APPLICATION OF THE FREQUENCY-CONTRAST CHARACTERISTIC METHOD TO THE CALCULATION OF A RADIATION INTROSCOPE

V. I. Gorbunov, A. P. Shpagin

(Presented by the scientific seminar of the Research Institute of Nuclear Physics)

In the present work we shall consider one of the common block diagrams of a radiation introscope, consisting of a radiation source, the object under inspection, a converter, an objective lens, an electro-optical amplifier (EOA), an objective lens, a transmitting television tube of the superorthicon type, a video-signal transmission channel, and a receiving tube. There are as yet no methods for calculating such introscopes; in selecting optimal operating conditions, experimental dependences are usually used.

An attempt to apply the method of frequency-contrast characteristics (FCC) to the calculation of such an introscope is of undoubted interest. Since the possibility of applying this method to the evaluation of optical and electro-optical systems has been considered in the literature [1, 2], here we shall dwell in more detail on the television system. It is known that, in the general case, a transmitting television tube is not a linear element; however, in the region of low illumination (and it is precisely in this range that the transmitting tube in a radiation introscope operates) the light characteristic is close to linear. Taking this into account, and also the fact that the transmitted contrast is usually small, we conclude that application of the frequency-contrast characteristic method is also possible for the transmitting tube. The converter is usually linear. The nonlinearity of the modulation characteristic of the kinescope can be corrected in the transmission channel, and since the principal role in the noise at the output is played by the noise of the transmitting tube, this does not change the output signal-to-noise ratio. Thus, we may assume that all the links of the introscope are linear; then the overall FCC of the introscope can be found by multiplying the FCCs of the characteristics of the links. In order to eliminate the influence of scales, it is only necessary to reduce them to the frame format. All links were assumed to be isotropic, which makes it possible to characterize each link by a single FCC. Figure 1 presents the FCCs of the individual links of the introscope. Here 1 is the FCC of a PIM-3 type EOA at a frame size of \(12 \times 16\ \mathrm{mm}^2\) [2]; 2 is the FCC of the “Tair-11” television objective lens, recalculated to a \(12 \times 16\ \mathrm{mm}\) format [3]; 3 is the FCC of a 40LK-16 kinescope [4]; 4 is the FCC of an LI-17 transmitting television tube, reduced to a \(24 \times 32\ \mathrm{mm}^2\) format [5]; 5 is the overall FCC of the entire path, beginning with the EOA. For an estimate of the entire...

of the introscope it is also necessary to know the FCC of the converter. Owing to the influence of scattered radiation, the FCC of the converter will vary with changes in the transillumination geometry and, in essence, will be a characteristic of the transillumination geometry for the particular converter in question. There are no data in the literature on the FCC of the converter, and their direct measurement is impossible because of the absence of test objects; therefore, to determine the FCC we used the following

Fig. 1. Frequency-contrast characteristics

method. The image of a narrow slit was photographed on the converter and microphotometered in the direction perpendicular to the slit. The curve obtained, for a sufficiently narrow slit, was the scattering function \(B(x)\) [1]. Using the known relation between the scattering function and the FCC,

\[ K(\omega)=\int_{-\infty}^{\infty} B(x)e^{-j\omega x}\,dx, \tag{1} \]

one can find the FCC.

For a NaJ(Tl) converter 10 mm thick and with an area of \(120\times160\ \text{mm}^2\), located 2 m from the betatron target, it was found that the scattering function can be approximated by a Gaussian function:

\[ B(x)=\frac{1}{v\sqrt{2\pi}}\,e^{-\frac{x^2}{2v^2}}, \tag{2} \]

where \(v\) is the half-width of the slit image at the level 0.61, expressed in fractions of the horizontal frame size,

\[ v=0.206\cdot10^{-2}. \]

Then

\[ K(\omega)=e^{-\frac{(v\omega)^2}{2}}. \tag{3} \]

Substituting the value of \(v\) and taking into account that \(\omega = 2\pi N\), we obtain

\[ K(N)=e^{-0.86\cdot 10^{-4}N^2}. \tag{4} \]

Using formula (4), curve 6 (Fig. 1)—the FCR of the converter—was calculated.

