Modulation Transfer Function (MTF)

The ideal lens is one that faithfully reproduces all the coarse and fine structures in an object. This means that differences in the brightness of the object must be reflected in the image in the same way for a wide variety of structures.
    
For physical reasons alone, this requirement cannot be strictly met. Contrast losses occur to a greater or lesser extent in the entire range from coarse to fine structures. This entire structure range is now measured numerically by the number of line pairs per millimeter (spatial frequency R [1/mm]).

The MTF describes the contrast remaining in the image plane as a function of the number of line pairs per millimeter.

The abbreviation MTF stands for Modulation Transfer Function. This results in a graph as shown on the left.

As you can see, the modulationdeteriorates for finer structures and drops to zero at a certain number of line pairs per millimeter. The impression of sharpness is therefore not determined by this highest spatial frequency, but by the highest possible contrast reproduction over the entire spatial frequency range, up to a maximum spatial frequency depending on the application.

A specific MTF curve is also only valid for a specific point of view. In general, the MTF curves for different image points are different, so a large number of curves would be required to describe the entire image field. Therefore, another representation is used in which the modulation is plotted over the image height, starting from the center of the image, for reasonably selected spatial frequencies. This is the basis of the data sheets.

In addition, the beam path in the lens becomes increasingly asymmetric for pixels outside the center. Therefore, the MTF also depends on the orientation of the line pairs in the image field. Two perpendicular orientations are selected from the large number of possible orientations, as shown in the adjacent figure.

The MTF curves valid for the two orientations are shown in the diagrams as dashed (tangential) and solid (sagittal) in the diagrams.

In addition, the modulation transfer function depends on the f-number set, the image scale used, and the weighting of the individual spectral components. Therefore, additional MTF curves are needed to characterize the image quality. Insensitivity to changes in scale is also a quality criterion that must be taken into account.

For extreme wide-angle designs with large image angles, it should be noted that the modulation in the image field decreases with the cosine for sagittal structures and with the cube of the cosine of the image angle for tangential structures.

The figure on the left illustrates this using the example of a diffraction-limited (i.e. perfect) lens for 20 line pairs per mm and f-number k = 22 at a wavelength of 546 nm. The relative illuminance is plotted as a function of image angle.