Optical transfer function


The optical transfer function of an optical system such as a camera, microscope, human eye, or projector is a scale-dependent description of their imaging contrast. Its magnitude is the image contrast of the harmonic intensity pattern,, as a function of the spatial frequency,, while its complex argument indicates a phase shift in the periodic pattern. The optical transfer function is used by optical engineers to describe how the optics project light from the object or scene onto a photographic film, detector array, retina, screen, or simply the next item in the optical transmission chain.
Formally, the optical transfer function is defined as the Fourier transform of the point spread function. As a Fourier transform, the OTF is generally complex-valued; however, it is real-valued in the common case of a PSF that is symmetric about its center. In practice, the imaging contrast, as given by the magnitude or modulus of the optical-transfer function, is of primary importance. This derived function is commonly referred to as the modulation transfer function.
The image on the right shows the optical transfer functions for two different optical systems in panels and. The former corresponds to the ideal, diffraction-limited, imaging system with a circular pupil. Its transfer function decreases approximately gradually with spatial frequency until it reaches the diffraction-limit, in this case at 500 cycles per millimeter or a period of 2 μm. Since periodic features as small as this period are captured by this imaging system, it could be said that its resolution is 2 μm. Panel shows an optical system that is out of focus. This leads to a sharp reduction in contrast compared to the diffraction-limited imaging system. It can be seen that the contrast is zero around 250 cycles/mm, or periods of 4 μm. This explains why the images for the out-of-focus system are more blurry than those of the diffraction-limited system. Note that although the out-of-focus system has very low contrast at spatial frequencies around 250 cycles/mm, the contrast at spatial frequencies just below the diffraction limit of 500 cycles/mm is comparable to that of the ideal system. Close observation of the image in panel shows that the image of the large spoke densities near the center of the spoke target is relatively sharp.

Definition and related concepts

Since the optical transfer function is defined as the Fourier transform of the point-spread function, it is generally speaking a complex-valued function of spatial frequency. The projection of a specific periodic pattern is represented by a complex number with absolute value and complex argument proportional to the relative contrast and translation of the projected projection, respectively.
DimensionsSpatial functionFourier transform
1DLine-spread function
1D section of 2D optical-transfer function
2DPoint-spread function Optical transfer function
3D3D point-spread function3D optical-transfer function

Often the contrast reduction is of most interest and the translation of the pattern can be ignored. The relative contrast is given by the absolute value of the optical transfer function, a function commonly referred to as the modulation transfer function. Its values indicate how much of the object's contrast is captured in the image as a function of spatial frequency. The MTF tends to decrease with increasing spatial frequency from 1 to 0 ; however, the function is often not monotonic. On the other hand, when also the pattern translation is important, the complex argument of the optical transfer function can be depicted as a second real-valued function, commonly referred to as the phase transfer function. The complex-valued optical transfer function can be seen as a combination of these two real-valued functions:
where
and represents the complex argument function, while is the spatial frequency of the periodic pattern. In general is a vector with a spatial frequency for each dimension, i.e. it indicates also the direction of the periodic pattern.
The impulse response of a well-focused optical system is a three-dimensional intensity distribution with a maximum at the focal plane, and could thus be measured by recording a stack of images while displacing the detector axially. By consequence, the three-dimensional optical transfer function can be defined as the three-dimensional Fourier transform of the impulse response. Although typically only a one-dimensional, or sometimes a two-dimensional section is used, the three-dimensional optical transfer function can improve the understanding of microscopes such as the structured illumination microscope.
True to the definition of transfer function, should indicate the fraction of light that was detected from the point source object. However, typically the contrast relative to the total amount of detected light is most important. It is thus common practice to normalize the optical transfer function to the detected intensity, hence.
Generally, the optical transfer function depends on factors such as the spectrum and polarization of the emitted light and the position of the point source. E.g. the image contrast and resolution are typically optimal at the center of the image, and deteriorate toward the edges of the field-of-view. When significant variation occurs, the optical transfer function may be calculated for a set of representative positions or colors.
Sometimes it is more practical to define the transfer functions based on a binary black-white stripe pattern. The transfer function for an equal-width black-white periodic pattern is referred to as the contrast transfer function .

