Probability density cloud as a geometrical tool to describe of scattered light

NATALIA YAITSKOVA Hermann-Weinhauser straße 33, 81673 München Corresponding author: [email protected] Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX

First-order statistics of scattered light is described using the representation of density cloud which visualizes a two-dimensional distribution for complex amplitude. The geometric parameters of the cloud are studied in detail and are connected to the statistical properties of phase. The moment-generating function for intensity is obtained in a closed form through these parameters. An example of exponentially modified is provided to illustrate the functioning of this geometrical approach. © 2017 Optical Society of America OCIS codes: (030.6600) Statistical optics; (030.6140) Speckle; (000.5490) and statistics. http://dx.doi.org/10.1364/AO.99.099999

1. INTRODUCTION materials and different polishing processes produce various types of surface height distribution. It can be either symmetrical or non- In the title of the historical article by Lord Rayleigh: “On the resultant of symmetrical, skewed positively or negatively [7, 8]. Some artificially a large number of vibrations of the same pitch and arbitrary phase” [1], generated biological surfaces have negative skewed height distribution the emphasis is placed on the word “arbitrary”, for it is always with one or two peaks [9]. In spite of the availability of various tempting for a scientist to find a model, an equation, describing the statistical distributions, the good old Gaussian model remains the widest range of phenomena. Unfortunately, the larger the range of favorite when it comes to analyzing experimental data. phenomena the model covers, the heavier it is mathematically, while a Petr Beckmann [10] first drew his attention to this deficiency and compact, ready-for-use expression is often derived for a particular suggested the tool of orthogonal polynomials to model the case. The Rayleigh distribution we know is valid only when the phase phenomenon of light scattering from surfaces with arbitrary height spans equiprobably the whole range of the possible values from zero distribution. In his approach, the choice of polynomials (Jacobi, Laguerre or Hermite) is however predefined by the type of surface to 2. The generalization, which Lord Rayleigh makes in the last part of height statistics. The major constraint of scattering theory remains: his paper, comprises only certain class of phase distributions particular model of height distribution must be postulated prior to any possessing a /2 periodicity. In spite of this limitation, the neat result analytical development. he has obtained (in his turn being inspired by the book of Émile Verdet To remove this constraint we suggest a geometrical approach, [2]) has become the mold for the theory of light statistics for over one alternative to the pure analytical method, to describe statistics of the century since [3]. scattered light. We develop a tool of probability cloud: graphical The model considered by Lord Rayleigh refers to the so-called representation of a two-dimensional probability density function (2D- strong diffuser. There is another group of random media and reflecting PDF) for complex amplitude. This representation is implicitly utilized smooth surfaces combined in a model of weak diffuser [4]. In this by Lord Rayleigh in his paper. The plots of the 2D-PDF are presented in model the phase is confined within a range smaller than 2 and the many canonical books on light statistics (for example, in the book by distribution is assumed to be normal (Gaussian). For volume scattering Joseph Goodman [3], paragraph 2.6), but to our knowledge, little this assumption can be justified: multiple phase shifts along a attention has been payed to the geometric properties of the cloud: its propagation path permit implementation of the central limit theorem. position, extension, elongation and orientation. A notable exception is The ubiquitous use of normal distribution is rather due to the the study by Jun Uozumi and Toshimitsu Asakura [11], where this tool mathematical convenience than to a physical insight: any deviation is employed to describe on- and off-axis statistics of light in the near- from the normal distribution brings extensive mathematical difficulties field. The tool of probability cloud provides a cognitive link between related to complicated expressions involving special functions [5]. stochastic events in the pupil and image planes. On the one side, the Applications often require simplified mathematics; for example, the geometry of the cloud is related to the distribution of phase and, on the determination of surface roughness by optical methods relies on the other side, it allows the quantitative prediction of the first order Gaussian model of phase [6]. statistics of intensity. Nevertheless, extensive experimental evidence shows that different From an arbitrary phase distribution, we derive expressions for the depend neither on direction nor on coordinate in the pupil. Fourth, the geometric parameters of the cloud: its position, extension, elongation illumination is regarded to be uniform: 푃(풓) is constant inside the and orientation. The focus of our study is the relation of these pupil (within the illumination spot) and zero outside. These parameters to phase statistics. Section 2 describes this relation in the assumptions allow splitting the pupil into 푛 identical zones and the most general way by setting the task into a physical context and then diffraction integral after some normalization turns into a random by introducing a mathematical formalism. Section 3 illustrates an phasor sum with 푛 phasors: application of our approach based on an example of the exponentially 푛 modified normal phase distribution. This is only an example, not a 1 푨 = ∑ 푒푗휑푖. (4) restriction of our method. In Section 4 we return to a general 푛 √ 푖=1 description of Section 2 and work with statistics of intensity. We give an expression for the moment-generating function in a closed form The random values 휑푖 are identically distributed which results from through the geometric parameters of the 2D-PDF. As the main subject the assumption of spatial stationarity. They are also mutually of the paper is the cloud itself, the material of Section 4 is condensed. uncorrelated which is ensured by the size of the zone. The size of the Our study strives to obtain the results for truly arbitrary phase zone defines also the number of phasors – an important parameter in distribution. The only “Gaussian” limitation we preserve is the the study. The definition of the zone and therefore the determination of assumption of a large number of contributors, which means that we the number of phasors involves second-order statistics of phase: its deal with a Gaussian complex . Non-Gaussian complex autocorrelation function. Some authors prefer working directly with random variable and the theory of non-Gaussian speckles were the autocorrelation function, assuming a Gaussian-correlated phase discussed in our earlier work [12]. [11, 14]. Nevertheless, this approach does not remove the mathematical complexity of the expressions involved. We prefer to work with the random phasor sum, keeping the number of phasors n 2. PROBABILITY CLOUD as a free parameter. If the source of randomness is telescope segmentation the issue of computing 푛 is trivial (it equals to the A. From a diffraction integral to a random phasor sum number of segments). We start by setting our task in the context of diffractive optics. Consider a monochromatic optical wave with a distorted wavefront 휑(풓). The B. Two-dimensional probability density function for the complex complex amplitude in the pupil plane of an imaging system is given by amplitude 푭(풓) = 푃(풓)푒푗휑(풓), (1) In the theory of diffused light various parameters and phenomena are said to be Gaussian: Gaussian speckles, Gaussian distribution of phase, ( ) where 풓 is a position in the pupil plane and 푃 풓 is a pupil function, Gaussian autocorrelation function of phase and even Gaussian beam. which is one inside the pupil and zero outside it. The amplitude of the Explicit modelling of the beam shape and of the autocorrelation wave is constant everywhere within the pupil. Phase randomness function can be avoided by introducing the notion of phasors. Although might be caused by a turbulent media through which the wave has we have assumed a uniform illumination, Eq.