The Modulation Transfer Function MTF

Many of the audio-to-visual comparisons above are interesting but have no practical application. Spatial frequency response of optics is a core issue, however, and there is a very important reason it is used. This response is the most objective measure of the quality of an optical system.

Spatial frequency response is generally written using the filtering concept of a transfer function. If the target displays a sinusoidal modulation-changing gradually from bright to dark—and we view the optical system as a black box filter, then it exits with a lesser variation. Perhaps the exit signal varies only from light gray to dark gray.

The spatial frequency target is not a light and dark bar pattern as in the picket fence example above but a smoothly varying pattern such as that shown in Fig. 3-4a. Strictly, such a pattern should extend infinitely to either side, but practical use limits it to a few bars. The common 3- or 4-bar resolution chart is not a valid target field in the sense of either being infinite or sinusoidal, but it is such an easy target to use that it commonly serves to estimate optical quality anyway. One other requirement is that the illumination of the bar pattern is completely incoherent, usually a very easy requirement to fill.

If C stands for contrast and v is the spatial frequency, the way we will define the modulation transfer function (MTF) is

where MTF(v) is always less than 1.

Let's look at this simple equation and see what it means. If one has a target pattern with a certain value of contrast, the transfer function of the optical system always acts on that contrast to lessen it. Readers familiar with filtering concepts know that, in general, the transfer function can change the phase of the signal (suggesting that the most general transfer function is complex). Here, we will be concerned only with its amplitude, the modulation transfer function. The contrast (both before and after) is measured from the intensity levels at the darkest place on a dark bar and the brightest location on the light bar.

where the intensities are measured as in Fig. 3-4. Note that if the "before" pattern has a dark intensity of zero, the modulation transfer function becomes the image contrast itself (Hecht 1987, p. 507).

The MTF is all-encompassing and powerful. Even optical difficulties not originating in wavefront errors but in obstruction and non-uniform transmission find a way of being expressed in the modulation transfer function.

What form does it take? One would think that in perfect optical systems, the value of the MTF would be 1 for all spatial frequencies. No such optical system exists although a large planar mirror comes very close. Consulting Fig. 3-1 depicting the "wobbly stack" of filters, we can ask which filters are active even under ideal conditions. Clearly the atmosphere can be neglected— assume that the telescope is under perfect skies, or in space. The eye and its processing errors also will be ignored. Assume alignment and cleanliness are perfect and that the vignetting can be ignored. This process can be continued until the stack is made as short as possible.

What is left? Remember, even a perfect optical system has these two filters: 1) the wave nature of the light used to make the image, and 2) the limited aperture. This irreducible minimum is enough to force the MTF to a value less than unity and determine where it goes to zero. The perfect circular aperture's MTF is depicted in Fig. 3-5.

Admittedly, this curve doesn't much resemble the flat curve of the typical audio system, because it is plotted on a linear scale and extends somewhat beyond the spatial frequency bandwidth. The logarithmic decibel scale used for music tends to compress the spectrum to a wide, flat-topped bright dark

bright dark bright dark

bright dark

before