**4. Piezo-optical transducer model for numerical simulation**

The mathematical models of the transducer were created for the accurate numerical simulation of its operation. The first model is for the simulation of the light parameters transformation as it passes through the optical elements of the piezo-optical transducer. The second model is for the simulation of stresses spatial distribution in the PE body and its deformation under applied force. The models, the equations used are described in detail in [30]. The initial data and the main results obtained are briefly presented here.

#### **4.1 Simulation of the light propagation**

The optical scheme of the piezo-optical transducer showed in **Figure 3** on the left was used for the simulations. The simulations were performed using the Wolfram Mathematica™ package and took into account the design of the developed transducer: the radiation diagram of LEDs used, the dimensions of the photosensitive areas of photodiodes used, light refraction in the PE, the distances between the transducer elements.

A lot of different computing methods have been developed allowing coping with cumbersome quantitative methods that have to be used to determine the output states of the electromagnetic wave amplitude and polarization. The most successful and obvious is the Muller formalism, where matrix algebra is used to describe the amplitude and polarization transformations [38–41]. The optical elements are represented in the form of a T*ij* matrices 4 � 4 that describe the polarizing elements, delay elements, and rotation matrix [39]. All values in this approach are real numbers.

The connection between light intensity Φ, the degree of polarization (*p*), and the polarization ellipse parameters (*ψ*, *χ*) (insert in the center of **Figure 3**) with the Stokes light parameters (vector **s**) is described with the algebraic expression:

$$\mathbf{s} = \begin{bmatrix} s\_0 \\ s\_1 \\ s\_2 \\ s\_3 \end{bmatrix} = \begin{bmatrix} \Phi \\ \Phi \, p \cos 2\psi \cos 2\chi \\ \Phi \, p \sin 2\psi \cos 2\chi \\ \Phi \, p \sin 2\chi \end{bmatrix}. \tag{9}$$

In case where polarizers are arranged perpendicularly to the incident light rays and the rays are parallel to the Z-axis, the polarizer and the analyzer are described by the linear transformation according to the Muller formalism [39]:

$$\mathbf{P}(\theta) = \frac{1}{2} \begin{bmatrix} 1 & \cos 2\theta & \sin 2\theta & 0 \\ \cos 2\theta & \cos^2 2\theta & \cos 2\theta \sin 2\theta & 0 \\ \sin 2\theta & \cos 2\theta \sin 2\theta & \sin^2 2\theta & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix},\tag{10}$$

#### *Optoelectronics*

where *θ* is the angle of the fast axis of the polarizer measured from the X-axis to the Y-axis (**Figure 3**).

The quarter-wave plate and the photoelastic element are described by a matrix for linear delay [38]:

$$\mathbf{R}(\theta,\delta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos^2 2\theta + \cos \delta \sin^2 2\theta & (1 - \cos \delta)\cos 2\theta \sin 2\theta & -\cos \delta \sin 2\theta \\ 0 & (1 - \cos \delta)\cos 2\theta \sin 2\theta & \cos \delta \cos^2 2\theta + \sin^2 2\theta & \cos 2\theta \sin \delta \\ 0 & \sin \delta \sin 2\theta & -\cos 2\theta \sin \delta & \cos \delta \end{bmatrix} \tag{11}$$

The elements are arranged perpendicularly to the incident light rays, and the rays are parallel to the Z-axis. *θ* is the angle of the fast axis measured from axis X to axis Y (**Figure 3**). *δ* ¼ *δ<sup>y</sup>* � *δ<sup>x</sup>* is the phase difference between the fast and the slow axes (the delay).

The Mueller matrix for refraction [41] in a PE is:

$$\mathbf{T}(\boldsymbol{\phi},\boldsymbol{\psi}) = \frac{\sin 2\boldsymbol{\phi}\sin 2\boldsymbol{\psi}}{2\left(\sin \boldsymbol{\phi}\_{+} \cos \boldsymbol{\phi}\_{-}\right)^{2}} \begin{bmatrix} \cos^{2}\boldsymbol{\phi}\_{-}+1 & \cos^{2}\boldsymbol{\phi}\_{-}-1 & 0 & 0\\ \cos^{2}\boldsymbol{\phi}\_{-}-1 & \cos^{2}\boldsymbol{\phi}\_{-}+1 & 0 & 0\\ 0 & 0 & 2\cos \boldsymbol{\phi}\_{-} & 0\\ 0 & 0 & 0 & 2\cos \boldsymbol{\phi}\_{-} \end{bmatrix} . \tag{12}$$

where *ϕ* – incidence angle, *ψ* – refraction angle, *ϕ*� ¼ *ϕ* � *ψ*.

These matrices make it easy to study the dependence of the light intensity and polarization on the angles of all transducer elements optical axes. The results obtained make it possible to determine the tolerances for the inaccuracy of the optical elements installation. Here, for simplicity, the light rays were considered as plane wave rays that fall at right angles to the surface of each element of the optical layout. We neglected the point source of light. After substituting all Muller matrixes and taking the first components, the photocurrents *I*<sup>1</sup> and *I*<sup>2</sup> of the photodetectors PD1 and PD2 (**Figure 3**) take the form:

$$\begin{aligned} I\_1 &= q \frac{\left(4n\right)^2}{\left(n+1\right)^4} \Phi\_0 \langle \mathbf{e}\_1, \mathbf{L}\_1 \mathbf{e}\_1 \rangle = 4n^2 q \Phi\_0 \frac{\mathbf{1} + \sin \Delta\_{\rm PE}}{\left(n+1\right)^4}, \\\ I\_2 &= q \frac{\left(4n\right)^2}{\left(n+1\right)^4} \Phi\_0 \langle \mathbf{e}\_1, \mathbf{L}\_2 \mathbf{e}\_1 \rangle = 4n^2 q \Phi\_0 \frac{\mathbf{1} - \sin \Delta\_{\rm PE}}{\left(n+1\right)^4}, \end{aligned} \tag{13}$$

where *k* is a proportionality factor determined by the transducer electrical circuit parameters. As a result, the signal, after the electronic circuit [36], takes the

