2.3. Interference of multiple waves with random amplitudes and phases

The creation of laser and discovery of the speckle structure of scattered radiation were immediately followed by a number of theoretical papers on statistical properties of speckles in free space [1] and in the image area of scattering surface [2]. Figure 1 presents a typical speckle pattern observed in the image plane of a rough surface. The picture was obtained by the authors of this manuscript during its preparation. In this section, we discuss some main features of speckle fields obtained by Goodman [3, 4] on a simple model describing interference of multiple waves.

Figure 1. Speckles in the image plane of a rough surface.

Iðx, y, zÞ ¼ lim

2.2. Interference of two waves

I ¼ A×A� ¼

h A0e ið k ! 1 r

Eq. (4), we have:

106 Optical Interferometry

where <sup>I</sup><sup>0</sup> <sup>¼</sup> <sup>A</sup><sup>2</sup>

point r

(Eq. (6)):

Δ ¼ λ.

contrast is less than 1.

T!∞ 1 T ð T=2

!þ<sup>ϕ</sup>1<sup>Þ</sup> <sup>þ</sup> <sup>A</sup>0<sup>e</sup>

¼ 2I<sup>0</sup> þ 2I<sup>0</sup> cos

the interferometer to "endless" band. If angle θ between vectors k

ið k ! 2 r !þ<sup>ϕ</sup>2<sup>Þ</sup> i × h A0e −ið k ! 1 r

h ð k ! <sup>1</sup>− k ! <sup>2</sup>Þ r

−T=2

<sup>j</sup>Eðx,y, <sup>z</sup>,tÞj<sup>2</sup>

Thus, the radiation intensity at a point of space equals the squared complex amplitude module.

Let us discuss the light intensity distribution in superposition of two monochrome waves. Suppose that two waves of the same length λ were emitted by one point source in various directions, then two plane waves 1 and 2 crossing at angle θ were shaped by the optical systems. Let us take some point in the area of beam superposition. For certainty, let the wave amplitudes be the same equaling A0, but their initial phases ϕ differ. In compliance with

dt ¼ jAðx,y, <sup>z</sup>Þj<sup>2</sup>

!þ<sup>ϕ</sup>1<sup>Þ</sup> <sup>þ</sup> <sup>A</sup>0<sup>e</sup>

! <sup>1</sup> and k !

i

! <sup>þ</sup> <sup>ϕ</sup><sup>1</sup> <sup>þ</sup> <sup>ϕ</sup><sup>2</sup>

<sup>0</sup>. Eq. (5) describes periodic light intensity distribution in the neighborhood of

!, which was called light interference by T. Jung. Elementary calculations can demonstrate [17, 18] that minimum distance Δ between neighboring intensity maximums or minimums called bandwidth or period of the interference fringes is determined by formula

> <sup>Δ</sup> <sup>¼</sup> <sup>λ</sup> 2 sin <sup>θ</sup> 2

It follows from Eq. (6) that if θ tends to zero, Δ tends to infinity, which corresponds to tuning of

Δ ¼ λ=2, which corresponds to the wave interference in colliding beams. For θ = 60° value,

Now let us discuss the contrast of the interference fringes γ introduced by Michelson and determined by formula γ ¼ ðImax−IminÞ=ðImax þ IminÞ, where Imin and I max are the minimum and the maximum intensity values, respectively. From Eq. (5), it follows that in the case of a point light source discussed here and constant wavelength λ, contrast γ ¼ 1. Experience shows that if the light source is not point and (or) it emits light in some wavelength interval, the fringes

It is commonly believed that case γ ¼ 0 corresponds to completely incoherent light; if 0 < γ < 1, the light is partially coherent, and coherence is the ability of waves to interfere. Interference of partially coherent light can be studied in Ref. [19]. In the text below, we will suppose that the light waves discussed here are completely coherent, i.e., two waves of the

−ið k ! 2 r !þ<sup>ϕ</sup>2<sup>Þ</sup> i

: (6)

, (5)

<sup>2</sup> equals 180°, then

: (4)

According to Goodman's model, the waves that arrived at an arbitrary point of the free space from elementary areas of surface can be regarded as plane monochrome waves with random amplitudes aj= ffiffiffiffi N <sup>p</sup> and phases <sup>ϕ</sup><sup>j</sup> , where j is the wave number, j ¼ 1, 2, … N. It was supposed that the amplitude and phase of the same wave and the amplitudes and phases of different waves are independent, and the values of a<sup>2</sup> <sup>j</sup> averaged by the object ensemble are nonzero. It was considered that phases ϕ<sup>j</sup> were homogeneously distributed in the area from −π to þπ. The presence of the object ensemble means the presence of numerous macroscopically identical scattering objects, each object generating N plane and monochrome waves with random amplitudes and phases. Any value averaged by the object ensemble is found by means of fixation for every object of the ensemble with subsequent calculation of its mean value. In the text below, we will denote the ensembleaverage with angular parenthesis. Note that mathematically, object ensemble-averaged value of some function f of random arguments x1, x2, …, xm, is determined in the following way:

$$
\langle f(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_m) \rangle = \int\_{-\infty}^{+\infty} \dots \int\_{-\infty}^{+\infty} f(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_m) \rho(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_m) d\mathbf{x}\_1 d\mathbf{x}\_2 \dots d\mathbf{x}\_m,\tag{7}
$$

where ρð x1, x2, …, xmÞ is joint probability density of random values x1, x2, …, xm. If these values are independent, ρðx1, x2, …, xmÞ ¼ ρðx1Þρðx2Þ…ρðxmÞ, and the calculation of integral (Eq. (7)) may simplify substantially.

