5.2. Optical and television systems

When the laboratory thermostat was used, the transparent substrate with cells was placed at the bottom of a special cuvette. In turn, the cuvette was placed on the stage of the microscope. The transparent lid of the cuvette was placed so that (1) the nutrient solution about 1 mm thick was placed above the cells and (2) there was no free liquid surface. Above the lid, there was matte glass that was illuminated by a diverging beam from the semiconductor laser module. The speckle image of the cells was entered into the computer using a monochrome TV camera of Videoscan—415/P/C-USB type. The camera had a photosensor array of 6.5 × 4.8 mm size with 780 × 572 cells (pixels) of 8.3 × 8.3 μm size. The frame input frequency was up to 25 Hz. The signals from the TV camera entered the laptop computer of Aspire 3692 WLMi 8 type by Acer firm via a USB port. A semiconductor laser module of KLM-D532-20-5 type with wavelength λ = 0.532 μm and 20 mW power was used as the light source. Note that the above TV camera and laser were used in all of our optical systems.

In the liquid thermostat and in the air bath, optical systems with the upright position of the substrate with cells in the optical cuvette were used. The logic chart of the optical systems is given in Figure 4, and its photograph is presented in Figure 5.

Figure 4. Logic chart of optical device: (1) laser module, (2) illuminating beam, (3) matte glass, (4) scattered radiation, (5) cuvette, (6) substrate with cells, (7) lens with diagram, (8) photosensor array, (9) computer.

### 5.3. Software

Input of the image frames into the computer was made using the software coming standard with camera Videoscan—415/P/C-USB. The frames entered into the computer were processed to obtain dependences <sup>η</sup>ðt<sup>Þ</sup> and <sup>~</sup>Iðt<sup>Þ</sup> using two original computer programs. The first program was intended to process the frames already entered into the computer. The operator displayed the first frame of the speckle image onto the monitor and selected the fragment of the frame to determine η using the mouse or the keyboard. Then it gave the second frame or a mass of frames. In the first case, the program displayed value η using Eq. (34) and digital value ~I, in the second case, it displayed dependences <sup>η</sup>ðt<sup>Þ</sup> and <sup>~</sup>Iðt<sup>Þ</sup> onto the monitor. The first program also allowed to obtain the distribution of values η. The operator selected two frames corresponding to time moments t<sup>1</sup> and t2. Then the operator selected a segment on the displayed image and divided it into subsegments. The program digitized the fragment and determined value η in every subsegment using Eq. (34). The obtained values were recorded into the computer memory as a matrix in text format to be processed.

Figure 5. Photography of the optical device: (1) laser module with microobjective, (2) matte glass, (3) lock of the object on the platform of the motorized translator, (4) camera with lens.

Value η was determined by Eq. (34):

The laboratory thermostat maintained the temperature of the cuvette to ±0.1°С precision for several hours, and the liquid thermostat ЗЦ-1125М did so for several days. An air bath was

When the laboratory thermostat was used, the transparent substrate with cells was placed at the bottom of a special cuvette. In turn, the cuvette was placed on the stage of the microscope. The transparent lid of the cuvette was placed so that (1) the nutrient solution about 1 mm thick was placed above the cells and (2) there was no free liquid surface. Above the lid, there was matte glass that was illuminated by a diverging beam from the semiconductor laser module. The speckle image of the cells was entered into the computer using a monochrome TV camera of Videoscan—415/P/C-USB type. The camera had a photosensor array of 6.5 × 4.8 mm size with 780 × 572 cells (pixels) of 8.3 × 8.3 μm size. The frame input frequency was up to 25 Hz. The signals from the TV camera entered the laptop computer of Aspire 3692 WLMi 8 type by Acer firm via a USB port. A semiconductor laser module of KLM-D532-20-5 type with wavelength λ = 0.532 μm and 20 mW power was used as the light source. Note that the above TV

In the liquid thermostat and in the air bath, optical systems with the upright position of the substrate with cells in the optical cuvette were used. The logic chart of the optical systems is

Input of the image frames into the computer was made using the software coming standard with camera Videoscan—415/P/C-USB. The frames entered into the computer were processed to obtain dependences <sup>η</sup>ðt<sup>Þ</sup> and <sup>~</sup>Iðt<sup>Þ</sup> using two original computer programs. The first program was intended to process the frames already entered into the computer. The operator displayed the first frame of the speckle image onto the monitor and selected the fragment of the frame to determine η using the mouse or the keyboard. Then it gave the second frame or a mass of frames. In the first case, the program displayed value η using Eq. (34) and digital value ~I, in the second case, it displayed dependences <sup>η</sup>ðt<sup>Þ</sup> and <sup>~</sup>Iðt<sup>Þ</sup> onto the monitor. The first program also

Figure 4. Logic chart of optical device: (1) laser module, (2) illuminating beam, (3) matte glass, (4) scattered radiation, (5)

used to study the cell reaction to temperature.

camera and laser were used in all of our optical systems.

given in Figure 4, and its photograph is presented in Figure 5.

cuvette, (6) substrate with cells, (7) lens with diagram, (8) photosensor array, (9) computer.