*(a) 3D model of the photoelastic element in the guard ring made of steel. Force F is applied to the top and bottom of the conical elements; (b) 1/8 part of the model; (c) model dimensions are indicated in*

*Piezo-Optical Transducers in High Sensitive Strain Measurements*

*DOI: http://dx.doi.org/10.5772/intechopen.94082*

The equation shows that the change of output signal *dI*out is directly proportional

We used the COMSOL Multiphysics™ package and the finite-element method for the simulations of the spatial distribution of Δ*σ* ¼ *σ<sup>y</sup>* � *σ<sup>x</sup>* in the PE, and PE

The 3D model of the PE encased in the steel guard ring is shown in **Figure 5a** [30, 31]. The upper and lower steel conical elements transmitted the external force to the guard ring. The embedded in COMSOL Multiphysics parameters of the material needed for the calculation of the deformation were used. Due to the PE symmetry, the simulation was only for the 1/8 part of it as shown in **Figure 5b**. The

The results of accurate numerical simulations and experiments are also detailed

The strain gauge sensitivity determination was based on the experimental determination of the minimum detectable force and numerical simulation of the deformation occurring under the force action. We used the device "rhombus" with

frictionless hinges to apply a calibrated force to the photoelastic element

to the phase shift change *d*ΔPE that is caused by the change of stresses in the

**4.3 Simulations of stresses and deformations in the photoelastic element**

dimensions of this model part are shown in detail in **Figure 5c**.

**5. Simulation and experimental results and analysis**

*I*out ¼ *k* sin ΔPE ≈ *k*ΔPE, at ΔPE ≪ 1 (16)

following form:

*millimeters [30].*

**Figure 5.**

photoelastic element.

deformation under the force *F*.

in [30, 31], here is a summary of them.

(**Figure 6a**).

**45**

where *q* – photodetector quantum efficiency, *n* ¼ *n*2*=n*<sup>1</sup> –relative refractive index, Φ<sup>0</sup> – light intensity, ΔPE – phase shift caused by the light ray passed through the photoelastic element, **e**<sup>1</sup> is the identity matrix, **L**<sup>1</sup> and **L**<sup>2</sup> are linear transformations:

$$\begin{aligned} \mathbf{L}\_1 &= \mathbf{P} \left( -\frac{\pi}{4} \right) \mathbf{R}(\mathbf{0}, \Delta\_{\text{PE}}) \mathbf{R} \left( -\frac{\pi}{4}, \frac{\pi}{2} \right) \mathbf{P}(\mathbf{0}),\\ \mathbf{L}\_2 &= \mathbf{P} \left( \frac{\pi}{4} \right) \mathbf{R}(\mathbf{0}, \Delta\_{\text{PE}}) \mathbf{R} \left( -\frac{\pi}{4}, \frac{\pi}{2} \right) \mathbf{P}(\mathbf{0}). \end{aligned} \tag{14}$$

#### **4.2 Transducer output signal**

The output signal *I*out of the electronic circuit is

$$I\_{\rm out} = k \frac{I\_1 - I\_2}{I\_1 + I\_2},\tag{15}$$

*Piezo-Optical Transducers in High Sensitive Strain Measurements DOI: http://dx.doi.org/10.5772/intechopen.94082*

#### **Figure 5.**

where *θ* is the angle of the fast axis of the polarizer measured from the X-axis to

The quarter-wave plate and the photoelastic element are described by a matrix

2*θ* ð Þ 1 � cos *δ* cos 2*θ* sin 2*θ* � cos *δ* sin 2*θ*

cos <sup>2</sup>*ϕ*� <sup>þ</sup> 1 cos <sup>2</sup>*ϕ*� � 10 0 cos <sup>2</sup>*ϕ*� � 1 cos <sup>2</sup>*ϕ*� <sup>þ</sup> 10 0 0 0 2 cos *ϕ*� 0 0 0 0 2 cos *ϕ*�

*q*Φ<sup>0</sup>

*q*Φ<sup>0</sup>

4 , *π* 2 � �

4 , *π* 2 � �

1 þ sin ΔPE ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>4</sup> ,

1 � sin ΔPE ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>4</sup> ,

**P**ð Þ 0 ,

**P**ð Þ 0 *:*

, (15)

2*θ* cos 2*θ* sin *δ*

*:* (12)

(13)

(14)

10 0 0

0 sin *δ* sin 2*θ* � cos 2*θ* sin *δ* cos *δ*

The elements are arranged perpendicularly to the incident light rays, and the rays are parallel to the Z-axis. *θ* is the angle of the fast axis measured from axis X to axis Y (**Figure 3**). *δ* ¼ *δ<sup>y</sup>* � *δ<sup>x</sup>* is the phase difference between the fast and the slow

These matrices make it easy to study the dependence of the light intensity and

polarization on the angles of all transducer elements optical axes. The results obtained make it possible to determine the tolerances for the inaccuracy of the optical elements installation. Here, for simplicity, the light rays were considered as plane wave rays that fall at right angles to the surface of each element of the optical layout. We neglected the point source of light. After substituting all Muller matrixes and taking the first components, the photocurrents *I*<sup>1</sup> and *I*<sup>2</sup> of the photodetectors

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>4</sup> <sup>Φ</sup>0h i¼ **<sup>e</sup>**1, **<sup>L</sup>**1**e**<sup>1</sup> <sup>4</sup>*n*<sup>2</sup>

ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>4</sup> <sup>Φ</sup>0h i¼ **<sup>e</sup>**1, **<sup>L</sup>**2**e**<sup>1</sup> <sup>4</sup>*n*<sup>2</sup>

where *q* – photodetector quantum efficiency, *n* ¼ *n*2*=n*<sup>1</sup> –relative refractive index,