Examples

Ideal lens system

A perfect lens system will provide a high contrast projection without shifting the periodic pattern, hence the optical transfer function is identical to the modulation transfer function. Typically the contrast will reduce gradually towards zero at a point defined by the resolution of the optics. For example, a perfect, non-aberrated, f/4 optical imaging system used, at the visible wavelength of 500 nm, would have the optical transfer function depicted in the right hand figure.
It can be read from the plot that the contrast gradually reduces and reaches zero at the spatial frequency of 500 cycles per millimeter. In other words, the optical resolution of the image projection is 1/500 of a millimeter, which corresponds to a feature size of 2 micrometer. Beyond 500 cycles per millimeter, the contrast of this imaging system, and therefore its modulation transfer function, is zero. Correspondingly, for this particular imaging device, the spokes become more and more blurred towards the center until they merge into a gray, unresolved, disc.
Note that sometimes the optical transfer function is given in units of the object or sample space, observation angle, film width, or normalized to the theoretical maximum. Conversion between units is typically a matter of a multiplication or division. E.g. a microscope typically magnifies everything 10 to 100-fold, and a reflex camera will generally demagnify objects at a distance of 5 meter by a factor of 100 to 200.
The resolution of a digital imaging device is not only limited by the optics, but also by the number of pixels, more in particular by their separation distance. As explained by the Nyquist–Shannon sampling theorem, to match the optical resolution of the given example, the pixels of each color channel should be separated by 1 micrometer, half the period of 500 cycles per millimeter. A higher number of pixels on the same sensor size will not allow the resolution of finer detail. On the other hand, when the pixel spacing is larger than 1 micrometer, the resolution will be limited by the separation between pixels; moreover, aliasing may lead to a further reduction of the image fidelity.

Imperfect lens system

An imperfect, aberrated imaging system could possess the optical transfer function depicted in the following figure.
As the ideal lens system, the contrast reaches zero at the spatial frequency of 500 cycles per millimeter. However, at lower spatial frequencies the contrast is considerably lower than that of the perfect system in the previous example. In fact, the contrast becomes zero on several occasions even for spatial frequencies lower than 500 cycles per millimeter. This explains the gray circular bands in the spoke image shown in the above figure. In between the gray bands, the spokes appear to invert from black to white and vice versa, this is referred to as contrast inversion, directly related to the sign reversal in the real part of the optical transfer function, and represents itself as a shift by half a period for some periodic patterns.
While it could be argued that the resolution of both the ideal and the imperfect system is 2 μm, or 500 LP/mm, it is clear that the images of the latter example are less sharp. A definition of resolution that is more in line with the perceived quality would instead use the spatial frequency at which the first zero occurs, 10 μm, or 100 LP/mm. Definitions of resolution, even for perfect imaging systems, vary widely. A more complete, unambiguous picture is provided by the optical transfer function.

Optical system with a non-rotational symmetric aberration

Optical systems, and in particular optical aberrations are not always rotationally symmetric. Periodic patterns that have a different orientation can thus be imaged with different contrast even if their periodicity is the same. Optical transfer function or modulation transfer functions are thus generally two-dimensional functions. The following figures shows the two-dimensional equivalent of the ideal and the imperfect system discussed earlier, for an optical system with trefoil, a non-rotational-symmetric aberration.
Optical transfer functions are not always real-valued. Period patterns can be shifted by any amount, depending on the aberration in the system. This is generally the case with non-rotational-symmetric aberrations. The hue of the colors of the surface plots in the above figure indicate phase. It can be seen that, while for the rotational symmetric aberrations the phase is either 0 or π and thus the transfer function is real valued, for the non-rotational symmetric aberration the transfer function has an imaginary component and the phase varies continuously.