3. Theory of dynamic speckle interferometry of thin phase objects

To study various properties of transparent objects, a variety of interference, shadow and speckle techniques are conventionally used [24–26]. As a rule, these techniques are oriented toward the analysis of macroscopic processes. With that, the logic of advancement in speckle optics and the practical needs pose the problem of studying microscopic processes occurring at the structure level. In particular, in biology, the problem of therapeutic drug management poses the problem of studying the processes in cells and their membranes. As at the structure level, the properties of biological media are random, when waves pass through various parts of the cell, their phase randomly varies in time. Therefore, a variation of the radiation intensity at the observation point is also a random process. The complexity of theoretical analysis of similar phenomena is that in the general case, there is necessity for dispersive ratios characterizing the wave phase variation in space and time.

In practice, there is an option when random values of the wave phases varying in space and time are independent. In particular, this option may be generated if the area of spatial correlation of a physical value causing the wave phase variations is less than the wavelength of light λ. In this case, the solution to the problem of establishing a relation between the wave phase dynamics in a thin transparent (phase) object and the dynamics of the light intensity in its image plane simplifies considerably. The solutions to this problem attempted for studying the properties of particular objects are found in the literature [27, 28]. In the para below, we are giving the general solution to the problem obtained by the authors of this paper.

### 3.1. Model of the object

At the first stage, the aim of the theoretical analysis is to obtain the expression for radiation intensity Ið q !Þ at some point <sup>q</sup> ! of the observation plane, and then for temporal autocorrelation function of a random process I ¼ IðtÞ. We will obtain the expression for value Ið q !Þ using the model of a three-dimensional diffuser published in the abovementioned paper [22]. Let a point source of coherent radiation with wavelength λ located at point 1 illuminate point scattering centers randomly located in thin diffuser 2 near (хоу) plane as is shown in Figure 2. Let the position of the point source be given by radius vector s !. To simplify the transformations, let us admit that the refraction indexes of the medium inside and outside the diffuser are the same and equal 1. At distance L<sup>0</sup> from (хоу) plane, in plane (βx0βy) there is a thin lens with focal distance f and diaphragm diameter D. Planes (хоу) and (qх0qу) are conjugate. We consider all the waves discussed linearly polarized in the same direction. Let us admit that phase ϕ<sup>j</sup> of the wave scattered by the jth center is random, and the waves from all the scattering centers arrive at an arbitrary point of (βx0βy) plane.

Figure 2. Optical system taken in the theory: (1) light source, (2) diffuser, (3) thin transparent object, (4) the lens with diaphragm, (5) the image plane, (6,7) conjugate points.

Let thin phase object 3 whose refraction index varies in time (Figure 2) be located near the diffuser to its right. Let us admit that the longitudinal resolution of the lens exceeds the sum of the diffuser thickness, the object thickness, and the distance from the object to the diffuser. We also suggest that the point scattering centers are fairly rare, so the random phases of the waves that have passed through the object are independent.

### 3.2. Radiation intensity

at the observation point is also a random process. The complexity of theoretical analysis of similar phenomena is that in the general case, there is necessity for dispersive ratios character-

In practice, there is an option when random values of the wave phases varying in space and time are independent. In particular, this option may be generated if the area of spatial correlation of a physical value causing the wave phase variations is less than the wavelength of light λ. In this case, the solution to the problem of establishing a relation between the wave phase dynamics in a thin transparent (phase) object and the dynamics of the light intensity in its image plane simplifies considerably. The solutions to this problem attempted for studying the properties of particular objects are found in the literature [27, 28]. In the para below, we are

At the first stage, the aim of the theoretical analysis is to obtain the expression for radiation

model of a three-dimensional diffuser published in the abovementioned paper [22]. Let a point source of coherent radiation with wavelength λ located at point 1 illuminate point scattering centers randomly located in thin diffuser 2 near (хоу) plane as is shown in Figure 2. Let the

admit that the refraction indexes of the medium inside and outside the diffuser are the same and equal 1. At distance L<sup>0</sup> from (хоу) plane, in plane (βx0βy) there is a thin lens with focal distance f and diaphragm diameter D. Planes (хоу) and (qх0qу) are conjugate. We consider all the waves discussed linearly polarized in the same direction. Let us admit that phase ϕ<sup>j</sup> of the wave scattered by the jth center is random, and the waves from all the scattering centers arrive

Figure 2. Optical system taken in the theory: (1) light source, (2) diffuser, (3) thin transparent object, (4) the lens with

! of the observation plane, and then for temporal autocorrelation

!. To simplify the transformations, let us

!Þ using the

giving the general solution to the problem obtained by the authors of this paper.

function of a random process I ¼ IðtÞ. We will obtain the expression for value Ið q

izing the wave phase variation in space and time.