**<sup>R</sup>**ð Þ 0, <sup>Δ</sup>PE **<sup>R</sup>** � *<sup>π</sup>*

**<sup>R</sup>**ð Þ 0, <sup>Δ</sup>PE **<sup>R</sup>** � *<sup>π</sup>*

*<sup>I</sup>*out <sup>¼</sup> *<sup>k</sup> <sup>I</sup>*<sup>1</sup> � *<sup>I</sup>*<sup>2</sup> *I*<sup>1</sup> þ *I*<sup>2</sup>

Φ<sup>0</sup> – light intensity, ΔPE – phase shift caused by the light ray passed through the photoelastic element, **e**<sup>1</sup> is the identity matrix, **L**<sup>1</sup> and **L**<sup>2</sup> are linear transformations:

0 1ð Þ � cos *<sup>δ</sup>* cos 2*<sup>θ</sup>* sin 2*<sup>θ</sup>* cos *<sup>δ</sup>* cos 22*<sup>θ</sup>* <sup>þ</sup> sin <sup>2</sup>

the Y-axis (**Figure 3**).

for linear delay [38]:

0 cos 22*<sup>θ</sup>* <sup>þ</sup> cos *<sup>δ</sup>* sin <sup>2</sup>

sin 2*ϕ* sin 2*ψ* 2 sin *ϕ*<sup>þ</sup> cos *ϕ*� � �<sup>2</sup>

PD1 and PD2 (**Figure 3**) take the form:

**4.2 Transducer output signal**

**44**

*<sup>I</sup>*<sup>1</sup> <sup>¼</sup> *<sup>q</sup>* ð Þ <sup>4</sup>*<sup>n</sup>*

*<sup>I</sup>*<sup>2</sup> <sup>¼</sup> *<sup>q</sup>* ð Þ <sup>4</sup>*<sup>n</sup>*

2

2

**<sup>L</sup>**<sup>1</sup> <sup>¼</sup> **<sup>P</sup>** � *<sup>π</sup>*

**<sup>L</sup>**<sup>2</sup> <sup>¼</sup> **<sup>P</sup>** *<sup>π</sup>*

The output signal *I*out of the electronic circuit is

4 � �

4 � �

The Mueller matrix for refraction [41] in a PE is:

where *ϕ* – incidence angle, *ψ* – refraction angle, *ϕ*� ¼ *ϕ* � *ψ*.

**R**ð Þ¼ *θ*, *δ*

*Optoelectronics*

axes (the delay).

**T**ð Þ¼ *ϕ*, *ψ*

*(a) 3D model of the photoelastic element in the guard ring made of steel. Force F is applied to the top and bottom of the conical elements; (b) 1/8 part of the model; (c) model dimensions are indicated in millimeters [30].*

where *k* is a proportionality factor determined by the transducer electrical circuit parameters. As a result, the signal, after the electronic circuit [36], takes the following form:

$$I\_{\rm out} = k \sin \Delta\_{\rm PE} \approx k \Delta\_{\rm PE}, \text{ at } \Delta\_{\rm PE} \ll 1 \tag{16}$$

The equation shows that the change of output signal *dI*out is directly proportional to the phase shift change *d*ΔPE that is caused by the change of stresses in the photoelastic element.

#### **4.3 Simulations of stresses and deformations in the photoelastic element**

We used the COMSOL Multiphysics™ package and the finite-element method for the simulations of the spatial distribution of Δ*σ* ¼ *σ<sup>y</sup>* � *σ<sup>x</sup>* in the PE, and PE deformation under the force *F*.

The 3D model of the PE encased in the steel guard ring is shown in **Figure 5a** [30, 31]. The upper and lower steel conical elements transmitted the external force to the guard ring. The embedded in COMSOL Multiphysics parameters of the material needed for the calculation of the deformation were used. Due to the PE symmetry, the simulation was only for the 1/8 part of it as shown in **Figure 5b**. The dimensions of this model part are shown in detail in **Figure 5c**.

#### **5. Simulation and experimental results and analysis**

The results of accurate numerical simulations and experiments are also detailed in [30, 31], here is a summary of them.

The strain gauge sensitivity determination was based on the experimental determination of the minimum detectable force and numerical simulation of the deformation occurring under the force action. We used the device "rhombus" with frictionless hinges to apply a calibrated force to the photoelastic element (**Figure 6a**).

Evaluating *F* we get:

*DOI: http://dx.doi.org/10.5772/intechopen.94082*

determined as well.

300 microamperes.

calculated.

**47**

**5.1 Minimum detectable force**

*F* ¼ *Fa*tg*α* ¼ *Fa*

setup. The value obtained was used to determine the minimum detectable PE deformation *ε*min ¼ *dL=L*. Further, the gauge factor was obtained by numerical simulation (GFsim) and was compared with experimentally measured GFexp � ). The dynamic range DR and transducer sensitivity *S* to force and deformation were

magnitude of the force applied to the photoelastic element:

*Piezo-Optical Transducers in High Sensitive Strain Measurements*

record it to a computer using the multimeter software.

measurement accuracy or the minimum detectable force is:

**5.2 Photoelastic element deformations**

Substituting the values used: *F*<sup>a</sup> = 7.848 N, *r* = 6.5 mm, *l* = 75 mm, we get the

The minimum detectable force *F*minwas measured using the described experimental

We used an analogue loop interface with current 20 mA according to standard IEC 62056–21/DIN 66258. The electric current in the analogue loop is independent of the cable resistance (its length), load resistance, EMF inductive interference, and supply voltage. Therefore, such an interface is more preferable for information transfer with remote control. The circuit allowed us to simultaneously power the transducer and generate the output signal in range 4–20 milliamps using a two-wire cable. The multimeter Agilent 34461A was used to measure the output signal and

Typical time dependence of the sensor output current *I*out under the rhombus loading by the calibrated weights is shown in **Figure 7** [30, 31]. The sensor load was as follows. First, the initial preload was applied to remove the backlash. Then the rhombus was sequentially loaded with four equal calibrated weights, each providing the force of *F* = 0.68 N. The output signal was averaged for the four weights. The averaged output signal magnitude corresponding to force *F* = 0.68 N was around

