**2.4 Piezo-optical effect and piezo-optical transducers**

The piezo-optical effect (also called "photoelasticity") used for precision stress (deformation) measurements is known since the 1930s [28]. If a light wave with a linear (circular) polarization (**Figure 2**) is incident upon transparent material


#### **Table 1.**

GF <sup>¼</sup> *dR=<sup>R</sup>*

**2.3 Piezoelectric effect and piezoelectric transducers**

Using Hooke's law *σ* = *E*(*dL/L*) and Eq. (4) we get:

GF <sup>¼</sup> <sup>Δ</sup>*<sup>q</sup>*

*dL=<sup>L</sup>* <sup>¼</sup> *dij<sup>σ</sup>*

is described by the Equation [26].

range of 2–6 [7].

*Optoelectronics*

resistive sensors [25]:

material.

**38**

**2.2 Fiber-optic strain gauge sensors**

*dL=<sup>L</sup>* <sup>¼</sup> *<sup>d</sup>ρ=<sup>ρ</sup>*

*dL=<sup>L</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>ν</sup>* � *<sup>ν</sup>*<sup>2</sup> *dL*

It can be seen from Eq. (2) that the GF magnitude depends only on the properties of the conductor material (*dρ=ρ*) and is *independent* of the strain gauge design. Typical gauge factor values for the materials used to create the sensors lie in the

The sensitive element of the fiber-optic sensors is a Bragg fiber grating which is an optical fiber with a periodically changing refractive index (**Figure 1b**). When light passes through such a structure, part of it with a specific Bragg wavelength (*λ*B) is reflected, and the rest is transmitted further. The reflected light wavelength *λ*<sup>B</sup> is given by the relation *λ*<sup>B</sup> = 2*nL*, where *n* is the effective refractive index of the optical fiber and *L* is the distance between the gratings or the grating period. Stretching/compression of the fiber changes the distance *L* and the refractive index *n* resulting in a change in Bragg wavelength (*λ*<sup>B</sup> � *dλ*B), which is recorded by the optical system. The fiber-optic strain sensor GF is found by analogy with strain-

GF <sup>¼</sup> *<sup>d</sup>λ*B*=λ*<sup>B</sup>

where *dλ* <sup>B</sup> is the change in the reflected-light wavelength during deformation of the fiber grating and *dL/L* is the relative deformation of the grating. According to [25], the fiber-optic strain sensor GF is about 0.78, which is markedly lower than the strain-resistive sensor GFs. As in case of strain-resistive sensors, it does not depend on the sensor design and is determined by the properties of the fiber-optic

A piezoelectric transducer converts a mechanical force into an electric charge. Its

where Δ*q* is the surface charge density; *dij* is the piezoelectric modulus described by a 3 � 6 matrix with typical component values in the range of 10�10–10�<sup>12</sup> C/N; *<sup>σ</sup>* is the stress in the material under the external load. The sensitivity of these sensors is described by the piezoelectric modulus which complicates their comparison with strain-resistive and fiber-optic sensors. Similarly to Eqs. (2) and (3), the piezoelectric sensor GF should be inversely proportional to the relative deformation *dL/L*.

*dL=<sup>L</sup>* <sup>¼</sup> *dijE dL* ð Þ *<sup>=</sup><sup>L</sup>*

operation is based on the piezoelectric effect which entails the occurrence of dielectric polarization under mechanical stresses (**Figure 1c**). The density of the electric charge induced on the piezoelectric element surface under an external load

*L* 

*dL=<sup>L</sup>* , (3)

Δ*q* ¼ *dijσ*, (4)

*dL=<sup>L</sup>* <sup>¼</sup> *dijE:* (5)

*:* (2)

*Parameters of some piezoelectric materials.*

**Figure 2.** *Effect of photoelasticity.*

(photoelastic element) under the load *F*, an additional phase difference Δ between the polarization components, perpendicular and parallel to the stress axis, arises due to the double refraction [20]. As a result, the light polarization in the general case becomes elliptical. The magnitude of the phase difference Δ is determined by the expression

$$
\Delta = \frac{2\pi d}{\lambda} (n\_\text{o} - n\_\text{e}) = \frac{2\pi d}{\lambda} K (\sigma\_\text{y} - \sigma\_\text{x}) = \frac{2\pi d}{\lambda} K \text{E}\varepsilon,\tag{6}
$$

2.We showed that cruciform photoelastic element (PE) allows us to significantly increase the stresses magnitude in the PE working area under the external force action and, thereby, increase the sensitivity to the force [32].