3.1. Model of the object

!Þ at some point <sup>q</sup>

at an arbitrary point of (βx0βy) plane.

diaphragm, (5) the image plane, (6,7) conjugate points.

position of the point source be given by radius vector s

intensity Ið q

110 Optical Interferometry

First, let us obtain the expression for radiation intensity Ið q !Þ at some point <sup>q</sup> ! of plane (qх0qу) in the absence of the phase object. We suppose that the optical system does not permit separate scattering centers, and that the number of the scattering centers is fairly large in the area of the transverse lens resolution. For total complex amplitude Aðβ ! Þ at arbitrary point β ! of plane (βx0βy) we have:

$$A(\overrightarrow{\beta}) = \sum\_{j=1}^{M} a\_j(\overrightarrow{\beta}),\tag{17}$$

where M is the count of scattering centers, ajðβ ! Þ is the complex amplitude of the jth wave at point β ! . We will obtain the complex amplitude of light Að q !Þ at point <sup>q</sup> ! adding the amplitudes of waves that arrived from the points of plane (βx0βy) to point q !, taking amplitude <sup>P</sup>ð<sup>β</sup> ! Þ and phase exp ikj β ! j=2f lens transmission into consideration [7]:

$$A(\overrightarrow{q}) = \int\_{-\infty}^{+\infty} \int\_{-\infty}^{\cdot + \infty} [P(\overrightarrow{\beta})e^{\frac{ik|\overrightarrow{\beta}|}{2\overrightarrow{\beta}}}e^{ik|\overrightarrow{L}\_q(\overrightarrow{\beta})|} \sum\_{j=1}^{M} a\_j(\overrightarrow{\beta})d\beta\_x d\beta\_y,\tag{18}$$

where i is an imaginary unit, k ¼ 2π=λ is the wave number, and L ! <sup>q</sup>ðβ ! Þ is the vector connecting points β ! and q !. Henceforward, the inferior index of the vector denotes the position of the vector head.

Let us take the relation between the complex amplitude of light in proximity to point r ! <sup>j</sup> and at point β ! in the same form as in Ref. [7]:

$$a\_{\vec{\boldsymbol{\beta}}}(\overrightarrow{\boldsymbol{\beta}}) = \sqrt{I\_0(\overrightarrow{\boldsymbol{r}\_j})} \xi(\overrightarrow{\boldsymbol{r}\_j}) \boldsymbol{e}^{\boldsymbol{i}\left\{k[|\overrightarrow{L}\_s(\overrightarrow{\boldsymbol{r}\_j})| + |\overrightarrow{L}\_{\boldsymbol{\beta}}(\overrightarrow{\boldsymbol{r}\_j})|] + \varphi\_{\vec{\boldsymbol{j}}}\right\}},\tag{19}$$

where I<sup>0</sup> ¼ I0ðr !Þ is a distribution of the illuminating radiation intensity, <sup>ξ</sup> <sup>¼</sup> <sup>ξ</sup>ð<sup>r</sup> !Þ in the general case is a complex coefficient accounting the share of the radiation going from point r ! to point β ! , L ! sðr !Þ is the vector connecting points <sup>r</sup> ! and s !, and L ! <sup>β</sup>ðr !Þ is the vector connecting points r ! and β ! .

Let us take arbitrary point 6 in plane (хоу) and its conjugate point 7 in plane (qx0qy) that are given by radius vectors r ! <sup>q</sup> and q ! <sup>r</sup>, respectively (figure 2). It is known that the wave going from point 6 generates an Airy pattern with the center in point 7 as the result of light diffraction on the diaphragm of diameter D. The radius of the central spot bs of the pattern equals 1:22λL′ <sup>0</sup>=D, where L′ <sup>0</sup> is the distance from the lens to plane (qx0qy). The areas of radius bs in plane (хоу) correspond to the area of radius as ¼ bs=m, where m is the magnification generated by the lens. It is known that 85% of the energy of the wave that passed through the lens falls on the central speckle of the Airy pattern. We are going to neglect the energy of the waves beyond the area of radius bs. This in turn means that we suppose that the waves only from the scattering centers in the area of radius as with the center at point 6 arrive at point 7. Let N be the number of these centers.