The random noise Δ*I*out of the output current *I*out was analyzed to calculate the minimum sensor sensitivity to the applied force (noise equivalent force). We used the first 20 seconds of the time dependence, before the preload (insert in **Figure 7**) to calculate the standard output signal magnitude deviation according to the normal probability distribution for the random error (Gaussian distribution). The experimental data processing yields the standard deviation magnitude *σ* = 0.1278 μA, which is the commonly accepted measurement accuracy. As a result, the force

*F*min ¼ 0*:*1278 ½ �� μA 0*:*68 N½ �*=*300½ � μA ≈0*:*00029 N ¼ 0*:*29 mN*:* (20)

Furthermore, the magnitude of the PE deformation under the force *F*min was

exerted force is estimated 1/4 of the experimentally applied that is 0.29 mN/4 = 0.073

To simplifying the simulation, due to the symmetry of the model, the

mN. The accurate simulations of applied static force 0.073 mN to the model

*r* ffiffiffiffiffiffiffiffiffiffiffiffiffi *l* <sup>2</sup> � *<sup>r</sup>*<sup>2</sup>

<sup>p</sup> *:* (18)

*F* ¼ 0*:*68 N*:* (19)

**Figure 6.**

*(a) Rhombus photo with the transducer; (b) Force application scheme; (c) Diagram of forces in the rhombus with a fixed transducer upon the application of calibrated force Fa.*

The rhombus with a fixed sensor was placed into the device for applying a calibrated force to the rhombus (**Figure 6b**). The rhombus was firmly restrained from one side and a calibrated force *F*<sup>a</sup> was applied to the other side along the main axis of the rhombus symmetry pattern. The force was created by a lever mechanism with calibrated weights. The lever mechanism has the force transmission ratio of 1:8. The calibrated weight was 0.1 kgf. Thus, the weight applied to the rhombus was 0.8 kgf (7.848 N). The rhombus design ensured the force was applied to the PE in the direction perpendicular to the optical axis of the piezo-optical transducer (**Figure 6b**). To this end, the force from the rhombus was transmitting on the PE by means of conical tips that rested against the tapered holes in the guard ring (**Figure 5**). This joint provides weak stress distribution dependence in the photoelastic element on the deviation of the optical axis from the rhombus axis, due to the mobility of these elements relative to each other. In the experiments, this deviation did not exceed 1 degree, which gives the stress magnitude deviation in the PE working area (**Figure 3**) obtained by the numerical simulation is less than 0.02% and can be neglected.

A diagram of the forces generated in the rhombus with a fixed sensor upon calibrated force *F*<sup>a</sup> application is shown in **Figure 6c**. *T* – tension force of the rhombus shoulder, *l* – length of the rhombus shoulder, *r* – radius of the photoelastic element in the guard ring, *α* – the angle between the shoulder and the vertical axis of the rhombus, *F* – the sought force applied to the photoelastic element perpendicular to the optical axis of the piezo-optical transducer.

The equations for the static forces are:

$$\begin{aligned} F\_{\mathfrak{a}} - 2T \text{cos}\mathfrak{a} &= \mathbf{0}, \\ F - 2T \text{sin}\mathfrak{a} &= \mathbf{0}. \end{aligned} \tag{17}$$

*Piezo-Optical Transducers in High Sensitive Strain Measurements DOI: http://dx.doi.org/10.5772/intechopen.94082*

Evaluating *F* we get:

$$F = F\_a \text{tg} a = F\_a \frac{r}{\sqrt{l^2 - r^2}}.\tag{18}$$

Substituting the values used: *F*<sup>a</sup> = 7.848 N, *r* = 6.5 mm, *l* = 75 mm, we get the magnitude of the force applied to the photoelastic element:

$$F = 0.68 \text{ N.}\tag{19}$$

The minimum detectable force *F*minwas measured using the described experimental setup. The value obtained was used to determine the minimum detectable PE deformation *ε*min ¼ *dL=L*. Further, the gauge factor was obtained by numerical simulation (GFsim) and was compared with experimentally measured GFexp � ). The dynamic range DR and transducer sensitivity *S* to force and deformation were determined as well.

#### **5.1 Minimum detectable force**

We used an analogue loop interface with current 20 mA according to standard IEC 62056–21/DIN 66258. The electric current in the analogue loop is independent of the cable resistance (its length), load resistance, EMF inductive interference, and supply voltage. Therefore, such an interface is more preferable for information transfer with remote control. The circuit allowed us to simultaneously power the transducer and generate the output signal in range 4–20 milliamps using a two-wire cable. The multimeter Agilent 34461A was used to measure the output signal and record it to a computer using the multimeter software.

Typical time dependence of the sensor output current *I*out under the rhombus loading by the calibrated weights is shown in **Figure 7** [30, 31]. The sensor load was as follows. First, the initial preload was applied to remove the backlash. Then the rhombus was sequentially loaded with four equal calibrated weights, each providing the force of *F* = 0.68 N. The output signal was averaged for the four weights. The averaged output signal magnitude corresponding to force *F* = 0.68 N was around 300 microamperes.

The random noise Δ*I*out of the output current *I*out was analyzed to calculate the minimum sensor sensitivity to the applied force (noise equivalent force). We used the first 20 seconds of the time dependence, before the preload (insert in **Figure 7**) to calculate the standard output signal magnitude deviation according to the normal probability distribution for the random error (Gaussian distribution). The experimental data processing yields the standard deviation magnitude *σ* = 0.1278 μA, which is the commonly accepted measurement accuracy. As a result, the force measurement accuracy or the minimum detectable force is:

$$F\_{\rm min} = 0.1278 \,\mathrm{[\mu A]} \times 0.68 \,\mathrm{[N]} / \Re 00 \,\mathrm{[\mu A]} \approx 0.00029 \,\mathrm{N} = 0.29 \,\mathrm{mN}.\tag{20}$$

Furthermore, the magnitude of the PE deformation under the force *F*min was calculated.