If all the waves are linearly polarized in the same mode, then, according to J. Goodman, the following ratio holds for total complex amplitude A at some point q ! of the free space:

$$A = ae^{-i\theta} = A^r + iA^i = \frac{1}{\sqrt{N}} \sum\_{j=1}^{N} a\_j e^{i\phi\_j},\tag{8}$$

where A<sup>r</sup> and Ai are real and imaginary parts of the total complex amplitude, respectively. Eqs. (9)–(11) obtained by Goodman on the basis of the discussed model that characterize the statistical properties of speckles are cited as follows:

$$
\langle A^r \rangle = \langle A^i \rangle = \langle A^r A^i \rangle = 0,\tag{9}
$$

$$
\langle A^\prime A^\prime \rangle = \langle A^i A^i \rangle = \frac{1}{2N} \sum\_{j=1}^N \langle a\_j^2 \rangle,\tag{10}
$$

$$\rho(A^r A^i) = \frac{1}{2\pi\sigma^2} e^{-\frac{(A^r)^2 + (A^i)^2}{2\sigma^2}},\tag{11}$$

$$\rho(\mathbf{I}, \boldsymbol{\Theta}) = \rho(\mathbf{I}) \rtimes \rho(\boldsymbol{\Theta}) = \frac{1}{\langle \mathbf{I} \rangle} e^{\frac{i}{\langle \boldsymbol{\theta} \rangle} \mathbf{x}} \frac{1}{2\pi} \text{, } \mathbf{I} \boldsymbol{\varepsilon} \mathbf{0} \text{, } -\pi \boldsymbol{\varepsilon} \boldsymbol{\Theta} \boldsymbol{\varepsilon} \pi \text{, } \boldsymbol{\Delta} \sigma^2 = \langle \mathbf{I} \rangle,\tag{12}$$

$$
\langle I^{\boldsymbol{n}} \rangle = \boldsymbol{n}! \langle I \rangle^{\boldsymbol{n}}.\tag{13}
$$

From Eqs. (9)–(13), it follows that at an arbitrary point of a free field, the real and imaginary parts of total complex amplitudes are independent, uncorrelated, and distributed according to the Gauss' law. Radiation intensity I and resulting phase θ are independent, value θ is homogeneously distributed in the range from −π to þπ. From Eq. (12), it follows that probability PI exceeding some threshold equal to I by the light intensity that is given by Eq. (14):

$$P\_I = e^{\frac{\Delta}{|\gamma|}} \tag{14}$$

Thus, in a speckle field, the most probable intensity value is value I equal to zero. With increasing intensity, the probability of its detection decreases exponentially. From Eq. (13), it also follows that speckle contrast С equal to the ratio of mean square deviation intensity to the mean intensity equals 1.

Experience shows that experimental dependence PIðIÞ agrees well with theoretical dependence Eq. (14) for scattering surfaces that lack the mirror constituent of scattered radiation, and whose height of heterogeneity of the surface relief is comparable with wavelength λ. The statistical properties of the speckles corresponding to other models of rough surface can be studied, for example, in Refs. [20, 21].

〈fðx1, x2, …, xmÞ〉 ¼

may simplify substantially.

108 Optical Interferometry

mean intensity equals 1.

þ ð∞

−∞ … þ ð∞

following ratio holds for total complex amplitude A at some point q

〈A<sup>r</sup>

<sup>ρ</sup>ðA<sup>r</sup> Ai Þ ¼ <sup>1</sup> <sup>2</sup>πσ<sup>2</sup> <sup>e</sup> − <sup>ð</sup>ArÞ2þðAiÞ<sup>2</sup>

> 〈I〉 e − I 〈I〉<sup>×</sup> <sup>1</sup>

> > 〈I n

exceeding some threshold equal to I by the light intensity that is given by Eq. (14):

〉 ¼ n!〈I〉

From Eqs. (9)–(13), it follows that at an arbitrary point of a free field, the real and imaginary parts of total complex amplitudes are independent, uncorrelated, and distributed according to the Gauss' law. Radiation intensity I and resulting phase θ are independent, value θ is homogeneously distributed in the range from −π to þπ. From Eq. (12), it follows that probability PI