*Piezo-Optical Transducers in High Sensitive Strain Measurements*

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

3.Fused quartz [33] was chosen as the photoelastic element material although it has a smaller stress-optical coefficient *K* compared to any crystals or solid polymers. However, fused quartz technology is inexpensive and well-developed. No plastic deformation exists in fused quartz and its elastic properties do not change with time. It offers a high compression damage threshold, thus, ensuring an overload resistance and a wide dynamic range of deformation measurement. Thus, there are no hysteresis and parameters degradation in such strain gauges.

4.Due to the cruciform PE, the remaining optical elements can be located within the PE dimension, and, consequently, the transducer can have its own unified casing and its technology is separated from the load cell technology. The attachment to the monitored object is carried out with the help of load elements, the design of which depends on the mounting method [34].

5. In the mounted state, the PE is under the preliminary compressive stresses along two orthogonal directions [35], which ensures: i) a reliable glueless force-closure between the PE and the load element; ii) the operation both in compression and in tension; iii) the temperature independence of the device output signal since

6.The output signal electrical circuit is located inside the transducer shielded housing and has any desired interfaces. As a result, the transducer is a complete device that does not require a secondary signal transducer as is the

As a result, we were able to optimize the transducer design and significantly reduce the production cost while maintaining high field-performance data. To confirm this, we compared its parameters with the parameters of most sensitive strain-resistive sensor used to calibrate the deadweight machines (see Section 6). The optical scheme of the piezo-optical transducer is shown in **Figure 3** and consists of an optically coupled light-emitting diode (LED), a polarizer (Pol), a quarter-wave plate (*λ*/4), a photoelastic element (PE), two analyzers (An1, An2) and two photodetectors (PD1, PD2) [30]. The measured force *F* is applied to the PE.

temperature changes do not change the pre-stressed isotropy.

case with strain-resistive sensors [36].

*Optical scheme of piezo-optical transducer (left) and its design (right).*

**Figure 3.**

**41**

here *d* is the path length of a light beam in the stressed material (photoelastic element thickness in the piezo-optical transducer). *λ* – working wavelength. *n*<sup>o</sup> and *n*<sup>e</sup> – refractive indexes for ordinary and extraordinary rays. *K* – stress-optical coefficient with typical value 10�11–10�<sup>12</sup> m<sup>2</sup> /N. *σ<sup>x</sup>* and *σ<sup>y</sup>* – tensions along and across the applied force in a plane perpendicular to the direction of light propagation. *E* – Young modulus of the optical material. *ε* ¼ *dL=L* – relative deformation of the optical material. In a general case, the stresses in a photoelastic element are described by the Cauchy stress tensor *σij*. Eq. (6) describes the effect of photoelasticity (**Figure 2**).

The GF for the piezo-optical transducer can be determined by analogy with the strain-resistive one, where the measuring parameter is *dU=U*, which is equivalent to the *dR=R* at constant current (Eq. (2)).

The measuring parameter for the piezo-optical effect is the phase difference Δ (Eq. (6)), which is measured by the ellipsometry techniques, so the expression for the piezo-optical GF takes the form:

$$\text{GF} = \frac{\Delta}{\varepsilon} = \frac{2\pi d}{\lambda} \text{KE}.\tag{7}$$

The GF magnitude depends *not only* on the material constants (*K*, *E*) but also on the design of the piezo-optical transducer (*d* and *λ*) (see Eq. (7)). In addition, the stresses magnitude (*σ<sup>y</sup>* � *σx*) in the photoelastic element depends strongly on the photoelastic element design to which the force is applied. This opens up the possibilities for optimizing the piezo-optical transducer parameters to increase its sensitivity to the applied force.

If fused quartz is used as the optical material, the gauge factor value GFtheor can be calculated according to Eq. (7) taking into account *<sup>K</sup>* = 3.5 � <sup>10</sup>�<sup>12</sup> <sup>m</sup><sup>2</sup> /N [29], *E* = 70 GPa, photoelastic element thickness *d* = 4 mm and *λ* = 660 nm at 20°C (conventional LED):

$$\mathrm{GF\_{theor}} = \frac{2\pi \cdot 4 \cdot 10^{-3}}{0.66 \cdot 10^{-6}} 3.5 \cdot 10^{-12} \cdot 7 \cdot 10^{10} = 9330. \tag{8}$$

The GF value is more than *three orders* of magnitude higher than the strain-resistive effect values [30].