#### **5.2 Photoelastic element deformations**

To simplifying the simulation, due to the symmetry of the model, the exerted force is estimated 1/4 of the experimentally applied that is 0.29 mN/4 = 0.073 mN. The accurate simulations of applied static force 0.073 mN to the model

The rhombus with a fixed sensor was placed into the device for applying a calibrated force to the rhombus (**Figure 6b**). The rhombus was firmly restrained from one side and a calibrated force *F*<sup>a</sup> was applied to the other side along the main axis of the rhombus symmetry pattern. The force was created by a lever mechanism with calibrated weights. The lever mechanism has the force transmission ratio of 1:8. The calibrated weight was 0.1 kgf. Thus, the weight applied to the rhombus was 0.8 kgf (7.848 N). The rhombus design ensured the force was applied to the PE in the direction perpendicular to the optical axis of the piezo-optical transducer (**Figure 6b**). To this end, the force from the rhombus was transmitting on the PE by

*with a fixed transducer upon the application of calibrated force Fa.*

*(a) Rhombus photo with the transducer; (b) Force application scheme; (c) Diagram of forces in the rhombus*

means of conical tips that rested against the tapered holes in the guard ring (**Figure 5**). This joint provides weak stress distribution dependence in the

and can be neglected.

**46**

**Figure 6.**

*Optoelectronics*

photoelastic element on the deviation of the optical axis from the rhombus axis, due to the mobility of these elements relative to each other. In the experiments, this deviation did not exceed 1 degree, which gives the stress magnitude deviation in the PE working area (**Figure 3**) obtained by the numerical simulation is less than 0.02%

A diagram of the forces generated in the rhombus with a fixed sensor upon calibrated force *F*<sup>a</sup> application is shown in **Figure 6c**. *T* – tension force of the rhombus shoulder, *l* – length of the rhombus shoulder, *r* – radius of the photoelastic element in the guard ring, *α* – the angle between the shoulder and the vertical axis of the rhombus, *F* – the sought force applied to the photoelastic element

*F*<sup>a</sup> � 2*T*cosα ¼ 0,

*<sup>F</sup>* � <sup>2</sup>*T*sin<sup>α</sup> <sup>¼</sup> <sup>0</sup>*:* (17)

perpendicular to the optical axis of the piezo-optical transducer.

The equations for the static forces are:

**Figure 7.** *Time dependence of the transducer output signal Iout when the load is applied consistently by means of identical calibrated weights.*

(**Figure 5b**) yield the magnitude of the model deformation along the radius and along the axis of applied load *dL*1*=*<sup>2</sup> = �0.00175 nm (i.e., this value by which the radius of the PE decreases along the applied force axis). To determine the PE diameter deformation, the result must be multiplied by 2. The resulting deformation is

$$dL = 0.0035 \text{ nm} = 3.5 \times 10^{-12} \text{ m.} \tag{21}$$

The *E*<sup>∗</sup> value is somewhat smaller than the Young modulus value of fused quartz

Now we need to define the gauge factor that works directly with photodetector output signals. If we take into account the output signal *I*out with proportionality

> <sup>¼</sup> *<sup>I</sup>*<sup>1</sup> � *<sup>I</sup>*<sup>2</sup> *I*<sup>1</sup> þ *I*<sup>2</sup>

Taking into account the precise quarter-wave plate parameters (thickness *d*λ*=*<sup>4</sup> = 40 μm, Δ*n* ¼ *n*<sup>o</sup> � *n*<sup>e</sup> ¼0.038 – characteristic of the quarter-wave plate birefringence), phase difference <sup>Δ</sup> <sup>¼</sup> <sup>Δ</sup>PE and relative deformation magnitude *<sup>ε</sup>*min <sup>¼</sup>2.92 � <sup>10</sup>�10, the

> 2*πd*λ*=*<sup>4</sup> *λ*

The simulated GFsim value is somewhat smaller than the theoretical value GFtheor (Eq. (8)). This is due to the selected PE design, which determines the magnitude of the stresses (*σ<sup>y</sup>* � *σx*) in the PE working area for a given applied force value, and the

In order to determine the GF dependence on the PE shape (value of *h* in **Figure 4**), the relative deformation *ε* = *dL/L* in Eq. (7) must be fixed in contrast to

When varying the geometric parameters of the PE, the magnitude of the force was chosen so as to provide the same PE deformation in the direction of force application (see **Figure 4**), namely, *dL* = 100 nm. **Figure 8** shows the resulting dependence of GF on the parameter *h* [31]. It can be seen from **Figure 8** that the

Δ*n* sin <sup>2</sup>*π<sup>d</sup>*

were *I*<sup>0</sup> is the *I*<sup>1</sup> signal from the photodetector PD1 without applied force

¼ GFsim

*dL*

*λ K*Δ*σ* 

*KE*<sup>∗</sup> , (25)

*<sup>L</sup>* , (26)

¼ 7389 (27)

*E* = 70 GPa, due to the chosen PE design. Thus, Eq. (7) takes the form:

where the *E*<sup>∗</sup> value is determined by the PE design.

*Piezo-Optical Transducers in High Sensitive Strain Measurements*

*DOI: http://dx.doi.org/10.5772/intechopen.94082*

<sup>¼</sup> *<sup>I</sup>*<sup>1</sup> � *<sup>I</sup>*<sup>0</sup> *I*0

simulated piezo-optical transducer gauge factor GFsim can be calculated:

*dI*out *I*out

(ΔPE ¼ 0) and GFsim is the simulated gauge factor value.

<sup>¼</sup> <sup>1</sup> *ε*min sin

sin *δλ<sup>=</sup>*<sup>4</sup> ¼ 0*:*947 (not 1*:*0 for perfect quarter-wave plate).

