The Thermal Image Receiver Realized in the Image Intensifier Tube Architecture

Abstract

An innovative design of thermal imaging is considered. The results of analysis and calculations of the characteristics of a thermal image receiver (3–15 μm), made in the electron-optical converter architecture, are presented. The spatial dependences of the spontaneous polarization the electric field strengths and the electric potentials on the surface of pyroelectric film are calculated. The characteristics of thermal-field-induced polarization of various pyroelectric films are obtained. The temperature dependences of various pyroelectric films polarizations are calculated by the COMSOL Multiphysics software package based on the finite element method. The possible influences of the piezoelectric effect to the images of the distribution of electric potentials of pyroelectric films are taken into account. The estimates of the values of the main characteristics of the image intensifier tube architecture are obtained.

Appendices

APPENDIX 1

The field intensity of the upper plane is

$${{E}_{{{\text{upper}}\,\,{\text{plane}}}}} = - \frac{{{{P}_{S}}z}}{{4\pi {{\varepsilon }_{0}}}}\int\limits_0^R {\int\limits_0^{2\pi } {\frac{r}{{{{{({{z}^{2}} + {{r}^{2}})}}^{{3/2}}}}}drd\varphi } } .$$

(A1.1)

Therefore,

$${{E}_{{{\text{upper}}\,\,{\text{plane}}}}} = - \frac{{{{P}_{S}}z}}{{2{{\varepsilon }_{0}}}}\int\limits_0^R {\frac{r}{{{{{({{z}^{2}} + {{r}^{2}})}}^{{3/2}}}}}dr} $$

(A1.2)

and, performing a substitution of variables, we obtain

$${{E}_{{{\text{upper}}\,\,{\text{plane}}}}} = - \frac{{{{P}_{S}}z}}{{4{{\varepsilon }_{0}}}}\int\limits_0^R {\frac{{d({{r}^{2}} + {{z}^{2}})}}{{{{{({{r}^{2}} + {{z}^{2}})}}^{{3/2}}}}}} .$$

(A1.3)

Thus, we derive the functional dependence of electric-field intensity E of the upper negatively charged plane of the dipole layer from expression (5):

$$\begin{gathered} {{E}_{{{\text{upper}}\,\,{\text{plane}}}}} = \left. {\left( {\frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\frac{z}{{\sqrt {{{z}^{2}} + {{r}^{2}}} }}} \right)} \right|_{0}^{R} \\ = - \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\left( {1 - \frac{z}{{\sqrt {{{z}^{2}} + {{R}^{2}}} }}} \right). \\ \end{gathered} $$

(A1.4)

APPENDIX 2

The integral of a superposition of electric-field intensities of both planes is

$$\varphi (z) = - \int\limits_\infty ^z {\frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\left( {\frac{z}{{\sqrt {{{z}^{2}} + {{R}^{2}}} }} + \frac{{z + d}}{{\sqrt {{{{(z + d)}}^{2}} + {{R}^{2}}} }}} \right)dz.} $$

(A2.1)

Its antiderivative is

$$\varphi (z) = \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}(\sqrt {{{{(z + d)}}^{2}} + {{R}^{2}}} - \sqrt {{{z}^{2}} + {{R}^{2}}} ){\text{|}}_{\infty }^{z}.$$

(A2.2)

Let us consider the lower limit of function (A2.2):

$$\mathop {\lim }\limits_{z \to \infty } \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}(\sqrt {{{{(z + d)}}^{2}} + {{R}^{2}}} - \sqrt {{{z}^{2}} + {{R}^{2}}} ).$$

(A2.3)

Transforming (A2.3) and expanding the obtained expression into a Taylor series (utilizing the smallness of  $\frac{{2zd + {{d}^{2}}}}{{{{z}^{2}} + {{R}^{2}}}}$ at z → ∞), we find

$$\mathop {\lim }\limits_{z \to \infty } \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\sqrt {{{z}^{2}} + {{R}^{2}}} \left( {1 + \frac{{2zd + {{d}^{2}}}}{{2({{z}^{2}} + {{R}^{2}})}} - 1} \right) = \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}d.$$

(A2.4)

Inserting (A2.4) into expression (A2.2), we obtain a functional dependence for the electric potential of arbitrary point z:

$$\varphi (z) = \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}(\sqrt {{{{(z + d)}}^{2}} + {{R}^{2}}} - \sqrt {{{z}^{2}} + {{R}^{2}}} - d).$$