#### **3. Piezo-optical transducer of new design**

In order to achieve the set goal of the work, the following was done [30].

1.We have studied the process of piezo-optical transducer output signal generating in detail with the help of accurate numerical simulation. We determined the piezo-optical sensor GF and compared it with other types [30, 31].

(photoelastic element) under the load *F*, an additional phase difference Δ between the polarization components, perpendicular and parallel to the stress axis, arises due to the double refraction [20]. As a result, the light polarization in the general case becomes elliptical. The magnitude of the phase difference Δ is determined by the expression

> 2*πd λ*

here *d* is the path length of a light beam in the stressed material (photoelastic element thickness in the piezo-optical transducer). *λ* – working wavelength. *n*<sup>o</sup> and *n*<sup>e</sup> – refractive indexes for ordinary and extraordinary rays. *K* – stress-optical

across the applied force in a plane perpendicular to the direction of light propagation. *E* – Young modulus of the optical material. *ε* ¼ *dL=L* – relative deformation of the optical material. In a general case, the stresses in a photoelastic element are

The GF for the piezo-optical transducer can be determined by analogy with the strain-resistive one, where the measuring parameter is *dU=U*, which is equivalent to

The measuring parameter for the piezo-optical effect is the phase difference Δ (Eq. (6)), which is measured by the ellipsometry techniques, so the expression for

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

The GF magnitude depends *not only* on the material constants (*K*, *E*) but also on the design of the piezo-optical transducer (*d* and *λ*) (see Eq. (7)). In addition, the stresses magnitude (*σ<sup>y</sup>* � *σx*) in the photoelastic element depends strongly on the photoelastic element design to which the force is applied. This opens up the possibilities for optimizing the piezo-optical transducer parameters to increase its

If fused quartz is used as the optical material, the gauge factor value GFtheor can be calculated according to Eq. (7) taking into account *<sup>K</sup>* = 3.5 � <sup>10</sup>�<sup>12</sup> <sup>m</sup><sup>2</sup>

[29], *E* = 70 GPa, photoelastic element thickness *d* = 4 mm and *λ* = 660 nm at 20°C

The GF value is more than *three orders* of magnitude higher than the

In order to achieve the set goal of the work, the following was done [30].

1.We have studied the process of piezo-optical transducer output signal generating in detail with the help of accurate numerical simulation. We determined the piezo-optical sensor GF and compared it with other types

<sup>0</sup>*:*<sup>66</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>12</sup> � <sup>7</sup> � <sup>10</sup><sup>10</sup> <sup>¼</sup> <sup>9330</sup>*:* (8)

GFtheor <sup>¼</sup> <sup>2</sup>*<sup>π</sup>* � <sup>4</sup> � <sup>10</sup>�<sup>3</sup>

**3. Piezo-optical transducer of new design**

described by the Cauchy stress tensor *σij*. Eq. (6) describes the effect of

GF <sup>¼</sup> <sup>Δ</sup>

*K σ<sup>y</sup>* � *σ<sup>x</sup>*

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

*λ*

/N. *σ<sup>x</sup>* and *σ<sup>y</sup>* – tensions along and

*KE:* (7)

/N

*KEε*, (6)

<sup>Δ</sup> <sup>¼</sup> <sup>2</sup>*π<sup>d</sup>*

coefficient with typical value 10�11–10�<sup>12</sup> m<sup>2</sup>

the *dR=R* at constant current (Eq. (2)).

the piezo-optical GF takes the form:

sensitivity to the applied force.

strain-resistive effect values [30].

(conventional LED):

[30, 31].

**40**

photoelasticity (**Figure 2**).

*Optoelectronics*

*<sup>λ</sup>* ð Þ¼ *<sup>n</sup>*<sup>o</sup> � *<sup>n</sup>*<sup>e</sup>


As a result, we were able to optimize the transducer design and significantly reduce the production cost while maintaining high field-performance data. To confirm this, we compared its parameters with the parameters of most sensitive strain-resistive sensor used to calibrate the deadweight machines (see Section 6).

