) as horizontal line (with error bands) in Fig. 11 and the intersection point with the corresponding modelling curve is determined (including the error bars) for each detector width.• The abscissa values of the intersection points yield the source diameter d.For the hypothetical sketched experiment, we obtain a value of d = 0.063 +0.07 −0.07 demonstrating that in fact the source diameter can be extracted by this measurement idea and configuration.Thus, this configuration idea based on the application of Eq. 8 represents a suggestion for a novel method for determining spatial coherence parameters, as e.g. the source dimensions in the spirit of a renascence of the HBT experiment.However, the limitations regarding the resolution of our scheme are equivalently challenging as the original HBT [71].Moreover, our scheme represents a different approach and probably even today, and particularly stimulated by new activities in "quantum metrology" new metrology aspects,  Fig. 11 Example experiment for the determination of an unknown source diameter: The experimental data of three g (2) measurements (including error bars) of an experimental situation as listed in Table 2 for three detector widths Δx are plotted in connection with modelling results according to (Eq. 8).The intersection between the g (2) values and the corresponding modelling results yields the source diameter d with the corresponding error bars 162 Page 8 of 14 either for very large or for very small thermal light emitter scales can be envisioned, i.e. "determining the size of a thermal emitter without measuring it directly" only by measuring the photon statistics of its emitted light alone with a single detector and analyzing the photon statistics.

Summary
We calculated the second order correlation coefficient g (2)  of light emitted by a real-world thermal light source and absorbed by a photodetector resulting in click events using a factorial moment-generating function for the probability distribution.For good temporal coherence we find that g (2)  reflects predominantly the spatial coherence function of the source, thus yielding an experimental technique to determine spatial coherence properties from single-detector g (2)  measurements.The predicted dependencies have then been experimentally studied and discussed for three configurations of a thermal Xe arc light source and for the sun light as true thermal light sources.The good agreement between the experimental results for the four light source configurations and the calculated g (2) based on the factorial moment-gener- ating function opens real-world perspectives for a novel type of HBT experiment allowing, e.g. to determine a light source diameter with a single-detector scheme.This progress gives fresh insight into the Hanbury Brown and Twiss experiment thus enabling a renascence even after 67 years.

Appendix
In these appendices we derive the second order correlation function g (2) based on a combination of a binomial detection process and a generalized Poisson statistics of the impinging radiation based on a Planck-Einstein approach for the photon partial systems and a classical probability approach for the a-priori probability, reflecting spatial and temporal coherence.Starting with a Poisson distribution, we develop the probability distribution of the detected photons in terms of a generalized Negative Binomial Distribution depending on the detected modes, their number and the a-priori detection probability.The a-priori-probability is then calculated for two selected geometries resulting in an expression for g (2) as function of detector area, source area, wavelength and distance between source and detector yielding finally an expression applicable on the experimental data.Finally, the influence of temporal incoherent light contributions will be considered.

A Partial sub-systems
The foundation of the thermodynamics of electromagnetic radiation is the quantized harmonic oscillator going back in its origins to Planck [72][73][74] where its energy E can be written as where = ℏ k B T with k B being the Boltzmann constant.We note that this has the form a geometric series, namely where r=1, 2, 3,.. take here the role of quantum numbers.By Eq. (A.2) the system of the electromagnetic radiation is factorized into denumerably infinite many sub systems which are statistically independent.This allows computing the observed photon distribution from that of the individual r-subsystems [66], a big advantage in the statistical treatment of radiation.The factorial moment generating function of the observed photon distribution can be written as the product of the fmgfs of the single r-systems.The partial systems can be each regarded as an ideal gas as has been already done by Einstein [75].

