## 2 Principles and Designs  
The analysis of any pyroelectric detector requires a careful consideration of both its thermal characteristics and its electrical response. Assume that infrared light with a power \(W(t)\) sinusoidally modulated at a frequency \(\omega\) is incident on the surface of a pyroelectric sensing element with an electrode area \(A\), thickness \(d\) and surrounding temperature \(T\). The incident power can be formulated mathematically as follows [31]:  
\[W(t)=W_{0}e^{iwt} \tag{1}\]  
Assume further that the sensing element has a thermal capacity \(H_{T}\), and the thermal conductance to the surroundings is denoted as \(G_{T}\), giving a thermal time constant of \(\tau_{T}=H_{T}\div G_{T}\). Given an emissivity \(\eta\), the temperature difference between the element and its surroundings, \(\theta\), can be described as [28]:  
\[\eta W(t)=H_{T}\frac{d\theta}{dt}+G_{T}\theta \tag{2}\]  
which has the solution [31]:  
\[\theta=\frac{\eta W_{0}e^{iet}}{G_{T}+i\omega H_{T}} \tag{3}\]  
Differentiating Equation (3), the rate of the temperature difference between the element and its surrounding is obtained as:  
\[\frac{d\theta}{dt}=\frac{\omega\eta W_{0}}{G_{T}\big{(}1+\omega^{2}\tau_{T}^{ 2}\big{)}^{\frac{1}{2}}} \tag{4}\]  
Quantitatively, the pyroelectric effect can be evaluated as the product of the pyroelectric coefficient \(p\) of the sensing material and the rate of change of the temperature difference, i.e.,  
\[\Delta P_{s}=p\times\Delta\theta \tag{5}\]  
When the top surface of a thin-film pyroelectric sensor is exposed to a heat source, the resulting temperature variation in the pyroelectric film manifests itself as a release of charge at the surface of the material and the subsequent production of a small voltage. Assuming that the pyroelectric coefficient \(p\) of the sensing element has a component \(p^{\prime}\) in the direction perpendicular to the electrode surface (with area \(A\)), the resulting charge can be detected as a current, \(i_{p}\), flowing through the external circuit (see Figure 1). The magnitude of this charge is given by [31] as:  
\[i_{p}=A\times\frac{dP_{\text{S}}}{dt}=A\times p^{\prime}\times\frac{d\theta}{ dt} \tag{6}\]  
In other words, the sensor is \"AC coupled\" to any input energy flux which generates a change in temperature, where this input may take the form of either electromagnetic radiation absorbed in the pyroelectric material or heat generated by the reaction of chemical species on a suitable catalyst deposited on the element surface.  
Figure 2 illustrates the basic structure and dimensions of the ZnO thin-film pyroelectric sensor considered in the present study. From Equation (6), it can be seen that a higher temperature variation rate in the ZnO layer leads to a higher response current. Moreover, a  
Figure 1: Principle of pyroelectric effect [31].  
partially covered top electrode yields a greater responsivity than a fully covered electrode since the uncovered part of the ZnO sensing element is directly exposed to the heat source and, therefore, experiences a larger heat absorption. Intuitively, a larger covered top electrode area reduces the temperature variation rate in the ZnO layer. By contrast, a larger uncovered top electrode area increases the temperature variation rate but decreases the responsivity since the resulting smaller top electrode area has a reduced ability to hold electric charge. Consequently, as shown in Figure 2c, the ZnO pyroelectric sensor considered in the present study was designed with a mesh-type partially covered top electrode in order to achieve a trade-off between the temperature variation rate and the charge storage ability of the electrode.