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<math>\tilde{V}_{out}(\omega)=\frac{\alpha}{a\omega^2+ib\omega+c}\tilde{I}_{in}(\omega)</math>
<math>\tilde{V}_{out}(\omega)=\frac{\alpha}{a\omega^2+ib\omega+c}\tilde{I}_{in}(\omega)</math>
where the parameters in the gain pre-factor are:
where the parameters in the gain pre-factor (g(ω)) are:
* <math>\alpha = -G/C_F</math>
* <math>\alpha = -G/C_F</math>
* <math>a = -(1+C_{HPD}/C_F)</math>
* <math>a = -(1+C_{HPD}/C_F)</math>
The subscript 'F' in the above expressions stands for op-amp feedback components. G is the gain-bandwidth product - nominally 1.6 GHz but was adjusted to 21 GHz to produce physical results. (Applying this response to the measured responses of LED pulses with the narrow nominal bandwidth produced negative LED intensities!)
The subscript 'F' in the above expressions stands for op-amp feedback components. G is the gain-bandwidth product - nominally 1.6 GHz but was adjusted to 21 GHz to produce physical results. (Applying this response to the measured responses of LED pulses with the narrow nominal bandwidth produced negative LED intensities!)
The time domain
The HPD response in the time domain is a straightforward inverse Fourier Transform:
<math>g(t)=\int\limits_{-\infty}^{\infty}\frac{\alpha}{a\omega^2+ib\omega+c}\, e^{i\omega t}\, d\omega =
\int\limits_{-\infty}^{\infty}\frac{\alpha}{(\omega-\omega_+)(\omega-\omega_-)}\, e^{i\omega t}\, d\omega</math>
Depending on the constants provided, this model produces two poles situated on the positive imaginary axis or along the -ib/2a line. For the physical constant of our circuit, the former set of poles are relevant.
where <math>\omega_\pm = \frac{-ib}{2a} \pm \frac{ \sqrt{-b^2-4ac} }{2a} = \omega_1 \pm i\omega_2</math>
Thus, in the complex space, this integral is equivalent to the residues of two poles. Depending on the constants provided, the poles are situated either on the positive imaginary axis or along the -ib/2a line. For the physical constant of our circuit, the former set of poles is relevant. The resulting response function is:
<math>g(t) = \frac{2\pi\alpha}{\omega_2} e^{i\omega_1 t} \sinh \omega_2 t </math>
Since the response seen on the HPD circuit output is the convolution of the light pulse shape (f<sub>L</sub>) with the response function of the HPD, the pulse shape can be recovered by deconvolving the measured signal (f<sub>M</sub>) with the g(ω). By the convolution theorem:
<math>f_L(t) = \mathcal{F}^{-1} \left\{ \frac{f_M(\omega)}{g(\omega)} \right\} </math>
This is a delicate procedure, in practice, as the division is sensitive to the small high-frequency errors. Direct application of this results in very noisy results. A good remedy is a low-pass filter on the spectrum of the measured signal (taking care to preserve the shape of the signal by keeping the cutoff well above the highest relevant frequencies).