(A2.5)

APPENDIX 3

Let us present expression (5) from the main text in the form

$$E = \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\frac{2}{{\sqrt {{{z}^{2}} + {{R}^{2}}} }}\left( {z - \frac{{z + d}}{{\sqrt {1 + \frac{{2zd + {{d}^{2}}}}{{{{z}^{2}} + {{R}^{2}}}}} }}} \right).$$

(A3.1)

Expanding (A3.1) into a Taylor series, we obtain the following after several transformations at R $\gg$ z:

$$\begin{gathered} E = \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\frac{1}{R}\left( {z - \frac{{2(z + d){{R}^{2}}}}{{2{{R}^{2}} + 2zd + {{d}^{2}}}}} \right) \\ = - \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\frac{{2(z + d)R}}{{2{{R}^{2}} + 2zd + {{d}^{2}}}} = - \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\frac{{(z + d)}}{R}. \\ \end{gathered} $$

(A3.2)

If dipole arm d $\gg$ z, it follows from expression (A3.2)

$$E = - \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\frac{d}{R}.$$

(A3.3)

Proceeding in a similar way, we derive an approximation for the electric potential from Eq. (A2.5) at R $\gg$ z:

$$\begin{gathered} \varphi (z) = \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\sqrt {{{z}^{2}} + {{R}^{2}}} \\ \times \,\,\left( {\sqrt {1 + \frac{{2zd + {{d}^{2}}}}{{{{z}^{2}} + {{R}^{2}}}}} - 1 - \frac{d}{{\sqrt {{{z}^{2}} + {{R}^{2}}} }}} \right). \\ \end{gathered} $$

(A3.4)

Using the Taylor series expansion, we find from (A3.4) that

$$\begin{gathered} \varphi (z) = \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\sqrt {{{z}^{2}} + {{R}^{2}}} \\ \times \,\,\left( {1 + \frac{{2{{z}_{0}}d + {{d}^{2}}}}{{2({{z}^{2}} + {{R}^{2}})}} - 1 - \frac{d}{{\sqrt {{{z}^{2}} + {{R}^{2}}} }}} \right). \\ \end{gathered} $$

(A3.5)

Since R $\gg$ z, expression (A3.5) may be transformed into

$$\varphi (z) = \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}R\left( {\frac{{2zd + {{d}^{2}}}}{{2{{R}^{2}}}} - \frac{d}{R}} \right).$$

(A3.6)

In the 2zd $\gg$ d² case, expression (A3.6) is transformed into (A3.7):

$$\varphi ({{z}_{0}}) = \frac{{{{P}_{S}}}}{{2{{\varepsilon }_{0}}}}\left( {\frac{{zd}}{R} - d} \right).$$

(A3.7)

APPENDIX 4

In order to determine the functional dependence for the electric potential at z $\gg$ R, we integrate expression (13) from the main text:

$$\Delta {{\varphi }_{{{\text{upper}}\,\,{\text{plane}}}}} = - \int\limits_\infty ^z { - \frac{{{{P}_{S}}{{R}^{2}}}}{{2{{\varepsilon }_{0}}}}\frac{d}{{{{z}^{3}}}}dz = - \left. {\frac{{{{P}_{S}}{{R}^{2}}d}}{{4{{\varepsilon }_{0}}{{z}^{2}}}}} \right|} _{\infty }^{z}.$$

(A4.1)

Thus, the potential of the dipole layer at arbitrary point z at z $\gg$ R is

$$\varphi (z) = - \frac{{{{P}_{S}}{{R}^{2}}}}{{4{{\varepsilon }_{0}}}}\frac{d}{{{{z}^{2}}}}.$$

(A4.2)

APPENDIX 5

Let us use the expression for the mechanical stress tensor to estimate the probable distortion of the distribution of electric potentials induced by the piezoeffect:

$${{S}_{{ij}}} = \sum\limits_{kl}^{} {{{C}_{{ijkl}}}{{\varepsilon }_{{kl}}}} - \sum\limits_{mn}^{} {{{e}_{{ijmn}}}{{E}_{{mn}}}.} $$

(A5.1)

Here, S_ij is the mechanical stress tensor, C_ijkl is the tensor of elastic constants, ε_kl is the deformation tensor (ε_kl = α_klΔT), e_ijmn is the tensor of piezoelectric coefficients, and E_mn is the electric-field intensity tensor; α_j is the matrix of thermal expansion coefficients and ΔT is the change in temperature of the studied material.

