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KnowHow > Modulation Transfer Function (MTF)

KnowHow

MTF - Modulation Transfer Function

 

Relevant for: ImageMaster®

MTF - Expressing the imaging quality

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 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. The 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 the so called cut off frequency (see figure below). When the contrast value reaches zero, the image becomes a uniform shade of grey.

A perfect grid pattern is imaged through the sample into the image plane
Fig. 1: A perfect grid pattern is imaged through the sample into the image plane

New method - single illuminated slit

The grids shown in Figure 1 are actually no longer used in order to measure the MTF. Modern MTF-Testers like the ImageMaster® use a single illuminated slit on 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 so called Line Spread Function (LSF). On the basis of Fourier analysis the contribution of each spatial frequency to the LSF can be calculated. Actually the amplitude of each spatial frequency is equal to the contrast at this frequency. The Fourier analysis of the Lines 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 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 not only related to the spatial frequency but 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.

 

Fig. 2 - MTF = Contrast vs. spatial frequency
Fig. 2: MTF = Contrast vs. spatial frequency. This figure shows the MTF of a diffraction limited optics and a measured MTF reffering to Fig. 1

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 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 tangential azimuth.

 

The image of a perfect slit is called Line Spread Function
Fig. 3: The image of a perfect slit is called "Line Spread Function"
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. Figure 4 shows the typical setup for this imaging condition.
typical setup


Fig. 4: Typcial setup for off-axis MTF measurements. The tangential (T) and sagittal (S) measurement sections are indicated.

 

 
Read more about MTF measurement and autocollimators.
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