Theory of the Photocount Statistics for Multi-Mode Multi-Frequency

arXiv:0905.0657v1 [physics.optics] 5 May 2009 re htti speieytergm sdi modern in used regime the [5]. precisely times lithography con- is optical short have this studies or Recent that large firmed regime. of intermediate the the limit in that the but show at most will neither are we radiation pronounced multi-frequency paper multi-mode this of the effects in of efficient Later uncertainties as for photocount. intensity acting detectors light thereby unintentional) the nanome- sensors, developed (albeit of few technical been changes a have most to resists than of sensitive photo more precision end, are a this that To with wave- [4]. light the tres than the smaller of structures in length printing precisely operates regime, lithography this optical Modern the scales. of dynamics quantum the to [3] due speckle” is laser field. “dynamic that radiation to effect refer over an [2] we changing moving – paper is is this perturbation that in the of but pattern source speckle the because static it time Frequently a literature. to the com- refers in two “dy- denote concepts term to different opposing used pletely The is unfortunately camera. speckle” a namic the on after creates named is it [1] effect speckle pattern latter static The struc- before as to detector. way mode referred the uncontrolled frequently the of some location but the in reaches source perturbed it radi- the then the at is if known ture appear is will field uncertainty ation Another the field. to properties radiation spectral the contribution distri- of the correlations on energy quantum and the and modes, including the modes, among of bution on number depends variance, the its both by quantified photocount, the of bu h ucm fa xeietweeasuitable position a some where at experiment placed an is of detector outcome the about naprtsta rvdsa“etr =oedefined) (=more point build the “better” to at a possible illumination provides not that is be it apparatus also way: an can inverse statement the finite This in a formulated still outcome. is the there possible, in as uncertainty precisely as known is field aito edfrsm time some for field radiation hoyo h htcutSaitc o ut-oeMulti-F Multi-Mode for Statistics Photocount the of Theory unu ffcsaems rmnn nsallength small on prominent most are effects Quantum uncertainty the that paper this in demonstrate will We ie oerdainfil,apeito a emade be can prediction a field, radiation some Given tnu o hs pc oue htlnstenme fmode of number effec the this links of that size volume) quantities. the space predict phase To (or appears. mod etendue speckle the static contributi la When another as the detector, to speckle. radiation, and laser source the dynamic between of disturbed as spectrum to frequency referred th the sometimes on and depends modes result ual The setup. multi-frequency multi-mode edrv ntelvlo unu pisepesosfrteu the for expressions optics quantum of level the on derive We .INTRODUCTION I. alZisSTA,Rdl-brSrß ,747Oekce,G Oberkochen, 73447 2, Rudolf-Eber-Straße AG, SMT Zeiss Carl R ~ hnti limit. this than T vnwe h radiation the when Even . 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( n ) fΓ( of ) ~ r ( E ~ r E ~ 2 ~ ~ r E ω , ~ ( ) n time and u h extension the but ~ r . 2 srpae ya by replaced is ) t , α , ~ r 2 1 ) ~r, i . 2 t , t n 1 na In . ( − ω ) (1) (2) (3) t 2 ≥ ) 0 . 2

Since φn is independent of t, the φn are the modes instead for the electric field at some position R~ is given by [6] of being some arbitrary base of the electric field. Compar- F (t)= φn(R~)an(t) . (4) ison of Eqs. (2) and (3) shows that the magnitude of an n can be computed from αn but not its phase. A semiclas- X sical treatment of the photocount statistics would assume Later it will prove helpful to switch from the time repre- that αn is fluctuating in time whereas within quantum sentation an(t) to the spectral representation an(ω), optics, which we will apply in this paper, fluctuations are 1 −iωt inherent to the operator description. an(t)= an(ω)e dω . (5) √2π This paper is organised as follows. In Sec. III we com- Z pute the variance of the photocount on a quantum-optical since in time representation, operators taken at different level when the radiation field is completely known. We times do not necessarily commute whereas they do in the will find that there is a shot noise term, a quantum cor- spectral representation, † ′ ′ relation term and a term describing the beating of differ- [an(ω),am(ω )] = δnmδ(ω ω ) . (6) ent frequencies. Frequently, by design or unintentionally, − When we label the number of photons in the n-the mode the mode structure emitted by a known light source is with I , which basically amounts to the total intensity completely changed and unknown when it reaches the n of that mode, this gives the expectation value point of the detector. As will be demonstrated in Sec. V, † ′ random-matrix theory allows a compact and exact treat- a (ω)am(ω) =2πδnmδ(ω ω )InGn(ω) , (7) h n i − ment of this problem. The resulting “static speckle” be- where Gn(ω) is the (normalised) spectrum of the radi- comes the smaller the more modes of the radiation field ation in the n-th mode, and the prefactor 2π has been are excited, and the more uniform the energy distribu- introduced for later convenience. tion among the modes is. Depending on the “size” of the Photodetection within some time interval T is then radiation field, there is thus a minimum amount of static described by the quantity [6] speckle, which is computed in Sec. IV. In Sec. VI we ex- T tend this question to computing the most likely amount W = η F †(t)F (t)dt , (8) of static speckle. Since we will demonstrate this quantity Z0 to be self-averaging, the computed average is more than where η marks the detection efficiency and includes in- just an average in that it is characteristic for almost all formation on the size of the detector. This amounts to individual speckle values. a perturbative description of the interaction between the detector and the radiation field. The perturbative approach neglects the effect that every detected photon de- III. QUANTUM THEORY OF creases the number of photons remaining in the radiation PHOTODETECTION field [9] but this is mainly an issue for microcavities where only a small number of photons are excited at one point in time and the dynamics are then studied. For the pur- While in “general” electrodynamics the electric field ~ pose of this paper, this is no relevant restriction. E(~r, t) can be probed directly, this is not possible in the The factorial moment n(k) := n(n 1) (n k + 1) ~ realm of optics as E(~r, t) changes too quickly in time of the photodetection count distributionh − is··· given− by i and space to allow a direct measurement. Rather, the n(k) = : W k : , (9) radiation field is probed by means of photodetection, i. e., h i by absorbing photons inside some device and counting where the colons denote normal-ordering of the operator the number of photons absorbed. This principle applies within, and the brackets ... denote the average which to technical machines as well as to the human eye or to has to be taken over bothh thei quantum fluctuations of photo resists. the operators an and the frequency distribution Gn(ω). On small length and time scales, quantum effects be- Mean and variance of the photo count then follow from come important. For this reason, and because all effects 2 n = n(1) , var n = n(1) + n(2) n(1) . (10) can then be treated in a more compact way, we will use − the quantum theory of photodetection in the following. For the first moment, this yields the result This theory [7, 8] is usually formulated for single-mode T detection. Since multi-mode fields [9] are at the core of (1) ∗ † n = η dt φ (R~)φm(R~)a (t)am(t) = this paper, we will present an extension of the theory to h n n i Z0 nm multi-mode photodetection here. We will treat the multi- X η T frequency aspect explicitly and not express the different dt dω φ∗ (R~)φ (R~) a† (ω )a (ω ) ei(ω1−ω2)t 2π 1,2 n m h n 1 m 2 i frequencies by different modes as is frequently done in Z0 ZZ nm textbooks. T X ~ 2 ~ 2 We label the modes of the electromagnetic field by = η dt dω φn(R) InGn(ω)= T η φn(R) In . 0 | | | | φn(~r). The annihilation operator associated with this Z Z n n X X (11) mode is an. Using this notation, the quantum operator 3

