Most x-ray imaging detectors employ a phosphor in the initial stage to absorb the x rays and produce light, which is then coupled to an optical sensor (photodetector). The use of relatively high-atomic-number phosphor materials causes the photoelectric effect to be the dominant type of x-ray interaction. The photoelectron produced in these interactions is given a substantial fraction of the energy of the x-ray. This energy is much larger than the bandgap of the crystal and, therefore, in being stopped, a single interacting x-ray has the potential to cause the excitation of many electrons in the phosphor and thereby the production of many light quanta. We describe this “quantum amplification” as the conversion gain, g_1. For example, in a Gd₂O₂S phosphor, the energy carried by a 25-keV x-ray quantum is equivalent to that of 10,400 green-light quanta(E_g=2.4 e V).Because of competing energy-loss processes and the need to conserve momentum, the con-version efficiency is only about 15%, so that, on average, it requires approximately 13 eV per light quantum created in this phosphor. The conversion gain is then approximately 1560 light quanta per interacting x-ray quantum.

The energy-loss process is stochastic and, therefore, g has a probability distribution, with standard deviation, σ_g, about its mean value as illustrated in Figure 1(a). Swank described this effect and the “Swank factor,” A_s, characterizes this additional noise source. The Swank factor is calculated in terms of the moments of the distribution of g as where M; indicates the ith moment of the distribution.

The number of light quanta produced when an x-ray interacts in a phosphor depends both on its incident energy and the mechanism of interaction with the phosphor crystal. The most likely type of interaction, the photoelectric effect, will result in both an energetic photoelectron and either a second (Auger) electron or a fluorescent x-ray quantum being produced. The energy of fluorescence depends on the shell in which the photoelectric interaction took place. The threshold K-shell energy for these interactions is shown for some detector materials in Table below:

Also in the table is the K-fluorescence yield, ω_k the probability of emission of x-ray fluorescence given that a K-shell photoelectric interaction has occurred. The fluorescent quanta are either reabsorbed in the phosphor or escape. In either case, if they are not absorbed locally, the apparent energy deposited in the phosphor from the x-ray quantum is reduced, giving rise to a second peak in the distribution with a lower value of g. The effect of fluorescence loss is to broaden the overall distribution of g (Figure 1(b)), thus decreasing As and causing an increase in σ_g, For many detector materials (e.g., Gd₂O₂S), their K-shell interactions lie above the energy range used for mammography and therefore, K fluorescence effects are not an issue.

There are both advantages and disadvantages in imaging with an x-ray spectrum that exceeds the K edge of the phosphor. Clearly, the value of η increases, however the “Swank noise”does also. In addition, deposition of energy from the fluorescence at some distance from the point of initial x-ray interaction causes the point spread function of the detector to increase, resulting in decreased spatial resolution.

Fig. 1 Distribution of initial gain of a detector stage. The Swank factor characterizes the distribution in terms of its moments:a) energy below K edge of detector, b)energy above K edge results in K fluorescence escape peak

After their formation, the light quanta must successfully escape the phosphor and be effectively coupled to the next stage for conversion to an electronic signal and readout. It is desirable to ensure that the created light quanta escape the phosphor efficiently and as near as possible to their point of formation.

Figure 2 illustrates the effect of phosphor thickness and the depth of x-ray interaction on spatial resolution of a phosphor detector. The probability of x-ray interaction is exponential so that the number of interacting quanta and the amount of light created will be proportionally greater near the x-ray entrance surface.

While travelling within the phosphor, the light will spread—the amount of diffusion being proportional to the path length required to escape the phosphor. X rays interacting close to the photodetector give rise to a sharper (less blurred) optical signal than those which interact more distantly. The paths of most optical quanta will be shortest if the photodetector is placed on the x-ray entrance side of the phosphor. It is often only possible, however, to record the photons which exit on the opposite face of the phosphor screen, that is, those which have had a greater opportunity to spread. If the phosphor layer is made thicker to improve quantum efficiency, the spreading becomes more severe. This imposes a fundamental compromise between spatial resolution and η. Methods to channel the optical photons out of the phosphor without spreading can significantly improve phosphor performance. This is accomplished, in part, with the use of CsI phosphors in detectors for digital mammography.

Figure 3 illustrates the propagation of signal through the various energy-conversion stages of an imaging system. In the diagram, N₀ quanta are incident on a specified area of the detector surface (Stage 0). A fraction of these, given by the quantum detection efficiency, η, interact with the detector(Stage I). In a perfect imaging system, η would be equal to 1.0.The mean number, N₁ of quanta interacting represents the “primary quantum sink” of the detector. The fluctuation about N₁ is σ_N1=(N1)^1/2. This defines the signal-to-noise ratio, SNR, of the imaging system which increases as the square root of the number of quanta interacting with the detector.

