AIP Conference Proceedings 2356, 020002 (2021); https://doi.org/10.1063/5.0053654 2356, 020002
© 2021 Author(s).
Methods for focal spot characteristics
applied measurement of microfocus x-ray
Cite as: AIP Conference Proceedings 2356, 020002 (2021); https://doi.org/10.1063/5.0053654
Published Online: 15 June 2021
A. K. Avakyan
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Methods for Focal Spot Characteristics Applied
Measurement of Microfocus X-Ray Sources
A. K. Avakyana)
Prodis.NDT, Moscow, Russia
a)Corresponding author: firstname.lastname@example.org
Abstract. The paper presents measurement methods of focal spot (FS) characteristics for X-ray sources with a nominal
FS size of less than 100 micrometers. The work is based on applied problems and is aimed for developers of X-ray
imaging systems, for whom it is important to control the real performance of FS in a wide range of source operating
parameters. An overview of existing approaches to the measurement of FS characteristics is presented. The developed
measurement technique using an X-ray spatial resolution bar-pattern test object and a digital flat-panel detector is
proposed. The mathematical model on which the developed method is based is shown. The interpretation of the measured
linear size of the FS as the intensity profile over the tube target surface and the modulation transfer function (MTF) of the
imaging system is justified.
One of the most important quality criteria for both tomographic and projection X-ray systems is the size of
visualized objects of interest. The possibility to visualize an object is determined by the modulation transfer function
(MTF) of the system, which depends on a number of technical parameters, including the actual focal spot (FS)
characteristics of an X-ray source, i.e., an X-ray intensity spatial distribution over the tube anode surface.
Manufacturers of X-ray sources usually specify the FS linear size (with reference to the measurement standard
used), less often – the pinhole projection image of the FS. Modern problems of nondestructive testing require a
source with a nominal FS size from 1 to 50 micrometers combined with a digital receiver to achieve the relevant
level of detail. The following tasks arose during 2D (printed circuit board inspection) and 3D (microtomography)
imaging systems development:
choice of a microfocus X-ray source among commercially available ones;
analytical MTF calculation for arbitrary geometric magnification and known characteristics of FS and
In practice, when solving these tasks, there is a problem of converting the FS linear size declared by the
manufacturer into the MTF function due to the lack of documented information about the shape of FS X-ray
intensity profile. Uncertainties arise for interpretation of the declared size if the measurement was performed only
in one operational mode of the source, or there is no documented information about the used focusing system and
possible FS drift, or the measurement was performed on an X-ray film. The accuracy of the measurement
performed by the manufacturer is questionable if the diameter of the pinhole used is comparable to the size of the
Thus, imaging systems development requires dynamic control of the actual FS operational characteristics in a
wide parameters range of the source. This kind of control is not a profile task for systems developer's laboratory,
therefore, for the used FS research method, it is possible to shift the priority from absolute accuracy to convenience
and high speed without necessity to use of specialized and expensive measuring equipment. Also, in accordance
with the current level of technology development, measurements should be performed on a digital flat-panel
7th International Conference on X-ray, Electrovacuum and Biomedical Technique
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MATERIALS AND METHODS
There are a number of standards for measuring the microfocus (less than 100 microns) linear size of the FS [1-2].
Based on the physical and mathematical principle of obtaining information about the FS, conventional standardized
approaches can be divided into three groups.
The first group includes a research method based on a projection imaging of a pinhole in a plate made of high-
contrast material, the obvious advantage of which is obtaining visual information about the two-dimensional spatial
distribution of the actual FS intensity. In practice, a number of difficulties arise when implementing the method.
Real pinhole-phantom has a finite diameter, but the assumption of a pinhole is valid only if its diameter is an order
of magnitude smaller than the size of the measured FS. In order to achieve sufficient X-ray contrast, the thickness of
the metal plate has to be at least 100 microns in the equivalent of tungsten. The technological complexity of
production makes such pinhole-phantom expensive. Without using additional tools, it is impossible to accurately
measure the geometric magnification just from the projection image. There is a difficulty in the phantom positioning
when the magnification exceeds ten times. Also, due to the low level of the useful signal on the image of the
phantom hole, a sufficient for accurate measurement signal-to-noise ratio can only be achieved by accumulating and
averaging the projections, which makes studies in dynamic mode impossible due to possible FS drift.