This is the same result as expected by classical theory mode at position R~. since φ (R~) 2I corresponds to the intensity of the n-th The second factorial moment is | n | n

T n(2) = η2 : dt φ∗ (R~)φ (R~)φ∗ (R~)φ (R~)a† (t )a (t )a† (t )a (t ): h 1,2 n1 n2 n3 n4 n1 1 n2 1 n3 2 n4 2 i 0 n1,...,n4 ZZ X η2 T = dω dt φ∗ (R~)φ (R~)φ∗ (R~)φ (R~) a† (ω )a† (ω )a (ω )a (ω ) 4π2 1,...,4 1,2 n1 n2 n3 n4 h n1 1 n3 3 n2 2 n4 4 i 0 n1,...,n4 ZZZZ ZZ X ei (ω1−ω2)t1+(ω3−ω4)t2 . (12) × Taking the average over the quantum mechanical expectation operators gives nonzero contributions for three cases of the indices n: n = n = n = n , n = n = n = n , and n = n = n = n , thus 1 2 3 4 1 2 6 3 4 1 4 6 2 3 η2 T n(2) = dt dω φ (R~) 4 [a† (ω)]2[a (ω)]2 4π2 1,2 | n | h n n i 0 n ZZ Z X 2 T η ~ 2 ~ 2 † † + 2 dt1,2 dω1,3 φn1 (R) φn3 (R) an1 (ω1)an1 (ω1)an3 (ω3)an3 (ω3) 4π 0 | | | | h i ZZ Z nX16=n3 2 T η ~ 2 ~ 2 † † i (ω1−ω2)t1+(ω2−ω1)t2 + 2 dω1,2 dt1,2 φn1 (R) φn2 (R) an1 (ω1)an1 (ω1)an2 (ω2)an2 (ω2) e . (13) 4π 0 | | | | h i ZZ ZZ nX16=n2

This equation can be simplified by inserting the expecta- subtracting in the first summation the terms just added. tion values from Eq. (7). The double integral over t1 and There now appears the quantity t in the last line of Eq. (13) can be reduced to a single 2 1 integral via f I2 := dω [a† (ω)]2[a (ω)]2 a† (ω)a (ω) 2 , n n 4π2 h n n i − h n n i Z T T (17) dt f(t t )= dtf(t)[T t ] . (14) that quantifies the deviation of the radiation in the n- 1,2 1 − 2 − | | ZZ0 Z−T th mode from a coherent state. For coherent radiation, † 2 2 † 2 fn = 0 since then [an(ω)] [an(ω)] = an(ω)an(ω) The remaining two integrations over ω1 and ω2 in that whereas for any Gaussianh light source,i inh particular anyi line each yield the Fourier transform Gˆ(t) of G(ω), † 2 2 object emitting thermal radiation, [an(ω)] [an(ω)] = † 2 h i 2 an(ω)an(ω) , and fn = 1. For classical radiation, 1 −iωt h i Gˆ(t)= G(ω)e dt . (15) fn 0 whereas for certain nonclassical radiation fn < 0 √2π ≥ Z is possible. Collecting results, using Eq. (10), gives the variance of This thus gives the photocount

2 2 (2) η T 4 † 2 2 2 2 ~ 4 2 n = dω φ (R~) [a (ω)] [a (ω)] var n = n + η T φn(R) Infn+ 4π2 | n | h n n i | | Z n n X 2 X 2 2 ~ 2 ~ 2 η 2 2 + η T φn1 (R) φn3 (R) In1 In3 φ (R~) φ (R~) I I | | | | 2π n1 n2 n1 n2 n16=n3 | | | | X nX16=n2 η2 T T ~ 2 ~ 2 ∗ + dt φ 1 (R) φ 2 (R) ˆ ˆ n n dtGn1 (t)Gn2 (t)[T t ] . (18) 2π −T | | | | × −T − | | Z nX16=n2 Z ˆ ˆ∗ In1 In2 Gn1 (t)Gn2 (t)[T t ] . (16) Equation (18) constitutes a core result of this paper. × − | | It gives the general expression for the noise that is intrin- The summation over n = n is rather inconvenient. sic to any measurement of a multi-mode multi-frequency 1 6 3 Thus, we include the missing terms n1 = n3 in that sum- radiation field over some time T . The first term is the mation, yielding the value [n(1)]2, and correct for this by well-known shot noise term that appears naturally in the 4 quantum treatment: electric field and photocount are 1.0 quantum-optically related via the factorial moment, see 0.9 without the -|t| in the integrand Eq. (9), whereas in classical optics the relation is via the 0.8 Gaussian “plain” moment, with the difference between factorial and plain moments being precisely the shot-noise term. 0.7 τ

T 0.6 The second term, quantified by fn from Eq. (17), marks the excess noise when the radiation field is not 0.5 Lorentzian 0.4 in a coherent state. As mentioned above, fn = 0 for co- integral / herent radiation, and fn = 1 for thermal radiation. Even 0.3 when fn is small, any nonzero fn will eventually become 0.2 the dominating effect for the photonoise as its contri- 0.1 bution increases quadratically with time, in contrast to 0.0 all other contributions. The cross-over point, where this 0 2 4 6 8 10 term becomes larger than the shot-noise term, is easily T / τ estimated from Eq. (18) as ηTfnIn & 1. For any thermal radiation this regime thus is already entered once FIG. 1: Value of the integral over time in Eq. (18) that quan- the measurement time is long enough for one photon per tifies the strength of the dynamic speckle. The two curves at mode to be counted on average. In can be computed from the bottom follow by inserting the expressions for a Gaussian the Bose-Einstein factor, evaluated at the temperature θ and a Lorentzian spectrum from Eq. (24). When the term of the source, (T − |t|) in Eq. (18) is replaced by T , and an exponentially decreasing Gˆ(t) is assumed, the third curve follows. 1 b(ω,θ)= . (19) exp(~ω/k θ) 1 B − curves. Their Fourier transforms are This value depends on the ratio of frequency and tem- 2 ˆ 1 t perature, and thus has the same value b 0.004 for both GGauss(t)= exp , (23) ≈ √2π −8πτ 2 a light bulb emitting visible light and an EUV plasma h i source operating at up to 200000 K to emit radiation 1 t GˆLorentz(t)= exp | | . (24) with λ 13 nm. √2π −2πτ ≈ The third and final contribution in Eq. (18) quantifies h i The results for entering these expressions into the integral the beating of different frequency components when the in Eq. (18) are shown in Fig. 1. There is a significant measurement time is finite. It is sometimes referred to as dependence on the spectral shape, with the longer tails of dynamic laser speckle since it is most relevant for mea- the Lorentzian leading to a slower approach to saturation. surement times T not much larger than the coherence Please note that the graph corresponds to the variance of time τ of the radiation, the photocount. When the square root of the variance is ∞ ∞ scaled by the mean intensity – this quantity is called the τ = dt Gˆ(t) 2 = dω G(ω) 2 , (20) contrast – the saturating curve shown in the figure turns | | | | Z−∞ Z−∞ into a curve decreasing to zero as T becomes larger. GˆLorentz is a plain exponential and thus conforms best hence the name “temporal degree of coherence” fre- to the simple model of how to model finite temporal co- quently given to Gˆ(t). However, the factor [T t ] is also −| | herence. We thus have used this curve for contrasting relevant, and it would be wrong to replace the integral Eq. (18) by an approximation where the factor T t simply by exp( t/τ)2dt, as we will now demonstrate. − | | − is replaced by T , thereby ignoring the additional depen- The extra factor [T t ] , which seems to be largely ig- dence on t . For all values of T , the integral increases nored in phenomenologicalR − | | literature, gives less weight to | | ˆ by about 40 %, making the factor T t important for the frequently surprisingly large tails of G(t). correct modelling of the photon counting− | | statistics. Two frequently encountered spectral distributions are A final word about the cross-over from “dynamics” to Gaussian and Lorentzian, “statics” seems to be in order. Since the temporal effect