Then, supposing that the area of radius as, the thickness of the diffuser and value D are small compared to the distances from the object to the radiation source and to the lens, and also from the lens to the image plane, we can obtain the expression for complex amplitude Að q !Þ:

$$A(\overrightarrow{q}) = \sqrt{I\_0} e^{i\psi} \sum\_{j=1}^{N} e^{i\theta\_j},\tag{20}$$

where <sup>I</sup>0,<sup>ψ</sup> are constants, <sup>θ</sup><sup>j</sup> <sup>¼</sup> k r! jð l ! <sup>s</sup>þ l ! Þ þ ϕ<sup>j</sup> , l ! <sup>s</sup> ¼ l ! <sup>s</sup>ðlsx, lsy, lszÞ and l ! ¼ l ! ðlx, ly, lzÞ are single vectors directed from point r ! <sup>q</sup> toward the radiation center and to the observer, respectively, complex amplitude ffiffiffiffi I0 <sup>p</sup> <sup>e</sup><sup>i</sup><sup>ψ</sup> determines the complex expression preceding the summation sign. A detailed output of Eq. (20) can be found in Ref. [29].

Let us insert a thin phase object between the diffuser and the lens, as shown in Figure 2. Let us suppose that the object will alter only the phase of the jth wave, and there is no light refraction. In this case value, θ<sup>j</sup> will change by value ζj, where

$$\zeta\_j = \frac{2\pi}{\lambda} \left\{ \left[ n\_j(l) - n\_0 \right] dl \right\} = \frac{2\pi}{\lambda} u\_j,\tag{21}$$

njðlÞ is a distribution of the refraction index in the phase object along the path of the jth wave, lj is the path length of the jth wave in the object; integrals are found along the wave path, uj is the optical difference of the jth wave travel path in the phase object.

So instead of Eq. (20), we have:

$$A(\overrightarrow{q}) = \sqrt{I\_0} e^{i\psi} \sum\_{j=1}^{N} e^{i(\zeta\_j + \theta\_j)}\tag{22}$$

For radiation intensity at point q ! we have the following:

$$I(\overrightarrow{q'}) = A(\overrightarrow{q})A^\*(\overrightarrow{q}) = I\_0 \sum\_{j=1}^{N} \sum\_{m=1}^{N} e^{i[k(u-u\_n) + \theta\_\uparrow - \theta\_n]} = I\_0N + 2I\_0 \sum\_{\kappa=1}^{K} \cos\left[k\Delta u\_\kappa + \Delta \theta\_\kappa\right],\tag{23}$$

where Δu<sup>κ</sup> is the relative optical difference of the travel path of the κth pair of scattering centers, Δθκ ¼ θj−θm, j≠m, κ ¼ 1, 2…K, K ¼ NðN−1Þ=2.

### 3.3. Temporal autocorrelation function

generates an Airy pattern with the center in point 7 as the result of light diffraction on the

<sup>0</sup> is the distance from the lens to plane (qx0qy). The areas of radius bs in plane (хоу) correspond to the area of radius as ¼ bs=m, where m is the magnification generated by the lens. It is known that 85% of the energy of the wave that passed through the lens falls on the central speckle of the Airy pattern. We are going to neglect the energy of the waves beyond the area of radius bs. This in turn means that we suppose that the waves only from the scattering centers in the area of radius as with

Then, supposing that the area of radius as, the thickness of the diffuser and value D are small compared to the distances from the object to the radiation source and to the lens, and also from

Let us insert a thin phase object between the diffuser and the lens, as shown in Figure 2. Let us suppose that the object will alter only the phase of the jth wave, and there is no light refraction.