Neglecting the FCR of the first objective, which for a screen size of \(120\times160\ \mathrm{mm}\) is close to unity, we find the FCR of the introscope 7 by multiplying curves 5 and 6 and approximating it by the function

\[ K_1(\omega)=e^{-\frac{(v_1\omega)^2}{2}}, \tag{5} \]

we find \(v_1=0.266\cdot 10^{-2}\) and the scattering function

\[ B_1(x)=\frac{1}{v_1\sqrt{2\pi}}\,e^{-\frac{x^2}{2v_1^2}}. \tag{6} \]

Using the obtained FCR and the scattering function of the introscope, one can calculate the image parameters at the output of the introscope if the thickness-distribution function of the defectometer is known (in the same way as pulses are calculated at the output of electrical circuits). Above we assume that the transmission coefficient of the links at zero frequency is equal to 1. In the general case, all expressions for \(K_1(\omega)\) should be multiplied by the transmission coefficient at zero frequency.

Let us consider a specific example. Suppose we have a defect in the form of a rectangular groove (a defectometer with grooves); denote the width of the groove by \(e\), its depth by \(\Delta\), and assume its length to be equal to the frame height. The distribution function of such a defect will take the form

\[ f(x)= \begin{cases} d, & (x\le 0),\\ d-\Delta, & \left(0<x<\dfrac{e}{160}\right),\\ d, & (x\ge e/160). \end{cases} \]

If an analogy is drawn with electrical circuits, then such a defect-distribution function is equivalent to the presence at the circuit input of a constant component

\[ I_0=n_0ES\tau e^{-\mu d} \tag{8} \]

and of a rectangular pulse with amplitude

\[ C=n_0ES\tau e^{-\mu d}\mu\Delta, \tag{9} \]

where: \(E\) is the energy of one quantum;
\(n_0\) is the number of gamma quanta incident on \(1\ \mathrm{cm}^2\) of the converter in the absence of the inspected article;
\(S\) is the area of the converter in \(\mathrm{cm}^2\), \(\tau\) is the frame time, \(\mu\) is the linear attenuation coefficient of the “narrow beam” of radiation in the material of the inspected article.

If the transmission coefficient of the introscope at zero frequency is \(K_0\), then, using the usual methods for calculating pulses at the output of electrical circuits [6], we obtain:

\[ f_1(x)=K_0C\left[\Phi\left(\frac{x}{v_1\sqrt{2}}\right)-\Phi\left(\frac{x-\frac{e}{160}}{v\sqrt{2}}\right)\right]. \tag{10} \]

where \(\Phi(z_0)\)—the Laplace integral—has the form \(\Phi(z_0)=\dfrac{1}{\sqrt{2\pi}}\displaystyle\int_0^{z_0} e^{-z^2/2}\,dz\); the values of \(\Phi(z_0)\) are found from tables.

It is not difficult to see that the function \(f_1(x)\) has a maximum at \(x=\dfrac{l}{320}\); substituting this value of the argument into expression (10), we obtain

\[ f_{1\max}(x)=2K_0 C\Phi\left(\frac{l}{450v_1}\right). \tag{11} \]

In determining the noise at the output of the introscope, we shall proceed from the two-dimensional FCF of the introscope, which, owing to the isotropy of the system, can be written as follows:

\[ K(\omega,\Omega)=K_0 e^{-\frac{v_1^2}{2}(\omega^2+\Omega^2)}. \tag{12} \]

The noise at the output is due to a large number of uniformly distributed flashes. Such noise has a uniform energy spectrum (at least for frequencies satisfying the condition \(\omega\Omega S_0 \ll 1\), where \(S_0\) is the area of one flash).