B Statistics basics: Poisson distribution
One might expect that the photons in thermal radiation are independent from each other.In this case the repeated photon counting experiment as described in the main part of the manuscript would be described by a Poisson statistics with the mean value M = ⟨m⟩ and the reduced autocorrela- tion g (2) would always be 1: Our measurements (and many others, see, e.g.[15,16,67,68]) have shown, that this is not true, thermal radiation is more interesting than that and g (2)  thermal light is exceeding one. (A.1)

C Towards Negative Binomial distribution
Instead, thermal light can be described using the independent sub-systems numbered here with an integer r denoting the number of photons within an "elementary bundle" or "radiation unit" of the radiation [76].This elementary bundle replaces the all-too-simple concept of a light ray, taking the wave properties of the radiation into account.It is defined by the emitter properties, that is, its area and shape and its wavelength spectrum.According to the uncertainty principle, the direction of the energy transport is not defined precisely, but will instead be angularly spread out in inverse relation to the source dimension, roughly within ΔΩ diffraction .Similarly, the timing of absorption events is again spread out in inverse relation to the width of the spectrum, roughly within ΔT spectrum .Both these effects are quantitatively described by the Fourier Transform of the respective source properties, which we shall exploit below performing the actual calculations for specifically selected special cases.
Here, we define a spatio-temporal "measurement volume" V measure = A detector ⋅ c ⋅ Δt consisting out of the prod- uct of the detector area A detector and the propagation length c ⋅ Δt within the measurement time.We also define a "coherence volume": Dividing the measurement volume V measure by this coherence volume V coh yields the number of elementary bundles Comparing this result to our definition of the coherence parameter Z according to Eq. ( 1) we find that the coherence parameter Z is just the number of elementary bundles N elementary bundles within our measurement.
We note that the measurement volume can have arbitrarily small or large values for both Δt and/or A Detector which together with R 2 defines a solid angle as seen from the emitter ΔΩ = A Detector ∕R 2 .Values Z < 1 just mean, that (C.1) the measurement is well inside the coherence length of the radiation, at least for one dimension.We also remark, that this number of elementary bundles is also called number of phase cells of phase space, degrees of freedom of light, number of propagating modes or Jean's number [77,78].
These elementary bundles or radiation units for propagating radiation are in some aspects comparable to the "oscillators" in the description of radiation in a cavity.Both can be "excited" to contain an integer number of energy packets, the photons, and the excited elementary bundle can be considered as non-interacting radiation units.They therefore obey a Poisson distribution.
For the probability distribution of the total radiation, the number N r of the radiation units in a single r-system is proportional to 1/r, and they follow a Poisson distributions with a unit number N r = Z where in the last step the photon number per coherence volume has been inserted according to Eq. A.1: Finally, we observe that our detector -since it is potentially smaller than the "size" of the radiation units (calculation 162 Page 10 of 14 below) -will only "see" an a-priori-fraction p Z with its quantum efficiency of and so it will only register this fraction of photons and we obtain for the by clicks registered photons For the mean value of the clicks we correctly obtain: From the expression of the factorial moments, derived in the general case from G(s) or in our case from the photodetected clicks G clicks (s) (Eq.C.8) the probability distribution w(k) is obtained according to Ref. [5,27]: which is the well-known Negative Binomial or Polya Distribution [27] with the second order correlation coefficient