**5.4 Gauge factor dependence on photoelastic element shape**

factor *k* ¼ 1, we get

GFsim <sup>¼</sup> sin *δλ<sup>=</sup>*<sup>4</sup> sin <sup>Δ</sup>PE

the method shown in **Figure 4**.

*Piezo-optical sensor gauge factor versus the parameter h.*

**Figure 8.**

**49**

*ε*min

GF <sup>¼</sup> <sup>2</sup>*π<sup>d</sup> λ*

Thus, it is assumed that it is the minimum of the absolute deformation detectable by this transducer. And it corresponds to the relative deformation

$$e\_{\rm min} = dL/L = 3.5 \times 10^{-12} \text{ m} / 12 \times 10^{-3} \text{ m} \approx 2.92 \times 10^{-10},\tag{22}$$

where 12 � <sup>10</sup>�<sup>3</sup> <sup>m</sup> – the photoelastic element diameter.

This result is significantly better than that for the known industrial deformation sensors (*ε*min > 10�<sup>8</sup> ).

#### **5.3 Piezo-optical transducer gauge factor**

The accurate numerical simulation of the stresses which are rising in the PE working area under the applied force *F*min = 0.29 mN yields the magnitude:

$$
\Delta \sigma = \sigma\_\circ - \sigma\_\times = 17.11 \text{ Pa.} \tag{23}
$$

The "effective" elasticity modulus *E*<sup>∗</sup> for present PE design can be calculated according to Hooke law:

$$E^\* = \frac{\Delta \sigma}{\varepsilon} = \frac{17.11 \text{ Pa}}{2.92 \times 10^{-10}} = 58.6 \text{ GPa.} \tag{24}$$

*Piezo-Optical Transducers in High Sensitive Strain Measurements DOI: http://dx.doi.org/10.5772/intechopen.94082*

The *E*<sup>∗</sup> value is somewhat smaller than the Young modulus value of fused quartz *E* = 70 GPa, due to the chosen PE design. Thus, Eq. (7) takes the form:

$$\text{GF} = \frac{2\pi d}{\lambda} KE^\*,\tag{25}$$

where the *E*<sup>∗</sup> value is determined by the PE design.

Now we need to define the gauge factor that works directly with photodetector output signals. If we take into account the output signal *I*out with proportionality factor *k* ¼ 1, we get

$$\frac{dI\_{\text{out}}}{I\_{\text{out}}} = \frac{I\_1 - I\_0}{I\_0} = \frac{I\_1 - I\_2}{I\_1 + I\_2} = \text{GF}\_{\text{sim}} \frac{dL}{L},\tag{26}$$

were *I*<sup>0</sup> is the *I*<sup>1</sup> signal from the photodetector PD1 without applied force (ΔPE ¼ 0) and GFsim is the simulated gauge factor value.

Taking into account the precise quarter-wave plate parameters (thickness *d*λ*=*<sup>4</sup> = 40 μm, Δ*n* ¼ *n*<sup>o</sup> � *n*<sup>e</sup> ¼0.038 – characteristic of the quarter-wave plate birefringence), phase difference <sup>Δ</sup> <sup>¼</sup> <sup>Δ</sup>PE and relative deformation magnitude *<sup>ε</sup>*min <sup>¼</sup>2.92 � <sup>10</sup>�10, the simulated piezo-optical transducer gauge factor GFsim can be calculated:

$$\text{GF}\_{\text{sim}} = \frac{\sin \delta\_{\lambda/4} \sin \Delta\_{\text{PE}}}{\varepsilon\_{\text{min}}} = \frac{1}{\varepsilon\_{\text{min}}} \sin \left( \frac{2 \pi d\_{\lambda/4}}{\lambda} \Delta n \right) \sin \left( \frac{2 \pi d}{\lambda} K \Delta \sigma \right) = 7389 \tag{27}$$

The simulated GFsim value is somewhat smaller than the theoretical value GFtheor (Eq. (8)). This is due to the selected PE design, which determines the magnitude of the stresses (*σ<sup>y</sup>* � *σx*) in the PE working area for a given applied force value, and the sin *δλ<sup>=</sup>*<sup>4</sup> ¼ 0*:*947 (not 1*:*0 for perfect quarter-wave plate).

#### **5.4 Gauge factor dependence on photoelastic element shape**

In order to determine the GF dependence on the PE shape (value of *h* in **Figure 4**), the relative deformation *ε* = *dL/L* in Eq. (7) must be fixed in contrast to the method shown in **Figure 4**.

When varying the geometric parameters of the PE, the magnitude of the force was chosen so as to provide the same PE deformation in the direction of force application (see **Figure 4**), namely, *dL* = 100 nm. **Figure 8** shows the resulting dependence of GF on the parameter *h* [31]. It can be seen from **Figure 8** that the

**Figure 8.** *Piezo-optical sensor gauge factor versus the parameter h.*

(**Figure 5b**) yield the magnitude of the model deformation along the radius and along the axis of applied load *dL*1*=*<sup>2</sup> = �0.00175 nm (i.e., this value by which the radius of the PE decreases along the applied force axis). To determine the PE diameter defor-

*Time dependence of the transducer output signal Iout when the load is applied consistently by means of identical*

Thus, it is assumed that it is the minimum of the absolute deformation detect-

This result is significantly better than that for the known industrial deformation

The accurate numerical simulation of the stresses which are rising in the PE working area under the applied force *F*min = 0.29 mN yields the magnitude:

The "effective" elasticity modulus *E*<sup>∗</sup> for present PE design can be calculated

<sup>ε</sup> <sup>¼</sup> <sup>17</sup>*:*11 Pa

*<sup>ε</sup>*min <sup>¼</sup> *dL=<sup>L</sup>* <sup>¼</sup> <sup>3</sup>*:*<sup>5</sup> � <sup>10</sup>–<sup>12</sup> <sup>m</sup>*=*<sup>12</sup> � <sup>10</sup>–<sup>3</sup> <sup>m</sup> <sup>≈</sup>2*:*<sup>92</sup> � <sup>10</sup>�10, (22)