2 2 begins to saturate at time τ it is frequently assumed that −2πτ ω GGauss(ω)= √2τe , (21) the nontemporal regime is entered once T τ. While 2τ this statement is correct for single-mode radiation≫ ﬁelds, G (ω)= . (22) Lorentz 1+4π2τ 2ω2 it is incorrect here since the shot-noise term n, setting the reference value, scales linearly with the number N of The curves above are already normalised to 1, i. e., their modes whereas the prefactor of the temporal term scales integral (not their square integral) is 1. The widths of as N (N 1) such that the condition T τ has to be the spectral distributions have been chosen such that the replaced· by− T Nτ. Incidentally, since it≫ is the regime coherence time, computed from Eq. (20), is the same Nτ T τ that≫ current high-power excimer lasers are value τ, allowing for a direct comparison of these two operating≫ ≫ in [5], this distinction is of actual importance. 5

IV. ETENDUE we restrict ourselves to one spatial dimension but the extension is straight-forward and becomes trivial when the Similar to classical mechanics, theoretical optics knows radiation field factorises in its spatial dimensions. Sec- the concept of phase space [10]. The optical phase space ond, we assume that the intensity distribution I(x, k) (x) is spanned by position and spatial frequency, informally factorises into a spatial term I and an angular term (k) called k-vector, and the radiation field is completely de- I . Since the etendue accepted by practically any real- scribed by the (pseudo)-density function W (~r,~k) known life apparatus factorises, these assumptions do not limit as the Wigner function. Any radiation field then occupies the applicability of this approach. a certain volume in phase space. Unfortunately, there is We start with N +1 orthonormal base functions φn(x), no good mathematical metric to actually measure or at n = 0,...,N. The orthogonality is essential since this least define such a volume. prevents the base functions from being too similar and In experimental and technological areas of optics, in thus collapsing into a very small volume of phase space. contrast, the volume of phase space occupied is well- Furthermore, all modes of the radiation field have to be defined and is referred to as etendue. The energy density mutually orthogonal anyhow as they are eigenfunctions I(~r, ~α) of the light, given as a function of position and of a Hermitian operator, cf. Eq. (2). We assume them to angle, is assumed to be larger than zero only inside a be normalised for the ease of the calculation. For every finite and well-defined region. This allows for an easy φn(x), there also exists its Fourier transform and direct definition of the etendue of the radiation field. ∞ ˆ 1 −ikx Etendue is an important quantity since, at least within φn(k)= φn(x)e dx . (26) √2π geometrical optics, it can only be changed by vignetting Z−∞ the radiation field. The etendue supplied by some radia- To progress, we need to choose a certain set of base tion field incident on some optical apparatus should thus functions. The aim is to pack the base functions as be no larger than the etendue that this apparatus can tightly as possible in phase space, and without loss of accept. generality we try to pack them around the origin. If all Convolving the Wigner function W (~r,~k) with a small φn, n =0,...,N, are assumed to fulfil the conditions kernel, its size given by an uncertainty-relation [11], 2 N −a|x| yields I(~r, ~α) when the relation φn(x) C(1 + x ) e , (27a) | |≤ | | 2 φˆ (k) C(1 + k )N e−b|k| , (27b) α = kλ , (25) | n |≤ | | then it can be shown [13] that ab = 1/4 is the most between angle and k-vector is utilised. Thus on first sight condensed situation for which there are solutions. Fur- it might seem that the relation between microscopic and thermore, all solutions can then be written as macroscopic definition would be straight-forward. 2 2 −a|x| ˆ −b|k| The essential difference is that W (~r,~k) has no finite φn(x)= PN (x)e , φn(k)= PÑ (k)e (28) support whereas I(~r, ~α) is assumed to have. It can be ˜ shown that due to the uncertainty principle, every mode where PN (x), PN (k) are polynomials of order not higher than N. Only the product ab is fixed by this but not a or φn needs to have infinite tails in at least either real space (position) or k-space (angle) [12], and this is reflected in b on their own. This freedom is equivalent to changing ˆ W (~r,~k). Of course, also I(~r, ~α) has the same infinite tails φn(x) φn(x/r) with the simultaneous change φn(k) ˆ → → but in the macroscopic world, the function values inside φn(kr). This also reflects that, while phase space volume those tails drop down so quickly that they are “assumed” is well-defined and preserved as volume, its projection to be zero once they are so low that they have become onto a particular axis is allowed to change. irrelevant for practical purposes. The set of solutions φn(x) of Eq. (28) is N + 1 dimen- We thus need a definition for phase space volume that sional, and a convenient orthonormal set consists of the is based on a microscopic radiation field but reduces to Hermite functions ψn(x), defined as the macroscopic etendue for “large” radiation fields. For 1 x2 the purpose of this paper, the relevant way of quantifying ψn(x/r)= exp Hn(x/r) , (29) n −2r2 phase space volume is by the number of modes that can n!2 √π be fit into it. We will tackle the problem in the opposite where H (x) arep the Hermite polynomials, way: We pick a certain number of “sensible” base func- n tions and compute all possible radiation fields that can H0(x)=1 , H1(x)=2x , be formed from them. We will then identify for the in- H (x)=2xH (x) 2nH (x) . (30) tensity profiles I(~r) as function of position and I(~α)asa n+1 n − n−1 function of direction the properties that allow to deduce The prefactor in Eq. (29) ensures normalisation of ψn(x). the number of base functions used. The Hermite functions are eigenfunctions of the Fourier To have shorter mathematical expressions, we will ex- transform, press directions in terms of the k-vector instead of the angle α. Furthermore we make two assumptions. First, ψˆ (ξr) = ( i)nψ (ξ/r) , (31) n − n 6 such that the derivation presented in the following for 2.5 real space also applies to k-space. )| 2.0 kr 2 All normalised functions φ(x) inside the solution space φ Ã( N=13 of Eq. (27) can be written as 1.5 )|, max | r /