(4) can be modified propagated or by a surface from which the light has reflected. In case of taking into account a non-uniform beam shape. However, we must free-space propagation, Eq.(1) describes the complex field directly clarify the difference between Gaussian speckles and Gaussian phases. after the surface and 푃(풓) – the shape of an illumination spot. The Although in many studies both assumptions are often made together, imaging system might also introduce phase distortions of its own, like they are two different issues. The former concerns the number of the multi-segmented primary mirror of a giant optical telescope [13]. phasors: if 푛 is large enough, the central limit theorem can be applied The complex amplitude in the focal plane of the imaging system to the complex random variable A and speckles formed by the (coordinate 풘) is given by the Fourier transform: resulting intensity are called Gaussian. The latter refers to the shape of 2휋 1 −푗 풓풘 (2) the probability density function of phase 휑 (P-PDF). The number of 푼(풘) = ∬ 푭(풓) 푒 휆푓 푑2푟, 푖 휆푓 phasors is defined by the pupil size and the correlation radius of the where 휆 is the wavelength and 푓 is the focal distance. The same complex field, i.e. by the second order statistics. The P-PDF is the first expression is valid for the complex amplitude in the far-field if the order statistics. Here we adopt the first assumption of a large number wave reflects from a surface and freely propagates assuming that of phasors but allow the P-PDF to be arbitrary. scattering angles are small, i.e. in the paraxial approximation. Then the In this Subsection we expound the basic mathematical facts about focal distance 푓 is replaced by a propagation distance 푧. two-dimensional random Gaussian values, conic sections in general The intensity in the image plane |푼(풘)|2 is a random field, which and ellipses in particular. If the reader finds this part trivial, he (she) statistical properties depend on the following factors: can skip it and go directly to the Subsection 2C. 1. First-order statistics of the wavefront distortions 휑(풓) including The assumption of a large number of phasors allows for employing spatial stationarity or non-stationarity; of the central limit theorem and for writing down an expression of the 2. The ratio between the size of the pupil (or of the illumination joint probability density function for the real (R=Re A) and imaginary (I=Im A) parts of the resultant complex amplitude A. From now on we spot) and the correlation radius of the complex field 푒푗휑(풓); refer to this function as 2D-PDF. It includes the following statistical 3. Observation point (coordinate 풘). moments of R and I (we use 〈… 〉 to denote statistical averaging): Further conditions must be set. First, the observation point is chosen to be on-axis, so the diffraction integral becomes 퐸푅 = 〈푅〉, 퐸퐼 = 〈퐼〉, 2 2 2 휎푅 = 〈푅 〉 − 〈푅〉 , 1 푼(0) = ∬ 푭(풓)푑2푟, (3) 휎2 = 〈퐼2〉 − 〈퐼〉2, (5) 휆푓 퐼 퐶푅퐼 = 〈(푅 − 퐸푅)(퐼 − 퐸퐼)〉. Second, it is assumed that the correlation radius of the complex field 푒푗휑(풓) is smaller than the size of the pupil so that the pupil includes These quantities form the : many coherent cells. Third, it is assumed that the random field 휑(풓) is isotropic and spatially stationary, so that its statistical properties Consider a new system of coordinates with the origin in the center of 휎2 퐶 (6) 푅 푅퐼 the ellipse and the axis coinciding with its axis (Figure 2). Through 풄̂ = [ 2 ] 퐶푅퐼 휎퐼 translation and rotation of the axes, a coordinate (푅, 퐼) transforms into with the determinant a coordinate (푥, 푦) and the equation of the conic section assumes the canonic form. The 2D-PDF becomes: 2 2 2 퐷푐 = 휎푅 휎퐼 − 퐶푅퐼. (7) 1 1 푇 ̂ (13) 푝퐴(푥, 푦) = ′ exp (− ′ 풗 풔′풗), Now we write the 2D-PDF through the covariance matrix: 2휋√퐷푐 2퐷푐 1 1 푇 −1 (8) where 풗푇 = {푥, 푦}. Matrix 푝퐴(푅, 퐼) = exp (− 풖 풄̂ 풖), 2휋√퐷푐 2 2 휎푦 0 (14) where 풖푇 = {푅 − 퐸 , 퐼 − 퐸 } and 풄̂−1 is the inverse covariance 풔̂′ = [ ] 푅 퐼 0 휎2 matrix. The product 풖푇풄̂−1풖 can be written as 푥 is obtained from the matrix 풔̂ by translation and rotation. It has 푇 −1 1 푇 풖 풄̂ 풖 = 풖 풔̂풖, (9) ′ 2 2 퐷푐 determinant 퐷푐 = 휎푥 휎푦 . Two quantities are invariant to the change of where the system of coordinates: the trace of the matrix and its determinant. The trace 2 (10) 휎퐼 −퐶푅퐼 2 2 2 2 풔̂ = [ 2 ]. 훴 = 휎푅 + 휎퐼 = 휎푥 + 휎푦 (15) −퐶푅퐼 휎푅 can be regarded as the total size of the ellipse: spread of the 2D-PDF. The determinant of the matrix 풔̂ also equals 퐷푐. The quantity ′ The discriminant is also preserved: 퐷푐 = 퐷푐. However, we prefer to 풖푇풔̂풖 ≝ 푄(푅, 퐼) = (11) work with another invariant: the eccentricity. We use the definition of eccentricity slightly different from the usual one: 2( )2 2( )2 ( )( ) 휎퐼 푅 − 퐸푅 + 휎푅 퐼 − 퐸퐼 − 2퐶푅퐼 푅 − 퐸푅 퐼 − 퐸퐼 2 2 휎푥 − 휎푦 (16) is a second-order polynomial, and the line 푄(푅, 퐼) = 푐표푛푠푡 is a conic 푒 = . 휎2 + 휎2 section. Therefore the contour of constant probability density, 푥 푦 푝퐴(푅, 퐼) = 푐표푛푠푡, is also a conic section. It gives the degree of elongation and it can be expressed through the The conic section can be classifies either by the sign of the total size and the determinant as discriminant of the quadratic part of 푄(푅, 퐼) or by the sign of the (17) determinant of the matrix 풔̂. These two quantities differ only by the 4퐷 푒 = √1 − 푐. factor -4. We use the determinant of the matrix 풔̂, which as we have 훴2 seen coincides with the determinant 퐷푐 of the covariance matrix from 2 2 Eq.(7). Due to the Schwarz’s inequality the determinant is non- Solving Eqs. (15) and (16) with respect to 휎푥 and 휎푦 yields in negative. We first consider the case when 퐷 > 0. The contours of 푐 2 훴 (18) constant probability density are ellipses or circles. We select one 휎푥 = (1 + 푒), 2 contour satisfying the equation 휎2 = 훴(1 − 푒). 푦 2 , (12) 푄(푅, 퐼) = 퐷푐 The eccentricity changes between zero and one. It equals zero when and associate the geometric properties of the 2D-PDF with the the semi-axes are identical: 휎푥 = 휎푦 = √훴⁄2. The cloud is then properties of this particular ellipse. The properties are: position of the circular, but it does not necessary mean that it is centered at the center center (휌, 휃휌), lengths of semi-major and semi-minor axis of coordinates. In general, position and extension of the cloud do not (휎 , 휎 ) and the rotation angle 훩 (Figure 1). depend on the eccentricity. When the cloud is circular, the determinant 푥 푦 2 is maximal: 퐷푐 = 훴 ⁄4. Consider now the case when 퐷푐 = 0. The argument of the exponential function in Eq.(8) is a negative infinitely large value everywhere except for the locus 풖푇풔̂풖 = 푄(푅, 퐼) = 0. The probability density function is zero everywhere except on this locus, where it is infinite. The locus satisfies the equation