½njðlÞ−n0�dl

njðlÞ is a distribution of the refraction index in the phase object along the path of the jth wave, lj is the path length of the jth wave in the object; integrals are found along the wave path, uj is the

> iψX N

j¼1 e

<sup>i</sup>½kðuj−umÞþθj−θm� <sup>¼</sup> <sup>I</sup>0<sup>N</sup> <sup>þ</sup> <sup>2</sup>I<sup>0</sup>

o <sup>¼</sup> <sup>2</sup><sup>π</sup>

iψX N

j¼1 e iθj

the lens to the image plane, we can obtain the expression for complex amplitude Að q

!Þ ¼ ffiffiffiffi I0 p e <sup>0</sup>=D, where

!Þ:

ðlx, ly, lzÞ are sin-

, (20)

<sup>λ</sup> uj, (21)

<sup>i</sup>ðζjþθj<sup>Þ</sup> (22)

cos ½kΔu<sup>κ</sup> þ Δθκ�, (23)

X K

κ¼1

! ¼ l !

<sup>s</sup>ðlsx, lsy, lszÞ and l

<sup>q</sup> toward the radiation center and to the observer, respectively,

<sup>p</sup> <sup>e</sup><sup>i</sup><sup>ψ</sup> determines the complex expression preceding the summation sign.

diaphragm of diameter D. The radius of the central spot bs of the pattern equals 1:22λL′

the center at point 6 arrive at point 7. Let N be the number of these centers.

Að q

! jð l ! <sup>s</sup>þ l ! Þ þ ϕ<sup>j</sup> , l ! <sup>s</sup> ¼ l !

<sup>ζ</sup><sup>j</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> λ nð

optical difference of the jth wave travel path in the phase object.

Að q

X N

X N

m¼1 e

j¼1

centers, Δθκ ¼ θj−θm, j≠m, κ ¼ 1, 2…K, K ¼ NðN−1Þ=2.

lj

!Þ ¼ ffiffiffiffi I0 p e

! we have the following:

where Δu<sup>κ</sup> is the relative optical difference of the travel path of the κth pair of scattering

!

where I0,ψ are constants, θ<sup>j</sup> ¼ k r

gle vectors directed from point r

So instead of Eq. (20), we have:

For radiation intensity at point q

! Þ ¼ Að q

!ÞA� ð q !Þ ¼ <sup>I</sup><sup>0</sup>

Ið q

I0

A detailed output of Eq. (20) can be found in Ref. [29].

In this case value, θ<sup>j</sup> will change by value ζj, where

complex amplitude ffiffiffiffi

L′

112 Optical Interferometry

First, let us obtain the expression for temporal autocorrelation function of the radiation intensity at point q !, i. е., Eq. (24):

$$R\_{1,2}(t\_1, t\_2) = \langle [I\_1 - \langle I\_1 \rangle] \times [I\_2 - \langle I\_2 \rangle] \rangle = \langle I\_1 I\_2 \rangle - \langle I\_1 \rangle \langle I\_2 \rangle,\tag{24}$$

where inferior indexes 1 and 2 denote time points t<sup>1</sup> and t2, angle parentheses denote averaging by the object (model) ensemble. Let us suggest that at different κ random values Δu<sup>κ</sup> are independent, and at the same κ, their time correlation occurs. Suppose also that joint probability density ρðΔuκ<sup>1</sup>, Δuκ<sup>2</sup>Þ is a two-dimensional Gaussian function that is the same for different κ. So further we are going to omit inferior index κ in expressions Δuκ. Using the suggestions made in Ref. [29], we obtained the expression for R1, <sup>2</sup>ðt1, t2Þ:

$$R\_{1,2}(t\_1, t\_2) = I\_0^2 N(N-1) \cos[(\langle \mathbf{x}\_2 \rangle - \langle \mathbf{x}\_1 \rangle)] \times e^{-\frac{1}{2}k\_{11} - \frac{1}{2}k\_{22} + k\_{12}},\tag{25}$$

where 〈x1〉 and 〈x2〉 are the object ensemble-averaged values x ¼ kΔu at time points t<sup>1</sup> and t2, respectively, k<sup>11</sup> and k<sup>22</sup> are dispersions of value х at time moments t<sup>1</sup> and t2, respectively, k<sup>12</sup> ¼ 〈ðx1−〈x1〉Þðx2−〈x2〉Þ〉. For the normalized autocorrelation function η<sup>12</sup> ¼ R12ðt1, t2Þ=R<sup>12</sup> ðt1, t1Þ we have the following:

$$\eta\_{12} = \cos[(\langle \mathbf{x}\_2 \rangle - \langle \mathbf{x}\_1 \rangle)] \times e^{\frac{\mathbf{k}\_{11}}{2} + \frac{\mathbf{k}}{2} \cdot \mathbf{z}\_2 + k\_{12}}.\tag{26}$$