The second group of methods is based on measuring the width of image blur (gradient) of an edge of high-
contrast test object (TO) according to principles of geometric optics. Phantoms with a circle in cross-section (ball,
wire), slit phantoms, and "sharp edge" type phantoms made of tungsten, platinum, and other metals are used as TO.
The general advantage of these methods (except for the "sharp edge") is a relatively accurate measurement of the
magnification ratio as the relationship of the projection width to the physical size of the phantom. Therefore, it is
possible to calculate an effective FS size within the required range of magnification factors out the results of just one
On the other hand, methods from second group have a number of significant disadvantages. First, according to a
single projection, the size of the FS can be measured only in one direction. Second, none of the existing standards
clearly interpret the measured size in terms of the shape of the FS intensity profile, which makes it difficult to
calculate MTF from the measured linear size. Third, in source-detector direction phantoms have finite thickness that
is significantly larger than the size of FS being measured. Due to this fact, shape of the projection of the object edge,
which is theoretically determined solely by the finite size of the FS, is distorted under the influence of uneven
attenuation in the phantom material and the emerging scattered radiation.
The third group includes a method of investigation using precision spatial resolution phantoms like JIMA with a
stroke size in the range of 0.4-15 micrometers . Phantom allows you to uniquely determine the minimum size of
the visualized object (defect), while the information obtained from the results of the study characterizes the
visualization system as a whole, but not directly the characteristics of the FS, so the results of a single measurement
cannot be analytically interpreted for an arbitrary geometric magnification.
Taking into consideration the advantages and disadvantages of the analyzed standards, it was decided to develop
a proprietary method for measuring the characteristics of FS that meets the following requirements:
possibility to interpret the measured FS size as the shape of the radiation intensity profile;
possibility of the MTF analytical calculation for an arbitrary magnification;
applicability in the tube voltage range from 40 to 150 kV;
availability and low cost of TO;
accurate magnification measurement from the projection of the TO;
the thickness of the TO is uniform and does not significantly exceed the size of the FS being measured;
projections should be acquired via digital flat-panel detector.
RESULTS AND DISCUSSION
The developed method is based on the edge gradient width measurement on the projection of a high-
contrast TO. In contrast to the existing standards, a theoretical proof of the relationship of the measured gradient
width with the shape of the FS intensity profile and the source's MTF is presented.
The most available TO that satisfies the specified conditions is a standardized X-ray bar-pattern test-object
(BPT) with a parallel direction of bars. BPT is made of lead or tungsten foil with a thickness of 20-50 microns and is
available in size ranges of up to 5, up to 10 and up to 20 line pairs per mm . Since the thickness of the BPT foil
does not exceed the size of the FS being measured, it is appropriate to assume that there are no undesirable
distortions at the gradient signal. During the projections acquisition, it is possible to choose such combination of the
geometric magnification and the BPT bars frequency, in which the gradients from neighboring bars do not overlap
with each other, which allows us to consider the edge of each bar as an ideal "sharp edge". In addition, it is possible
to accurately measure the magnification factor as the ratio of the projection width to the known physical size of the
bar. The BPT image at high magnification and the signal profile selected across the image are shown in Fig. 1, every
designation further in the text is done in accordance with this figure.
FIGURE 1. BPT image at high magnification and the signal across this image.
The measurement of the FS characteristics is performed using the signal profile in the direction perpendicular to
the orientation of the bars. Measurement interpretation is based on the following statement: if the FS intensity
profile F(x) has only one maximum, then when imaging the "sharp edge", regardless of the shape of F(x), the
following equality takes place:
where ǻx – is the distance between the intersection coordinates of the tangent, drawn to the signal at the signal's
derivative extremum point, with the signal levels of the dark and light "plateaus”; E – the total radiated energy from
the FS surface; Ⱥ – the pick magnitude of the FS radiation intensity; Ɇ – magnification scale factor during imaging.