2 (19) 푄(푅, 퐼) = [휎퐼(푅 − 퐸푅) − 휎푅(퐼 − 퐸퐼)] = 0

and therefore represents a degenerated conic. If at least one of 휎푅 and 휎퐼 is non-zero (훴 ≠ 0), the degenerated conic is a line; if 휎푅 = 휎퐼 = 0, (훴 = 0), it is a point at the coordinate {푅 = 퐸푅, 퐼 = 퐸퐼}. For the both types of degenerated conic the eccentricity equals one. Rotation and translation brings the ellipse to the canonic form, therefore one can find the expressions, relating the coefficients of the covariance matrix and the geometric parameters of the conic section. We present them without the derivation: 휎2 = 훴(1 + 푒 cos 훩), 푅 2 Fig. 1. Two-dimensional probability density cloud and its geometric 훴 휎2 = (1 − 푒 cos 훩), parameters: distance to the center, semiaxes and the angles. The semi- 퐼 2 (20) major axis is always 휎푥 regardless of the orientation of the ellipse. 퐶 = 푒훴 sin 훩 , 푅퐼 2 where 훩 is the rotation angle. (28) The geometry of the 2D-PDF is fully described if the following five 휌 = √푛(1 − 훴). parameters are known: 휌, 휃 , 훴, 푒 and 훩. In the next sub-section we 휌 The total size varies between zero and one. If the number of phasors is relate them to the statistical properties of the phase. preserved, the cloud shrinks while moving away from the center of the coordinate system. At the farthest distance equal to √푛 the total size is C. Statistics of phase and geometry of the cloud zero and the cloud shrinks into a point. When the center of the cloud The principal input is the characteristic function of phase 푴휑(푘). It is a coincides with the center of the system of coordinates, the total size statistical average of the complex value 푒푗푘휑 and it is related to the P- equals one and the cloud is at its maximal spread. PDF by the Fourier transform: To compute the determinant we combine Eq.(25 c, d, e): 2 2 2 1 2 2 2 1 2 2 2휋 [( ) ] ( ) (29) (21) 퐷푐 = 휎푅 휎퐼 − 퐶푅퐼 = 4 1 − 푉 − 푊 = 4 훴 − 푊 . 푗푘휑 푗푘휑 푴휑(푘) = 〈푒 〉 = ∫ 푝(휑)푒 푑휑. 0 Substituting this expression into Eq.(17) for the eccentricity we obtain