Let process x ¼ xðtÞ be stationary. Then 〈x1〉 ¼ 〈x2〉, k<sup>11</sup> ¼ k<sup>22</sup> and, therefore,

$$
\eta(\tau) = e^{-k\_{11} + k\_{11}\rho\_{12}(\tau)},
\tag{27}
$$

where τ ¼ t2−t1, ρ12ðτÞ is a normalized temporal correlation function of random value kΔu. Let ρ12ðτÞ ! 0, τ ! ∞. For example, this is a feature of normalized Lorentzian and Gaussian correlation functions. Then function ηðτÞ levels off to η� equal to expð−k11Þ. So by value η leveling off with time, we can determine dispersion k<sup>11</sup> of phase differences varying in time and variation <sup>σ</sup><sup>u</sup> <sup>¼</sup> <sup>λ</sup> 2π ffiffiffiffiffiffi <sup>k</sup><sup>11</sup> <sup>p</sup> of value <sup>Δ</sup>u. We used this fact in experiments studying the processes occurring in live cells. These experiments will be discussed in Sections 5 and 6.

#### 3.4. Temporal spectral function

Subtracting constant component η� from Eq. (27) and renormalizing it, we obtain a new temporal autocorrelation function of radiation intensity fluctuation η 0 ðτÞ:

$$
\eta'(\tau) = \frac{\eta(\tau) - \eta^\*}{1 - \eta^\*}. \tag{28}
$$

Let Δu ≪ λ. Then it easy to demonstrate that η′ ðτÞ≅ρ12ðτÞ. Therefore, the temporal autocorrelation function of intensity fluctuations corresponds to the temporal autocorrelation function of the wave pair optical path differences. Let us further suppose that random process Δu ¼ ΔuðtÞ is not only stationary but also ergodic. As the normalized temporal energetic spectrum for these processes is Fourier's transformation from the normal autocorrelation function, the corresponding normalized temporal spectral functions of intensity fluctuations g<sup>Δ</sup>IðωÞ and optical path differences g<sup>Δ</sup>uðωÞ are equal.

It was also demonstrated in Ref. [29] that if Δu≥λ and ρ12ðτÞ is a Gaussian function, functions g<sup>Δ</sup>uðωÞ and g<sup>Δ</sup><sup>I</sup> ðωÞ are also Gaussian functions, but spectrum width g<sup>Δ</sup><sup>I</sup> ðωÞ is k11-fold spectrum width g<sup>Δ</sup>uðωÞ. So at k<sup>11</sup> increasing spectrum g<sup>Δ</sup>IðωÞ widens, and at Δu≥λ it widens k11-fold.

#### 3.5. Time averaging technique

A disadvantage of the theory presented in the Section 3.4 is the difficulty of application in the case when the wave phase variations in time happen due to existence of various processes occurring simultaneously at different scale levels. For example, when the target of research is a cell, sounding wave phase variation can occur due to passage of ions via the membrane, to capture large molecules by endocytosis (local variation of the cell shape), due to chemical processes during protein synthesis in the cytoplasm and nucleus of the cell, so in Ref. [30] the technique was upgraded to overcome this disadvantage. The idea was in the application of time-averaging procedure for speckle dynamics. If characteristic time τ<sup>0</sup> of wave phase variation corresponding to the most rapid process is known, averaging time T of the recorded optical signals can be taken as a value exceeding τ0. In this case, the speckle dynamics will result from slower processes, and interpretation of the experimental data can get simplified. In the para below the results obtained in Ref. [30] are presented in brief.

Using the model discussed in Section 3.1, we obtained expressions for time-average intensity ~Ið q !Þ at point <sup>q</sup> ! <sup>r</sup> (Figure 2). Having substantiated the possibility of discussing continuous function <sup>~</sup><sup>I</sup> <sup>¼</sup> <sup>~</sup>Iðt<sup>Þ</sup> at point <sup>ð</sup> <sup>q</sup> !Þ, we obtained the expression for temporal autocorrelation function R1,2ðt1,t2Þ of time-average intensity:

$$\tilde{I}(\vec{q}) = I\_1 + I\_2 e^{-\vec{k}^2 \sigma^2 / 2} \cos \left(k\mu + \alpha\right),\tag{29}$$

$$R\_{1,2}(t\_1, t\_2) = I^2 N(N-1) \mathbb{C}\_0^2 \cos[\langle \mathbf{x}\_2 \rangle - \langle \mathbf{x}\_1 \rangle] \times e^{\frac{\hbar}{2}k\_{11} + \frac{\hbar}{2}k\_{22} + k\_{12}}.\tag{30}$$