The optical scheme of the piezo-optical transducer is shown in **Figure 3** and consists of an optically coupled light-emitting diode (LED), a polarizer (Pol), a quarter-wave plate (*λ*/4), a photoelastic element (PE), two analyzers (An1, An2) and two photodetectors (PD1, PD2) [30]. The measured force *F* is applied to the PE.

**Figure 3.** *Optical scheme of piezo-optical transducer (left) and its design (right).*

The analyzers axes are oriented at the angle of 90°. The photoelastic element is the main component of the piezo-optical transducer. The working area of the PE is limited by the part (dashed circle in **Figure 3**) passing through which the light rays hit the photosensitive areas of the photodetectors. The rest of the PE does not participate in photodetectors signals. A feature of the transducer's optical scheme is the separation of the light beam along the front of the incident wave into two beams before falling onto the photoelastic element. This solution allows the use of film polarizers (Polaroid) reducing the optical path of light beams and, consequently, the dimensions of the converter and also allowing the use of an incoherent light source with low power consumption. The size of the optical scheme does not exceed a cubic centimeter.

compared to the ultimate strength of quartz (51.7 MPa) [37]. The dependence (*2*) in **Figure 4b** shows the threshold force *f*th under which the PE breakdown occurs. It can be seen that the damage threshold increases with a change in PE form from square to circular then to rhombus and even continues to grow with an increase in the "dent" depth *h* up to 2 mm. Further increase in *h* resulted in a reduction in *f*th. The threshold begins to decrease rapidly only when *h* > 3.5 mm. Thus, at the same damage threshold for *h* ≈ 3.5 mm, we have a significant increase in stresses in the PE working area.

*Piezo-Optical Transducers in High Sensitive Strain Measurements*

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

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

results obtained are briefly presented here.

**4.1 Simulation of the light propagation**

**s** ¼

**P**ð Þ¼ *θ*

**43**

1 2

*s*0 *s*1 *s*2 *s*3

by the linear transformation according to the Muller formalism [39]:

transducer elements.

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

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

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:

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

sin 2*θ* cos 2*θ* sin 2*θ* sin <sup>2</sup>

1 cos 2*θ* sin 2*θ* 0 cos 2*θ* cos 22*θ* cos 2*θ* sin 2*θ* 0

0 0 00

Φ Φ *p* cos 2*ψ* cos 2*χ* Φ *p* sin 2*ψ* cos 2*χ* Φ *p* sin 2*χ*

2*θ* 0

, (10)

*:* (9)

The phase difference Δ between two orthogonal components of the beam polarization caused by stresses in the PE working area leads to the change in light beams intensities (Φ1, Φ<sup>2</sup> in **Figure 3** on the left) incident on the photodetectors. Accordingly, it leads to the change in the output of electrical signals (*I*1, *I*<sup>2</sup> in **Figure 3** on the left). The transducer electronic circuit generates a differential output signal � ð Þ *I*<sup>1</sup> � *I*<sup>2</sup> *=*ð Þ� *I*<sup>1</sup> þ *I*<sup>2</sup> Δ which is proportional to the Δ*σ* ¼ *σ<sup>y</sup>* � *σ<sup>x</sup>* (Eq. (6)) and to the applied load value.

We have optimized the PE shape and showed that the cruciform PE allows us to significantly increase the stresses magnitude in the PE working area for a given applied force [32]. The results of numerical modeling for stresses in PEs of various shapes (square, circular, rhombic, and cruciform) subjected to the same external force *F* = 4 N are shown in **Figure 4** [30–32]. The Δ*σ* ¼ *σ<sup>y</sup>* � *σ<sup>x</sup>* isolines show the stresses distribution in PEs (Δ*σ* magnitudes are expressed in MPa). The PE working area which light passes when falling on photodetectors is shown with dashed circles. The overall dimension of all PEs was 12 � 12 mm. The calculated points are connected by straight lines just for convenience. It can be seen that the stresses are concentrated near the force application points and they are reducing considerably toward the PE center. Thus, the PE working area falls into the PE part where the stresses are minimal.