*dL* <sup>¼</sup> <sup>0</sup>*:*0035 nm <sup>¼</sup> <sup>3</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>12</sup> <sup>m</sup>*:* (21)

Δ*σ* ¼ *σ<sup>y</sup>* � *σ<sup>x</sup>* ¼ 17*:*11 Pa*:* (23)

<sup>2</sup>*:*<sup>92</sup> � <sup>10</sup>�<sup>10</sup> <sup>¼</sup> <sup>58</sup>*:*6 GPa*:* (24)

mation, the result must be multiplied by 2. The resulting deformation is

able by this transducer. And it corresponds to the relative deformation

where 12 � <sup>10</sup>�<sup>3</sup> <sup>m</sup> – the photoelastic element diameter.

sensors (*ε*min > 10�<sup>8</sup>

**Figure 7.**

*calibrated weights.*

*Optoelectronics*

according to Hooke law:

**48**

).

**5.3 Piezo-optical transducer gauge factor**

*<sup>E</sup>*<sup>∗</sup> <sup>¼</sup> <sup>Δ</sup><sup>σ</sup>

dependence of GF on *h* is non-monotonic and contains two local maxima apparently due to the contribution of the nonlinearly changing shape of the PE side surfaces into its elastic properties. The changes in GF in the whole range of *h* were about 5.4% of the initial value, which is significantly less than the change in the stress difference Δσav obtained in [32] and shown in **Figure 4**, which was almost 100%. This is due to the fact that as *h* increases, the PE stiffness (effective Young's modulus *E*<sup>∗</sup> ) decreases in the direction of force application, which, in turn, leads to an increase in the relative deformation *dL/L* at the given force and a decrease in GF.

#### **5.5 Piezo-optical transducer parameters**

*Experimental gauge factor.* The direct measurement of the photocurrents (*I*PD1, *I*PD2) from the photodetectors (PD1, PD2 in **Figure 2**) yielded the experimental gauge factor GFexp value

$$\text{GF}\_{\text{exp}} = \frac{I\_{\text{PD1}} - I\_{\text{PD2}}}{I\_{\text{PD1}} + I\_{\text{PD2}}} = 7340. \tag{28}$$

with the help of small weights, many times less than the nominal load value. The preload for both sensors was 110 lbs. and then the sensors were subsequently loaded with the calibrated weights from 1 gram to 100 grams The results are shown in **Figure 10**. The upper part of **Figure 10** corresponds to the presented piezo-optical transducer and the lower part – to the Load Cell Interface Force™. It can be seen that the piezo-optical transducer accuracy is approximately an order of magnitude higher than that for the Load Cell. This is less than the predicted calculations, and it

*Time dependence of the piezo-optical transducer (upper) and Load Cell (lower) output signals at the sequential*

*(a)Ultra precision LowProfile™ Load Cell Interface force™; (b) Photos of our Load Element (left) with the*

*piezo-optical transducer (right) and installed into a deadweight machine.*

*Piezo-Optical Transducers in High Sensitive Strain Measurements*

*DOI: http://dx.doi.org/10.5772/intechopen.94082*

**Figure 9.**

**Figure 10.**

**51**

*load with calibrated weights 1, 3, 5, 10, 20, 50, 100 g.*

This agrees well with the simulated gauge factor GFsim and confirms the accuracy of the created transducer model.

*Dynamic range.* In our transducer design, as mentioned above, the PE has been affixed to the loading element in the initially stressed state that ensures the transducer operates at compressing and stretching deformation.

The transducer output signal varied from 4 to 20 mA. We set the initial output current value of 12 mA, corresponding to zero strain. The maximum change in the output signal *I*range equal to �8 mA, then the dynamic range DR will be

$$\text{DR} = I\_{\text{range}} / \sigma = 8 \times 10^{-3} \text{ A} / 0.1278 \times 10^{-6} \text{ A} \approx 6.2 \times 10^{4} \text{.} \tag{29}$$

The resulting dynamic range value is much higher than the known values for strain gauges.

S*ensitivity.* The sensitivity *S* (the transfer function slope) was as follows:

$$\mathbf{S}\_{\mathrm{F}} = \mathbf{300}\,\mathrm{\mu A/0.68 N} \approx 441\,\mathrm{\mu A/N} \text{ to the force and} \tag{30}$$

*SL* ¼ 0*:*1278 μA*=*0*:*0035 nm ≈ 36*:*6 μA*=*nm to the deformation*:*

#### **6. Testing the piezo-optical transducer**

For experimental verification of the claimed parameters of the piezo-optical sensor, we carried out comparative testing with the most sensitive of the strainresistive gauge sensor Ultra Precision LowProfile™ Load Cell Interface Force™ (**Figure 9b**) [42] used only to calibrate the deadweight machines due to its complexity and high cost. The Load Cell, selected for comparison, had the nominal load of 2000 lbs. (907.185 kg) and is based on a strain-resistive sensor. Our Load Element was a parallelepiped made of hardened steel with a transducer installed in it as shown in **Figure 9a** and had a nominal load of 1000 kg, which is close the Load Cell nominal load. Tests took place in a certified laboratory Detroit Calibration Lab Trescal [43] laboratory using a deadweight machine of the National Institute of Standards and Technology (NIST) [44].

The Load Element and then Load Cell Interface Force™ were installed in a deadweight machine where they were consistently loaded by means of calibration weights. The most striking results of comparative tests were obtained under load

*Piezo-Optical Transducers in High Sensitive Strain Measurements DOI: http://dx.doi.org/10.5772/intechopen.94082*

#### **Figure 9.**

dependence of GF on *h* is non-monotonic and contains two local maxima apparently due to the contribution of the nonlinearly changing shape of the PE side surfaces into its elastic properties. The changes in GF in the whole range of *h* were about 5.4% of the initial value, which is significantly less than the change in the stress difference Δσav obtained in [32] and shown in **Figure 4**, which was almost 100%. This is due to the fact that as *h* increases, the PE stiffness (effective Young's

modulus *E*<sup>∗</sup> ) decreases in the direction of force application, which, in turn, leads to an increase in the relative deformation *dL/L* at the given force and a decrease in GF.