We use the Cauchy−Green form of the deformation tensor:

$${{\varepsilon }_{{kl}}} = \frac{1}{2}\left( {\frac{{\partial {{u}_{k}}}}{{\partial {{x}_{l}}}} + \frac{{\partial {{u}_{l}}}}{{\partial {{x}_{k}}}}} \right),$$

(A5.2)

where u is the vector characterizing displacements.

A simultaneous solution of Eqs. (A5.1), (A5.2) with the use of the finite element method allows one to calculate deformations ε_kl as functions of temperature T(t). The obtained deformation values at different temperatures ε_kl(T(t)) are used to calculate the electric flux density:

$${{D}_{{ij}}} = \sum\limits_{kl}^{} {{{e}_{{ijkl}}}{{\varepsilon }_{{kl}}}} + \sum\limits_{mn}^{} {{{\varepsilon }_{{ijmn}}}{{E}_{{mn}}},} $$

(A5.3)

where ε_ijmn is the permittivity tensor of the piezoelectric material. Assuming that the external field is zero (E_exteranl = 0), the following relation is obtained for electric flux density (A5.3):

$${\mathbf{D}} = {{{\mathbf{P}}}_{{{\text{piezo}}}}}.$$

(A5.4)

Inserting the values of piezoelectric polarization P_piezo into Eq. (19) from the main text, which characterizes the overall pattern of the potential distribution regardless of the cause of polarization, we determine the surface potential induced by elastic deformations.

In doing this, we use

tensor of piezoelectric coefficients e_ijkl for PZT

$${{e}_{{ijkl}}} = \left( {\begin{array}{*{20}{c}} 0&0&0&{17.034}&0&0 \\ 0&0&0&0&0&0 \\ { - 6.62281}&{ - 6.62281}&{23.24}&0&0&0 \end{array}} \right)\,\left[ {\frac{{\text{C}}}{{{{{\text{m}}}^{2}}}}} \right],$$

(A5.5)

tensor of elastic constants C_ijkl for PZT

$${{C}_{{ijkl}}} = \left( {\begin{array}{*{20}{c}} {127.2}&{80.21}&{84.67}&0&0&0 \\ {80.21}&{127.2}&{84.67}&0&0&0 \\ {84.67}&{84.67}&{117.43}&0&0&0 \\ 0&0&0&{22.98}&0&0 \\ 0&0&0&0&{22.98}&0 \\ 0&0&0&0&0&{23.47} \end{array}} \right)\,\,[{\text{GPa}}]$$

(A5.6)

and tensor of elastic constants C_ijkl for Si

$${{C}_{{ijkl}}} = \left( {\begin{array}{*{20}{c}} {166}&{64}&{64}&0&0&0 \\ {64}&{166}&{64}&0&0&0 \\ {64}&{64}&{166}&0&0&0 \\ 0&0&0&{80}&0&0 \\ 0&0&0&0&{80}&0 \\ 0&0&0&0&0&{80} \end{array}} \right)\,\,[{\text{GPa}}].$$

(A5.7)

APPENDIX 6

Let us use literature data on resistivities of the examined films (Table A6.1).

Table A6.1. Resistivities of pyroelectric films used in calculations for the double-layer Si membrane/pyroelectric structure

Since Johnson noise is the dominant type of noise in thermal imaging [39], we use the following expression to estimate the thermal noise level:

$${{V}_{N}} = \sqrt {4kTR\Delta f} ,$$

(A6.1)

where k is the Boltzmann constant, T is the temperature of the sensing and conversion sample, R is the resistance of the sensing and conversion sample, and Δf is the equivalent frequency passband.

Having calculated the thermal noise level, we obtain the following for the ultimate sensitivity of pyroelectric image receivers:

$$\mathcal{R} = \frac{{\Delta \phi }}{P}.$$

(A6.2)

The results of calculation of the key parameters of sensing and conversion materials, which were determined in this study for specific types of pyroelectric films, are presented in Table 4 in the main text.