In Eq. (29) I1, I2, α are constants, μ and σ<sup>2</sup> are mean value and dispersion of variable Δu obtained by time-averaging and averaging by a region of radius as (see Figure 2). Eq. (30) coincides with Eq. (25) to a precision of insufficient coefficients I <sup>2</sup> and C<sup>2</sup> <sup>0</sup>. But now x ¼ kμ, so the arguments of the cosine and the exponent contain mean values, dispersions, and the correlation moment of a new value x ¼ kμ.

The peculiarity of Eq. (30) is that if averaging time T of the radiation intensity exceeds the correlation time of value μ, normalized function, Eq. (30) takes on the following appearance:

Dynamic Speckle Interferometry of Thin Biological Objects: Theory, Experiments, and Practical Perspectives http://dx.doi.org/10.5772/66712 115

$$\eta(\overrightarrow{\boldsymbol{\eta}}^{\*},t\_{1},t\_{2}) = \frac{R\_{1,2}(\overrightarrow{\boldsymbol{\eta}},t\_{1},t\_{2})}{R\_{1,2}(\overrightarrow{\boldsymbol{\eta}},t\_{1},t\_{2})} = \cos[\langle \mathbf{x}\_{2} \rangle - \langle \mathbf{x}\_{1} \rangle]e^{-k\_{22}/2 + k\_{11}/2}.\tag{31}$$

Supposing in Eq. (31) that values k11, k22, and kμ are small compared with 1, let us decompose Eq. (31) into Taylor's series in the neighborhood of points k<sup>22</sup> and kμ equal to zero, having retained the first-order derivatives. We obtain

Δu ¼ ΔuðtÞ is not only stationary but also ergodic. As the normalized temporal energetic spectrum for these processes is Fourier's transformation from the normal autocorrelation function, the corresponding normalized temporal spectral functions of intensity fluctuations

It was also demonstrated in Ref. [29] that if Δu≥λ and ρ12ðτÞ is a Gaussian function,

k11-fold spectrum width g<sup>Δ</sup>uðωÞ. So at k<sup>11</sup> increasing spectrum g<sup>Δ</sup>IðωÞ widens, and at Δu≥λ

A disadvantage of the theory presented in the Section 3.4 is the difficulty of application in the case when the wave phase variations in time happen due to existence of various processes occurring simultaneously at different scale levels. For example, when the target of research is a cell, sounding wave phase variation can occur due to passage of ions via the membrane, to capture large molecules by endocytosis (local variation of the cell shape), due to chemical processes during protein synthesis in the cytoplasm and nucleus of the cell, so in Ref. [30] the technique was upgraded to overcome this disadvantage. The idea was in the application of time-averaging procedure for speckle dynamics. If characteristic time τ<sup>0</sup> of wave phase variation corresponding to the most rapid process is known, averaging time T of the recorded optical signals can be taken as a value exceeding τ0. In this case, the speckle dynamics will result from slower processes, and interpretation of the experimental data can get simplified. In the para below the results obtained in Ref. [30]

Using the model discussed in Section 3.1, we obtained expressions for time-average intensity

In Eq. (29) I1, I2, α are constants, μ and σ<sup>2</sup> are mean value and dispersion of variable Δu obtained by time-averaging and averaging by a region of radius as (see Figure 2). Eq. (30)

the arguments of the cosine and the exponent contain mean values, dispersions, and the

The peculiarity of Eq. (30) is that if averaging time T of the radiation intensity exceeds the correlation time of value μ, normalized function, Eq. (30) takes on the following appearance:

<sup>r</sup> (Figure 2). Having substantiated the possibility of discussing continuous

0cos½〈x2〉−〈x1〉�×e<sup>−</sup><sup>1</sup>

!Þ, we obtained the expression for temporal autocorrelation func-

<sup>−</sup>k2σ2=<sup>2</sup> cos <sup>ð</sup>k<sup>μ</sup> <sup>þ</sup> <sup>α</sup>Þ, (29)

<sup>2</sup> and C<sup>2</sup>

<sup>2</sup>k22þk<sup>12</sup> : (30)

<sup>0</sup>. But now x ¼ kμ, so

2k11−<sup>1</sup>

ðωÞ are also Gaussian functions, but spectrum width g<sup>Δ</sup><sup>I</sup>

ðωÞ is

g<sup>Δ</sup>IðωÞ and optical path differences g<sup>Δ</sup>uðωÞ are equal.

functions g<sup>Δ</sup>uðωÞ and g<sup>Δ</sup><sup>I</sup>

3.5. Time averaging technique

it widens k11-fold.