D Calculation of p Z
To calculate the effective a-priori-probability p Z we again observe that the radiation units or elementary bundles are extended in space, shaped by the properties of the light source due to the uncertainty principle.They may travel in any direction: with their center of gravity right towards the center of the detector, off-center, towards the detector edge, or even outside the detector.We use a coordinate pair (u, v) within the detector plane to designate this direction from the light source, meaning the center of gravity of a radiation unit while hitting the detector would be at (u, v).
The photons of this radiation unit may then found scattered around this spot.A specific photon may at position (u � , v � ) with the normalized probability density q(u � − u, v � − v) .This is the absolute square of the Fourier transform of the light source shape S(x, y), normalized to ∫ ΔZ q(u � − u, v � − v)du � dv � = 1 .This can be understood in some sense as reminescence of the Van Zittert-Zernike theorem [27,[79][80][81].In case of a rectangular light source with A source = x source ⋅ y source = 2h ⋅ 2w the Fourier transform separates into x and y components, and each are sinc-functions.In case of a circular light source the shape of q is the well-known Airy-disk.In the general case, the 2-dimensional Fourier transform has to be calculated numerically.
The a-priori-probability p Z can now be calculated by inte- grating the probability density of photons that fall into the detector sensitive area Z: The factor 1 Z ensures that for Z x >> 1 and Z y >> 1 (very large detector) we get the normalization p Z ≈ 1 .It follows that in the case Z x << 1 and Z y << 1 we get p Z ≈ Z .By inserting Eq.D.1 into C.11 we can calculate the desired g (2) from the parameters of the experimental setup.
E. Calculation of g (2) for selected geometries where can be recognized as the module of the normalized degree of coherence (for the case D >>  ) between two points P 1 and P 2 in a plane (observation plane of r 1 and r 2 ) located at a distance R of an incoherent monochro- matic rectangular source with the dimensions height 2h and width 2w and with wavelength and uniform brilliance [26].2. For a circular source (diameter d or radius ) of uniform brilliance and a rectangular detector with surface S at a distance R we obtain (D.1) with r 12 = √ (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2 , the distance R between source and detector, and wavelength .Also here, the function under the integral can be recognized as the module of the normalized degree of coherence (for the case R >>  ) between two points P 1 and P 2 in the observation plane.We define which is the basis of the calculations and discussions in the main part.

F The effect of incoherent light contributions
In principle the temporal coherence Z t could be discussed the same way as the spatial coherence above.However, in our measurement setup (see section 5 Experimental set-up) the spectral distribution of the light is quite complex as it is realized by a combination of an interference filter and three different Etalons.We carefully matched the central wavelengths of the Etalons to the same values to get a very narrow Δ central line, which together with the temporal resolution of our detector corresponds to a temporal coherence parameter Z t = 0.26 .We consider this to be sufficiently small so that p t ≈ Z t .However, it was unavoidable that a lot of further transmission peaks were also transmitted by the filter combination at small transmission ratios.Fig. 12 shows the theoretical transmission curves of the three etalons separately.In total we get the product of all three curves multiplied again by the transmission of the interference filter.
With this we have about = 62% of the transmitted inten- sity in the central peak, while the residual light is distributed over a wide spectral range.For simplicity, we consider this residual light as an incoherent background intensity, which due to the wide spectral range (large Z t ) for itself exhibits a g (2)  background = 1 .In addition, photons from this background are also not correlated with the photons from the central peak, which reduces the measured g (2) further.A comprehensive treatment yields: Therefore, for our realized experimental conditions we expect the value of g (2) in the limit Z x , Z y → 0 to be around 1 + 2 ≈ 1.38 .The largest measured value is still a little bit smaller due to the finite Z x and Z y .However, this agrees very well with our actual measurement and modelling results shown in Fig. 9 and Fig. 10.

Fig. 1
Fig. 1 Schematical depiction of the scheme for the investigations of the photon statistics with the source and the detector

Fig. 2 Fig. 3
Fig. 2 3D-Plot of the second order correlation coefficient g (2) as a function of source diameter d and detector dimensions Δx = Δy for a source-detector distance R = 180 mm

Fig. 5 Fig. 6 Fig. 7
Fig.5 Second order correlation coefficient g(2) as a function of rectangular detector width Δx for five rectangular detector heights Δy for a circular source diameter d = 85 m and a source-detector distance R = 500 mm rp Z e −r .With Eq. (B.1) we obtain for the factorial moment-generating function We are interested in the statistics of the photons and not in that of the radiation units which requires a re-scaling of the v a r i a b l e s k radiation units ⋅ r = k photons a n d t h u s G r,photons (s) = ⟨s k photon,r ⟩ = ⟨s r⋅k radiation units ⟩ .This transforms to and considering we obtain

Table 1
Etalon and interference filter spectral characteristic data

Table 2
Hypothetical measurement data for a source diameter determination