5.2. Optical and television systems

120 Optical Interferometry

5.3. Software

$$\eta = \frac{\frac{1}{m \star n} \sum\_{i=0}^{m-1} \sum\_{j=0}^{n-1} [A\_{i,j} - \overline{A}][B\_{i,j} - \overline{B}]}{\sqrt{\frac{1}{m \star n} \sum\_{i=0}^{m-1} \sum\_{j=0}^{n-1} [A\_{i,j} - \overline{A}]^2} \times \sqrt{\frac{1}{m \star n} \sum\_{i=0}^{m-1} \sum\_{j=0}^{n-1} [B\_{i,j} - \overline{B}]^2}},\tag{34}$$

where Ai,<sup>j</sup> are digitized signals at a segment of m × n-pixel size at initial time point t1, Bi,<sup>j</sup> are the signals in the same segment at a different time point t2, i, and j are the segment pixel numbers xd and yd, respectively, A is the mean signal value in the segment at the start time, and B is the mean signal value in the segment at time moment t2.

The second program in DOS medium permitted real-time determination of digital values η and ~I. First, the operator set x and y coordinates of the pixels (up to 40 pieces) on the program interface. By the operator's command, the program determined digital values of mean intensity ~I in the above pixels, and it obtained η values in the neighborhood of the pixels using Eq. (34). Masses η and ~I were saved in txt format in preset files. The program could work for several days continuously.

### 5.4. Errors and calibration of optical measurements

We assessed the error of value η determination by Eq. (34). In compliance with the indirect measurement error assessment techniques recommended in Russia [31], the mean square deviation of random error Sðη~Þ in the indirect measurement result is determined by Eq. (35):

$$\mathcal{S}(\tilde{\eta}) = \sqrt{\sum\_{i}^{m} \left(\frac{\partial \eta}{\partial a\_{i}}\right)^{2} \ast \mathcal{S}^{2}(\tilde{a}\_{i})},\tag{35}$$

where η~ is experimentally found value of η, aiði ¼ 1, …, mÞ are values Ai,<sup>j</sup> and Bi,<sup>j</sup> featuring Eq. (34), <sup>∂</sup><sup>η</sup> ∂ai is the first derivative of function η by argument ai, calculated at point ~a1, …, ~am, ~ai is the result of measuring value ai, and Sð~aiÞ the mean square deviation of random errors in the result of measuring the ai-th argument.

We performed the transformations by Eq. (35) and assessed the error of value η determination in a typical experiment. We selected the variant with 8-bit digitization of the radiation intensity averaging-out half the dynamic range, the minimum speckle size slightly exceeding the TV camera pixel size, and Sð~aiÞ ¼ 0:7. Calculations showed that value ∂η=∂ai featuring Eq. (35) consists of sum m of random 10-2-order values of different signs. The random sign value appears due to randomness of intensity deviation from the average value in the speckle field. In the model experiment, a reflecting rough object in the form of a metal plate was used, and value η varied due to its shift. For a 10 × 10-pixel fragment of the speckle image, we obtained that Sðη~Þ decreases steadily with increase of η in the range from 0.3 to 0.999, and the relative error of η determination does not exceed 1%.

Essentially, the calibration technique for the optical systems intended to determine the optical path dispersion value was developed in Ref. [29]. In this technique, batched random variations of the wave phase difference were set by means of shifting a 1-mm thick transparent plate. The plate shift ux was performed with a 0.12-μm pitch. The plate roughness was prechecked with a WYKO NT-1100 optical profilometre. Figure 6 shows experimental and theoretical dependences of η on the plate shift ux.

Figure 6. Theoretical (---) and experimental dependences η(ux).

As is seen from the graphs, when the plate is shifted for a value exceeding the characteristic surface roughness size, the correlation coefficient levels off to η�. The theoretical curve was obtained for the Gaussian function by Eq. (27). The difference between the theory and the experiment was in the range of 2.5%. Roughness parameter difference Ra obtained by level η� from that measured by the profilometre was in the range of 5%. The experiment details and the digital derivations can be found in Ref. [29]. This experimental technique can be used to calibrate the equipment used to determine value k22. As for the calibration of the device for determination of value 〈x2〉, further research is needed to perform this procedure.

Good coincidence of the data obtained by the speckle dynamics and with the optical profilometre (Figure 6) is to a great extent determined by small errors of determining the speckle image correlation coefficient η and with high sensitivity of the technique. Let us assess the sensitivity limits of the equipment for the determination of values 〈x2〉 and σu. Let us admit that in Eq. (26) values 〈x1〉, k11, k22, and k<sup>12</sup> are equal to zero. Typical values η<sup>12</sup> equaled 0.99 in the absence of the object. Then for wavelength λ ¼ 0:532 μm, we obtain that Δu ¼ ðλ=2πÞarccos0:99 ¼ 12 nm. Hence, it follows that the limit sensitivity of the device related to the optical path difference generated in the range of the linear resolution of the lens equals 12 nm. Let us admit that the object thickness is invariable, and the optical paths vary due to variation of the mean refraction index. For instance, for the 10-μm cell thickness, we obtain that the refraction index will vary by 1:2×10<sup>−</sup><sup>3</sup> . Now let us find the limit sensitivity of the equipment in determination of mean square deviation <sup>σ</sup> ¼ ½λ=ð2πÞ� ffiffiffiffiffiffi <sup>k</sup><sup>11</sup> <sup>p</sup> of value <sup>Δ</sup><sup>u</sup> by value <sup>η</sup>�. Supposing again in formula η� ¼ expð−k11Þ that η� ¼ 0:99, and λ ¼ 0:532 μm, we obtain that <sup>σ</sup> ¼ ½λ=ð2πÞ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>−</sup>ln0:<sup>99</sup> <sup>p</sup> <sup>¼</sup> 8nm.

It is noteworthy that values η<sup>12</sup> and η� equal to 0.99 appeared due to application of mediumquality equipment. If limit values η<sup>12</sup> and η� equal to 0.999 are reached due to noise decrease, then the sensitivity to the mean value and dispersion of value Δu will equal 4 and 3 nm, respectively.