x 1.0 (

N N φ 2 0.5 φ(x)= anψn(x) , an =1 . (32) max | | | N=1 n=0 n=0 0.0 X X 0 5 10 15 20 The maximum intensity φ(x) 2 possible for such a state x/r, kr can be computed using the| method| of Lagrange multipli- φ x φˆ k ers. Finding an extremum of φ(x) is equivalent to finding FIG. 2: Maximum values max| ( )| and max| ( )|, computed 2 from Eq. (35), as a function of N where N is the number of one of φ(x) , and we will pick either of these quantities basis functions minus 1. Depicted are the curves for N = | | 2 depending on which one is more convenient. The con- 1, 4, 9,..., 13 . If the corresponding curves for a normalised dition ∂Λ/∂a0 = ... = ∂Λ/∂aN = ∂Λ/∂µ = 0 of the radiation field are below a given line, the radiation field cannot function have more modes than indicated by the number N on the curve. N N Λ(a ,...,a ,µ)= a ψ (x) µ(1 a 2) , (33) 0 N n n − − | n| n=0 n=0 X X is done [14] such that the result depends more on this arbitrary decision than on the radiation field. Also the yields the intermediary result ψ (x)+2µa = 0. By k k method of “almost bandwidth-limited functions” (which multiplying this with ψ (x) respectively a and summing, k k is usually formulated for time-dependent electric signals, this gives the two conditions hence the term “bandwidth” instead of “angular range”) N N suffers from a similar problem [15]. ψ (x) 2 = 2µ a ψ (x)= 2µφ(x) , (34a) Luckily, there exists a strong link between the micro- | n | − n n − n=0 n=0 scopic and macroscopic worlds because the Hermite func- X X N N tions ψn(x) are the solutions of the eigenvalue equation 2 2µ an = anψn(x)= φ(x) . (34b) 2 − | | d ψn(x) n=0 n=0 + (2n +1 x2)ψ (x)=0 , (36) X X dx2 − n Solving for µ by eliminating φ(x) and inserting µ into that can be found in any quantum mechanics textbook Eq. (34b), remembering that N a 2 = 1, one arrives n=0| n| as it describes the harmonic oscillator. The eigenfunction at the result for the maximum intensity, namely ψ has eigenvalue (=energy) of n +1/2, and a classical P n N oscillator with the same energy is restricted to the in- 2 2 terval √2n +1 x √2n +1. The extremal curve max φ(x) = ψn(x) . (35) − ≤ ≤ | | | | φ(x) computed above thus is the extremal superposition n=0 X of harmonic-oscillator solutions with energy level n up to The method of Lagrange multipliers only gives necessary N. Hence, the classical edge beyond which the oscillator but not sufficient conditions for extrema, meaning that cannot be found is given by √2N + 1. there cannot be any additional extrema different from Summarising, a microscopic radiation field has at most Eq. (35) but this equation might not describe an ex- N +1 modes when for all of its modes φm(x) the condi- tremum in the first place. Equation (35) actually de- tions scribes two solutions, φ(x)=+ ... and φ(x) = .... It − N is obvious from physics reasons that at least two extrema, φ (x) 2 ψ (x/r) 2 , (37a) | m | ≤ | n | one being the most positive and one the most negative, n=0 X exist. Hence, Eq. (35) uniquely describes these two ex- N trema. 2 2 φˆm(k) ψn(kr) , (37b) A graphical display of the solutions max φ(x) and | | ≤ | | | | n=0 max φˆ(x) can be found in Fig. 2 as a function of N. The X right| edge| of max φ(x) becomes progressively steeper as are fulfilled for some value r identical for all modes. Since | | φ (x) 2 and φˆ (k) 2 are the intensities as a function of N increases. This means that in the macroscopic limit | m | | m | N 1 there exist a sharp edge that can be associ- position and angle, respectively, these are quantities that ated≫ with the macroscopic etendue. To define the po- can in principle be measured. sition of this edge, one could pick the x-coordinate where A macroscopic radiation field with a half-diameter X max φ(x) 2 =1/2 or where it is equal some other particu- in real space and a half-diameter K in angle space is thus lar value| but| this would lead to an arbitrary result. It can spanned by modes ψn with n =0,...,N determined by be shown that the number of basis functions is then pro- 1 r√2N +1= X, √2N +1= K. (38) portional to the square of the width at which the “cut” r 7

S'' The scale factor r drops out when expressing results in S' terms of the macroscopic etendue. For given etendue basis transformation E XK, the number N of modes (minus 1) is thus "apparatus" to deform ≡ mode structure E 1 detector N = − . (39) light source 2 , φ , φ' , φ' If the etendue is speciﬁed in angle (radians) instead of a a n n a'n n b n n k-vector, this becomes n=1,...,N n=1,...,M