The proof of statement (1) is based on the following mathematical model of an X-ray imaging physical process.
Radiation intensity profile F(x) is a function of the coordinate x in the FS plane. Thin object G(x) is modeled as a
relative transmission function of the coordinate in the object plane, where the transmission is determined by the ratio
of the intensity passed through the object to the intensity at the entrance. Signal S(x) in the detector plane is the
result of convolution of the scaled functions F(x) and G(x), where scaling factors are determined by the mutual
position of the FS, the object and the detector:
where ADD – the distance between TO and detector, FAD – the distance between FS and TO.
An ideal "sharp edge" object is described by G(x) being the unit step function (Heaviside function). By the
properties of the Heaviside function, expression (2) is converted to the following form:
The tangent line y(x) to the signal S(x) at an arbitrary point with the coordinate x0, taking into consideration the
statement (3), is described by the following equation:
On the image of the "sharp edge" (Fig. 1), a dark "plateau" with the signal level Smin and a light "plateau" with
the signal level Smax are distinguished, and:
௫ ൌሺλሻൌ නሺݔሻ݀ݔ ൌڄሺെͳሻ
where E – the total radiated energy from the FS surface.
The value of the tangent line y(x) equals Smin and Smax at the corresponding coordinates x1 and x2, and the
difference between these coordinates is determined by the expression:
In the special case, when x0 is the coordinate of the single maximum of the function Fpr(x), i.e.
Fpr(x0) = max(Fpr) = A, expression (6) transforms to statement (1). In accordance with the expression (3), the
derivative Sc(x) coincides with the projection of the FS intensity profile Fpr(x), and x0 is the coordinate of the Sc(x)
On the basis of the criterion stated in (2) about the total FS energy being constant, the FS intensity profile F(x) is
described by experimentally measured values ǻx and Ɇ, and is considered to have the curve form in the range from
uniform (7) to gaussian (8) with an accuracy of up to the amplitude. Then the MTF function can be calculated
analytically for an arbitrary magnification by definition as a direct Fourier transform of F(x). The graphical
interpretation is shown in Fig. 2
FIGURE 2. The FS intensity profile curves in range from uniform to gaussian with condition
of total emitted energy being
constant (left) and the corresponding MTF series (right).
The usage of BPT as a TO allows to perform measurement of FS characteristics only in one direction at a single
X-ray image. Before conducting the study, one should choose a combination of the bar pattern frequency and the
geometric magnification factor, in which the gradients from adjacent bars do not overlap, the full projection of at
least one pair of bars completely fits into the detector boundaries, and the gradient width covers as many detectors
channels as possible. It is also necessary to control the influence of noise when calculating the numerical derivative
of the signal.
The method was tested on the following x-ray sources with a nominal FS size of less than 100 micrometers:
RAP-100 and RAP-150 (ELTECH-Med, Russia), XRB011 (Spellman, USA), L9181-02 (Hamamatsu, Japan).
Images were acquired on the PRODIS.Mark 1215SS flat-panel detector with a pixel size of 49.5 microns. The BPT
frequency from 0.5 to 2.0 l.p. per mm was applied depending on the size of the FS being measure, and the
magnification factor was chosen so that the edge gradient covers least 40 pixels. The MTF calculated by the
proposed method with high accuracy coincided with the measurements under similar conditions with the JIMA
phantom. The derivative of the signal, in accordance with expression (3), allows us to qualitatively estimate the
shape of the FS intensity profile. In addition, it is possible to measure the relative drift of the FS in time or in
relation to the source parameters using the BPT images.
1. M. Salamon et al., Nuclear Instruments and Methods in Physics Research Section A: Accelerators,
Spectrometers, Detectors and Associated Equipment 591 (1), 54–58 (2008).
2. Standard ASTM E2903-13. Standard test method for measurement of the effective focal spot size of mini and
micro focus X-ray tubes.
3. Phantom manufacturer website. URL: https://www.jima.jp/english/assen-e.html
4. Russian distributer website. URL: http://www.doza.ru/catalog/common_phantom/260/