Modulation Transfer Function (MTF) has been associated with the measurement of the performance of optical systems from the initial introduction of linear system analysis to this field. As the demand for higher quality, higher resolution optical systems has become prevalent, both designers and metrology scientists have begun investigating MTF as a mutual mode of optical system characterization. This article serves to identify the reasons for specification and measurement of MTF as a system characterization tool.
MTF is a direct and quantitative measure of image quality.
Most optical systems are expected to perform to a predetermined level of image integrity. Photographic optics, photolithographic optics, contact lenses, video systems, fax and copy optics, and compact disk lenses only sample the list of such optical systems. A convenient measure of this quality level is the ability of the optical system to transfer various levels of detail from object to image. Performance is measured in terms of contrast (degrees of gray) or modulation, and is related to the degradation of the image of a perfect source produced by the lens.
The MTF describes the image structure as a function of its spatial frequencies, most commonly produced by Fourier transforming the image spatial distribution or spread function. Therefore, the MTF provides simple presentation of image structure information similar in form and interpretation to audio frequency response. The various frequency components can be isolated for specific evaluation.
MTF can be related to end use applications.
Frequently, imaging systems are designed to project or capture detailed components in the object or image. Applications which rely upon image integrity or resolution ability can utilize MTF as a measure of performance at a critical dimension, such as a line width or pixel resolution or retinal sensor spacing. Optical systems from low resolution hand magnifiers to the most demanding photographic or lithographic lenses relate image size and structure to the end application requirement.
For example, video imaging systems must be designed to consider the image size pro- duced by the lens relative to the array pixel size and location. An array pixel width of 6 microns corresponds to a cut-off frequency of 83 lp/mm. In most cases, attempting to resolve beyond this limit is impossible; therefore designing a lens which maintains the high MTF out to this cut-off frequency is appropriate for this application. Specifying performance of a lens beyond this frequency is superfluous.
MTF is ideal for modeling concatenated systems.
MTF is analogous to electrical frequency response, and therefore allows for modeling of optical systems using linear system theory. Optical systems employing numerous stages (i.e. lenses, film, the eye) have a system MTF equal to the product of the MTF of the individual stages, allowing the expected system performance to be gauged by subsystem characterization. Proper concatenation requires that certain mathematical validity conditions, such as pupil matching and image relaying, be met, however.
MTF testing is objective and universal.
The influence of the test engineer is minimized with MTF testing, as human judgments of contrast, resolution or image quality are not required. Under the same test conditions, the polychromatic MTF of a lens can be compared to the polychromatic MTF of a design, or to another instrument. Consequently, no standardization or interpretation difficulties arise with MTF specification and testing.
MTF allows system testing in the exact application environment.
Since MTF testing is performed on the image or wavefront produced by an optical system, the parameters which influence lens performance and design can be recreated exactly in the test of the lens. Field angle positions, conjugate ratios, spectral regions, and image plane architecture can all be modeled in the test of an optical system. For example, the characterization of a double Gauss lens at finite conjugates can be easily performed and compared to design expectations or application requirements.
MTF provides the most direct method of measuring integrated polychromatic performance of optical systems.
Most commercial instruments dedicated to measuring MTF readily determine polychromatic MTF. The test spectrum can be tuned to the polychromatic design spectrum using photopic filters, infrared bandpass filters, or color filters, for example. Glass substitutions which share the same index of refraction at one wavelength but have different dispersions can severely affect performance of a polychromatic optical system.
Monochromatic testing of such a system may not uncover the substitution; polychromatic. MTF testing will expose the fabrication error.
MTFs can be accurately predicted and toleranced with lens design software.
Virtually all lens design software today allows graphical depiction and subsequent tolerancing of polychromatic MTF. Lens design software programs calculate the system MTF either through the autocorrelation of the exit pupil wavefront function or by Fourier transforming the point spread function, which is calculated by Fourier transforming the pupil wavefront. Polychromatic MTF is computed by vector addition of the mono- chromatic MTF and phase function. These methods for calculating MTF are also used with phase measuring interferometers to calculate the MTF of relatively well corrected monochromatic optical systems.
MTF measurement instruments provide testing versatility.
Numerous other tests can be performed on a standard MTF test instrument. Field curvature, distortion, and blur spot size are parameters relating to image characterization, and are therefore capable of quantification with MTF metrology equipment. MTF testing offers the engineer and test technician a method for directly measuring the image features that are related to overall system performance. It is a well-developed and understood concept and bridges the gap between the lens designers, optical fabricators, and metrology engineers.
© Optikos Corporation, 1999
All Rights Reserved
Adapted from an article appearing in Optics and Photonics News, June 1990