E/λ 1 FIG. 3: Graphical display of the transformation steps to ar- N = − . (40) rive at static speckle. By means of a basis transformation, 2 ′ described by S , one switches from the modes φn that are optimal to describe the light source, to new base functions ′ V. STATIC SPECKLE φn that are optimal to describe the photodetector. Along ′ with the transformation φ → φ , the annihilation operators also need to be transformed, a → a′. The additional action The variance of the photocount computed in Sec. III ′′ of scattering or mixing is then described by the matrix S , describes the uncertainty of the photocount if the ra- ′ transforming a into b. diation field at the detector is completely known. Fre- quently, the mode structure is known at the light source but not at the detector. The standard undergraduate basis set, at the expense of replacing Eq. (7) by some- example is a HeNe-laser pointed at the wall of the lec- thing more complicated. ture room where the observer can see a speckle pattern. Considering only the radiation source and the detec- When the eyes are then moved, the direction of motion tor, hence ignoring the scattering between them for the of the observed speckle pattern depends on whether the ′ moment, we pick new basis functions φn(~r) and associ- viewer is near-sighted or far-sighted [1]. The HeNe-laser ′ ated annihilation operators an such that at the location emits a very regular mode structure but the modes are ~ ′ ′ R of the detector all φn except for φ1 vanish. Such a then changed by scattering at the rough wall. The influ- basis set can always be found and offers the advantage ence of the eye movement demonstrates that scattering that photodetection at the point R~ has been reduced to at the wall does not result in local intensity changes at the photocount of the operator a′ . Since both bases are the place of scattering but rather in a deformation of 1 orthonormal, the transformation between ~a = a ,a ,... the mode structure that translates into intensity changes 1 2 and ~a′ = a′ ,a′ ,... can be described by a unitary matrix only by propagation to and focusing in the eye. 1 2 S′, The change of the mode structure on its way from the radiation source to the detector can be unintentional, just ~a′ = S′~a, (41) as in the example above, or it can be intentional due to some apparatus designed to mix the modes of the radi- with SS† = 11. Computing S′ explicitly would be a very ation field. Unless optical systems are manufactured to formidable task but it will turn out that knowledge of S′ the highest standards, they will have surface roughness is not needed. ′ in excess of the wavelengths. This results in uncontrolled We will fix the number M of new basis functions φn – thus in a certain sense random – mode deformations. shortly but need to allow for the case that M is larger In addition, any apparatus designed to provide a uni- than the number N of excited modes of the radiation form and stable illumination at its exit can only do this field. This is easily achieved by simply adding M N by means of “mixing” the incoming light. Illumination vacuum states to the input for Eq. (41). − systems used in optical lithography are the supreme ex- We now assume that the radiation field is perturbed ample of such an apparatus as the local intensity at the between the light source and the place of the detector by output varies only by about 1 % even when the form of means of some “virtual device” – either intentionally by the illumination of its entrance is completely changed. a properly designed apparatus or, usually unintention- While the light paths inside such an apparatus might be ally, by quasi-random scattering. This can be described very controlled, for an outside observer there (intention- in two opposite but physically equivalent approaches. Ei- ally) is no recognisable connection between the light at ther, one assumes that the mode structure is perturbed, ′ ′ the entrance and the exit. In other words, an apparently thus changing φn while keeping an unchanged, or { } { }′ random change of the mode structure occurs in this case one assumes that the mode structure φn is unchanged as well. while energy is transferred between modes,{ } thus modify- ′ In Sec. III we have used the modes φn(~r) of the radi- ing an . We use the second approach, also known as the ation field to expand the operator for the electric field, method{ } of input-output relations [16], since the annihila- cf. Eq. (4). Any orthonormal basis could have been used tion operators bn describing the radiation leaving the instead but using the modes offered the advantage that device can then{ be} expressed in terms of the annihilation ′ the associated annihilation operators an(ω) become un- operators an entering device. In the absence of non- correlated, cf. Eq. (7). We will now switch to a different linear media,{ this} relation is linear, and can be described 8 by a matrix S′′ is equal to 1/M where M is the order of the matrix, yielding ~b = S′′~a′ . (42) ηT n(1) = I . (45) If there is no absorption, S′′ is unitary. Apart from en- M n n ergy conservation, already the commutation relation (6) X demands the unitarity of S. In the presence of absorp- This is the expected result, namely that the mean intention, Eq. (42) would need to be supplemented by noise sity at the detector is equal to the properly scaled total sources [16] to ensure these commutation relations. intensity of the original light field. In the absence of further information, the unitary ma- Computing the first moment demonstrated that to trix S′ describing the distortion of the mode structure translate the equations from Sec. III, all terms of the is uniformly distributed in the space of unitary matri- form φn(R~) have to be replaced by S1n, and all terms ∗ ~ † ces [17]. This concept of “maximum uncertainty” is φn(R) by Sn1. The correct starting point is Eq. (16), known from other areas of physics as well, most notably which transforms into from thermodynamics where the basic lemma is that ev- 2 2 ery microstate compatible with the macroscopic bound- (2) η T 4 † 2 2 n = dω S1n [a (ω)] [an(ω)] ary conditions is equally likely. 4π2 h| | ih n i Z n The photocount distribution at some point R~ has thus X + η2T 2 S 2 S 2 I I been reduced to the photocount distribution for the mode h| 1n1 | | 1n3 | i n1 n3 n16=n3 b1, related to the original incident mode structure of the X 2 T radiation source by η 2 2 + dt S1n1 S1n2 2π −T h| | | | i ~ ′ ′′ n16=n2 b = S~a, S = S S . (43) Z X ˆ ˆ∗ In1 In2 Gn1 (t)Gn2 (t)[T t ] . (46) Since for a given position of detector and light source R~, × − | | S′ is a fixed unitary matrix, the product S = S′S′′ is also It only remains to compute the necessary averages over distributed uniformly [18], and so no knowledge of S′ is the unitary group in a similar spirit to above where we † needed, as promised above. found that S1nS = 1/M. The average of the square h n1i The only information about what happens between of this expression is frequently needed and can be found light source and detector enters in the form of the number in many papers, e. g. in Ref. 19, with the result M of basis functions. This number is not arbitrary but 2 is determined by the physics of the mode deformation [S S† ]2 = . (47) 1n n1 M(M + 1) process described by the matrix S′′. Please remember h i 2 2 that M is the number of base functions that are mixed The average S1n S1m for n = m is not identical ′′ 2 h| 2 | | | i 6 † by the transformation S . If the mode deformation is to S1n S1m as the unitary condition SS = 11 in- due to some intentional mixing in an apparatus accept- troducesh| | correlationsih| | i among the elements of S. Starting ing and emitting some etendue E, the number M follows 2 2 from the exact relation 1 = nm S1m S1n allows us from the results of Sec. IV and has a finite, well-defined to write | | | | value. If the mode deformation is due to scattering or a P 2 2 4 diffuser, M is determined by the number of basis func- S1m S1n + S1n =1 , (48) h| | | | i h| | i n tions that could in principle end up in the detector due to nX6=m X scattering or diffusing. Depending on maximum scatter- which, together with Eq. (47), gives us the desired aver- ing angle, this can be a very large number and is always age, larger than N. Computing the first factorial moment analogue to 1 2 2 M(M+1) n = m Eq. (11) yields S1n S1m = 2 6 (49) h| | | | i ( M(M+1) n = m T (1) † This allows us to transform Eq. (46) into n = η dtb1(t)b1(t) h 0 i Z 2 2 T (2) η T † 2 2 † † † n = dω [an(ω)] [an(ω)] = η dt an(t)Sn1S1mam(t) = ηT S1nSn1 In , 2π2M(M + 1) h i h i h i n Z0 nm n Z X X X (44) η2T 2 + I I M(M + 1) n1 n3 where in the final equality sign all averages over the fluc- nX16=n3 tuations of electromagnetic field have been taken, just as 2 T η ˆ ˆ∗ described in Sec. III, and an average over the ensemble of + dt In1 In2 Gn1 (t)Gn2 (t)[T t ] . 2πM(M + 1) −T −| | random unitary matrices remains. The average is easily Z nX16=n2 computed from SS† = 11, hence the average in Eq. (44) (50) 9