114 Optical Interferometry

are presented in brief.

!

tion R1,2ðt1,t2Þ of time-average intensity:

correlation moment of a new value x ¼ kμ.

~Ið q

coincides with Eq. (25) to a precision of insufficient coefficients I

R1, <sup>2</sup>ðt1, t2Þ ¼ I

!Þ ¼ <sup>I</sup><sup>1</sup> <sup>þ</sup> <sup>I</sup>2<sup>e</sup>

<sup>N</sup>ðN−1ÞC<sup>2</sup>

2

function <sup>~</sup><sup>I</sup> <sup>¼</sup> <sup>~</sup>Iðt<sup>Þ</sup> at point <sup>ð</sup> <sup>q</sup>

!Þ at point <sup>q</sup>

~Ið q

$$\eta(t) = 1 - \frac{(k\_{22}(t) - k\_{11})}{2},\tag{32}$$

where t ¼ t2−t1. It is seen from the formula that if averaging time T exceeds the correlation time of random value kμ, the relation between η and k<sup>22</sup> is linear.

Now among N waves let us have two wave groups with random optical wave path variations un ¼ unðtÞ occurring homogeneously in the statistical sense. Let count n of such waves in groups 1 and 2, respectively, equal N<sup>1</sup> and N2. In practice, groups 1 and 2, for example, lie inside and outside a live cell. At large magnifications, such groups can lie within the cell nucleus and in its cytoplasm. In Ref. [30], it was shown that in this case time-average radiation intensity at conjugate point q ! <sup>r</sup> is determined by Eq. (33):

$$\tilde{I}(\stackrel{\frown}{q}) = \tilde{I}\_1 + \tilde{I}\_2 + 2\tilde{I}\_{12}\cos[k\Delta\mu + \theta],\tag{33}$$

where ~I <sup>1</sup> and ~I <sup>2</sup> are time-average intensities generated by groups 1 and 2 individually, <sup>~</sup><sup>I</sup> <sup>12</sup> <sup>¼</sup> <sup>I</sup>3e<sup>−</sup>σ<sup>2</sup> 1=2−σ<sup>2</sup> <sup>2</sup>=<sup>2</sup> and I3, θ are constants, Δμ is the difference of time-average values un in groups 1 and 2, σ<sup>2</sup> <sup>1</sup> and σ<sup>2</sup> <sup>2</sup> are dispersions of values un in groups 1 and 2, respectively.

#### 3.6. Relation between the object features and the parameters of speckle dynamics

We used the results of the theory presented in Section 3 to conduct experiments with live cells cultured or precipitated on a transparent substrate. To determine the value η, we took segments of diameter 2as in the object plane. We regarded a region containing a large number of such segments as an object ensemble. The corresponding segments of the speckle image in the conjugate region were recorded at time points t<sup>1</sup> and t2, and then they were used to determine correlation coefficient η of digital speckle images.

Analysis of the formulas obtained demonstrates that experimentally obtained dependences <sup>η</sup>ðt<sup>Þ</sup> and (or) <sup>~</sup>Iðt<sup>Þ</sup> can in principle be used to determine the mean value, variation, and correlation time of the medium refraction index in small regions of the transparent object. In turn, the refraction index is related to medium density ρ and its specific refractivity ^r via Lorentz-Lorenz formula for liquids. For multicomponent media, the latter is equal to the sum of the products ^r of single molecules on their relative concentration.

It is known that the density of liquids depends on their temperature. Therefore, if the time range where the composition of the medium can be considered constant is selected, under certain conditions the spectrum of intensity fluctuations can be regarded as the energetic spectrum of chemical reactions occurring in the cell areas under study. Similarly, if a time range or the object segments with the temperature (density) that can be considered invariable is selected, the processes of mass transfer in live cells can be studied by variation of correlation coefficient <sup>η</sup> or average intensity <sup>~</sup>IðtÞ.