The transition to the cruciform PE and the increase in "dent" depth *h* (**Figure 4**) result in the stresses redistribution toward the PE center and the increase in the stress in the PE working area. For the PE shape shown in **Figure 4a** (bottom right), the increase in stresses averaged over the PE working area was 2.1 times higher compared to the square and circular shapes (dependence (1) in **Figure 4b**).

However, it is evident that the mechanical strength of a PE should go down as the "dent" *h* gets deeper. This issue was investigated by calculating the PE damage threshold for various PE shapes. The magnitude of stress arising in PEs of various shapes was

**Figure 4.**

*(a) Isolines of stress difference* Δ*σ for the photoelastic elements of different shapes; (b) The dependences of the stress difference* Δ*σav, averaged over the PE working area (1), and damage threshold fth (2) on depth h [32].*

The analyzers axes are oriented at the angle of 90°. The photoelastic element is the main component of the piezo-optical transducer. The working area of the PE is limited by the part (dashed circle in **Figure 3**) passing through which the light rays hit the photosensitive areas of the photodetectors. The rest of the PE does not participate in photodetectors signals. A feature of the transducer's optical scheme is the separation of the light beam along the front of the incident wave into two beams before falling onto the photoelastic element. This solution allows the use of film polarizers (Polaroid) reducing the optical path of light beams and, consequently, the dimensions of the converter and also allowing the use of an incoherent light source with low power consumption. The size of the optical scheme does not exceed a cubic centimeter.

The phase difference Δ between two orthogonal components of the beam polarization caused by stresses in the PE working area leads to the change in light beams intensities (Φ1, Φ<sup>2</sup> in **Figure 3** on the left) incident on the photodetectors. Accordingly, it leads to the change in the output of electrical signals (*I*1, *I*<sup>2</sup> in **Figure 3** on the left). The transducer electronic circuit generates a differential output signal � ð Þ *I*<sup>1</sup> � *I*<sup>2</sup> *=*ð Þ� *I*<sup>1</sup> þ *I*<sup>2</sup> Δ which is proportional to the Δ*σ* ¼ *σ<sup>y</sup>* � *σ<sup>x</sup>* (Eq. (6))

We have optimized the PE shape and showed that the cruciform PE allows us to significantly increase the stresses magnitude in the PE working area for a given applied force [32]. The results of numerical modeling for stresses in PEs of various shapes (square, circular, rhombic, and cruciform) subjected to the same external force *F* = 4 N are shown in **Figure 4** [30–32]. The Δ*σ* ¼ *σ<sup>y</sup>* � *σ<sup>x</sup>* isolines show the stresses distribution in PEs (Δ*σ* magnitudes are expressed in MPa). The PE working area which light passes when falling on photodetectors is shown with dashed circles. The overall dimension of all PEs was 12 � 12 mm. The calculated points are connected by straight lines just for convenience. It can be seen that the stresses are concentrated near the force application points and they are reducing considerably toward the PE center. Thus, the PE working area falls into the PE part where the stresses are minimal.

The transition to the cruciform PE and the increase in "dent" depth *h* (**Figure 4**) result in the stresses redistribution toward the PE center and the increase in the stress in the PE working area. For the PE shape shown in **Figure 4a** (bottom right), the increase in stresses averaged over the PE working area was 2.1 times higher compared to the square and circular shapes (dependence (1) in **Figure 4b**).

However, it is evident that the mechanical strength of a PE should go down as the "dent" *h* gets deeper. This issue was investigated by calculating the PE damage threshold for various PE shapes. The magnitude of stress arising in PEs of various shapes was

*(a) Isolines of stress difference* Δ*σ for the photoelastic elements of different shapes; (b) The dependences of the stress difference* Δ*σav, averaged over the PE working area (1), and damage threshold fth (2) on depth h [32].*

and to the applied load value.

*Optoelectronics*

**Figure 4.**

**42**

compared to the ultimate strength of quartz (51.7 MPa) [37]. The dependence (*2*) in **Figure 4b** shows the threshold force *f*th under which the PE breakdown occurs. It can be seen that the damage threshold increases with a change in PE form from square to circular then to rhombus and even continues to grow with an increase in the "dent" depth *h* up to 2 mm. Further increase in *h* resulted in a reduction in *f*th. The threshold begins to decrease rapidly only when *h* > 3.5 mm. Thus, at the same damage threshold for *h* ≈ 3.5 mm, we have a significant increase in stresses in the PE working area.