When computing the variance in Sec. III, the term with source and detector is different for a every member of (1) 2 In1 In3 dropped out when [n ] was subtracted from the ensemble. However, from Eq. (43) it follows that n(2). This is no longer the case here, and one arrives this variance can equally well be computed or measured at by keeping the mode formations constant while changing the position of the detector. The static speckle contrast 1 η2T 2 c, var n = n n2 + (2f + 1)I2 − (M + 1) M(M + 1) n n n 2 1/2 X √var nstatic n In] η2 T c , (56) ˆ ˆ∗ ≡ n ≈ n In + dt In1 In2 Gn1 (t)Gn2 (t)[T t ] . P 2πM(M + 1) −T −| | Z nX16=n2 can thus equally well be determinedP by doing a (long) (51) measurement at different detector positions R~. In prac- Before discussing this result, we will analyse an approx- tise, one uses a camera instead of a point-like photode- imation that is frequently done: The exact averages from tector, and c is calculated from the observed contrast of Eq. (49) are replaced by their Gaussian approximation, the image taken by the camera. The image resembles the equivalent to a phasor representation [1], speckle pattern found on the fur or skin of many animals, hence the name “speckle”. The static speckle contrast 1 can, in this approximation, conveniently be expressed in 2 n = m 2 2 M terms of the effective number of modes N , S1n S1m 2 6 (52) eff h| | | | i≈ ( M 2 n = m 2 In] N = n , (57) The denominator in Eq. (49) was M(M + 1) so that eff 2 n In the mixing of the modes would conserve energy. Equa- P tion (52) neglects this constraint. While this error might that is equal to the actual numberP of modes if all intensi- seem to be negligible for large M, we will demonstrate ties are equal, and smaller (implying more static speckle) that its effect is surprisingly large. First, we compute otherwise. For the exact result (51), static speckle is not so easy to η2T 2 define, as the difference between Eq. (51) and the results (2) † 2 2 2 n dω [a (ω)] [an(ω)] from Sec. III is not a simple term proportional to T . ≈ 2π2M 2 h n i n Z X However, taking the limit T , hence living up to the η2T 2 name “static”, the difference→ between ∞ these two terms + I I M 2 n1 n3 gives n16=n3 X 1/2 η2 T var n var n ˆ ˆ∗ Eq. (51) − Eq. (18) + 2 dt In1 In2 Gn1 (t)Gn2 (t)[T t ] . c = 2πM −T − | | n n16=n2 Z X 1/2 (53) 2 2 M n In M 1 n fnIn 1 = 2 + − 2 "M +1 In M +1 In − M +1# In contrast to Eq. (50), the term with In1 In2 now cancels Pn P n (1) 2 1/2 against [n ] . The variance then becomes P P 2 M 1 1 M 1 n fnIn = + − 2 . 2 2 M +1 Neff − M M +1 η T " P n In # var n n + (f + 1)I2 ≈ M 2 n n (58) n P X 2 T There thus is a very significant difference between the η ˆ ˆ∗ + 2 dt In1 In2 Gn1 (t)Gn2 (t)[T t ] . exact result and the approximation. In the latter, the 2πM −T − | | Z nX16=n2 contribution (55) is minimal but still nonzero if all modes (54) carry the same intensity In whereas for the exact result Eq. (51), assuming coherent radiation, this contribution This is basically Eq. (18) with an additional contribution vanishes if all In are identical since then Neff = M. The approximation becomes valid in the limit that η2T 2 var n I2 , (55) there are many modes but only a small number of them static ≈ M 2 n n are actually excited: Equation (55) depends on the av- X 2 erage squared intensity In n of the modes of the light that is called “static” as its scaling with T 2 implies that source, where for a fixedh lighti source the average is taken its strength, when scaled by the mean squared intensity, over its modes. This kind of average is indicated by the is independent of measurement time T . notation ... n. In contrast, the exact result Eq. (51) We arrived at the ensemble of random S for calculating depends onh M/i (M +1)[ I2 I 2 ] but the term I 2 h nin −h nin h nin the variance var nstatic by assuming a fixed position of the scales with the square of the fraction of excited modes detector while the mode deformations between radiation and may thus be neglected in the limit stated above. 10