*Experimental gauge factor.* The direct measurement of the photocurrents (*I*PD1, *I*PD2) from the photodetectors (PD1, PD2 in **Figure 2**) yielded the experimental

*I*PD1 þ *I*PD2

*Dynamic range.* In our transducer design, as mentioned above, the PE has been

The transducer output signal varied from 4 to 20 mA. We set the initial output current value of 12 mA, corresponding to zero strain. The maximum change in the

The resulting dynamic range value is much higher than the known values for

*SL* ¼ 0*:*1278 μA*=*0*:*0035 nm ≈ 36*:*6 μA*=*nm to the deformation*:*

For experimental verification of the claimed parameters of the piezo-optical sensor, we carried out comparative testing with the most sensitive of the strainresistive gauge sensor Ultra Precision LowProfile™ Load Cell Interface Force™ (**Figure 9b**) [42] used only to calibrate the deadweight machines due to its complexity and high cost. The Load Cell, selected for comparison, had the nominal load of 2000 lbs. (907.185 kg) and is based on a strain-resistive sensor. Our Load Element was a parallelepiped made of hardened steel with a transducer installed in it as shown in **Figure 9a** and had a nominal load of 1000 kg, which is close the Load Cell nominal load. Tests took place in a certified laboratory Detroit Calibration Lab Trescal [43] laboratory using a deadweight machine of the National Institute of

The Load Element and then Load Cell Interface Force™ were installed in a deadweight machine where they were consistently loaded by means of calibration weights. The most striking results of comparative tests were obtained under load

S*ensitivity.* The sensitivity *S* (the transfer function slope) was as follows:

DR <sup>¼</sup> *<sup>I</sup>*range*=<sup>σ</sup>* <sup>¼</sup> <sup>8</sup> � <sup>10</sup>�<sup>3</sup> <sup>A</sup>*=*0*:*<sup>1278</sup> � <sup>10</sup>�<sup>6</sup> <sup>A</sup> <sup>≈</sup>6*:*<sup>2</sup> � <sup>10</sup><sup>4</sup>*:* (29)

*SF* ¼ 300 μA*=*0*:*68 N ≈441 μA*=*N to the force and (30)

This agrees well with the simulated gauge factor GFsim and confirms the

¼ 7340*:* (28)

GFexp <sup>¼</sup> *<sup>I</sup>*PD1 � *<sup>I</sup>*PD2

affixed to the loading element in the initially stressed state that ensures the

output signal *I*range equal to �8 mA, then the dynamic range DR will be

transducer operates at compressing and stretching deformation.

**5.5 Piezo-optical transducer parameters**

accuracy of the created transducer model.

**6. Testing the piezo-optical transducer**

Standards and Technology (NIST) [44].

gauge factor GFexp value

*Optoelectronics*

strain gauges.

**50**

*(a)Ultra precision LowProfile™ Load Cell Interface force™; (b) Photos of our Load Element (left) with the piezo-optical transducer (right) and installed into a deadweight machine.*

with the help of small weights, many times less than the nominal load value. The preload for both sensors was 110 lbs. and then the sensors were subsequently loaded with the calibrated weights from 1 gram to 100 grams The results are shown in **Figure 10**. The upper part of **Figure 10** corresponds to the presented piezo-optical transducer and the lower part – to the Load Cell Interface Force™. It can be seen that the piezo-optical transducer accuracy is approximately an order of magnitude higher than that for the Load Cell. This is less than the predicted calculations, and it

#### **Figure 10.**

*Time dependence of the piezo-optical transducer (upper) and Load Cell (lower) output signals at the sequential load with calibrated weights 1, 3, 5, 10, 20, 50, 100 g.*

is due to the fact that the Load Cell contains a vacuum chamber where a complex and expensive circuit is located to stabilize the output signal and reduce the noises. In our sensor, we used a design which was as simple as possible since the sensor is designed for a wide range of consumers. Nevertheless, this design showed higher sensitivity compared to the calibration Load Cell.

The sensor can be used in all cases where winch mechanisms are used, for example,

The theoretical, technological and design foundations for the highly sensitive piezo-optical transducers creation for strain gauges have been developed. It has

• high sensitivity to mechanical stresses, significantly exceeding the sensitivity of strain-resistive, piezoelectric and fiber-optic gauges and allowing to register the value of force less than 3 <sup>10</sup><sup>4</sup> N, with a transfer function slope of <sup>≈</sup>

;

• new functionalities corresponding to the sensitivity to relative deformations

calibration laboratory Trescal (Detroit, USA) and other testing laboratories.

The authors are grateful to Alex Zaguskin, AZ Enterprise L.L.C. for his help in organizing the transducer test in the Detroit Calibration Lab Trescal, as well as to

; the specified sensitivity is documented by tests in certified

• absence of hysteresis within <sup>≈</sup> 1.7 <sup>10</sup><sup>5</sup> of the nominal load;

Due to its high sensitivity, the sensor can be used for remote deformation monitoring by mounting at a certain distance away from the measured deformations zone: in bridge structures, cars and railway wagons weight remote control,

in mines, escalators, moving walks, conveyors, cranes, etc.

*Piezo-Optical Transducers in High Sensitive Strain Measurements*

*DOI: http://dx.doi.org/10.5772/intechopen.94082*

been shown experimentally that such sensors have:

440 μA/N and ≈37 μA/nm;

• resistance to overload;

less than 10<sup>9</sup>

**Acknowledgements**

**53**

• wide dynamic range, up to 6 104

Robert Bolthouse for his help with the tests.

liquids and gases flow control.

**8. Conclusions**