It can be shown that the distribution density of such noise is equal to [7]

\[ p(\omega,\Omega)=4n_0E^2S\tau e^{-\mu d}. \tag{13} \]

Neglecting the effect of the defect on the noise, the correlation function of the noise at the output of the introscope can be found from expressions (7), (8):

\[ \psi(x,y)=\frac{1}{4\pi^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} p(\omega,\Omega)\,[K(\omega,\Omega)]^2 e^{-j(\omega x+\Omega y)}\,d\omega d\Omega . \tag{14} \]

Substituting the values of \(p(\omega,\Omega)\) and \(K(\omega,\Omega)\) and carrying out the integration, we obtain

\[ \psi(x,y)=\frac{K_0^2EI_0}{\pi v_1}e^{-\left(\frac{x^2+y^2}{4v_1^2}\right)}. \tag{15} \]

The value of the correlation function at \(x=0,\ y=0\) gives the variance [8]

\[ \sigma^2=\psi(0,0)=\frac{K_0EI}{\pi v_1}. \tag{16} \]

The value of the noise determined by us represents only one component of the noise at the output of the introscope, namely the noise due to the statistical nature of the radiation; in a real introscope, each link will generate its own noise, the variance of which at the output can be determined by repeating the reasoning given above, only substituting into formula (14), instead of the FCF of the entire introscope, the FCF of the subsequent links. The total noise at the output will be equal to the square root of the sum of the variances of all components. Here we shall confine ourselves to determining the sensitivity of an introscope in which the noises are due only to the statistical nature of the radiation.

For detecting a signal in noise it is necessary that the condition

\[ f_{1\max}(x)=\varphi\sigma, \tag{17} \]

be satisfied, where \(\varphi\) is the signal-to-noise ratio required to ensure the prescribed probability of detection and false alarm. If the values \(f_{1\max}(x)\) and \(\sigma\) are substituted into expression (17) and transformed taking (8) and (9) into account, one obtains

\[ \Delta_{\min}=\frac{\varphi e^{0.5\mu d}}{2\vartheta_1\mu\sqrt{\pi n_0 S\tau}\,\Phi\left(\dfrac{l}{450\vartheta_1}\right)}. \tag{18} \]

Using formula (18), the values of \(\Delta_{\min}\) were calculated for grooves of width \(l=0.1\) mm, \(l=1\) mm, and \(l=10\) mm for three thicknesses of the inspected object.

Table 1

In the calculation the following values of the coefficients were adopted:

\[ \varphi=2,\quad \vartheta_1=0.266\cdot10^{-2},\quad n_0=10^{-9}\ \frac{1}{\mathrm{sec}\cdot\mathrm{cm}},\quad S=192\ \mathrm{cm}^2,\quad \mu=0.233\ \frac{1}{\mathrm{cm}}, \]

\[ \tau=0.04\ \mathrm{sec}. \]

The results of the calculations are given in Table 1.

It is not difficult to see that \(\vartheta_1\) is, in a certain sense, a parameter of the introscope characterizing the ability of the introscope to average over the surface. In order to trace how the sensitivity of the introscope changes when this parameter is varied, values of \(\Delta_{\min}\) were calculated for \(d=10\) cm and several values of \(\vartheta_1\). The results are given in Table 2.

Table 2

\[ (\vartheta_0=0.266\cdot10^{-2}) \]

In some cases the value of the minimum detectable defect is limited not by the signal-to-noise ratio, but by the minimum contrast of the defect. In this connection it is of interest to determine

contrast transfer coefficient of the introscope. If the contrast is defined by the formula

\[ \chi=\frac{W_{\max}-W_{\min}}{W_{\min}}, \tag{19} \]

where \(W_{\max}\) and \(W_{\min}\) are, respectively, the minimum and maximum energy densities, then it is not difficult to find that the contrast transfer coefficient of the introscope is determined by the expression

\[ \frac{\chi}{\chi_0}=2\Phi\left(\frac{l}{450v_1}\right). \tag{20} \]

Table 2 gives the values of \(\chi/\chi_0\) for \(d=10\) cm and for several values of \(v_1\), as a function of the groove width.

Thus, it follows from the above that application of the FCH method makes it possible to carry out a sufficiently complete calculation of introscopes.

From the results of Table 2 the following conclusions may be drawn. With increasing \(v_1\), the sensitivity of the introscope increases and practically reaches its maximum value at \(v_1=(3\div 4)l\); the contrast transfer coefficient, on the contrary, decreases with increasing \(v\).

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