This important conclusion can also be formulated in way depending on whether the exact result or the ap- terms of the etendue introduced in Sec. IV. The approx- proximation is used, imation is valid in the limit that the light source fills only 4 a small fraction of the etendue that the “device” used for 1 n an = | | 2 . (60) mixing can accept. It also means that mixing will not Neff 2 Pn an introduce static speckle as long as the entire etendue of | | the mixer is uniformly filled by the light source (assuming In Sec. IV the linear spanP corresponding to some given coherent radiation). etendue E was computed. In the absence of dynamic This might seem to contradict phenomenological theo- information, one cannot select the modes out of all pos- ries that claim that there should aways be some pattern sible sets of basis vectors, and any set of basis vectors in the detector plane because light from the same point could thus represent the modes. We assume that the de- of the light source can reach the detector along differ- composition (59) represents the actual modes φn(x) of ent paths, thereby resulting in uncontrolled interference the radiation field. Since, as just explained, we cannot that inevitably creates intensity fluctuations. However, know φn(x), the equations for the static speckle contrast the unitary condition SS† = 11 ensures that, whenever are instead evaluated using some other orthonormal basis ′ a mode of the light source is brighter at the detector, φn(~r). The two bases are related by a unitary transfor- another mode has to be dimmer. This property is ful- mation, φ~(~r) = Uφ~′(~r), and consequently ~a′ = U~a, but 2 filled for any individual S since n S1n = 1, and no U is unknown for the reasons just explained. averaging over S is necessary to arrive| at this| conclusion. If no additional information is available, the best avail- Technologically, mixing apparatusesP have a well- able estimate is the average of defined etendue, hence a well-defined value of M. When ′ 4 the etendue that they can accept, is filled to a large ex- 1 n an ′ = | | 2 tend by the incident radiation, the effect of the unitary Neff a′ 2 Pn| n| condition is strong, and the exact formula (51) has to be † † ∗ ∗ Pn,i1,...,i4Un,i1 Un,i2 Un,i3 Un,i4 ai1 ai2 ai3 ai4 used. For random scattering, on the other hand, in par- = , 2 2 ticular by a diffuser, frequently N M, and therefore P an ≪ n| | the approximate formula (55) might suffice. (61) P after integrating over the unitary group, i. e., over all ′ VI. STATIC SPECKLE AND ETENDUE possible unitary matrices. We study the average of 1/Neff ′ instead of Neff since it both is easier to compute and is the relevant quantity for the speckle contrast. The variance In Sec. V we demonstrated that the static speckle de- [1/N ′ ]2 1/N ′ 2 then specifies how well the average pends on the mean squared intensity in each mode of the eff eff his representativei−h fori all possible members of the ensemble. radiation source, hence on I2, cf. Eqs. (51) and (55). n n Averaging over the unitary group, the nonvanishing The effect is minimal if all modes carry equal weight, terms are of the form and the maximum numberP of modes possible follows from the etendue via Sec. IV. The minimum amount of static ∗ ∗ U U U ′ U ′ speckle possible can then be computed – it is zero if the h i1,j1 ··· iM ,jM P (i1),P (j1) ··· P (iM ),P (jM )i etendue is not increased by the scattering and mixing = Vs(P −1 P ′) , (62) processes, and finite otherwise. ′ This raises the following question: is it possible not where P and P are permutations of the numbers only to compute the minimum amount of static speckle 1,...,M, and s denotes the cycle structure of a per- mutation. The coefficients V depend only on the cycle for given etendue of the radiation field but also the “typ- −1 ′ ical” amount of speckle? We will demonstrate that static structure of P P and have been tabulated [19]. speckle contrast is a self-averaging quantity, i. e., in the Labelling the numerator in Eq. (61) as Z, limit of many modes M, the relative fluctuation while Z = a′ 4 = U † U U † U a∗ a a∗ a , taking the average over the unitary group becomes small. | n| n,i1 n,i2 n,i3 n,i4 i1 i2 i3 i4 n n,i1,...,i4 This implies that the ensemble average is characteristic X X (63) for almost all realisation of the ensemble. and counting all the permutations of the indices in it When φ (~r) are the modes of the electric field, the n yields electric field E(~r) is uniquely described by coefficients an, Z = 2(V + V ) a 2 a 2 . (64) h i 1,1 2 | i1 | | i3 | n i1,i3 E(~r)= anφn(~r) , (59) X X n For the square, X and the static speckle contrast is determined by the effec- Z2 = a′ 4 a′ 4 , (65) | n| | m| tive number Neff of modes in a more or less complicated n,m X 11 one finds time-dependence of the electromagnetic field is included in the frequency-dependence of the annihilation opera- Z2 = (4V +24V +32V +12V +24V ) h i 1,1,1,1 2,1,1 3,1 2,2 4 × tors. However, frequently it is rather inconvenient to de- 2 scribe time-dependence in this way. An extreme example ai1 ai2 ai3 ai4 | | are pulsed lasers where it is more appropriate to describe n6=m i1,...,i4 X X the mode structure of each pulse separately. This comes + (24V + 144V + 192V + 72V + 144V ) 1,1,1,1 2,1,1 3,1 2,2 4 × at a price, though, as the modes of one pulse are then a a a a 2 . (66) not necessarily orthogonal to the modes of another pulse. | i1 i2 i3 i4 | n i1,...,i4 This is primarily an issue for the static speckle computed X X in Sec. V as the mutual orthogonality of the modes was The terms a ,...,a appearing in the averages above i1 i4 essential for the derivation presented there. cancel against the denominator of Eq. (61). This is as The static speckle contrast in Eq. (58) has an easy in- expected since, as explained, these coefficients depend terpretation: except for a minor correction due to the on the base that was used to determine them, i. e., they unitary of the scattering matrix, every mode creates a depend on the choice of one particular unitary matrix. mutually uncorrelated speckle pattern in the detector Averaging over the unitary group, this choice must be plane. The effect of a superposition of different modes irrelevant, thus the coefficients drop out, and the results is easily understood from this, and for the static part it depends only on the number M of modes. is irrelevant if all the modes are excited simultaneously or Inserting Eq. (64) into Eq. (61) and using the tabulated rather one after the other. In this section, we will show coefficients from Ref. 19 gives the result that the speckle patterns of two nonorthogonal modes 1 2 (i. e., necessarily at two different times) are correlated. ′ = , (67) hNeff i M +1 We will, without loss of generality, refer to these two times as first and second pulse, respectively. which has an easy interpretation. The term M +1 instead of the naive term M in the denominator is due to the The first pulse consists of modes φn and annihilation operators a . For a fixed arrangement of mode structure correlations imposed on U as it is unitary, cf. the factor n M + 1 in Eq. (58). The 2 in the numerator implies that of the light source, scattering between source and detector, and detector position, thus only averaging over the every mode is occupied with an effective intensity of 1/2 relative to the maximum intensity of any mode – a very fluctuations of the radiation field, the mean photocount is, completely analogue to Eq. (44), plausible result since the intensity is a random quantity between 0 and that maximum value. † n = ηT S S I . (70) A similar calculation using Eq. (66) gives for the square 1n n1 n n 1 M +5 X =4 , (68) We now consider a second set φ′ of modes of the light h ′ 2 i (M + 1)(M + 2)(M + 3) n Neff source but keep the scattering between source and detec- so that the variance becomes tor and the detector position fixed. The transformation ′ 1 M 1 from the φn to the φn is linear and can thus be described ′ var ′ =4 2 − . (69) by a matrix S , Neff (M + 1) (M + 2)(M + 3) ′ ′ ′ S = φk φ , (71) The variance decreases much faster with M than 1/Neff kl h | li ′ 2 or 1/Neff as the lowest-orders in 1/M cancel when com- where the notation ...... denotes the overlap integral. ′ h | i puting the variance. This implies that 1/Neff is a self- The annihilation operators transform as averaging quantity, and its value computed from the ′ ′ number M and hence from the etendue of the radiation an = S a , (72) ′ nk k field gives a good estimation of the actual value of 1/Neff . k One should be aware, however, that this is a statistical X statement: in “almost all cases” this statement is correct. and the mean photocount thus becomes By use of particular light sources it still is possible to have ′ ′ ′ † ′ ′ n¯ = ηT (SS )1n(SS )n1In . (73) a very different value for 1/Neff. This is also reflected in n the calculation presented here. Equation (60) gives the X actual value of 1/Neff which, of course, depends on the The essential quantity is the cross-correlation between n ′ values of the an. The average value from Eq. (67) no andn ¯ after averaging over all possible scattering config- ′ longer depends on the an and can thus deviate. urations, hence over all S, and only the matrix S may remain in the result. A positive correlation means that the light patterns caused by the two radiation fields are VII. NONORTHOGONAL MODES similar. When the photocounts of two such fields are subsequently integrated on the same detector, the static The modes φn(~r) of the electromagnetic field are mu- speckle contrast will be higher than naively expected by tually orthogonal and stationary, cf. Eq. (2), and all the the inverse square-root of number-of-modes law. 12