Fig. 2 Effect of the thickness of a conventional phosphor layer on spatial resolution (line-spread function)

Regardless of the value of η, the maximum possible SNR of the imaging system will occur at this point and if the SNR of the imaging system is essentially deter-mined there, the system will be x-ray quantum limited in its performance. However, the SNR will, in general, become reduced in passage of the signal through the imaging system because of losses and additional sources of fluctuation. To avoid losses that can occur at subsequent stages, it is important that the detector provide adequate quantum gain, g1, directly following the initial x-ray interaction. Stages II and IIl illustrate the processes of creation of many light photons from a single interacting x-ray (conversion gain) and the escape of quanta from the phosphor with mean probability g2.Here, light absorption, scattering and reflection processes are important.

Further losses occur in the coupling of the light to the photodetector which converts light to electronic charge (Stage IV) and in the spectral sensitivity and optical quantum efficiency of the photodetector (Stage V).If the conversion gain of the phosphor is not sufficiently high to overcome these losses and the number of light quanta or electronic charges at a subsequent stage falls below that at the primary quantum sink, then a secondary quantum sink is formed. In this case the statistical fluctuation of the light or charge at this point becomes an additional important noise source. Even when an actual secondary sink does not exist, a low value of light or charge will cause increased noise. This becomes evident in a spatial-frequency-dependent analysis of SNR as a reduction of the detective quantum efficiency with increasing spatial frequency. Figure 4 illustrates the effect of optical coupling efficiency of light from a phosphor to a photodetector on DQE(f) for an optically coupled system.

Fig. 3 Quantum accounting diagram can be used to identify quantum sinks in a multi-stage imaging system

Figure 5 illustrates two approaches for coupling a phosphor to a photodetector. In Figure 5(a) a lens is used to collect light emitted from the surface of the phosphor material. Because the size of available photodetectors such as CCDs is limited by manufacturing considerations to a maximum dimension of only 2 to 5 cm, it is often necessary to demagnify the image from the phosphor to allow coverage of the required field size in the patient. The efficiency of lens coupling is determined largely by the solid angle subtended by the collecting optics. For a single lens system, the coupling efficiency is given by

where 𝜏 is the optical transmission factor for the lens, F is the f-number of the lens (ratio of the focal length to its limiting aperture diameter), and m is the demagnification factor from the phosphor to the photodetector. For a lens with F=1.2，𝜏=0.8 and m=10，ξ will be 0.1%. Because of this low efficiency, the SNR of systems employing lens coupling is often limited by a secondary quantum sink, especially where the demagnification factor is large and/or g1 is small. It is also possible to use fiber optics to affect the coupling. These can be in the form of fiber-optic bundles (Figure 5), where optical fibers of constant diameter are fused to form a light guide. The fibers form an orderly array so that there is a one-to-one correspondence between the elements of the optical image at the exit of the phosphor and at the entrance to the photodetector. Where demagnification is required, the fiber-optic bundle can be tapered by drawing it under heat. While facilitating the construction of a detector to cover the required anatomy in the patient, demagnification by tapering also reduces coupling efficiency by limiting the acceptance angle at the fiber-optic input.

Fig. 4 Effect of coupling efficiency on spatial-frequency dependent DQE. Numbers on each curve represent the coupling efficiency in terms of the number of electrons produced at the photodetector per x-ray quantum interacting in the phosphor

Fig. 5 Methods to achieve coupling with demagnification between a phosphor and a photodetector(a)lens, (b)fiber-optic

A simplified expression for the coupling efficiency of a fiber optic taper is

where a is the fraction of the entrance surface that comprises the core glass of the optical fibers, 𝜏(θ) is the transmission factor for the core glass, NA is the numerical aperture of the untapered fiber, and m is the demagnification factor caused by tapering. For example, a taper with 2 times demagnification (m=2), with α =0.8, 𝜏 =0.9, and NA=1.0, has an efficiency of 18%, about seven times higher than a lens with F=1.2 with the same demagnification factor and about 2.5 times higher than a lens with F =0.7. It should be noted that for both lenses and fiber optics, the transmission efficiency is dependent on the angle of incidence, θ of the light and, therefore, a complete analysis involves an integral of the angular distribution of emission of the phosphor over θ. A comparison of the efficiency of lens versus fibreoptic coupling is shown in Figure 6.

Fiber-optic bundles are subject to geometric distortion which must be mini-mized. To maintain high resolution, the crosstalk of signal between fibers must be controlled and this is accomplished, in part, by the use of an extramural absorber (EMA), i.e., an optically attenuating material incorporated between individual fibers in the bundle to absorb light that escapes from the fibers or that directly enters the fiber cladding material on the entrance surface of the bundle. Both fiber-optic and lens designs are used in small-field-of-view cameras for digital mammography to couple a phosphor to a full-frame CCD photodetector. These systems are used for guiding needle biopsy and for localization of suspicious lesions. Typically only about 2× demagnification is employed, resulting in acceptable coupling efficiency.

Fig. 6 Efficiency of coupling with lenses and fiber-optics. In coth cases, efficiency falls as the demagnification factor between the input and output increases