Modulation Transfer Function (MTF) of detectors in EM

Chapter/Index: Introduction | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | Appendix

Modulation Transfer Function (MTF) of Detectors in EM

The Modulation Transfer Function (MTF) measures how well a detector records spatial frequencies in an image, which is key for achieving accurate, high-resolution imaging in electron microscopy. MTF is impacted by the pixel size and the scattering of electrons in the detector’s active layer. An MTF value of 1 indicates perfect retention of spatial frequency details, though this is challenging to achieve at higher frequencies. The spatial frequency range that a detector can effectively capture is limited by factors like pixel spacing, which sets the Nyquist limit (the maximum spatial frequency that can be accurately recorded without aliasing).

Advances in direct electron detectors, especially with CMOS/MAPS technology, have enhanced MTF by reducing electron scattering through techniques like backthinning. This reduces noise and improves resolution. High frame rates and electron counting mode also improve MTF by minimizing overlapping electron events, resulting in clearer imaging and higher spatial accuracy. Note that backthinning is a process applied to semiconductor detectors, where the detector’s substrate thickness is reduced by removing material from the back. This technique minimizes the amount of backscattered electrons, which can otherwise introduce noise and blur into the images, especially in high-resolution imaging applications like electron microscopy.

For example, Gatan camera Modulation Transfer Function (MTF) at 1/2 Nyquist frequency is about 0.1 at 200keV, which suggests relatively poor performance in terms of resolving detail at this spatial frequency:

MTF Value Interpretation: The MTF measures a detector's ability to transfer contrast at different spatial frequencies. A value of 1.0 would indicate perfect contrast retention at that frequency, while a value near zero indicates poor retention, meaning that the details are largely blurred or lost.
1/2 Nyquist Frequency: This frequency is relatively high in the spatial frequency spectrum, but it’s well within the range of details that many high-resolution electron microscopy applications aim to capture, especially at 200 keV where high-resolution imaging is crucial.
MTF of 0.1: An MTF of 0.1 at 1/2 Nyquist frequency indicates that only 10% of the original contrast is retained, meaning the detector struggles to capture fine details at that spatial frequency. For high-resolution applications like single-particle cryo-electron microscopy (cryo-EM) or structural biology, an MTF of 0.1 would generally be considered inadequate for capturing sharp, high-contrast details.

While an MTF of 0.1 at 1/2 Nyquist might be acceptable for applications that do not require extreme detail, it would typically be considered a limitation for high-resolution imaging. Ideally, detectors used in such applications should have a higher MTF at 1/2 Nyquist, closer to 0.3 or higher, to ensure better image quality and detail retention.

MTF is normally plotted against spatial frequency, where a value of one indicates complete retention of the relative amplitude for that spatial frequency. Figure 0076 illustrates how MTF changes with increasing energy, showing a shift from a single Gaussian MTF shape at low energies to distinct low and high spatial frequency components at higher energies. At 300 keV, results are shown for configurations with a standard holder, a cut-away holder, and a cut-away holder with light-absorbing ink to reduce backscatter. The densitometer’s influence on Nyquist frequency MTF values has not been corrected, so intrinsic MTF values for film at Nyquist are approximately twice those shown.

MTF

Figure 0076. Variation of the MTF for a detector with Kodak SO-163 film across different incident electron energies (20, 50, 120, 200, and 300 keV). Images were scanned at a 6 µm pixel step. The Fraction of Nyquist refers to a normalized measure of spatial frequency relative to the Nyquist frequency, which is defined as half of the sampling rate. The sampling rate is the number of pixels (samples) used per unit distance (e.g., per millimeter) to capture the detail of an object. Adapted from [1].

A higher sampling rate means more detail can be captured, allowing the image to contain finer features and higher resolution. However, according to the Nyquist theorem, to capture all the information without aliasing, the sampling rate must be at least twice the highest spatial frequency (or detail level) present in the object. This threshold is known as the Nyquist frequency. If the sampling rate is below this threshold, aliasing can occur, where higher spatial frequencies are misrepresented in the sampled data, leading to loss or distortion of detail.

The MTF at different spatial frequencies can be considered with contributions from both deterministic blur (Gaussian component) and stochastic scattering (backscattering effect) as electron energy increases. A general equation that combines these effects can be represented as,

---------------------------------------- [0076a]

where,
           is the spatial frequency (often in units of Nyquist frequency).
           represents the deterministic blur from the emulsion or primary scattering in the sensitive layer.
           represents the additional reduction in MTF due to backscattering effects.

The Gaussian blur component can be modeled as,

---------------------------------------- [0076b]

where,
is the standard deviation associated with the initial scattering in the emulsion layer or sensitive layer.

The backscatter component accounts for the low-frequency tail caused by scattering from the substrate and other layers,

---------------------------------------- [0076c]

where,
is a parameter related to the extent of backscatter, which increases with electron energy and depends on substrate material and thickness.

Combining these equations above, the overall MTF can be given by,

---------------------------------------- [0076d]

This model reflects how the Gaussian component affects the MTF more strongly at high spatial frequencies, while the backscatter effect mainly degrades low spatial frequencies. The parameters and would be empirically fitted to the specific electron energies and detector configurations.

[1] McMullan, G., Chen, S., Henderson, R., & Faruqi, A. R. (2009). Detective quantum efficiency of electron area detectors in electron microscopy. Ultramicroscopy, 109(9), 1126–1143, https://doi.org/10.1016/j.ultramic.2009.04.002.

Practical Electron Microscopy and Database

An Online Book, Second Edition by Dr. Yougui Liao (2006)

Modulation Transfer Function (MTF) of Detectors in EM