The cross-correlation between n andn ¯′ is given by that decreases exponentially with separation but is always larger than zero. Two plane waves restricted to c := nn¯′ n n¯′ , (74) a finite interval have an overlap that decreases exponen- h i − h ih i tially with the angle between the two waves but is always and the average has to be taken over the matrix S keeping finite. ′ the matrix S fixed. Writing out the first term explicitly, Depending on the way that this expansion is done, some coherence information might be lost, namely when ′ ∗ ∗ ′ ′∗ ′ nn¯ = S1nS1nS1kS1lSkmSlmInIm , (75) doing the expansion such that only the time-averaged in- nmkl X tensities at all field-points are matched, and each φn is then assigned an intensity I . This might sound like a the average yields nonzero terms only for k = l, and the n bad approximation but effectively amounts to using the two cases k = n and k = have to be distinguished. The assumptions from Sec. VI. Inserting the I into Eq. (51) necessary averages have6 already been given in Eq. (49), n would yield a low static speckle contrast since the number and the result is of In’s is large. However, the correct equation to use is Eq. (78) where every term is evaluated for a single func- ′ 2 ′ 2 ′ nn¯ = Snm InIm tion φ , and the nonorthogonality of the φ implies that h i M(M + 1) | | n n nm X the cross-correlation terms are large. This is as expected 1 + S′ 2I I′ . (76) since introducing additional functions φn to describe the M(M + 1) | km| n m same system should not change its computed properties. m X nX6=k This can be simplified by noting that S′ 2 = n6=k| km| 1 S′ 2. The cross-correlator then becomes VIII. DISCUSSION − | nm| P 1 1 The purpose of this paper has been to give a con- c = φ φ′ 2 I I′ . (77) M(M + 1) |h n| mi| − M n m sistent and concise description of multi-frequency and nm X multi-mode effects on photodetection on a quantum level This equation gives the expected result in the two ex- trying to avoid the use of uncontrolled approximations. treme situations. If all intensities are equal, the cross- Its focus lies on exactness, not on ease of presentation correlation becomes zero since, as explained earlier in or application to some particular device. It thus comple- the context of Eq. (58), the static speckle itself van- ments previous work, for example by Joseph W. Good- ishes, and thus also the cross-correlation does. If only a man or the group around Christer Rydberg [1, 3]. single mode n is excited, the cross-correlation becomes In Sec. III we derived how the uncertainty of the pho- ′ 2 proportional to φn φn 1/M. The average value tocount depends on the measurement time and the tem- of φ φ′ 2 when|h assuming| i| − a random S′ is equal to poral coherence function. Among others, it was shown |h n| ni| 1/M, and the cross-correlator is again zero. If φn and that a description via the coherence time of the radiation ′ φn are identical, the cross-correlator becomes maximal. is insufficient to arrive at an exact result. Similarly, in ′ The cross-correlator thus correctly quantifies if φn and φn Sec. V not only previously-known approximations of the are stronger correlated than in a random configuration, static speckle contrast were retrieved, namely an inverse ′ 2 and the overlap integral φn φm is directly related to square-root law with the effective number of modes, but the static speckle pattern.|h | i| in addition also effects of finite etendue accepted by the When a photodetector sums over N pulses, the vari- “mixing device” or of noncoherent, e. g. thermal, radi- ance var n of the integrated photocount follows from the ation followed from the chosen mathematical formalism variances var nk of the k-th pulse, given by Eq. (51), and without additional effort. from the cross-correlator ckl, given by Eq. (77), between Some of the results of this paper can be explained, al- the k-th and the l-th pulse, beit only on a qualitative level, by the “brick wall” model: The radiation field is like a brick wall (horizontally: time, var n = var nk + ckl . (78) vertically: position), and only bricks in the same col- k k6=l umn or row can cause interference effects. Counting the X X fraction of such bricks gives an intuitive explanation of Apart from the rather obvious relation (78), the re- the scaling of dynamic and static speckle contrast upon sults from this section have another application. Fre- changing system parameters. If this is sufficient, the quently, radiation fields are expanded not in their modes methods and results presented in this paper are a bit but rather in a set of “convenient” functions, such as of an overkill. If, on the other hand, exact agreement Gaussian beams or plane waves. Drawback of this ap- between predication and performance of some device is proach is that this set then usually has to be overcom- essential, this paper offers the advantage of giving results plete, hence it has more elements than there had been using only well-defined input parameters, thus not rely- modes, and not all its elements are mutually orthogo- ing on effective parameters that need to be tuned until nal. For example, two Gaussian beams have an overlap the desired result is retrieved. 13

Main function of any theory is to predict whether term once the measurement time is sufficiently long, this some device or experimental setup will work as expected lack of knowledge is an actual issue. and/or required. For state-of-the-art technological ap- The static speckle, computed in Sec. V, depends on plications, the performance might improve by about 25 % the number of excited modes of the radiation field (and from one generation of a device or setup to the next – the energy distribution among them) but, in addition, but rarely more. To predict if something actually is an also on the number of modes that could in principle be improvement, the error due to approximations thus has excited. This is closely related to the concept of etendue, to be much smaller this number. As an example, Fig. 1 cf. Sec. IV. Even though etendue is a very basic quantity, gives an indication of the error made when using the co- its microscopic definition does not seem to have been ad- herence time instead of the exact shape of the frequency dressed before. The relation presented between number spectrum. If improvement of dynamic speckle is a topic, of basis function and macroscopic etendue has applica- one should thus use the exact formulae and refrain from tion beyond this paper as any modelling of a radiation approximations. field – for the purpose of simulation or for an analytic Depending on the desired application, either the study – needs to start with a given number of modes, and present paper or one of the previous works by other au- knowing this number a priori makes this process much thors will thus be more appropriate. Furthermore, we more efficient. restrict us to universal problems. For example, we study While the applicability of the relation between etendue in Sec. V only the variance of the photocount, hence the and maximum number of modes is obvious, this is less the contrast of the dark and bright “spots” in the plane of case for Sec. VI discussing the relation between etendue the detector, but not their spatial correlations, i. e., the and average effective number of modes. We demon- size of these spots. Reason is that, while the contrast strated that this average is identical to the actual number is universal, the size of the spots is not and depends on for almost all radiation fields. However, that is “only” a the setup. The book of Goodman [1] is to a large extend statistical statement: it assumes that all possible radia- devoted to studying this question (on the classical level) tion fields with given etendue are equally likely, and it for a large number of setups of technological or scientific is still allowed that a few cases (albeit of measure zero) importance. deviate strongly. Different light sources have different In Sec. III the uncertainty of the photocount for known properties, and for any technological application, one will radiation field has been computed, yielding in addition choose the most appropriate light source. Selecting a to the well-known shot noise term two terms depending single-mode laser that operates in a high-order Hermite- on the frequency spectrum and on the photon correla- Gauss mode would give the most extreme disagreement tions of the radiation source, respectively. The impor- between the actual number of modes (=one) and the pre- tance of the frequency spectrum is obvious, in particular diction from Sec. VI. The results from this section can in view of the two examples presented in this paper. For thus only be applied when the choice for a particular light the importance of the effects of noncoherent radiation, source does not adversely impair the freedom of the mode the situation is less clear. For traditional thermal light structure of the generated light field. sources, the necessary information can be found in this paper; for traditional lasers, the emitted radiation is coherent, and the question becomes trivial. For modern light sources, such as excimer lasers [20] that are increas- Acknowledgments ingly used for high-power applications, insufficient data is available. Technological advances are fast, and no good characterisation of the photon correlations for modern The author would like to acknowledge valuable dis- devices seems to have been published. Since this con- cussions with Olaf Dittmann, Norbert Kerwien and Jo- tribution to the noise inevitably becomes the dominant hannes Wangler.

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