Two values of the characteristic function, 푴휑(1) and 푴휑(2), give the 푊 푊 (30) 푒 = = . mean and the variance of the random complex variable 푒푗휑: 1 − 푉2 훴

푽 = 푴 (1), 푾 = 푴 (2) − 푴2 (1). (22) Comparing Eq.(25) with Eq.(20) we conclude that 휑 휑 휑 휃 (31) The entities of Eq.(22) are complex values and we represent them 훩 = 푊 . through the moduli and the angles: 2

푽 = 푉푒푗휃푉, 푾 = 푊푒푗휃푊. (23) For the application another angle is the more important: the angle between the major axis and the line passing through the center of the To relate them to the moments from Eq.(5) we use the equations (2- ellipse and the center of the coordinate system (Figure 1). We refer to it 47) from the Ref. (3) and the properties of complex variable: as inclination angle 휃푇. The inclination angle is the sum ∗ 푴휑(1) + 푴휑(−1) = 푴휑(1) + 푴휑(1) (32) 휃푊 = 2Re[푴 (1)] = 2푉 cos 휃 , 휃푇 = 휃휌 + 훩 = 휃푉 + . 휑 푉 2 2 2 푴휑(2) + 푴휑(−2) − 푴휑(1) − 푴휑(−1) As it is indicated in Figure 1, we assume that the angles 휃휌 (휃푉) and (24) 휃 increase anticlockwise, but 휃 and 훩 – clockwise. = 2Re[푴 (2)−푴2 (1)] = 2푊 cos 휃 , 푇 푊 휑 휑 푊 Eqs. (26), (27), (30) and (31) relate the five parameters of the cloud: 2 2 푴휑(2) − 푴휑(−2) − 푴휑(1) + 푴휑(−1) 휌, 휃휌, 훴, 푒 and 훩 to the mean and the variance of the random complex 2 variable 푒푗휑. In the following, we demonstrate how this approach can = 2푗Im[푴휑(2)−푴휑(1)] = 2푗푊 sin 휃푊 , be used for a given P-PDF, but before we return shortly to the canonical 2 푴휑(1)푴휑(−1) = 푉 . equation of ellipse. The moments become

퐸푅 = √푛푉 cos 휃푉 (a) (b) 퐸퐼 = √푛푉 sin 휃푉 , (25) 휎2 = 1(1 − 푉2 + 푊 cos 휃 ), (c) 푅 2 푊 휎2 = 1(1 − 푉2 − 푊 cos 휃 ), (d) 퐼 2 푊 퐶 = 1푊 sin 휃 . (e) 푅퐼 2 푊 Note, that the structure of Eq.(25) can be generalized beyond the model of random phasor sum. Expressions for the moments, derived directly from the diffraction integrals [11, 15], have the same structure, although 푽 and 푾 have a physical meaning other the mean and the variance of 푒푗휑. In this sense, the results of the next sub-section are not limited to the framework of random phasor sum model. We relate the parameters of the ellipse to 푽 and 푾. As the center of the ellipse is located at {푅 = 퐸푅, 퐼 = 퐸퐼} in the (푅, 퐼)-system of coordinates, the first two parameters come directly from Eq.(25 a, b):

(26) 휌 = √푛푉, Fig. 2. Two-dimensional probability density cloud in different systems 휃휌 = 휃푉. of coordinates: (푅, 퐼) – initial system of coordinates, (푥′, 푦′) - rotated by the angle 휃 , (푥, 푦) – related to the ellipse center and the semi-axes. The total size is obtain by summing Eq.(25 c) and Eq.(25 d): 푇 2 2 2 To conclude this Section we present equation of the ellipse in the 훴 = 휎푅 + 휎퐼 = 1 − 푉 , (27) canonic form through the geometric parameters. Consider a system of from where we conclude that the distance to the center of the ellipse coordinates (푥′, 푦′) which is obtained from (푅, 퐼)-system of and its total size are related: coordinates by rotation by the angle 훩 = 휃푊⁄2 (Figure 2). Its center coincides with the center of the system (푅, 퐼) and the axes are parallel to the axes of the system (푥, 푦), i. e. are parallel to the axes of the independent processes. The third included in ellipse. The coordinates transformation is skewness depends only on 휏 and its sign is determined by 휀. 휃 휃 푥′ = 푅 cos 푊 − 퐼 sin 푊 , 2 2 휃푊 휃푊 (33) 푦′ = 푅 sin + 퐼 cos . 2 2 Introducing the polar coordinates (푅 = 퐴 cos 휃퐴 , 퐼 = 퐴 sin 휃퐴) and performing the trigonometric combining, we obtain: 푥′ = 퐴 cos 휃′ , (34) 푦′ = 퐴 sin 휃′ , ′ where 휃 = 휃퐴 + 휃푊⁄2. On the other hand, the system of coordinates (푥′, 푦′) relates to the system of coordinate (푥, 푦) by a parallel shift:

′ (35) 푥 = 푥 − 휌 cos 휃푇 , ′ 푦 = 푦 − 휌 sin 휃푇 . The equation of the ellipse becomes

2(퐴 cos 휃′ − 휌 cos 휃 )2 2(퐴 sin 휃′ − 휌 sin 휃 )2 (36) 푇 + 푇 = 1. 훴(1 + 푒) 훴(1 − 푒)

The inclination angle 휃푇 is given by Eq.(32) and 휌 is given by Eq. (28). Eq.(36) describes the same curve as Eq. (12), but written in polar coordinates through the geometric parameters of the cloud. We will come back to this result in Section 4.

3. EXAMPLE Fig. 3. Family of ellipses showing the geometry of the 2D-PDF for exponentially modified normal phase distribution. Each ellipse corresponds to certain combination (휏, 휎). Rays are the lines of A. Exponentially modified normal P-PDF constant 휏 = 0, 0.12, 0.3, 0.5, 0.7, 1, 1.5. Circles are the lines of constant Suppose that phase distortions are governed by two independent 휎 = 0, 0.4, 0.6, 0.8, 1, 1.2, 1.5 (from the outer circle inwards). Upper random processes: the first process is normally distributed with zero half-plane: positively skewed P-PDF (휀 = 1), lower half-plane: mean and standard deviation 휎 ≥ 0: negatively skewed P-PDF (휀 = −1). Arrows indicate orientation of ellipses. The angle between an arrow and a ray is inclination angle. The 2 (37) 1 휑 number of phasor 푛 = 500. 푝1(휑) = exp (− ) , √2휋휎2 2휎2 To describe the geometric properties of the cloud we need to know and the second process is exponentially distributed with 휏 ≥ 0: only two values of the characteristic function. Expression for a 1 휀휑 (38) characteristic function is often much easier to obtain than expression exp (− ) , 휀휑 ≥ 0, for a PDF because the characteristic function of a sum of independent 푝2(휑) = {휏 휏 0, 휀휑 < 0, random processes is the product of the characteristic functions. For the exponentially modified normal distribution the characteristic function 휀 = ±1. The resultant phase is a sum of these two processes and is: follows exponentially modified normal distribution law. From the basic theory of random process, we know that the residual probability 푘2휎2 (41) 푴 (푘) = (1 − 푗푘휀휏)−1exp (− ) . density function is a convolution of two initial probability density 휑 2 functions. Although for our purpose it is not necessary to know the P- PDF itself, for the completeness we write it down. The exponentially The complex values we need to calculate are modified normal P-PDF has the following shape: 2 (42) −1 휎 2 2 (39) 푽 = (1 − 푗휀휏) exp (− ) , 1 1 휎 1 휎 2 푝(휑) = exp [ ( − 2휀휑)] erfc [ ( − 휀휑)] , 2휏 2휏 휏 √2휎2 휏 푾 = (1 − 푗2휀휏)−1exp(−2휎2) − (1 − 푗휀휏)−2exp(−휎2). where erfc(… ) is complementary error function and parameter 휀 In order to extract the parameters of the cloud some algebraic determines the direction of the modification. When 휀 = 1 the P-PDF is manipulations with complex variables must be performed. It can be positively skewed, when 휀 = −1 the P-PDF is negatively skewed. done using any symbolic manipulation software. We omit the details of The first three central moments of the phase are this step and present in the next Sub-section only the result.

〈휑〉 = 휀휏, (40) B. Parameters of the 2D-PDF 〈(휑 − 〈휑〉)2〉 = 휎2 + 휏2, We remain in the domain of Gaussian values and there is no surprise 〈( 〈 〉)3〉 3 휑 − 휑 = 2휀휏 . that 푽, 푾 and the geometric parameters of the cloud are some The zero mean of the initial normal distribution is shifted by the functions of means and variances of two initial random processes, i.e. exponential process in the direction defined by 휀. The variance is the of 휏, 휏2 and 휎2. To present these functions in a compact form we sum of the two variances as we adopted the assumption of introduce four simple expressions:

2 2 1 2 푓 = (1 + 휏 ) , − 휎 1 푉 = 푓 4 exp (− ) , 푓 = 1 + 4휏2, 1 2 (43) 2 (44) 2 푓3 = 1 − 휏 , 휃푉 = atan(휀휏) . 2 1 푓4 = 1 + 3휏 . − 1 1 4 2 − 2 − Note, that 푓 = (1 + 휏 ) 2 = (1 + tan 휃 ) 2 = cos 휃 and All 푓 equal one when 휏 = 0, i.e. when the exponentially distributed 1 푉 푉 푖 2 second process is absent and phase is a Gaussian variable with zero 푉 = exp(− 휎 ⁄2) cos 휃푉. To find the cloud center position (휌, 휃휌) mean. we substitute Eq.(44) into Eq.(26). When 휏 = 0 the angular position of the center is zero. The cloud is 2 centered on the real axis at the distance 휌0 = √푛 exp(− 휎 ⁄2) from the center of coordinates. For small 휏, when the contribution of the second process is weak, the angular position equals to the phase mean value: 휃휌 ≈ 휀휏 = 〈휑〉. When 휀휏 changes but 휎 remains constant, the center of the cloud moves along a circular line, connecting the point (휌0, 0) and the center of coordinates(Figure 3): anticlockwise if the P-PDF is positively skewed (휀 = 1) or clockwise if the P-PDF is negatively skewed (휀 = −1). The circular shape of the line can be deduced from Eq. (26) and Eq. (44): 휌 = 휌0 cos 휃휌. When, on the contrary, 휎 increases but 휀휏 remains constant, the center of the cloud moves along a ray making angle 휃휌 with the abscissa inwards. While moving along a ray or a circle towards the center of coordinates the cloud expands, because increasing 휏 or 휎 means that the P-PDF widens approaching a uniform distribution. The spread of the cloud is characterized by the total size which in this example is 훴 = 1 − 푉2 = 1 − exp(−휎2) cos2 휃 (45) 푉 ≈ 1 − exp(−휎2) + 휏2exp(−휎2). The last approximation is given for small 휏. Now we study orientation of the ellipse and its eccentricity. For this we need to calculate 푊 and 휃푊. The real and the imaginary parts of 푾 are

푓 exp(−휎2) − 푓 푓 Re 푾 = exp(−휎2) 1 2 3 , 푓 푓 1 2 (46) 푓 exp(−휎2) − 푓 Im 푾 = 2휀휏exp(−휎2) 1 2 . 푓1푓2 When 휏 = 0 the imaginary part is zero, but the real part is negative and therefore the angle 휃푊 is 휋. If 휀 = 1 and 휏 is sufficiently small both, Re 푾 and Im 푾, are small negative values. In order to be consistent with the definition of angles presented in Figure 1 and to avoid uncertainty of atan(… ) we define a piecewise function:

휋 − atan 푇푊 , 휏 ≤ 휏0,

휃푊 = {− atan 푇푊 , 휏 > 휏0, 휀 = 1, (47)

− atan 푇푊 + 2휋 , 휏 > 휏0, 휀 = −1, where

Im 푾 푓 exp(−휎2) − 푓 (48) 1 2 휃푇 = = 2휀휏 2 Re 푾 푓1 exp(−휎 ) − 푓2푓3

and 휏0 is a point when Re 푾 = 0, i.e. a positive root of quadratic with 2 2 respect to 휏 equation: 푓1 exp(−휎 ) − 푓2푓3 = 0 . In Figure 3 an arrow indicated the orientation of an ellipse. There Fig. 4. Inclination angle (a) and eccentricity (b) as functions of ellipse exists certain combination (휏, 휎) for which the cloud is oriented center: (휏, 휎) – mapping. The number of phasor 푛 = 500. horizontally. This combination satisfies the condition 휃푊 = 0 or 휃 = 휃 for the upper half-plane and 휃 = 2휋 or 휃 = 휋 − 휃 for First, we study the location of the center and the size of the cloud. 푇 푉 푊 푇 푉 the lower half-plane. The combination satisfies the condition when For this we need to find 푉 and 휃 : 푉 Im 푾 = 0, and can be found as the positive root of quadratic, with 2 2 respect to 휏 equation: 푓1exp(−휎 ) − 푓2 = 0. The inclination angle 휃푇 is calculated according to Eq. (32). When the contribution of the second process is weak (휏 is small) the statistics of the first random value does not change), while  increases inclination angle can be estimated by the following approximation: from 0 to 10. To compare with the theory we plotted an analytical ellipse given by Eq.(36) for every 휏 and the circle 휌 = 21.9 cos 휃 . The 휋 휀휏3 (49) 휌 behavior is as it is described above: when 휏 increases the ellipse moves 휃푇 ≈ + 2 2 exp(−휎 ) − 1 along the circle towards zero, widens, rounds and rotates. It starts from

When 휏 = 0, the cloud is vertical (휃푇 = 휋⁄2). The deviation of the the vertical orientation and rotates slower than the tangential to the inclination angle from 휋⁄2 is proportional to 휀휏3. In this term we circle, therefore it assumes the horizontal orientation not at 휏 = 1, but recognize the third moment of 휑 (Eq. (40)). Although the equation later at 휏ℎ = 1.48. above cannot serve as a mathematical proof of connection between the deviation of the inclination angle from 휋⁄2 and an asymmetry of the P- PDF, it gives a feeling of this existing connection. In Figure 3, 휃푇 is the angle between an arrow and a ray. Eccentricity is computed according to Eq.(30) and after some algebraic manipulations becomes:

(50) √푓 exp(−2휎2) − 2푓 exp(−휎2) + 푓 푒 = exp(−휎2) 1 4 2 2 √푓1푓2 − √푓2 exp(−휎 )

In absence of the second process, i.e. when 휏 = 0 and 휃휌 = 0 the eccentricity equals 푒 = exp(−휎2) = 1 − 훴: the vertical cloud with its center on the abscissa has a maximal elongation and a minimal spread. For small 휏 the following approximation is valid: 푒 ≈ exp(−휎2) − 휏exp(−휎2), and the relation 푒 = 1 − 훴 is preserved. With an increase of 휏 it is not anymore the case. Figure 3 shows the set of 2D-PDFs plotted with use of Eq.(36) on a grid of constant 휏 (rays) versus constant휎(circles). Crossing point between a ray and a circle is a center of an ellipse. The center can be located only within the largest circle with diameter √푛 (boundary circle, 휎 = 0). Every point on the (푅, 퐼)-plane within the boundary circle is a center of an ellipse and uniquely corresponds to cetain pare (휏, 휎). So we speak about (휏, 휎)-mapping:

Fig. 5. Two-dimensional PDF of complex amplitude for exponentially 휎2 1 푅 = √푛 exp (− ) , modified normal phase distribution. The number of phasors equals 2 1 + 휏2 (51) 500; 휎 equals 0.2 rad. The theoretical ellipses are plotted in solid lines. 휎2 휀휏 The results of the simulation for the same set of parameters are shown 퐼 = √푛 exp (− ) . 2 1 + 휏2 in dots. The pare (휏, 휎) gives a values for ellipse parameter. This allows plotting this parameter on the (푅, 퐼)-plane as a two-dimensional 4. STATISTICS OF INTENSITY function. Figure 4 shows the inclinations angle and the eccentricity in The main subject of this paper is joint distribution of the real and the this representation. Not only the geometric parameters of ellipse but imaginary parts of the complex amplitude, which we refer to as also all values derived from them can be presented on the (푅, 퐼)-plane probability cloud or 2D-PDF. Although we wish to keep the focus on through (휏, 휎)-mapping. Figure 6 in Section 4 illustrates it on example the geometry of the cloud itself, it is important to show how the of intensity standard deviation. knowledge of geometry helps to predict the statistical behavior of the measurable quantity, i.e. intensity. The standard procedure is as C. Numerical simulation following. First step is finding the joint distribution of modulus and We simulate the random phasor sum with MATLAB R2014. A phase angle of the complex amplitude by moving to the polar coordinates value is generated as a sum of two random numbers: the first one with (퐴, 휃퐴) from the mutual distribution of the real and the imaginary the normal distribution 푁(0, 휎) and the second one with the parts. Second step is passing to a marginal statistics 푝퐴(퐴) by exponential distribution 퐸(휏). The phase is used to generate a integrating over 휃퐴. The final step is obtaining a distribution of complex phasor with unit length and the random phasor sum contains intensity by substituting = √퐼푠. (We use notation 퐼푠 to distinguish 500 independent phasors. The case 휀 = −1 is simulated by taking the intensity from the imaginary part of the complex amplitude.) Except phase value as a difference between 푁(0, 휎)-distributed number and for a small number of cases, the integration over 휃퐴 is either insoluble 퐸(휏)-distributed number. Every is plotted as a point on the analytically or the result is complicated and inapplicable. For example, (푅, 퐼)-plane. For every combination (휏, 휎) up to 600 points are Equation 9 from [4] gives the probability of intensity through an generated to create a cloud. The smaller clouds are created with the infinite sum of products modified Bessel functions; and the sum is smaller number of points. We have chosen this simple way to generate multiplied by an exponential function. It is a very neat formula, but to the complex amplitude and not the full Fourier optics propagation to our knowledge no mathematical software can process it. avoid complications related to generation of rough surface with given The representation of the cloud through its geometric parameters statistics. In other words, we deal here only with the second part of the allows obtaining the statistics of intensity without passing through the optical task discussed in Section 2A. painful integration over the angle. To avoid overloading this paper, we Figure 5 shows the simulated clouds when 휎 is fixed to 0.2 (the give only some clues of the procedure leading to the result. We plan to present more details in the next paper. In Section 2, Figure 2, we introduced the coordinates (푥′, 푦′). The 2D-PDF written in these coordinates

푝(푥′, 푦′) = ′ 2 2 1 (푥 − 휇 ) 1 (푦′ − 휇푦) exp [− 푥 ] exp [− ] (52) 2 2휎2 2휎2 √2휋휎푥 푥 2 푦 √2휋휎푦 describes the join distribution of two independent normally distributed variables, 푥′ and 푦′ with the means 휇푥 = 휌 cos 휃푇 and 2 2 휇푦 = 휌 sin 휃푇 and the variances 휎푥 and 휎푦 . Eq.(52) is obtained by recognizing that Eq.(36) and Eq.(12) describe the same curve and that determinant 퐷푐 is invariant to any linear transformation of coordinates. These variables 푥′ and 푦′ are related to the intensity as

(53) 푥′ = √퐼 cos 휃′ , 푠 푦′ = √퐼푠 sin 휃′ . The intensity is a sum of squares of two independent normally distributed random variables with known means and variances:

2 2 (54) 퐼푠 = 푥′ + 푦′ . ′ We do not need to integrate over 휃 . Fig. 6. Intensity standard deviation through (휏, 휎)-mapping. The Now we can follow the standard methods of statistics to obtain a number of phasors 푛 = 500. moment generating function for intensity. We postpone the details of this calculation until the next paper and present only the outcone: 5. CONCLUSION 1 (55) 훯(푡) = Mutual distribution of the real and the imaginary parts of complex √1 − 2푡 + 푡2(1 − 푒2) amplitude – the resultant of a large number of vibrations, is best 휌2푡 1 − 푡(1 − 푒 cos 2휃 ) visualized as a probability cloud. The cloud is a powerful tool to × exp [ ∙ 푇 ] . 훴 1 − 2푡 + 푡2(1 − 푒2) describe statistics of light with phase fluctuation. The geometric parameters of the cloud: its position, extension, elongation and Due to the rotational symmetry (intensity is insensitive to the mean orientation, – are directly related to statistics of phase and involve only phase) the angular position of ellipse 휃휌 is not present in this result. two values of phase characteristic function (which in general are The moments of intensity are the partial derivatives of this function: complex values). There are common rules applied to the behavior of 휕푛훯(푡) (56) these parameters independently of a particular phase distribution: the 〈퐼푛〉 = 훴푛 | . cloud always widens when its center approaches the center of 휕푡푛 푡=0 coordinates; deviation of the inclination angle from 휋⁄2 indicates and yield in polynomial expressions of 휌, 훴, 푒 and cos 2휃푇. These certain asymmetry in initial phase distribution; eccentricity tends to values, as we have studied in the paper, are linked to statistics of phase. one when the cloud shrinks into a point. The demonstration of the last 푛 푛 rule (not presented in the paper) involves expanding 푽 in series at The central moments ℳ퐼 = 〈(퐼 − 〈퐼〉) 〉 are therefore also some polynomials of the same values. We present here the expressions for 훴 = 0. the first three: We illustrate the method of geometrical description on example of exponentially modified normal distribution and introduce the principle 1 2 ℳ퐼 = 휌 + 훴, of (휏, 휎) -mapping. This principle is based on the fact that there is a 2 [ 2( ) ( 2)] one-to-one correspondence between the vector (휏, 휎) and a vector ℳ퐼 = 훴 2휌 1 + 푒 cos 2휃푇 + 훴 1 + 푒 , (57) 3 2 2 2 2 pointing to ellipse center. If for another phase distribution there is no ℳ퐼 = 2훴 [3휌 (1 + 2푒 cos 2휃푇 + 푒 ) + 훴(1 + 3푒 )]. unique relation between parameters of the P-PDF and a point on the 2 complex plane, it nevertheless can be found by combining the Figure 6 shows intensity standard deviation 휎푠 = √ℳ퐼 for exponentially modified normal P-PDF from the previous section parameters. In the considered example of exponentially modified normal distribution we assumed that the mean of normally distributed through (휏, 휎) -mapping. This function has two local minima: the first component is zero. If it is not zero but equals certain 휇, the mapping is one is located at the center of coordinates and the second one – at the still possible through combined parameter 휏′ = 휀휏 + 휇. opposite extreme of the boundary circle. The first minimum (휎 = 1) 푠 The geometrical representation of the two-dimensional probability corresponds to the case studied by Lord Rayleigh of well-developed density function of complex amplitude allows for obtaining the speckles. The second minimum (휎 = 0) corresponds to the situation 푠 moment-generating function of intensity in a closed form and hence (also discussed in his paper) when there is no randomness whatsoever expressions for all statistical moments of intensity through the and all phasors are identical. Between these two minima there is a parameters of ellipse. All central moments of intensity are positive ridge along which 휎 is maximal. 푠 regardless of the statistics of phase. Nevertheless, simulations and experimental data [16] give negative values for some odd moments of intensity or the values based on these moments (skewness, for example). This discrepancy is not due to an error in our calculation, but to limitations of the postulate lying in its base: assuming a large 13. N. Yaitskova, K. Dohlen, and P. Dierickx, “Analytical study of diffraction number of phasor, applying the central limit theorem and writing the effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 2D-PDF as two-dimensional Gaussian function we enforce an elliptical 1563–1575 (2003) shape of the cloud. The odd moments of intensity are sensitive to cloud 14. H. Escamilla and E. Mendez, “Speckle statistics from gamma-distributed non-ellipticity, i.e. to the number of phasors, and therefore cannot be random-phase screens,” JOSAA 8, 1929-1935, (1991) estimated within the frame of the central limit theorem. 15. J. Uozumi and T. Asakura, “Statisticsl properties of laser speckle produced Continuation of this work has three directions. First task is to relate in the diffraction field,” Applied Optics 16, 1742-1753, (1977) the number of phasors (푛) to the second-order statistics 16. S. Gladysz, J. Christou, M. Redfern, "Characterization of the Lick adaptive (autocorrelation function) of phase. Our preliminary calculations show optics point spread function", Proc. SPIE 62720, (2006) that assigning diameter of a zone to a coherence length yields in 17. J. Uozumi and T. Asakura, “The first-order statistics of partially developed unrealistically small 푛. However, this result requires a more thorough non-Gaussian speckle patterns” J. Opt. 12 177-186 (1981) investigation. Second task is to unfold Section 4 of the present paper: having obtained the general expression for moment generating function we must study this expression in detail. In particular, we shall see if it gives the correct result for the known intensity statistics included in it: exponential decay, Rician and modified Rayleigh distributions. Third task is to explore the domain of non-Gaussian complex random variables and try to find not only expressions for intensity moments by means of combinatorics (these expressions have been obtained in [17]), but the shape of the 2D-PDF. One way to do this is to assume 푛 to be large but not infinite and retain the second term in Euler’s theorem.

Acknowledgement. I wish to thank my colleagues: Szymon Gładysz from Fraunhofer Institute for Optronics and Jérôme Paufique from European Organisation for Astronomical Research in the Southern Hemisphere, whose constructive remarks and corrections helped improving the quality of the paper.

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