> PI ¼ e − I

Thus, in a speckle field, the most probable intensity value is value I equal to zero. With increasing intensity, the probability of its detection decreases exponentially. From Eq. (13), it also follows that speckle contrast С equal to the ratio of mean square deviation intensity to the

Experience shows that experimental dependence PIðIÞ agrees well with theoretical dependence Eq. (14) for scattering surfaces that lack the mirror constituent of scattered radiation, and whose height of heterogeneity of the surface relief is comparable with wavelength λ. The

〈A<sup>r</sup> Ar 〉 <sup>¼</sup> 〈A<sup>i</sup> Ai 〉 <sup>¼</sup> <sup>1</sup> 2N X N

statistical properties of speckles are cited as follows:

<sup>ρ</sup>ðI, <sup>θ</sup>Þ ¼ <sup>ρ</sup>ðIÞ×ρðθÞ ¼ <sup>1</sup>

−∞

where ρð x1, x2, …, xmÞ is joint probability density of random values x1, x2, …, xm. If these values are independent, ρðx1, x2, …, xmÞ ¼ ρðx1Þρðx2Þ…ρðxmÞ, and the calculation of integral (Eq. (7))

If all the waves are linearly polarized in the same mode, then, according to J. Goodman, the

where A<sup>r</sup> and Ai are real and imaginary parts of the total complex amplitude, respectively. Eqs. (9)–(11) obtained by Goodman on the basis of the discussed model that characterize the

> 〉 <sup>¼</sup> 〈Ar Ai

ffiffiffiffi N <sup>p</sup> <sup>X</sup> N

> j¼1 〈a2

j¼1 aje

<sup>A</sup> <sup>¼</sup> ae<sup>−</sup>i<sup>θ</sup> <sup>¼</sup> <sup>A</sup><sup>r</sup> <sup>þ</sup> iAi <sup>¼</sup> <sup>1</sup>

〉 <sup>¼</sup> 〈A<sup>i</sup>

fðx1, x2, …, xmÞρðx1, x2, …, xmÞdx1dx2…dxm, (7)

! of the free space:

〉 ¼ 0, (9)

<sup>2</sup>σ<sup>2</sup> , (11)

<sup>n</sup>: (13)

〈I〉 (14)

<sup>2</sup><sup>π</sup> , <sup>I</sup>≥<sup>0</sup> , <sup>−</sup><sup>π</sup> <sup>≤</sup> <sup>θ</sup> <sup>≤</sup> <sup>π</sup> , <sup>2</sup>σ<sup>2</sup> <sup>¼</sup> 〈I〉, (12)

<sup>j</sup> 〉, (10)

<sup>i</sup>ϕ<sup>j</sup> , (8)

The model proposed by Goodman was further developed in Ref. [22] to obtain the formula allowing determination of three-dimensional speckle sizes. It was supposed that point scattering centers were located in some three-dimensional area transparent for radiation. A formula was obtained that allowed determination of three-dimensional speckle sizes for an area of an arbitrary shape with random location of the radiation source, the object, and the observation site by the width of a spatial autocorrelation function of intensity of scattered radiation. The formulas determining the transverse and longitudinal speckle sizes for two objects of a simple shape are given below. Let us examine a transparent area shaped like a right-angle parallelepiped of size 2X and 2Y on ох and оу axes, respectively, and of size 2Z on oz axis. Point scatterers are located within the area. Let the coordinate origin be located in the center of the area. Then, if the direction of illumination is arbitrary at distance ρ<sup>q</sup> on oz axis, speckles with minimum sizes Δ~qx, Δ~qy, Δ~qz are generated as xd, yd, and zd, respectively:

$$
\Delta \tilde{q}\_x = \frac{\lambda \rho\_q}{2X}, \ \Delta \tilde{q}\_y = \frac{\lambda \rho\_q}{2Y}, \ \Delta \tilde{q}\_z = \frac{5\lambda \rho\_q^2}{\pi (X^4 + Y^4)^{1/2}}.\tag{15}
$$

If the object is cylindrical, axis oz coincides with the axis of the cylinder, and the coordinate origin is located in the center of the object, then in similar observation and illuminating conditions

$$
\Delta \ddot{r} = 1.22 \frac{\lambda \rho\_q}{D}, \ \Delta \ddot{q}\_z = 2 \frac{\lambda \rho\_q^2}{R^2}, \ \Delta r = \left(\Delta q\_x^2 + \Delta q\_y^2\right)^{1/2}, \tag{16}
$$

where R is the radius of the cylinder, D =2R. In the literature [5, 23], it was shown that the mean xd, yd, and zd speckle sizes are threefold compared with the minimum.

In the preceding text above, we confined ourselves to the main features of speckle fields in a free field. The speckles generated in the scattering image plane have very similar properties. For the speckles in the area of the object images in Eqs. (15) and (16), values 2X, 2Y, and R equal the size of a diaphragm of the relevant shape located near the lens.
