The Modulation Transfer Function (MTF) is an important aid to objective evaluation of the image-forming capability of optical systems. Not only that, the MTF also provides a means of expressing the imaging quality of optical systems objectively and quantitatively, but it can be calculated from the lens design data. In this way it allows optical and systems designers to predict reliably the performance of the optical systems. Manufacturers can compare the image quality of the manufactured lenses with the design expectations.
The Modulation Transfer Function (MTF), describing the resolution and performance of an optical system, is the ratio of relative image contrast divided by relative object contrast.
MTF = Relative Image Contrast/Relative Object Contrast.
When an object (illuminated target or reticle) is observed with an optical system, the resulting image will be somewhat degraded due to inevitable aberrations and diffraction phenomena. In addition, a real lens will not fully conform with the design data. Manufacturing errors, assembly and alignment errors in the optics will deteriorate the overall imaging performance of the system.
As a result, in the image, bright highlights will not appear as bright as they do in the object, and dark or shadowed areas will not be as black as those observed in the original patterns. In general, an illuminated target can be defined by its spatial frequency (number of bright and dark areas per millimeter) and the contrast (the apparent difference in brightness between bright and dark areas of the image).
By convention, the modulation transfer function is normalized to unity at zero spatial frequency. For low spatial frequencies, the modulation transfer function is close to 1 (or 100%) and generally falls as the spatial frequency increases until it reaches zero. The contrast values are lower for higher spatial frequencies as shown above. As spatial frequency increases, the MTF curve falls until it reaches zero. This is the limit of resolution for a given optical system or what is known as the cut off frequency (see figure below). When the contrast value reaches zero, the image becomes a uniform shade of grey.
Fig.1: The grids shown in the figure are actually no longer used in order to measure the MTF.
Modern MTF- Testers like the ImageMaster® use a single illuminated slit on an opaque background as the object. From a mathematical point of view a single slit can be regarded as the sum over all spatial frequencies (Fourier synthesis). All frequencies contribute with the same amplitude (=1) to this slit, not taking the finite slit width into account for this description. This single slit will be imaged into the image plane of the sample. Due to diffraction and aberrations, there will be no perfect slit image in this plane, instead the slit image is broadened. It represents the Line Spread Function (LSF).
The contribution of each spatial frequency to the LSF can be calculated on the basis of the Fourier analysis. Actually, the amplitude of each spatial frequency is equal to the contrast at this frequency. The Fourier analysis of the Line Spread Function corresponds to the MTF of the sample. Taking a single image of the LSF unveils the complete MTF.
Alternatively, it is also possible to use a cross (i.e. two perpendicular slits) for the target. This enables the ImageMaster® to measure the MTF in two image directions simultaneously provided a CCD camera is used for the image analyzer. And finally, a pinhole target can be used as the object, too. The image of a pinhole target is called the Point Spread Function. This function contains the complete MTF information in all image directions. The basic terms and mathematical relations used for MTF are described in the ISO 9334 standard.
The modulation transfer function varies in relation to the spatial frequency and also with the position in the field of view. The MTF measurement along the axis of symmetry of the optical system is known as on-axis measurement.
To completely characterize the imaging performance of an optical system, the MTF must be measured at different positions within the field of view. The MTF measurement within the field of view is known as off-axis measurement. In order to achieve an off-axis measurement, the target is moved in the field of view at the desired object position and the image analyzer to the corresponding image position.
The MTF measurement can be accomplished at a single wavelength or in a spectral range covering a finite band of wavelengths. The resulting measurement data are known as monochromatic or polychromatic MTF values, respectively.
Usually, the MTF is used in its one-dimensional form, calculated for one azimuthal section through the image plane. The azimuth (section plane) of the object pattern is called the sagittal azimuth when the prolongation of the slit or object passes through the reference axis. When the prolongation of the slit pattern is perpendicular to the reference axis, the azimuth is called the tangential azimuth.
In this so-called finite-finite imaging condition the illuminated slit or crosshair target is directly moved in the object plane of the sample. In the more common infinite-finite imaging condition, the illuminated slit or crosshair is part of a collimator projecting the target to infinity. The collimator is then oriented at different offaxis angles for characterizing the MTF at the corresponding image fields.