Abstract.
Author developed the onedimensional mathematical model of photoacoustic effect with piezoelectric registration in thermal thick optical opaque sample with subsurface defect. The model is developed in view of the assumptions of ideally rigid connection piezosensor with sample and free fastening of sample in a microscope cell. This model allows to calculate a amplitude and phase of the photoacoustic signal, to restore the information about physical properties of object under the experimentally received image, to describe thermal and elastic properties of solids and subsurface defects, to investigate multylayered and heterogeneous structures, subsurface of semiconductor materials, ceramics and metals.
INTRODUCTION
The photoacoustic nondestructive control of solids is a powerful technology, which is applied to diagnostics of products and materials of electronics. The photoacoustic microscopy (PAM) enables to receive the images of objects subsurface structure, which cannot be received by the traditional NDEmethods. The PAM is used for detection and analysis of subsurface heterogeneities, such as defects, cracks, inclusion, delaminations [13]. This method makes it possible to investigate the product during the manufacture without influence on properties, quality and parameters of it.
Theoretical model
The principle of the PAM is based on the phenomenon of generation and distribution of both thermal and acoustic waves excited by modulated in intensity laser radiation. The acoustic fluctuations spreading in the object are detected by the piezosensor and converted by it into electrical signal. The computer processing of this signal enables to receive the photoacoustic image of internal structure of a researched material under the scanning of a laser beam along the sample surface.
For interpretation and decoding of the photoacoustic images it is necessary to establish connection between parameters of photoacoustic signal (PAS) and characteristics of subsurface heterogeneities. For this purpose author developed the mathematical model of photoacoustic effect and calculated an amplitude and a phase of photoacoustic signal.
An infinite multylayer system (Fig. 1) modeling the isotropic crystal with defect was studied. To calculate a photoacoustic signal registered by piezoelectric transducer, which has the acoustic contact with object, full thermoelasticity equations in onedimensional geometry were solved with account of the following assumptions [6]:
 a sample is optically opaque (the absorption of radiation is described by the exponential law);
 a sample is optically and thermally thick;
 the heat exchange with air on the object surface is neglected;
 the contact of the sample surface with the piezosensor is considered as the ideal one and acoustic reflection from interface is neglected;
 a sample is thermally and acoustically homogeneous on each layer;
 the mechanical pressure and deformations are continuous;
 interfaces at x =0 and x = l_{1} are free.
Fig 1: Geometry of the task of PAS piezoelectrical detection: 0£x£l is the sample size;
h£x£h_{1} is the defect size;
l£x£l_{1} is piezosensor size.

The photoacoustic signal V is determined as opencircuit voltage on the piezosensor plates and proportional to a difference of the bias amplitudes of displacement u on its boundaries:
where ; e is piezomidulus, e^{s}is dielectric constant of squeezed crystal.
Let us enter the following designations:
d_{1} = h;

d_{2} = h_{1}  h;

d_{3} = l h_{1};

d_{4} = l_{1} 1; 
The indexes i=1..4 used also for others parameters in text are related to these boundaries.
For the PAS calculation it is necessary to solve jointly equations of a thermal conduction and wave equation for each model layers.
For 0 £ x £ h:
 (1) 
 (2) 
For h £x £h_{1}:
 (3) 
 (4) 
For h_{1} £x £ l:
 (5) 
 (6) 
For l_{1} £x £ l_{1}:
 (7) 
where q is temperature; m_{i} is displacement; is intensity of irradiation; a_{i} is specific heat; u_{i} is speed of longitudinal bulk waves distribution;
, B is the bulk modulus of elasticity; c_{i}^{T} = l + 2m,l m is Lame constants; k_{i} is thermal diffusivity.
The equations (1)(7) are solved with the following boundary conditions:
On the boundaries " air  sample" x=0:
 (8) 
 (8a) 
On the upper side "homogeneous layer defect" x = h:
 (9) 
 (9a) 
 (9b) 
 (9c) 
On the back side "defect  homogeneous layer" x = h_{1}
 (10) 
 (10a) 
 (10b) 
 (10c) 
On the boundary "sample  piezosensor" x = l:
T_{3}(l) = T_{4}(l),
 (11a) 
u_{3}(l) = u_{4}(l),
 (11c) 
On the piezosensor free surface x = l_{1}
where k_{i} is thermal conductivity; T_{i} are strains.
The strains T_{i} and deformation m_{i} are related as follows:
 (13)

 (14)

 (15)

 (16)

where ,c^{E} is coefficient of rigidity of the piezosensor.
Omitting cumbersome evaluations we shall write the solution for photoacoustic signal:
 (17) 
where is the wave vector.
Amplitude and phase of a photoacoustic signal are accordingly:
 (18) 
 (19) 
where D_{41} is the complex constant which is included in the common solution of the wave equation for spreading the longitudinal bulk waves in the piezoelectric plate and depends on modulation frequency, optically, thermally and elastic properties of object, depth and size of defect [6].
Analyzing the received expressions it is possible to see, that amplitude and phase of a photoacoustic signal depend on modulation frequency of a laser beam determining depth of distribution of thermal fluctuations in object, on depth and size of defect, on physical properties of defect. To analyze mathematical model and to estimate a degree of influence this parameters on a photoacoustic signal a numerical estimation for particular model was made.
Discussion
The structure Si  GaAs  Si was chosen as a model. The piezosensor was made from PZT8 piezoceramic. The thickness of the simple was 1mm, the thickness of piezosensor was 0,8 mm. A size and a depth of GaAslayer simulating defect were different for various conditions of experiment.


Fig 2:The dependence of amplitude (a) and phase (b) of the PAS on size and depth of defect at modulation frequency 1 kHz. 
The dependences of amplitude and phase of the PAS on modulation frequency, size and depth of defect under irradiation of a surface of a sample by the Ar+ laser with a waves length 0, 52 mm and capacity 4 W are shown in figures 24. Figures 2 and 3 illustrate that amplitude and phase are constant for great depths of defect and have smooth recession for small depths. The area of distortion corresponds to interval of depths, in which photoacoustic microscope will be sensitive to defect. The most depth of defect for a sample Si  GaAs  Si is 0,4 mm for frequency 1 kHz.


Fig 3: The dependence of amplitude (a) and phase (b) of the PAS on depth of defect and modulation frequency at the size of defect 0,01 mm. 
It is possible to notice, that the minimum of amplitude and phase of the PAS is displaced to the left under the increase of modulation frequency. The maximal for displaying depth of defect decreases. This principle is base of visualization of subsurface layers. Changing modulation frequency we receive the levelbylevel information about properties of object.
Figure 4 illustrates that there is a minimal size of defect, which can be detect by PAM. This value corresponds to the sharp recession on curve of PAS amplitude. It is 25 mm for frequency 1 kHz. The minimal size of defect, which influences on a photoacoustic signal, determines a sensitivity of PAM.


Fig 4: The dependence of amplitude (a) and phase (b) of the PAS on size of defect and modulation frequency at depth of defect 20 m m . 
CONCLUSION
Author developed the onedimensional mathematical model of photoacoustic effect with piezoelectric registration in thermal thick and optical opaque sample with subsurface defect. This model allows to restore the information about physical properties of researched object under the experimentally received amplitude and phase characteristics, to estimate the minimal size and maximal depth of defect, that determines sensitivity and depth of visualization of the PAM, and also to choose optimum modulation frequency for deep profiling of researched object. The model can be applied for decoding of computer photoacoustic images and their further analysis.
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